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User:Unque/22-TET Plagal Theory

2025-02-03T14:27:57Z

Moremajorthanmajor: /* Notation */

{{Novelty}}

"Plagal Theory" (Eleutheric: ''Χωρη Πλαγικη'', ''Xoree Plagikee'' /ʃori plaʝici/) is a fictitious alternative music theory popularized across Europe in an alternate history. As such, the concepts discussed here, and the history thereof, should be understood to be fictitious, and therefore not necessarily applicable to the real world; specifically, the scales and their modes are named in accordance with the in-universe traditions, and are not legitimate proposals for conventions to use in real-world xenharmonic theory.

== Overview ==
Plagal Theory is a system of functionality based on several scales of [[22edo|22-EDO]]. The main defining feature of compositions in this tradition are the modulation from one mode of a scale to another by sharpening a single note; for instance, a piece written in the 5|1 mode of 5L2s would traditionally modulate to the 6|0 mode (with the same root pitch) to add a sense of grandeur and finality to the last movement of a piece. These modes are often grouped into "plagal pairs" that indicate how scales are related and how modulation tends to occur between pairs of scales.

The main scales considered in Plagal Theory are the [[MOS scale|MOS]] scales [[5L 2s|5L2s]] (known as ''Monotonic'' or ''Unitonic''), [[3L 4s|3L4s]] (known as ''Bitonic''), and [[2L 5s|2L5s]] (known as ''Enarmonic'').

== Monotonic (5L 2s) ==

=== Modes of Monotonic ===
The modes of the Monotonic scale are grouped into three plagal pairs, with one additional mode that lacks a plagal partner. Shown below is the general UDP notation and step sizes for each mode, as well as the degrees of 22-EDO contained by that mode (with the root implied at 0\22).
{| class="wikitable"
|+Modes of Monotonic
!Mode Pair
!Standard Mode (general)
!Standard Mode (22-EDO)
!Plagal Mode (general)
!Plagal Mode (22-EDO)
|-
|Lydian
|<nowiki>5|1 (LLsLLLs)</nowiki>
|4 8 [[Tel:9 13 17 21 22|9 13 17 21 22]]
|<nowiki>6|0 (LLLsLLs)</nowiki>
|4 [[Tel:8 12 13 17 21 22|8 12 13 17 21 22]]
|-
|Dorian
|<nowiki>3|3 (LsLLLsL)</nowiki>
|4 5 [[Tel:9 13 17 18 22|9 13 17 18 22]]
|<nowiki>4|2 (LLsLLsL)</nowiki>
|4 8 [[Tel:9 13 17 18 22|9 13 17 18 22]]
|-
|Phrygian
|<nowiki>1|5 (sLLLsLL)</nowiki>
|1 5 [[Tel:9 13 14 18 22|9 13 14 18 22]]
|<nowiki>2|4 (LsLLsLL)</nowiki>
|4 5 [[Tel:9 13 14 18 22|9 13 14 18 22]]
|-
|Locrian
|<nowiki>0|6 (sLLsLLL)</nowiki>
|1 5 [[Tel:9 10 14 18 22|9 10 14 18 22]]
|
|
|}
The Locrian mode is considered to be a "mixed" mode with no plagal form, and is seldom considered in traditional composition.

== Bitonic (3L 4s) ==

=== Modes of Bitonic ===
The modes of Bitonic number a total of four plagal pairs, and an additional "mixed" mode.
{| class="wikitable"
|+Modes of Bitonic
!Mode Pair
!Standard Mode (general)
!Standard Mode (22-EDO)
!Plagal Mode (general)
!Plagal Mode (22-EDO)
|-
|Lydian
|<nowiki>4|2 LssLsLs</nowiki>
|6 7 [[Tel:8 14 15 21 22|8 14 15 21 22]]
|<nowiki>5|1 LsLssLs</nowiki>
|6 [[Tel:7 13 14 15 16 22|7 13 14 15 16 22]]
|-
|Phrygian
|<nowiki>0|6 ssLsLsL</nowiki>
|1 2 8 9 15 16 22
|<nowiki>1|5 sLssLsL</nowiki>
|1 7 8 9 15 16 22
|-
|Dorian
|<nowiki>3|3 sLsLsLs</nowiki>
|1 7 [[Tel:8 14 15 21 22|8 14 15 21 22]]
|<nowiki>4|2 LssLsLs</nowiki>
|6 7 [[Tel:8 14 15 21 22|8 14 15 21 22]]
|-
|Mixodorian
|<nowiki>2|4 sLsLssL</nowiki>
|1 7 [[Tel:8 14 15 16 22|8 14 15 16 22]]
|<nowiki>3|3 sLsLsLs</nowiki>
|1 7 [[Tel:8 14 15 21 22|8 14 15 21 22]]
|-
|Locrian
|<nowiki>6|0 LsLsLss</nowiki>
|6 [[Tel:7 13 14 20 21 22|7 13 14 20 21 22]]
|
|
|}
The so-called "Mixodorian" mode is a later invention by theorists, especially experimentalists, in an attempt to "fix" the issues presented by the traditional Dorian pair: traditional Plagal Dorian was the same pattern as Standard Lydian, which left two "mixed" modes instead of just one. The Mixodorian mode does not see widespread usage, but it is not entirely unheard of either.

Just like with Monotonic, the "mixed" modes do not see widespread usage.

== Enarmonic (2L5s) ==

=== Modes of Enarmonic ===
Like monotonic, the Enarmonic scale has three non-overlapping plagal pairs and only one "mixed" mode.
{| class="wikitable"
|+Modes of Enarmonic
!Mode Pair
!Standard Mode (general)
!Standard Mode (22-EDO)
!Plagal Mode (general)
!Plagal Mode (22-EDO)
|-
|Lydian
|<nowiki>5|1 LsssLss</nowiki>
|6 [[Tel:8 10 12 18 20 22|8 10 12 18 20 22]]
|<nowiki>6|0 LssLsss</nowiki>
|6 [[Tel:8 10 16 18 20 22|8 10 16 18 20 22]]
|-
|Phrygian
|<nowiki>0|6 sssLssL</nowiki>
|2 4 [[Tel:6 12 14 16 22|6 12 14 16 22]]
|<nowiki>1|5 ssLsssL</nowiki>
|2 [[Tel:4 10 12 14 16 22|4 10 12 14 16 22]]
|-
|Dorian
|<nowiki>2|4 ssLssLs</nowiki>
|2 [[Tel:4 10 12 14 20 22|4 10 12 14 20 22]]
|<nowiki>3|3 sLsssLs</nowiki>
|2 [[Tel:8 10 12 14 20 22|8 10 12 14 20 22]]
|-
|Locrian
|<nowiki>4|2 sLssLss</nowiki>
|2 [[Tel:8 10 12 18 20|8 10 12 18 20]]
|
|
|}
Once again, the "mixed" Locrian mode does not see widespread usage in composition.

== Notation ==
The notation system used in Plagal Theory utilizes three types of clefs to describe the three primary MOS scales: the G-clef (𝄞) is used for the Monotonic Scale, the F-clef (𝄢) is used for the Bitonic scale, and the C-clef (𝄡) is used for the Enarmonic scale. Each scale has seven nominals and uses accidentals to alter those nominals, just like in common-practice Western theory in the real world.

The accidentals ♭ and ♯ respectively lower and raise the note by the chroma of the current scale; in other words, they are the difference between a large and small step. A circular accidental placed above or below a note also occurs in later iterations of musical notation, and is used to respectively raise or lower the note by a single step of 22edo, regardless of the scale being used.

Note that this notation system arbitrarily treats G as the "default" starting point, akin to how real-world descriptions of musical structure often treat C as the "default" tonic.
{| class="wikitable"
|+Notation
!EDO steps
!Monotonic (G Dorian)
!Bitonic (G Dorian)
!Enarmonic (G Dorian)
!Harmonic Category (See below)
|-
|0
|G
|G
|G
|
|-
|1
|A♭
|A
|
|Hard Dissonance
|-
|2
|B♭
|B♭
|A ~ F♯
|Hard Dissonance
|-
|3
|G♯
|C♭
|
|Soft Dissonance
|-
|4
|A
|F♯
|B ~ G♯
|Soft Consonance
|-
|5
|B
|G♯
|
|Soft Consonance
|-
|6
|C♭
|A♯
|A♯ ~ C♭
|Soft Consonance
|-
|7
|A♯
|B
|
|Hard Consonance
|-
|8
|B♯
|C
|B♯ ~ D♭
|Soft Dissonance
|-
|9
|C
|D♭
|
|Hard Consonance
|-
|10
|D♭
|E♭
|C ~ E♭
|Soft Consonance
|-
|11
|E♭
|
|
|Hard Dissonance
|-
|12
|C♯
|B♯
|D
|Soft Consonance
|-
|13
|D
|C♯
|
|Hard Consonance
|-
|14
|E
|D
|E ~ C♯
|Soft Dissonance
|-
|15
|F♭
|E
|
|Hard Consonance
|-
|16
|D♯
|F♭
|D♯ ~ F♭
|Soft Consonance
|-
|17
|E♯
|G♭
|
|Soft Consonance
|-
|18
|F
|A♭
|G♭
|Soft Consonance
|-
|19
|G♭
|D♯
|
|Soft Dissonance
|-
|20
|
|E♯
|F
|Hard Dissonance
|-
|21
|F♯
|F
|
|Hard Dissonance
|-
|22
|G
|G
|G
|Hard Consonance
|}
Note that, unlike in real-world musical notation, the semitones of 5L2s occur between A-B and D-E, rather than B-C and E-F. Additionally, the Enarmonic notation is only productive for notating 11edo, because the intervals of the 2L5s scale can only be used to reach every other note of 22edo.

Finally, the Monotonic and Ditonic scales each have a singular "blind spot" note that cannot be reached by a single alteration of the nominals. This note must instead be reached by use of double-accidentals or the over/under dot accidentals.

== Harmony ==
Over time, elements of chord-based harmony begin to take shape in the Plagal system, beginning with Parallel Generator dyads and later extending to triads and tetrads. Unlike in real-world Western classical theory, chords in Plagal Theory do not have a conventional method of construction by stacking flavors of thirds, and instead are built off of the mode that they begin on.

All intervals are categorized as "soft" and "hard" consonances and dissonances, with "hard" categories being the strongest resolutions and tensions, and "soft" categories being passing intervals that are used more often, without extreme weight placed on their harmonic content. These categories arose from rules of counterpoint composition, with the most concordant intervals being favored as ending points and the least concordant intervals being used as controlled dissonances to provide tension and set up for resolutions.

[[Category:Worldbuilding]]

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-02-03T06:15:02Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth, nineteenth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth, nineteenth
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis'', beside which the weakness of ''lenis'' is that its desired (sub)harmonics mostly form [[wolf interval]]<nowiki/>s. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads|mean minor mode]] with a flexible sixth, which is primarily considered to apply temperament, especially of a superparticular interval of the 2.3.5.7.11.13.17.19.43 subgroup up to [[28/27]] such as [[129/128]] or [[136/135]], directly to the fourth.

[[Category:Worldbuilding]]

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-02-03T02:00:38Z

Moremajorthanmajor: /* Ideal, tempering out 81/80 */

{{Editable user page}}Here are all mean minor tunings that can be written in the form "m/n-comma mean minor", where the '''43-limit Johnston''' comma ([[129/128]]) or the '''diatisma''' ([[136/135]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or 13 or smaller. The former range is almost the same as the range of m/n-comma Archytas and reverse Archytas temperaments and often confused for it in modern practice.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 [often “confused for 3402:4096:5103 vs. 4096:5103:6144 or 3510:4096:5265 vs. 4096:5265:6144”] and 34:40:51 vs. 40:51:60), the mean minor triad is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]] (as this comma is already unnoticeable, the Boethian interpretation is more commonly used as a joke); but with more complex thirds, and tempering out the quarter tone of [[1053/1024]] (often confused for the simpler [[36/35]]) or [[33/32]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds. That is why doing the latter is considered to generate a perverse temperament, the “perversion” being that a recognizable quarter tone is greater than 36.09 cents, though only the former quarter tone is considered to generate the “real” one.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, and, by extension, any comma is considered as primarily applied to the generator, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 (often “confused for 567/512 or 72/65”) or 30/17 and 16/9.

== The table==
=== Beyond historically-defined mean minor (most often approached as oneirotonic or superdiatonic)===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings 2/1-comma to 1/1-comma meantone
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[2/1-comma meantone|2/1-comma]]
|423.173||658.942||Close to [[51edo]].
|-
|[[43/22-comma meantone|43/22-comma]]
|420.240
|659.920
|
|-
|[[41/21-comma meantone|41/21-comma]]
|420.100
|659.967
|
|-
|[[39/20-comma meantone|39/20-comma]]
|419.947
|660.018
|Close to [[30edo|20edo]]
|-
|[[37/19-comma meantone|37/19-comma]]
|419.777
|660.074
|
|-
|[[35/18-comma meantone|35/18-comma]]
|419.588
|660.137
|
|-
|[[33/17-comma meantone|33/17-comma]]
|419.378
|660.207
|
|-
|[[31/16-comma meantone|31/16-comma]]
|419.140||660.287||
|-
|[[29/15-comma meantone|29/15-comma]]
|418.871||660.376||
|-
|[[27/14-comma meantone|27/14-comma]]
|418.564||660.479||
|-
|[[25/13-comma meantone|25/13-comma]]
|418.210||660.597||
|-
|[[23/12-comma meantone|23/12-comma]]
|417.796||660.735||Close to [[89edo]].
|-
|[[21/11-comma meantone|21/11-comma]]
|417.307||660.898||Close to [[69edo]].
|-
|[[40/21-comma meantone|40/21-comma]]
|417.028
|660.990
|
|-
|[[19/10-comma meantone|19/10-comma]]
|416.721||661.093||
|-
|[[36/19-comma meantone|36/19-comma]]
|416.381
|661.206
|Close to [[49edo]].
|-
|[[17/9-comma meantone|17/9-comma]]
|416.004||661.332||Close to [[58edo]].
|-
|[[32/17-comma meantone|32/17-comma]]
|415.582
|661.473
|Close to [[78edo]].
|-
|[[15/8-comma meantone|15/8-comma]]
|415.108||661.631||
|-
|[[28/15-comma meantone|28/15-comma]]
|414.570||661.810||
|-
|[[41/22-comma meantone|41/22-comma]]
|414.375
|661.875
|
|-
|[[13/7-comma meantone|13/7-comma]]
|413.956||662.015||Close to [[29edo]].
|-
|[[37/20-comma meantone|37/20-comma]]
|413.495
|662.168
|
|-
|[[11/13-comma meantone|24/13-comma]]
|413.247||662.251||
|-
|[[35/19-comma meantone|35/19-comma]]
|412.986
|662.338
|
|-
|[[46/25-comma meantone|46/25-comma]]
|412.850
|662.383
|
|-
|[[11/6-comma meantone|11/6-comma]]
|412.420||662.527||Close to [[96edo]].
|-
|[[31/17-comma meantone|31/17-comma]]
|411.787
|662.738
|Close to [[67edo]].
|-
|[[20/11-comma meantone|20/11-comma]]
|411.442||662.853||
|-
|[[13/16-comma meantone|29/16-comma]]
|411.075||662.975||Close to [[100edo]].
|-
|[[38/21-comma meantone|38/21-comma]]
|410.883
|663.039
|
|-
|[[9/5-comma meantone|9/5-comma]]
|410.269||663.244||Close to [[38edo]]
|-
|[[34/19-comma meantone|34/19-comma]]
|409.590
|663.470
|
|-
|[[25/14-comma meantone|25/14-comma]]
|409.347||663.551||Close to [[85edo]]
|-
|[[16/9-comma meantone|16/9-comma]]
|408.835||663.722||
|-
|[[39/22-comma meantone|39/22-comma]]
|408.509
|663.830
|Almost exactly [[47edo]]
|-
|[[10/13-comma meantone|23/13-comma]]
|408.284||663.905||
|-
|[[30/17-comma meantone|30/17-comma]]
|407.992
|664.003
|
|-
|[[16/21-comma meantone|37/21-comma]]
|407.811
|664.063
|
|-
|[[7/4-comma meantone|7/4-comma]]
|407.043||664.319||Close to [[56edo]]
|-
|[[33/19-comma meantone|33/19-comma]]
|406.194
|664.602
|Close to [[65edo]].
|-
|[[26/15-comma meantone|26/15-comma]]
|405.978||664.677||
|-
|[[45/26-comma meantone|45/26-comma]]
|405.802
|664.733
|
|-
|[[8/11-comma meantone|19/11-comma]]
|405.577||664.808||
|-
|[[13/18-comma meantone|31/18-comma]]
|405.251
|664.916
|Close to [[74edo]].
|-
|[[12/7-comma meantone|12/7-comma]]
|404.739||665.087||Close to [[83edo]].
|-
|[[41/24-comma meantone|41/24-comma]]
|404.355
|665.215
|Almost exactly [[92edo]].
|-
|[[29/17-comma meantone|29/17-comma]]
|404.197
|665.268
|
|-
|[[7/10-comma meantone|17/10-comma]]
|403.817||665.394||
|-
|[[22/13-comma meantone|22/13-comma]]
| 403.321||665.560||
|-
|[[27/16-comma meantone|27/16-comma]]
| 403.011||665.663||
|-
|[[32/19-comma meantone|32/19-comma]]
|402.798
|665.734
|
|-
|[[15/22-comma meantone|37/22-comma]]
|402.644
|665.785
|
|-
|[[42/25-comma meantone|42/25-comma]]
|402.527
|665.824
|
|-
|[[47/28-comma meantone|47/28-comma]]
|402.435
|665.855
|
|-
|[[5/3-comma meantone|5/3-comma]]
| 401.666||666.111||
|-
|[[43/26-comma meantone|43/26-comma]]
|400.839
|666.387
|Close to [[82edo]].
|-
|[[15/23-comma meantone|38/23-comma]]
|400.731
|666.423
|
|-
|[[33/20-comma meantone|33/20-comma]]
|400.591
|666.470
|
|-
|[[11/17-comma meantone|28/17-comma]]
|400.401
|666.533
|Close to [[75edo]]
|-
|[[23/14-comma meantone|23/14-comma]]
| 400.130||666.623||Everything up to this point generates 20 and 29 tone MOS scales.
|-
|[[9edo]]
|400.000||666.667||The largest MOS scale this can generate is 9 tone. '''Lower boundary of proper superdiatonic MOS scales.'''
|-
|[[18/11-comma meantone|18/11-comma]]
| 399.711||666.763||Everything from this point onwards generates 25 and 41 tone MOS scales.
|-
|[[31/19-comma meantone|31/19-comma]]
|399.403
|666.866
|
|-
|[[5/8-comma meantone|13/8-comma]]
| 398.978||667.007||Close to 215/162.
|-
|[[34/21-comma meantone|34/21-comma]]
|398.594
|667.135
|
|-
|[[Φ-comma meantone|ϕ-comma]]
|398.529
|667.157
|
|-
|[[21/13-comma meantone|21/13-comma]]
| 398.358||667.214||
|-
|[[29/18-comma meantone|29/18-comma]]
|398.082
|667.306
|
|-
|[[37/23-comma meantone|37/23-comma]]
|397.926
|667.358
|
|-
|[[3/5-comma meantone|8/5-comma]]
| 397.365||667.545||
|-
|[[35/22-comma meantone|35/22-comma]]
|396.779
|667.740
|
|-
|[[27/17-comma meantone|27/17-comma]]
|396.606
|667.798
|
|-
|[[19/12-comma meantone|19/12-comma]]
| 396.290||667.903||
|-
|[[11/19-comma meantone|30/19-comma]]
|396.007
|667.998
|Close to [[97edo]].
|-
|[[4/7-comma meantone|11/7-comma]]
| 395.522||668.159||Close to [[88edo]].
|-
|[[25/16-comma meantone|25/16-comma]]
| 395.946||668.351|| Close to [[73edo|79edo]].
|-
|[[14/9-comma meantone|14/9-comma]]
| 394.498||668.501||Close to [[81edo]].
|-
|[[31/20-comma meantone|31/20-comma]]
|394.139
|668.620
|Close to [[70edo]].
|-
|[[17/11-comma meantone|17/11-comma]]
| 393.846||668.718||
|-
|[[20/13-comma meantone|20/13-comma]]
| 393.395||668.868||Close to [[61edo]].
|-
|[[23/15-comma meantone|23/15-comma]]
| 393.064||668.979||
|-
|[[26/17-comma meantone|26/17-comma]]
|392.811
|669.063
|
|-
|[[29/19-comma meantone|29/19-comma]]
|392.611
|669.130
|
|-
|[[32/21-comma meantone|32/21-comma]]
|392.449
|669.184
| Close to [[52edo]].
|-
|[[3/2-comma meantone|3/2-comma]]
| 390.913||669.696||Close to [[43edo]]
|-
|[[10/21-comma meantone|31/21-comma]]
|389.377
|670.208
|
|-
|[[28/19-comma meantone|28/19-comma]]
|389.215
|670.261
|Close to [[85edo]].
|-
|[[25/17-comma meantone|25/17-comma]]
|389.016
|670.328
|
|-
|[[22/15-comma meantone|22/15-comma]]
|388.763
|670.412
|
|-
|[[19/13-comma meantone|19/13-comma]]
|388.432
|670.522
|Close to [[34edo]]
|-
|[[16/11-comma meantone|16/11-comma]]
|387.981
|670.673
|
|-
|[[29/20-comma meantone|29/20-comma]]
|387.673
|670.771
|Close to [[26edo]].
|-
|[[13/9-comma meantone|13/9-comma]]
|387.329
|670.890
|
|-
|[[23/16-comma meantone|23/16-comma]]
|386.881
|671.040
|Close to [[93edo]]
|-
|[[10/7-comma meantone|10/7-comma]]
|386.305
|671.232
|Close to [[59edo]], 1/3-[[135/128|limma]] Pelogic
|-
|[[27/19-comma meantone|27/19-comma]]
|385.820
|671.393
|Close to [[84edo]].
|-
|[[17/12-comma meantone|17/12-comma]]
|385.537
|671.488
|Close to [[71edo]].
|-
|[[7/17-comma meantone|24/17-comma]]
|385.220
|671.593
|
|-
|[[31/22-comma meantone|31/22-comma]]
|385.048
|671.651
|
|-
|[[7/5-comma meantone|7/5-comma]]
|384.461
|671.841
|
|-
|[[32/23-comma meantone|32/23-comma]]
|383.900
|672.033
|Close to [[25edo]].
|-
|[[25/18-comma meantone|25/18-comma]]
|383.745
|672.085
|
|-
|[[18/13-comma meantone|18/13-comma]]
|383.469
|672.177
|
|-
|[[(ϕ+2)/(φ+1)-comma meantone|(ϕ+2)]] [[1/(φ+1)-comma meantone|/(ϕ+1)-comma]]
|383.298
|672.234
|Close to [[64edo]].
|-
|[[29/21-comma meantone|29/21-comma]]
|383.232
|672.256
|
|-
|[[11/8-comma meantone|11/8-comma]]
|382.848
|672.384
|Close to [[83edo]].
|-
|[[26/19-comma meantone|26/19-comma]]
|382.424
|672.525
|Close to [[91edo]].
|-
|[[15/11-comma meantone|15/11-comma]]
|382.115
|672.628
|Close to [[66edo]].
|-
|[[19/14-comma meantone|19/14-comma]]
|381.696
|672.768
|
|-
|[[23/17-comma meantone|23/17-comma]]
|381.425
|672.858
|
|-
|[[27/20-comma meantone|27/20-comma]]
|381.235
|672.922
|
|-
|[[31/23-comma meantone|31/23-comma]]
|381.095
|672.968
|
|-
|[[35/26-comma meantone|35/26-comma]]
|380.987
|673.004
|
|-
|[[4/3-comma meantone|4/3-comma]]
|380.160
|673.280
|Close to [[41edo]].
|-
|[[37/28-comma meantone|37/28-comma]]
|379.392
|673.536
|Close to [[98edo]]
|-
|[[33/25-comma meantone|33/25-comma]]
|379.300
|673.567
|
|-
|[[29/22-comma meantone|29/22-comma]]
|379.185
|673.606
|
|-
|[[25/19-comma meantone|25/19-comma]]
|379.028
|673.657
|Close to [[57edo]].
|-
|[[5/16-comma meantone|21/16-comma]]
|378.816
|673.728
|
|-
|[[17/13-comma meantone|17/13-comma]]
|378.505
|673.831
|
|-
|[[13/10-comma meantone|13/10-comma]]
|378.010
|673.998
|Close to [[73edo]].
|-
|[[22/17-comma meantone|22/17-comma]]
|377.630
|674.123
|
|-
|[[31/24-comma meantone|31/24-comma]]
|377.472
|674.176
|Close to [[89edo]].
|-
|[[9/7-comma meantone|9/7-comma]]
|377.088
|674.304
|
|-
|[[23/18-comma meantone|23/18-comma]]
|376.576
|674.475
|
|-
|[[14/11-comma meantone|14/11-comma]]
|376.250
|674.583
|
|-
|[[33/26-comma meantone|33/26-comma]]
|376.024
|674.659
|
|-
|[[19/15-comma meantone|19/15-comma]]
|375.859
|674.714
|
|-
|[[24/19-comma meantone|24/19-comma]]
|375.633
|674.789
|Everything up to this point generates 25 and 41 tone MOS scales.
|-
|[[16edo]]
|375.000
|675.000
|The largest MOS scale this can generate is 9 tone. '''Lower boundary of proper superdiatonic MOS scales.'''
|-
|[[5/4-comma meantone|5/4-comma]]
|374.783
|675.072
|Everything from this point onwards generates 23 and 39 tone MOS scales.
|-
|[[26/21-comma meantone|26/21-comma]]
|374.016
|675.328
|
|-
|[[21/17-comma meantone|21/17-comma]]
|373.835
|675.388
|
|-
|[[16/13-comma meantone|16/13-comma]]
|373.543
|675.486
|
|-
|[[27/22-comma meantone|27/22-comma]]
|373.317
|675.561
|
|-
|[[11/9-comma meantone|11/9-comma]]
|372.991
|675.670
|
|-
|[[17/14-comma meantone|17/14-comma]]
|372.479
|675.840
|
|-
|[[23/19-comma meantone|23/19-comma]]
|372.237
|675.921
|
|-
|[[6/5-comma meantone|6/5-comma]]
|371.558
|676.147
|Close to [[71edo]], [[87edo]].
|-
|[[25/21-comma meantone|25/21-comma]]
|370.943
|676.352
|
|-
|[[3/16-comma meantone|19/16-comma]]
|370.751
|676.416
|Close to [[55edo]].
|-
|[[13/11-comma meantone|13/11-comma]]
|370.385
|676.538
|Close to [[94edo]].
|-
|[[20/17-comma meantone|20/17-comma]]
|370.040
|676.653
|
|-
|[[27/23-comma meantone|27/23-comma]]
|369.875
|676.708
|
|-
|[[7/6-comma meantone|7/6-comma]]
|369.407
|676.864
|Close to [[39edo]].
|-
|[[4/25-comma meantone|29/25-comma]]
|368.977
|677.008
|
|-
|[[22/19-comma meantone|22/19-comma]]
|368.841
|677.053
|
|-
|[[15/13-comma meantone|15/13-comma]]
|368.580
|677.140
|
|-
|[[23/20-comma meantone|23/20-comma]]
|368.332
|677.223
|
|-
|[[8/7-comma meantone|8/7-comma]]
|367.871
|677.376
|Close to [[62edo]].
|-
|[[3/22-comma meantone|25/22-comma]]
|367.452
|677.516
|
|-
|[[2/15-comma meantone|17/15-comma]]
|367.256
|677.581
|Close to [[85edo]].
|-
|[[9/8-comma meantone|9/8-comma]]
|366.719
|677.760
|
|-
|[[19/17-comma meantone|19/17-comma]]
|366.244
|677.919
|
|-
|[[10/9-comma meantone|10/9-comma]]
|365.823
|678.059
|
|-
|[[21/19-comma meantone|21/19-comma]]
|365.445
|678.185
|
|-
|[[11/10-comma meantone|11/10-comma]]
|365.106
|678.298
|Close to [[23edo]].
|-
|[[23/21-comma meantone|23/21-comma]]
|364.799
|678.400
|
|-
|[[12/11-comma meantone|12/11-comma]]
|363.519
|678.494
|
|-
|[[1/12-comma meantone|13/12-comma]]
|364.034
|678.657
|
|-
|[[14/13-comma meantone|14/13-comma]]
|363.617
|678.794
|Close to [[99edo]].
|-
|[[1/14-comma meantone|15/14-comma]]
|363.262
|678.913
|Close to [[76edo]], 1/4-[[135/128|limma]] Pelogic
|-
|[[16/15-comma meantone|16/15-comma]]
|362.955
|679.015
|
|-
|[[17/16-comma meantone|17/16-comma]]
|362.686
|679.105
|
|-
|[[18/17-comma meantone|18/17-comma]]
|362.449
|679.184
|
|-
|[[19/18-comma meantone|19/18-comma]]
|362.238
|679.254
|Almost exactly [[53edo]]
|-
|[[1/19-comma meantone|20/19-comma]]
|362.050
|679.317
|
|-
|[[21/20-comma meantone|21/20-comma]]
|361.880
|679.373
|
|-
|[[22/21-comma meantone|22/21-comma]]
|361.726
|679.425
|
|-
|[[23/22-comma meantone|23/22-comma]]
|361.587
|679.471
|Close to [[83edo]].
|-
|[[1/1-comma meantone|1/1-comma]]
|358.654
|680.449
|Close to [[97edo]], [[67edo]], [[30edo]]
|}
====Tempering out [[136/135]]====
{| class="wikitable mw-collapsible"
|+Mean minor tunings from 2-comma to [[7edo|1-comma]]
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|2-comma
|217.475
|727.508
|
|-
|25/13-comma
|220.423
|726.526
|
|-
|23/12-comma
|220.669
|726.444
|
|-
|21/11-comma
|220.959
|726.347
|
|-
|19/10-comma
|221.308
|726.231
|
|-
|17/9-comma
|221.734
|726.089
|
|-
|15/8-comma
|222.266
|725.911
|
|-
|13/7-comma
|222.951
|725.683
|
|-
|24/13-comma
|223.371
|725.543
|
|-
|11/6-comma
|223.863
|725.378
|
|-
|20/11-comma
|224.444
|725.185
|
|-
|9/5-comma
|225.141
|724.953
|
|-
|16/9-comma
|225.993
|724.669
|
|-
|23/13-comma
|226.320
|724.560
|
|-
|7/4-comma
|227.057
|724.314
|
|-
|19/11-comma
|227.928
|724.024
|
|-
|12/7-comma
|228.426
|723.858
|
|-
|17/10-comma
|228.974
|723.675
|
|-
|22/13-comma
|229.269
|723.577
|
|-
|5/3-comma
|230.252
|723.249
|
|-
|18/11-comma
|231.413
|722.862
|
|-
|13/8-comma
|231.849
|722.717
|
|-
|ϕ-comma
|232.116
|722.628
|
|-
|21/13-comma
|232.217
|722.594
|
|-
|8/5-comma
|232.807
|722.398
|
|-
|19/12-comma
|233.446
|722.185
|
|-
|11/7-comma
|233.902
|722.933
|
|-
|14/9-comma
|234.510
|721.830
|
|-
|17/11-comma
|356.599
|721.701
|
|-
|20/13-comma
|235.166
|721.611
|
|-
|3/2-comma
|236.640
|721.120
|
|-
|19/13-comma
|238.114
|720.628
|
|-
|16/11-comma
|238.382
|720.539
|
|-
|13/9-comma
|238.769
|720.410
|
|-
|10/7-comma
|239.378
|720.207
|
|-
|17/12-comma
|239.834
|720.055
|
|-
|[[5edo]]
|240.000
|720.000
|
|-
|7/5-comma
|240.473
|719.842
|
|-
|18/13-comma
|241.063
|719.646
|
|-
|(ϕ+2)/(ϕ+1)-comma
|241.164
|719.612
|
|-
|11/8-comma
|241.431
|719.533
|
|-
|15/11-comma
|241.867
|719.378
|
|-
|4/3-comma
|243.028
|719.900
|
|-
|17/13-comma
|244.011
|718.663
|
|-
|13/10-comma
|244.306
|718.565
|
|-
|9/7-comma
|244.835
|718.382
|
|-
|14/11-comma
|245.352
|718.216
|
|-
|5/4-comma
|246.222
|717.926
|
|-
|16/13-comma
|246.960
|717.680
|
|-
|11/9-comma
|247.287
|717.571
|
|-
|6/5-comma
|248.139
|717.287
|
|-
|13/11-comma
|248.836
|717.055
|
|-
|7/6-comma
|249.417
|716.861
|
|-
|15/13-comma
|249.908
|716.697
|
|-
|8/7-comma
|250.329
|716.557
|
|-
|9/8-comma
|251.013
|716.329
|
|-
|10/9-comma
|251.546
|716.151
|
|-
|11/10-comma
|251.972
|716.009
|
|-
|12/11-comma
|252.320
|715.833
|
|-
|13/12-comma
|252.611
|715.796
|
|-
|14/13-comma
|252.856
|715.715
|
|-
|1-comma
|255.805
|714.732
|
|}

===Historically-defined mean minor===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings 1/1-comma meantone to Pythagorean
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[1/1-comma meantone|1/1-comma]]
| 358.654||680.449||Close to [[97edo]], [[67edo]], [[30edo]]
|-
|[[21/22-comma meantone|21/22-comma]]
|
|681.426
|Close to [[37edo]].
|-
|[[20/21-comma meantone|20/21-comma]]
|
|681.473
|
|-
|[[19/20-comma meantone|19/20-comma]]
|
|681.524
|
|-
|[[18/19-comma meantone|18/19-comma]]
|
|681.581
|
|-
|[[17/18-comma meantone|17/18-comma]]
|
|681.644
|
|-
|[[16/17-comma meantone|16/17-comma]]
|
|681.713
|
|-
|[[15/16-comma meantone|15/16-comma]]
| ||681.793||Close to [[44edo]].
|-
|[[14/15-comma meantone|14/15-comma]]
| ||681.883||
|-
|[[13/14-comma meantone|13/14-comma]]
| ||681.985||
|-
|[[12/13-comma meantone|12/13-comma]]
| ||682.103||
|-
|[[11/12-comma meantone|11/12-comma]]
| ||682.241||
|-
|[[10/11-comma meantone|10/11-comma]]
| ||682.404||Close to [[51edo]].
|-
|[[19/21-comma meantone|19/21-comma]]
|
|682.497
|
|-
|[[9/10-comma meantone|9/10-comma]]
| ||682.599||
|-
|[[17/19-comma meantone|17/19-comma]]
|
|682.713
|
|-
|[[8/9-comma meantone|8/9-comma]]
| ||682.838||Close to [[58edo]].
|-
|[[15/17-comma meantone|15/17-comma]]
|
|682.979
|
|-
|[[7/8-comma meantone|7/8-comma]]
| ||683.137||Close to [[65edo]].
|-
|[[13/15-comma meantone|13/15-comma]]
| ||683.316||Close to [[72edo]].
|-
|[[19/22-comma meantone|19/22-comma]]
|
|683.381
|
|-
|[[6/7-comma meantone|6/7-comma]]
| ||683.521||Close to [[79edo]].
|-
|[[17/20-comma meantone|17/20-comma]]
|
|683.675
|
|-
|[[11/13-comma meantone|11/13-comma]]
| ||683.757||Close to [[86edo]].
|-
|[[16/19-comma meantone|16/19-comma]]
|
|683.844
|
|-
|[[21/25-comma meantone|21/25-comma]]
|
|683.890
|Close to [[93edo]]
|-
|[[5/6-comma meantone|5/6-comma]]
| ||684.033||Close to [[100edo]].
|-
|[[14/17-comma meantone|14/17-comma]]
|
|684.244
|
|-
|[[9/11-comma meantone|9/11-comma]]
| ||684.359||
|-
|[[13/16-comma meantone|13/16-comma]]
| ||684.481||
|-
|[[17/21-comma meantone|17/21-comma]]
|
|684.545
|
|-
|[[4/5-comma meantone|4/5-comma]]
| ||684.750||
|-
|[[15/19-comma meantone|15/19-comma]]
|
|684.976
|
|-
|[[11/14-comma meantone|11/14-comma]]
| ||685.057||
|-
|[[7/9-comma meantone|7/9-comma]]
| ||685.228||
|-
|[[17/22-comma meantone|17/22-comma]]
|
|685.337
|
|-
|[[10/13-comma meantone|10/13-comma]]
| ||685.412||
|-
|[[13/17-comma meantone|13/17-comma]]
|
|685.509
|
|-
|[[16/21-comma meantone|16/21-comma]]
|
|685.569
|Everything up to this point generates 9 and 16 tone MOS scales.
|-
|[[7edo]]
| ||685.714||The largest MOS scale this can generate is 7 tone. '''Lower boundary of 5-limit diamond monotone.'''
|-
|[[3/4-comma meantone|3/4-comma]]
| ||685.825||Everything from this point onwards generates 12 and 19 tone MOS scales.
|-
|[[14/19-comma meantone|14/19-comma]]
|
|686.108
|
|-
|[[11/15-comma meantone|11/15-comma]]
| ||686.184||
|-
|[[19/26-comma meantone|19/26-comma]]
|
|686.239
|
|-
|[[8/11-comma meantone|8/11-comma]]
| ||686.314||
|-
|[[13/18-comma meantone|13/18-comma]]
|
|686.423
|
|-
|[[5/7-comma meantone|5/7-comma]]
| ||686.593||
|-
|[[17/24-comma meantone|17/24-comma]]
|
|686.721
|
|-
|[[12/17-comma meantone|12/17-comma]]
|
|686.774
|
|-
|[[7/10-comma meantone|7/10-comma]]
| ||686.901||
|-
|[[9/13-comma meantone|9/13-comma]]
| ||687.066||
|-
|[[11/16-comma meantone|11/16-comma]]
| ||687.169||
|-
|[[13/19-comma meantone|13/19-comma]]
|
|687.240
|
|-
|[[15/22-comma meantone|15/22-comma]]
|
|687.292
|
|-
|[[17/25-comma meantone|17/25-comma]]
|
|687.331
|
|-
|[[19/28-comma]]
|
|687.361
|
|-
|[[2/3-comma meantone|2/3-comma]]
| ||687.617|| Close to [[89edo]].
|-
|[[17/26-comma meantone|17/26-comma]]
|
|687.893
|Close to [[82edo]].
|-
|[[15/23-comma meantone|15/23-comma]]
|
|687.929
|
|-
|[[13/20-comma meantone|13/20-comma]]
|
|687.976
|
|-
|[[11/17-comma meantone|11/17-comma]]
|
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone|9/14-comma]]
| ||688.129||
|-
|[[7/11-comma meantone|7/11-comma]]
| ||688.269||Close to [[68edo]].
|-
|[[12/19-comma meantone|12/19-comma]]
|
|688.372
|
|-
|[[5/8-comma meantone|5/8-comma]]
| ||688.514||Close to [[61edo]] and [[43/32]].
|-
|[[13/21-comma meantone|13/21-comma]]
|
|688.641
|
|-
|[[1/φ-comma meantone|1/ϕ-comma]]
|
|688.663
|
|-
|[[8/13-comma meantone|8/13-comma]]
| ||688.720||
|-
|[[11/18-comma meantone|11/18-comma]]
|
|688.812
|Close to [[54edo]].
|-
|[[14/23-comma meantone|14/23-comma]]
|
|688.864
|
|-
|[[3/5-comma meantone|3/5-comma]]
| ||689.051||
|-
|[[13/22-comma meantone|13/22-comma]]
|
|689.247
|
|-
|[[10/17-comma meantone|10/17-comma]]
|
|689.304
|
|-
|[[7/12-comma meantone|7/12-comma]]
| ||689.410||Close to [[47edo]].
|-
|[[11/19-comma meantone|11/19-comma]]
|
|689.504
|
|-
|[[4/7-comma meantone|4/7-comma]]
| ||689.666||Close to [[87edo]].
|-
|[[9/16-comma meantone|9/16-comma]]
| ||689.858||
|-
|[[5/9-comma meantone|5/9-comma]]
| ||690.007||Close to [[40edo]].
|-
|[[11/20-comma meantone|11/20-comma]]
|
|690.127
|
|-
|[[6/11-comma meantone|6/11-comma]]
| ||690.224||
|-
|[[7/13-comma meantone|7/13-comma]]
| ||690.375||Close to [[73edo]].
|-
|[[8/15-comma meantone|8/15-comma]]
| ||690.485||
|-
|[[9/17-comma meantone|9/17-comma]]
|
|690.569
|
|-
|[[10/19-comma meantone|10/19-comma]]
|
|690.636
|
|-
|[[11/21-comma meantone|11/21-comma]]
|
|690.690
| Close to [[33edo]]
|-
|[[1/2-comma meantone|1/2-comma]]
| ||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not have a whole tone between 10/9 and 9/8.
|-
|[[10/21-comma meantone|10/21-comma]]
|
|691.714
|
|-
|[[9/19-comma meantone|9/19-comma]]
|
|691.768
|Close to [[85edo]].
|-
|[[8/17-comma meantone|8/17-comma]]
|
|691.834
|
|-
|[[7/15-comma meantone|7/15-comma]]
|
|691.919
|
|-
|[[6/13-comma meantone|6/13-comma]]
|
|692.029
|
|-
|[[5/11-comma meantone|5/11-comma]]
|
|692.179
|
|-
|[[9/20-comma meantone|9/20-comma]]
|
|692.277
|Close to [[26edo]].
|-
|[[4/9-comma meantone|4/9-comma]]
|
|692.397
|
|-
|[[7/16-comma meantone|7/16-comma]]
|
|692.546
|
|-
|[[3/7-comma meantone|3/7-comma]]
|
|692.738
|
|-
|[[8/19-comma meantone|8/19-comma]]
|
|692.899
|
|-
|[[5/12-comma meantone|5/12-comma]]
|
|692.994
|Close to [[71edo]].
|-
|[[7/17-comma meantone|7/17-comma]]
|
|693.099
|
|-
|[[9/22-comma meantone|9/22-comma]]
|
|693.157
|
|-
|[[2/5-comma meantone|2/5-comma]]
|
|693.352
|Close to [[45edo]].
|-
|[[9/23-comma meantone|9/23-comma]]
|
|693.539
|
|-
|[[7/18-comma meantone|7/18-comma]]
|
|693.591
|
|-
|[[5/13-comma meantone|5/13-comma]]
|
|693.683
|
|-
|[[1/(φ+1)-comma meantone|1/(ϕ+1)-comma]]
|
|693.740
|Close to [[64edo]].
|-
|[[8/21-comma meantone|8/21-comma]]
|
|693.762
|
|-
|[[3/8-comma meantone|3/8-comma]]
|
|693.890
|Close to [[83edo]].
|-
|[[7/19-comma meantone|7/19-comma]]
|
|694.032
|
|-
|[[4/11-comma meantone|4/11-comma]]
|
|694.134
|Almost exactly 1/3-''Pythagorean'' comma meantone.
|-
|[[5/14-comma meantone|5/14-comma]]
|
|694.274
|
|-
|[[6/17-comma meantone|6/17-comma]]
|
|694.365
|
|-
|[[7/20-comma meantone|7/20-comma]]
|
|694.428
|
|-
|[[8/23-comma meantone|8/23-comma]]
|
|694.475
|
|-
|[[9/26-comma meantone|9/26-comma]]
|
|694.511
|
|-
|[[1/3-comma meantone|1/3-comma]]
|
|694.786
|Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[9/28-comma meantone|9/28-comma]]
|
|695.042
|
|-
|[[8/25-comma meantone|8/25-comma]]
|
|695.073
|
|-
|[[7/22-comma meantone|7/22-comma]]
|
|695.112
|
|-
|[[6/19-comma meantone|6/19-comma]]
|
|695.164
|
|-
|[[5/16-comma meantone|5/16-comma]]
|
|695.234
|
|-
|[[4/13-comma meantone|4/13-comma]]
|
|695.338
|
|-
|[[3/10-comma meantone|3/10-comma]]
|
|695.503
|Close to [[88edo]] and [[Lucy tuning]]. Historically significant (see [[historical temperaments]]).
|-
|[[5/17-comma meantone|5/17-comma]]
|
|695.630
|Close to [[69edo]].
|-
|[[7/24-comma meantone|7/24-comma]]
|
|695.682
|
|-
|[[2/7-comma meantone|2/7-comma]]
|
|695.810
|Historically significant (see [[historical temperaments]]).
|-
|[[5/18-comma meantone|5/18-comma]]
|
|695.981
|Close to [[50edo]].
|-
|[[3/11-comma meantone|3/11-comma]]
|
|696.090
|
|-
|[[7/26-comma meantone|7/26-comma]]
|
|696.165
|Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/15-comma meantone|4/15-comma]]
|
|696.220
|Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|[[5/19-comma meantone|5/19-comma]]
|
|696.295
|Close to [[81edo]].
|-
|[[Quarter-comma meantone|1/4-comma]]
|
|696.578
|Close to [[7-limit|septimal]] and [[tridecimal]] meantone POTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[5/21-comma meantone|5/21-comma]]
|
|696.834
|Close to [[31edo]].
|-
|[[4/17-comma meantone|4/17-comma]]
|
|696.895
|
|-
|[[3/13-comma meantone|3/13-comma]]
|
|696.992
|Close to [[7-limit|septimal]] & [[tridecimal]] meantone [[CTE]] tunings. Close to [[undecimal]] meantone POTE tuning.
|-
|[[5/22-comma meantone|5/22-comma]]
|
|697.067
|
|-
|[[2/9-comma meantone|2/9-comma]]
|
|697.176
|Close to [[5-limit]] and [[undecimal]] meantone CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone|3/14-comma]]
|
|697.346
|Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/19-comma meantone|4/19-comma]]
|
|697.427
|
|-
|[[1/5-comma meantone|1/5-comma]]
|
|697.654
|Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/21-comma meantone|4/21-comma]]
|
|697.859
|
|-
|[[3/16-comma meantone|3/16-comma]]
|
|697.923
|
|-
|[[2/11-comma meantone|2/11-comma]]
|
|698.045
|Close to [[55edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/17-comma meantone|3/17-comma]]
|
|698.159
|
|-
|[[4/23-comma meantone|4/23-comma]]
|
|698.215
|
|-
|[[1/6-comma meantone|1/6-comma]]
|
|698.371
|Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[4/25-comma meantone|4/25-comma]]
|
|698.514
|Close to [[67edo]].
|-
|[[3/19-comma meantone|3/19-comma]]
|
|698.559
|
|-
|[[2/13-comma meantone|2/13-comma]]
|
|698.646
|Close to [[79edo]].
|-
|[[3/20-comma meantone|3/20-comma]]
|
|698.729
|
|-
|[[1/7-comma meantone|1/7-comma]]
|
|698.883
|Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/22-comma meantone|3/22-comma]]
|
|699.022
|
|-
|[[2/15-comma meantone|2/15-comma]]
|
|699.088
|
|-
|[[1/8-comma meantone|1/8-comma]]
|
|699.267
|
|-
|[[2/17-comma meantone|2/17-comma]]
|
|699.425
|
|-
|[[1/9-comma meantone|1/9-comma]]
|
|699.565
|
|-
|[[2/19-comma meantone|2/19-comma]]
|
|699.691
|
|-
|[[1/10-comma meantone|1/10-comma]]
|
|699.804
|
|-
|[[2/21-comma meantone|2/21-comma]]
|
|699.907
|
|-
|[[1/11-comma meantone|1/11-comma]]
|
|700.000
|Everything up to this point generates 12 and 19 tone MOS scales.
|-
|[[12edo]]
|
|700.000
|The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|[[1/12-comma meantone|1/12-comma]]
|
|700.163
|Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|[[1/13-comma meantone|1/13-comma]]
|
|700.301
|
|-
|[[1/14-comma meantone|1/14-comma]]
|
|700.419
|
|-
|[[1/15-comma meantone|1/15-comma]]
|
|700.521
|
|-
|[[1/16-comma meantone|1/16-comma]]
|
|700.611
|
|-
|[[1/17-comma meantone|1/17-comma]]
|
|700.690
|
|-
|[[1/18-comma meantone|1/18-comma]]
|
|700.760
|
|-
|[[1/19-comma meantone|1/19-comma]]
|
|700.823
|
|-
|[[1/20-comma meantone|1/20-comma]]
|
|700.879
|
|-
|[[1/21-comma meantone|1/21-comma]]
|
|700.931
|
|-
|[[1/22-comma meantone|1/22-comma]]
|
|700.977
|
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Historically significant (see [[historical temperaments]].) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|}
===Tempering out [[136/135]]===
{| class="wikitable mw-collapsible"
|+Mean minor tunings from 1-comma to Pythagorean
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|1-comma
|255.805
|714.732
|
|-
|12/13-comma
|258.753
|713.749
|
|-
|11/12-comma
|259.000
|713.667
|
|-
|10/11-comma
|259.289
|713.570
|
|-
|9/10-comma
|259.638
|713.455
|
|-
|8/9-comma
|260.064
|713.312
|
|-
|7/8-comma
|260.597
|713.135
|
|-
|6/7-comma
|261.281
|712.906
|
|-
|11/13-comma
|261.702
|712.766
|
|-
|5/6-comma
|262.193
|712.602
|
|-
|9/11-comma
|262.774
|712.409
|
|-
|4/5-comma
|263.471
|712.176
|
|-
|7/9-comma
|264.322
|711.892
|
|-
|10/13-comma
|264.650
|711.783
|
|-
|3/4-comma
|264.387
|711.538
|
|-
|8/11-comma
|266.259
|711.247
|
|-
|5/7-comma
|266.756
|711.081
|Even closer to 1/3-comma superpyth than 27edo
|-
|7/10-comma
|267.304
|710.899
|
|-
|9/13-comma
|267.599
|710.800
|
|-
|2/3-comma
|268.582
|710.473
|
|-
|7/11-comma
|269.743
|710.086
|
|-
|5/8-comma
|270.179
|709.940
|
|-
|1/ϕ-comma
|270.446
|709.851
|
|-
|8/13-comma
|270.547
|709.818
|
|-
|3/5-comma
|271.137
|709.621
|
|-
|7/12-comma
|271.776
|709.408
|
|-
|4/7-comma
|272.232
|709.256
|
|-
|5/9-comma
|272.841
|709.053
|Very close to [[22edo]]
|-
|6/11-comma
|273.228
|708.924
|
|-
|7/13-comma
|273.496
|708.835
|Close to 1/4-comma superpyth
|-
|1/2-comma
|274.970
|708.343
|Everything from this point onwards has a minor seventh between 30/17 and 16/9. This is the canonical mean minor tuning in universe
|-
|6/13-comma
|276.444
|707.851
|
|-
|5/11-comma
|276.712
|707.763
|
|-
|4/9-comma
|277.099
|707.634
|
|-
|3/7-comma
|277.708
|707.431
|
|-
|5/12-comma
|278.164
|707.279
|
|-
|2/5-comma
|278.803
|707.066
|
|-
|5/13-comma
|279.393
|706.869
|
|-
|1/(ϕ+1)-comma
|279.494
|706.836
|
|-
|3/8-comma
|279.716
|706.746
|
|-
|4/11-comma
|280.197
|706.601
|
|-
|1/3-comma
|281.358
|706.214
|
|-
|4/13-comma
|282.341
|705.886
|
|-
|3/10-comma
|282.636
|705.788
|
|-
|2/7-comma
|283.184
|705.605
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|3/11-comma
|283.681
|705.440
|
|-
|1/4-comma
|284.552
|705.149
|
|-
|3/13-comma
|285.290
|704.903
|
|-
|2/9-comma
|285.617
|704.794
|
|-
|1/5-comma
|286.469
|704.510
|
|-
|2/11-comma
|287.166
|704.278
|
|-
|1/6-comma
|287.747
|704.084
|
|-
|2/13-comma
|288.238
|703.921
|
|-
|1/7-comma
|288.659
|703.780
|
|-
|1/8-comma
|289.344
|703.552
|
|-
|1/9-comma
|289.876
|703.375
|
|-
|1/10-comma
|290.302
|703.233
|
|-
|1/11-comma
|290.650
|703.117
|
|-
|1/12-comma
|290.941
|703.020
|
|-
|1/13-comma
|291.187
|702.938
|
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 30/17 and 16/9
|}

===Negative harmony theory-defined mean minor (most often approached as Reversed Archytas)===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings Pythagorean to -1/1-comma meantone
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[Pythagorean tuning]]
|294.135
|701.955||Historically significant (see [[historical temperaments]].) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|-
|[[-1/22-comma meantone|-1/22-comma]]
|
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|
|705.538
|Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]]
|
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|
|706.563
|About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]]
|
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[-1/4-comma meantone|-1/4-comma]]
|
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|
|710.148
|
|-
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]]
|
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|
|712.196
|
|-
|[[-1/2-comma meantone|-1/2-comma]]
|
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|
|715.190
|
|-
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|
|715.538
|
|-
|[[-7/11-comma meantone|-7/11-comma]]
|
|715.641
|
|-
|[[-9/14-comma meantone|-9/14-comma]]
|
|715.780
|Close to [[57edo]].
|-
|[[-11/17-comma meantone|-11/17-comma]]
|
|715.871
|
|-
|[[-13/20-comma meantone|-13/20-comma]]
|
|715.934
|
|-
|[[-2/3-comma meantone|-2/3-comma]]
|
|716.293
|Close to [[62edo]].
|-
|[[-15/22 comma meantone|-15/22 comma]]
|
|716.618
|Close to [[67edo]].
|-
|[[-13/19 comma meantone|-13/19 comma]]
|
|716.669
|Close to [[72edo]].
|-
|[[-11/16-comma meantone|-11/16-comma]]
|
|716.741
|
|-
|[[-9/13-comma meantone|-9/13-comma]]
|
|716.844
|Close to [[77edo]].
|-
|[[-7/10-comma meantone|-7/10-comma]]
|
|717.009
|
|-
|[[-12/17-comma meantone|-12/17-comma]]
|
|717.136
|Close to [[82edo]].
|-
|[[-17/24-comma meantone|-17/24-comma]]
|
|717.188
|Close to [[87edo]].
|-
|[[-5/7-comma meantone|-5/7-comma]]
|
|717.317
|Close to [[92edo]].
|-
|[[-13/18-comma meantone|-13/18-comma]]
|
|717.487
|Close to [[97edo]].
|-
|[[-8/11-comma meantone|-8/11-comma]]
|
|717.596
|
|-
|[[-19/26-comma meantone|-19/26-comma]]
|
|717.671
|
|-
|[[-11/15-comma meantone|-11/15-comma]]
|
|717.726
|
|-
|[[-14/19-comma meantone|-14/19-comma]]
|
|717.802
|
|-
|[[-3/4-comma meantone|-3/4-comma]]
|
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|[[-21/26-comma meantone|-21/26-comma]]
|
|718.325
|
|-
|[[-16/21-comma meantone|-16/21-comma]]
|
|718.341
|
|-
|[[-13/17-comma meantone|-13/17-comma]]
|
|718.401
|
|-
|[[-10/13-comma meantone|-10/13-comma]]
|
|718.498
|
|-
|[[-17/22-comma meantone|-17/22-comma]]
|
|718.574
|
|-
|[[-7/9-comma meantone|-7/9-comma]]
|
|718.682
|
|-
|[[-11/14-comma meantone|-11/14-comma]]
|
|718.853
|
|-
|[[-15/19-comma meantone|-15/19-comma]]
|
|718.934
|
|-
|[[-4/5-comma meantone|-4/5-comma]]
|
|719.160
|
|-
|[[-17/21-comma meantone|-17/21-comma]]
|
|719.365
|
|-
|[[-13/16-comma meantone|-13/16-comma]]
|
|719.429
|
|-
|[[-9/11-comma meantone|-9/11-comma]]
|
|719.551
|
|-
|[[-14/17-comma meantone|-14/17-comma]]
|
|719.666
|
|-
|[[-5/6-comma meantone|-5/6-comma]]
|
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|[[5edo]]
|
|720.000
|The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|[[-21/25-comma meantone|-21/25-comma]]
|
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|[[-16/19-comma meantone|-16/19-comma]]
|
|720.066
|
|-
|[[-11/13-comma meantone|-11/13-comma]]
|
|720.153
|
|-
|[[-17/20-comma meantone|-17/20-comma]]
|
|720.235
|
|-
|[[-6/7-comma meantone|-6/7-comma]]
|
|720.399
|
|-
|[[-19/22-comma meantone|-19/22-comma]]
|
|720.529
|
|-
|[[-13/15-comma meantone|-13/15-comma]]
|
|720.594
|
|-
| -[[7/8-comma meantone|7/8-comma]]
|
|720.773
|
|-
|[[-15/17-comma meantone|-15/17-comma]]
|
|720.931
|
|-
|[[-8/9-comma meantone|-8/9-comma]]
|
|721.017
|
|-
|[[-17/19-comma meantone|-17/19-comma]]
|
|721.197
|
|-
|[[-9/10-comma meantone|-9/10-comma]]
|
|721.311
|
|-
|[[-19/21-comma meantone|-19/21-comma]]
|
|721.413
|
|-
|[[-10/11-comma meantone|-10/11-comma]]
|
|721.506
|
|-
|[[-11/12-comma meantone|-11/12-comma]]
|
|721.669
|
|-
|[[-12/13-comma meantone|-12/13-comma]]
|
|721.807
|
|-
|[[-13/14-comma meantone|-13/14-comma]]
|
|721.925
|
|-
|[[-14/15-comma meantone|-14/15-comma]]
|
|722.028
|
|-
|[[-15/16-comma meantone|-15/16-comma]]
|
|722.117
|
|-
|[[-16/17-comma meantone|-16/17-comma]]
|
|722.196
|
|-
|[[-17/18-comma meantone|-17/18-comma]]
|
|722.266
|
|-
|[[-18/19-comma meantone|-18/19-comma]]
|
|722.329
|
|-
|[[-19/20-comma meantone|-19/20-comma]]
|
|722.386
|
|-
|[[-20/21-comma meantone|-20/21-comma]]
|
|722.437
|
|-
|[[-21/22-comma meantone|-21/22-comma]]
|
|722.484
|
|-
|[[-1/1-comma meantone|-1/1-comma]]
|
|723.461
|Close to [[68edo]].
|}

==== Tempering out [[136/135]] ====
{| class="wikitable mw-collapsible"
|+Mean minor tunings from Pythagorean to -1-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 30/17 and 16/9
|-
| -1/13-comma
|297.083
|700.972
|
|-
|<nowiki>-1/12-comma</nowiki>
|297.329
|700.890
|
|-
|<nowiki>-1/11-comma</nowiki>
|297.620
|700.793
|
|-
| -1/10-comma
|297.968
|700.677
|
|-
| -1/9-comma
|298.394
|700.535
|
|-
| -1/8-comma
|298.926
|700.358
|
|-
| -1/7-comma
|299.611
|700.130
|
|-
| -2/13-comma
|300.032
|699.989
|
|-
| -1/6-comma
|300.523
|699.826
|
|-
|<nowiki>-2/11-comma</nowiki>
|301.104
|699.632
|
|-
| -1/5-comma
|301.801
|699.400
|
|-
| -2/9-comma
|302.653
|699.116
|
|-
| -3/13-comma
|302.980
|699.007
|
|-
| -1/4-comma
|303.718
|698.761
|
|-
|<nowiki>-3/11-comma</nowiki>
|304.589
|698.470
|
|-
| -2/7-comma
|305.086
|698.305
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -3/10-comma
|305.634
|698.122
|
|-
| -4/13-comma
|305.929
|698.024
|
|-
| -1/3-comma
|306.911
|697.696
|
|-
|<nowiki>-4/11-comma</nowiki>
|308.073
|697.309
|
|-
| -3/8-comma
|308.509
|697.164
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|308.776
|697.075
|
|-
| -5/13-comma
|308.877
|697.041
|
|-
| -2/5-comma
|309.467
|696.844
|
|-
|<nowiki>-5/12-comma</nowiki>
|310.106
|696.631
|Almost [[quarter-comma meantone]] tuning
|-
| -3/7-comma
|310.562
|696.479
|
|-
| -4/9-comma
|311.171
|696.276
|
|-
|<nowiki>-5/11-comma</nowiki>
|311.558
|696.147
|
|-
| -6/13-comma
|311.826
|696.058
|
|-
| -1/2-comma
|313.300
|695.567
|Everything from this point onwards does not have a minor seventh between 16/9 and 2176/1215.
|-
| -7/13-comma
|314.774
|695.075
|
|-
|<nowiki>-6/11-comma</nowiki>
|315.042
|694.986
|
|-
| -5/9-comma
|315.429
|694.857
|
|-
| -4/7-comma
|316.038
|694.654
|
|-
|<nowiki>-7/12-comma</nowiki>
|316.494
|694.502
|
|-
| -3/5-comma
|317.133
|694.289
|
|-
| -8/13-comma
|317.723
|694.092
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|317.824
|694.058
|
|-
| -5/8-comma
|318.091
|693.970
|
|-
|<nowiki>-7/11-comma</nowiki>
|318.527
|693.824
|
|-
| -2/3-comma
|319.688
|693.437
|
|-
| -9/13-comma
|320.671
|693.110
|
|-
| -7/10-comma
|320.966
|693.011
|
|-
| -5/7-comma
|321.514
|692.829
|
|-
|<nowiki>-8/11-comma</nowiki>
|322.011
|692.663
|
|-
| -3/4-comma
|322.883
|692.372
|
|-
| -10/13-comma
|323.620
|692.127
|
|-
| -7/9-comma
|323.947
|692.018
|
|-
| -4/5-comma
|324.799
|691.734
|
|-
|<nowiki>-9/11-comma</nowiki>
|325.496
|691.501
|
|-
| -5/6-comma
|326.077
|691.308
|
|-
| -11/13-comma
|326.568
|691.145
|
|-
| -6/7-comma
|326.989
|691.004
|
|-
| -7/8-comma
|327.674
|690.775
|
|-
| -8/9-comma
|328.206
|690.598
|
|-
| -9/10-comma
|328.632
|690.456
|
|-
|<nowiki>-10/11-comma</nowiki>
|328.980
|690.340
|
|-
| -11/12-comma
|329.271
|690.243
|
|-
| -12/13-comma
|329.517
|690.161
|
|-
|-
| -1-comma
|332.465
|689.178
|}

===Beyond Negative harmony theory-defined mean minor (most often approached as superdiatonic)===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings -1/1-comma to -2/1-comma meantone
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[-1/1-comma meantone|-1/1-comma]]
|
|723.461||Close to [[68edo]].
|-
|[[-1/22-comma meantone|-1/22-comma]]
|
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|
|705.538
|Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]]
|
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|
|706.563
|About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]]
|
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[-1/4-comma meantone|-1/4-comma]]
|
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|
|710.148
|
|-
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]]
|
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|
|712.196
|
|-
|[[-1/2-comma meantone|-1/2-comma]]
|
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|
|715.190
|
|-
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|
|715.538
|
|-
|[[-7/11-comma meantone|-7/11-comma]]
|
|715.641
|
|-
|[[-9/14-comma meantone|-9/14-comma]]
|
|715.780
|Close to [[57edo]].
|-
|[[-11/17-comma meantone|-11/17-comma]]
|
|715.871
|
|-
|[[-13/20-comma meantone|-13/20-comma]]
|
|715.934
|
|-
|[[-2/3-comma meantone|-2/3-comma]]
|
|716.293
|Close to [[62edo]].
|-
|[[-15/22 comma meantone|-15/22 comma]]
|
|716.618
|Close to [[67edo]].
|-
|[[-13/19 comma meantone|-13/19 comma]]
|
|716.669
|Close to [[72edo]].
|-
|[[-11/16-comma meantone|-11/16-comma]]
|
|716.741
|
|-
|[[-9/13-comma meantone|-9/13-comma]]
|
|716.844
|Close to [[77edo]].
|-
|[[-7/10-comma meantone|-7/10-comma]]
|
|717.009
|
|-
|[[-12/17-comma meantone|-12/17-comma]]
|
|717.136
|Close to [[82edo]].
|-
|[[-17/24-comma meantone|-17/24-comma]]
|
|717.188
|Close to [[87edo]].
|-
|[[-5/7-comma meantone|-5/7-comma]]
|
|717.317
|Close to [[92edo]].
|-
|[[-13/18-comma meantone|-13/18-comma]]
|
|717.487
|Close to [[97edo]].
|-
|[[-8/11-comma meantone|-8/11-comma]]
|
|717.596
|
|-
|[[-19/26-comma meantone|-19/26-comma]]
|
|717.671
|
|-
|[[-11/15-comma meantone|-11/15-comma]]
|
|717.726
|
|-
|[[-14/19-comma meantone|-14/19-comma]]
|
|717.802
|
|-
|[[-3/4-comma meantone|-3/4-comma]]
|
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|[[-21/26-comma meantone|-21/26-comma]]
|
|718.325
|
|-
|[[-16/21-comma meantone|-16/21-comma]]
|
|718.341
|
|-
|[[-13/17-comma meantone|-13/17-comma]]
|
|718.401
|
|-
|[[-10/13-comma meantone|-10/13-comma]]
|
|718.498
|
|-
|[[-17/22-comma meantone|-17/22-comma]]
|
|718.574
|
|-
|[[-7/9-comma meantone|-7/9-comma]]
|
|718.682
|
|-
|[[-11/14-comma meantone|-11/14-comma]]
|
|718.853
|
|-
|[[-15/19-comma meantone|-15/19-comma]]
|
|718.934
|
|-
|[[-4/5-comma meantone|-4/5-comma]]
|
|719.160
|
|-
|[[-17/21-comma meantone|-17/21-comma]]
|
|719.365
|
|-
|[[-13/16-comma meantone|-13/16-comma]]
|
|719.429
|
|-
|[[-9/11-comma meantone|-9/11-comma]]
|
|719.551
|
|-
|[[-14/17-comma meantone|-14/17-comma]]
|
|719.666
|
|-
|[[-5/6-comma meantone|-5/6-comma]]
|
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|[[5edo]]
|
|720.000
|The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|[[-21/25-comma meantone|-21/25-comma]]
|
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|[[-16/19-comma meantone|-16/19-comma]]
|
|720.066
|
|-
|[[-11/13-comma meantone|-11/13-comma]]
|
|720.153
|
|-
|[[-17/20-comma meantone|-17/20-comma]]
|
|720.235
|
|-
|[[-6/7-comma meantone|-6/7-comma]]
|
|720.399
|
|-
|[[-19/22-comma meantone|-19/22-comma]]
|
|720.529
|
|-
|[[-13/15-comma meantone|-13/15-comma]]
|
|720.594
|
|-
| -[[7/8-comma meantone|7/8-comma]]
|
|720.773
|
|-
|[[-15/17-comma meantone|-15/17-comma]]
|
|720.931
|
|-
|[[-8/9-comma meantone|-8/9-comma]]
|
|721.017
|
|-
|[[-17/19-comma meantone|-17/19-comma]]
|
|721.197
|
|-
|[[-9/10-comma meantone|-9/10-comma]]
|
|721.311
|
|-
|[[-19/21-comma meantone|-19/21-comma]]
|
|721.413
|
|-
|[[-10/11-comma meantone|-10/11-comma]]
|
|721.506
|
|-
|[[-11/12-comma meantone|-11/12-comma]]
|
|721.669
|
|-
|[[-12/13-comma meantone|-12/13-comma]]
|
|721.807
|
|-
|[[-13/14-comma meantone|-13/14-comma]]
|
|721.925
|
|-
|[[-14/15-comma meantone|-14/15-comma]]
|
|722.028
|
|-
|[[-15/16-comma meantone|-15/16-comma]]
|
|722.117
|
|-
|[[-16/17-comma meantone|-16/17-comma]]
|
|722.196
|
|-
|[[-17/18-comma meantone|-17/18-comma]]
|
|722.266
|
|-
|[[-18/19-comma meantone|-18/19-comma]]
|
|722.329
|
|-
|[[-19/20-comma meantone|-19/20-comma]]
|
|722.386
|
|-
|[[-20/21-comma meantone|-20/21-comma]]
|
|722.437
|
|-
|[[-21/22-comma meantone|-21/22-comma]]
|
|722.484
|
|-
|[[-1/1-comma meantone|-1/1-comma]]
|
|723.461
|Close to [[68edo]].
|}

==== Tempering out [[136/135]] ====
{| class="wikitable mw-collapsible"
|+Mean minor tunings from -1-comma to -2-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
| -1-comma
|332.465
|689.178
|
|-
| -14/13-comma
|335.414
|688.195
|
|-
| -13/12-comma
|335.659
|688.114
|
|-
| -12/11-comma
|335.950
|688.017
|
|-
| -11/10-comma
|336.298
|687.901
|
|-
| -10/9-comma
|336.724
|687.759
|
|-
| -9/8-comma
|337.256
|687.581
|
|-
| -8/7-comma
|337.941
|687.353
|
|-
| -15/13-comma
|338.362
|687.213
|
|-
| -7/6-comma
|338.853
|687.049
|
|-
| -13/11-comma
|339.434
|686.855
|
|-
| -6/5-comma
|340.131
|686.623
|
|-
| -11/9-comma
|340.983
|686.339
|
|-
| -16/13-comma
|341.340
|686.230
|
|-
| -5/4-comma
|342.048
|685.984
|
|-
|[[7edo]]
|342.857
|685,714
|
|-
| -14/11-comma
|342.919
|685.694
|
|-
| -9/7-comma
|343.417
|685.528
|
|-
| -13/10-comma
|343.964
|685.345
|
|-
| -17/13-comma
|344.259
|685.247
|
|-
| -4/3-comma
|345.242
|684.919
|
|-
| -15/11-comma
|346.403
|684.532
|
|-
| -11/8-comma
|346.839
|684.387
|
|-
| -(ϕ+2)/(ϕ+1)-comma
|347.106
|684.298
|
|-
| -18/13-comma
|347.207
|684.264
|
|-
| -7/5-comma
|347.797
|684.068
|
|-
| -17/12-comma
|348.436
|683.855
|
|-
| -10/7-comma
|348.892
|683.703
|
|-
| -13/9-comma
|349.501
|683.500
|
|-
| -16/11-comma
|349.888
|683.371
|
|-
| -19/13-comma
|350.156
|683.281
|
|-
| -3/2-comma
|351.630
|682.790
|
|-
| -20/13-comma
|353.104
|682.299
|Close to [[93edo]]
|-
| -17/11-comma
|353.372
|682.209
|Close to [[88edo]]
|-
| -14/9-comma
|353.760
|682.080
|Close to [[83edo]]
|-
| -11/7-comma
|354.368
|681.877
|Close to [[78edo]]
|-
| -19/12-comma
|354.824
|681.725
|Close to [[73edo]]
|-
| -8/5-comma
|355.463
|681.512
|Close to [[68edo]].
|-
| -21/13-comma
|356.053
|681.315
|
|-
| -ϕ-comma
|356.154
|681.282
|
|-
| -13/8-comma
|356.421
|681.193
|Close to [[63edo]]
|-
| -18/11-comma
|356.857
|681.048
|
|-
| -5/3-comma
|358.018
|680.661
|Close to [[53edo]]
|-
| -22/13-comma
|359.001
|680.333
|
|-
| -17/10-comma
|359.296
|680.235
|
|-
| -12/7-comma
|359.844
|680.052
|Close to [[30edo]]
|-
| -19/11-comma
|360.341
|679.886
|
|-
| -7/4-comma
|361.213
|679.596
|Close to [[83edo]]
|-
| -23/13-comma
|361.950
|679.350
|Close to [[53edo]]
|-
| -16/9-comma
|362.277
|679.241
|
|-
| -9/5-comma
|363.129
|678.957
|Close to [[76edo]]
|-
| -20/11-comma
|363.826
|678.725
|Close to [[99edo]]
|-
| -11/6-comma
|364.407
|678.531
|
|-
| -24/13-comma
|364.898
|678.367
|
|-
| -13/7-comma
|365.319
|678.227
|Close to [[23edo]]
|-
| -15/8-comma
|366.004
|677.999
|
|-
| -17/9-comma
|366.536
|677.821
|
|-
| -19/10-comma
|366.962
|677.679
|Close to [[85edo]]
|-
| -21/11-comma
|367.311
|677.563
|
|-
| -23/12-comma
|367.601
|677.466
|Close to [[62edo]]
|-
| -25/13-comma
|367.847
|677.384
|
|-
|-
| -2-comma
|370.795
|676.402
|Close to [[28edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-02-02T19:49:57Z

Moremajorthanmajor: /* The table */

{{Editable user page}}Here are all mean minor tunings that can be written in the form "m/n-comma mean minor", where the '''43-limit Johnston''' comma ([[129/128]]) or the '''diatisma''' ([[136/135]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or 13 or smaller. The former range is almost the same as the range of m/n-comma Archytas and reverse Archytas temperaments and often confused for it in modern practice.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 [often “confused for 3402:4096:5103 vs. 4096:5103:6144 or 3510:4096:5265 vs. 4096:5265:6144”] and 34:40:51 vs. 40:51:60), the mean minor triad is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]] (as this comma is already unnoticeable, the Boethian interpretation is more commonly used as a joke); but with more complex thirds, and tempering out the quarter tone of [[1053/1024]] (often confused for the simpler [[36/35]]) or [[33/32]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds. That is why doing the latter is considered to generate a perverse temperament, the “perversion” being that a recognizable quarter tone is greater than 36.09 cents, though only the former quarter tone is considered to generate the “real” one.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, and, by extension, any comma is considered as primarily applied to the generator, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 (often “confused for 567/512 or 72/65”) or 30/17 and 16/9.

== The table==
=== Beyond historically-defined mean minor (most often approached as oneirotonic or superdiatonic)===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings 2/1-comma to 1/1-comma meantone
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[2/1-comma meantone|2/1-comma]]
|423.173||658.942||Close to [[51edo]].
|-
|[[43/22-comma meantone|43/22-comma]]
|420.240
|659.920
|
|-
|[[41/21-comma meantone|41/21-comma]]
|420.100
|659.967
|
|-
|[[39/20-comma meantone|39/20-comma]]
|419.947
|660.018
|Close to [[30edo|20edo]]
|-
|[[37/19-comma meantone|37/19-comma]]
|419.777
|660.074
|
|-
|[[35/18-comma meantone|35/18-comma]]
|419.588
|660.137
|
|-
|[[33/17-comma meantone|33/17-comma]]
|419.378
|660.207
|
|-
|[[31/16-comma meantone|31/16-comma]]
|419.140||660.287||
|-
|[[29/15-comma meantone|29/15-comma]]
|418.871||660.376||
|-
|[[27/14-comma meantone|27/14-comma]]
|418.564||660.479||
|-
|[[25/13-comma meantone|25/13-comma]]
|418.210||660.597||
|-
|[[23/12-comma meantone|23/12-comma]]
|417.796||660.735||Close to [[89edo]].
|-
|[[21/11-comma meantone|21/11-comma]]
|417.307||660.898||Close to [[69edo]].
|-
|[[40/21-comma meantone|40/21-comma]]
|417.028
|660.990
|
|-
|[[19/10-comma meantone|19/10-comma]]
|416.721||661.093||
|-
|[[36/19-comma meantone|36/19-comma]]
|416.381
|661.206
|Close to [[49edo]].
|-
|[[17/9-comma meantone|17/9-comma]]
|416.004||661.332||Close to [[58edo]].
|-
|[[32/17-comma meantone|32/17-comma]]
|415.582
|661.473
|Close to [[78edo]].
|-
|[[15/8-comma meantone|15/8-comma]]
|415.108||661.631||
|-
|[[28/15-comma meantone|28/15-comma]]
|414.570||661.810||
|-
|[[41/22-comma meantone|41/22-comma]]
|414.375
|661.875
|
|-
|[[13/7-comma meantone|13/7-comma]]
|413.956||662.015||Close to [[29edo]].
|-
|[[37/20-comma meantone|37/20-comma]]
|413.495
|662.168
|
|-
|[[11/13-comma meantone|24/13-comma]]
|413.247||662.251||
|-
|[[35/19-comma meantone|35/19-comma]]
|412.986
|662.338
|
|-
|[[46/25-comma meantone|46/25-comma]]
|412.850
|662.383
|
|-
|[[11/6-comma meantone|11/6-comma]]
|412.420||662.527||Close to [[96edo]].
|-
|[[31/17-comma meantone|31/17-comma]]
|411.787
|662.738
|Close to [[67edo]].
|-
|[[20/11-comma meantone|20/11-comma]]
|411.442||662.853||
|-
|[[13/16-comma meantone|29/16-comma]]
|411.075||662.975||Close to [[100edo]].
|-
|[[38/21-comma meantone|38/21-comma]]
|410.883
|663.039
|
|-
|[[9/5-comma meantone|9/5-comma]]
|410.269||663.244||Close to [[38edo]]
|-
|[[34/19-comma meantone|34/19-comma]]
|409.590
|663.470
|
|-
|[[25/14-comma meantone|25/14-comma]]
|409.347||663.551||Close to [[85edo]]
|-
|[[16/9-comma meantone|16/9-comma]]
|408.835||663.722||
|-
|[[39/22-comma meantone|39/22-comma]]
|408.509
|663.830
|Almost exactly [[47edo]]
|-
|[[10/13-comma meantone|23/13-comma]]
|408.284||663.905||
|-
|[[30/17-comma meantone|30/17-comma]]
|407.992
|664.003
|
|-
|[[16/21-comma meantone|37/21-comma]]
|407.811
|664.063
|
|-
|[[7/4-comma meantone|7/4-comma]]
|407.043||664.319||Close to [[56edo]]
|-
|[[33/19-comma meantone|33/19-comma]]
|406.194
|664.602
|Close to [[65edo]].
|-
|[[26/15-comma meantone|26/15-comma]]
|405.978||664.677||Everything up to this point generates 20 and 29 tone MOS scales.
|-
|[[45/26-comma meantone|45/26-comma]]
|405.802
|664.733
|Everything from this point onwards generates 16 and 25 tone MOS scales.
|-
|[[8/11-comma meantone|19/11-comma]]
|405.577||664.808||
|-
|[[13/18-comma meantone|31/18-comma]]
|405.251
|664.916
|Close to [[74edo]].
|-
|[[12/7-comma meantone|12/7-comma]]
|404.739||665.087||Close to [[83edo]].
|-
|[[41/24-comma meantone|41/24-comma]]
|404.355
|665.215
|Almost exactly [[92edo]].
|-
|[[29/17-comma meantone|29/17-comma]]
|404.197
|665.268
|
|-
|[[7/10-comma meantone|17/10-comma]]
|403.817||665.394||
|-
|[[9edo]]
|400.000||666.667||The largest MOS scale this can generate is 9 tone. '''Lower boundary of proper superdiatonic MOS scales.'''
|-
|[[9/13-comma meantone|9/13-comma]]
| ||687.066||
|-
|[[11/16-comma meantone|11/16-comma]]
| ||687.169||
|-
|[[13/19-comma meantone|13/19-comma]]
|
|687.240
|
|-
|[[15/22-comma meantone|15/22-comma]]
|
|687.292
|
|-
|[[17/25-comma meantone|17/25-comma]]
|
|687.331
|
|-
|[[19/28-comma]]
|
|687.361
|
|-
|[[2/3-comma meantone|2/3-comma]]
| ||687.617||
|-
|[[17/26-comma meantone|17/26-comma]]
|
|687.893
|Close to [[82edo]].
|-
|[[15/23-comma meantone|15/23-comma]]
|
|687.929
|
|-
|[[13/20-comma meantone|13/20-comma]]
|
|687.976
|
|-
|[[11/17-comma meantone|11/17-comma]]
|
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone|9/14-comma]]
| ||688.129||
|-
|[[7/11-comma meantone|7/11-comma]]
| ||688.269||Close to [[68edo]].
|-
|[[12/19-comma meantone|12/19-comma]]
|
|688.372
|
|-
|[[5/8-comma meantone|5/8-comma]]
| ||688.514||Close to [[61edo]] and [[43/32]].
|-
|[[13/21-comma meantone|13/21-comma]]
|
|688.641
|
|-
|[[1/φ-comma meantone|1/ϕ-comma]]
|
|688.663
|
|-
|[[8/13-comma meantone|8/13-comma]]
| ||688.720||
|-
|[[11/18-comma meantone|11/18-comma]]
|
|688.812
|Close to [[54edo]].
|-
|[[14/23-comma meantone|14/23-comma]]
|
|688.864
|
|-
|[[3/5-comma meantone|3/5-comma]]
| ||689.051||
|-
|[[13/22-comma meantone|13/22-comma]]
|
|689.247
|
|-
|[[10/17-comma meantone|10/17-comma]]
|
|689.304
|
|-
|[[7/12-comma meantone|7/12-comma]]
| ||689.410||Close to [[47edo]].
|-
|[[11/19-comma meantone|11/19-comma]]
|
|689.504
|
|-
|[[4/7-comma meantone|4/7-comma]]
| ||689.666||Close to [[87edo]].
|-
|[[9/16-comma meantone|9/16-comma]]
| ||689.858||
|-
|[[5/9-comma meantone|5/9-comma]]
| ||690.007||Close to [[40edo]].
|-
|[[11/20-comma meantone|11/20-comma]]
|
|690.127
|
|-
|[[6/11-comma meantone|6/11-comma]]
| ||690.224||
|-
|[[7/13-comma meantone|7/13-comma]]
| ||690.375||Close to [[73edo]].
|-
|[[8/15-comma meantone|8/15-comma]]
| ||690.485||
|-
|[[9/17-comma meantone|9/17-comma]]
|
|690.569
|
|-
|[[10/19-comma meantone|10/19-comma]]
|
|690.636
|
|-
|[[11/21-comma meantone|11/21-comma]]
|
|690.690
| Close to [[33edo]]
|-
|[[1/2-comma meantone|1/2-comma]]
| ||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not have a whole tone between 10/9 and 9/8.
|-
|[[10/21-comma meantone|10/21-comma]]
|
|691.714
|
|-
|[[9/19-comma meantone|9/19-comma]]
|
|691.768
|Close to [[85edo]].
|-
|[[8/17-comma meantone|8/17-comma]]
|
|691.834
|
|-
|[[7/15-comma meantone|7/15-comma]]
|
|691.919
|
|-
|[[6/13-comma meantone|6/13-comma]]
|
|692.029
|
|-
|[[5/11-comma meantone|5/11-comma]]
|
|692.179
|
|-
|[[9/20-comma meantone|9/20-comma]]
|
|692.277
|Close to [[26edo]].
|-
|[[4/9-comma meantone|4/9-comma]]
|
|692.397
|
|-
|[[7/16-comma meantone|7/16-comma]]
|
|692.546
|
|-
|[[3/7-comma meantone|3/7-comma]]
|
|692.738
|
|-
|[[8/19-comma meantone|8/19-comma]]
|
|692.899
|
|-
|[[5/12-comma meantone|5/12-comma]]
|
|692.994
|Close to [[71edo]].
|-
|[[7/17-comma meantone|7/17-comma]]
|
|693.099
|
|-
|[[9/22-comma meantone|9/22-comma]]
|
|693.157
|
|-
|[[2/5-comma meantone|2/5-comma]]
|
|693.352
|Close to [[45edo]].
|-
|[[9/23-comma meantone|9/23-comma]]
|
|693.539
|
|-
|[[7/18-comma meantone|7/18-comma]]
|
|693.591
|
|-
|[[5/13-comma meantone|5/13-comma]]
|
|693.683
|
|-
|[[1/(φ+1)-comma meantone|1/(ϕ+1)-comma]]
|
|693.740
|Close to [[64edo]].
|-
|[[8/21-comma meantone|8/21-comma]]
|
|693.762
|
|-
|[[3/8-comma meantone|3/8-comma]]
|
|693.890
|Close to [[83edo]].
|-
|[[7/19-comma meantone|7/19-comma]]
|
|694.032
|
|-
|[[4/11-comma meantone|4/11-comma]]
|
|694.134
|Almost exactly 1/3-''Pythagorean'' comma meantone.
|-
|[[5/14-comma meantone|5/14-comma]]
|
|694.274
|
|-
|[[6/17-comma meantone|6/17-comma]]
|
|694.365
|
|-
|[[7/20-comma meantone|7/20-comma]]
|
|694.428
|
|-
|[[8/23-comma meantone|8/23-comma]]
|
|694.475
|
|-
|[[9/26-comma meantone|9/26-comma]]
|
|694.511
|
|-
|[[1/3-comma meantone|1/3-comma]]
|
|694.786
|Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[9/28-comma meantone|9/28-comma]]
|
|695.042
|
|-
|[[8/25-comma meantone|8/25-comma]]
|
|695.073
|
|-
|[[7/22-comma meantone|7/22-comma]]
|
|695.112
|
|-
|[[6/19-comma meantone|6/19-comma]]
|
|695.164
|
|-
|[[5/16-comma meantone|5/16-comma]]
|
|695.234
|
|-
|[[4/13-comma meantone|4/13-comma]]
|
|695.338
|
|-
|[[3/10-comma meantone|3/10-comma]]
|
|695.503
|Close to [[88edo]] and [[Lucy tuning]]. Historically significant (see [[historical temperaments]]).
|-
|[[5/17-comma meantone|5/17-comma]]
|
|695.630
|Close to [[69edo]].
|-
|[[7/24-comma meantone|7/24-comma]]
|
|695.682
|
|-
|[[2/7-comma meantone|2/7-comma]]
|
|695.810
|Historically significant (see [[historical temperaments]]).
|-
|[[5/18-comma meantone|5/18-comma]]
|
|695.981
|Close to [[50edo]].
|-
|[[3/11-comma meantone|3/11-comma]]
|
|696.090
|
|-
|[[7/26-comma meantone|7/26-comma]]
|
|696.165
|Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/15-comma meantone|4/15-comma]]
|
|696.220
|Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|[[5/19-comma meantone|5/19-comma]]
|
|696.295
|Close to [[81edo]].
|-
|[[Quarter-comma meantone|1/4-comma]]
|
|696.578
|Close to [[7-limit|septimal]] and [[tridecimal]] meantone POTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[5/21-comma meantone|5/21-comma]]
|
|696.834
|Close to [[31edo]].
|-
|[[4/17-comma meantone|4/17-comma]]
|
|696.895
|
|-
|[[3/13-comma meantone|3/13-comma]]
|
|696.992
|Close to [[7-limit|septimal]] & [[tridecimal]] meantone [[CTE]] tunings. Close to [[undecimal]] meantone POTE tuning.
|-
|[[5/22-comma meantone|5/22-comma]]
|
|697.067
|
|-
|[[2/9-comma meantone|2/9-comma]]
|
|697.176
|Close to [[5-limit]] and [[undecimal]] meantone CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone|3/14-comma]]
|
|697.346
|Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/19-comma meantone|4/19-comma]]
|
|697.427
|
|-
|[[1/5-comma meantone|1/5-comma]]
|
|697.654
|Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/21-comma meantone|4/21-comma]]
|
|697.859
|
|-
|[[3/16-comma meantone|3/16-comma]]
|
|697.923
|
|-
|[[2/11-comma meantone|2/11-comma]]
|
|698.045
|Close to [[55edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/17-comma meantone|3/17-comma]]
|
|698.159
|
|-
|[[4/23-comma meantone|4/23-comma]]
|
|698.215
|
|-
|[[1/6-comma meantone|1/6-comma]]
|
|698.371
|Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[4/25-comma meantone|4/25-comma]]
|
|698.514
|Close to [[67edo]].
|-
|[[3/19-comma meantone|3/19-comma]]
|
|698.559
|
|-
|[[2/13-comma meantone|2/13-comma]]
|
|698.646
|Close to [[79edo]].
|-
|[[3/20-comma meantone|3/20-comma]]
|
|698.729
|
|-
|[[1/7-comma meantone|1/7-comma]]
|
|698.883
|Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/22-comma meantone|3/22-comma]]
|
|699.022
|
|-
|[[2/15-comma meantone|2/15-comma]]
|
|699.088
|
|-
|[[1/8-comma meantone|1/8-comma]]
|
|699.267
|
|-
|[[2/17-comma meantone|2/17-comma]]
|
|699.425
|
|-
|[[1/9-comma meantone|1/9-comma]]
|
|699.565
|
|-
|[[2/19-comma meantone|2/19-comma]]
|
|699.691
|
|-
|[[1/10-comma meantone|1/10-comma]]
|
|699.804
|
|-
|[[2/21-comma meantone|2/21-comma]]
|
|699.907
|
|-
|[[1/11-comma meantone|1/11-comma]]
|
|700.000
|Everything up to this point generates 12 and 19 tone MOS scales.
|-
|[[12edo]]
|
|700.000
|The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|[[1/12-comma meantone|1/12-comma]]
|
|700.163
|Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|[[1/13-comma meantone|1/13-comma]]
|
|700.301
|
|-
|[[1/14-comma meantone|1/14-comma]]
|
|700.419
|
|-
|[[1/15-comma meantone|1/15-comma]]
|
|700.521
|
|-
|[[1/16-comma meantone|1/16-comma]]
|
|700.611
|
|-
|[[1/17-comma meantone|1/17-comma]]
|
|700.690
|
|-
|[[1/18-comma meantone|1/18-comma]]
|
|700.760
|
|-
|[[1/19-comma meantone|1/19-comma]]
|
|700.823
|
|-
|[[1/20-comma meantone|1/20-comma]]
|
|700.879
|
|-
|[[1/21-comma meantone|1/21-comma]]
|
|700.931
|
|-
|[[1/22-comma meantone|1/22-comma]]
|
|700.977
|
|-
|[[1/1-comma meantone|1/1-comma]]
|
|680.449
|Close to [[30edo]]
|}
====Tempering out [[136/135]]====
{| class="wikitable mw-collapsible"
|+Mean minor tunings from 2-comma to [[7edo|1-comma]]
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|2-comma
|217.475
|727.508
|
|-
|25/13-comma
|220.423
|726.526
|
|-
|23/12-comma
|220.669
|726.444
|
|-
|21/11-comma
|220.959
|726.347
|
|-
|19/10-comma
|221.308
|726.231
|
|-
|17/9-comma
|221.734
|726.089
|
|-
|15/8-comma
|222.266
|725.911
|
|-
|13/7-comma
|222.951
|725.683
|
|-
|24/13-comma
|223.371
|725.543
|
|-
|11/6-comma
|223.863
|725.378
|
|-
|20/11-comma
|224.444
|725.185
|
|-
|9/5-comma
|225.141
|724.953
|
|-
|16/9-comma
|225.993
|724.669
|
|-
|23/13-comma
|226.320
|724.560
|
|-
|7/4-comma
|227.057
|724.314
|
|-
|19/11-comma
|227.928
|724.024
|
|-
|12/7-comma
|228.426
|723.858
|
|-
|17/10-comma
|228.974
|723.675
|
|-
|22/13-comma
|229.269
|723.577
|
|-
|5/3-comma
|230.252
|723.249
|
|-
|18/11-comma
|231.413
|722.862
|
|-
|13/8-comma
|231.849
|722.717
|
|-
|ϕ-comma
|232.116
|722.628
|
|-
|21/13-comma
|232.217
|722.594
|
|-
|8/5-comma
|232.807
|722.398
|
|-
|19/12-comma
|233.446
|722.185
|
|-
|11/7-comma
|233.902
|722.933
|
|-
|14/9-comma
|234.510
|721.830
|
|-
|17/11-comma
|356.599
|721.701
|
|-
|20/13-comma
|235.166
|721.611
|
|-
|3/2-comma
|236.640
|721.120
|
|-
|19/13-comma
|238.114
|720.628
|
|-
|16/11-comma
|238.382
|720.539
|
|-
|13/9-comma
|238.769
|720.410
|
|-
|10/7-comma
|239.378
|720.207
|
|-
|17/12-comma
|239.834
|720.055
|
|-
|[[5edo]]
|240.000
|720.000
|
|-
|7/5-comma
|240.473
|719.842
|
|-
|18/13-comma
|241.063
|719.646
|
|-
|(ϕ+2)/(ϕ+1)-comma
|241.164
|719.612
|
|-
|11/8-comma
|241.431
|719.533
|
|-
|15/11-comma
|241.867
|719.378
|
|-
|4/3-comma
|243.028
|719.900
|
|-
|17/13-comma
|244.011
|718.663
|
|-
|13/10-comma
|244.306
|718.565
|
|-
|9/7-comma
|244.835
|718.382
|
|-
|14/11-comma
|245.352
|718.216
|
|-
|5/4-comma
|246.222
|717.926
|
|-
|16/13-comma
|246.960
|717.680
|
|-
|11/9-comma
|247.287
|717.571
|
|-
|6/5-comma
|248.139
|717.287
|
|-
|13/11-comma
|248.836
|717.055
|
|-
|7/6-comma
|249.417
|716.861
|
|-
|15/13-comma
|249.908
|716.697
|
|-
|8/7-comma
|250.329
|716.557
|
|-
|9/8-comma
|251.013
|716.329
|
|-
|10/9-comma
|251.546
|716.151
|
|-
|11/10-comma
|251.972
|716.009
|
|-
|12/11-comma
|252.320
|715.833
|
|-
|13/12-comma
|252.611
|715.796
|
|-
|14/13-comma
|252.856
|715.715
|
|-
|1-comma
|255.805
|714.732
|
|}

===Historically-defined mean minor===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings 1/1-comma meantone to Pythagorean
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[1/1-comma meantone|1/1-comma]]
| ||680.449||Close to [[30edo]]
|-
|[[21/22-comma meantone|21/22-comma]]
|
|681.426
|Close to [[37edo]].
|-
|[[20/21-comma meantone|20/21-comma]]
|
|681.473
|
|-
|[[19/20-comma meantone|19/20-comma]]
|
|681.524
|
|-
|[[18/19-comma meantone|18/19-comma]]
|
|681.581
|
|-
|[[17/18-comma meantone|17/18-comma]]
|
|681.644
|
|-
|[[16/17-comma meantone|16/17-comma]]
|
|681.713
|
|-
|[[15/16-comma meantone|15/16-comma]]
| ||681.793||Close to [[44edo]].
|-
|[[14/15-comma meantone|14/15-comma]]
| ||681.883||
|-
|[[13/14-comma meantone|13/14-comma]]
| ||681.985||
|-
|[[12/13-comma meantone|12/13-comma]]
| ||682.103||
|-
|[[11/12-comma meantone|11/12-comma]]
| ||682.241||
|-
|[[10/11-comma meantone|10/11-comma]]
| ||682.404||Close to [[51edo]].
|-
|[[19/21-comma meantone|19/21-comma]]
|
|682.497
|
|-
|[[9/10-comma meantone|9/10-comma]]
| ||682.599||
|-
|[[17/19-comma meantone|17/19-comma]]
|
|682.713
|
|-
|[[8/9-comma meantone|8/9-comma]]
| ||682.838||Close to [[58edo]].
|-
|[[15/17-comma meantone|15/17-comma]]
|
|682.979
|
|-
|[[7/8-comma meantone|7/8-comma]]
| ||683.137||Close to [[65edo]].
|-
|[[13/15-comma meantone|13/15-comma]]
| ||683.316||Close to [[72edo]].
|-
|[[19/22-comma meantone|19/22-comma]]
|
|683.381
|
|-
|[[6/7-comma meantone|6/7-comma]]
| ||683.521||Close to [[79edo]].
|-
|[[17/20-comma meantone|17/20-comma]]
|
|683.675
|
|-
|[[11/13-comma meantone|11/13-comma]]
| ||683.757||Close to [[86edo]].
|-
|[[16/19-comma meantone|16/19-comma]]
|
|683.844
|
|-
|[[21/25-comma meantone|21/25-comma]]
|
|683.890
|Close to [[93edo]]
|-
|[[5/6-comma meantone|5/6-comma]]
| ||684.033||Close to [[100edo]].
|-
|[[14/17-comma meantone|14/17-comma]]
|
|684.244
|
|-
|[[9/11-comma meantone|9/11-comma]]
| ||684.359||
|-
|[[13/16-comma meantone|13/16-comma]]
| ||684.481||
|-
|[[17/21-comma meantone|17/21-comma]]
|
|684.545
|
|-
|[[4/5-comma meantone|4/5-comma]]
| ||684.750||
|-
|[[15/19-comma meantone|15/19-comma]]
|
|684.976
|
|-
|[[11/14-comma meantone|11/14-comma]]
| ||685.057||
|-
|[[7/9-comma meantone|7/9-comma]]
| ||685.228||
|-
|[[17/22-comma meantone|17/22-comma]]
|
|685.337
|
|-
|[[10/13-comma meantone|10/13-comma]]
| ||685.412||
|-
|[[13/17-comma meantone|13/17-comma]]
|
|685.509
|
|-
|[[16/21-comma meantone|16/21-comma]]
|
|685.569
|Everything up to this point generates 9 and 16 tone MOS scales.
|-
|[[7edo]]
| ||685.714||The largest MOS scale this can generate is 7 tone. '''Lower boundary of 5-limit diamond monotone.'''
|-
|[[3/4-comma meantone|3/4-comma]]
| ||685.825||Everything from this point onwards generates 12 and 19 tone MOS scales.
|-
|[[14/19-comma meantone|14/19-comma]]
|
|686.108
|
|-
|[[11/15-comma meantone|11/15-comma]]
| ||686.184||
|-
|[[19/26-comma meantone|19/26-comma]]
|
|686.239
|
|-
|[[8/11-comma meantone|8/11-comma]]
| ||686.314||
|-
|[[13/18-comma meantone|13/18-comma]]
|
|686.423
|
|-
|[[5/7-comma meantone|5/7-comma]]
| ||686.593||
|-
|[[17/24-comma meantone|17/24-comma]]
|
|686.721
|
|-
|[[12/17-comma meantone|12/17-comma]]
|
|686.774
|
|-
|[[7/10-comma meantone|7/10-comma]]
| ||686.901||
|-
|[[9/13-comma meantone|9/13-comma]]
| ||687.066||
|-
|[[11/16-comma meantone|11/16-comma]]
| ||687.169||
|-
|[[13/19-comma meantone|13/19-comma]]
|
|687.240
|
|-
|[[15/22-comma meantone|15/22-comma]]
|
|687.292
|
|-
|[[17/25-comma meantone|17/25-comma]]
|
|687.331
|
|-
|[[19/28-comma]]
|
|687.361
|
|-
|[[2/3-comma meantone|2/3-comma]]
| ||687.617|| Close to [[89edo]].
|-
|[[17/26-comma meantone|17/26-comma]]
|
|687.893
|Close to [[82edo]].
|-
|[[15/23-comma meantone|15/23-comma]]
|
|687.929
|
|-
|[[13/20-comma meantone|13/20-comma]]
|
|687.976
|
|-
|[[11/17-comma meantone|11/17-comma]]
|
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone|9/14-comma]]
| ||688.129||
|-
|[[7/11-comma meantone|7/11-comma]]
| ||688.269||Close to [[68edo]].
|-
|[[12/19-comma meantone|12/19-comma]]
|
|688.372
|
|-
|[[5/8-comma meantone|5/8-comma]]
| ||688.514||Close to [[61edo]] and [[43/32]].
|-
|[[13/21-comma meantone|13/21-comma]]
|
|688.641
|
|-
|[[1/φ-comma meantone|1/ϕ-comma]]
|
|688.663
|
|-
|[[8/13-comma meantone|8/13-comma]]
| ||688.720||
|-
|[[11/18-comma meantone|11/18-comma]]
|
|688.812
|Close to [[54edo]].
|-
|[[14/23-comma meantone|14/23-comma]]
|
|688.864
|
|-
|[[3/5-comma meantone|3/5-comma]]
| ||689.051||
|-
|[[13/22-comma meantone|13/22-comma]]
|
|689.247
|
|-
|[[10/17-comma meantone|10/17-comma]]
|
|689.304
|
|-
|[[7/12-comma meantone|7/12-comma]]
| ||689.410||Close to [[47edo]].
|-
|[[11/19-comma meantone|11/19-comma]]
|
|689.504
|
|-
|[[4/7-comma meantone|4/7-comma]]
| ||689.666||Close to [[87edo]].
|-
|[[9/16-comma meantone|9/16-comma]]
| ||689.858||
|-
|[[5/9-comma meantone|5/9-comma]]
| ||690.007||Close to [[40edo]].
|-
|[[11/20-comma meantone|11/20-comma]]
|
|690.127
|
|-
|[[6/11-comma meantone|6/11-comma]]
| ||690.224||
|-
|[[7/13-comma meantone|7/13-comma]]
| ||690.375||Close to [[73edo]].
|-
|[[8/15-comma meantone|8/15-comma]]
| ||690.485||
|-
|[[9/17-comma meantone|9/17-comma]]
|
|690.569
|
|-
|[[10/19-comma meantone|10/19-comma]]
|
|690.636
|
|-
|[[11/21-comma meantone|11/21-comma]]
|
|690.690
| Close to [[33edo]]
|-
|[[1/2-comma meantone|1/2-comma]]
| ||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not have a whole tone between 10/9 and 9/8.
|-
|[[10/21-comma meantone|10/21-comma]]
|
|691.714
|
|-
|[[9/19-comma meantone|9/19-comma]]
|
|691.768
|Close to [[85edo]].
|-
|[[8/17-comma meantone|8/17-comma]]
|
|691.834
|
|-
|[[7/15-comma meantone|7/15-comma]]
|
|691.919
|
|-
|[[6/13-comma meantone|6/13-comma]]
|
|692.029
|
|-
|[[5/11-comma meantone|5/11-comma]]
|
|692.179
|
|-
|[[9/20-comma meantone|9/20-comma]]
|
|692.277
|Close to [[26edo]].
|-
|[[4/9-comma meantone|4/9-comma]]
|
|692.397
|
|-
|[[7/16-comma meantone|7/16-comma]]
|
|692.546
|
|-
|[[3/7-comma meantone|3/7-comma]]
|
|692.738
|
|-
|[[8/19-comma meantone|8/19-comma]]
|
|692.899
|
|-
|[[5/12-comma meantone|5/12-comma]]
|
|692.994
|Close to [[71edo]].
|-
|[[7/17-comma meantone|7/17-comma]]
|
|693.099
|
|-
|[[9/22-comma meantone|9/22-comma]]
|
|693.157
|
|-
|[[2/5-comma meantone|2/5-comma]]
|
|693.352
|Close to [[45edo]].
|-
|[[9/23-comma meantone|9/23-comma]]
|
|693.539
|
|-
|[[7/18-comma meantone|7/18-comma]]
|
|693.591
|
|-
|[[5/13-comma meantone|5/13-comma]]
|
|693.683
|
|-
|[[1/(φ+1)-comma meantone|1/(ϕ+1)-comma]]
|
|693.740
|Close to [[64edo]].
|-
|[[8/21-comma meantone|8/21-comma]]
|
|693.762
|
|-
|[[3/8-comma meantone|3/8-comma]]
|
|693.890
|Close to [[83edo]].
|-
|[[7/19-comma meantone|7/19-comma]]
|
|694.032
|
|-
|[[4/11-comma meantone|4/11-comma]]
|
|694.134
|Almost exactly 1/3-''Pythagorean'' comma meantone.
|-
|[[5/14-comma meantone|5/14-comma]]
|
|694.274
|
|-
|[[6/17-comma meantone|6/17-comma]]
|
|694.365
|
|-
|[[7/20-comma meantone|7/20-comma]]
|
|694.428
|
|-
|[[8/23-comma meantone|8/23-comma]]
|
|694.475
|
|-
|[[9/26-comma meantone|9/26-comma]]
|
|694.511
|
|-
|[[1/3-comma meantone|1/3-comma]]
|
|694.786
|Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[9/28-comma meantone|9/28-comma]]
|
|695.042
|
|-
|[[8/25-comma meantone|8/25-comma]]
|
|695.073
|
|-
|[[7/22-comma meantone|7/22-comma]]
|
|695.112
|
|-
|[[6/19-comma meantone|6/19-comma]]
|
|695.164
|
|-
|[[5/16-comma meantone|5/16-comma]]
|
|695.234
|
|-
|[[4/13-comma meantone|4/13-comma]]
|
|695.338
|
|-
|[[3/10-comma meantone|3/10-comma]]
|
|695.503
|Close to [[88edo]] and [[Lucy tuning]]. Historically significant (see [[historical temperaments]]).
|-
|[[5/17-comma meantone|5/17-comma]]
|
|695.630
|Close to [[69edo]].
|-
|[[7/24-comma meantone|7/24-comma]]
|
|695.682
|
|-
|[[2/7-comma meantone|2/7-comma]]
|
|695.810
|Historically significant (see [[historical temperaments]]).
|-
|[[5/18-comma meantone|5/18-comma]]
|
|695.981
|Close to [[50edo]].
|-
|[[3/11-comma meantone|3/11-comma]]
|
|696.090
|
|-
|[[7/26-comma meantone|7/26-comma]]
|
|696.165
|Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/15-comma meantone|4/15-comma]]
|
|696.220
|Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|[[5/19-comma meantone|5/19-comma]]
|
|696.295
|Close to [[81edo]].
|-
|[[Quarter-comma meantone|1/4-comma]]
|
|696.578
|Close to [[7-limit|septimal]] and [[tridecimal]] meantone POTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[5/21-comma meantone|5/21-comma]]
|
|696.834
|Close to [[31edo]].
|-
|[[4/17-comma meantone|4/17-comma]]
|
|696.895
|
|-
|[[3/13-comma meantone|3/13-comma]]
|
|696.992
|Close to [[7-limit|septimal]] & [[tridecimal]] meantone [[CTE]] tunings. Close to [[undecimal]] meantone POTE tuning.
|-
|[[5/22-comma meantone|5/22-comma]]
|
|697.067
|
|-
|[[2/9-comma meantone|2/9-comma]]
|
|697.176
|Close to [[5-limit]] and [[undecimal]] meantone CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone|3/14-comma]]
|
|697.346
|Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/19-comma meantone|4/19-comma]]
|
|697.427
|
|-
|[[1/5-comma meantone|1/5-comma]]
|
|697.654
|Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/21-comma meantone|4/21-comma]]
|
|697.859
|
|-
|[[3/16-comma meantone|3/16-comma]]
|
|697.923
|
|-
|[[2/11-comma meantone|2/11-comma]]
|
|698.045
|Close to [[55edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/17-comma meantone|3/17-comma]]
|
|698.159
|
|-
|[[4/23-comma meantone|4/23-comma]]
|
|698.215
|
|-
|[[1/6-comma meantone|1/6-comma]]
|
|698.371
|Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[4/25-comma meantone|4/25-comma]]
|
|698.514
|Close to [[67edo]].
|-
|[[3/19-comma meantone|3/19-comma]]
|
|698.559
|
|-
|[[2/13-comma meantone|2/13-comma]]
|
|698.646
|Close to [[79edo]].
|-
|[[3/20-comma meantone|3/20-comma]]
|
|698.729
|
|-
|[[1/7-comma meantone|1/7-comma]]
|
|698.883
|Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/22-comma meantone|3/22-comma]]
|
|699.022
|
|-
|[[2/15-comma meantone|2/15-comma]]
|
|699.088
|
|-
|[[1/8-comma meantone|1/8-comma]]
|
|699.267
|
|-
|[[2/17-comma meantone|2/17-comma]]
|
|699.425
|
|-
|[[1/9-comma meantone|1/9-comma]]
|
|699.565
|
|-
|[[2/19-comma meantone|2/19-comma]]
|
|699.691
|
|-
|[[1/10-comma meantone|1/10-comma]]
|
|699.804
|
|-
|[[2/21-comma meantone|2/21-comma]]
|
|699.907
|
|-
|[[1/11-comma meantone|1/11-comma]]
|
|700.000
|Everything up to this point generates 12 and 19 tone MOS scales.
|-
|[[12edo]]
|
|700.000
|The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|[[1/12-comma meantone|1/12-comma]]
|
|700.163
|Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|[[1/13-comma meantone|1/13-comma]]
|
|700.301
|
|-
|[[1/14-comma meantone|1/14-comma]]
|
|700.419
|
|-
|[[1/15-comma meantone|1/15-comma]]
|
|700.521
|
|-
|[[1/16-comma meantone|1/16-comma]]
|
|700.611
|
|-
|[[1/17-comma meantone|1/17-comma]]
|
|700.690
|
|-
|[[1/18-comma meantone|1/18-comma]]
|
|700.760
|
|-
|[[1/19-comma meantone|1/19-comma]]
|
|700.823
|
|-
|[[1/20-comma meantone|1/20-comma]]
|
|700.879
|
|-
|[[1/21-comma meantone|1/21-comma]]
|
|700.931
|
|-
|[[1/22-comma meantone|1/22-comma]]
|
|700.977
|
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Historically significant (see [[historical temperaments]].) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|}
===Tempering out [[136/135]]===
{| class="wikitable mw-collapsible"
|+Mean minor tunings from 1-comma to Pythagorean
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|1-comma
|255.805
|714.732
|
|-
|12/13-comma
|258.753
|713.749
|
|-
|11/12-comma
|259.000
|713.667
|
|-
|10/11-comma
|259.289
|713.570
|
|-
|9/10-comma
|259.638
|713.455
|
|-
|8/9-comma
|260.064
|713.312
|
|-
|7/8-comma
|260.597
|713.135
|
|-
|6/7-comma
|261.281
|712.906
|
|-
|11/13-comma
|261.702
|712.766
|
|-
|5/6-comma
|262.193
|712.602
|
|-
|9/11-comma
|262.774
|712.409
|
|-
|4/5-comma
|263.471
|712.176
|
|-
|7/9-comma
|264.322
|711.892
|
|-
|10/13-comma
|264.650
|711.783
|
|-
|3/4-comma
|264.387
|711.538
|
|-
|8/11-comma
|266.259
|711.247
|
|-
|5/7-comma
|266.756
|711.081
|Even closer to 1/3-comma superpyth than 27edo
|-
|7/10-comma
|267.304
|710.899
|
|-
|9/13-comma
|267.599
|710.800
|
|-
|2/3-comma
|268.582
|710.473
|
|-
|7/11-comma
|269.743
|710.086
|
|-
|5/8-comma
|270.179
|709.940
|
|-
|1/ϕ-comma
|270.446
|709.851
|
|-
|8/13-comma
|270.547
|709.818
|
|-
|3/5-comma
|271.137
|709.621
|
|-
|7/12-comma
|271.776
|709.408
|
|-
|4/7-comma
|272.232
|709.256
|
|-
|5/9-comma
|272.841
|709.053
|Very close to [[22edo]]
|-
|6/11-comma
|273.228
|708.924
|
|-
|7/13-comma
|273.496
|708.835
|Close to 1/4-comma superpyth
|-
|1/2-comma
|274.970
|708.343
|Everything from this point onwards has a minor seventh between 30/17 and 16/9. This is the canonical mean minor tuning in universe
|-
|6/13-comma
|276.444
|707.851
|
|-
|5/11-comma
|276.712
|707.763
|
|-
|4/9-comma
|277.099
|707.634
|
|-
|3/7-comma
|277.708
|707.431
|
|-
|5/12-comma
|278.164
|707.279
|
|-
|2/5-comma
|278.803
|707.066
|
|-
|5/13-comma
|279.393
|706.869
|
|-
|1/(ϕ+1)-comma
|279.494
|706.836
|
|-
|3/8-comma
|279.716
|706.746
|
|-
|4/11-comma
|280.197
|706.601
|
|-
|1/3-comma
|281.358
|706.214
|
|-
|4/13-comma
|282.341
|705.886
|
|-
|3/10-comma
|282.636
|705.788
|
|-
|2/7-comma
|283.184
|705.605
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|3/11-comma
|283.681
|705.440
|
|-
|1/4-comma
|284.552
|705.149
|
|-
|3/13-comma
|285.290
|704.903
|
|-
|2/9-comma
|285.617
|704.794
|
|-
|1/5-comma
|286.469
|704.510
|
|-
|2/11-comma
|287.166
|704.278
|
|-
|1/6-comma
|287.747
|704.084
|
|-
|2/13-comma
|288.238
|703.921
|
|-
|1/7-comma
|288.659
|703.780
|
|-
|1/8-comma
|289.344
|703.552
|
|-
|1/9-comma
|289.876
|703.375
|
|-
|1/10-comma
|290.302
|703.233
|
|-
|1/11-comma
|290.650
|703.117
|
|-
|1/12-comma
|290.941
|703.020
|
|-
|1/13-comma
|291.187
|702.938
|
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 30/17 and 16/9
|}

===Negative harmony theory-defined mean minor (most often approached as Reversed Archytas)===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings Pythagorean to -1/1-comma meantone
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[Pythagorean tuning]]
|294.135
|701.955||Historically significant (see [[historical temperaments]].) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|-
|[[-1/22-comma meantone|-1/22-comma]]
|
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|
|705.538
|Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]]
|
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|
|706.563
|About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]]
|
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[-1/4-comma meantone|-1/4-comma]]
|
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|
|710.148
|
|-
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]]
|
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|
|712.196
|
|-
|[[-1/2-comma meantone|-1/2-comma]]
|
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|
|715.190
|
|-
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|
|715.538
|
|-
|[[-7/11-comma meantone|-7/11-comma]]
|
|715.641
|
|-
|[[-9/14-comma meantone|-9/14-comma]]
|
|715.780
|Close to [[57edo]].
|-
|[[-11/17-comma meantone|-11/17-comma]]
|
|715.871
|
|-
|[[-13/20-comma meantone|-13/20-comma]]
|
|715.934
|
|-
|[[-2/3-comma meantone|-2/3-comma]]
|
|716.293
|Close to [[62edo]].
|-
|[[-15/22 comma meantone|-15/22 comma]]
|
|716.618
|Close to [[67edo]].
|-
|[[-13/19 comma meantone|-13/19 comma]]
|
|716.669
|Close to [[72edo]].
|-
|[[-11/16-comma meantone|-11/16-comma]]
|
|716.741
|
|-
|[[-9/13-comma meantone|-9/13-comma]]
|
|716.844
|Close to [[77edo]].
|-
|[[-7/10-comma meantone|-7/10-comma]]
|
|717.009
|
|-
|[[-12/17-comma meantone|-12/17-comma]]
|
|717.136
|Close to [[82edo]].
|-
|[[-17/24-comma meantone|-17/24-comma]]
|
|717.188
|Close to [[87edo]].
|-
|[[-5/7-comma meantone|-5/7-comma]]
|
|717.317
|Close to [[92edo]].
|-
|[[-13/18-comma meantone|-13/18-comma]]
|
|717.487
|Close to [[97edo]].
|-
|[[-8/11-comma meantone|-8/11-comma]]
|
|717.596
|
|-
|[[-19/26-comma meantone|-19/26-comma]]
|
|717.671
|
|-
|[[-11/15-comma meantone|-11/15-comma]]
|
|717.726
|
|-
|[[-14/19-comma meantone|-14/19-comma]]
|
|717.802
|
|-
|[[-3/4-comma meantone|-3/4-comma]]
|
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|[[-21/26-comma meantone|-21/26-comma]]
|
|718.325
|
|-
|[[-16/21-comma meantone|-16/21-comma]]
|
|718.341
|
|-
|[[-13/17-comma meantone|-13/17-comma]]
|
|718.401
|
|-
|[[-10/13-comma meantone|-10/13-comma]]
|
|718.498
|
|-
|[[-17/22-comma meantone|-17/22-comma]]
|
|718.574
|
|-
|[[-7/9-comma meantone|-7/9-comma]]
|
|718.682
|
|-
|[[-11/14-comma meantone|-11/14-comma]]
|
|718.853
|
|-
|[[-15/19-comma meantone|-15/19-comma]]
|
|718.934
|
|-
|[[-4/5-comma meantone|-4/5-comma]]
|
|719.160
|
|-
|[[-17/21-comma meantone|-17/21-comma]]
|
|719.365
|
|-
|[[-13/16-comma meantone|-13/16-comma]]
|
|719.429
|
|-
|[[-9/11-comma meantone|-9/11-comma]]
|
|719.551
|
|-
|[[-14/17-comma meantone|-14/17-comma]]
|
|719.666
|
|-
|[[-5/6-comma meantone|-5/6-comma]]
|
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|[[5edo]]
|
|720.000
|The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|[[-21/25-comma meantone|-21/25-comma]]
|
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|[[-16/19-comma meantone|-16/19-comma]]
|
|720.066
|
|-
|[[-11/13-comma meantone|-11/13-comma]]
|
|720.153
|
|-
|[[-17/20-comma meantone|-17/20-comma]]
|
|720.235
|
|-
|[[-6/7-comma meantone|-6/7-comma]]
|
|720.399
|
|-
|[[-19/22-comma meantone|-19/22-comma]]
|
|720.529
|
|-
|[[-13/15-comma meantone|-13/15-comma]]
|
|720.594
|
|-
| -[[7/8-comma meantone|7/8-comma]]
|
|720.773
|
|-
|[[-15/17-comma meantone|-15/17-comma]]
|
|720.931
|
|-
|[[-8/9-comma meantone|-8/9-comma]]
|
|721.017
|
|-
|[[-17/19-comma meantone|-17/19-comma]]
|
|721.197
|
|-
|[[-9/10-comma meantone|-9/10-comma]]
|
|721.311
|
|-
|[[-19/21-comma meantone|-19/21-comma]]
|
|721.413
|
|-
|[[-10/11-comma meantone|-10/11-comma]]
|
|721.506
|
|-
|[[-11/12-comma meantone|-11/12-comma]]
|
|721.669
|
|-
|[[-12/13-comma meantone|-12/13-comma]]
|
|721.807
|
|-
|[[-13/14-comma meantone|-13/14-comma]]
|
|721.925
|
|-
|[[-14/15-comma meantone|-14/15-comma]]
|
|722.028
|
|-
|[[-15/16-comma meantone|-15/16-comma]]
|
|722.117
|
|-
|[[-16/17-comma meantone|-16/17-comma]]
|
|722.196
|
|-
|[[-17/18-comma meantone|-17/18-comma]]
|
|722.266
|
|-
|[[-18/19-comma meantone|-18/19-comma]]
|
|722.329
|
|-
|[[-19/20-comma meantone|-19/20-comma]]
|
|722.386
|
|-
|[[-20/21-comma meantone|-20/21-comma]]
|
|722.437
|
|-
|[[-21/22-comma meantone|-21/22-comma]]
|
|722.484
|
|-
|[[-1/1-comma meantone|-1/1-comma]]
|
|723.461
|Close to [[68edo]].
|}

==== Tempering out [[136/135]] ====
{| class="wikitable mw-collapsible"
|+Mean minor tunings from Pythagorean to -1-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 30/17 and 16/9
|-
| -1/13-comma
|297.083
|700.972
|
|-
|<nowiki>-1/12-comma</nowiki>
|297.329
|700.890
|
|-
|<nowiki>-1/11-comma</nowiki>
|297.620
|700.793
|
|-
| -1/10-comma
|297.968
|700.677
|
|-
| -1/9-comma
|298.394
|700.535
|
|-
| -1/8-comma
|298.926
|700.358
|
|-
| -1/7-comma
|299.611
|700.130
|
|-
| -2/13-comma
|300.032
|699.989
|
|-
| -1/6-comma
|300.523
|699.826
|
|-
|<nowiki>-2/11-comma</nowiki>
|301.104
|699.632
|
|-
| -1/5-comma
|301.801
|699.400
|
|-
| -2/9-comma
|302.653
|699.116
|
|-
| -3/13-comma
|302.980
|699.007
|
|-
| -1/4-comma
|303.718
|698.761
|
|-
|<nowiki>-3/11-comma</nowiki>
|304.589
|698.470
|
|-
| -2/7-comma
|305.086
|698.305
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -3/10-comma
|305.634
|698.122
|
|-
| -4/13-comma
|305.929
|698.024
|
|-
| -1/3-comma
|306.911
|697.696
|
|-
|<nowiki>-4/11-comma</nowiki>
|308.073
|697.309
|
|-
| -3/8-comma
|308.509
|697.164
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|308.776
|697.075
|
|-
| -5/13-comma
|308.877
|697.041
|
|-
| -2/5-comma
|309.467
|696.844
|
|-
|<nowiki>-5/12-comma</nowiki>
|310.106
|696.631
|Almost [[quarter-comma meantone]] tuning
|-
| -3/7-comma
|310.562
|696.479
|
|-
| -4/9-comma
|311.171
|696.276
|
|-
|<nowiki>-5/11-comma</nowiki>
|311.558
|696.147
|
|-
| -6/13-comma
|311.826
|696.058
|
|-
| -1/2-comma
|313.300
|695.567
|Everything from this point onwards does not have a minor seventh between 16/9 and 2176/1215.
|-
| -7/13-comma
|314.774
|695.075
|
|-
|<nowiki>-6/11-comma</nowiki>
|315.042
|694.986
|
|-
| -5/9-comma
|315.429
|694.857
|
|-
| -4/7-comma
|316.038
|694.654
|
|-
|<nowiki>-7/12-comma</nowiki>
|316.494
|694.502
|
|-
| -3/5-comma
|317.133
|694.289
|
|-
| -8/13-comma
|317.723
|694.092
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|317.824
|694.058
|
|-
| -5/8-comma
|318.091
|693.970
|
|-
|<nowiki>-7/11-comma</nowiki>
|318.527
|693.824
|
|-
| -2/3-comma
|319.688
|693.437
|
|-
| -9/13-comma
|320.671
|693.110
|
|-
| -7/10-comma
|320.966
|693.011
|
|-
| -5/7-comma
|321.514
|692.829
|
|-
|<nowiki>-8/11-comma</nowiki>
|322.011
|692.663
|
|-
| -3/4-comma
|322.883
|692.372
|
|-
| -10/13-comma
|323.620
|692.127
|
|-
| -7/9-comma
|323.947
|692.018
|
|-
| -4/5-comma
|324.799
|691.734
|
|-
|<nowiki>-9/11-comma</nowiki>
|325.496
|691.501
|
|-
| -5/6-comma
|326.077
|691.308
|
|-
| -11/13-comma
|326.568
|691.145
|
|-
| -6/7-comma
|326.989
|691.004
|
|-
| -7/8-comma
|327.674
|690.775
|
|-
| -8/9-comma
|328.206
|690.598
|
|-
| -9/10-comma
|328.632
|690.456
|
|-
|<nowiki>-10/11-comma</nowiki>
|328.980
|690.340
|
|-
| -11/12-comma
|329.271
|690.243
|
|-
| -12/13-comma
|329.517
|690.161
|
|-
|-
| -1-comma
|332.465
|689.178
|}

===Beyond Negative harmony theory-defined mean minor (most often approached as superdiatonic)===

==== Ideal, tempering out [[81/80]] ====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings -1/1-comma to -2/1-comma meantone
!mean minor Temperament
!third!!Generator (cents)!!Comments
|-
|[[-1/1-comma meantone|-1/1-comma]]
|
|723.461||Close to [[68edo]].
|-
|[[-1/22-comma meantone|-1/22-comma]]
|
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|
|705.538
|Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]]
|
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|
|706.563
|About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]]
|
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[-1/4-comma meantone|-1/4-comma]]
|
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|
|710.148
|
|-
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]]
|
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|
|712.196
|
|-
|[[-1/2-comma meantone|-1/2-comma]]
|
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|
|715.190
|
|-
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|
|715.538
|
|-
|[[-7/11-comma meantone|-7/11-comma]]
|
|715.641
|
|-
|[[-9/14-comma meantone|-9/14-comma]]
|
|715.780
|Close to [[57edo]].
|-
|[[-11/17-comma meantone|-11/17-comma]]
|
|715.871
|
|-
|[[-13/20-comma meantone|-13/20-comma]]
|
|715.934
|
|-
|[[-2/3-comma meantone|-2/3-comma]]
|
|716.293
|Close to [[62edo]].
|-
|[[-15/22 comma meantone|-15/22 comma]]
|
|716.618
|Close to [[67edo]].
|-
|[[-13/19 comma meantone|-13/19 comma]]
|
|716.669
|Close to [[72edo]].
|-
|[[-11/16-comma meantone|-11/16-comma]]
|
|716.741
|
|-
|[[-9/13-comma meantone|-9/13-comma]]
|
|716.844
|Close to [[77edo]].
|-
|[[-7/10-comma meantone|-7/10-comma]]
|
|717.009
|
|-
|[[-12/17-comma meantone|-12/17-comma]]
|
|717.136
|Close to [[82edo]].
|-
|[[-17/24-comma meantone|-17/24-comma]]
|
|717.188
|Close to [[87edo]].
|-
|[[-5/7-comma meantone|-5/7-comma]]
|
|717.317
|Close to [[92edo]].
|-
|[[-13/18-comma meantone|-13/18-comma]]
|
|717.487
|Close to [[97edo]].
|-
|[[-8/11-comma meantone|-8/11-comma]]
|
|717.596
|
|-
|[[-19/26-comma meantone|-19/26-comma]]
|
|717.671
|
|-
|[[-11/15-comma meantone|-11/15-comma]]
|
|717.726
|
|-
|[[-14/19-comma meantone|-14/19-comma]]
|
|717.802
|
|-
|[[-3/4-comma meantone|-3/4-comma]]
|
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|[[-21/26-comma meantone|-21/26-comma]]
|
|718.325
|
|-
|[[-16/21-comma meantone|-16/21-comma]]
|
|718.341
|
|-
|[[-13/17-comma meantone|-13/17-comma]]
|
|718.401
|
|-
|[[-10/13-comma meantone|-10/13-comma]]
|
|718.498
|
|-
|[[-17/22-comma meantone|-17/22-comma]]
|
|718.574
|
|-
|[[-7/9-comma meantone|-7/9-comma]]
|
|718.682
|
|-
|[[-11/14-comma meantone|-11/14-comma]]
|
|718.853
|
|-
|[[-15/19-comma meantone|-15/19-comma]]
|
|718.934
|
|-
|[[-4/5-comma meantone|-4/5-comma]]
|
|719.160
|
|-
|[[-17/21-comma meantone|-17/21-comma]]
|
|719.365
|
|-
|[[-13/16-comma meantone|-13/16-comma]]
|
|719.429
|
|-
|[[-9/11-comma meantone|-9/11-comma]]
|
|719.551
|
|-
|[[-14/17-comma meantone|-14/17-comma]]
|
|719.666
|
|-
|[[-5/6-comma meantone|-5/6-comma]]
|
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|[[5edo]]
|
|720.000
|The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|[[-21/25-comma meantone|-21/25-comma]]
|
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|[[-16/19-comma meantone|-16/19-comma]]
|
|720.066
|
|-
|[[-11/13-comma meantone|-11/13-comma]]
|
|720.153
|
|-
|[[-17/20-comma meantone|-17/20-comma]]
|
|720.235
|
|-
|[[-6/7-comma meantone|-6/7-comma]]
|
|720.399
|
|-
|[[-19/22-comma meantone|-19/22-comma]]
|
|720.529
|
|-
|[[-13/15-comma meantone|-13/15-comma]]
|
|720.594
|
|-
| -[[7/8-comma meantone|7/8-comma]]
|
|720.773
|
|-
|[[-15/17-comma meantone|-15/17-comma]]
|
|720.931
|
|-
|[[-8/9-comma meantone|-8/9-comma]]
|
|721.017
|
|-
|[[-17/19-comma meantone|-17/19-comma]]
|
|721.197
|
|-
|[[-9/10-comma meantone|-9/10-comma]]
|
|721.311
|
|-
|[[-19/21-comma meantone|-19/21-comma]]
|
|721.413
|
|-
|[[-10/11-comma meantone|-10/11-comma]]
|
|721.506
|
|-
|[[-11/12-comma meantone|-11/12-comma]]
|
|721.669
|
|-
|[[-12/13-comma meantone|-12/13-comma]]
|
|721.807
|
|-
|[[-13/14-comma meantone|-13/14-comma]]
|
|721.925
|
|-
|[[-14/15-comma meantone|-14/15-comma]]
|
|722.028
|
|-
|[[-15/16-comma meantone|-15/16-comma]]
|
|722.117
|
|-
|[[-16/17-comma meantone|-16/17-comma]]
|
|722.196
|
|-
|[[-17/18-comma meantone|-17/18-comma]]
|
|722.266
|
|-
|[[-18/19-comma meantone|-18/19-comma]]
|
|722.329
|
|-
|[[-19/20-comma meantone|-19/20-comma]]
|
|722.386
|
|-
|[[-20/21-comma meantone|-20/21-comma]]
|
|722.437
|
|-
|[[-21/22-comma meantone|-21/22-comma]]
|
|722.484
|
|-
|[[-1/1-comma meantone|-1/1-comma]]
|
|723.461
|Close to [[68edo]].
|}

==== Tempering out [[136/135]] ====
{| class="wikitable mw-collapsible"
|+Mean minor tunings from -1-comma to -2-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
| -1-comma
|332.465
|689.178
|
|-
| -14/13-comma
|335.414
|688.195
|
|-
| -13/12-comma
|335.659
|688.114
|
|-
| -12/11-comma
|335.950
|688.017
|
|-
| -11/10-comma
|336.298
|687.901
|
|-
| -10/9-comma
|336.724
|687.759
|
|-
| -9/8-comma
|337.256
|687.581
|
|-
| -8/7-comma
|337.941
|687.353
|
|-
| -15/13-comma
|338.362
|687.213
|
|-
| -7/6-comma
|338.853
|687.049
|
|-
| -13/11-comma
|339.434
|686.855
|
|-
| -6/5-comma
|340.131
|686.623
|
|-
| -11/9-comma
|340.983
|686.339
|
|-
| -16/13-comma
|341.340
|686.230
|
|-
| -5/4-comma
|342.048
|685.984
|
|-
|[[7edo]]
|342.857
|685,714
|
|-
| -14/11-comma
|342.919
|685.694
|
|-
| -9/7-comma
|343.417
|685.528
|
|-
| -13/10-comma
|343.964
|685.345
|
|-
| -17/13-comma
|344.259
|685.247
|
|-
| -4/3-comma
|345.242
|684.919
|
|-
| -15/11-comma
|346.403
|684.532
|
|-
| -11/8-comma
|346.839
|684.387
|
|-
| -(ϕ+2)/(ϕ+1)-comma
|347.106
|684.298
|
|-
| -18/13-comma
|347.207
|684.264
|
|-
| -7/5-comma
|347.797
|684.068
|
|-
| -17/12-comma
|348.436
|683.855
|
|-
| -10/7-comma
|348.892
|683.703
|
|-
| -13/9-comma
|349.501
|683.500
|
|-
| -16/11-comma
|349.888
|683.371
|
|-
| -19/13-comma
|350.156
|683.281
|
|-
| -3/2-comma
|351.630
|682.790
|
|-
| -20/13-comma
|353.104
|682.299
|Close to [[93edo]]
|-
| -17/11-comma
|353.372
|682.209
|Close to [[88edo]]
|-
| -14/9-comma
|353.760
|682.080
|Close to [[83edo]]
|-
| -11/7-comma
|354.368
|681.877
|Close to [[78edo]]
|-
| -19/12-comma
|354.824
|681.725
|Close to [[73edo]]
|-
| -8/5-comma
|355.463
|681.512
|Close to [[68edo]].
|-
| -21/13-comma
|356.053
|681.315
|
|-
| -ϕ-comma
|356.154
|681.282
|
|-
| -13/8-comma
|356.421
|681.193
|Close to [[63edo]]
|-
| -18/11-comma
|356.857
|681.048
|
|-
| -5/3-comma
|358.018
|680.661
|Close to [[53edo]]
|-
| -22/13-comma
|359.001
|680.333
|
|-
| -17/10-comma
|359.296
|680.235
|
|-
| -12/7-comma
|359.844
|680.052
|Close to [[30edo]]
|-
| -19/11-comma
|360.341
|679.886
|
|-
| -7/4-comma
|361.213
|679.596
|Close to [[83edo]]
|-
| -23/13-comma
|361.950
|679.350
|Close to [[53edo]]
|-
| -16/9-comma
|362.277
|679.241
|
|-
| -9/5-comma
|363.129
|678.957
|Close to [[76edo]]
|-
| -20/11-comma
|363.826
|678.725
|Close to [[99edo]]
|-
| -11/6-comma
|364.407
|678.531
|
|-
| -24/13-comma
|364.898
|678.367
|
|-
| -13/7-comma
|365.319
|678.227
|Close to [[23edo]]
|-
| -15/8-comma
|366.004
|677.999
|
|-
| -17/9-comma
|366.536
|677.821
|
|-
| -19/10-comma
|366.962
|677.679
|Close to [[85edo]]
|-
| -21/11-comma
|367.311
|677.563
|
|-
| -23/12-comma
|367.601
|677.466
|Close to [[62edo]]
|-
| -25/13-comma
|367.847
|677.384
|
|-
|-
| -2-comma
|370.795
|676.402
|Close to [[28edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-02-02T17:06:41Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth, nineteenth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth, nineteenth
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis'', beside which the weakness of ''lenis'' is that its desired (sub)harmonics mostly form [[wolf interval]]<nowiki/>s. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads|mean minor mode]] with a flexible sixth, which is primarily considered to apply temperament, especially of a superparticular interval of the 2.3.5.7.11.13.17.19.43 subgroup up to [[21/20]] such as [[129/128]] or [[136/135]], directly to the fourth.

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-02-01T22:53:00Z

Moremajorthanmajor:

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-02-01T22:51:50Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth, nineteenth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth, nineteenth
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis'', beside which the weakness of ''lenis'' is that its desired (sub)harmonics mostly form [[wolf interval]]<nowiki/>s. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads|mean minor mode]], which is primarily considered to apply temperament, especially of a superparticular interval of the 2.3.5.7.11.13.17.19.43 subgroup such as [[129/128]] or [[136/135]], directly to the fourth.

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-30T05:28:20Z

Moremajorthanmajor: /* The table */

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-30T05:23:34Z

Moremajorthanmajor: /* Beyond historically-defined mean minor (most often approached as superdiatonic and oneirotonic) */

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-29T23:27:18Z

Moremajorthanmajor: /* The table */

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-29T05:10:00Z

Moremajorthanmajor:

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-29T03:43:17Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth, nineteenth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth, nineteenth (technically)
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis'', beside which the weakness of ''lenis'' is that its desired (sub)harmonics mostly form [[wolf interval]]<nowiki/>s. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads|mean minor mode]], which is primarily considered to apply temperament, especially of [[129/128]] or [[256/255]], directly to the fourth.

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean hexachords

2025-01-29T03:36:41Z

Moremajorthanmajor: Moremajorthanmajor moved page User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean hexachords to User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

#REDIRECT [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads]]

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-29T03:36:41Z

{{Editable user page}}Here are all mean minor tunings that can be written in the form "m/n-comma mean minor", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments and often confused for it.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor triad is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]] (as this comma is already unnoticeable, the Boethian interpretation is more commonly used as a joke); but with more complex thirds, and tempering out the quarter tone of [[1053/1024]] (often confused for the simpler [[36/35]]) or [[33/32]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds. That is why doing the latter is considered to generate a perverse temperament, the “perversion” being that a recognizable quarter tone is greater than 36.09 cents, though only the former quarter tone is considered to generate the “real” one.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, and, by extension, any comma is considered as primarily applied to the generator, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean minor (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from 2-comma to [[7edo|1-comma]]
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|2-comma
|374.971
|675.010
|
|-
|27/14-comma
|372.084
|675.972
|
|-
|25/13-comma
|371.862
|676.046
|
|-
|23/12-comma
|371.603
|676.132
|
|-
|21/11-comma
|371.297
|676.234
|
|-
|19/10-comma
|370.929
|676.357
|
|-
|17/9-comma
|370.480
|676.507
|
|-
|15/8-comma
|369.919
|676.694
|
|-
|13/7-comma
|369.197
|676.934
|
|-
|24/13-comma
|368.753
|677.082
|
|-
|11/6-comma
|368.235
|677.255
|
|-
|20/11-comma
|367.622
|677.459
|
|-
|9/5-comma
|366.888
|677.704
|
|-
|25/14-comma
|366.310
|677.897
|
|-
|16/9-comma
|365.989
|678.004
|
|-
|23/13-comma
|365.644
|678.119
|
|-
|7/4-comma
|678.378
|364.867
|
|-
|19/11-comma
|363.948
|678.684
|
|-
|12/7-comma
|363.423
|678.859
|
|-
|17/10-comma
|679.051
|362.846
|
|-
|22/13-comma
|362.535
|679.155
|
|-
|5/3-comma
|361.499
|679.500
|
|-
|23/14-comma
|360.536
|679.821
|
|-
|18/11-comma
|360.274
|679.909
|
|-
|13/8-comma
|359.814
|680.062
|
|-
|ϕ-comma
|359.533
|680.156
|
|-
|21/13-comma
|359.426
|680.191
|
|-
|8/5-comma
|358.804
|680.399
|
|-
|19/12-comma
|358.130
|680.623
|
|-
|11/7-comma
|357.649
|680.784
|
|-
|14/9-comma
|357.008
|680.997
|
|-
|17/11-comma
|356.599
|681.134
|
|-
|20/13-comma
|356.317
|681.228
|
|-
|3/2-comma
|354.762
|681.746
|
|-
|19/13-comma
|353.208
|682.264
|
|-
|16/11-comma
|352.925
|682.358
|
|-
|13/9-comma
|352.517
|682.494
|
|-
|10/7-comma
|351.875
|682.718
|
|-
|17/12-comma
|351.393
|682.869
|
|-
|7/5-comma
|350.720
|682.093
|
|-
|18/13-comma
|350.099
|683.300
|
|-
|(ϕ+2)/(ϕ+1)-comma
|349.991
|683.336
|
|-
|11/8-comma
|349.710
|683.430
|
|-
|15/11-comma
|349.251
|683.583
|
|-
|19/14-comma
|348.988
|683.671
|
|-
|4/3-comma
|348.026
|683.991
|
|-
|17/13-comma
|346.989
|684.337
|
|-
|13/10-comma
|346.679
|684.440
|
|-
|9/7-comma
|346.101
|684.633
|
|-
|14/11-comma
|345.576
|684.808
|
|-
|5/4-comma
|344.658
|685.114
|
|-
|16/13-comma
|343.880
|685.373
|
|-
|11/9-comma
|343.535
|685.488
|
|-
|17/14-comma
|343.214
|685.596
|
|-
|[[7edo]]
|342.857
|685.714
|
|-
|6/5-comma
|342.637
|685.788
|
|-
|13/11-comma
|341.902
|686.033
|
|-
|7/6-comma
|341.289
|686.237
|
|-
|15/13-comma
|340.771
|686.410
|
|-
|8/7-comma
|340.327
|686.578
|
|-
|9/8-comma
|339.605
|686.798
|
|-
|10/9-comma
|339.044
|686.985
|
|-
|11/10-comma
|338.595
|687.135
|
|-
|12/11-comma
|338.227
|687.258
|
|-
|13/12-comma
|337.921
|687.360
|
|-
|14/13-comma
|337.662
|687.456
|
|-
|15/14-comma
|337.440
|687.520
|
|-
|1-comma
|334.553
|688.482
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from 4-comma to [[5edo|2-comma]]
|-
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|4-comma
|212.824
|729.051
|
|-
|27/7-comma
|215.728
|728.091
|
|-
|23/6-comma
|216.212
|727.929
|
|-
|19/5-comma
|216.890
|727.703
|
|-
|15/4-comma
|217.906
|727.365
|
|-
|26/7-comma
|218.632
|727.123
|
|-
|11/3-comma
| 219.600
|726.800
|
|-
|(2+ϕ)-comma
|220.589
|726.470
|
|-
|18/5-comma
|220.956
|726.348
|
|-
|25/7-comma
|221.536
|726.155
|
|-
|7/2-comma
|222.988
|725.671
|
|-
|24/7-comma
|224.440
|725.187
|
|-
|17/5-comma
|225.021
|724.993
|
|-
|10/3-comma
|226.376
|724.541
|
|-
|23/7-comma
|227.344
|724.219
|
|-
|13/4-comma
|228.070
|723.977
|
|-
|16/5-comma
|229.087
|723.638
|
|-
|19/6-comma
|229.764
|723.412
|
|-
|22/7-comma
|230.248
|723.251
|
|-
|3-comma
|233.152
|722.283
|
|-
|20/7-comma
|236.056
|721.315
|
|-
|17/6-comma
|236.540
|721.153
|
|-
|14/5-comma
|237.218
|720.927
|
|-
|11/4-comma
|238.234
|720.589
|
|-
|19/7-comma
|238.960
|720.347
|
|-
|8/3-comma
|239.928
|720.024
|
|-
|[[5edo]]
|240.000
|720.000
|
|-
|13/5-comma
|241.283
|719.572
|
|-
|18/7-comma
|241.864
|719.379
|
|-
|5/2-comma
|243.316
|718.895
|
|-
|17/7-comma
|244.768
|718.411
|
|-
|12/5-comma
|245.349
|718.217
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|245.715
|718.095
|
|-
|7/3-comma
|246.704
|717.765
|
|-
|16/7-comma
|247.672
|717.423
|
|-
|9/4-comma
|248.398
|717.201
|
|-
|11/5-comma
|249.414
|716.817
|
|-
|13/6-comma
|250.092
|716.636
|
|-
|15/7-comma
|250.576
|716.475
|
|-
|2-comma
|253.480
|715.507
|
|}

===Historically-defined mean minor===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from 1-comma to Pythagorean
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|1-comma
|334.553
|688.482
|
|-
|13/14-comma
|331.666
|689.445
|
|-
|12/13-comma
|331.444
|689.519
|
|-
|11/12-comma
|331.185
|689.605
|
|-
|10/11-comma
|330.879
|689.707
|
|-
|9/10-comma
|330.511
|689.830
|
|-
|8/9-comma
|330.062
|689.979
|
|-
|7/8-comma
|329.501
|690.166
|
|-
|6/7-comma
|328.779
|690.407
|
|-
|11/13-comma
|328.335
|690.555
|
|-
|5/6-comma
|327.817
|690.728
|
|-
|9/11-comma
|327.204
|690.932
|
|-
|4/5-comma
|326.469
|691.177
|
|-
|11/14-comma
|325.892
|691.370
|
|-
|7/9-comma
|325.571
|691.477
|
|-
|10/13-comma
|325.226
|691.592
|
|-
|3/4-comma
|324.449
|691.850
|
|-
|8/11-comma
|323.530
|692.157
|
|-
|5/7-comma
|323.005
|692.362
|
|-
|7/10-comma
|322.428
|692.524
|
|-
|9/13-comma
|322.117
|692.628
|
|-
|2/3-comma
|321.080
|692.973
|
|-
|9/14-comma
|320.118
|693.294
|
|-
|7/11-comma
|319.856
|693.381
|
|-
|5/8-comma
|319.396
|693.535
|
|-
|1/ϕ-comma
|319.115
|693.628
|
|-
|8/13-comma
|319.008
|693.664
|
|-
|3/5-comma
|318.386
|693.871
|
|-
|7/12-comma
|317.712
|694.096
|
|-
|4/7-comma
|317.231
|694.256
|
|-
|5/9-comma
|316.590
|694.470
|
|-
|6/11-comma
|316.181
|694.606
|
|-
|7/13-comma
|315.899
|694.700
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|314.344
|695.219
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean minor tuning in universe
|-
|6/13-comma
|312.790
|695.737
|
|-
|5/11-comma
|312.507
|695.831
|
|-
|4/9-comma
|312.099
|695.967
|
|-
|3/7-comma
|311.457
|696.181
|
|-
|5/12-comma
|310.976
|696.341
|
|-
|2/5-comma
|310.302
|696.566
|Almost exactly meantone
|-
|5/13-comma
|309.680
|696.773
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|309.573
|696.801
|
|-
|3/8-comma
|309.291
|696.904
|
|-
|4/11-comma
|308.832
|697.956
|
|-
|5/14-comma
|308.570
|697.144
|
|-
|1/3-comma
|307.608
|697.424
|
|-
|4/13-comma
|306.571
|697.810
|
|-
|3/10-comma
|306.260
|697.913
|
|-
|2/7-comma
|305.683
|698.106
|
|-
|3/11-comma
|305.158
|698.282
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|304.240
|698.587
|
|-
|3/13-comma
|303.462
|698.846
|
|-
|2/9-comma
|303.117
|698.961
|
|-
|3/14-comma
|302.796
|699.068
|
|-
|1/5-comma
|302.219
|699.260
|
|-
|2/11-comma
|301.484
|699.505
|
|-
|1/6-comma
|300.871
|699.810
|
|-
|2/13-comma
|300.353
|699.882
|
|-
|1/7-comma
|299.909
|700.030
|
|-
|1/8-comma
|299.187
|700.271
|
|-
|1/9-comma
|298.626
|700.558
|
|-
|1/10-comma
|298.177
|700.608
|
|-
|1/11-comma
|297.810
|700.730
|
|-
|1/12-comma
|297.503
|700.832
|
|-
|1/13-comma
|297.244
|700.019
|
|-
|1/14-comma
|297.022
|700.993
|
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from 2-comma to Pythagorean
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|2-comma
|253.480
|715.507
|
|-
|13/7-comma
|256.384
|714.539
|
|-
|11/6-comma
|256.868
|714.377
|
|-
|9/5-comma
|257.545
|714.156
|
|-
| 7/4-comma
|258.562
|713.813
|
|-
|12/7-comma
|259.288
|713.571
|
|-
|5/3-comma
|260.253
|713.248
|
|-
|ϕ-comma
|261.244
|712.919
|
|-
|8/5-comma
|261.611
|712.796
|
|-
|11/7-comma
|262.192
|712.603
|
|-
|3/2-comma
|263.644
|712.189
|
|-
|10/7-comma
|265.096
|711.645
|
|-
|7/5-comma
|265.676
|711.441
|
|-
|4/3-comma
|267.031
|710.990
|
|-
|9/7-comma
|267.999
|710.667
|
|-
|5/4-comma
|268.725
|710.425
|
|-
| 6/5-comma
|269.742
|710.086
|
|-
|7/6-comma
|270.419
|709.860
|
|-
|8/7-comma
|270.903
|709.699
|
|-
|1-comma
|273.807
|708.731
|
|-
|6/7-comma
|276.711
|707.762
|
|-
|5/6-comma
|277.195
|707.602
|
|-
| 4/5-comma
|277.873
|707.376
|
|-
|3/4-comma
|278.889
|707.037
|.
|-
|5/7-comma
|279.615
|706.795
|
|-
|2/3-comma
|280.583
|706.472
|
|-
|3/5-comma
|281.938
|706.021
|
|-
|4/7-comma
|282.519
|705.827
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|283.971
|705.343
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean hexachord tuning in universe.
|-
|3/7-comma
|285.423
|704.859
|
|-
|2/5-comma
|286.004
|704.665
|
|-
|1/(ϕ+1)-comma
|286.371
|704.543
|
|-
|1/3-comma
|287.359
|704.214
|
|-
|2/7-comma
|289.372
|703.891
|
|-
|1/4-comma
|289.053
|703.649
|
|-
|1/5-comma
|290.069
|703.310
|
|-
|1/6-comma
|290.747
|703.084
|
|-
|1/7-comma
|291.231
|702.923
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from Pythagorean to -1-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|291.248
|702.917
|
|-
| -1/13-comma
|291.026
|702.993
|
|-
|<nowiki>-1/12-comma</nowiki>
|290.767
|703.078
|
|-
|<nowiki>-1/11-comma</nowiki>
|290.460
|703.180
|
|-
| -1/10-comma
|290.093
|703.302
|
|-
| -1/9-comma
|289.644
|703.452
|
|-
| -1/8-comma
|289.083
|703.639
|
|-
| -1/7-comma
|288.361
|703.880
|
|-
| -2/13-comma
|287.917
|704.028
|
|-
| -1/6-comma
|287.399
|704.200
|
|-
|<nowiki>-2/11-comma</nowiki>
|286.786
|704.405
|
|-
| -1/5-comma
|286.051
|704.650
|
|-
| -3/14-comma
|285.474
|704.842
|
|-
| -2/9-comma
|285.153
|704.949
|
|-
| -3/13-comma
|284.808
|705.064
|
|-
| -1/4-comma
|284.030
|705.323
|
|-
|<nowiki>-3/11-comma</nowiki>
|283.111
|705.629
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|282.587
|705.804
|
|-
| -3/10-comma
|282.010
|705.997
|
|-
| -4/13-comma
|281.699
|706.100
|
|-
| -1/3-comma
|280.662
|706.446
|
|-
| -5/14-comma
|279.700
|706.767
|
|-
|<nowiki>-4/11-comma</nowiki>
|279.437
|706.854
|
|-
| -3/8-comma
|278.979
|707.007
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|278.697
|707.101
|
|-
| -5/13-comma
|278.590
|707.137
|
|-
| -2/5-comma
|277.968
|707.344
|
|-
|<nowiki>-5/12-comma</nowiki>
|277.294
|707.569
|
|-
| -3/7-comma
|276.813
|707.729
|
|-
| -4/9-comma
|276.171
|707.943
|
|-
|<nowiki>-5/11-comma</nowiki>
|275.763
|708.079
|
|-
| -6/13-comma
|275.480
|708.173
|
|-
| -1/2-comma
|273.926
|708.691
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|272.371
|709.210
|
|-
|<nowiki>-6/11-comma</nowiki>
|272.089
|709.304
|
|-
| -5/9-comma
|271.680
|709.440
|
|-
| -4/7-comma
|271.039
|709.654
|
|-
|<nowiki>-7/12-comma</nowiki>
|270.558
|709.814
|
|-
| -3/5-comma
|269.884
|710.039
|
|-
| -8/13-comma
|269.262
|710.246
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|269.155
|710.284
|
|-
| -5/8-comma
|268.874
|710.375
|
|-
|<nowiki>-7/11-comma</nowiki>
|268.414
|710.529
|
|-
| -9/14-comma
|268.152
|710.616
|
|-
| -2/3-comma
|267.190
|710.939
|
|-
| -9/13-comma
|266.153
|711.282
|
|-
| -7/10-comma
|265.842
|711.386
|
|-
| -5/7-comma
|265.265
|711.376
|
|-
|<nowiki>-8/11-comma</nowiki>
|264.740
|711.753
|
|-
| -3/4-comma
|263.821
|712.060
|
|-
| -10/13-comma
|263.044
|712.319
|
|-
| -7/9-comma
|263.044
|712.434
|
|-
| -11/14-comma
|262.378
|712.541
|
|-
| -4/5-comma
|261.801
|712.723
|
|-
|<nowiki>-9/11-comma</nowiki>
|261.066
|712.978
|
|-
| -5/6-comma
|260.453
|713.182
|
|-
| -11/13-comma
|259.935
|713.355
|
|-
| -6/7-comma
|259.491
|713.503
|
|-
| -7/8-comma
|258.769
|713.744
|
|-
| -8/9-comma
|258.208
|713.931
|
|-
| -9/10-comma
|257.759
|714.080
|
|-
|<nowiki>-10/11-comma</nowiki>
|257.391
|714.203
|
|-
| -11/12-comma
|257.085
|714.305
|
|-
| -12/13-comma
|256.826
|714.391
|
|-
| -13/14-comma
|256.604
|714.465
|
|-
|-
| -1-comma
|253.717
|715.248
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from Pythagorean to -2-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
|[[Pythagorean tuning]]
|294.135
|701.955
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|297.039
|700.987
|
|-
| -1/6-comma
|297.523
|700.826
|
|-
| -1/5-comma
|298.201
|700.600
|
|-
| -1/4-comma
|299.217
|700.261
|
|-
| -2/7-comma
|299.942
|700.019
|
|-
| -1/3-comma
|300.911
|699.697
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|301.900
|699.367
|
|-
| -2/5-comma
|302.266
|699.245
|
|-
| -3/7-comma
|302.847
|699.051
|
|-
| -1/2-comma
|304.299
|699.567
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295. Close to [[67edo]]
|-
| -4/7-comma
|305.751
|698.083
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -3/5-comma
|306.332
|697.889
|
|-
| -2/3-comma
|307.687
|697.438
|
|-
| -5/7-comma
|308.655
|697.115
|
|-
| -4/5-comma
|310.397
|696.534
|
|-
| -5/6-comma
|311.075
|696.308
|
|-
|-
| -6/7-comma
|311.556
|696.147
|
|-
| -1-comma
|314.463
|695.179
|
|-
| -8/7-comma
|317.367
|694.211
|
|-
| -7/6-comma
|317.851
|694.050
|
|-
| -6/5-comma
|318.528
|693.824
|
|-
| -5/4-comma
|319.545
|693.485
|
|-
| -9/7-comma
|320.271
|693.243
|
|-
| -4/3-comma
|321.239
|692.920
|
|-
| -7/5-comma
|322.594
|692.469
|
|-
| -10/7-comma
|323.174
|692.275
|
|-
| -3/2-comma
|324.626
|691.791
|
|-
| -11/7-comma
|326.078
|691.307
|
|-
| -8/5-comma
|326.659
|691.114
|
|-
| -ϕ-comma
|327.026
|690.991
|
|-
| -5/3-comma
|328.014
|690.662
|
|-
| -12/7-comma
|328.982
|690.339
|
|-
| -7/4-comma
|329.708
|690.097
|
|-
| -9/5-comma
|330.725
|689.758
|
|-
| -11/6-comma
|331.402
|689.533
|
|-
| -13/7-comma
|331.886
|689.371
|
|-
| -2-comma
|334.790
|688.403
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from -1-comma to -2-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
| -1-comma
|253.717
|715.248
|
|-
| -15/14-comma
|250.830
|716.390
|
|-
| -14/13-comma
|250.608
|716.464
|
|-
| -13/12-comma
|250.349
|716.550
|
|-
| -12/11-comma
|250.043
|716.642
|
|-
| -11/10-comma
|249.675
|716.775
|
|-
| -10/9-comma
|249.226
|716.925
|
|-
| -9/8-comma
|248.665
|717.112
|
|-
| -8/7-comma
|247.943
|717.352
|
|-
| -15/13-comma
|247.499
|717.500
|
|-
| -7/6-comma
|246.981
|717.673
|
|-
| -13/11-comma
|246.368
|717.877
|
|-
| -6/5-comma
|245.633
|718.122
|
|-
| -17/14-comma
|245.056
|718.315
|
|-
| -11/9-comma
|244.735
|718.422
|
|-
| -16/13-comma
|244.390
|718.537
|
|-
| -5/4-comma
|243.612
|718.796
|
|-
| -14/11-comma
|242.694
|719.102
|
|-
| -9/7-comma
|242.169
|719.277
|
|-
| -13/10-comma
|241.591
|719.470
|
|-
| -17/13-comma
|241.280
|719.573
|
|-
| -4/3-comma
|240.244
|719.919
|
|-
|[[5edo]]
|240.000
|720.000
|
|-
| -19/14-comma
|239.282
|720.239
|
|-
| -15/11-comma
|239.019
|720.327
|
|-
| -11/8-comma
|238.560
|720.480
|
|-
| -(ϕ+2)/(ϕ+1)-comma
|238.279
|720.574
|
|-
| -18/13-comma
|238.171
|720.610
|
|-
| -7/5-comma
|237.550
|720.817
|
|-
| -17/12-comma
|236.876
|721.041
|
|-
| -10/7-comma
|236.395
|721.202
|
|-
| -13/9-comma
|235.753
|721.416
|
|-
| -16/11-comma
|235.345
|721.552
|
|-
| -19/13-comma
|235.062
|721.646
|
|-
| -3/2-comma
|233.508
|722.164
|
|-
| -20/13-comma
|231.953
|722.682
|Close to [[93edo]]
|-
| -17/11-comma
|231.671
|722.776
|Close to [[88edo]]
|-
| -14/9-comma
|231.262
|722.913
|Close to [[83edo]]
|-
| -11/7-comma
|230.621
|723.127
|Close to [[78edo]]
|-
| -19/12-comma
|230.140
|723.287
|Close to [[73edo]]
|-
| -8/5-comma
|229.466
|723.511
|Close to [[68edo]].
|-
| -21/13-comma
|228.844
|723.719
|
|-
| -ϕ-comma
|228.737
|723.754
|
|-
| -13/8-comma
|228.456
|723.848
|Close to [[63edo]]
|-
| -18/11-comma
|227.996
|724.001
|
|-
| -23/14-comma
|227.734
|724.089
|Close to [[58edo]]
|-
| -5/3-comma
|226.771
|724.410
|Close to [[53edo]]
|-
| -22/13-comma
|225.735
|724.755
|
|-
| -17/10-comma
|225.424
|724.859
|
|-
| -12/7-comma
|224.847
|725.051
|Close to [[48edo]]
|-
| -19/11-comma
|224.322
|725.226
|
|-
| -7/4-comma
|223.403
|725.532
|Close to [[43edo]]
|-
| -23/13-comma
|222.626
|725.791
|
|-
| -16/9-comma
|222.281
|725.906
|
|-
| -25/14-comma
|221.960
|726.013
|
|-
| -9/5-comma
|221.382
|726.206
|Close to [[38edo]]
|-
| -20/11-comma
|220.648
|726.451
|
|-
| -11/6-comma
|220.035
|726.655
|
|-
| -24/13-comma
|219.517
|726.828
|
|-
| -13/7-comma
|219.073
|726.076
|
|-
| -15/8-comma
|218.351
|727.216
|Close to [[33edo]]
|-
| -17/9-comma
|217.790
|727.403
|
|-
| -19/10-comma
|217.341
|727.553
|
|-
| -21/11-comma
|216.973
|727.676
|
|-
| -23/12-comma
|216.667
|727.778
|
|-
| -25/13-comma
|216.408
|727.865
|
|-
| -27/14-comma
|216.186
|727.948
|
|-
|-
| -2-comma
|213.299
|728.900
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean minor tunings from -2 to -4-comma
!Mean minor temperament
!third
!g (cents)
!Comments
|-
| -2-comma
|334.790
|688.403
|
|-
| -15/7-comma
|337.694
|687.435
|
|-
| -13/6-comma
|338.178
|687.274
|
|-
| -11/5-comma
|338.856
|687.048
|
|-
| -9/4-comma
|339.872
|686.709
|
|-
| -16/7-comma
|340.598
|686.467
|
|-
| -7/3-comma
|341.566
|686.145
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|342.555
|685.815
|
|-
| -12/5-comma
|342.921
|685.693
|Close to [[7edo]].
|-
| -17/7-comma
|343.502
|685.499
|
|-
| -5/2-comma
|344.954
|685.016
|
|-
| -18/7-comma
|346.406
|684.531
|
|-
| -13/5-comma
|346.987
|684.378
|
|-
| -8/3-comma
|348.342
|683.886
|
|-
| -19/7-comma
|349.310
|683.563
|
|-
| -11/4-comma
|350.034
|683.321
|
|-
| -14/5-comma
|351.052
|682.983
|Very close to [[6ed6]]
|-
| -17/6-comma
|351.730
|682.757
|
|-
|-
| -20/7-comma
|352.214
|682.596
|
|-
| -3-comma
|355.118
|681.727
|
|-
| -22/7-comma
|358.022
|680.362
|
|-
| -19/6-comma
|358.501
|680.498
|
|-
| -16/5-comma
|359.183
|680.278
|
|-
| -13/4-comma
|360.200
|679.933
|Close to [[30edo]]
|-
| -23/7-comma
|360.926
|679.691
|Close to [[83edo]]
|-
| -10/3-comma
|361.894
|679.369
|Close to [[53edo]]
|-
| -17/5-comma
|363.249
|678.917
|Close to [[76edo]]
|-
| -24/7-comma
|363.830
|678.723
|Close to [[99edo]]
|-
| -7/2-comma
|365.282
|678.239
|Close to [[23edo]]
|-
| -25/7-comma
|366.734
|677.755
|
|-
| -18/5-comma
|367.315
|677.562
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|367.681
|677.440
|Close to [[62edo]]
|-
| -11/3-comma
|368.670
|677.110
|
|-
| -26/7-comma
|369.638
|676.787
|Close to [[39edo]]
|-
| -15/4-comma
|370.364
|676.545
|Close to [[94edo]]
|-
| -19/5-comma
|371.380
|676.217
|Close to [[55edo]]
|-
| -23/6-comma
|372.058
|675.980
|Close to [[71edo]]
|-
| -27/7-comma
|372.542
|675.819
|Close to [[87edo]]
|-
| -4-comma
|375.446
|674.851
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-29T03:35:41Z

Moremajorthanmajor:

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-28T23:29:37Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth, nineteenth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth, nineteenth (technically)
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis'', beside which the weakness of ''lenis'' is that its desired (sub)harmonics mostly form [[wolf interval]]<nowiki/>s. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean hexachords|mean minor mode]], which is primarily considered to temper out [[129/128]] or [[256/255]].

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-27T19:35:32Z

Moremajorthanmajor: /* Tempering out 129/128 */

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]] (as this comma is already unnoticeable, the Boethian “mean minor third” is more commonly used as a joke); but with more complex thirds, and tempering out the quarter tone of [[1053/1024]] (often confused for the simpler [[36/35]]) or [[33/32]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds. That is why doing the latter is considered to generate a perverse temperament, though only the former quarter tone is considered to generate the “real” one.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|212.824
|470.941
|729.051
|683.766
|987.176
|
|-
|27/7-comma
|256.181
|215.728
|471.909
|728.091
|687.637
|984.272
|
|-
|23/6-comma
|255.858
|216.212
|472.071
|727.929
|688.283
|983.788
|
|-
|19/5-comma
|255.407
|216.890
|472.297
|727.703
|689.187
|983.110
|
|-
|15/4-comma
|254.769
|217.906
|472.635
|727.365
|690.542
|982.094
|
|-
|26/7-comma
|254.243
|218.632
|472.877
|727.123
|691.510
|981.378
|
|-
|11/3-comma
| 253.600
|219.600
|473.200
|726.800
|692.800
|980.400
|
|-
|(2+ϕ)-comma
|252.940
|220.589
|473.530
|726.470
|694.119
|979.411
|
|-
|18/5-comma
|252.696
|220.956
|473.652
|726.348
|694.607
|979.044
|
|-
|25/7-comma
|252.309
|221.536
|473.845
|726.155
|695.382
|978.464
|
|-
|7/2-comma
|251.341
|222.988
|474.329
|725.671
|697.318
|977.012
|
|-
|24/7-comma
|250.373
|224.440
|474.813
|725.187
|699.253
|975.560
|
|-
|17/5-comma
|249.986
|225.021
|475.007
|724.993
|700.028
|974.979
|
|-
|10/3-comma
|249.083
|226.376
|475.459
|724.541
|701.835
|973.624
|
|-
|23/7-comma
|248.437
|227.344
|475.781
|724.219
|703.126
|972.656
|
|-
|13/4-comma
|247.953
|228.070
|476.023
|723.977
|704.094
|971.930
|
|-
|16/5-comma
|247.258
|229.087
|476.362
|723.638
|705.449
|970.913
|
|-
|19/6-comma
|246.824
|229.764
|476.588
|723.412
|706.352
|970.236
|
|-
|22/7-comma
|246.501
|230.248
|476.749
|723.251
|706.998
|969.752
|
|-
|3-comma
|244.565
|233.152
|477.717
|722.283
|710.870
|966.848
|
|-
|20/7-comma
|242.629
|236.056
|478.685
|721.315
|714.741
|963.944
|
|-
|17/6-comma
|242.307
|236.540
|478.847
|721.153
|715.387
|963.460
|
|-
|14/5-comma
|241.855
|237.218
|479.073
|720.927
|716.290
|962.782
|
|-
|11/4-comma
|241.177
|238.234
|479.411
|720.589
|717.645
|961.766
|
|-
|19/7-comma
|240.693
|238.960
|479.653
|720.347
|718.613
|961.040
|
|-
|8/3-comma
|240.048
|239.928
|479.976
|720.024
|719.904
|960.072
|
|-
|13/5-comma
|239.145
|241.283
|480.428
|719.572
|721.711
|958.717
|
|-
|18/7-comma
|238.757
|241.864
|480.621
|719.379
|722.485
|958.136
|
|-
|5/2-comma
| 237.789
|243.316
|481.105
|718.895
|724.421
|956.684
|
|-
|17/7-comma
|236.821
|244.768
|481.589
|718.411
|726.357
|955.232
|
|-
|12/5-comma
|236.434
|245.349
|481.783
|718.217
|727.132
|954.651
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|245.715
|481.905
|718.095
|727.620
|954.285
|
|-
|7/3-comma
|235.531
|246.704
|482.235
|717.765
|728.938
|953.296
|
|-
|16/7-comma
|234.115
|247.672
|482.557
|717.423
|730.229
|952.328
|
|-
|9/4-comma
|234.401
|248.398
|482.799
|717.201
|731.197
|951.602
|
|-
|11/5-comma
|233.276
|249.414
|483.183
|716.817
|732.552
|950.596
|
|-
|13/6-comma
|233.272
|250.092
|483.364
|716.636
|733.456
|949.909
|
|-
|15/7-comma
|232.051
|250.576
|483.525
|716.475
|734.101
|949.424
|
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|846.071
| 865.447
|
|-
|13/14-comma
|178.890
|331.666
|510.555
|689.445
|842.221
|868.334
|
|-
|12/13-comma
|179.037
|331.444
|510.481
|689.519
|841.925
| 868.556
|
|-
|11/12-comma
|179.210
|331.185
|510.395
|689.605
|841.580
|868.815
|
|-
|10/11-comma
| 179.414
|330.879
| 510.293
|689.707
|841.172
|869.121
|
|-
|9/10-comma
|179.659
|330.511
| 510.170
|689.830
|840.682
|869.489
|
|-
|8/9-comma
|179.959
|330.062
|510.021
|689.979
|840.083
|869.038
|
|-
|7/8-comma
|180.333
|329.501
|509.834
|690.166
|839.334
|870.499
|
|-
|6/7-comma
|180.814
|328.779
|509.593
|690.407
|838.372
|871.221
|
|-
|11/13-comma
|181.110
|328.335
|509.445
|690.555
|837.780
|871.665
|
|-
|5/6-comma
|181.455
|327.817
|509.272
|690.728
|837.089
|872.193
|
|-
|9/11-comma
|181.864
|327.204
|509.068
|690.932
|836.272
|872.796
|
|-
|4/5-comma
|182.354
|326.469
|508.823
|691.177
|835.293
|873.531
|
|-
|11/14-comma
|182.739
|325.892
|508.630
|691.370
|834.523
|874.108
|
|-
|7/9-comma
|182.952
|325.571
|508.523
|691.477
|834.095
| 874.429
|
|-
|10/13-comma
|183.183
|325.226
|508.408
|691.592
|833.634
|874.774
|
|-
|3/4-comma
|183.701
|324.449
|508.150
|691.850
|832.598
|875.551
|
|-
|8/11-comma
|184.687
|323.530
|507.843
|692.157
|831.373
|876.470
|
|-
|5/7-comma
|184.633
|323.005
|507.638
|692.362
|830.673
|876.995
|
|-
|7/10-comma
|184.952
|322.428
|507.476
|692.524
|829.904
|877.572
|
|-
|9/13-comma
|185.255
|322.117
|507.372
|692.628
|829.489
|877.883
|
|-
|2/3-comma
|185.946
|321.080
|507.027
|692.973
|828.107
|878.920
|
|-
|9/14-comma
|186.588
|320.118
|506.706
|693.294
|828.824
|879.882
|
|-
|7/11-comma
|186.763
|319.856
|506.619
|693.381
|826.474
|880.144
|
|-
|5/8-comma
|187.069
|319.396
|506.465
|693.535
|825.862
|880.604
|
|-
|1/ϕ-comma
|187.257
|319.115
|506.372
|693.628
|825.486
|880.885
|
|-
|8/13-comma
|187.320
|319.008
|506.336
|693.664
|825.344
|880.992
|
|-
|3/5-comma
|187.743
|318.386
|506.129
|693.871
|824.514
|881.614
|
|-
|7/12-comma
|188.194
|317.712
|505.904
|694.096
|823.616
|882.288
|
|-
|4/7-comma
|188.512
|317.231
|505.744
|694.256
|822.975
|882.769
|
|-
|5/9-comma
|188.940
|316.590
|505.530
|694.470
|822.119
|883.410
|
|-
|6/11-comma
|189.213
|316.181
|505.394
|694.606
|821.575
|883.891
|
|-
|7/13-comma
|189.401
|315.899
|505.300
|694.700
|821.198
|884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|314.344
|504.781
|695.219
|819.125
|885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|312.790
|504.263
|695.737
|817.053
|887.210
|
|-
|5/11-comma
|191.338
|312.507
|504.169
|695.831
|816.676
|887.493
|
|-
|4/9-comma
|191.934
|312.099
|504.033
|695.967
|816.131
|877.901
|
|-
|3/7-comma
|192.362
|311.457
|503.819
|696.181
|815.276
|388.443
|
|-
|5/12-comma
|192.683
|310.976
|503.659
|696.341
|814.635
|889.024
|
|-
|2/5-comma
|193.132
|310.302
|503.434
|696.566
|813.736
|889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|309.680
|503.227
|696.773
|812.907
|890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|309.573
|503.191
|696.801
|812.764
| 890.427
|
|-
|3/8-comma
|193.805
|309.291
| 503.096
|696.904
|812.389
|890.709
|
|-
|4/11-comma
|194.112
|308.832
|502.944
|697.956
|811.776
|891.168
|
|-
|5/14-comma
|194.287
|308.570
|502.856
|697.144
|811.427
|891.430
|
|-
|1/3-comma
|194.928
|307.608
|502.536
|697.424
|810.144
|892.392
|
|-
|4/13-comma
|195.619
|306.571
|502.190
|697.810
|808.762
|893.429
|
|-
|3/10-comma
|195.174
|306.260
|502.087
|697.913
|808.347
|893.740
|
|-
|2/7-comma
|196.211
|305.683
|501.894
|698.106
|807.577
|894.317
|
|-
|3/11-comma
|196.561
|305.158
|501.718
|698.282
|806.877
|894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|304.240
|501.413
|698.587
|805.653
|895.760
|
|-
|3/13-comma
|197.692
|303.462
|501.154
|698.846
|804.616
|896.538
|
|-
|2/9-comma
|197.922
|303.117
|501.039
|698.961
|804.155
|896.883
|
|-
|3/14-comma
|198.136
|302.796
|500.932
|699.068
|803.728
|897.204
|
|-
|1/5-comma
|198.521
|302.219
|500.740
|699.260
|802.958
|897.781
|
|-
|2/11-comma
|199.011
|301.484
|500.495
|699.505
|801.978
|898.516
|
|-
|1/6-comma
|199.419
|300.871
|500.290
|699.810
|801.162
|899.129
|
|-
|2/13-comma
|199.765
|300.353
|500.118
|699.882
|800.471
|899.647
|
|-
|1/7-comma
|200.061
|299.909
|499.970
|700.030
|799.879
|900.091
|
|-
|1/8-comma
|200.542
|299.187
| 499.729
|700.271
|798.916
|900.823
|
|-
|1/9-comma
|200.916
|298.626
|499.542
|700.558
|798.168
|901.374
|
|-
|1/10-comma
|201.785
|298.177
|499.392
|700.608
|797.569
|901.823
|
|-
|1/11-comma
|201.460
|297.810
|499.270
|700.730
|797.079
|902.190
|
|-
|1/12-comma
|201.665
|297.503
|499.168
|700.832
|796.671
|902.497
|
|-
|1/13-comma
|201.837
|297.244
|499.081
|700.019
|796.325
|902.756
|
|-
|1/14-comma
|201.953
|297.022
|499.007
|700.993
|796.029
|902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|-
|13/7-comma
|229.078
|256.384
|485.461
|714.539
|741.845
|943.616
|
|-
|11/6-comma
|228.755
|256.868
|485.623
|714.377
|742.490
|943.132
|
|-
|9/5-comma
|228.697
|257.545
|485.848
|714.156
|743.394
|942.455
|
|-
| 7/4-comma
|227.626
|258.562
|486.187
|713.813
|744.749
|941.438
|
|-
|12/7-comma
|227.142
|259.288
|486.429
|713.571
|745.717
|940.712
|
|-
|5/3-comma
|226.496
|260.253
|486.752
|713.248
|747.007
|939.747
|
|-
|ϕ-comma
|225.837
|261.244
|487.081
|712.919
|748.326
|938.756
|
|-
|8/5-comma
|225.593
|261.611
|487.204
|712.796
|748.814
|938.389
|
|-
|11/7-comma
|225.206
|262.192
| 487.397
|712.603
|749.589
|937.808
|
|-
|3/2-comma
| 224.762
|263.644
|487.881
|712.189
|751.525
|936.356
|
|-
|10/7-comma
|223.270
|265.096
|488.365
|711.645
|753.461
|964.904
|
|-
|7/5-comma
|222.882
|265.676
|488.559
|711.441
|754.235
|964.324
|
|-
|4/3-comma
|221.979
|267.031
|489.010
|710.990
|756.042
|932.969
|
|-
|9/7-comma
|221.334
|267.999
|489.333
|710.667
|757.333
|932.001
|
|-
|5/4-comma
|220.850
|268.725
|489.575
|710.425
|758.301
|931.275
|
|-
| 6/5-comma
|220.172
|269.742
|489.914
|710.086
|759.656
|930.258
|
|-
|7/6-comma
|219.720
|270.419
|490.140
|709.860
|760.559
|929.581
|
|-
|8/7-comma
|219.398
|270.903
|490.301
|709.699
|761.205
|929.297
|
|-
|1-comma
|217.538
|273.807
|491.269
|708.731
|765.076
| 926.193
|
|-
|6/7-comma
|215.526
|276.711
|492.237
|707.762
|768.948
|923.289
|
|-
|5/6-comma
|215.203
|277.195
|492.398
|707.602
|769.593
|922.805
|
|-
| 4/5-comma
|214.751
|277.873
| 492.624
|707.376
|770.497
|922.167
|
|-
|3/4-comma
|214.926
|278.889
|492.963
|707.037
|771.852
|921.111
|.
|-
|5/7-comma
|213.590
|279.615
|493.205
|706.795
|772.820
|920.795
|
|-
|2/3-comma
|212.945
|280.583
|493.528
|706.472
|774.111
|919.417
|
|-
|3/5-comma
|212.041
|281.938
|493.979
|706.021
|775.918
|918.062
|
|-
|4/7-comma
|211.346
|282.519
|494.173
|705.827
|776.692
|917.401
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|283.971
|494.657
|705.343
|778.628
|916.021
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean hexachord tuning in universe.
|-
|3/7-comma
|209.718
|285.423
|495.141
|704.859
|780.564
|914.577
|
|-
|2/5-comma
|209.331
|286.004
|495.335
|704.665
|781.339
|913.996
|
|-
|1/(ϕ+1)-comma
|209.086
|286.371
|495.457
|704.543
|781.827
|913.629
|
|-
|1/3-comma
|208.573
|287.359
|495.786
|704.214
|783.145
|912.641
|
|-
|2/7-comma
|207.782
|289.372
|496.109
|703.891
|784.436
|910.628
|
|-
|1/4-comma
|207.293
|289.053
|496.351
|703.649
|785.404
|910.947
|
|-
|1/5-comma
|206.620
|290.069
|496.690
|703.310
|786.759
|909.931
|
|-
|1/6-comma
|206.169
|290.747
|496.916
|703.084
|787.663
|909.253
|
|-
|1/7-comma
|205.846
|291.231
|497.077
|702.923
|788.308
|908.769
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|291.248
|497.083
|702.917
|788.331
|908.752
|
|-
| -1/13-comma
|205.983
|291.026
|497.009
|702.993
|788.035
|908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|290.767
|496.922
|703.078
|787.689
|909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|290.460
|496.820
|703.180
|787.280
|909.540
|
|-
| -1/10-comma
|206.605
|290.093
|496.698
|703.302
|786.791
|909.907
|
|-
| -1/9-comma
|206.904
|289.644
|496.548
|703.452
|786.192
|910.356
|
|-
| -1/8-comma
|207.278
|289.083
|496.361
|703.639
|785.444
|910.917
|
|-
| -1/7-comma
|207.759
|288.361
|496.120
|703.880
|784.481
|911.639
|
|-
| -2/13-comma
|208.055
|287.917
|495.972
|704.028
|783.889
|912.083
|
|-
| -1/6-comma
|208.401
|287.399
|495.800
|704.200
|783.198
|912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|286.786
|495.595
|704.405
|782.382
|913.214
|
|-
| -1/5-comma
|209.299
|286.051
|495.350
|704.650
|781.401
|913.949
|
|-
| -3/14-comma
|209.684
|285.474
|495.158
|704.842
|780.632
|914.526
|
|-
| -2/9-comma
|209.898
|285.153
|495.051
|704.949
|780.204
|914.847
|
|-
| -3/13-comma
|210.128
|284.808
|494.936
|705.064
|779.744
|915.192
|
|-
| -1/4-comma
|210.646
|284.030
|494.677
|705.323
|778.707
|915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|283.111
|494.371
|705.629
|777.482
|916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|282.587
|494.196
|705.804
|776.783
|917.413
|
|-
| -3/10-comma
|211.994
|282.010
|494.003
|705.997
|776.013
|917.990
|
|-
| -4/13-comma
|212.799
|281.699
|493.900
|706.100
|775.598
|918.301
|
|-
| -1/3-comma
|212.892
|280.662
|493.554
|706.446
|774.216
|919.338
|
|-
| -5/14-comma
|213.537
|279.700
|493.233
|706.767
|772.933
|920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|279.437
|493.146
|706.854
|772.583
|920.563
|
|-
| -3/8-comma
|214.014
|278.979
|492.993
|707.007
|771.971
|921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|278.697
|492.899
|707.101
|771.596
|921.303
|
|-
| -5/13-comma
|214.274
|278.590
|492.863
|707.137
|771.453
|921.410
|
|-
| -2/5-comma
|214.688
|277.968
|492.656
|707.344
|770.624
|922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|277.294
|492.431
|707.569
|769.725
|922.706
|
|-
| -3/7-comma
|215.458
|276.813
|492.271
|707.729
|769.084
|923.187
|
|-
| -4/9-comma
|215.886
|276.171
|492.057
|707.943
|768.229
|923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|275.763
|491.921
|708.079
|767.684
|924.237
|
|-
| -6/13-comma
|216.346
|275.480
|491.827
|708.173
|767.307
|924.520
|
|-
| -1/2-comma
|217.383
|273.926
|491.309
|708.691
|765.235
|926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|272.371
|490.790
|709.210
|763.161
|927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|272.089
|490.696
|709.304
|762.785
|927.911
|
|-
| -5/9-comma
|218.880
|271.680
|490.560
|709.440
|762.241
|928.320
|
|-
| -4/7-comma
|219.307
|271.039
|490.346
|709.654
|761.385
|928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|270.558
|490.186
|709.814
|760.744
|929.442
|
|-
| -3/5-comma
|220.077
|269.884
|489.961
|710.039
|759.846
|930.116
|
|-
| -8/13-comma
|220.492
|269.262
|489.754
|710.246
|759.016
|930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|269.155
|489.716
|710.284
|758.874
|930.845
|
|-
| -5/8-comma
|220.751
|268.874
|489.625
|710.375
|758.498
|931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|268.414
|489.471
|710.529
|757.886
|931.586
|
|-
| -9/14-comma
|221.232
|268.152
|489.384
|710.616
|757.536
|931.848
|
|-
| -2/3-comma
|221.874
|267.190
|489.063
|710.939
|756.253
|932.810
|
|-
| -9/13-comma
|222.565
|266.153
|488.718
|711.282
|754.871
|933.847
|
|-
| -7/10-comma
|222.772
|265.842
|488.614
|711.386
|754.456
|934.158
|
|-
| -5/7-comma
|223.157
|265.265
|488.422
|711.376
|753.687
|934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|264.740
|488.247
|711.753
|752.987
|935.260
|
|-
| -3/4-comma
|224.119
|263.821
|487.940
|712.060
|751.762
|936.189
|
|-
| -10/13-comma
|224.637
|263.044
|487.681
|712.319
|750.726
|936.956
|
|-
| -7/9-comma
|224.868
|263.044
|487.566
|712.434
|750.265
|937.302
|
|-
| -11/14-comma
|225.081
|262.378
|487.459
|712.541
|749.837
|937.622
|
|-
| -4/5-comma
|225.466
|261.801
|487.267
|712.723
|749.067
|938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|261.066
|487.022
|712.978
|748.088
|938.934
|
|-
| -5/6-comma
|226.365
|260.453
|486.818
|713.182
|747.271
|939.447
|
|-
| -11/13-comma
|226.710
|259.935
|486.645
|713.355
|746.580
|940.065
|
|-
| -6/7-comma
|227.006
|259.491
|486.497
|713.503
|745.988
|940.509
|
|-
| -7/8-comma
|227.487
|258.769
|486.256
|713.744
|745.026
|941.231
|
|-
| -8/9-comma
|227.861
|258.208
|486.069
|713.931
|744.277
|941.792
|
|-
| -9/10-comma
|228.161
|257.759
|485.920
|714.080
|743.678
|942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|257.391
|485.797
|714.203
|743.188
|942.609
|
|-
| -11/12-comma
|228.610
|257.085
|485.695
|714.305
|742.780
|942.915
|
|-
| -12/13-comma
|228.783
|256.826
|485.609
|714.391
|742.435
|943.174
|
|-
| -13/14-comma
|228.931
|256.604
|485.535
|714.465
|742.139
|943.396
|
|-
|-
| -1-comma
|230.855
|253.717
|484.752
|715.248
|738.289
|946.283
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|297.039
|499.013
|700.987
|796.052
|902.961
|
|-
| -1/6-comma
|201.652
|297.523
|499.174
|700.826
|796.697
|902.477
|
|-
| -1/5-comma
|201.200
|298.201
|499.400
|700.600
|797.601
|901.799
|
|-
| -1/4-comma
|200.522
|299.217
|499.739
|700.261
|798.956
|900.783
|
|-
| -2/7-comma
|200.038
|299.942
|499.981
|700.019
|799.924
|900.058
|
|-
| -1/3-comma
|199.393
|300.911
|500.303
|699.697
|801.214
|899.089
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|301.900
|500.633
|699.367
|802.533
|898.100
|
|-
| -2/5-comma
|198.499
|302.266
|500.755
|699.245
|803.021
|897.634
|
|-
| -3/7-comma
|198.102
|302.847
|500.949
|699.051
|803.796
|897.153
|
|-
| -1/2-comma
|197.134
|304.299
|501.433
|699.567
|805.732
|895.701
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295. Close to [[67edo]]
|-
| -4/7-comma
|196.166
|305.751
|501.917
|698.083
|807.668
|894.249
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -3/5-comma
|195.779
|306.332
|502.111
|697.889
|808.442
|893.668
|
|-
| -2/3-comma
|194.876
|307.687
|502.562
|697.438
|810.249
|892.313
|
|-
| -5/7-comma
|194.230
|308.655
|502.885
|697.115
|811.540
|891.345
|
|-
| -4/5-comma
|193.069
|310.397
|503.466
|696.534
|813.863
|889.603
|
|-
| -5/6-comma
|192.617
|311.075
|503.692
|696.308
|814.766
|888.925
|
|-
|-
| -6/7-comma
|192.294
|311.556
|503.853
|696.147
|815.412
|888.444
|
|-
| -1-comma
|190.352
|314.463
|504.821
|695.179
|819.283
|885.537
|
|-
| -8/7-comma
|188.422
|317.367
|505.789
|694.211
|823.155
|882.633
|
|-
| -7/6-comma
|188.100
|317.851
|505.950
|694.050
|823.801
|882.149
|
|-
| -6/5-comma
|187.648
|318.528
|506.176
|693.824
|824.704
|881.472
|
|-
| -5/4-comma
|186.970
|319.545
|506.515
|693.485
|826.059
|880.455
|
|-
| -9/7-comma
|186.486
|320.271
|506.757
|693.243
|827.027
|879.730
|
|-
| -4/3-comma
|185.841
|321.239
|507.080
|692.920
|828.318
|878.761
|
|-
| -7/5-comma
|184.937
|322.594
|507.531
|692.469
|830.125
|877.406
|
|-
| -10/7-comma
|184.550
|323.174
|507.725
|692.275
|830.899
|876.826
|
|-
| -3/2-comma
|183.582
|324.626
|508.209
|691.791
|832.835
|875.374
|
|-
| -11/7-comma
|182.614
|326.078
|508.693
|691.307
|834.771
|873.922
|
|-
| -8/5-comma
|182.228
|326.659
|508.886
|691.114
|835.546
|873.341
|
|-
| -ϕ-comma
|181.983
|327.026
|509.009
|690.991
|836.034
|872.974
|
|-
| -5/3-comma
|181.324
|328.014
|509.338
|690.662
|837.353
|871.986
|
|-
| -12/7-comma
|180.678
|328.982
|509.661
|690.339
|838.643
|871.018
|
|-
| -7/4-comma
|180.194
|329.708
|509.903
|690.097
|839.611
|870.292
|
|-
| -9/5-comma
|179.517
|330.725
|510.242
|689.758
|840.966
|869.275
|
|-
| -11/6-comma
|179.065
|331.402
|510.467
|689.533
|841.870
|868.598
|
|-
| -13/7-comma
|178.742
|331.886
|510.629
|689.371
|842.515
|868.114
|
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|846.387
|865.210
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|253.717
|484.752
|715.248
|738.289
|946.283
|
|-
| -15/14-comma
|232.780
|250.830
|483.610
|716.390
|734.440
|949.170
|
|-
| -14/13-comma
|232.928
|250.608
|483.536
|716.464
|734.133
|949.392
|
|-
| -13/12-comma
|233.101
|250.349
|483.450
|716.550
|733.798
|949.651
|
|-
| -12/11-comma
|233.305
|250.043
|483.348
|716.642
|733.390
|949.957
|
|-
| -11/10-comma
|233.550
|249.675
|483.225
|716.775
|732.900
|950.325
|
|-
| -10/9-comma
|233.151
|249.226
|483.075
|716.925
|732.301
|950.774
|
|-
| -9/8-comma
|234.234
|248.665
|482.888
|717.112
|731.553
|951.335
|
|-
| -8/7-comma
|234.295
|247.943
|482.648
|717.352
|730.590
|952.352
|
|-
| -15/13-comma
|235.001
|247.499
|482.500
|717.500
|729.998
|952.501
|
|-
| -7/6-comma
|235.346
|246.981
|482.327
|717.673
|729.307
|953.019
|
|-
| -13/11-comma
|235.755
|246.368
|482.123
|717.877
|728.491
|953.632
|
|-
| -6/5-comma
|236.244
|245.633
|481.878
|718.122
|727.511
|954.367
|
|-
| -17/14-comma
|236.629
|245.056
|481.685
|718.315
|726.741
|954.943
|
|-
| -11/9-comma
|236.843
|244.735
|481.578
|718.422
|726.313
|955.265
|
|-
| -16/13-comma
|237.926
|244.390
|481.463
|718.537
|725.853
|955.610
|
|-
| -5/4-comma
|237.592
|243.612
|481.204
|718.796
|724.816
|956.388
|
|-
| -14/11-comma
|238.204
|242.694
|480.898
|719.102
|723.592
|957.694
|
|-
| -9/7-comma
|238.554
|242.169
|480.723
|719.277
|722.892
|957.831
|
|-
| -13/10-comma
|238.939
|241.591
|480.530
|719.470
|722.122
|957.409
|
|-
| -17/13-comma
|239.146
|241.280
|480.427
|719.573
|721.707
|958.720
|
|-
| -4/3-comma
|239.837
|240.244
|480.081
|719.919
|720.326
|959.756
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|239.282
|479.761
|720.239
|719.042
|960.718
|
|-
| -15/11-comma
|240.634
|239.019
|479.673
|720.327
|718.693
|960.981
|
|-
| -11/8-comma
|240.960
|238.560
|479.520
|720.480
|718.080
|961.440
|
|-
| -(ϕ+2)/(ϕ+1)-comma
|241.148
|238.279
|479.426
|720.574
|717.705
|961.721
|
|-
| -18/13-comma
|241.219
|238.171
|479.390
|720.610
|717.561
|961.829
|
|-
| -7/5-comma
|241.634
|237.550
|479.183
|720.817
|716.733
|962.450
|
|-
| -17/12-comma
|242.917
|236.876
|478.959
|721.041
|715.835
|962.124
|
|-
| -10/7-comma
|242.403
|236.395
|478.798
|721.202
|715.193
|963.605
|
|-
| -13/9-comma
|242.831
|235.753
|478.584
|721.416
|714.338
|964.247
|
|-
| -16/11-comma
|243.103
|235.345
|478.448
|721.552
|713.793
|964.655
|
|-
| -19/13-comma
|243.708
|235.062
|478.354
|721.646
|713.416
|964.938
|
|-
| -3/2-comma
|244.328
|233.508
|477.836
|722.164
|711.343
|966.492
|
|-
| -20/13-comma
|245.344
|231.953
|477.318
|722.682
|709.271
|968.047
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|231.671
|477.224
|722.776
|708.894
|968.329
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|231.262
|477.087
|722.913
|708.350
|968.738
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|230.621
|476.873
|723.127
|707.493
|969.379
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|230.140
|476.713
|723.287
|706.853
|969.860
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|229.466
|476.489
|723.511
|705.955
|970.534
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|228.844
|476.281
|723.719
|705.156
|971.156
|
|-
| -ϕ-comma
|247.491
|228.737
|476.246
|723.754
|704.983
|971.263
|
|-
| -13/8-comma
|247.696
|228.456
|476.152
|723.848
|704.607
|971.544
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|227.996
|475.999
|724.001
|703.995
|972.004
|
|-
| -23/14-comma
|248.823
|227.734
|475.911
|724.089
|703.645
|972.266
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|226.771
|475.590
|724.410
|702.362
|973.229
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|225.735
|475.245
|724.755
|700.980
|974.265
|
|-
| -17/10-comma
|249.717
|225.424
|475.141
|724.859
|700.566
|974.576
|
|-
| -12/7-comma
|250.105
|224.847
|474.949
|725.051
|699.796
|975.153
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|224.322
|474.774
|725.226
|699.096
|975.678
|
|-
| -7/4-comma
|251.064
|223.403
|474.468
|725.532
|697.871
|976.597
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|222.626
|474.209
|725.791
|696.835
|977.374
|
|-
| -16/9-comma
|251.823
|222.281
|474.094
|725.906
|696.374
|977.719
|
|-
| -25/14-comma
|252.027
|221.960
|473.987
|726.013
|695.946
|978.040
|
|-
| -9/5-comma
|252.412
|221.382
|473.794
|726.206
|695.177
|978.618
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|220.648
|473.549
|726.451
|694.197
|979.352
|
|-
| -11/6-comma
|253.610
|220.035
|473.345
|726.655
|693.380
|979.965
|
|-
| -24/13-comma
|253.345
|219.517
|473.172
|726.828
|692.689
|980.483
|
|-
| -13/7-comma
|253.951
|219.073
|473.924
|726.076
|692.097
|980.927
|
|-
| -15/8-comma
|254.433
|218.351
|472.784
|727.216
|691.135
|981.649
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|217.790
|472.597
|727.403
|690.386
|982.210
|
|-
| -19/10-comma
|255.106
|217.341
|472.447
|727.553
|689.787
|982.659
|
|-
| -21/11-comma
|255.351
|216.973
|472.324
|727.676
|689.296
|983.027
|
|-
| -23/12-comma
|255.555
|216.667
|472.222
|727.778
|688.889
|983.333
|
|-
| -25/13-comma
|255.728
|216.408
|472.135
|727.865
|688.544
|983.592
|
|-
| -27/14-comma
|255.876
|216.186
|472.052
|727.948
|688.248
|983.814
|
|-
|-
| -2-comma
|258.801
|213.299
|471.100
|728.900
|684.398
|986.701
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-27T08:57:47Z

Moremajorthanmajor: /* The table */

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]] (as this comma is already unnoticeable, the Boethian “mean minor third” is more commonly used as a joke); but with more complex thirds, and tempering out the quarter tone of [[1053/1024]] (often confused for the simpler [[36/35]]) or [[33/32]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds. That is why doing the latter is considered to generate a perverse temperament, though only the former quarter tone is considered to generate the “real” one.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|212.824
|470.941
|729.051
|683.766
|987.176
|
|-
|27/7-comma
|256.181
|215.728
|471.909
|728.091
|687.637
|984.272
|
|-
|23/6-comma
|255.858
|216.212
|472.071
|727.929
|688.283
|983.788
|
|-
|19/5-comma
|255.407
|216.890
|472.297
|727.703
|689.187
|983.110
|
|-
|15/4-comma
|254.769
|217.906
|472.635
|727.365
|690.542
|982.094
|
|-
|26/7-comma
|254.243
|218.632
|472.877
|727.123
|691.510
|981.378
|
|-
|11/3-comma
| 253.600
|219.600
|473.200
|726.800
|692.800
|980.400
|
|-
|(2+ϕ)-comma
|252.940
|220.589
|473.530
|726.470
|694.119
|979.411
|
|-
|18/5-comma
|252.696
|220.956
|473.652
|726.348
|694.607
|979.044
|
|-
|25/7-comma
|252.309
|221.536
|473.845
|726.155
|695.382
|978.464
|
|-
|7/2-comma
|251.341
|222.988
|474.329
|725.671
|697.318
|977.012
|
|-
|24/7-comma
|250.373
|224.440
|474.813
|725.187
|699.253
|975.560
|
|-
|17/5-comma
|249.986
|225.021
|475.007
|724.993
|700.028
|974.979
|
|-
|10/3-comma
|249.083
|226.376
|475.459
|724.541
|701.835
|973.624
|
|-
|23/7-comma
|248.437
|227.344
|475.781
|724.219
|703.126
|972.656
|
|-
|13/4-comma
|247.953
|228.070
|476.023
|723.977
|704.094
|971.930
|
|-
|16/5-comma
|247.258
|229.087
|476.362
|723.638
|705.449
|970.913
|
|-
|19/6-comma
|246.824
|229.764
|476.588
|723.412
|706.352
|970.236
|
|-
|22/7-comma
|246.501
|230.248
|476.749
|723.251
|706.998
|969.752
|
|-
|3-comma
|244.565
|233.152
|477.717
|722.283
|710.870
|966.848
|
|-
|20/7-comma
|242.629
|236.056
|478.685
|721.315
|714.741
|963.944
|
|-
|17/6-comma
|242.307
|236.540
|478.847
|721.153
|715.387
|963.460
|
|-
|14/5-comma
|241.855
|237.218
|479.073
|720.927
|716.290
|962.782
|
|-
|11/4-comma
|241.177
|238.234
|479.411
|720.589
|717.645
|961.766
|
|-
|19/7-comma
|240.693
|238.960
|479.653
|720.347
|718.613
|961.040
|
|-
|8/3-comma
|240.048
|239.928
|479.976
|720.024
|719.904
|960.072
|
|-
|13/5-comma
|239.145
|241.283
|480.428
|719.572
|721.711
|958.717
|
|-
|18/7-comma
|238.757
|241.864
|480.621
|719.379
|722.485
|958.136
|
|-
|5/2-comma
| 237.789
|243.316
|481.105
|718.895
|724.421
|956.684
|
|-
|17/7-comma
|236.821
|244.768
|481.589
|718.411
|726.357
|955.232
|
|-
|12/5-comma
|236.434
|245.349
|481.783
|718.217
|727.132
|954.651
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|245.715
|481.905
|718.095
|727.620
|954.285
|
|-
|7/3-comma
|235.531
|246.704
|482.235
|717.765
|728.938
|953.296
|
|-
|16/7-comma
|234.115
|247.672
|482.557
|717.423
|730.229
|952.328
|
|-
|9/4-comma
|234.401
|248.398
|482.799
|717.201
|731.197
|951.602
|
|-
|11/5-comma
|233.276
|249.414
|483.183
|716.817
|732.552
|950.596
|
|-
|13/6-comma
|233.272
|250.092
|483.364
|716.636
|733.456
|949.909
|
|-
|15/7-comma
|232.051
|250.576
|483.525
|716.475
|734.101
|949.424
|
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|846.071
| 865.447
|
|-
|13/14-comma
|178.890
|331.666
|510.555
|689.445
|842.221
|868.334
|
|-
|12/13-comma
|179.037
|331.444
|510.481
|689.519
|841.925
| 868.556
|
|-
|11/12-comma
|179.210
|331.185
|510.395
|689.605
|841.580
|868.815
|
|-
|10/11-comma
| 179.414
|330.879
| 510.293
|689.707
|841.172
|869.121
|
|-
|9/10-comma
|179.659
|330.511
| 510.170
|689.830
|840.682
|869.489
|
|-
|8/9-comma
|179.959
|330.062
|510.021
|689.979
|840.083
|869.038
|
|-
|7/8-comma
|180.333
|329.501
|509.834
|690.166
|839.334
|870.499
|
|-
|6/7-comma
|180.814
|328.779
|509.593
|690.407
|838.372
|871.221
|
|-
|11/13-comma
|181.110
|328.335
|509.445
|690.555
|837.780
|871.665
|
|-
|5/6-comma
|181.455
|327.817
|509.272
|690.728
|837.089
|872.193
|
|-
|9/11-comma
|181.864
|327.204
|509.068
|690.932
|836.272
|872.796
|
|-
|4/5-comma
|182.354
|326.469
|508.823
|691.177
|835.293
|873.531
|
|-
|11/14-comma
|182.739
|325.892
|508.630
|691.370
|834.523
|874.108
|
|-
|7/9-comma
|182.952
|325.571
|508.523
|691.477
|834.095
| 874.429
|
|-
|10/13-comma
|183.183
|325.226
|508.408
|691.592
|833.634
|874.774
|
|-
|3/4-comma
|183.701
|324.449
|508.150
|691.850
|832.598
|875.551
|
|-
|8/11-comma
|184.687
|323.530
|507.843
|692.157
|831.373
|876.470
|
|-
|5/7-comma
|184.633
|323.005
|507.638
|692.362
|830.673
|876.995
|
|-
|7/10-comma
|184.952
|322.428
|507.476
|692.524
|829.904
|877.572
|
|-
|9/13-comma
|185.255
|322.117
|507.372
|692.628
|829.489
|877.883
|
|-
|2/3-comma
|185.946
|321.080
|507.027
|692.973
|828.107
|878.920
|
|-
|9/14-comma
|186.588
|320.118
|506.706
|693.294
|828.824
|879.882
|
|-
|7/11-comma
|186.763
|319.856
|506.619
|693.381
|826.474
|880.144
|
|-
|5/8-comma
|187.069
|319.396
|506.465
|693.535
|825.862
|880.604
|
|-
|1/ϕ-comma
|187.257
|319.115
|506.372
|693.628
|825.486
|880.885
|
|-
|8/13-comma
|187.320
|319.008
|506.336
|693.664
|825.344
|880.992
|
|-
|3/5-comma
|187.743
|318.386
|506.129
|693.871
|824.514
|881.614
|
|-
|7/12-comma
|188.194
|317.712
|505.904
|694.096
|823.616
|882.288
|
|-
|4/7-comma
|188.512
|317.231
|505.744
|694.256
|822.975
|882.769
|
|-
|5/9-comma
|188.940
|316.590
|505.530
|694.470
|822.119
|883.410
|
|-
|6/11-comma
|189.213
|316.181
|505.394
|694.606
|821.575
|883.891
|
|-
|7/13-comma
|189.401
|315.899
|505.300
|694.700
|821.198
|884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|314.344
|504.781
|695.219
|819.125
|885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|312.790
|504.263
|695.737
|817.053
|887.210
|
|-
|5/11-comma
|191.338
|312.507
|504.169
|695.831
|816.676
|887.493
|
|-
|4/9-comma
|191.934
|312.099
|504.033
|695.967
|816.131
|877.901
|
|-
|3/7-comma
|192.362
|311.457
|503.819
|696.181
|815.276
|388.443
|
|-
|5/12-comma
|192.683
|310.976
|503.659
|696.341
|814.635
|889.024
|
|-
|2/5-comma
|193.132
|310.302
|503.434
|696.566
|813.736
|889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|309.680
|503.227
|696.773
|812.907
|890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|309.573
|503.191
|696.801
|812.764
| 890.427
|
|-
|3/8-comma
|193.805
|309.291
| 503.096
|696.904
|812.389
|890.709
|
|-
|4/11-comma
|194.112
|308.832
|502.944
|697.956
|811.776
|891.168
|
|-
|5/14-comma
|194.287
|308.570
|502.856
|697.144
|811.427
|891.430
|
|-
|1/3-comma
|194.928
|307.608
|502.536
|697.424
|810.144
|892.392
|
|-
|4/13-comma
|195.619
|306.571
|502.190
|697.810
|808.762
|893.429
|
|-
|3/10-comma
|195.174
|306.260
|502.087
|697.913
|808.347
|893.740
|
|-
|2/7-comma
|196.211
|305.683
|501.894
|698.106
|807.577
|894.317
|
|-
|3/11-comma
|196.561
|305.158
|501.718
|698.282
|806.877
|894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|304.240
|501.413
|698.587
|805.653
|895.760
|
|-
|3/13-comma
|197.692
|303.462
|501.154
|698.846
|804.616
|896.538
|
|-
|2/9-comma
|197.922
|303.117
|501.039
|698.961
|804.155
|896.883
|
|-
|3/14-comma
|198.136
|302.796
|500.932
|699.068
|803.728
|897.204
|
|-
|1/5-comma
|198.521
|302.219
|500.740
|699.260
|802.958
|897.781
|
|-
|2/11-comma
|199.011
|301.484
|500.495
|699.505
|801.978
|898.516
|
|-
|1/6-comma
|199.419
|300.871
|500.290
|699.810
|801.162
|899.129
|
|-
|2/13-comma
|199.765
|300.353
|500.118
|699.882
|800.471
|899.647
|
|-
|1/7-comma
|200.061
|299.909
|499.970
|700.030
|799.879
|900.091
|
|-
|1/8-comma
|200.542
|299.187
| 499.729
|700.271
|798.916
|900.823
|
|-
|1/9-comma
|200.916
|298.626
|499.542
|700.558
|798.168
|901.374
|
|-
|1/10-comma
|201.785
|298.177
|499.392
|700.608
|797.569
|901.823
|
|-
|1/11-comma
|201.460
|297.810
|499.270
|700.730
|797.079
|902.190
|
|-
|1/12-comma
|201.665
|297.503
|499.168
|700.832
|796.671
|902.497
|
|-
|1/13-comma
|201.837
|297.244
|499.081
|700.019
|796.325
|902.756
|
|-
|1/14-comma
|201.953
|297.022
|499.007
|700.993
|796.029
|902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|-
|13/7-comma
|229.078
|256.384
|485.461
|714.539
|741.845
|943.616
|
|-
|11/6-comma
|228.755
|256.868
|485.623
|714.377
|742.490
|943.132
|
|-
|9/5-comma
|228.697
|257.545
|485.848
|714.156
|743.394
|942.455
|
|-
| 7/4-comma
|227.626
|258.562
|486.187
|713.813
|744.749
|941.438
|
|-
|12/7-comma
|227.142
|259.288
|486.429
|713.571
|745.717
|940.712
|
|-
|5/3-comma
|226.496
|260.253
|486.752
|713.248
|747.007
|939.747
|
|-
|ϕ-comma
|225.837
|261.244
|487.081
|712.919
|748.326
|938.756
|
|-
|8/5-comma
|225.593
|261.611
|487.204
|712.796
|748.814
|938.389
|
|-
|11/7-comma
|225.206
|262.192
| 487.397
|712.603
|749.589
|937.808
|
|-
|3/2-comma
| 224.762
|263.644
|487.881
|712.189
|751.525
|936.356
|
|-
|10/7-comma
|223.270
|265.096
|488.365
|711.645
|753.461
|964.904
|
|-
|7/5-comma
|222.882
|265.676
|488.559
|711.441
|754.235
|964.324
|
|-
|4/3-comma
|221.979
|267.031
|489.010
|710.990
|756.042
|932.969
|
|-
|9/7-comma
|221.334
|267.999
|489.333
|710.667
|757.333
|932.001
|
|-
|5/4-comma
|220.850
|268.725
|489.575
|710.425
|758.301
|931.275
|
|-
| 6/5-comma
|220.172
|269.742
|489.914
|710.086
|759.656
|930.258
|
|-
|7/6-comma
|219.720
|270.419
|490.140
|709.860
|760.559
|929.581
|
|-
|8/7-comma
|219.398
|270.903
|490.301
|709.699
|761.205
|929.297
|
|-
|1-comma
|217.538
|273.807
|491.269
|708.731
|765.076
| 926.193
|
|-
|6/7-comma
|215.526
|276.711
|492.237
|707.762
|768.948
|923.289
|
|-
|5/6-comma
|215.203
|277.195
|492.398
|707.602
|769.593
|922.805
|
|-
| 4/5-comma
|214.751
|277.873
| 492.624
|707.376
|770.497
|922.167
|
|-
|3/4-comma
|214.926
|278.889
|492.963
|707.037
|771.852
|921.111
|.
|-
|5/7-comma
|213.590
|279.615
|493.205
|706.795
|772.820
|920.795
|
|-
|2/3-comma
|212.945
|280.583
|493.528
|706.472
|774.111
|919.417
|
|-
|3/5-comma
|212.041
|281.938
|493.979
|706.021
|775.918
|918.062
|
|-
|4/7-comma
|211.346
|282.519
|494.173
|705.827
|776.692
|917.401
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|283.971
|494.657
|705.343
|778.628
|916.021
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean hexachord tuning in universe.
|-
|3/7-comma
|209.718
|285.423
|495.141
|704.859
|780.564
|914.577
|
|-
|2/5-comma
|209.331
|286.004
|495.335
|704.665
|781.339
|913.996
|
|-
|1/(ϕ+1)-comma
|209.086
|286.371
|495.457
|704.543
|781.827
|913.629
|
|-
|1/3-comma
|208.573
|287.359
|495.786
|704.214
|783.145
|912.641
|
|-
|2/7-comma
|207.782
|289.372
|496.109
|703.891
|784.436
|910.628
|
|-
|1/4-comma
|207.293
|289.053
|496.351
|703.649
|785.404
|910.947
|
|-
|1/5-comma
|206.620
|290.069
|496.690
|703.310
|786.759
|909.931
|
|-
|1/6-comma
|206.169
|290.747
|496.916
|703.084
|787.663
|909.253
|
|-
|1/7-comma
|205.846
|291.231
|497.077
|702.923
|788.308
|908.769
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|291.248
|497.083
|702.917
|788.331
|908.752
|
|-
| -1/13-comma
|205.983
|291.026
|497.009
|702.993
|788.035
|908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|290.767
|496.922
|703.078
|787.689
|909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|290.460
|496.820
|703.180
|787.280
|909.540
|
|-
| -1/10-comma
|206.605
|290.093
|496.698
|703.302
|786.791
|909.907
|
|-
| -1/9-comma
|206.904
|289.644
|496.548
|703.452
|786.192
|910.356
|
|-
| -1/8-comma
|207.278
|289.083
|496.361
|703.639
|785.444
|910.917
|
|-
| -1/7-comma
|207.759
|288.361
|496.120
|703.880
|784.481
|911.639
|
|-
| -2/13-comma
|208.055
|287.917
|495.972
|704.028
|783.889
|912.083
|
|-
| -1/6-comma
|208.401
|287.399
|495.800
|704.200
|783.198
|912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|286.786
|495.595
|704.405
|782.382
|913.214
|
|-
| -1/5-comma
|209.299
|286.051
|495.350
|704.650
|781.401
|913.949
|
|-
| -3/14-comma
|209.684
|285.474
|495.158
|704.842
|780.632
|914.526
|
|-
| -2/9-comma
|209.898
|285.153
|495.051
|704.949
|780.204
|914.847
|
|-
| -3/13-comma
|210.128
|284.808
|494.936
|705.064
|779.744
|915.192
|
|-
| -1/4-comma
|210.646
|284.030
|494.677
|705.323
|778.707
|915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|283.111
|494.371
|705.629
|777.482
|916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|282.587
|494.196
|705.804
|776.783
|917.413
|
|-
| -3/10-comma
|211.994
|282.010
|494.003
|705.997
|776.013
|917.990
|
|-
| -4/13-comma
|212.799
|281.699
|493.900
|706.100
|775.598
|918.301
|
|-
| -1/3-comma
|212.892
|280.662
|493.554
|706.446
|774.216
|919.338
|
|-
| -5/14-comma
|213.537
|279.700
|493.233
|706.767
|772.933
|920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|279.437
|493.146
|706.854
|772.583
|920.563
|
|-
| -3/8-comma
|214.014
|278.979
|492.993
|707.007
|771.971
|921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|278.697
|492.899
|707.101
|771.596
|921.303
|
|-
| -5/13-comma
|214.274
|278.590
|492.863
|707.137
|771.453
|921.410
|
|-
| -2/5-comma
|214.688
|277.968
|492.656
|707.344
|770.624
|922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|277.294
|492.431
|707.569
|769.725
|922.706
|
|-
| -3/7-comma
|215.458
|276.813
|492.271
|707.729
|769.084
|923.187
|
|-
| -4/9-comma
|215.886
|276.171
|492.057
|707.943
|768.229
|923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|275.763
|491.921
|708.079
|767.684
|924.237
|
|-
| -6/13-comma
|216.346
|275.480
|491.827
|708.173
|767.307
|924.520
|
|-
| -1/2-comma
|217.383
|273.926
|491.309
|708.691
|765.235
|926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|272.371
|490.790
|709.210
|763.161
|927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|272.089
|490.696
|709.304
|762.785
|927.911
|
|-
| -5/9-comma
|218.880
|271.680
|490.560
|709.440
|762.241
|928.320
|
|-
| -4/7-comma
|219.307
|271.039
|490.346
|709.654
|761.385
|928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|270.558
|490.186
|709.814
|760.744
|929.442
|
|-
| -3/5-comma
|220.077
|269.884
|489.961
|710.039
|759.846
|930.116
|
|-
| -8/13-comma
|220.492
|269.262
|489.754
|710.246
|759.016
|930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|269.155
|489.716
|710.284
|758.874
|930.845
|
|-
| -5/8-comma
|220.751
|268.874
|489.625
|710.375
|758.498
|931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|268.414
|489.471
|710.529
|757.886
|931.586
|
|-
| -9/14-comma
|221.232
|268.152
|489.384
|710.616
|757.536
|931.848
|
|-
| -2/3-comma
|221.874
|267.190
|489.063
|710.939
|756.253
|932.810
|
|-
| -9/13-comma
|222.565
|266.153
|488.718
|711.282
|754.871
|933.847
|
|-
| -7/10-comma
|222.772
|265.842
|488.614
|711.386
|754.456
|934.158
|
|-
| -5/7-comma
|223.157
|265.265
|488.422
|711.376
|753.687
|934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|264.740
|488.247
|711.753
|752.987
|935.260
|
|-
| -3/4-comma
|224.119
|263.821
|487.940
|712.060
|751.762
|936.189
|
|-
| -10/13-comma
|224.637
|263.044
|487.681
|712.319
|750.726
|936.956
|
|-
| -7/9-comma
|224.868
|263.044
|487.566
|712.434
|750.265
|937.302
|
|-
| -11/14-comma
|225.081
|262.378
|487.459
|712.541
|749.837
|937.622
|
|-
| -4/5-comma
|225.466
|261.801
|487.267
|712.723
|749.067
|938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|261.066
|487.022
|712.978
|748.088
|938.934
|
|-
| -5/6-comma
|226.365
|260.453
|486.818
|713.182
|747.271
|939.447
|
|-
| -11/13-comma
|226.710
|259.935
|486.645
|713.355
|746.580
|940.065
|
|-
| -6/7-comma
|227.006
|259.491
|486.497
|713.503
|745.988
|940.509
|
|-
| -7/8-comma
|227.487
|258.769
|486.256
|713.744
|745.026
|941.231
|
|-
| -8/9-comma
|227.861
|258.208
|486.069
|713.931
|744.277
|941.792
|
|-
| -9/10-comma
|228.161
|257.759
|485.920
|714.080
|743.678
|942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|257.391
|485.797
|714.203
|743.188
|942.609
|
|-
| -11/12-comma
|228.610
|257.085
|485.695
|714.305
|742.780
|942.915
|
|-
| -12/13-comma
|228.783
|256.826
|485.609
|714.391
|742.435
|943.174
|
|-
| -13/14-comma
|228.931
|256.604
|485.535
|714.465
|742.139
|943.396
|
|-
|-
| -1-comma
|230.855
|253.717
|484.752
|715.248
|738.289
|946.283
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|297.039
|499.013
|700.987
|796.052
|902.961
|
|-
| -1/6-comma
|201.652
|297.523
|499.174
|700.826
|796.697
|902.477
|
|-
| -1/5-comma
|201.200
|298.201
|499.400
|700.600
|797.601
|901.799
|
|-
| -1/4-comma
|200.522
|299.217
|499.739
|700.261
|798.956
|900.783
|
|-
| -2/7-comma
|200.038
|299.942
|499.981
|700.019
|799.924
|900.058
|
|-
| -1/3-comma
|199.393
|300.911
|500.303
|699.697
|801.214
|899.089
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|301.900
|500.633
|699.367
|802.533
|898.100
|
|-
| -2/5-comma
|198.499
|302.266
|500.755
|699.245
|803.021
|897.634
|
|-
| -3/7-comma
|198.102
|302.847
|500.949
|699.051
|803.796
|897.153
|
|-
| -1/2-comma
|197.134
|304.299
|501.433
|699.567
|805.732
|895.701
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|196.166
|305.751
|501.917
|698.083
|807.668
|894.249
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -3/5-comma
|195.779
|306.332
|502.111
|697.889
|808.442
|893.668
|
|-
| -2/3-comma
|194.876
|307.687
|502.562
|697.438
|810.249
|892.313
|
|-
| -5/7-comma
|194.230
|308.655
|502.885
|697.115
|811.540
|891.345
|
|-
| -4/5-comma
|193.069
|310.397
|503.466
|696.534
|813.863
|889.603
|
|-
| -5/6-comma
|192.617
|311.075
|503.692
|696.308
|814.766
|888.925
|
|-
|-
| -6/7-comma
|192.294
|311.556
|503.853
|696.147
|815.412
|888.444
|
|-
| -1-comma
|190.352
|314.463
|504.821
|695.179
|819.283
|885.537
|
|-
| -8/7-comma
|188.422
|317.367
|505.789
|694.211
|823.155
|882.633
|
|-
| -7/6-comma
|188.100
|317.851
|505.950
|694.050
|823.801
|882.149
|
|-
| -6/5-comma
|187.648
|318.528
|506.176
|693.824
|824.704
|881.472
|
|-
| -5/4-comma
|186.970
|319.545
|506.515
|693.485
|826.059
|880.455
|
|-
| -9/7-comma
|186.486
|320.271
|506.757
|693.243
|827.027
|879.730
|
|-
| -4/3-comma
|185.841
|321.239
|507.080
|692.920
|828.318
|878.761
|
|-
| -7/5-comma
|184.937
|322.594
|507.531
|692.469
|830.125
|877.406
|
|-
| -10/7-comma
|184.550
|323.174
|507.725
|692.275
|830.899
|876.826
|
|-
| -3/2-comma
|183.582
|324.626
|508.209
|691.791
|832.835
|875.374
|
|-
| -11/7-comma
|182.614
|326.078
|508.693
|691.307
|834.771
|873.922
|
|-
| -8/5-comma
|182.228
|326.659
|508.886
|691.114
|835.546
|873.341
|
|-
| -ϕ-comma
|181.983
|327.026
|509.009
|690.991
|836.034
|872.974
|
|-
| -5/3-comma
|181.324
|328.014
|509.338
|690.662
|837.353
|871.986
|
|-
| -12/7-comma
|180.678
|328.982
|509.661
|690.339
|838.643
|871.018
|
|-
| -7/4-comma
|180.194
|329.708
|509.903
|690.097
|839.611
|870.292
|
|-
| -9/5-comma
|179.517
|330.725
|510.242
|689.758
|840.966
|869.275
|
|-
| -11/6-comma
|179.065
|331.402
|510.467
|689.533
|841.870
|868.598
|
|-
| -13/7-comma
|178.742
|331.886
|510.629
|689.371
|842.515
|868.114
|
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|846.387
|865.210
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|253.717
|484.752
|715.248
|738.289
|946.283
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|232.780
|250.830
|483.610
|716.390
|734.440
|949.170
|
|-
| -14/13-comma
|232.928
|250.608
|483.536
|716.464
|734.133
|949.392
|
|-
| -13/12-comma
|233.101
|250.349
|483.450
|716.550
|733.798
|949.651
|
|-
| -12/11-comma
|233.305
|250.043
|483.348
|716.642
|733.390
|949.957
|
|-
| -11/10-comma
|233.550
|249.675
|483.225
|716.775
|732.900
|950.325
|
|-
| -10/9-comma
|233.151
|249.226
|483.075
|716.925
|732.301
|950.774
|
|-
| -9/8-comma
|234.234
|248.665
|482.888
|717.112
|731.553
|951.335
|
|-
| -8/7-comma
|234.295
|247.943
|482.648
|717.352
|730.590
|952.352
|
|-
| -15/13-comma
|235.001
|247.499
|482.500
|717.500
|729.998
|952.501
|
|-
| -7/6-comma
|235.346
|246.981
|482.327
|717.673
|729.307
|953.019
|
|-
| -13/11-comma
|235.755
|246.368
|482.123
|717.877
|728.491
|953.632
|
|-
| -6/5-comma
|236.244
|245.633
|481.878
|718.122
|727.511
|954.367
|
|-
| -17/14-comma
|236.629
|245.056
|481.685
|718.315
|726.741
|954.943
|
|-
| -11/9-comma
|236.843
|244.735
|481.578
|718.422
|726.313
|955.265
|
|-
| -16/13-comma
|237.926
|244.390
|481.463
|718.537
|725.853
|955.610
|
|-
| -5/4-comma
|237.592
|243.612
|481.204
|718.796
|724.816
|956.388
|
|-
| -14/11-comma
|238.204
|242.694
|480.898
|719.102
|723.592
|957.694
|
|-
| -9/7-comma
|238.554
|242.169
|480.723
|719.277
|722.892
|957.831
|
|-
| -13/10-comma
|238.939
|241.591
|480.530
|719.470
|722.122
|957.409
|
|-
| -17/13-comma
|239.146
|241.280
|480.427
|719.573
|721.707
|958.720
|
|-
| -4/3-comma
|239.837
|240.244
|480.081
|719.919
|720.326
|959.756
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|239.282
|479.761
|720.239
|719.042
|960.718
|
|-
| -15/11-comma
|240.634
|239.019
|479.673
|720.327
|718.693
|960.981
|
|-
| -11/8-comma
|240.960
|238.560
|479.520
|720.480
|718.080
|961.440
|
|-
| -(ϕ+2)/(ϕ+1)-comma
|241.148
|238.279
|479.426
|720.574
|717.705
|961.721
|
|-
| -18/13-comma
|241.219
|238.171
|479.390
|720.610
|717.561
|961.829
|
|-
| -7/5-comma
|241.634
|237.550
|479.183
|720.817
|716.733
|962.450
|
|-
| -17/12-comma
|242.917
|236.876
|478.959
|721.041
|715.835
|962.124
|
|-
| -10/7-comma
|242.403
|236.395
|478.798
|721.202
|715.193
|963.605
|
|-
| -13/9-comma
|242.831
|235.753
|478.584
|721.416
|714.338
|964.247
|
|-
| -16/11-comma
|243.103
|235.345
|478.448
|721.552
|713.793
|964.655
|
|-
| -19/13-comma
|243.708
|235.062
|478.354
|721.646
|713.416
|964.938
|
|-
| -3/2-comma
|244.328
|233.508
|477.836
|722.164
|711.343
|966.492
|
|-
| -20/13-comma
|245.344
|231.953
|477.318
|722.682
|709.271
|968.047
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|231.671
|477.224
|722.776
|708.894
|968.329
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|231.262
|477.087
|722.913
|708.350
|968.738
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|230.621
|476.873
|723.127
|707.493
|969.379
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|230.140
|476.713
|723.287
|706.853
|969.860
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|229.466
|476.489
|723.511
|705.955
|970.534
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|228.844
|476.281
|723.719
|705.156
|971.156
|
|-
| -ϕ-comma
|247.491
|228.737
|476.246
|723.754
|704.983
|971.263
|
|-
| -13/8-comma
|247.696
|228.456
|476.152
|723.848
|704.607
|971.544
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|227.996
|475.999
|724.001
|703.995
|972.004
|
|-
| -23/14-comma
|248.823
|227.734
|475.911
|724.089
|703.645
|972.266
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|226.771
|475.590
|724.410
|702.362
|973.229
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|
|475.245
|
|
|225.735
|
|-
| -17/10-comma
|249.717
|
|475.141
|
|
|225.424
|
|-
| -12/7-comma
|250.105
|
|474.949
|
|
|224.847
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|
|474.774
|
|
|224.322
|
|-
| -7/4-comma
|251.064
|
|474.468
|
|
|223.403
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|
|474.209
|
|
|222.626
|
|-
| -16/9-comma
|251.823
|
|474.094
|
|
|222.281
|
|-
| -25/14-comma
|252.027
|
|473.987
|
|
|221.960
|
|-
| -9/5-comma
|252.412
|
|473.794
|
|
|221.382
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|
|473.549
|
|
|220.648
|
|-
| -11/6-comma
|253.610
|
|473.345
|
|
|220.035
|
|-
| -24/13-comma
|253.345
|
|473.172
|
|
|219.517
|
|-
| -13/7-comma
|253.951
|
|473.924
|
|
|219.073
|
|-
| -15/8-comma
|254.433
|
|472.784
|
|
|218.351
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|
|472.597
|
|
|217.790
|
|-
| -19/10-comma
|255.106
|
|472.447
|
|
|217.341
|
|-
| -21/11-comma
|255.351
|
|472.324
|
|
|216.973
|
|-
| -23/12-comma
|255.555
|
|472.222
|
|
|216.667
|
|-
| -25/13-comma
|255.728
|
|472.135
|
|
|216.408
|
|-
| -27/14-comma
|255.876
|
|472.052
|
|
|216.186
|
|-
|-
| -2-comma
|258.801
|
|471.100
|
|
|213.299
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-24T04:18:15Z

Moremajorthanmajor:

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]] (as this comma is already unnoticeable, the Boethian “mean minor third” is more commonly used as a joke); but with more complex thirds, and tempering out the quarter tone of [[1053/1024]] (often confused for the simpler [[36/35]]) or [[33/32]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds. That is why doing the latter is considered to generate a perverse temperament, though only the former quarter tone is considered to generate the “real” one.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|212.824
|470.941
|729.051
|683.766
|987.176
|
|-
|27/7-comma
|256.181
|215.728
|471.909
|728.091
|687.637
|984.272
|
|-
|23/6-comma
|255.858
|216.212
|472.071
|727.929
|688.283
|983.788
|
|-
|19/5-comma
|255.407
|216.890
|472.297
|727.703
|689.187
|983.110
|
|-
|15/4-comma
|254.769
|217.906
|472.635
|727.365
|690.542
|982.094
|
|-
|26/7-comma
|254.243
|218.632
|472.877
|727.123
|691.510
|981.378
|
|-
|11/3-comma
| 253.600
|219.600
|473.200
|726.800
|692.800
|980.400
|
|-
|(2+ϕ)-comma
|252.940
|220.589
|473.530
|726.470
|694.119
|979.411
|
|-
|18/5-comma
|252.696
|220.956
|473.652
|726.348
|694.607
|979.044
|
|-
|25/7-comma
|252.309
|221.536
|473.845
|726.155
|695.382
|978.464
|
|-
|7/2-comma
|251.341
|222.988
|474.329
|725.671
|697.318
|977.012
|
|-
|24/7-comma
|250.373
|224.440
|474.813
|725.187
|699.253
|975.560
|
|-
|17/5-comma
|249.986
|225.021
|475.007
|724.993
|700.028
|974.979
|
|-
|10/3-comma
|249.083
|226.376
|475.459
|724.541
|701.835
|973.624
|
|-
|23/7-comma
|248.437
|227.344
|475.781
|724.219
|703.126
|972.656
|
|-
|13/4-comma
|247.953
|228.070
|476.023
|723.977
|704.094
|971.930
|
|-
|16/5-comma
|247.258
|229.087
|476.362
|723.638
|705.449
|970.913
|
|-
|19/6-comma
|246.824
|229.764
|476.588
|723.412
|706.352
|970.236
|
|-
|22/7-comma
|246.501
|230.248
|476.749
|723.251
|706.998
|969.752
|
|-
|3-comma
|244.565
|233.152
|477.717
|722.283
|710.870
|966.848
|
|-
|20/7-comma
|242.629
|236.056
|478.685
|721.315
|714.741
|963.944
|
|-
|17/6-comma
|242.307
|236.540
|478.847
|721.153
|715.387
|963.460
|
|-
|14/5-comma
|241.855
|237.218
|479.073
|720.927
|716.290
|962.782
|
|-
|11/4-comma
|241.177
|238.234
|479.411
|720.589
|717.645
|961.766
|
|-
|19/7-comma
|240.693
|238.960
|479.653
|720.347
|718.613
|961.040
|
|-
|8/3-comma
|240.048
|239.928
|479.976
|720.024
|719.904
|960.072
|
|-
|13/5-comma
|239.145
|241.283
|480.428
|719.572
|721.711
|958.717
|
|-
|18/7-comma
|238.757
|241.864
|480.621
|719.379
|722.485
|958.136
|
|-
|5/2-comma
| 237.789
|243.316
|481.105
|718.895
|724.421
|956.684
|
|-
|17/7-comma
|236.821
|244.768
|481.589
|718.411
|726.357
|955.232
|
|-
|12/5-comma
|236.434
|245.349
|481.783
|718.217
|727.132
|954.651
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|245.715
|481.905
|718.095
|727.620
|954.285
|
|-
|7/3-comma
|235.531
|246.704
|482.235
|717.765
|728.938
|953.296
|
|-
|16/7-comma
|234.115
|247.672
|482.557
|717.423
|730.229
|952.328
|
|-
|9/4-comma
|234.401
|248.398
|482.799
|717.201
|731.197
|951.602
|
|-
|11/5-comma
|233.276
|249.414
|483.183
|716.817
|732.552
|950.596
|
|-
|13/6-comma
|233.272
|250.092
|483.364
|716.636
|733.456
|949.909
|
|-
|15/7-comma
|232.051
|250.576
|483.525
|716.475
|734.101
|949.424
|
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|846.071
| 865.447
|
|-
|13/14-comma
|178.890
|331.666
|510.555
|689.445
|842.221
|868.334
|
|-
|12/13-comma
|179.037
|331.444
|510.481
|689.519
|841.925
| 868.556
|
|-
|11/12-comma
|179.210
|331.185
|510.395
|689.605
|841.580
|868.815
|
|-
|10/11-comma
| 179.414
|330.879
| 510.293
|689.707
|841.172
|869.121
|
|-
|9/10-comma
|179.659
|330.511
| 510.170
|689.830
|840.682
|869.489
|
|-
|8/9-comma
|179.959
|330.062
|510.021
|689.979
|840.083
|869.038
|
|-
|7/8-comma
|180.333
|329.501
|509.834
|690.166
|839.334
|870.499
|
|-
|6/7-comma
|180.814
|328.779
|509.593
|690.407
|838.372
|871.221
|
|-
|11/13-comma
|181.110
|328.335
|509.445
|690.555
|837.780
|871.665
|
|-
|5/6-comma
|181.455
|327.817
|509.272
|690.728
|837.089
|872.193
|
|-
|9/11-comma
|181.864
|327.204
|509.068
|690.932
|836.272
|872.796
|
|-
|4/5-comma
|182.354
|326.469
|508.823
|691.177
|835.293
|873.531
|
|-
|11/14-comma
|182.739
|325.892
|508.630
|691.370
|834.523
|874.108
|
|-
|7/9-comma
|182.952
|325.571
|508.523
|691.477
|834.095
| 874.429
|
|-
|10/13-comma
|183.183
|325.226
|508.408
|691.592
|833.634
|874.774
|
|-
|3/4-comma
|183.701
|324.449
|508.150
|691.850
|832.598
|875.551
|
|-
|8/11-comma
|184.687
|323.530
|507.843
|692.157
|831.373
|876.470
|
|-
|5/7-comma
|184.633
|323.005
|507.638
|692.362
|830.673
|876.995
|
|-
|7/10-comma
|184.952
|322.428
|507.476
|692.524
|829.904
|877.572
|
|-
|9/13-comma
|185.255
|322.117
|507.372
|692.628
|829.489
|877.883
|
|-
|2/3-comma
|185.946
|321.080
|507.027
|692.973
|828.107
|878.920
|
|-
|9/14-comma
|186.588
|320.118
|506.706
|693.294
|828.824
|879.882
|
|-
|7/11-comma
|186.763
|319.856
|506.619
|693.381
|826.474
|880.144
|
|-
|5/8-comma
|187.069
|319.396
|506.465
|693.535
|825.862
|880.604
|
|-
|1/ϕ-comma
|187.257
|319.115
|506.372
|693.628
|825.486
|880.885
|
|-
|8/13-comma
|187.320
|319.008
|506.336
|693.664
|825.344
|880.992
|
|-
|3/5-comma
|187.743
|318.386
|506.129
|693.871
|824.514
|881.614
|
|-
|7/12-comma
|188.194
|317.712
|505.904
|694.096
|823.616
|882.288
|
|-
|4/7-comma
|188.512
|317.231
|505.744
|694.256
|822.975
|882.769
|
|-
|5/9-comma
|188.940
|316.590
|505.530
|694.470
|822.119
|883.410
|
|-
|6/11-comma
|189.213
|316.181
|505.394
|694.606
|821.575
|883.891
|
|-
|7/13-comma
|189.401
|315.899
|505.300
|694.700
|821.198
|884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|314.344
|504.781
|695.219
|819.125
|885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|312.790
|504.263
|695.737
|817.053
|887.210
|
|-
|5/11-comma
|191.338
|312.507
|504.169
|695.831
|816.676
|887.493
|
|-
|4/9-comma
|191.934
|312.099
|504.033
|695.967
|816.131
|877.901
|
|-
|3/7-comma
|192.362
|311.457
|503.819
|696.181
|815.276
|388.443
|
|-
|5/12-comma
|192.683
|310.976
|503.659
|696.341
|814.635
|889.024
|
|-
|2/5-comma
|193.132
|310.302
|503.434
|696.566
|813.736
|889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|309.680
|503.227
|696.773
|812.907
|890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|309.573
|503.191
|696.801
|812.764
| 890.427
|
|-
|3/8-comma
|193.805
|309.291
| 503.096
|696.904
|812.389
|890.709
|
|-
|4/11-comma
|194.112
|308.832
|502.944
|697.956
|811.776
|891.168
|
|-
|5/14-comma
|194.287
|308.570
|502.856
|697.144
|811.427
|891.430
|
|-
|1/3-comma
|194.928
|307.608
|502.536
|697.424
|810.144
|892.392
|
|-
|4/13-comma
|195.619
|306.571
|502.190
|697.810
|808.762
|893.429
|
|-
|3/10-comma
|195.174
|306.260
|502.087
|697.913
|808.347
|893.740
|
|-
|2/7-comma
|196.211
|305.683
|501.894
|698.106
|807.577
|894.317
|
|-
|3/11-comma
|196.561
|305.158
|501.718
|698.282
|806.877
|894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|304.240
|501.413
|698.587
|805.653
|895.760
|
|-
|3/13-comma
|197.692
|303.462
|501.154
|698.846
|804.616
|896.538
|
|-
|2/9-comma
|197.922
|303.117
|501.039
|698.961
|804.155
|896.883
|
|-
|3/14-comma
|198.136
|302.796
|500.932
|699.068
|803.728
|897.204
|
|-
|1/5-comma
|198.521
|302.219
|500.740
|699.260
|802.958
|897.781
|
|-
|2/11-comma
|199.011
|301.484
|500.495
|699.505
|801.978
|898.516
|
|-
|1/6-comma
|199.419
|300.871
|500.290
|699.810
|801.162
|899.129
|
|-
|2/13-comma
|199.765
|300.353
|500.118
|699.882
|800.471
|899.647
|
|-
|1/7-comma
|200.061
|299.909
|499.970
|700.030
|799.879
|900.091
|
|-
|1/8-comma
|200.542
|299.187
| 499.729
|700.271
|798.916
|900.823
|
|-
|1/9-comma
|200.916
|298.626
|499.542
|700.558
|798.168
|901.374
|
|-
|1/10-comma
|201.785
|298.177
|499.392
|700.608
|797.569
|901.823
|
|-
|1/11-comma
|201.460
|297.810
|499.270
|700.730
|797.079
|902.190
|
|-
|1/12-comma
|201.665
|297.503
|499.168
|700.832
|796.671
|902.497
|
|-
|1/13-comma
|201.837
|297.244
|499.081
|700.019
|796.325
|902.756
|
|-
|1/14-comma
|201.953
|297.022
|499.007
|700.993
|796.029
|902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|-
|13/7-comma
|229.078
|256.384
|485.461
|
|
|
|
|-
|11/6-comma
|228.755
|256.868
|485.623
|
|
|
|
|-
|9/5-comma
|228.697
|257.545
|485.848
|
|
|
|
|-
| 7/4-comma
|227.626
|258.562
|486.187
|
|
|
|
|-
|12/7-comma
|227.142
|259.288
|486.429
|
|
|
|
|-
|5/3-comma
|226.496
|260.253
|486.752
|
|
|
|
|-
|ϕ-comma
|225.837
|261.244
|487.081
|
|
|
|
|-
|8/5-comma
|225.593
|261.611
|487.204
|
|
|
|
|-
|11/7-comma
|225.206
|262.192
| 487.397
|
|
|
|
|-
|3/2-comma
| 224.762
|263.644
|487.881
|
|
|
|
|-
|10/7-comma
|223.270
|265.096
|488.365
|
|
|
|
|-
|7/5-comma
|222.882
|265.676
|488.559
|
|
|
|
|-
|4/3-comma
|221.979
|267.031
|489.010
|
|
|
|
|-
|9/7-comma
|221.334
|267.999
|489.333
|
|
|
|
|-
|5/4-comma
|220.850
|268.725
|489.575
|
|
|
|
|-
| 6/5-comma
|220.172
|269.742
|489.914
|
|
|
|
|-
|7/6-comma
|219.720
|270.419
|490.140
|
|
|
|
|-
|8/7-comma
|219.398
|270.903
|490.301
|
|
|
|
|-
|1-comma
|217.538
|273.807
|491.269
|
|
|
|
|-
|6/7-comma
|215.526
|276.711
|492.237
|
|
|
|
|-
|5/6-comma
|215.203
|277.195
|492.398
|
|
|
|
|-
| 4/5-comma
|214.751
|277.873
| 492.624
|
|
|
|
|-
|3/4-comma
|214.926
|278.889
|492.963
|
|
|
|
|-
|5/7-comma
|213.590
|279.615
|493.205
|
|
|
|
|-
|2/3-comma
|212.945
|280.583
|493.528
|
|
|
|
|-
|3/5-comma
|212.041
|281.938
|493.979
|
|
|
|
|-
|4/7-comma
|211.346
|282.519
|494.173
|
|
|
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|283.971
|494.657
|
|
|
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|209.718
|285.423
|495.141
|
|
|
|
|-
|2/5-comma
|209.331
|286.004
|495.335
|
|
|
|
|-
|1/(ϕ+1)-comma
|209.086
|286.371
|495.457
|
|
|
|
|-
|1/3-comma
|208.573
|287.359
|495.786
|
|
|
|
|-
|2/7-comma
|207.782
|289.372
|496.109
|
|
|
|
|-
|1/4-comma
|207.293
|289.053
|496.351
|
|
|
|
|-
|1/5-comma
|206.620
|290.069
|496.690
|
|
|
|
|-
|1/6-comma
|206.169
|290.747
|496.916
|
|
|
|
|-
|1/7-comma
|205.846
|291.231
|497.077
|
|
|
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|291.248
|497.083
|702.917
|788.331
|908.752
|
|-
| -1/13-comma
|205.983
|291.026
|497.009
|702.993
|788.035
|908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|290.767
|496.922
|703.078
|787.689
|909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|290.460
|496.820
|703.180
|787.280
|909.540
|
|-
| -1/10-comma
|206.605
|290.093
|496.698
|703.302
|786.791
|909.907
|
|-
| -1/9-comma
|206.904
|289.644
|496.548
|703.452
|786.192
|910.356
|
|-
| -1/8-comma
|207.278
|289.083
|496.361
|703.639
|785.444
|910.917
|
|-
| -1/7-comma
|207.759
|288.361
|496.120
|703.880
|784.481
|911.639
|
|-
| -2/13-comma
|208.055
|287.917
|495.972
|704.028
|783.889
|912.083
|
|-
| -1/6-comma
|208.401
|287.399
|495.800
|704.200
|783.198
|912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|286.786
|495.595
|704.405
|782.382
|913.214
|
|-
| -1/5-comma
|209.299
|286.051
|495.350
|704.650
|781.401
|913.949
|
|-
| -3/14-comma
|209.684
|285.474
|495.158
|704.842
|780.632
|914.526
|
|-
| -2/9-comma
|209.898
|285.153
|495.051
|704.949
|780.204
|914.847
|
|-
| -3/13-comma
|210.128
|284.808
|494.936
|705.064
|779.744
|915.192
|
|-
| -1/4-comma
|210.646
|284.030
|494.677
|705.323
|778.707
|915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|283.111
|494.371
|705.629
|777.482
|916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|282.587
|494.196
|705.804
|776.783
|917.413
|
|-
| -3/10-comma
|211.994
|282.010
|494.003
|705.997
|776.013
|917.990
|
|-
| -4/13-comma
|212.799
|281.699
|493.900
|706.100
|775.598
|918.301
|
|-
| -1/3-comma
|212.892
|280.662
|493.554
|706.446
|774.216
|919.338
|
|-
| -5/14-comma
|213.537
|279.700
|493.233
|706.767
|772.933
|920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|279.437
|493.146
|706.854
|772.583
|920.563
|
|-
| -3/8-comma
|214.014
|278.979
|492.993
|707.007
|771.971
|921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|278.697
|492.899
|707.101
|771.596
|921.303
|
|-
| -5/13-comma
|214.274
|278.590
|492.863
|707.137
|771.453
|921.410
|
|-
| -2/5-comma
|214.688
|277.968
|492.656
|707.344
|770.624
|922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|277.294
|492.431
|707.569
|769.725
|922.706
|
|-
| -3/7-comma
|215.458
|276.813
|492.271
|707.729
|769.084
|923.187
|
|-
| -4/9-comma
|215.886
|276.171
|492.057
|707.943
|768.229
|923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|275.763
|491.921
|708.079
|767.684
|924.237
|
|-
| -6/13-comma
|216.346
|275.480
|491.827
|708.173
|767.307
|924.520
|
|-
| -1/2-comma
|217.383
|273.926
|491.309
|708.691
|765.235
|926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|272.371
|490.790
|709.210
|763.161
|927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|272.089
|490.696
|709.304
|762.785
|927.911
|
|-
| -5/9-comma
|218.880
|271.680
|490.560
|709.440
|762.241
|928.320
|
|-
| -4/7-comma
|219.307
|271.039
|490.346
|709.654
|761.385
|928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|270.558
|490.186
|709.814
|760.744
|929.442
|
|-
| -3/5-comma
|220.077
|269.884
|489.961
|710.039
|759.846
|930.116
|
|-
| -8/13-comma
|220.492
|269.262
|489.754
|710.246
|759.016
|930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|269.155
|489.716
|710.284
|758.874
|930.845
|
|-
| -5/8-comma
|220.751
|268.874
|489.625
|710.375
|758.498
|931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|268.414
|489.471
|710.529
|757.886
|931.586
|
|-
| -9/14-comma
|221.232
|268.152
|489.384
|710.616
|757.536
|931.848
|
|-
| -2/3-comma
|221.874
|267.190
|489.063
|710.939
|756.253
|932.810
|
|-
| -9/13-comma
|222.565
|266.153
|488.718
|711.282
|754.871
|933.847
|
|-
| -7/10-comma
|222.772
|265.842
|488.614
|711.386
|754.456
|934.158
|
|-
| -5/7-comma
|223.157
|265.265
|488.422
|711.376
|753.687
|934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|264.740
|488.247
|711.753
|752.987
|935.260
|
|-
| -3/4-comma
|224.119
|263.821
|487.940
|712.060
|751.762
|936.189
|
|-
| -10/13-comma
|224.637
|263.044
|487.681
|712.319
|750.726
|936.956
|
|-
| -7/9-comma
|224.868
|263.044
|487.566
|712.434
|750.265
|937.302
|
|-
| -11/14-comma
|225.081
|262.378
|487.459
|712.541
|749.837
|937.622
|
|-
| -4/5-comma
|225.466
|261.801
|487.267
|712.723
|749.067
|938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|261.066
|487.022
|712.978
|748.088
|938.934
|
|-
| -5/6-comma
|226.365
|260.453
|486.818
|713.182
|747.271
|939.447
|
|-
| -11/13-comma
|226.710
|259.935
|486.645
|713.355
|746.580
|940.065
|
|-
| -6/7-comma
|227.006
|259.491
|486.497
|713.503
|745.988
|940.509
|
|-
| -7/8-comma
|227.487
|258.769
|486.256
|713.744
|745.026
|941.231
|
|-
| -8/9-comma
|227.861
|258.208
|486.069
|713.931
|744.277
|941.792
|
|-
| -9/10-comma
|228.161
|257.759
|485.920
|714.080
|743.678
|942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|257.391
|485.797
|714.203
|743.188
|942.609
|
|-
| -11/12-comma
|228.610
|257.085
|485.695
|714.305
|742.780
|942.915
|
|-
| -12/13-comma
|228.783
|256.826
|485.609
|714.391
|742.435
|943.174
|
|-
| -13/14-comma
|228.931
|256.604
|485.535
|714.465
|742.139
|943.396
|
|-
|-
| -1-comma
|230.855
|253.717
|484.752
|715,248
|738.289
|946.283
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|
|499.013
|
|
|297.039
|
|-
| -1/6-comma
|201.652
|
|499.174
|
|
|297.523
|
|-
| -1/5-comma
|201.200
|
|499.400
|
|
|298.201
|
|-
| -1/4-comma
|200.522
|
|499.739
|
|
|299.217
|
|-
| -2/7-comma
|200.038
|
|499.981
|
|
|299.942
|
|-
| -1/3-comma
|199.393
|
|500.303
|
|
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|
|500.633
|
|
|301.900
|
|-
| -2/5-comma
|198.499
|
|500.755
|
|
|302.266
|
|-
| -3/7-comma
|198.102
|
|500.949
|
|
|302.847
|
|-
| -1/2-comma
|197.134
|
|501.433
|
|
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|196.166
|
|501.917
|
|
|305.751
|
|-
| -3/5-comma
|195.779
|
|502.111
|
|
|306.332
|
|-
| -2/3-comma
|194.876
|
|502.562
|
|
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|194.230
|
|502.885
|
|
|308.655
|
|-
| -4/5-comma
|193.069
|
|503.466
|
|
|310.397
|
|-
| -5/6-comma
|192.617
|
|503.692
|
|
|311.075
|
|-
|-
| -6/7-comma
|192.294
|
|503.853
|
|
|311.556
|
|-
| -1-comma
|190.352
|
|504.821
|
|
|314.463
|
|-
| -8/7-comma
|188.422
|
|505.789
|
|
|317.367
|
|-
| -7/6-comma
|188.100
|
|505.950
|
|
|317.851
|
|-
| -6/5-comma
|187.648
|
|506.176
|
|
|318.528
|
|-
| -5/4-comma
|186.970
|
|506.515
|
|
|319.545
|
|-
| -9/7-comma
|186.486
|
|506.757
|
|
|320.271
|
|-
| -4/3-comma
|185.841
|
|507.080
|
|
|321.239
|
|-
| -7/5-comma
|184.937
|
|507.531
|
|
|322.594
|
|-
| -10/7-comma
|184.550
|
|507.725
|
|
|323.174
|
|-
| -3/2-comma
|183.582
|
|508.209
|
|
|324.626
|
|-
| -11/7-comma
|182.614
|
|508.693
|
|
|326.078
|
|-
| -8/5-comma
|182.228
|
|508.886
|
|
|326.659
|
|-
| -ϕ-comma
|181.983
|
|509.009
|
|
|327.026
|
|-
| -5/3-comma
|181.324
|
|509.338
|
|
|328.014
|
|-
| -12/7-comma
|180.678
|
|509.661
|
|
|328.982
|
|-
| -7/4-comma
|180.194
|
|509.903
|
|
|329.708
|
|-
| -9/5-comma
|179.517
|
|510.242
|
|
|330.725
|
|-
| -11/6-comma
|179.065
|
|510.467
|
|
|331.402
|
|-
| -13/7-comma
|178.742
|
|510.629
|
|
|331.886
|
|-
| -2-comma
|176.807
|
|511.597
|
|
|334.790
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|253.717
|484.752
|715,248
|738.289
|946.283
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|232.780
|
|483.610
|
|
|250.830
|
|-
| -14/13-comma
|232.928
|
|483.536
|
|
|250.608
|
|-
| -13/12-comma
|233.101
|
|483.450
|
|
|250.349
|
|-
| -12/11-comma
|233.305
|
|483.348
|
|
|250.043
|
|-
| -11/10-comma
|233.550
|
|483.225
|
|
|249.675
|
|-
| -10/9-comma
|233.151
|
|483.075
|
|
|249.226
|
|-
| -9/8-comma
|234.234
|
|482.888
|
|
|248.665
|
|-
| -8/7-comma
|234.295
|
|482.648
|
|
|247.943
|
|-
| -15/13-comma
|235.001
|
|482.500
|
|
|247.499
|
|-
| -7/6-comma
|235.346
|
|482.327
|
|
|246.981
|
|-
| -13/11-comma
|235.755
|
|482.123
|
|
|246.368
|
|-
| -6/5-comma
|236.244
|
|481.878
|
|
|245.633
|
|-
| -17/14-comma
|236.629
|
|481.685
|
|
|245.056
|
|-
| -11/9-comma
|236.843
|
|481.578
|
|
|244.735
|
|-
| -16/13-comma
|237.926
|
|481.463
|
|
|244.390
|
|-
| -5/4-comma
|237.592
|
|481.204
|
|
|243.612
|
|-
| -14/11-comma
|238.204
|
|480.898
|
|
|242.694
|
|-
| -9/7-comma
|238.554
|
|480.723
|
|
|242.169
|
|-
| -13/10-comma
|238.939
|
|480.530
|
|
|241.591
|
|-
| -17/13-comma
|239.146
|
|480.427
|
|
|241.280
|
|-
| -4/3-comma
|239.837
|
|480.081
|
|
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|
|479.761
|
|
|239.282
|
|-
| -15/11-comma
|240.634
|
|479.673
|
|
|239.019
|
|-
| -11/8-comma
|240.960
|
|479.520
|
|
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|241.148
|
|479.426
|
|
|238.279
|
|-
| -18/13-comma
|241.219
|
|479.390
|
|
|238.171
|
|-
| -7/5-comma
|241.634
|
|479.183
|
|
|237.550
|
|-
| -17/12-comma
|242.917
|
|478.959
|
|
|236.876
|
|-
| -10/7-comma
|242.403
|
|478.798
|
|
|236.395
|
|-
| -13/9-comma
|242.831
|
|478.584
|
|
|235.753
|
|-
| -16/11-comma
|243.103
|
|478.448
|
|
|235.345
|
|-
| -19/13-comma
|243.708
|
|478.354
|
|
|235.062
|
|-
| -3/2-comma
|244.328
|
|477.836
|
|
|233.508
|
|-
| -20/13-comma
|245.344
|
|477.318
|
|
|231.953
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|
|477.224
|
|
|231.671
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|
|477.087
|
|
|231.262
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|
|476.873
|
|
|230.621
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|
|476.713
|
|
|230.140
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|
|476.489
|
|
|229.466
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|
|476.281
|
|
|228.844
|
|-
| -ϕ-comma
|247.491
|
|476.246
|
|
|228.737
|
|-
| -13/8-comma
|247.696
|
|476.152
|
|
|228.456
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|
|475.999
|
|
|227.996
|
|-
| -23/14-comma
|248.823
|
|475.911
|
|
|227.734
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|
|475.590
|
|
|226.771
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|
|475.245
|
|
|225.735
|
|-
| -17/10-comma
|249.717
|
|475.141
|
|
|225.424
|
|-
| -12/7-comma
|250.105
|
|474.949
|
|
|224.847
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|
|474.774
|
|
|224.322
|
|-
| -7/4-comma
|251.064
|
|474.468
|
|
|223.403
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|
|474.209
|
|
|222.626
|
|-
| -16/9-comma
|251.823
|
|474.094
|
|
|222.281
|
|-
| -25/14-comma
|252.027
|
|473.987
|
|
|221.960
|
|-
| -9/5-comma
|252.412
|
|473.794
|
|
|221.382
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|
|473.549
|
|
|220.648
|
|-
| -11/6-comma
|253.610
|
|473.345
|
|
|220.035
|
|-
| -24/13-comma
|253.345
|
|473.172
|
|
|219.517
|
|-
| -13/7-comma
|253.951
|
|473.924
|
|
|219.073
|
|-
| -15/8-comma
|254.433
|
|472.784
|
|
|218.351
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|
|472.597
|
|
|217.790
|
|-
| -19/10-comma
|255.106
|
|472.447
|
|
|217.341
|
|-
| -21/11-comma
|255.351
|
|472.324
|
|
|216.973
|
|-
| -23/12-comma
|255.555
|
|472.222
|
|
|216.667
|
|-
| -25/13-comma
|255.728
|
|472.135
|
|
|216.408
|
|-
| -27/14-comma
|255.876
|
|472.052
|
|
|216.186
|
|-
|-
| -2-comma
|258.801
|
|471.100
|
|
|213.299
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-23T15:20:32Z

Moremajorthanmajor: /* Cautions */

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]] (as this comma is already unnoticeable, the Boethian “mean minor third” is more commonly used as a joke); but with more complex thirds, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|212.824
|470.941
|729.051
|683.766
|987.176
|
|-
|27/7-comma
|256.181
|215.728
|471.909
|728.091
|687.637
|984.272
|
|-
|23/6-comma
|255.858
|216.212
|472.071
|727.929
|688.283
|983.788
|
|-
|19/5-comma
|255.407
|216.890
|472.297
|727.703
|689.187
|983.110
|
|-
|15/4-comma
|254.769
|217.906
|472.635
|727.365
|690.542
|982.094
|
|-
|26/7-comma
|254.243
|218.632
|472.877
|727.123
|691.510
|981.378
|
|-
|11/3-comma
| 253.600
|219.600
|473.200
|726.800
|692.800
|980.400
|
|-
|(2+ϕ)-comma
|252.940
|220.589
|473.530
|726.470
|694.119
|979.411
|
|-
|18/5-comma
|252.696
|220.956
|473.652
|726.348
|694.607
|979.044
|
|-
|25/7-comma
|252.309
|221.536
|473.845
|726.155
|695.382
|978.464
|
|-
|7/2-comma
|251.341
|222.988
|474.329
|725.671
|697.318
|977.012
|
|-
|24/7-comma
|250.373
|224.440
|474.813
|725.187
|699.253
|975.560
|
|-
|17/5-comma
|249.986
|225.021
|475.007
|724.993
|700.028
|974.979
|
|-
|10/3-comma
|249.083
|226.376
|475.459
|724.541
|701.835
|973.624
|
|-
|23/7-comma
|248.437
|227.344
|475.781
|724.219
|703.126
|972.656
|
|-
|13/4-comma
|247.953
|228.070
|476.023
|723.977
|704.094
|971.930
|
|-
|16/5-comma
|247.258
|229.087
|476.362
|723.638
|705.449
|970.913
|
|-
|19/6-comma
|246.824
|229.764
|476.588
|723.412
|706.352
|970.236
|
|-
|22/7-comma
|246.501
|230.248
|476.749
|723.251
|706.998
|969.752
|
|-
|3-comma
|244.565
|233.152
|477.717
|722.283
|710.870
|966.848
|
|-
|20/7-comma
|242.629
|236.056
|478.685
|721.315
|714.741
|963.944
|
|-
|17/6-comma
|242.307
|236.540
|478.847
|721.153
|715.387
|963.460
|
|-
|14/5-comma
|241.855
|237.218
|479.073
|720.927
|716.290
|962.782
|
|-
|11/4-comma
|241.177
|238.234
|479.411
|720.589
|717.645
|961.766
|
|-
|19/7-comma
|240.693
|238.960
|479.653
|720.347
|718.613
|961.040
|
|-
|8/3-comma
|240.048
|239.928
|479.976
|720.024
|719.904
|960.072
|
|-
|13/5-comma
|239.145
|241.283
|480.428
|719.572
|721.711
|958.717
|
|-
|18/7-comma
|238.757
|241.864
|480.621
|719.379
|722.485
|958.136
|
|-
|5/2-comma
| 237.789
|243.316
|481.105
|718.895
|724.421
|956.684
|
|-
|17/7-comma
|236.821
|244.768
|481.589
|718.411
|726.357
|955.232
|
|-
|12/5-comma
|236.434
|245.349
|481.783
|718.217
|727.132
|954.651
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|245.715
|481.905
|718.095
|727.620
|954.285
|
|-
|7/3-comma
|235.531
|246.704
|482.235
|717.765
|728.938
|953.296
|
|-
|16/7-comma
|234.115
|247.672
|482.557
|717.423
|730.229
|952.328
|
|-
|9/4-comma
|234.401
|248.398
|482.799
|717.201
|731.197
|951.602
|
|-
|11/5-comma
|233.276
|249.414
|483.183
|716.817
|732.552
|950.596
|
|-
|13/6-comma
|233.272
|250.092
|483.364
|716.636
|733.456
|949.909
|
|-
|15/7-comma
|232.051
|250.576
|483.525
|716.475
|734.101
|949.424
|
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|846.071
| 865.447
|
|-
|13/14-comma
|178.890
|331.666
|510.555
|689.445
|842.221
|868.334
|
|-
|12/13-comma
|179.037
|331.444
|510.481
|689.519
|841.925
| 868.556
|
|-
|11/12-comma
|179.210
|331.185
|510.395
|689.605
|841.580
|868.815
|
|-
|10/11-comma
| 179.414
|330.879
| 510.293
|689.707
|841.172
|869.121
|
|-
|9/10-comma
|179.659
|330.511
| 510.170
|689.830
|840.682
|869.489
|
|-
|8/9-comma
|179.959
|330.062
|510.021
|689.979
|840.083
|869.038
|
|-
|7/8-comma
|180.333
|329.501
|509.834
|690.166
|839.334
|870.499
|
|-
|6/7-comma
|180.814
|328.779
|509.593
|690.407
|838.372
|871.221
|
|-
|11/13-comma
|181.110
|328.335
|509.445
|690.555
|837.780
|871.665
|
|-
|5/6-comma
|181.455
|327.817
|509.272
|690.728
|837.089
|872.193
|
|-
|9/11-comma
|181.864
|327.204
|509.068
|690.932
|836.272
|872.796
|
|-
|4/5-comma
|182.354
|326.469
|508.823
|691.177
|835.293
|873.531
|
|-
|11/14-comma
|182.739
|325.892
|508.630
|691.370
|834.523
|874.108
|
|-
|7/9-comma
|182.952
|325.571
|508.523
|691.477
|834.095
| 874.429
|
|-
|10/13-comma
|183.183
|325.226
|508.408
|691.592
|833.634
|874.774
|
|-
|3/4-comma
|183.701
|324.449
|508.150
|691.850
|832.598
|875.551
|
|-
|8/11-comma
|184.687
|323.530
|507.843
|692.157
|831.373
|876.470
|
|-
|5/7-comma
|184.633
|323.005
|507.638
|692.362
|830.673
|876.995
|
|-
|7/10-comma
|184.952
|322.428
|507.476
|692.524
|829.904
|877.572
|
|-
|9/13-comma
|185.255
|322.117
|507.372
|692.628
|829.489
|877.883
|
|-
|2/3-comma
|185.946
|321.080
|507.027
|692.973
|828.107
|878.920
|
|-
|9/14-comma
|186.588
|320.118
|506.706
|693.294
|828.824
|879.882
|
|-
|7/11-comma
|186.763
|319.856
|506.619
|693.381
|826.474
|880.144
|
|-
|5/8-comma
|187.069
|319.396
|506.465
|693.535
|825.862
|880.604
|
|-
|1/ϕ-comma
|187.257
|319.115
|506.372
|693.628
|825.486
|880.885
|
|-
|8/13-comma
|187.320
|319.008
|506.336
|693.664
|825.344
|880.992
|
|-
|3/5-comma
|187.743
|318.386
|506.129
|693.871
|824.514
|881.614
|
|-
|7/12-comma
|188.194
|317.712
|505.904
|694.096
|823.616
|882.288
|
|-
|4/7-comma
|188.512
|317.231
|505.744
|694.256
|822.975
|882.769
|
|-
|5/9-comma
|188.940
|316.590
|505.530
|694.470
|822.119
|883.410
|
|-
|6/11-comma
|189.213
|316.181
|505.394
|694.606
|821.575
|883.891
|
|-
|7/13-comma
|189.401
|315.899
|505.300
|694.700
|821.198
|884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|314.344
|504.781
|695.219
|819.125
|885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|312.790
|504.263
|695.737
|817.053
|887.210
|
|-
|5/11-comma
|191.338
|312.507
|504.169
|695.831
|816.676
|887.493
|
|-
|4/9-comma
|191.934
|312.099
|504.033
|695.967
|816.131
|877.901
|
|-
|3/7-comma
|192.362
|311.457
|503.819
|696.181
|815.276
|388.443
|
|-
|5/12-comma
|192.683
|310.976
|503.659
|696.341
|814.635
|889.024
|
|-
|2/5-comma
|193.132
|310.302
|503.434
|696.566
|813.736
|889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|309.680
|503.227
|696.773
|812.907
|890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|309.573
|503.191
|696.801
|812.764
| 890.427
|
|-
|3/8-comma
|193.805
|309.291
| 503.096
|696.904
|812.389
|890.709
|
|-
|4/11-comma
|194.112
|308.832
|502.944
|697.956
|811.776
|891.168
|
|-
|5/14-comma
|194.287
|308.570
|502.856
|697.144
|811.427
|891.430
|
|-
|1/3-comma
|194.928
|307.608
|502.536
|697.424
|810.144
|892.392
|
|-
|4/13-comma
|195.619
|306.571
|502.190
|697.810
|808.762
|893.429
|
|-
|3/10-comma
|195.174
|306.260
|502.087
|697.913
|808.347
|893.740
|
|-
|2/7-comma
|196.211
|305.683
|501.894
|698.106
|807.577
|894.317
|
|-
|3/11-comma
|196.561
|305.158
|501.718
|698.282
|806.877
|894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|304.240
|501.413
|698.587
|805.653
|895.760
|
|-
|3/13-comma
|197.692
|303.462
|501.154
|698.846
|804.616
|896.538
|
|-
|2/9-comma
|197.922
|303.117
|501.039
|698.961
|804.155
|896.883
|
|-
|3/14-comma
|198.136
|302.796
|500.932
|699.068
|803.728
|897.204
|
|-
|1/5-comma
|198.521
|302.219
|500.740
|699.260
|802.958
|897.781
|
|-
|2/11-comma
|199.011
|301.484
|500.495
|699.505
|801.978
|898.516
|
|-
|1/6-comma
|199.419
|300.871
|500.290
|699.810
|801.162
|899.129
|
|-
|2/13-comma
|199.765
|300.353
|500.118
|699.882
|800.471
|899.647
|
|-
|1/7-comma
|200.061
|299.909
|499.970
|700.030
|799.879
|900.091
|
|-
|1/8-comma
|200.542
|299.187
| 499.729
|700.271
|798.916
|900.823
|
|-
|1/9-comma
|200.916
|298.626
|499.542
|700.558
|798.168
|901.374
|
|-
|1/10-comma
|201.785
|298.177
|499.392
|700.608
|797.569
|901.823
|
|-
|1/11-comma
|201.460
|297.810
|499.270
|700.730
|797.079
|902.190
|
|-
|1/12-comma
|201.665
|297.503
|499.168
|700.832
|796.671
|902.497
|
|-
|1/13-comma
|201.837
|297.244
|499.081
|700.019
|796.325
|902.756
|
|-
|1/14-comma
|201.953
|297.022
|499.007
|700.993
|796.029
|902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|-
|13/7-comma
|229.078
|256.384
|485.461
|
|
|
|
|-
|11/6-comma
|228.755
|256.868
|485.623
|
|
|
|
|-
|9/5-comma
|228.697
|257.545
|485.848
|
|
|
|
|-
| 7/4-comma
|227.626
|258.562
|486.187
|
|
|
|
|-
|12/7-comma
|227.142
|259.288
|486.429
|
|
|
|
|-
|5/3-comma
|226.496
|260.253
|486.752
|
|
|
|
|-
|ϕ-comma
|225.837
|261.244
|487.081
|
|
|
|
|-
|8/5-comma
|225.593
|261.611
|487.204
|
|
|
|
|-
|11/7-comma
|225.206
|262.192
| 487.397
|
|
|
|
|-
|3/2-comma
| 224.762
|263.644
|487.881
|
|
|
|
|-
|10/7-comma
|223.270
|265.096
|488.365
|
|
|
|
|-
|7/5-comma
|222.882
|265.676
|488.559
|
|
|
|
|-
|4/3-comma
|221.979
|267.031
|489.010
|
|
|
|
|-
|9/7-comma
|221.334
|267.999
|489.333
|
|
|
|
|-
|5/4-comma
|220.850
|268.725
|489.575
|
|
|
|
|-
| 6/5-comma
|220.172
|269.742
|489.914
|
|
|
|
|-
|7/6-comma
|219.720
|270.419
|490.140
|
|
|
|
|-
|8/7-comma
|219.398
|270.903
|490.301
|
|
|
|
|-
|1-comma
|217.538
|273.807
|491.269
|
|
|
|
|-
|6/7-comma
|215.526
|276.711
|492.237
|
|
|
|
|-
|5/6-comma
|215.203
|277.195
|492.398
|
|
|
|
|-
| 4/5-comma
|214.751
|277.873
| 492.624
|
|
|
|
|-
|3/4-comma
|214.926
|278.889
|492.963
|
|
|
|
|-
|5/7-comma
|213.590
|279.615
|493.205
|
|
|
|
|-
|2/3-comma
|212.945
|280.583
|493.528
|
|
|
|
|-
|3/5-comma
|212.041
|281.938
|493.979
|
|
|
|
|-
|4/7-comma
|211.346
|282.519
|494.173
|
|
|
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|283.971
|494.657
|
|
|
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|209.718
|285.423
|495.141
|
|
|
|
|-
|2/5-comma
|209.331
|286.004
|495.335
|
|
|
|
|-
|1/(ϕ+1)-comma
|209.086
|286.371
|495.457
|
|
|
|
|-
|1/3-comma
|208.573
|287.359
|495.786
|
|
|
|
|-
|2/7-comma
|207.782
|289.372
|496.109
|
|
|
|
|-
|1/4-comma
|207.293
|289.053
|496.351
|
|
|
|
|-
|1/5-comma
|206.620
|290.069
|496.690
|
|
|
|
|-
|1/6-comma
|206.169
|290.747
|496.916
|
|
|
|
|-
|1/7-comma
|205.846
|291.231
|497.077
|
|
|
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|
|
|
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|291.248
|497.083
|702.917
|788.331
|908.752
|
|-
| -1/13-comma
|205.983
|291.026
|497.009
|702.993
|788.035
|908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|290.767
|496.922
|703.078
|787.689
|909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|290.460
|496.820
|703.180
|787.280
|909.540
|
|-
| -1/10-comma
|206.605
|290.093
|496.698
|703.302
|786.791
|909.907
|
|-
| -1/9-comma
|206.904
|289.644
|496.548
|703.452
|786.192
|910.356
|
|-
| -1/8-comma
|207.278
|289.083
|496.361
|703.639
|785.444
|910.917
|
|-
| -1/7-comma
|207.759
|288.361
|496.120
|703.880
|784.481
|911.639
|
|-
| -2/13-comma
|208.055
|287.917
|495.972
|704.028
|783.889
|912.083
|
|-
| -1/6-comma
|208.401
|287.399
|495.800
|704.200
|783.198
|912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|286.786
|495.595
|704.405
|782.382
|913.214
|
|-
| -1/5-comma
|209.299
|286.051
|495.350
|704.650
|781.401
|913.949
|
|-
| -3/14-comma
|209.684
|285.474
|495.158
|704.842
|780.632
|914.526
|
|-
| -2/9-comma
|209.898
|285.153
|495.051
|704.949
|780.204
|914.847
|
|-
| -3/13-comma
|210.128
|284.808
|494.936
|705.064
|779.744
|915.192
|
|-
| -1/4-comma
|210.646
|284.030
|494.677
|705.323
|778.707
|915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|283.111
|494.371
|705.629
|777.482
|916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|282.587
|494.196
|705.804
|776.783
|917.413
|
|-
| -3/10-comma
|211.994
|282.010
|494.003
|705.997
|776.013
|917.990
|
|-
| -4/13-comma
|212.799
|281.699
|493.900
|706.100
|775.598
|918.301
|
|-
| -1/3-comma
|212.892
|280.662
|493.554
|706.446
|774.216
|919.338
|
|-
| -5/14-comma
|213.537
|279.700
|493.233
|706.767
|772.933
|920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|279.437
|493.146
|706.854
|772.583
|920.563
|
|-
| -3/8-comma
|214.014
|278.979
|492.993
|707.007
|771.971
|921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|278.697
|492.899
|707.101
|771.596
|921.303
|
|-
| -5/13-comma
|214.274
|278.590
|492.863
|707.137
|771.453
|921.410
|
|-
| -2/5-comma
|214.688
|277.968
|492.656
|707.344
|770.624
|922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|277.294
|492.431
|707.569
|769.725
|922.706
|
|-
| -3/7-comma
|215.458
|276.813
|492.271
|707.729
|769.084
|923.187
|
|-
| -4/9-comma
|215.886
|276.171
|492.057
|707.943
|768.229
|923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|275.763
|491.921
|708.079
|767.684
|924.237
|
|-
| -6/13-comma
|216.346
|275.480
|491.827
|708.173
|767.307
|924.520
|
|-
| -1/2-comma
|217.383
|273.926
|491.309
|708.691
|765.235
|926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|272.371
|490.790
|709.210
|763.161
|927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|272.089
|490.696
|709.304
|762.785
|927.911
|
|-
| -5/9-comma
|218.880
|271.680
|490.560
|709.440
|762.241
|928.320
|
|-
| -4/7-comma
|219.307
|271.039
|490.346
|709.654
|761.385
|928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|270.558
|490.186
|709.814
|760.744
|929.442
|
|-
| -3/5-comma
|220.077
|269.884
|489.961
|710.039
|759.846
|930.116
|
|-
| -8/13-comma
|220.492
|269.262
|489.754
|710.246
|759.016
|930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|269.155
|489.716
|710.284
|758.874
|930.845
|
|-
| -5/8-comma
|220.751
|268.874
|489.625
|710.375
|758.498
|931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|268.414
|489.471
|710.529
|757.886
|931.586
|
|-
| -9/14-comma
|221.232
|268.152
|489.384
|710.616
|757.536
|931.848
|
|-
| -2/3-comma
|221.874
|267.190
|489.063
|710.939
|756.253
|932.810
|
|-
| -9/13-comma
|222.565
|266.153
|488.718
|711.282
|754.871
|933.847
|
|-
| -7/10-comma
|222.772
|265.842
|488.614
|711.386
|754.456
|934.158
|
|-
| -5/7-comma
|223.157
|265.265
|488.422
|711.376
|753.687
|934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|264.740
|488.247
|711.753
|752.987
|935.260
|
|-
| -3/4-comma
|224.119
|263.821
|487.940
|712.060
|751.762
|936.189
|
|-
| -10/13-comma
|224.637
|263.044
|487.681
|712.319
|750.726
|936.956
|
|-
| -7/9-comma
|224.868
|263.044
|487.566
|712.434
|750.265
|937.302
|
|-
| -11/14-comma
|225.081
|262.378
|487.459
|712.541
|749.837
|937.622
|
|-
| -4/5-comma
|225.466
|261.801
|487.267
|712.723
|749.067
|938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|261.066
|487.022
|712.978
|748.088
|938.934
|
|-
| -5/6-comma
|226.365
|260.453
|486.818
|713.182
|747.271
|939.447
|
|-
| -11/13-comma
|226.710
|259.935
|486.645
|713.355
|746.580
|940.065
|
|-
| -6/7-comma
|227.006
|259.491
|486.497
|713.503
|745.988
|940.509
|
|-
| -7/8-comma
|227.487
|258.769
|486.256
|713.744
|745.026
|941.231
|
|-
| -8/9-comma
|227.861
|258.208
|486.069
|713.931
|744.277
|941.792
|
|-
| -9/10-comma
|228.161
|257.759
|485.920
|714.080
|743.678
|942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|257.391
|485.797
|714.203
|743.188
|942.609
|
|-
| -11/12-comma
|228.610
|257.085
|485.695
|714.305
|742.780
|942.915
|
|-
| -12/13-comma
|228.783
|256.826
|485.609
|714.391
|742.435
|943.174
|
|-
| -13/14-comma
|228.931
|256.604
|485.535
|714.465
|742.139
|943.396
|
|-
|-
| -1-comma
|230.855
|253.717
|484.752
|715,248
|738.289
|946.283
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|
|499.013
|
|
|297.039
|
|-
| -1/6-comma
|201.652
|
|499.174
|
|
|297.523
|
|-
| -1/5-comma
|201.200
|
|499.400
|
|
|298.201
|
|-
| -1/4-comma
|200.522
|
|499.739
|
|
|299.217
|
|-
| -2/7-comma
|200.038
|
|499.981
|
|
|299.942
|
|-
| -1/3-comma
|199.393
|
|500.303
|
|
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|
|500.633
|
|
|301.900
|
|-
| -2/5-comma
|198.499
|
|500.755
|
|
|302.266
|
|-
| -3/7-comma
|198.102
|
|500.949
|
|
|302.847
|
|-
| -1/2-comma
|197.134
|
|501.433
|
|
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|196.166
|
|501.917
|
|
|305.751
|
|-
| -3/5-comma
|195.779
|
|502.111
|
|
|306.332
|
|-
| -2/3-comma
|194.876
|
|502.562
|
|
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|194.230
|
|502.885
|
|
|308.655
|
|-
| -4/5-comma
|193.069
|
|503.466
|
|
|310.397
|
|-
| -5/6-comma
|192.617
|
|503.692
|
|
|311.075
|
|-
|-
| -6/7-comma
|192.294
|
|503.853
|
|
|311.556
|
|-
| -1-comma
|190.352
|
|504.821
|
|
|314.463
|
|-
| -8/7-comma
|188.422
|
|505.789
|
|
|317.367
|
|-
| -7/6-comma
|188.100
|
|505.950
|
|
|317.851
|
|-
| -6/5-comma
|187.648
|
|506.176
|
|
|318.528
|
|-
| -5/4-comma
|186.970
|
|506.515
|
|
|319.545
|
|-
| -9/7-comma
|186.486
|
|506.757
|
|
|320.271
|
|-
| -4/3-comma
|185.841
|
|507.080
|
|
|321.239
|
|-
| -7/5-comma
|184.937
|
|507.531
|
|
|322.594
|
|-
| -10/7-comma
|184.550
|
|507.725
|
|
|323.174
|
|-
| -3/2-comma
|183.582
|
|508.209
|
|
|324.626
|
|-
| -11/7-comma
|182.614
|
|508.693
|
|
|326.078
|
|-
| -8/5-comma
|182.228
|
|508.886
|
|
|326.659
|
|-
| -ϕ-comma
|181.983
|
|509.009
|
|
|327.026
|
|-
| -5/3-comma
|181.324
|
|509.338
|
|
|328.014
|
|-
| -12/7-comma
|180.678
|
|509.661
|
|
|328.982
|
|-
| -7/4-comma
|180.194
|
|509.903
|
|
|329.708
|
|-
| -9/5-comma
|179.517
|
|510.242
|
|
|330.725
|
|-
| -11/6-comma
|179.065
|
|510.467
|
|
|331.402
|
|-
| -13/7-comma
|178.742
|
|510.629
|
|
|331.886
|
|-
| -2-comma
|176.807
|
|511.597
|
|
|334.790
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|253.717
|484.752
|715,248
|738.289
|946.283
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|232.780
|
|483.610
|
|
|250.830
|
|-
| -14/13-comma
|232.928
|
|483.536
|
|
|250.608
|
|-
| -13/12-comma
|233.101
|
|483.450
|
|
|250.349
|
|-
| -12/11-comma
|233.305
|
|483.348
|
|
|250.043
|
|-
| -11/10-comma
|233.550
|
|483.225
|
|
|249.675
|
|-
| -10/9-comma
|233.151
|
|483.075
|
|
|249.226
|
|-
| -9/8-comma
|234.234
|
|482.888
|
|
|248.665
|
|-
| -8/7-comma
|234.295
|
|482.648
|
|
|247.943
|
|-
| -15/13-comma
|235.001
|
|482.500
|
|
|247.499
|
|-
| -7/6-comma
|235.346
|
|482.327
|
|
|246.981
|
|-
| -13/11-comma
|235.755
|
|482.123
|
|
|246.368
|
|-
| -6/5-comma
|236.244
|
|481.878
|
|
|245.633
|
|-
| -17/14-comma
|236.629
|
|481.685
|
|
|245.056
|
|-
| -11/9-comma
|236.843
|
|481.578
|
|
|244.735
|
|-
| -16/13-comma
|237.926
|
|481.463
|
|
|244.390
|
|-
| -5/4-comma
|237.592
|
|481.204
|
|
|243.612
|
|-
| -14/11-comma
|238.204
|
|480.898
|
|
|242.694
|
|-
| -9/7-comma
|238.554
|
|480.723
|
|
|242.169
|
|-
| -13/10-comma
|238.939
|
|480.530
|
|
|241.591
|
|-
| -17/13-comma
|239.146
|
|480.427
|
|
|241.280
|
|-
| -4/3-comma
|239.837
|
|480.081
|
|
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|
|479.761
|
|
|239.282
|
|-
| -15/11-comma
|240.634
|
|479.673
|
|
|239.019
|
|-
| -11/8-comma
|240.960
|
|479.520
|
|
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|241.148
|
|479.426
|
|
|238.279
|
|-
| -18/13-comma
|241.219
|
|479.390
|
|
|238.171
|
|-
| -7/5-comma
|241.634
|
|479.183
|
|
|237.550
|
|-
| -17/12-comma
|242.917
|
|478.959
|
|
|236.876
|
|-
| -10/7-comma
|242.403
|
|478.798
|
|
|236.395
|
|-
| -13/9-comma
|242.831
|
|478.584
|
|
|235.753
|
|-
| -16/11-comma
|243.103
|
|478.448
|
|
|235.345
|
|-
| -19/13-comma
|243.708
|
|478.354
|
|
|235.062
|
|-
| -3/2-comma
|244.328
|
|477.836
|
|
|233.508
|
|-
| -20/13-comma
|245.344
|
|477.318
|
|
|231.953
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|
|477.224
|
|
|231.671
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|
|477.087
|
|
|231.262
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|
|476.873
|
|
|230.621
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|
|476.713
|
|
|230.140
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|
|476.489
|
|
|229.466
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|
|476.281
|
|
|228.844
|
|-
| -ϕ-comma
|247.491
|
|476.246
|
|
|228.737
|
|-
| -13/8-comma
|247.696
|
|476.152
|
|
|228.456
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|
|475.999
|
|
|227.996
|
|-
| -23/14-comma
|248.823
|
|475.911
|
|
|227.734
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|
|475.590
|
|
|226.771
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|
|475.245
|
|
|225.735
|
|-
| -17/10-comma
|249.717
|
|475.141
|
|
|225.424
|
|-
| -12/7-comma
|250.105
|
|474.949
|
|
|224.847
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|
|474.774
|
|
|224.322
|
|-
| -7/4-comma
|251.064
|
|474.468
|
|
|223.403
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|
|474.209
|
|
|222.626
|
|-
| -16/9-comma
|251.823
|
|474.094
|
|
|222.281
|
|-
| -25/14-comma
|252.027
|
|473.987
|
|
|221.960
|
|-
| -9/5-comma
|252.412
|
|473.794
|
|
|221.382
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|
|473.549
|
|
|220.648
|
|-
| -11/6-comma
|253.610
|
|473.345
|
|
|220.035
|
|-
| -24/13-comma
|253.345
|
|473.172
|
|
|219.517
|
|-
| -13/7-comma
|253.951
|
|473.924
|
|
|219.073
|
|-
| -15/8-comma
|254.433
|
|472.784
|
|
|218.351
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|
|472.597
|
|
|217.790
|
|-
| -19/10-comma
|255.106
|
|472.447
|
|
|217.341
|
|-
| -21/11-comma
|255.351
|
|472.324
|
|
|216.973
|
|-
| -23/12-comma
|255.555
|
|472.222
|
|
|216.667
|
|-
| -25/13-comma
|255.728
|
|472.135
|
|
|216.408
|
|-
| -27/14-comma
|255.876
|
|472.052
|
|
|216.186
|
|-
|-
| -2-comma
|258.801
|
|471.100
|
|
|213.299
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-23T15:09:52Z

Moremajorthanmajor: /* Historically-defined mean tetrachord */

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with more complex thirds, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|212.824
|470.941
|729.051
|683.766
|987.176
|
|-
|27/7-comma
|256.181
|215.728
|471.909
|728.091
|687.637
|984.272
|
|-
|23/6-comma
|255.858
|216.212
|472.071
|727.929
|688.283
|983.788
|
|-
|19/5-comma
|255.407
|216.890
|472.297
|727.703
|689.187
|983.110
|
|-
|15/4-comma
|254.769
|217.906
|472.635
|727.365
|690.542
|982.094
|
|-
|26/7-comma
|254.243
|218.632
|472.877
|727.123
|691.510
|981.378
|
|-
|11/3-comma
| 253.600
|219.600
|473.200
|726.800
|692.800
|980.400
|
|-
|(2+ϕ)-comma
|252.940
|220.589
|473.530
|726.470
|694.119
|979.411
|
|-
|18/5-comma
|252.696
|220.956
|473.652
|726.348
|694.607
|979.044
|
|-
|25/7-comma
|252.309
|221.536
|473.845
|726.155
|695.382
|978.464
|
|-
|7/2-comma
|251.341
|222.988
|474.329
|725.671
|697.318
|977.012
|
|-
|24/7-comma
|250.373
|224.440
|474.813
|725.187
|699.253
|975.560
|
|-
|17/5-comma
|249.986
|225.021
|475.007
|724.993
|700.028
|974.979
|
|-
|10/3-comma
|249.083
|226.376
|475.459
|724.541
|701.835
|973.624
|
|-
|23/7-comma
|248.437
|227.344
|475.781
|724.219
|703.126
|972.656
|
|-
|13/4-comma
|247.953
|228.070
|476.023
|723.977
|704.094
|971.930
|
|-
|16/5-comma
|247.258
|229.087
|476.362
|723.638
|705.449
|970.913
|
|-
|19/6-comma
|246.824
|229.764
|476.588
|723.412
|706.352
|970.236
|
|-
|22/7-comma
|246.501
|230.248
|476.749
|723.251
|706.998
|969.752
|
|-
|3-comma
|244.565
|233.152
|477.717
|722.283
|710.870
|966.848
|
|-
|20/7-comma
|242.629
|236.056
|478.685
|721.315
|714.741
|963.944
|
|-
|17/6-comma
|242.307
|236.540
|478.847
|721.153
|715.387
|963.460
|
|-
|14/5-comma
|241.855
|237.218
|479.073
|720.927
|716.290
|962.782
|
|-
|11/4-comma
|241.177
|238.234
|479.411
|720.589
|717.645
|961.766
|
|-
|19/7-comma
|240.693
|238.960
|479.653
|720.347
|718.613
|961.040
|
|-
|8/3-comma
|240.048
|239.928
|479.976
|720.024
|719.904
|960.072
|
|-
|13/5-comma
|239.145
|241.283
|480.428
|719.572
|721.711
|958.717
|
|-
|18/7-comma
|238.757
|241.864
|480.621
|719.379
|722.485
|958.136
|
|-
|5/2-comma
| 237.789
|243.316
|481.105
|718.895
|724.421
|956.684
|
|-
|17/7-comma
|236.821
|244.768
|481.589
|718.411
|726.357
|955.232
|
|-
|12/5-comma
|236.434
|245.349
|481.783
|718.217
|727.132
|954.651
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|245.715
|481.905
|718.095
|727.620
|954.285
|
|-
|7/3-comma
|235.531
|246.704
|482.235
|717.765
|728.938
|953.296
|
|-
|16/7-comma
|234.115
|247.672
|482.557
|717.423
|730.229
|952.328
|
|-
|9/4-comma
|234.401
|248.398
|482.799
|717.201
|731.197
|951.602
|
|-
|11/5-comma
|233.276
|249.414
|483.183
|716.817
|732.552
|950.596
|
|-
|13/6-comma
|233.272
|250.092
|483.364
|716.636
|733.456
|949.909
|
|-
|15/7-comma
|232.051
|250.576
|483.525
|716.475
|734.101
|949.424
|
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|334.553
|511.518
|688.482
|846.071
| 865.447
|
|-
|13/14-comma
|178.890
|331.666
|510.555
|689.445
|842.221
|868.334
|
|-
|12/13-comma
|179.037
|331.444
|510.481
|689.519
|841.925
| 868.556
|
|-
|11/12-comma
|179.210
|331.185
|510.395
|689.605
|841.580
|868.815
|
|-
|10/11-comma
| 179.414
|330.879
| 510.293
|689.707
|841.172
|869.121
|
|-
|9/10-comma
|179.659
|330.511
| 510.170
|689.830
|840.682
|869.489
|
|-
|8/9-comma
|179.959
|330.062
|510.021
|689.979
|840.083
|869.038
|
|-
|7/8-comma
|180.333
|329.501
|509.834
|690.166
|839.334
|870.499
|
|-
|6/7-comma
|180.814
|328.779
|509.593
|690.407
|838.372
|871.221
|
|-
|11/13-comma
|181.110
|328.335
|509.445
|690.555
|837.780
|871.665
|
|-
|5/6-comma
|181.455
|327.817
|509.272
|690.728
|837.089
|872.193
|
|-
|9/11-comma
|181.864
|327.204
|509.068
|690.932
|836.272
|872.796
|
|-
|4/5-comma
|182.354
|326.469
|508.823
|691.177
|835.293
|873.531
|
|-
|11/14-comma
|182.739
|325.892
|508.630
|691.370
|834.523
|874.108
|
|-
|7/9-comma
|182.952
|325.571
|508.523
|691.477
|834.095
| 874.429
|
|-
|10/13-comma
|183.183
|325.226
|508.408
|691.592
|833.634
|874.774
|
|-
|3/4-comma
|183.701
|324.449
|508.150
|691.850
|832.598
|875.551
|
|-
|8/11-comma
|184.687
|323.530
|507.843
|692.157
|831.373
|876.470
|
|-
|5/7-comma
|184.633
|323.005
|507.638
|692.362
|830.673
|876.995
|
|-
|7/10-comma
|184.952
|322.428
|507.476
|692.524
|829.904
|877.572
|
|-
|9/13-comma
|185.255
|322.117
|507.372
|692.628
|829.489
|877.883
|
|-
|2/3-comma
|185.946
|321.080
|507.027
|692.973
|828.107
|878.920
|
|-
|9/14-comma
|186.588
|320.118
|506.706
|693.294
|828.824
|879.882
|
|-
|7/11-comma
|186.763
|319.856
|506.619
|693.381
|826.474
|880.144
|
|-
|5/8-comma
|187.069
|319.396
|506.465
|693.535
|825.862
|880.604
|
|-
|1/ϕ-comma
|187.257
|319.115
|506.372
|693.628
|825.486
|880.885
|
|-
|8/13-comma
|187.320
|319.008
|506.336
|693.664
|825.344
|880.992
|
|-
|3/5-comma
|187.743
|318.386
|506.129
|693.871
|824.514
|881.614
|
|-
|7/12-comma
|188.194
|317.712
|505.904
|694.096
|823.616
|882.288
|
|-
|4/7-comma
|188.512
|317.231
|505.744
|694.256
|822.975
|882.769
|
|-
|5/9-comma
|188.940
|316.590
|505.530
|694.470
|822.119
|883.410
|
|-
|6/11-comma
|189.213
|316.181
|505.394
|694.606
|821.575
|883.891
|
|-
|7/13-comma
|189.401
|315.899
|505.300
|694.700
|821.198
|884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|314.344
|504.781
|695.219
|819.125
|885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|312.790
|504.263
|695.737
|817.053
|887.210
|
|-
|5/11-comma
|191.338
|312.507
|504.169
|695.831
|816.676
|887.493
|
|-
|4/9-comma
|191.934
|312.099
|504.033
|695.967
|816.131
|877.901
|
|-
|3/7-comma
|192.362
|311.457
|503.819
|696.181
|815.276
|388.443
|
|-
|5/12-comma
|192.683
|310.976
|503.659
|696.341
|814.635
|889.024
|
|-
|2/5-comma
|193.132
|310.302
|503.434
|696.566
|813.736
|889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|309.680
|503.227
|696.773
|812.907
|890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|309.573
|503.191
|696.801
|812.764
| 890.427
|
|-
|3/8-comma
|193.805
|309.291
| 503.096
|696.904
|812.389
|890.709
|
|-
|4/11-comma
|194.112
|308.832
|502.944
|697.956
|811.776
|891.168
|
|-
|5/14-comma
|194.287
|308.570
|502.856
|697.144
|811.427
|891.430
|
|-
|1/3-comma
|194.928
|307.608
|502.536
|697.424
|810.144
|892.392
|
|-
|4/13-comma
|195.619
|306.571
|502.190
|697.810
|808.762
|893.429
|
|-
|3/10-comma
|195.174
|306.260
|502.087
|697.913
|808.347
|893.740
|
|-
|2/7-comma
|196.211
|305.683
|501.894
|698.106
|807.577
|894.317
|
|-
|3/11-comma
|196.561
|305.158
|501.718
|698.282
|806.877
|894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|304.240
|501.413
|698.587
|805.653
|895.760
|
|-
|3/13-comma
|197.692
|303.462
|501.154
|698.846
|804.616
|896.538
|
|-
|2/9-comma
|197.922
|303.117
|501.039
|698.961
|804.155
|896.883
|
|-
|3/14-comma
|198.136
|302.796
|500.932
|699.068
|803.728
|897.204
|
|-
|1/5-comma
|198.521
|302.219
|500.740
|699.260
|802.958
|897.781
|
|-
|2/11-comma
|199.011
|301.484
|500.495
|699.505
|801.978
|898.516
|
|-
|1/6-comma
|199.419
|300.871
|500.290
|699.810
|801.162
|899.129
|
|-
|2/13-comma
|199.765
|300.353
|500.118
|699.882
|800.471
|899.647
|
|-
|1/7-comma
|200.061
|299.909
|499.970
|700.030
|799.879
|900.091
|
|-
|1/8-comma
|200.542
|299.187
| 499.729
|700.271
|798.916
|900.823
|
|-
|1/9-comma
|200.916
|298.626
|499.542
|700.558
|798.168
|901.374
|
|-
|1/10-comma
|201.785
|298.177
|499.392
|700.608
|797.569
|901.823
|
|-
|1/11-comma
|201.460
|297.810
|499.270
|700.730
|797.079
|902.190
|
|-
|1/12-comma
|201.665
|297.503
|499.168
|700.832
|796.671
|902.497
|
|-
|1/13-comma
|201.837
|297.244
|499.081
|700.019
|796.325
|902.756
|
|-
|1/14-comma
|201.953
|297.022
|499.007
|700.993
|796.029
|902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|-
|13/7-comma
|229.078
|256.384
|485.461
|
|
|
|
|-
|11/6-comma
|228.755
|256.868
|485.623
|
|
|
|
|-
|9/5-comma
|228.697
|257.545
|485.848
|
|
|
|
|-
| 7/4-comma
|227.626
|258.562
|486.187
|
|
|
|
|-
|12/7-comma
|227.142
|259.288
|486.429
|
|
|
|
|-
|5/3-comma
|226.496
|260.253
|486.752
|
|
|
|
|-
|ϕ-comma
|225.837
|261.244
|487.081
|
|
|
|
|-
|8/5-comma
|225.593
|261.611
|487.204
|
|
|
|
|-
|11/7-comma
|225.206
|262.192
| 487.397
|
|
|
|
|-
|3/2-comma
| 224.762
|263.644
|487.881
|
|
|
|
|-
|10/7-comma
|223.270
|265.096
|488.365
|
|
|
|
|-
|7/5-comma
|222.882
|265.676
|488.559
|
|
|
|
|-
|4/3-comma
|221.979
|267.031
|489.010
|
|
|
|
|-
|9/7-comma
|221.334
|267.999
|489.333
|
|
|
|
|-
|5/4-comma
|220.850
|268.725
|489.575
|
|
|
|
|-
| 6/5-comma
|220.172
|269.742
|489.914
|
|
|
|
|-
|7/6-comma
|219.720
|270.419
|490.140
|
|
|
|
|-
|8/7-comma
|219.398
|270.903
|490.301
|
|
|
|
|-
|1-comma
|217.538
|273.807
|491.269
|
|
|
|
|-
|6/7-comma
|215.526
|276.711
|492.237
|
|
|
|
|-
|5/6-comma
|215.203
|277.195
|492.398
|
|
|
|
|-
| 4/5-comma
|214.751
|277.873
| 492.624
|
|
|
|
|-
|3/4-comma
|214.926
|278.889
|492.963
|
|
|
|
|-
|5/7-comma
|213.590
|279.615
|493.205
|
|
|
|
|-
|2/3-comma
|212.945
|280.583
|493.528
|
|
|
|
|-
|3/5-comma
|212.041
|281.938
|493.979
|
|
|
|
|-
|4/7-comma
|211.346
|282.519
|494.173
|
|
|
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|283.971
|494.657
|
|
|
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|209.718
|285.423
|495.141
|
|
|
|
|-
|2/5-comma
|209.331
|286.004
|495.335
|
|
|
|
|-
|1/(ϕ+1)-comma
|209.086
|286.371
|495.457
|
|
|
|
|-
|1/3-comma
|208.573
|287.359
|495.786
|
|
|
|
|-
|2/7-comma
|207.782
|289.372
|496.109
|
|
|
|
|-
|1/4-comma
|207.293
|289.053
|496.351
|
|
|
|
|-
|1/5-comma
|206.620
|290.069
|496.690
|
|
|
|
|-
|1/6-comma
|206.169
|290.747
|496.916
|
|
|
|
|-
|1/7-comma
|205.846
|291.231
|497.077
|
|
|
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|
|
|
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|291.248
|497.083
|702.917
|788.331
|908.752
|
|-
| -1/13-comma
|205.983
|291.026
|497.009
|702.993
|788.035
|908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|290.767
|496.922
|703.078
|787.689
|909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|290.460
|496.820
|703.180
|787.280
|909.540
|
|-
| -1/10-comma
|206.605
|290.093
|496.698
|703.302
|786.791
|909.907
|
|-
| -1/9-comma
|206.904
|289.644
|496.548
|703.452
|786.192
|910.356
|
|-
| -1/8-comma
|207.278
|289.083
|496.361
|703.639
|785.444
|910.917
|
|-
| -1/7-comma
|207.759
|288.361
|496.120
|703.880
|784.481
|911.639
|
|-
| -2/13-comma
|208.055
|287.917
|495.972
|704.028
|783.889
|912.083
|
|-
| -1/6-comma
|208.401
|287.399
|495.800
|704.200
|783.198
|912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|286.786
|495.595
|704.405
|782.382
|913.214
|
|-
| -1/5-comma
|209.299
|286.051
|495.350
|704.650
|781.401
|913.949
|
|-
| -3/14-comma
|209.684
|285.474
|495.158
|704.842
|780.632
|914.526
|
|-
| -2/9-comma
|209.898
|285.153
|495.051
|704.949
|780.204
|914.847
|
|-
| -3/13-comma
|210.128
|284.808
|494.936
|705.064
|779.744
|915.192
|
|-
| -1/4-comma
|210.646
|284.030
|494.677
|705.323
|778.707
|915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|283.111
|494.371
|705.629
|777.482
|916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|282.587
|494.196
|705.804
|776.783
|917.413
|
|-
| -3/10-comma
|211.994
|282.010
|494.003
|705.997
|776.013
|917.990
|
|-
| -4/13-comma
|212.799
|281.699
|493.900
|706.100
|775.598
|918.301
|
|-
| -1/3-comma
|212.892
|280.662
|493.554
|706.446
|774.216
|919.338
|
|-
| -5/14-comma
|213.537
|279.700
|493.233
|706.767
|772.933
|920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|279.437
|493.146
|706.854
|772.583
|920.563
|
|-
| -3/8-comma
|214.014
|278.979
|492.993
|707.007
|771.971
|921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|278.697
|492.899
|707.101
|771.596
|921.303
|
|-
| -5/13-comma
|214.274
|278.590
|492.863
|707.137
|771.453
|921.410
|
|-
| -2/5-comma
|214.688
|277.968
|492.656
|707.344
|770.624
|922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|277.294
|492.431
|707.569
|769.725
|922.706
|
|-
| -3/7-comma
|215.458
|276.813
|492.271
|707.729
|769.084
|923.187
|
|-
| -4/9-comma
|215.886
|276.171
|492.057
|707.943
|768.229
|923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|275.763
|491.921
|708.079
|767.684
|924.237
|
|-
| -6/13-comma
|216.346
|275.480
|491.827
|708.173
|767.307
|924.520
|
|-
| -1/2-comma
|217.383
|273.926
|491.309
|708.691
|765.235
|926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|272.371
|490.790
|709.210
|763.161
|927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|272.089
|490.696
|709.304
|762.785
|927.911
|
|-
| -5/9-comma
|218.880
|271.680
|490.560
|709.440
|762.241
|928.320
|
|-
| -4/7-comma
|219.307
|271.039
|490.346
|709.654
|761.385
|928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|270.558
|490.186
|709.814
|760.744
|929.442
|
|-
| -3/5-comma
|220.077
|269.884
|489.961
|710.039
|759.846
|930.116
|
|-
| -8/13-comma
|220.492
|269.262
|489.754
|710.246
|759.016
|930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|269.155
|489.716
|710.284
|758.874
|930.845
|
|-
| -5/8-comma
|220.751
|268.874
|489.625
|710.375
|758.498
|931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|268.414
|489.471
|710.529
|757.886
|931.586
|
|-
| -9/14-comma
|221.232
|268.152
|489.384
|710.616
|757.536
|931.848
|
|-
| -2/3-comma
|221.874
|267.190
|489.063
|710.939
|756.253
|932.810
|
|-
| -9/13-comma
|222.565
|266.153
|488.718
|711.282
|754.871
|933.847
|
|-
| -7/10-comma
|222.772
|265.842
|488.614
|711.386
|754.456
|934.158
|
|-
| -5/7-comma
|223.157
|265.265
|488.422
|711.376
|753.687
|934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|264.740
|488.247
|711.753
|752.987
|935.260
|
|-
| -3/4-comma
|224.119
|263.821
|487.940
|712.060
|751.762
|936.189
|
|-
| -10/13-comma
|224.637
|263.044
|487.681
|712.319
|750.726
|936.956
|
|-
| -7/9-comma
|224.868
|263.044
|487.566
|712.434
|750.265
|937.302
|
|-
| -11/14-comma
|225.081
|262.378
|487.459
|712.541
|749.837
|937.622
|
|-
| -4/5-comma
|225.466
|261.801
|487.267
|712.723
|749.067
|938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|261.066
|487.022
|712.978
|748.088
|938.934
|
|-
| -5/6-comma
|226.365
|260.453
|486.818
|713.182
|747.271
|939.447
|
|-
| -11/13-comma
|226.710
|259.935
|486.645
|713.355
|746.580
|940.065
|
|-
| -6/7-comma
|227.006
|259.491
|486.497
|713.503
|745.988
|940.509
|
|-
| -7/8-comma
|227.487
|258.769
|486.256
|713.744
|745.026
|941.231
|
|-
| -8/9-comma
|227.861
|258.208
|486.069
|713.931
|744.277
|941.792
|
|-
| -9/10-comma
|228.161
|257.759
|485.920
|714.080
|743.678
|942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|257.391
|485.797
|714.203
|743.188
|942.609
|
|-
| -11/12-comma
|228.610
|257.085
|485.695
|714.305
|742.780
|942.915
|
|-
| -12/13-comma
|228.783
|256.826
|485.609
|714.391
|742.435
|943.174
|
|-
| -13/14-comma
|228.931
|256.604
|485.535
|714.465
|742.139
|943.396
|
|-
|-
| -1-comma
|230.855
|253.717
|484.752
|715,248
|738.289
|946.283
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|701.955
|792.180
|905.865
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|
|499.013
|
|
|297.039
|
|-
| -1/6-comma
|201.652
|
|499.174
|
|
|297.523
|
|-
| -1/5-comma
|201.200
|
|499.400
|
|
|298.201
|
|-
| -1/4-comma
|200.522
|
|499.739
|
|
|299.217
|
|-
| -2/7-comma
|200.038
|
|499.981
|
|
|299.942
|
|-
| -1/3-comma
|199.393
|
|500.303
|
|
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|
|500.633
|
|
|301.900
|
|-
| -2/5-comma
|198.499
|
|500.755
|
|
|302.266
|
|-
| -3/7-comma
|198.102
|
|500.949
|
|
|302.847
|
|-
| -1/2-comma
|197.134
|
|501.433
|
|
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|196.166
|
|501.917
|
|
|305.751
|
|-
| -3/5-comma
|195.779
|
|502.111
|
|
|306.332
|
|-
| -2/3-comma
|194.876
|
|502.562
|
|
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|194.230
|
|502.885
|
|
|308.655
|
|-
| -4/5-comma
|193.069
|
|503.466
|
|
|310.397
|
|-
| -5/6-comma
|192.617
|
|503.692
|
|
|311.075
|
|-
|-
| -6/7-comma
|192.294
|
|503.853
|
|
|311.556
|
|-
| -1-comma
|190.352
|
|504.821
|
|
|314.463
|
|-
| -8/7-comma
|188.422
|
|505.789
|
|
|317.367
|
|-
| -7/6-comma
|188.100
|
|505.950
|
|
|317.851
|
|-
| -6/5-comma
|187.648
|
|506.176
|
|
|318.528
|
|-
| -5/4-comma
|186.970
|
|506.515
|
|
|319.545
|
|-
| -9/7-comma
|186.486
|
|506.757
|
|
|320.271
|
|-
| -4/3-comma
|185.841
|
|507.080
|
|
|321.239
|
|-
| -7/5-comma
|184.937
|
|507.531
|
|
|322.594
|
|-
| -10/7-comma
|184.550
|
|507.725
|
|
|323.174
|
|-
| -3/2-comma
|183.582
|
|508.209
|
|
|324.626
|
|-
| -11/7-comma
|182.614
|
|508.693
|
|
|326.078
|
|-
| -8/5-comma
|182.228
|
|508.886
|
|
|326.659
|
|-
| -ϕ-comma
|181.983
|
|509.009
|
|
|327.026
|
|-
| -5/3-comma
|181.324
|
|509.338
|
|
|328.014
|
|-
| -12/7-comma
|180.678
|
|509.661
|
|
|328.982
|
|-
| -7/4-comma
|180.194
|
|509.903
|
|
|329.708
|
|-
| -9/5-comma
|179.517
|
|510.242
|
|
|330.725
|
|-
| -11/6-comma
|179.065
|
|510.467
|
|
|331.402
|
|-
| -13/7-comma
|178.742
|
|510.629
|
|
|331.886
|
|-
| -2-comma
|176.807
|
|511.597
|
|
|334.790
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|253.717
|484.752
|715,248
|738.289
|946.283
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|232.780
|
|483.610
|
|
|250.830
|
|-
| -14/13-comma
|232.928
|
|483.536
|
|
|250.608
|
|-
| -13/12-comma
|233.101
|
|483.450
|
|
|250.349
|
|-
| -12/11-comma
|233.305
|
|483.348
|
|
|250.043
|
|-
| -11/10-comma
|233.550
|
|483.225
|
|
|249.675
|
|-
| -10/9-comma
|233.151
|
|483.075
|
|
|249.226
|
|-
| -9/8-comma
|234.234
|
|482.888
|
|
|248.665
|
|-
| -8/7-comma
|234.295
|
|482.648
|
|
|247.943
|
|-
| -15/13-comma
|235.001
|
|482.500
|
|
|247.499
|
|-
| -7/6-comma
|235.346
|
|482.327
|
|
|246.981
|
|-
| -13/11-comma
|235.755
|
|482.123
|
|
|246.368
|
|-
| -6/5-comma
|236.244
|
|481.878
|
|
|245.633
|
|-
| -17/14-comma
|236.629
|
|481.685
|
|
|245.056
|
|-
| -11/9-comma
|236.843
|
|481.578
|
|
|244.735
|
|-
| -16/13-comma
|237.926
|
|481.463
|
|
|244.390
|
|-
| -5/4-comma
|237.592
|
|481.204
|
|
|243.612
|
|-
| -14/11-comma
|238.204
|
|480.898
|
|
|242.694
|
|-
| -9/7-comma
|238.554
|
|480.723
|
|
|242.169
|
|-
| -13/10-comma
|238.939
|
|480.530
|
|
|241.591
|
|-
| -17/13-comma
|239.146
|
|480.427
|
|
|241.280
|
|-
| -4/3-comma
|239.837
|
|480.081
|
|
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|
|479.761
|
|
|239.282
|
|-
| -15/11-comma
|240.634
|
|479.673
|
|
|239.019
|
|-
| -11/8-comma
|240.960
|
|479.520
|
|
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|241.148
|
|479.426
|
|
|238.279
|
|-
| -18/13-comma
|241.219
|
|479.390
|
|
|238.171
|
|-
| -7/5-comma
|241.634
|
|479.183
|
|
|237.550
|
|-
| -17/12-comma
|242.917
|
|478.959
|
|
|236.876
|
|-
| -10/7-comma
|242.403
|
|478.798
|
|
|236.395
|
|-
| -13/9-comma
|242.831
|
|478.584
|
|
|235.753
|
|-
| -16/11-comma
|243.103
|
|478.448
|
|
|235.345
|
|-
| -19/13-comma
|243.708
|
|478.354
|
|
|235.062
|
|-
| -3/2-comma
|244.328
|
|477.836
|
|
|233.508
|
|-
| -20/13-comma
|245.344
|
|477.318
|
|
|231.953
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|
|477.224
|
|
|231.671
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|
|477.087
|
|
|231.262
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|
|476.873
|
|
|230.621
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|
|476.713
|
|
|230.140
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|
|476.489
|
|
|229.466
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|
|476.281
|
|
|228.844
|
|-
| -ϕ-comma
|247.491
|
|476.246
|
|
|228.737
|
|-
| -13/8-comma
|247.696
|
|476.152
|
|
|228.456
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|
|475.999
|
|
|227.996
|
|-
| -23/14-comma
|248.823
|
|475.911
|
|
|227.734
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|
|475.590
|
|
|226.771
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|
|475.245
|
|
|225.735
|
|-
| -17/10-comma
|249.717
|
|475.141
|
|
|225.424
|
|-
| -12/7-comma
|250.105
|
|474.949
|
|
|224.847
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|
|474.774
|
|
|224.322
|
|-
| -7/4-comma
|251.064
|
|474.468
|
|
|223.403
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|
|474.209
|
|
|222.626
|
|-
| -16/9-comma
|251.823
|
|474.094
|
|
|222.281
|
|-
| -25/14-comma
|252.027
|
|473.987
|
|
|221.960
|
|-
| -9/5-comma
|252.412
|
|473.794
|
|
|221.382
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|
|473.549
|
|
|220.648
|
|-
| -11/6-comma
|253.610
|
|473.345
|
|
|220.035
|
|-
| -24/13-comma
|253.345
|
|473.172
|
|
|219.517
|
|-
| -13/7-comma
|253.951
|
|473.924
|
|
|219.073
|
|-
| -15/8-comma
|254.433
|
|472.784
|
|
|218.351
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|
|472.597
|
|
|217.790
|
|-
| -19/10-comma
|255.106
|
|472.447
|
|
|217.341
|
|-
| -21/11-comma
|255.351
|
|472.324
|
|
|216.973
|
|-
| -23/12-comma
|255.555
|
|472.222
|
|
|216.667
|
|-
| -25/13-comma
|255.728
|
|472.135
|
|
|216.408
|
|-
| -27/14-comma
|255.876
|
|472.052
|
|
|216.186
|
|-
|-
| -2-comma
|258.801
|
|471.100
|
|
|213.299
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-23T05:36:43Z

Moremajorthanmajor: /* Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic) */

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with more complex thirds, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|588.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|212.824
|470.941
|729.051
|683.766
|987.176
|
|-
|27/7-comma
|256.181
|215.728
|471.909
|728.091
|687.637
|984.272
|
|-
|23/6-comma
|255.858
|216.212
|472.071
|727.929
|688.283
|983.788
|
|-
|19/5-comma
|255.407
|216.890
|472.297
|727.703
|689.187
|983.110
|
|-
|15/4-comma
|254.769
|217.906
|472.635
|727.365
|690.542
|982.094
|
|-
|26/7-comma
|254.243
|218.632
|472.877
|727.123
|691.510
|981.378
|
|-
|11/3-comma
| 253.600
|219.600
|473.200
|726.800
|692.800
|980.400
|
|-
|(2+ϕ)-comma
|252.940
|220.589
|473.530
|726.470
|694.119
|979.411
|
|-
|18/5-comma
|252.696
|220.956
|473.652
|726.348
|694.607
|979.044
|
|-
|25/7-comma
|252.309
|221.536
|473.845
|726.155
|695.382
|978.464
|
|-
|7/2-comma
|251.341
|222.988
|474.329
|725.671
|697.318
|977.012
|
|-
|24/7-comma
|250.373
|224.440
|474.813
|725.187
|699.253
|975.560
|
|-
|17/5-comma
|249.986
|225.021
|475.007
|724.993
|700.028
|974.979
|
|-
|10/3-comma
|249.083
|226.376
|475.459
|724.541
|701.835
|973.624
|
|-
|23/7-comma
|248.437
|227.344
|475.781
|724.219
|703.126
|972.656
|
|-
|13/4-comma
|247.953
|228.070
|476.023
|723.977
|704.094
|971.930
|
|-
|16/5-comma
|247.258
|229.087
|476.362
|723.638
|705.449
|970.913
|
|-
|19/6-comma
|246.824
|229.764
|476.588
|723.412
|706.352
|970.236
|
|-
|22/7-comma
|246.501
|230.248
|476.749
|723.251
|706.998
|969.752
|
|-
|3-comma
|244.565
|233.152
|477.717
|722.283
|710.870
|966.848
|
|-
|20/7-comma
|242.629
|236.056
|478.685
|721.315
|714.741
|963.944
|
|-
|17/6-comma
|242.307
|236.540
|478.847
|721.153
|715.387
|963.460
|
|-
|14/5-comma
|241.855
|237.218
|479.073
|720.927
|716.290
|962.782
|
|-
|11/4-comma
|241.177
|238.234
|479.411
|720.589
|717.645
|961.766
|
|-
|19/7-comma
|240.693
|238.960
|479.653
|720.347
|718.613
|961.040
|
|-
|8/3-comma
|240.048
|239.928
|479.976
|720.024
|719.904
|960.072
|
|-
|13/5-comma
|239.145
|241.283
|480.428
|719.572
|721.711
|958.717
|
|-
|18/7-comma
|238.757
|241.864
|480.621
|719.379
|722.485
|958.136
|
|-
|5/2-comma
| 237.789
|243.316
|481.105
|718.895
|724.421
|956.684
|
|-
|17/7-comma
|236.821
|244.768
|481.589
|718.411
|726.357
|955.232
|
|-
|12/5-comma
|236.434
|245.349
|481.783
|718.217
|727.132
|954.651
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|245.715
|481.905
|718.095
|727.620
|954.285
|
|-
|7/3-comma
|235.531
|246.704
|482.235
|717.765
|728.938
|953.296
|
|-
|16/7-comma
|234.115
|247.672
|482.557
|717.423
|730.229
|952.328
|
|-
|9/4-comma
|234.401
|248.398
|482.799
|717.201
|731.197
|951.602
|
|-
|11/5-comma
|233.276
|249.414
|483.183
|716.817
|732.552
|950.596
|
|-
|13/6-comma
|233.272
|250.092
|483.364
|716.636
|733.456
|949.909
|
|-
|15/7-comma
|232.051
|250.576
|483.525
|716.475
|734.101
|949.424
|
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|334.553
|511.518
|588.482
|846.071
| 865.447
|
|-
|13/14-comma
|178.890
|
|510.555
|
|842.221
|331.666, 868.334
|
|-
|12/13-comma
|179.037
|
|510.481
|
|841.925
| 331.444, 868.556
|
|-
|11/12-comma
|179.210
|
|510.395
|
|841.580
|331.185, 868.815
|
|-
|10/11-comma
| 179.414
|
| 510.293
|
|841.172
|330.879, 869.121
|
|-
|9/10-comma
|179.659
|
| 510.170
|
|840.682
|330.511, 869.489
|
|-
|8/9-comma
|179.959
|
|510.021
|
|840.083
|330.062, 869.038
|
|-
|7/8-comma
|180.333
|
|509.834
|
|839.334
|329.501, 870.499
|
|-
|6/7-comma
|180.814
|
|509.593
|
|838.372
|328.779, 871.221
|
|-
|11/13-comma
|181.110
|
|509.445
|
|837.780
|328.335, 871.665
|
|-
|5/6-comma
|181.455
|
|509.272
|
|837.089
|327.817, 872.193
|
|-
|9/11-comma
|181.864
|
|509.068
|
|836.272
|327.204, 872.796
|
|-
|4/5-comma
|182.354
|
|508.823
|
|835.293
|326.469, 873.531
|
|-
|11/14-comma
|182.739
|
|508.630
|
|834.523
|325.892, 874.108
|
|-
|7/9-comma
|182.952
|
|508.523
|
|834.095
| 325.571, 874.429
|
|-
|10/13-comma
|183.183
|
|508.408
|
|833.634
|325.226, 874.774
|
|-
|3/4-comma
|183.701
|
|508.150
|
|832.598
|324.449, 875.551
|
|-
|8/11-comma
|184.687
|
|507.843
|
|831.373
|323.530, 876.470
|
|-
|5/7-comma
|184.633
|
|507.638
|
|830.673
|323.005, 876.995
|
|-
|7/10-comma
|184.952
|
|507.476
|
|829.904
|322.428, 877.572
|
|-
|9/13-comma
|185.255
|
|507.372
|
|829.489
|322.117, 877.883
|
|-
|2/3-comma
|185.946
|
|507.027
|
|828.107
|321.080, 878.920
|
|-
|9/14-comma
|186.588
|
|506.706
|
|828.824
|320.118, 879.882
|
|-
|7/11-comma
|186.763
|
|506.619
|
|826.474
|319.856, 880.144
|
|-
|5/8-comma
|187.069
|
|506.465
|
|825.862
|319.396, 880.604
|
|-
|1/ϕ-comma
|187.257
|
|506.372
|
|825.486
|319.115, 880.885
|
|-
|8/13-comma
|187.320
|
|506.336
|
|825.344
|319.008, 880.992
|
|-
|3/5-comma
|187.743
|
|506.129
|
|824.514
|318.386, 881.614
|
|-
|7/12-comma
|188.194
|
|505.904
|
|823.616
|317.712, 882.288
|
|-
|4/7-comma
|188.512
|
|505.744
|
|822.975
|317.231, 882.769
|
|-
|5/9-comma
|188.940
|
|505.530
|
|822.119
|316.590, 883.410
|
|-
|6/11-comma
|189.213
|
|505.394
|
|821.575
|316.181, 883.891
|
|-
|7/13-comma
|189.401
|
|505.300
|
|821.198
|315.899, 884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|
|504.781
|
|819.125
|314.344, 885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|
|504.263
|
|817.053
|312.790, 887.210
|
|-
|5/11-comma
|191.338
|
|504.169
|
|816.676
|312.507, 887.493
|
|-
|4/9-comma
|191.934
|
|504.033
|
|816.131
|312.099, 877.901
|
|-
|3/7-comma
|192.362
|
|503.819
|
|815.276
|311.457, 388.443
|
|-
|5/12-comma
|192.683
|
|503.659
|
|814.635
|310.976, 889.024
|
|-
|2/5-comma
|193.132
|
|503.434
|
|813.736
|310.302. 889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|
|503.227
|
|812.907
|309.680, 890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|
|503.191
|
|812.764
| 309.573, 890.427
|
|-
|3/8-comma
|193.805
|
| 503.096
|
|812.389
|309.291, 890.709
|
|-
|4/11-comma
|194.112
|
|502.944
|
|811.776
|308.832, 891.168
|
|-
|5/14-comma
|194.287
|
|502.856
|
|811.427
|308.570, 891.430
|
|-
|1/3-comma
|194.928
|
|502.536
|
|810.144
|307.608, 892.392
|
|-
|4/13-comma
|195.619
|
|502.190
|
|808.762
|306.571, 893.429
|
|-
|3/10-comma
|195.174
|
|502.087
|
|808.347
|306.260, 893.740
|
|-
|2/7-comma
|196.211
|
|501.894
|
|807.577
|305.683, 894.317
|
|-
|3/11-comma
|196.561
|
|501.718
|
|806.877
|305.158, 894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|
|501.413
|
|805.653
|304.240, 895.760
|
|-
|3/13-comma
|197.692
|
|501.154
|
|804.616
|303.462, 896.538
|
|-
|2/9-comma
|197.922
|
|501.039
|
|804.155
|303.117, 896.883
|
|-
|3/14-comma
|198.136
|
|500.932
|
|803.728
|302.796, 897.204
|
|-
|1/5-comma
|198.521
|
|500.740
|
|802.958
|302.219, 897.781
|
|-
|2/11-comma
|199.011
|
|500.495
|
|801.978
|301.484, 898.516
|
|-
|1/6-comma
|199.419
|
|500.290
|
|801.162
|300.871, 899.129
|
|-
|2/13-comma
|199.765
|
|500.118
|
|800.471
|300.353, 899.647
|
|-
|1/7-comma
|200.061
|
|499.970
|
|799.879
|299.909, 900.091
|
|-
|1/8-comma
|200.542
|
| 499.729
|
|798.916
|299.187, 900.823
|
|-
|1/9-comma
|200.916
|
|499.542
|
|798.168
|298.626, 901.374
|
|-
|1/10-comma
|201.785
|
|499.392
|
|797.569
|298.177, 901.823
|
|-
|1/11-comma
|201.460
|
|499.270
|
|797.079
|297.810, 902.190
|
|-
|1/12-comma
|201.665
|
|499.168
|
|796.671
|297.503, 902.497
|
|-
|1/13-comma
|201.837
|
|499.081
|
|796.325
|297.244, 902.756
|
|-
|1/14-comma
|201.953
|
|499.007
|
|796.029
|297.022, 902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|792.180
|294.135, 905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|253.480
|484.493
|715.507
|737.973
|946.520
|
|-
|13/7-comma
|229.078
|256.384
|485.461
|
|
|
|
|-
|11/6-comma
|228.755
|256.868
|485.623
|
|
|
|
|-
|9/5-comma
|228.697
|257.545
|485.848
|
|
|
|
|-
| 7/4-comma
|227.626
|258.562
|486.187
|
|
|
|
|-
|12/7-comma
|227.142
|259.288
|486.429
|
|
|
|
|-
|5/3-comma
|226.496
|260.253
|486.752
|
|
|
|
|-
|ϕ-comma
|225.837
|261.244
|487.081
|
|
|
|
|-
|8/5-comma
|225.593
|261.611
|487.204
|
|
|
|
|-
|11/7-comma
|225.206
|262.192
| 487.397
|
|
|
|
|-
|3/2-comma
| 224.762
|263.644
|487.881
|
|
|
|
|-
|10/7-comma
|223.270
|265.096
|488.365
|
|
|
|
|-
|7/5-comma
|222.882
|265.676
|488.559
|
|
|
|
|-
|4/3-comma
|221.979
|267.031
|489.010
|
|
|
|
|-
|9/7-comma
|221.334
|267.999
|489.333
|
|
|
|
|-
|5/4-comma
|220.850
|268.725
|489.575
|
|
|
|
|-
| 6/5-comma
|220.172
|269.742
|489.914
|
|
|
|
|-
|7/6-comma
|219.720
|270.419
|490.140
|
|
|
|
|-
|8/7-comma
|219.398
|270.903
|490.301
|
|
|
|
|-
|1-comma
|217.538
|273.807
|491.269
|
|
|
|
|-
|6/7-comma
|215.526
|276.711
|492.237
|
|
|
|
|-
|5/6-comma
|215.203
|277.195
|492.398
|
|
|
|
|-
| 4/5-comma
|214.751
|277.873
| 492.624
|
|
|
|
|-
|3/4-comma
|214.926
|278.889
|492.963
|
|
|
|
|-
|5/7-comma
|213.590
|279.615
|493.205
|
|
|
|
|-
|2/3-comma
|212.945
|280.583
|493.528
|
|
|
|
|-
|3/5-comma
|212.041
|281.938
|493.979
|
|
|
|
|-
|4/7-comma
|211.346
|282.519
|494.173
|
|
|
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|283.971
|494.657
|
|
|
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|209.718
|285.423
|495.141
|
|
|
|
|-
|2/5-comma
|209.331
|286.004
|495.335
|
|
|
|
|-
|1/(ϕ+1)-comma
|209.086
|286.371
|495.457
|
|
|
|
|-
|1/3-comma
|208.573
|287.359
|495.786
|
|
|
|
|-
|2/7-comma
|207.782
|289.372
|496.109
|
|
|
|
|-
|1/4-comma
|207.293
|289.053
|496.351
|
|
|
|
|-
|1/5-comma
|206.620
|290.069
|496.690
|
|
|
|
|-
|1/6-comma
|206.169
|290.747
|496.916
|
|
|
|
|-
|1/7-comma
|205.846
|291.231
|497.077
|
|
|
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|294.135
|498.045
|
|
|
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|792.180
|294.135, 905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|
|497.083
|
|788.331
|291.248, 908.752
|
|-
| -1/13-comma
|205.983
|
|497.009
|
|788.035
|291.026, 908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|
|496.922
|
|787.689
|290.767, 909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|
|496.820
|
|787.280
|290.460, 909.540
|
|-
| -1/10-comma
|206.605
|
|496.698
|
|786.791
|290.093, 909.907
|
|-
| -1/9-comma
|206.904
|
|496.548
|
|786.192
|289.644, 910.356
|
|-
| -1/8-comma
|207.278
|
|496.361
|
|785.444
|289.083, 910.917
|
|-
| -1/7-comma
|207.759
|
|496.120
|
|784.481
|288.361, 911.639
|
|-
| -2/13-comma
|208.055
|
|495.972
|
|783.889
|287.917, 912.083
|
|-
| -1/6-comma
|208.401
|
|495.800
|
|783.198
|287.399, 912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|
|495.595
|
|782.382
|286.786, 913.214
|
|-
| -1/5-comma
|209.299
|
|495.350
|
|781.401
|286.051, 913.949
|
|-
| -3/14-comma
|209.684
|
|495.158
|
|780.632
|285.474, 914.526
|
|-
| -2/9-comma
|209.898
|
|495.051
|
|780.204
|285.153, 914.847
|
|-
| -3/13-comma
|210.128
|
|494.936
|
|779.744
|284.808, 915.192
|
|-
| -1/4-comma
|210.646
|
|494.677
|
|778.707
|284.030, 915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|
|494.371
|
|777.482
|283.111, 916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|
|494.196
|
|776.783
|282.587, 917.413
|
|-
| -3/10-comma
|211.994
|
|494.003
|
|776.013
|282.010, 917.990
|
|-
| -4/13-comma
|212.799
|
|493.900
|
|775.598
|281.699, 918.301
|
|-
| -1/3-comma
|212.892
|
|493.554
|
|774.216
|280.662, 919.338
|
|-
| -5/14-comma
|213.537
|
|493.233
|
|772.933
|279.700, 920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|
|493.146
|
|772.583
|279.437, 920.563
|
|-
| -3/8-comma
|214.014
|
|492.993
|
|771.971
|278.979, 921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|
|492.899
|
|771.596
|278.697, 921.303
|
|-
| -5/13-comma
|214.274
|
|492.863
|
|771.453
|278.590, 921.410
|
|-
| -2/5-comma
|214.688
|
|492.656
|
|770.624
|277.968, 922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|
|492.431
|
|769.725
|277.294, 922.706
|
|-
| -3/7-comma
|215.458
|
|492.271
|
|769.084
|276.813, 923.187
|
|-
| -4/9-comma
|215.886
|
|492.057
|
|768.229
|276.171, 923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|
|491.921
|
|767.684
|275.763, 924.237
|
|-
| -6/13-comma
|216.346
|
|491.827
|
|767.307
|275.480, 924.520
|
|-
| -1/2-comma
|217.383
|
|491.309
|
|765.235
|273.926, 926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|
|490.790
|
|763.161
|272.371, 927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|
|490.696
|
|762.785
|272.089, 927.911
|
|-
| -5/9-comma
|218.880
|
|490.560
|
|762.241
|271.680, 928.320
|
|-
| -4/7-comma
|219.307
|
|490.346
|
|761.385
|271.039, 928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|
|490.186
|
|760.744
|270.558, 929.442
|
|-
| -3/5-comma
|220.077
|
|489.961
|
|759.846
|269.884, 930.116
|
|-
| -8/13-comma
|220.492
|
|489.754
|
|759.016
|269.262, 930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|
|489.716
|
|758.874
|269.155, 930.845
|
|-
| -5/8-comma
|220.751
|
|489.625
|
|758.498
|268.874, 931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|
|489.471
|
|757.886
|268.414, 931.586
|
|-
| -9/14-comma
|221.232
|
|489.384
|
|757.536
|268.152, 931.848
|
|-
| -2/3-comma
|221.874
|
|489.063
|
|756.253
|267.190, 932.810
|
|-
| -9/13-comma
|222.565
|
|488.718
|
|754.871
|266.153, 933.847
|
|-
| -7/10-comma
|222.772
|
|488.614
|
|754.456
|265.842, 934.158
|
|-
| -5/7-comma
|223.157
|
|488.422
|
|753.687
|265.265, 934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|
|488.247
|
|752.987
|264.740, 935.260
|
|-
| -3/4-comma
|224.119
|
|487.940
|
|751.762
|263.821, 936.189
|
|-
| -10/13-comma
|224.637
|
|487.681
|
|750.726
|263.044, 936.956
|
|-
| -7/9-comma
|224.868
|
|487.566
|
|750.265
|262.698, 937.302
|
|-
| -11/14-comma
|225.081
|
|487.459
|
|749.837
|262.378, 937.622
|
|-
| -4/5-comma
|225.466
|
|487.267
|
|749.067
|261.801, 938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|
|487.022
|
|748.088
|261.066, 938.934
|
|-
| -5/6-comma
|226.365
|
|486.818
|
|747.271
|260.453, 939.447
|
|-
| -11/13-comma
|226.710
|
|486.645
|
|746.580
|259.935, 940.065
|
|-
| -6/7-comma
|227.006
|
|486.497
|
|745.988
|259.491, 940.509
|
|-
| -7/8-comma
|227.487
|
|486.256
|
|745.026
|258.769, 941.231
|
|-
| -8/9-comma
|227.861
|
|486.069
|
|744.277
|258.208, 941.792
|
|-
| -9/10-comma
|228.161
|
|485.920
|
|743.678
|257.759, 942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|
|485.797
|
|743.188
|257.391, 942.609
|
|-
| -11/12-comma
|228.610
|
|485.695
|
|
|257.085
|
|-
| -12/13-comma
|228.783
|
|485.609
|
|
|256.826
|
|-
| -13/14-comma
|228.931
|
|485.535
|
|
|256.604
|
|-
|-
| -1-comma
|230.855
|
|484.752
|
|
|253.717
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|
|499.013
|
|
|297.039
|
|-
| -1/6-comma
|201.652
|
|499.174
|
|
|297.523
|
|-
| -1/5-comma
|201.200
|
|499.400
|
|
|298.201
|
|-
| -1/4-comma
|200.522
|
|499.739
|
|
|299.217
|
|-
| -2/7-comma
|200.038
|
|499.981
|
|
|299.942
|
|-
| -1/3-comma
|199.393
|
|500.303
|
|
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|
|500.633
|
|
|301.900
|
|-
| -2/5-comma
|198.499
|
|500.755
|
|
|302.266
|
|-
| -3/7-comma
|198.102
|
|500.949
|
|
|302.847
|
|-
| -1/2-comma
|197.134
|
|501.433
|
|
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|196.166
|
|501.917
|
|
|305.751
|
|-
| -3/5-comma
|195.779
|
|502.111
|
|
|306.332
|
|-
| -2/3-comma
|194.876
|
|502.562
|
|
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|194.230
|
|502.885
|
|
|308.655
|
|-
| -4/5-comma
|193.069
|
|503.466
|
|
|310.397
|
|-
| -5/6-comma
|192.617
|
|503.692
|
|
|311.075
|
|-
|-
| -6/7-comma
|192.294
|
|503.853
|
|
|311.556
|
|-
| -1-comma
|190.352
|
|504.821
|
|
|314.463
|
|-
| -8/7-comma
|188.422
|
|505.789
|
|
|317.367
|
|-
| -7/6-comma
|188.100
|
|505.950
|
|
|317.851
|
|-
| -6/5-comma
|187.648
|
|506.176
|
|
|318.528
|
|-
| -5/4-comma
|186.970
|
|506.515
|
|
|319.545
|
|-
| -9/7-comma
|186.486
|
|506.757
|
|
|320.271
|
|-
| -4/3-comma
|185.841
|
|507.080
|
|
|321.239
|
|-
| -7/5-comma
|184.937
|
|507.531
|
|
|322.594
|
|-
| -10/7-comma
|184.550
|
|507.725
|
|
|323.174
|
|-
| -3/2-comma
|183.582
|
|508.209
|
|
|324.626
|
|-
| -11/7-comma
|182.614
|
|508.693
|
|
|326.078
|
|-
| -8/5-comma
|182.228
|
|508.886
|
|
|326.659
|
|-
| -ϕ-comma
|181.983
|
|509.009
|
|
|327.026
|
|-
| -5/3-comma
|181.324
|
|509.338
|
|
|328.014
|
|-
| -12/7-comma
|180.678
|
|509.661
|
|
|328.982
|
|-
| -7/4-comma
|180.194
|
|509.903
|
|
|329.708
|
|-
| -9/5-comma
|179.517
|
|510.242
|
|
|330.725
|
|-
| -11/6-comma
|179.065
|
|510.467
|
|
|331.402
|
|-
| -13/7-comma
|178.742
|
|510.629
|
|
|331.886
|
|-
| -2-comma
|176.807
|
|511.597
|
|
|334.790
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|
|484.752
|
|
|253.717
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|232.780
|
|483.610
|
|
|250.830
|
|-
| -14/13-comma
|232.928
|
|483.536
|
|
|250.608
|
|-
| -13/12-comma
|233.101
|
|483.450
|
|
|250.349
|
|-
| -12/11-comma
|233.305
|
|483.348
|
|
|250.043
|
|-
| -11/10-comma
|233.550
|
|483.225
|
|
|249.675
|
|-
| -10/9-comma
|233.151
|
|483.075
|
|
|249.226
|
|-
| -9/8-comma
|234.234
|
|482.888
|
|
|248.665
|
|-
| -8/7-comma
|234.295
|
|482.648
|
|
|247.943
|
|-
| -15/13-comma
|235.001
|
|482.500
|
|
|247.499
|
|-
| -7/6-comma
|235.346
|
|482.327
|
|
|246.981
|
|-
| -13/11-comma
|235.755
|
|482.123
|
|
|246.368
|
|-
| -6/5-comma
|236.244
|
|481.878
|
|
|245.633
|
|-
| -17/14-comma
|236.629
|
|481.685
|
|
|245.056
|
|-
| -11/9-comma
|236.843
|
|481.578
|
|
|244.735
|
|-
| -16/13-comma
|237.926
|
|481.463
|
|
|244.390
|
|-
| -5/4-comma
|237.592
|
|481.204
|
|
|243.612
|
|-
| -14/11-comma
|238.204
|
|480.898
|
|
|242.694
|
|-
| -9/7-comma
|238.554
|
|480.723
|
|
|242.169
|
|-
| -13/10-comma
|238.939
|
|480.530
|
|
|241.591
|
|-
| -17/13-comma
|239.146
|
|480.427
|
|
|241.280
|
|-
| -4/3-comma
|239.837
|
|480.081
|
|
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|
|479.761
|
|
|239.282
|
|-
| -15/11-comma
|240.634
|
|479.673
|
|
|239.019
|
|-
| -11/8-comma
|240.960
|
|479.520
|
|
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|241.148
|
|479.426
|
|
|238.279
|
|-
| -18/13-comma
|241.219
|
|479.390
|
|
|238.171
|
|-
| -7/5-comma
|241.634
|
|479.183
|
|
|237.550
|
|-
| -17/12-comma
|242.917
|
|478.959
|
|
|236.876
|
|-
| -10/7-comma
|242.403
|
|478.798
|
|
|236.395
|
|-
| -13/9-comma
|242.831
|
|478.584
|
|
|235.753
|
|-
| -16/11-comma
|243.103
|
|478.448
|
|
|235.345
|
|-
| -19/13-comma
|243.708
|
|478.354
|
|
|235.062
|
|-
| -3/2-comma
|244.328
|
|477.836
|
|
|233.508
|
|-
| -20/13-comma
|245.344
|
|477.318
|
|
|231.953
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|
|477.224
|
|
|231.671
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|
|477.087
|
|
|231.262
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|
|476.873
|
|
|230.621
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|
|476.713
|
|
|230.140
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|
|476.489
|
|
|229.466
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|
|476.281
|
|
|228.844
|
|-
| -ϕ-comma
|247.491
|
|476.246
|
|
|228.737
|
|-
| -13/8-comma
|247.696
|
|476.152
|
|
|228.456
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|
|475.999
|
|
|227.996
|
|-
| -23/14-comma
|248.823
|
|475.911
|
|
|227.734
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|
|475.590
|
|
|226.771
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|
|475.245
|
|
|225.735
|
|-
| -17/10-comma
|249.717
|
|475.141
|
|
|225.424
|
|-
| -12/7-comma
|250.105
|
|474.949
|
|
|224.847
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|
|474.774
|
|
|224.322
|
|-
| -7/4-comma
|251.064
|
|474.468
|
|
|223.403
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|
|474.209
|
|
|222.626
|
|-
| -16/9-comma
|251.823
|
|474.094
|
|
|222.281
|
|-
| -25/14-comma
|252.027
|
|473.987
|
|
|221.960
|
|-
| -9/5-comma
|252.412
|
|473.794
|
|
|221.382
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|
|473.549
|
|
|220.648
|
|-
| -11/6-comma
|253.610
|
|473.345
|
|
|220.035
|
|-
| -24/13-comma
|253.345
|
|473.172
|
|
|219.517
|
|-
| -13/7-comma
|253.951
|
|473.924
|
|
|219.073
|
|-
| -15/8-comma
|254.433
|
|472.784
|
|
|218.351
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|
|472.597
|
|
|217.790
|
|-
| -19/10-comma
|255.106
|
|472.447
|
|
|217.341
|
|-
| -21/11-comma
|255.351
|
|472.324
|
|
|216.973
|
|-
| -23/12-comma
|255.555
|
|472.222
|
|
|216.667
|
|-
| -25/13-comma
|255.728
|
|472.135
|
|
|216.408
|
|-
| -27/14-comma
|255.876
|
|472.052
|
|
|216.186
|
|-
|-
| -2-comma
|258.801
|
|471.100
|
|
|213.299
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-22T04:59:44Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth, nineteenth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth, nineteenth (technically)
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis'', beside which the weakness of ''lenis'' is that its desired (sub)harmonics mostly form [[wolf interval]]<nowiki/>s. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean hexachords|mean hexachord]], which is primarily considered to temper out [[129/128]] or [[256/255]].

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean tetrachords

2025-01-22T04:55:44Z

Moremajorthanmajor: Moremajorthanmajor moved page User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean tetrachords to User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean hexachords

#REDIRECT [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean hexachords]]

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-22T04:55:44Z

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with more complex thirds, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|588.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|
|470.941
|
|
|212.824
|
|-
|27/7-comma
|256.181
|
|471.909
|
|
|215.728
|
|-
|23/6-comma
|255.858
|
|472.071
|
|
|216.212
|
|-
|19/5-comma
|255.407
|
|472.297
|
|
|216.890
|
|-
|15/4-comma
|254.769
|
|472.635
|
|
|217.906
|
|-
|26/7-comma
|254.243
|
|472.877
|
|
|218.632
|
|-
|11/3-comma
| 253.600
|
|473.200
|
|
|216.600
|
|-
|(2+ϕ)-comma
|252.940
|
|473.530
|
|
|220.589
|
|-
|18/5-comma
|252.696
|
|473.652
|
|
|220.956
|
|-
|25/7-comma
|252.309
|
|473.845
|
|
|221.536
|
|-
|7/2-comma
|251.341
|
|474.329
|
|
|222.988
|
|-
|24/7-comma
|250.373
|
|474.813
|
|
|224.440
|
|-
|17/5-comma
|249.986
|
|475.007
|
|
|225.021
|
|-
|10/3-comma
|249.083
|
|475.459
|
|
|226.376
|
|-
|23/7-comma
|248.437
|
|475.781
|
|
|227.344
|
|-
|13/4-comma
|247.953
|
|476.023
|
|
|228.070
|
|-
|16/5-comma
|247.258
|
|476.362
|
|
|229.087
|
|-
|19/6-comma
|246.824
|
|476.588
|
|
|229.764
|
|-
|22/7-comma
|246.501
|
|476.749
|
|
|230.248
|
|-
|3-comma
|244.565
|
|477.717
|
|
|233.152
|
|-
|20/7-comma
|242.629
|
|478.685
|
|
|236.056
|
|-
|17/6-comma
|242.307
|
|478.847
|
|
|236.540
|
|-
|14/5-comma
|241.855
|
|479.073
|
|
|237.218
|
|-
|11/4-comma
|241.177
|
|479.411
|
|
|238.234
|
|-
|19/7-comma
|240.693
|
|479.653
|
|
|238.960
|
|-
|8/3-comma
|240.048
|
|479.976
|
|
|239.928
|
|-
|13/5-comma
|239.145
|
|480.428
|
|
|241.283
|
|-
|18/7-comma
|238.757
|
|480.621
|
|
|241.864
|
|-
|5/2-comma
| 237.789
|
|481.105
|
|
|243.316
|
|-
|17/7-comma
|236.821
|
|481.589
|
|
|244.768
|
|-
|12/5-comma
|236.434
|
|481.783
|
|
|245.349
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|
|481.905
|
|
|245.715
|
|-
|7/3-comma
|235.531
|
|482.235
|
|
|246.704
|
|-
|16/7-comma
|234.115
|
|482.557
|
|
|247.672
|
|-
|9/4-comma
|234.401
|
|482.799
|
|
|248.398
|
|-
|11/5-comma
|233.276
|
|483.183
|
|
|249.414
|
|-
|13/6-comma
|233.272
|
|483.364
|
|
|250.092
|
|-
|15/7-comma
|232.051
|
|483.525
|
|
|250.576
|
|-
|2-comma
|231.014
|
|484.493
|
|
|253.480
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|
|511.518
|
|846.071
| 334.553, 865.447
|
|-
|13/14-comma
|178.890
|
|510.555
|
|842.221
|331.666, 868.334
|
|-
|12/13-comma
|179.037
|
|510.481
|
|841.925
| 331.444, 868.556
|
|-
|11/12-comma
|179.210
|
|510.395
|
|841.580
|331.185, 868.815
|
|-
|10/11-comma
| 179.414
|
| 510.293
|
|841.172
|330.879, 869.121
|
|-
|9/10-comma
|179.659
|
| 510.170
|
|840.682
|330.511, 869.489
|
|-
|8/9-comma
|179.959
|
|510.021
|
|840.083
|330.062, 869.038
|
|-
|7/8-comma
|180.333
|
|509.834
|
|839.334
|329.501, 870.499
|
|-
|6/7-comma
|180.814
|
|509.593
|
|838.372
|328.779, 871.221
|
|-
|11/13-comma
|181.110
|
|509.445
|
|837.780
|328.335, 871.665
|
|-
|5/6-comma
|181.455
|
|509.272
|
|837.089
|327.817, 872.193
|
|-
|9/11-comma
|181.864
|
|509.068
|
|836.272
|327.204, 872.796
|
|-
|4/5-comma
|182.354
|
|508.823
|
|835.293
|326.469, 873.531
|
|-
|11/14-comma
|182.739
|
|508.630
|
|834.523
|325.892, 874.108
|
|-
|7/9-comma
|182.952
|
|508.523
|
|834.095
| 325.571, 874.429
|
|-
|10/13-comma
|183.183
|
|508.408
|
|833.634
|325.226, 874.774
|
|-
|3/4-comma
|183.701
|
|508.150
|
|832.598
|324.449, 875.551
|
|-
|8/11-comma
|184.687
|
|507.843
|
|831.373
|323.530, 876.470
|
|-
|5/7-comma
|184.633
|
|507.638
|
|830.673
|323.005, 876.995
|
|-
|7/10-comma
|184.952
|
|507.476
|
|829.904
|322.428, 877.572
|
|-
|9/13-comma
|185.255
|
|507.372
|
|829.489
|322.117, 877.883
|
|-
|2/3-comma
|185.946
|
|507.027
|
|828.107
|321.080, 878.920
|
|-
|9/14-comma
|186.588
|
|506.706
|
|828.824
|320.118, 879.882
|
|-
|7/11-comma
|186.763
|
|506.619
|
|826.474
|319.856, 880.144
|
|-
|5/8-comma
|187.069
|
|506.465
|
|825.862
|319.396, 880.604
|
|-
|1/ϕ-comma
|187.257
|
|506.372
|
|825.486
|319.115, 880.885
|
|-
|8/13-comma
|187.320
|
|506.336
|
|825.344
|319.008, 880.992
|
|-
|3/5-comma
|187.743
|
|506.129
|
|824.514
|318.386, 881.614
|
|-
|7/12-comma
|188.194
|
|505.904
|
|823.616
|317.712, 882.288
|
|-
|4/7-comma
|188.512
|
|505.744
|
|822.975
|317.231, 882.769
|
|-
|5/9-comma
|188.940
|
|505.530
|
|822.119
|316.590, 883.410
|
|-
|6/11-comma
|189.213
|
|505.394
|
|821.575
|316.181, 883.891
|
|-
|7/13-comma
|189.401
|
|505.300
|
|821.198
|315.899, 884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|
|504.781
|
|819.125
|314.344, 885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|
|504.263
|
|817.053
|312.790, 887.210
|
|-
|5/11-comma
|191.338
|
|504.169
|
|816.676
|312.507, 887.493
|
|-
|4/9-comma
|191.934
|
|504.033
|
|816.131
|312.099, 877.901
|
|-
|3/7-comma
|192.362
|
|503.819
|
|815.276
|311.457, 388.443
|
|-
|5/12-comma
|192.683
|
|503.659
|
|814.635
|310.976, 889.024
|
|-
|2/5-comma
|193.132
|
|503.434
|
|813.736
|310.302. 889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|
|503.227
|
|812.907
|309.680, 890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|
|503.191
|
|812.764
| 309.573, 890.427
|
|-
|3/8-comma
|193.805
|
| 503.096
|
|812.389
|309.291, 890.709
|
|-
|4/11-comma
|194.112
|
|502.944
|
|811.776
|308.832, 891.168
|
|-
|5/14-comma
|194.287
|
|502.856
|
|811.427
|308.570, 891.430
|
|-
|1/3-comma
|194.928
|
|502.536
|
|810.144
|307.608, 892.392
|
|-
|4/13-comma
|195.619
|
|502.190
|
|808.762
|306.571, 893.429
|
|-
|3/10-comma
|195.174
|
|502.087
|
|808.347
|306.260, 893.740
|
|-
|2/7-comma
|196.211
|
|501.894
|
|807.577
|305.683, 894.317
|
|-
|3/11-comma
|196.561
|
|501.718
|
|806.877
|305.158, 894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|
|501.413
|
|805.653
|304.240, 895.760
|
|-
|3/13-comma
|197.692
|
|501.154
|
|804.616
|303.462, 896.538
|
|-
|2/9-comma
|197.922
|
|501.039
|
|804.155
|303.117, 896.883
|
|-
|3/14-comma
|198.136
|
|500.932
|
|803.728
|302.796, 897.204
|
|-
|1/5-comma
|198.521
|
|500.740
|
|802.958
|302.219, 897.781
|
|-
|2/11-comma
|199.011
|
|500.495
|
|801.978
|301.484, 898.516
|
|-
|1/6-comma
|199.419
|
|500.290
|
|801.162
|300.871, 899.129
|
|-
|2/13-comma
|199.765
|
|500.118
|
|800.471
|300.353, 899.647
|
|-
|1/7-comma
|200.061
|
|499.970
|
|799.879
|299.909, 900.091
|
|-
|1/8-comma
|200.542
|
| 499.729
|
|798.916
|299.187, 900.823
|
|-
|1/9-comma
|200.916
|
|499.542
|
|798.168
|298.626, 901.374
|
|-
|1/10-comma
|201.785
|
|499.392
|
|797.569
|298.177, 901.823
|
|-
|1/11-comma
|201.460
|
|499.270
|
|797.079
|297.810, 902.190
|
|-
|1/12-comma
|201.665
|
|499.168
|
|796.671
|297.503, 902.497
|
|-
|1/13-comma
|201.837
|
|499.081
|
|796.325
|297.244, 902.756
|
|-
|1/14-comma
|201.953
|
|499.007
|
|796.029
|297.022, 902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|792.180
|294.135, 905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|
|484.493
|
|
|253.480
|
|-
|13/7-comma
|229.078
|
|485.461
|
|
|256.384
|
|-
|11/6-comma
|228.755
|
|485.623
|
|
|256.868
|
|-
|9/5-comma
|228.697
|
|485.848
|
|
|257.545
|
|-
| 7/4-comma
|227.626
|
|486.187
|
|
|258.562
|
|-
|12/7-comma
|227.142
|
|486.429
|
|
|259.288
|
|-
|5/3-comma
|226.496
|
|486.752
|
|
|260.253
|
|-
|ϕ-comma
|225.837
|
|487.081
|
|
|261.244
|
|-
|8/5-comma
|225.593
|
|487.204
|
|
|261.611
|
|-
|11/7-comma
|225.206
|
| 487.397
|
|
|262.192
|
|-
|3/2-comma
| 224.762
|
|487.881
|
|
|263.644
|
|-
|10/7-comma
|223.270
|
|488.365
|
|
|265.096
|
|-
|7/5-comma
|222.882
|
|488.559
|
|
|265.676
|
|-
|4/3-comma
|221.979
|
|489.010
|
|
|267.031
|
|-
|9/7-comma
|221.334
|
|489.333
|
|
|267.999
|
|-
|5/4-comma
|220.850
|
|489.575
|
|
|268.725
|
|-
| 6/5-comma
|220.172
|
|489.914
|
|
|269.742
|
|-
|7/6-comma
|219.720
|
|490.140
|
|
|270.419
|
|-
|8/7-comma
|219.398
|
|490.301
|
|
|270.903
|
|-
|1-comma
|217.538
|
|491.269
|
|
| 273.807
|
|-
|6/7-comma
|215.526
|
|492.237
|
|
|276.711
|
|-
|5/6-comma
|215.203
|
|492.398
|
|
|277.195
|
|-
| 4/5-comma
|214.751
|
| 492.624
|
|
|277.873
|
|-
|3/4-comma
|214.926
|
|492.963
|
|
|278.889
|
|-
|5/7-comma
|213.590
|
|493.205
|
|
|279.615
|
|-
|2/3-comma
|212.945
|
|493.528
|
|
|280.583
|
|-
|3/5-comma
|212.041
|
|493.979
|
|
|281.938
|
|-
|4/7-comma
|211.346
|
|494.173
|
|
|282.519
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|
|494.657
|
|
|283.971
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|209.718
|
|495.141
|
|
|285.423
|
|-
|2/5-comma
|209.331
|
|495.335
|
|
|286.004
|
|-
|1/(ϕ+1)-comma
|209.086
|
|495.457
|
|
|286.371
|
|-
|1/3-comma
|208.573
|
|495.786
|
|
|287.359
|
|-
|2/7-comma
|207.782
|
|496.109
|
|
|289.372
|
|-
|1/4-comma
|207.293
|
|496.351
|
|
|289.053
|
|-
|1/5-comma
|206.620
|
|496.690
|
|
|290.069
|
|-
|1/6-comma
|206.169
|
|496.916
|
|
|290.747
|
|-
|1/7-comma
|205.846
|
|497.077
|
|
|291.231
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|792.180
|294.135, 905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|
|497.083
|
|788.331
|291.248, 908.752
|
|-
| -1/13-comma
|205.983
|
|497.009
|
|788.035
|291.026, 908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|
|496.922
|
|787.689
|290.767, 909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|
|496.820
|
|787.280
|290.460, 909.540
|
|-
| -1/10-comma
|206.605
|
|496.698
|
|786.791
|290.093, 909.907
|
|-
| -1/9-comma
|206.904
|
|496.548
|
|786.192
|289.644, 910.356
|
|-
| -1/8-comma
|207.278
|
|496.361
|
|785.444
|289.083, 910.917
|
|-
| -1/7-comma
|207.759
|
|496.120
|
|784.481
|288.361, 911.639
|
|-
| -2/13-comma
|208.055
|
|495.972
|
|783.889
|287.917, 912.083
|
|-
| -1/6-comma
|208.401
|
|495.800
|
|783.198
|287.399, 912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|
|495.595
|
|782.382
|286.786, 913.214
|
|-
| -1/5-comma
|209.299
|
|495.350
|
|781.401
|286.051, 913.949
|
|-
| -3/14-comma
|209.684
|
|495.158
|
|780.632
|285.474, 914.526
|
|-
| -2/9-comma
|209.898
|
|495.051
|
|780.204
|285.153, 914.847
|
|-
| -3/13-comma
|210.128
|
|494.936
|
|779.744
|284.808, 915.192
|
|-
| -1/4-comma
|210.646
|
|494.677
|
|778.707
|284.030, 915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|
|494.371
|
|777.482
|283.111, 916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|
|494.196
|
|776.783
|282.587, 917.413
|
|-
| -3/10-comma
|211.994
|
|494.003
|
|776.013
|282.010, 917.990
|
|-
| -4/13-comma
|212.799
|
|493.900
|
|775.598
|281.699, 918.301
|
|-
| -1/3-comma
|212.892
|
|493.554
|
|774.216
|280.662, 919.338
|
|-
| -5/14-comma
|213.537
|
|493.233
|
|772.933
|279.700, 920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|
|493.146
|
|772.583
|279.437, 920.563
|
|-
| -3/8-comma
|214.014
|
|492.993
|
|771.971
|278.979, 921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|
|492.899
|
|771.596
|278.697, 921.303
|
|-
| -5/13-comma
|214.274
|
|492.863
|
|771.453
|278.590, 921.410
|
|-
| -2/5-comma
|214.688
|
|492.656
|
|770.624
|277.968, 922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|
|492.431
|
|769.725
|277.294, 922.706
|
|-
| -3/7-comma
|215.458
|
|492.271
|
|769.084
|276.813, 923.187
|
|-
| -4/9-comma
|215.886
|
|492.057
|
|768.229
|276.171, 923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|
|491.921
|
|767.684
|275.763, 924.237
|
|-
| -6/13-comma
|216.346
|
|491.827
|
|767.307
|275.480, 924.520
|
|-
| -1/2-comma
|217.383
|
|491.309
|
|765.235
|273.926, 926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|
|490.790
|
|763.161
|272.371, 927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|
|490.696
|
|762.785
|272.089, 927.911
|
|-
| -5/9-comma
|218.880
|
|490.560
|
|762.241
|271.680, 928.320
|
|-
| -4/7-comma
|219.307
|
|490.346
|
|761.385
|271.039, 928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|
|490.186
|
|760.744
|270.558, 929.442
|
|-
| -3/5-comma
|220.077
|
|489.961
|
|759.846
|269.884, 930.116
|
|-
| -8/13-comma
|220.492
|
|489.754
|
|759.016
|269.262, 930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|
|489.716
|
|758.874
|269.155, 930.845
|
|-
| -5/8-comma
|220.751
|
|489.625
|
|758.498
|268.874, 931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|
|489.471
|
|757.886
|268.414, 931.586
|
|-
| -9/14-comma
|221.232
|
|489.384
|
|757.536
|268.152, 931.848
|
|-
| -2/3-comma
|221.874
|
|489.063
|
|756.253
|267.190, 932.810
|
|-
| -9/13-comma
|222.565
|
|488.718
|
|754.871
|266.153, 933.847
|
|-
| -7/10-comma
|222.772
|
|488.614
|
|754.456
|265.842, 934.158
|
|-
| -5/7-comma
|223.157
|
|488.422
|
|753.687
|265.265, 934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|
|488.247
|
|752.987
|264.740, 935.260
|
|-
| -3/4-comma
|224.119
|
|487.940
|
|751.762
|263.821, 936.189
|
|-
| -10/13-comma
|224.637
|
|487.681
|
|750.726
|263.044, 936.956
|
|-
| -7/9-comma
|224.868
|
|487.566
|
|750.265
|262.698, 937.302
|
|-
| -11/14-comma
|225.081
|
|487.459
|
|749.837
|262.378, 937.622
|
|-
| -4/5-comma
|225.466
|
|487.267
|
|749.067
|261.801, 938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|
|487.022
|
|748.088
|261.066, 938.934
|
|-
| -5/6-comma
|226.365
|
|486.818
|
|747.271
|260.453, 939.447
|
|-
| -11/13-comma
|226.710
|
|486.645
|
|746.580
|259.935, 940.065
|
|-
| -6/7-comma
|227.006
|
|486.497
|
|745.988
|259.491, 940.509
|
|-
| -7/8-comma
|227.487
|
|486.256
|
|745.026
|258.769, 941.231
|
|-
| -8/9-comma
|227.861
|
|486.069
|
|744.277
|258.208, 941.792
|
|-
| -9/10-comma
|228.161
|
|485.920
|
|743.678
|257.759, 942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|
|485.797
|
|743.188
|257.391, 942.609
|
|-
| -11/12-comma
|228.610
|
|485.695
|
|
|257.085
|
|-
| -12/13-comma
|228.783
|
|485.609
|
|
|256.826
|
|-
| -13/14-comma
|228.931
|
|485.535
|
|
|256.604
|
|-
|-
| -1-comma
|230.855
|
|484.752
|
|
|253.717
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|
|499.013
|
|
|297.039
|
|-
| -1/6-comma
|201.652
|
|499.174
|
|
|297.523
|
|-
| -1/5-comma
|201.200
|
|499.400
|
|
|298.201
|
|-
| -1/4-comma
|200.522
|
|499.739
|
|
|299.217
|
|-
| -2/7-comma
|200.038
|
|499.981
|
|
|299.942
|
|-
| -1/3-comma
|199.393
|
|500.303
|
|
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|
|500.633
|
|
|301.900
|
|-
| -2/5-comma
|198.499
|
|500.755
|
|
|302.266
|
|-
| -3/7-comma
|198.102
|
|500.949
|
|
|302.847
|
|-
| -1/2-comma
|197.134
|
|501.433
|
|
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|196.166
|
|501.917
|
|
|305.751
|
|-
| -3/5-comma
|195.779
|
|502.111
|
|
|306.332
|
|-
| -2/3-comma
|194.876
|
|502.562
|
|
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|194.230
|
|502.885
|
|
|308.655
|
|-
| -4/5-comma
|193.069
|
|503.466
|
|
|310.397
|
|-
| -5/6-comma
|192.617
|
|503.692
|
|
|311.075
|
|-
|-
| -6/7-comma
|192.294
|
|503.853
|
|
|311.556
|
|-
| -1-comma
|190.352
|
|504.821
|
|
|314.463
|
|-
| -8/7-comma
|188.422
|
|505.789
|
|
|317.367
|
|-
| -7/6-comma
|188.100
|
|505.950
|
|
|317.851
|
|-
| -6/5-comma
|187.648
|
|506.176
|
|
|318.528
|
|-
| -5/4-comma
|186.970
|
|506.515
|
|
|319.545
|
|-
| -9/7-comma
|186.486
|
|506.757
|
|
|320.271
|
|-
| -4/3-comma
|185.841
|
|507.080
|
|
|321.239
|
|-
| -7/5-comma
|184.937
|
|507.531
|
|
|322.594
|
|-
| -10/7-comma
|184.550
|
|507.725
|
|
|323.174
|
|-
| -3/2-comma
|183.582
|
|508.209
|
|
|324.626
|
|-
| -11/7-comma
|182.614
|
|508.693
|
|
|326.078
|
|-
| -8/5-comma
|182.228
|
|508.886
|
|
|326.659
|
|-
| -ϕ-comma
|181.983
|
|509.009
|
|
|327.026
|
|-
| -5/3-comma
|181.324
|
|509.338
|
|
|328.014
|
|-
| -12/7-comma
|180.678
|
|509.661
|
|
|328.982
|
|-
| -7/4-comma
|180.194
|
|509.903
|
|
|329.708
|
|-
| -9/5-comma
|179.517
|
|510.242
|
|
|330.725
|
|-
| -11/6-comma
|179.065
|
|510.467
|
|
|331.402
|
|-
| -13/7-comma
|178.742
|
|510.629
|
|
|331.886
|
|-
| -2-comma
|176.807
|
|511.597
|
|
|334.790
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|
|484.752
|
|
|253.717
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|232.780
|
|483.610
|
|
|250.830
|
|-
| -14/13-comma
|232.928
|
|483.536
|
|
|250.608
|
|-
| -13/12-comma
|233.101
|
|483.450
|
|
|250.349
|
|-
| -12/11-comma
|233.305
|
|483.348
|
|
|250.043
|
|-
| -11/10-comma
|233.550
|
|483.225
|
|
|249.675
|
|-
| -10/9-comma
|233.151
|
|483.075
|
|
|249.226
|
|-
| -9/8-comma
|234.234
|
|482.888
|
|
|248.665
|
|-
| -8/7-comma
|234.295
|
|482.648
|
|
|247.943
|
|-
| -15/13-comma
|235.001
|
|482.500
|
|
|247.499
|
|-
| -7/6-comma
|235.346
|
|482.327
|
|
|246.981
|
|-
| -13/11-comma
|235.755
|
|482.123
|
|
|246.368
|
|-
| -6/5-comma
|236.244
|
|481.878
|
|
|245.633
|
|-
| -17/14-comma
|236.629
|
|481.685
|
|
|245.056
|
|-
| -11/9-comma
|236.843
|
|481.578
|
|
|244.735
|
|-
| -16/13-comma
|237.926
|
|481.463
|
|
|244.390
|
|-
| -5/4-comma
|237.592
|
|481.204
|
|
|243.612
|
|-
| -14/11-comma
|238.204
|
|480.898
|
|
|242.694
|
|-
| -9/7-comma
|238.554
|
|480.723
|
|
|242.169
|
|-
| -13/10-comma
|238.939
|
|480.530
|
|
|241.591
|
|-
| -17/13-comma
|239.146
|
|480.427
|
|
|241.280
|
|-
| -4/3-comma
|239.837
|
|480.081
|
|
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|
|479.761
|
|
|239.282
|
|-
| -15/11-comma
|240.634
|
|479.673
|
|
|239.019
|
|-
| -11/8-comma
|240.960
|
|479.520
|
|
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|241.148
|
|479.426
|
|
|238.279
|
|-
| -18/13-comma
|241.219
|
|479.390
|
|
|238.171
|
|-
| -7/5-comma
|241.634
|
|479.183
|
|
|237.550
|
|-
| -17/12-comma
|242.917
|
|478.959
|
|
|236.876
|
|-
| -10/7-comma
|242.403
|
|478.798
|
|
|236.395
|
|-
| -13/9-comma
|242.831
|
|478.584
|
|
|235.753
|
|-
| -16/11-comma
|243.103
|
|478.448
|
|
|235.345
|
|-
| -19/13-comma
|243.708
|
|478.354
|
|
|235.062
|
|-
| -3/2-comma
|244.328
|
|477.836
|
|
|233.508
|
|-
| -20/13-comma
|245.344
|
|477.318
|
|
|231.953
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|
|477.224
|
|
|231.671
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|
|477.087
|
|
|231.262
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|
|476.873
|
|
|230.621
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|
|476.713
|
|
|230.140
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|
|476.489
|
|
|229.466
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|
|476.281
|
|
|228.844
|
|-
| -ϕ-comma
|247.491
|
|476.246
|
|
|228.737
|
|-
| -13/8-comma
|247.696
|
|476.152
|
|
|228.456
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|
|475.999
|
|
|227.996
|
|-
| -23/14-comma
|248.823
|
|475.911
|
|
|227.734
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|
|475.590
|
|
|226.771
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|
|475.245
|
|
|225.735
|
|-
| -17/10-comma
|249.717
|
|475.141
|
|
|225.424
|
|-
| -12/7-comma
|250.105
|
|474.949
|
|
|224.847
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|
|474.774
|
|
|224.322
|
|-
| -7/4-comma
|251.064
|
|474.468
|
|
|223.403
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|
|474.209
|
|
|222.626
|
|-
| -16/9-comma
|251.823
|
|474.094
|
|
|222.281
|
|-
| -25/14-comma
|252.027
|
|473.987
|
|
|221.960
|
|-
| -9/5-comma
|252.412
|
|473.794
|
|
|221.382
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|
|473.549
|
|
|220.648
|
|-
| -11/6-comma
|253.610
|
|473.345
|
|
|220.035
|
|-
| -24/13-comma
|253.345
|
|473.172
|
|
|219.517
|
|-
| -13/7-comma
|253.951
|
|473.924
|
|
|219.073
|
|-
| -15/8-comma
|254.433
|
|472.784
|
|
|218.351
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|
|472.597
|
|
|217.790
|
|-
| -19/10-comma
|255.106
|
|472.447
|
|
|217.341
|
|-
| -21/11-comma
|255.351
|
|472.324
|
|
|216.973
|
|-
| -23/12-comma
|255.555
|
|472.222
|
|
|216.667
|
|-
| -25/13-comma
|255.728
|
|472.135
|
|
|216.408
|
|-
| -27/14-comma
|255.876
|
|472.052
|
|
|216.186
|
|-
|-
| -2-comma
|258.801
|
|471.100
|
|
|213.299
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-22T04:54:49Z

Moremajorthanmajor:

{{Editable user page}}Here are all mean hexachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor hexachord (root-whole tone-minor third-tempered fourth-tempered fifth-sixth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with more complex thirds, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to 1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
|2-comma
|150.019
|374.971
|524.990
|675.010
|825.029
|899.962
|
|-
|27/14-comma
|151.944
|372.084
|524.028
|675.972
|827.916
|896.112
|
|-
|25/13-comma
|152.092
|371.862
|523.954
|676.046
|828.138
|895.816
|
|-
|23/12-comma
|152.265
|371.603
|523.868
|676.132
|828.397
|895.471
|
|-
|21/11-comma
|152.469
|371.297
|523.766
|676.234
|828.703
|895.062
|
|-
|19/10-comma
|152.714
|370.929
|523.643
|676.357
|829.071
|894.573
|
|-
|17/9-comma
|153.013
|370.480
|523.493
|676.507
|829.520
|893.974
|
|-
|15/8-comma
| 153.387
|369.919
|523.306
|676.694
|830.081
|893.225
|
|-
|13/7-comma
|153.869
|369.197
|523.066
|676.934
|830.803
|892.263
|
|-
|24/13-comma
|154.165
|368.753
|522.918
|677.082
|831.247
|891.671
|
|-
|11/6-comma
|154.510
|368.235
|522.745
|677.255
|831.765
|890.980
|
|-
|20/11-comma
|154.918
|367.622
|522.541
|677.459
|832.378
|890.163
|
|-
|9/5-comma
|155.408
|366.888
|522.296
|677.704
|833.112
|889.183
|
|-
|25/14-comma
|155.793
|366.310
|522.103
|677.897
|833.690
|888.414
|
|-
|16/9-comma
|156.007
|365.989
|521.996
|678.004
|834.011
|887.986
|
|-
|23/13-comma
|156.237
|365.644
|521.881
|678.119
|834.356
|887.525
|
|-
|7/4-comma
|156.756
|678.378
|521.622
|364.867
|835.133
|886.489
|
|-
|19/11-comma
|157.632
|363.948
|521.316
|678.684
|836.052
|885.264
|
|-
|12/7-comma
|157.712
|363.423
|521.141
|678.859
|836.577
|884.564
|
|-
|17/10-comma
|158.103
|679.051
|520.949
|362.846
|837.154
|883.794
|
|-
|22/13-comma
|158.690
|362.535
|520.845
|679.155
|837.465
|883.380
|
|-
|5/3-comma
|159.001
|361.499
|520.500
|679.500
|838.501
|881.998
|
|-
|23/14-comma
|159.643
|360.536
|520.179
|679.821
|839.474
|880.715
|
|-
|18/11-comma
|159.818
|360.274
|520.091
|679.909
|839.726
|880.364
|
|-
|13/8-comma
|160.124
|359.814
|519.938
|680.062
|840.186
|879.753
|
|-
|ϕ-comma
|160.311
|359.533
|519.844
|680.156
|840.467
|879.377
|
|-
|21/13-comma
|160.383
|359.426
|519.809
|680.191
|840.574
|879.234
|
|-
|8/5-comma
|160.797
|358.804
|519.601
|680.399
|841.196
|878.405
|
|-
|19/12-comma
|161.246
|358.130
|519.377
|680.623
|841.870
|877.507
|
|-
|11/7-comma
|161.567
|357.649
|519.216
|680.784
|842.351
|876.855
|
|-
|14/9-comma
|161.995
|357.008
|519.003
|680.997
|842.922
|876.010
|
|-
|17/11-comma
|162.267
|356.599
|518.866
|681.134
|843.411
|875.466
|
|-
|20/13-comma
|162.456
|356.317
|518.772
|681.228
|843.683
|875.089
|
|-
|3/2-comma
|163.492
|354.762
|518.254
|681.746
|845.238
|873.016
|
|-
|19/13-comma
|164.528
|353.208
|517.736
|682.264
|846.792
|870.944
|
|-
|16/11-comma
|164.717
|352.925
|517.642
|682.358
|847.075
|870.567
|
|-
|13/9-comma
|164.989
|352.517
|517.506
|682.494
|847.483
|870.022
|
|-
|10/7-comma
|165.417
|351.875
|517.292
|682.718
|848.125
|869.167
|
|-
|17/12-comma
|165.737
|351.393
|517.131
|682.869
|848.607
|868.526
|
|-
|7/5-comma
|166.186
|350.720
|516.907
|682.093
|849.280
|867.627
|
|-
|18/13-comma
|166.600
|350.099
|516.700
|683.300
|849.901
|866.798
|
|-
|(ϕ+2)/(ϕ+1)-comma
|166.328
|349.991
|516.664
|683.336
|850.009
|866.655
|
|-
|11/8-comma
|166.860
|349.710
|516.570
|683.430
|850.290
|866.280
|
|-
|15/11-comma
|167.164
|349.251
|516.417
|683.583
|850.749
|865.667
|
|-
|19/14-comma
|167.341
|348.988
|516.329
|683.671
|851.012
|865.318
|
|-
|4/3-comma
|167.983
|348.026
|516.009
|683.991
|851.974
|864.034
|
|-
|17/13-comma
|168.674
|346.989
|515.663
|684.337
|853.011
|862.653
|
|-
|13/10-comma
|168.881
|346.679
|515.560
|684.440
|853.321
|862.238
|
|-
|9/7-comma
|169.266
|346.101
|515.367
|684.633
|853.899
|861.468
|
|-
|14/11-comma
|169.616
|345.576
|515.192
|684.808
|854.424
|860.768
|
|-
|5/4-comma
|170.228
|344.658
|514.886
|685.114
|855.342
|859.544
|
|-
|16/13-comma
|170.746
|343.880
|514.627
|685.373
|856.120
|858.507
|
|-
|11/9-comma
|170.977
|343.535
|514.512
|685.488
|856.465
|858.047
|
|-
|17/14-comma
|171.191
|343.214
|514.404
|685.596
|856.786
|857.619
|
|-
|6/5-comma
|171.576
|342.637
|514.212
|685.788
|857.363
|856.849
|
|-
|13/11-comma
|172.065
|341.902
|513.967
|686.033
|858.098
|855.869
|
|-
|7/6-comma
|172.474
|341.289
|513.763
|686.237
|858.711
|855.053
|
|-
|15/13-comma
|173.811
|340.771
|513.590
|686.410
|859.229
|854.362
|
|-
|8/7-comma
|173.115
|340.327
|513.422
|686.578
|859.673
|853.770
|
|-
|9/8-comma
|173.596
|339.605
|513.202
|686.798
|860.395
|852.807
|
|-
|10/9-comma
|173.971
|339.044
|513.015
|686.985
|860.956
|852.059
|
|-
|11/10-comma
|174.270
|338.595
|512.865
|687.135
|861.405
|851.469
|
|-
|12/11-comma
|174.515
|338.227
|512.742
|687.258
|861.773
|850.970
|
|-
|13/12-comma
|174.719
|337.921
|512.640
|687.360
|862.079
|850.562
|
|-
|14/13-comma
|174.892
|337.662
|512.554
|687.456
|862.378
|850.216
|
|-
|15/14-comma
|175.040
|337.440
|512.480
|687.520
|862.560
|849.920
|
|-
|1-comma
|176.965
|334.553
|511.518
|588.482
|865.447
|846.071
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 4-comma to 2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|4-comma
|258.178
|
|470.941
|
|
|212.824
|
|-
|27/7-comma
|256.181
|
|471.909
|
|
|215.728
|
|-
|23/6-comma
|255.858
|
|472.071
|
|
|216.212
|
|-
|19/5-comma
|255.407
|
|472.297
|
|
|216.890
|
|-
|15/4-comma
|254.769
|
|472.635
|
|
|217.906
|
|-
|26/7-comma
|254.243
|
|472.877
|
|
|218.632
|
|-
|11/3-comma
| 253.600
|
|473.200
|
|
|216.600
|
|-
|(2+ϕ)-comma
|252.940
|
|473.530
|
|
|220.589
|
|-
|18/5-comma
|252.696
|
|473.652
|
|
|220.956
|
|-
|25/7-comma
|252.309
|
|473.845
|
|
|221.536
|
|-
|7/2-comma
|251.341
|
|474.329
|
|
|222.988
|
|-
|24/7-comma
|250.373
|
|474.813
|
|
|224.440
|
|-
|17/5-comma
|249.986
|
|475.007
|
|
|225.021
|
|-
|10/3-comma
|249.083
|
|475.459
|
|
|226.376
|
|-
|23/7-comma
|248.437
|
|475.781
|
|
|227.344
|
|-
|13/4-comma
|247.953
|
|476.023
|
|
|228.070
|
|-
|16/5-comma
|247.258
|
|476.362
|
|
|229.087
|
|-
|19/6-comma
|246.824
|
|476.588
|
|
|229.764
|
|-
|22/7-comma
|246.501
|
|476.749
|
|
|230.248
|
|-
|3-comma
|244.565
|
|477.717
|
|
|233.152
|
|-
|20/7-comma
|242.629
|
|478.685
|
|
|236.056
|
|-
|17/6-comma
|242.307
|
|478.847
|
|
|236.540
|
|-
|14/5-comma
|241.855
|
|479.073
|
|
|237.218
|
|-
|11/4-comma
|241.177
|
|479.411
|
|
|238.234
|
|-
|19/7-comma
|240.693
|
|479.653
|
|
|238.960
|
|-
|8/3-comma
|240.048
|
|479.976
|
|
|239.928
|
|-
|13/5-comma
|239.145
|
|480.428
|
|
|241.283
|
|-
|18/7-comma
|238.757
|
|480.621
|
|
|241.864
|
|-
|5/2-comma
| 237.789
|
|481.105
|
|
|243.316
|
|-
|17/7-comma
|236.821
|
|481.589
|
|
|244.768
|
|-
|12/5-comma
|236.434
|
|481.783
|
|
|245.349
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|236.190
|
|481.905
|
|
|245.715
|
|-
|7/3-comma
|235.531
|
|482.235
|
|
|246.704
|
|-
|16/7-comma
|234.115
|
|482.557
|
|
|247.672
|
|-
|9/4-comma
|234.401
|
|482.799
|
|
|248.398
|
|-
|11/5-comma
|233.276
|
|483.183
|
|
|249.414
|
|-
|13/6-comma
|233.272
|
|483.364
|
|
|250.092
|
|-
|15/7-comma
|232.051
|
|483.525
|
|
|250.576
|
|-
|2-comma
|231.014
|
|484.493
|
|
|253.480
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 1-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|1-comma
|176.965
|
|511.518
|
|846.071
| 334.553, 865.447
|
|-
|13/14-comma
|178.890
|
|510.555
|
|842.221
|331.666, 868.334
|
|-
|12/13-comma
|179.037
|
|510.481
|
|841.925
| 331.444, 868.556
|
|-
|11/12-comma
|179.210
|
|510.395
|
|841.580
|331.185, 868.815
|
|-
|10/11-comma
| 179.414
|
| 510.293
|
|841.172
|330.879, 869.121
|
|-
|9/10-comma
|179.659
|
| 510.170
|
|840.682
|330.511, 869.489
|
|-
|8/9-comma
|179.959
|
|510.021
|
|840.083
|330.062, 869.038
|
|-
|7/8-comma
|180.333
|
|509.834
|
|839.334
|329.501, 870.499
|
|-
|6/7-comma
|180.814
|
|509.593
|
|838.372
|328.779, 871.221
|
|-
|11/13-comma
|181.110
|
|509.445
|
|837.780
|328.335, 871.665
|
|-
|5/6-comma
|181.455
|
|509.272
|
|837.089
|327.817, 872.193
|
|-
|9/11-comma
|181.864
|
|509.068
|
|836.272
|327.204, 872.796
|
|-
|4/5-comma
|182.354
|
|508.823
|
|835.293
|326.469, 873.531
|
|-
|11/14-comma
|182.739
|
|508.630
|
|834.523
|325.892, 874.108
|
|-
|7/9-comma
|182.952
|
|508.523
|
|834.095
| 325.571, 874.429
|
|-
|10/13-comma
|183.183
|
|508.408
|
|833.634
|325.226, 874.774
|
|-
|3/4-comma
|183.701
|
|508.150
|
|832.598
|324.449, 875.551
|
|-
|8/11-comma
|184.687
|
|507.843
|
|831.373
|323.530, 876.470
|
|-
|5/7-comma
|184.633
|
|507.638
|
|830.673
|323.005, 876.995
|
|-
|7/10-comma
|184.952
|
|507.476
|
|829.904
|322.428, 877.572
|
|-
|9/13-comma
|185.255
|
|507.372
|
|829.489
|322.117, 877.883
|
|-
|2/3-comma
|185.946
|
|507.027
|
|828.107
|321.080, 878.920
|
|-
|9/14-comma
|186.588
|
|506.706
|
|828.824
|320.118, 879.882
|
|-
|7/11-comma
|186.763
|
|506.619
|
|826.474
|319.856, 880.144
|
|-
|5/8-comma
|187.069
|
|506.465
|
|825.862
|319.396, 880.604
|
|-
|1/ϕ-comma
|187.257
|
|506.372
|
|825.486
|319.115, 880.885
|
|-
|8/13-comma
|187.320
|
|506.336
|
|825.344
|319.008, 880.992
|
|-
|3/5-comma
|187.743
|
|506.129
|
|824.514
|318.386, 881.614
|
|-
|7/12-comma
|188.194
|
|505.904
|
|823.616
|317.712, 882.288
|
|-
|4/7-comma
|188.512
|
|505.744
|
|822.975
|317.231, 882.769
|
|-
|5/9-comma
|188.940
|
|505.530
|
|822.119
|316.590, 883.410
|
|-
|6/11-comma
|189.213
|
|505.394
|
|821.575
|316.181, 883.891
|
|-
|7/13-comma
|189.401
|
|505.300
|
|821.198
|315.899, 884.101
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|190.437
|
|504.781
|
|819.125
|314.344, 885.656
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean hexachord tuning in universe
|-
|6/13-comma
|191.574
|
|504.263
|
|817.053
|312.790, 887.210
|
|-
|5/11-comma
|191.338
|
|504.169
|
|816.676
|312.507, 887.493
|
|-
|4/9-comma
|191.934
|
|504.033
|
|816.131
|312.099, 877.901
|
|-
|3/7-comma
|192.362
|
|503.819
|
|815.276
|311.457, 388.443
|
|-
|5/12-comma
|192.683
|
|503.659
|
|814.635
|310.976, 889.024
|
|-
|2/5-comma
|193.132
|
|503.434
|
|813.736
|310.302. 889.698
|Almost exactly meantone
|-
|5/13-comma
|193.546
|
|503.227
|
|812.907
|309.680, 890.320
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|193.618
|
|503.191
|
|812.764
| 309.573, 890.427
|
|-
|3/8-comma
|193.805
|
| 503.096
|
|812.389
|309.291, 890.709
|
|-
|4/11-comma
|194.112
|
|502.944
|
|811.776
|308.832, 891.168
|
|-
|5/14-comma
|194.287
|
|502.856
|
|811.427
|308.570, 891.430
|
|-
|1/3-comma
|194.928
|
|502.536
|
|810.144
|307.608, 892.392
|
|-
|4/13-comma
|195.619
|
|502.190
|
|808.762
|306.571, 893.429
|
|-
|3/10-comma
|195.174
|
|502.087
|
|808.347
|306.260, 893.740
|
|-
|2/7-comma
|196.211
|
|501.894
|
|807.577
|305.683, 894.317
|
|-
|3/11-comma
|196.561
|
|501.718
|
|806.877
|305.158, 894.842
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|197.174
|
|501.413
|
|805.653
|304.240, 895.760
|
|-
|3/13-comma
|197.692
|
|501.154
|
|804.616
|303.462, 896.538
|
|-
|2/9-comma
|197.922
|
|501.039
|
|804.155
|303.117, 896.883
|
|-
|3/14-comma
|198.136
|
|500.932
|
|803.728
|302.796, 897.204
|
|-
|1/5-comma
|198.521
|
|500.740
|
|802.958
|302.219, 897.781
|
|-
|2/11-comma
|199.011
|
|500.495
|
|801.978
|301.484, 898.516
|
|-
|1/6-comma
|199.419
|
|500.290
|
|801.162
|300.871, 899.129
|
|-
|2/13-comma
|199.765
|
|500.118
|
|800.471
|300.353, 899.647
|
|-
|1/7-comma
|200.061
|
|499.970
|
|799.879
|299.909, 900.091
|
|-
|1/8-comma
|200.542
|
| 499.729
|
|798.916
|299.187, 900.823
|
|-
|1/9-comma
|200.916
|
|499.542
|
|798.168
|298.626, 901.374
|
|-
|1/10-comma
|201.785
|
|499.392
|
|797.569
|298.177, 901.823
|
|-
|1/11-comma
|201.460
|
|499.270
|
|797.079
|297.810, 902.190
|
|-
|1/12-comma
|201.665
|
|499.168
|
|796.671
|297.503, 902.497
|
|-
|1/13-comma
|201.837
|
|499.081
|
|796.325
|297.244, 902.756
|
|-
|1/14-comma
|201.953
|
|499.007
|
|796.029
|297.022, 902.978
|
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|792.180
|294.135, 905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from 2-comma to Pythagorean
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|2-comma
|231.014
|
|484.493
|
|
|253.480
|
|-
|13/7-comma
|229.078
|
|485.461
|
|
|256.384
|
|-
|11/6-comma
|228.755
|
|485.623
|
|
|256.868
|
|-
|9/5-comma
|228.697
|
|485.848
|
|
|257.545
|
|-
| 7/4-comma
|227.626
|
|486.187
|
|
|258.562
|
|-
|12/7-comma
|227.142
|
|486.429
|
|
|259.288
|
|-
|5/3-comma
|226.496
|
|486.752
|
|
|260.253
|
|-
|ϕ-comma
|225.837
|
|487.081
|
|
|261.244
|
|-
|8/5-comma
|225.593
|
|487.204
|
|
|261.611
|
|-
|11/7-comma
|225.206
|
| 487.397
|
|
|262.192
|
|-
|3/2-comma
| 224.762
|
|487.881
|
|
|263.644
|
|-
|10/7-comma
|223.270
|
|488.365
|
|
|265.096
|
|-
|7/5-comma
|222.882
|
|488.559
|
|
|265.676
|
|-
|4/3-comma
|221.979
|
|489.010
|
|
|267.031
|
|-
|9/7-comma
|221.334
|
|489.333
|
|
|267.999
|
|-
|5/4-comma
|220.850
|
|489.575
|
|
|268.725
|
|-
| 6/5-comma
|220.172
|
|489.914
|
|
|269.742
|
|-
|7/6-comma
|219.720
|
|490.140
|
|
|270.419
|
|-
|8/7-comma
|219.398
|
|490.301
|
|
|270.903
|
|-
|1-comma
|217.538
|
|491.269
|
|
| 273.807
|
|-
|6/7-comma
|215.526
|
|492.237
|
|
|276.711
|
|-
|5/6-comma
|215.203
|
|492.398
|
|
|277.195
|
|-
| 4/5-comma
|214.751
|
| 492.624
|
|
|277.873
|
|-
|3/4-comma
|214.926
|
|492.963
|
|
|278.889
|
|-
|5/7-comma
|213.590
|
|493.205
|
|
|279.615
|
|-
|2/3-comma
|212.945
|
|493.528
|
|
|280.583
|
|-
|3/5-comma
|212.041
|
|493.979
|
|
|281.938
|
|-
|4/7-comma
|211.346
|
|494.173
|
|
|282.519
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|210.686
|
|494.657
|
|
|283.971
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|209.718
|
|495.141
|
|
|285.423
|
|-
|2/5-comma
|209.331
|
|495.335
|
|
|286.004
|
|-
|1/(ϕ+1)-comma
|209.086
|
|495.457
|
|
|286.371
|
|-
|1/3-comma
|208.573
|
|495.786
|
|
|287.359
|
|-
|2/7-comma
|207.782
|
|496.109
|
|
|289.372
|
|-
|1/4-comma
|207.293
|
|496.351
|
|
|289.053
|
|-
|1/5-comma
|206.620
|
|496.690
|
|
|290.069
|
|-
|1/6-comma
|206.169
|
|496.916
|
|
|290.747
|
|-
|1/7-comma
|205.846
|
|497.077
|
|
|291.231
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -1-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|792.180
|294.135, 905.865
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|205.835
|
|497.083
|
|788.331
|291.248, 908.752
|
|-
| -1/13-comma
|205.983
|
|497.009
|
|788.035
|291.026, 908.974
|
|-
|<nowiki>-1/12-comma</nowiki>
|206.155
|
|496.922
|
|787.689
|290.767, 909.233
|
|-
|<nowiki>-1/11-comma</nowiki>
|206.360
|
|496.820
|
|787.280
|290.460, 909.540
|
|-
| -1/10-comma
|206.605
|
|496.698
|
|786.791
|290.093, 909.907
|
|-
| -1/9-comma
|206.904
|
|496.548
|
|786.192
|289.644, 910.356
|
|-
| -1/8-comma
|207.278
|
|496.361
|
|785.444
|289.083, 910.917
|
|-
| -1/7-comma
|207.759
|
|496.120
|
|784.481
|288.361, 911.639
|
|-
| -2/13-comma
|208.055
|
|495.972
|
|783.889
|287.917, 912.083
|
|-
| -1/6-comma
|208.401
|
|495.800
|
|783.198
|287.399, 912.601
|
|-
|<nowiki>-2/11-comma</nowiki>
|208.809
|
|495.595
|
|782.382
|286.786, 913.214
|
|-
| -1/5-comma
|209.299
|
|495.350
|
|781.401
|286.051, 913.949
|
|-
| -3/14-comma
|209.684
|
|495.158
|
|780.632
|285.474, 914.526
|
|-
| -2/9-comma
|209.898
|
|495.051
|
|780.204
|285.153, 914.847
|
|-
| -3/13-comma
|210.128
|
|494.936
|
|779.744
|284.808, 915.192
|
|-
| -1/4-comma
|210.646
|
|494.677
|
|778.707
|284.030, 915.970
|
|-
|<nowiki>-3/11-comma</nowiki>
|211.259
|
|494.371
|
|777.482
|283.111, 916.889
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|211.609
|
|494.196
|
|776.783
|282.587, 917.413
|
|-
| -3/10-comma
|211.994
|
|494.003
|
|776.013
|282.010, 917.990
|
|-
| -4/13-comma
|212.799
|
|493.900
|
|775.598
|281.699, 918.301
|
|-
| -1/3-comma
|212.892
|
|493.554
|
|774.216
|280.662, 919.338
|
|-
| -5/14-comma
|213.537
|
|493.233
|
|772.933
|279.700, 920.300
|
|-
|<nowiki>-4/11-comma</nowiki>
|213.709
|
|493.146
|
|772.583
|279.437, 920.563
|
|-
| -3/8-comma
|214.014
|
|492.993
|
|771.971
|278.979, 921.021
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|214.203
|
|492.899
|
|771.596
|278.697, 921.303
|
|-
| -5/13-comma
|214.274
|
|492.863
|
|771.453
|278.590, 921.410
|
|-
| -2/5-comma
|214.688
|
|492.656
|
|770.624
|277.968, 922.032
|
|-
|<nowiki>-5/12-comma</nowiki>
|215.137
|
|492.431
|
|769.725
|277.294, 922.706
|
|-
| -3/7-comma
|215.458
|
|492.271
|
|769.084
|276.813, 923.187
|
|-
| -4/9-comma
|215.886
|
|492.057
|
|768.229
|276.171, 923.829
|
|-
|<nowiki>-5/11-comma</nowiki>
|216.158
|
|491.921
|
|767.684
|275.763, 924.237
|
|-
| -6/13-comma
|216.346
|
|491.827
|
|767.307
|275.480, 924.520
|
|-
| -1/2-comma
|217.383
|
|491.309
|
|765.235
|273.926, 926.274
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161. Almost quarter-comma Archytas tuning
|-
| -7/13-comma
|218.419
|
|490.790
|
|763.161
|272.371, 927.629
|
|-
|<nowiki>-6/11-comma</nowiki>
|218.607
|
|490.696
|
|762.785
|272.089, 927.911
|
|-
| -5/9-comma
|218.880
|
|490.560
|
|762.241
|271.680, 928.320
|
|-
| -4/7-comma
|219.307
|
|490.346
|
|761.385
|271.039, 928.951
|
|-
|<nowiki>-7/12-comma</nowiki>
|219.629
|
|490.186
|
|760.744
|270.558, 929.442
|
|-
| -3/5-comma
|220.077
|
|489.961
|
|759.846
|269.884, 930.116
|
|-
| -8/13-comma
|220.492
|
|489.754
|
|759.016
|269.262, 930.438
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|220.563
|
|489.716
|
|758.874
|269.155, 930.845
|
|-
| -5/8-comma
|220.751
|
|489.625
|
|758.498
|268.874, 931.124
|
|-
|<nowiki>-7/11-comma</nowiki>
|221.057
|
|489.471
|
|757.886
|268.414, 931.586
|
|-
| -9/14-comma
|221.232
|
|489.384
|
|757.536
|268.152, 931.848
|
|-
| -2/3-comma
|221.874
|
|489.063
|
|756.253
|267.190, 932.810
|
|-
| -9/13-comma
|222.565
|
|488.718
|
|754.871
|266.153, 933.847
|
|-
| -7/10-comma
|222.772
|
|488.614
|
|754.456
|265.842, 934.158
|
|-
| -5/7-comma
|223.157
|
|488.422
|
|753.687
|265.265, 934.935
|
|-
|<nowiki>-8/11-comma</nowiki>
|223.507
|
|488.247
|
|752.987
|264.740, 935.260
|
|-
| -3/4-comma
|224.119
|
|487.940
|
|751.762
|263.821, 936.189
|
|-
| -10/13-comma
|224.637
|
|487.681
|
|750.726
|263.044, 936.956
|
|-
| -7/9-comma
|224.868
|
|487.566
|
|750.265
|262.698, 937.302
|
|-
| -11/14-comma
|225.081
|
|487.459
|
|749.837
|262.378, 937.622
|
|-
| -4/5-comma
|225.466
|
|487.267
|
|749.067
|261.801, 938.199
|
|-
|<nowiki>-9/11-comma</nowiki>
|225.957
|
|487.022
|
|748.088
|261.066, 938.934
|
|-
| -5/6-comma
|226.365
|
|486.818
|
|747.271
|260.453, 939.447
|
|-
| -11/13-comma
|226.710
|
|486.645
|
|746.580
|259.935, 940.065
|
|-
| -6/7-comma
|227.006
|
|486.497
|
|745.988
|259.491, 940.509
|
|-
| -7/8-comma
|227.487
|
|486.256
|
|745.026
|258.769, 941.231
|
|-
| -8/9-comma
|227.861
|
|486.069
|
|744.277
|258.208, 941.792
|
|-
| -9/10-comma
|228.161
|
|485.920
|
|743.678
|257.759, 942.241
|
|-
|<nowiki>-10/11-comma</nowiki>
|228.406
|
|485.797
|
|743.188
|257.391, 942.609
|
|-
| -11/12-comma
|228.610
|
|485.695
|
|
|257.085
|
|-
| -12/13-comma
|228.783
|
|485.609
|
|
|256.826
|
|-
| -13/14-comma
|228.931
|
|485.535
|
|
|256.604
|
|-
|-
| -1-comma
|230.855
|
|484.752
|
|
|253.717
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from Pythagorean to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
|[[Pythagorean tuning]]
|203.910
|
|498.045
|
|
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|201.974
|
|499.013
|
|
|297.039
|
|-
| -1/6-comma
|201.652
|
|499.174
|
|
|297.523
|
|-
| -1/5-comma
|201.200
|
|499.400
|
|
|298.201
|
|-
| -1/4-comma
|200.522
|
|499.739
|
|
|299.217
|
|-
| -2/7-comma
|200.038
|
|499.981
|
|
|299.942
|
|-
| -1/3-comma
|199.393
|
|500.303
|
|
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|198.734
|
|500.633
|
|
|301.900
|
|-
| -2/5-comma
|198.499
|
|500.755
|
|
|302.266
|
|-
| -3/7-comma
|198.102
|
|500.949
|
|
|302.847
|
|-
| -1/2-comma
|197.134
|
|501.433
|
|
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|196.166
|
|501.917
|
|
|305.751
|
|-
| -3/5-comma
|195.779
|
|502.111
|
|
|306.332
|
|-
| -2/3-comma
|194.876
|
|502.562
|
|
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|194.230
|
|502.885
|
|
|308.655
|
|-
| -4/5-comma
|193.069
|
|503.466
|
|
|310.397
|
|-
| -5/6-comma
|192.617
|
|503.692
|
|
|311.075
|
|-
|-
| -6/7-comma
|192.294
|
|503.853
|
|
|311.556
|
|-
| -1-comma
|190.352
|
|504.821
|
|
|314.463
|
|-
| -8/7-comma
|188.422
|
|505.789
|
|
|317.367
|
|-
| -7/6-comma
|188.100
|
|505.950
|
|
|317.851
|
|-
| -6/5-comma
|187.648
|
|506.176
|
|
|318.528
|
|-
| -5/4-comma
|186.970
|
|506.515
|
|
|319.545
|
|-
| -9/7-comma
|186.486
|
|506.757
|
|
|320.271
|
|-
| -4/3-comma
|185.841
|
|507.080
|
|
|321.239
|
|-
| -7/5-comma
|184.937
|
|507.531
|
|
|322.594
|
|-
| -10/7-comma
|184.550
|
|507.725
|
|
|323.174
|
|-
| -3/2-comma
|183.582
|
|508.209
|
|
|324.626
|
|-
| -11/7-comma
|182.614
|
|508.693
|
|
|326.078
|
|-
| -8/5-comma
|182.228
|
|508.886
|
|
|326.659
|
|-
| -ϕ-comma
|181.983
|
|509.009
|
|
|327.026
|
|-
| -5/3-comma
|181.324
|
|509.338
|
|
|328.014
|
|-
| -12/7-comma
|180.678
|
|509.661
|
|
|328.982
|
|-
| -7/4-comma
|180.194
|
|509.903
|
|
|329.708
|
|-
| -9/5-comma
|179.517
|
|510.242
|
|
|330.725
|
|-
| -11/6-comma
|179.065
|
|510.467
|
|
|331.402
|
|-
| -13/7-comma
|178.742
|
|510.629
|
|
|331.886
|
|-
| -2-comma
|176.807
|
|511.597
|
|
|334.790
|
|}
===Beyond Negative harmony theory-defined mean hexachord (most often approached as superdiatonic and oneirotonic)===
===Tempering out [[129/128]]===
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -1-comma to -2-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!minor
!major
!
|-
| -1-comma
|230.855
|
|484.752
|
|
|253.717
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|232.780
|
|483.610
|
|
|250.830
|
|-
| -14/13-comma
|232.928
|
|483.536
|
|
|250.608
|
|-
| -13/12-comma
|233.101
|
|483.450
|
|
|250.349
|
|-
| -12/11-comma
|233.305
|
|483.348
|
|
|250.043
|
|-
| -11/10-comma
|233.550
|
|483.225
|
|
|249.675
|
|-
| -10/9-comma
|233.151
|
|483.075
|
|
|249.226
|
|-
| -9/8-comma
|234.234
|
|482.888
|
|
|248.665
|
|-
| -8/7-comma
|234.295
|
|482.648
|
|
|247.943
|
|-
| -15/13-comma
|235.001
|
|482.500
|
|
|247.499
|
|-
| -7/6-comma
|235.346
|
|482.327
|
|
|246.981
|
|-
| -13/11-comma
|235.755
|
|482.123
|
|
|246.368
|
|-
| -6/5-comma
|236.244
|
|481.878
|
|
|245.633
|
|-
| -17/14-comma
|236.629
|
|481.685
|
|
|245.056
|
|-
| -11/9-comma
|236.843
|
|481.578
|
|
|244.735
|
|-
| -16/13-comma
|237.926
|
|481.463
|
|
|244.390
|
|-
| -5/4-comma
|237.592
|
|481.204
|
|
|243.612
|
|-
| -14/11-comma
|238.204
|
|480.898
|
|
|242.694
|
|-
| -9/7-comma
|238.554
|
|480.723
|
|
|242.169
|
|-
| -13/10-comma
|238.939
|
|480.530
|
|
|241.591
|
|-
| -17/13-comma
|239.146
|
|480.427
|
|
|241.280
|
|-
| -4/3-comma
|239.837
|
|480.081
|
|
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|240.479
|
|479.761
|
|
|239.282
|
|-
| -15/11-comma
|240.634
|
|479.673
|
|
|239.019
|
|-
| -11/8-comma
|240.960
|
|479.520
|
|
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|241.148
|
|479.426
|
|
|238.279
|
|-
| -18/13-comma
|241.219
|
|479.390
|
|
|238.171
|
|-
| -7/5-comma
|241.634
|
|479.183
|
|
|237.550
|
|-
| -17/12-comma
|242.917
|
|478.959
|
|
|236.876
|
|-
| -10/7-comma
|242.403
|
|478.798
|
|
|236.395
|
|-
| -13/9-comma
|242.831
|
|478.584
|
|
|235.753
|
|-
| -16/11-comma
|243.103
|
|478.448
|
|
|235.345
|
|-
| -19/13-comma
|243.708
|
|478.354
|
|
|235.062
|
|-
| -3/2-comma
|244.328
|
|477.836
|
|
|233.508
|
|-
| -20/13-comma
|245.344
|
|477.318
|
|
|231.953
|Close to [[93edo]]
|-
| -17/11-comma
|245.553
|
|477.224
|
|
|231.671
|Close to [[88edo]]
|-
| -14/9-comma
|245.825
|
|477.087
|
|
|231.262
|Close to [[83edo]]
|-
| -11/7-comma
|246.747
|
|476.873
|
|
|230.621
|Close to [[78edo]]
|-
| -19/12-comma
|246.426
|
|476.713
|
|
|230.140
|Close to [[73edo]]
|-
| -8/5-comma
|247.023
|
|476.489
|
|
|229.466
|Close to [[68edo]].
|-
| -21/13-comma
|247.437
|
|476.281
|
|
|228.844
|
|-
| -ϕ-comma
|247.491
|
|476.246
|
|
|228.737
|
|-
| -13/8-comma
|247.696
|
|476.152
|
|
|228.456
|Close to [[63edo]]
|-
| -18/11-comma
|248.002
|
|475.999
|
|
|227.996
|
|-
| -23/14-comma
|248.823
|
|475.911
|
|
|227.734
|Close to [[58edo]]
|-
| -5/3-comma
|248.819
|
|475.590
|
|
|226.771
|Close to [[53edo]]
|-
| -22/13-comma
|249.510
|
|475.245
|
|
|225.735
|
|-
| -17/10-comma
|249.717
|
|475.141
|
|
|225.424
|
|-
| -12/7-comma
|250.105
|
|474.949
|
|
|224.847
|Close to [[48edo]]
|-
| -19/11-comma
|250.552
|
|474.774
|
|
|224.322
|
|-
| -7/4-comma
|251.064
|
|474.468
|
|
|223.403
|Close to [[43edo]]
|-
| -23/13-comma
|251.583
|
|474.209
|
|
|222.626
|
|-
| -16/9-comma
|251.823
|
|474.094
|
|
|222.281
|
|-
| -25/14-comma
|252.027
|
|473.987
|
|
|221.960
|
|-
| -9/5-comma
|252.412
|
|473.794
|
|
|221.382
|Close to [[38edo]]
|-
| -20/11-comma
|252.912
|
|473.549
|
|
|220.648
|
|-
| -11/6-comma
|253.610
|
|473.345
|
|
|220.035
|
|-
| -24/13-comma
|253.345
|
|473.172
|
|
|219.517
|
|-
| -13/7-comma
|253.951
|
|473.924
|
|
|219.073
|
|-
| -15/8-comma
|254.433
|
|472.784
|
|
|218.351
|Close to [[33edo]]
|-
| -17/9-comma
|254.807
|
|472.597
|
|
|217.790
|
|-
| -19/10-comma
|255.106
|
|472.447
|
|
|217.341
|
|-
| -21/11-comma
|255.351
|
|472.324
|
|
|216.973
|
|-
| -23/12-comma
|255.555
|
|472.222
|
|
|216.667
|
|-
| -25/13-comma
|255.728
|
|472.135
|
|
|216.408
|
|-
| -27/14-comma
|255.876
|
|472.052
|
|
|216.186
|
|-
|-
| -2-comma
|258.801
|
|471.100
|
|
|213.299
|Close to [[28edo]]
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean hexachord tunings from -2 to -4-comma
!Mean hexachord temperament
!Tone
!third
! colspan="2" |g (cents)
! colspan="2" |sixth
!Comments
|-
!
!
!
!Fourth
!Fifth
!major
!minor
!
|-
| -2-comma
|176.807
|334.790
|511.597
|688.403
|865.210
|846.387
|
|-
| -15/7-comma
|174.870
|337.694
|512.565
|687.435
|862.306
|850.258
|
|-
| -13/6-comma
|174.548
|338.178
|512.726
|687.274
|861.822
|850.904
|
|-
| -11/5-comma
|174.096
|338.856
|512.952
|687.048
|861.144
|851.808
|
|-
| -9/4-comma
|173.419
|339.872
|513.291
|686.709
|860.128
|853.163
|
|-
| -16/7-comma
|172.935
|340.598
|513.533
|686.467
|859.402
|854.131
|
|-
| -7/3-comma
|172.289
|341.566
|513.855
|686.145
|858.434
|855.422
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|171.630
|342.555
|514.185
|685.815
|857.445
|856.740
|
|-
| -12/5-comma
|171.386
|342.921
|514.307
|685.693
|857.079
|857.228
|Close to [[7edo]].
|-
| -17/7-comma
|170.999
|343.502
|514.501
|685.499
|856.498
|858.003
|
|-
| -5/2-comma
|170.031
|344.954
|514.984
|685.016
|855.046
|859.939
|
|-
| -18/7-comma
|169.063
|346.406
|515.469
|684.531
|853.594
|861.878
|
|-
| -13/5-comma
|168.675
|346.987
|515.662
|684.378
|853.013
|862.649
|
|-
| -8/3-comma
|167.772
|348.342
|516.114
|683.886
|851.658
|864.456
|
|-
| -19/7-comma
|167.167
|349.310
|516.437
|683.563
|850.490
|865.747
|
|-
| -11/4-comma
|166.643
|350.034
|516.679
|683.321
|849.966
|866.715
|
|-
| -14/5-comma
|165.965
|351.052
|517.017
|682.983
|848.948
|868.070
|Very close to [[6ed6]]
|-
| -17/6-comma
|165.513
|351.730
|517.243
|682.757
|848.270
|868.973
|
|-
|-
| -20/7-comma
|165.191
|352.214
|517.404
|682.596
|847.786
|869.619
|
|-
| -3-comma
|163.255
|355.118
|518.373
|681.727
|844.882
|873.491
|
|-
| -22/7-comma
|161.389
|358.022
|519.341
|680.362
|841.978
|877.362
|
|-
| -19/6-comma
|160.996
|358.501
|519.502
|680.498
|841.499
|878.008
|
|-
| -16/5-comma
|160.544
|359.183
|519.728
|680.278
|840.817
|878.911
|
|-
| -13/4-comma
|159.867
|360.200
|520.067
|679.933
|839.800
|880.266
|Close to [[30edo]]
|-
| -23/7-comma
|159.383
|360.926
|520.309
|679.691
|839.074
|881.234
|Close to [[83edo]]
|-
| -10/3-comma
|158.737
|361.894
|520.631
|679.369
|838.116
|882.525
|Close to [[53edo]]
|-
| -17/5-comma
|157.834
|363.249
|521.083
|678.917
|836.751
|884.332
|Close to [[76edo]]
|-
| -24/7-comma
|157.447
|363.830
|521.277
|678.723
|836.170
|885.106
|Close to [[99edo]]
|-
| -7/2-comma
|156.479
|365.282
|521.761
|678.239
|834.718
|887.042
|Close to [[23edo]]
|-
| -25/7-comma
|155.511
|366.734
|522.245
|677.755
|833.266
|888.978
|
|-
| -18/5-comma
|155.124
|367.315
|522.438
|677.562
|832.685
|889.753
|Close to [[85edo]]
|-
| -(ϕ+2)-comma
|154.879
|367.681
|522.560
|677.440
|832.319
|890.241
|Close to [[62edo]]
|-
| -11/3-comma
|154.220
|368.670
|522.890
|677.110
|831.330
|891.560
|
|-
| -26/7-comma
|153.575
|369.638
|523.213
|676.787
|830.213
|892.850
|Close to [[39edo]]
|-
| -15/4-comma
|153.091
|370.364
|523.455
|676.545
|829.636
|893.818
|Close to [[94edo]]
|-
| -19/5-comma
|152.433
|371.380
|523.793
|676.217
|828.620
|895.173
|Close to [[55edo]]
|-
| -23/6-comma
|151.962
|372.058
|524.020
|675.980
|827.942
|896.077
|Close to [[71edo]]
|-
| -27/7-comma
|151.639
|372.542
|524.181
|675.819
|827.458
|896.722
|Close to [[87edo]]
|-
| -4-comma
|149.703
|375.446
|525.149
|674.851
|824.554
|900.594
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-20T03:09:37Z

Moremajorthanmajor: /* The table */

{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor tetrachord (root-whole tone-minor third-tempered fourth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with more complex thirds, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 1-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|2-comma
|524.990
|150.019
|374.971
|
|-
|13/14-comma
|524.028
|151.954
|372.084
|
|-
|25/13-comma
|523.954
|152.092
|371.862
|
|-
|23/12-comma
|523.868
|152.265
|371.603
|
|-
|21/11-comma
|523.766
|152.469
|371.297
|
|-
|19/10-comma
|523.643
|152.286
|370.929
|
|-
|17/9-comma
|523.493
|153.013
|370.480
|
|-
|15/8-comma
|523.306
| 153.387
|369.919
|
|-
|13/7-comma
|523.066
|153.869
|369.197
|
|-
|24/13-comma
|522.918
|154.165
|368.753
|
|-
|11/6-comma
|522.745
|154.510
|368.235
|
|-
|20/11-comma
|522.541
|154.918
|367.622
|
|-
|9/5-comma
|522.296
|155.592
|366.888
|
|-
|25/14-comma
|522.103
|155.207
|366.310
|
|-
|16/9-comma
|521.996
|156.007
|365.989
|
|-
|23/13-comma
|521.881
|156.237
|365.644
|
|-
|7/4-comma
|521.622
|156.756
|364.867
|
|-
|19/11-comma
|521.316
|157.632
|363.948
|
|-
|12/7-comma
|521.141
|157.712
|363.423
|
|-
|17/10-comma
|520.949
|158.103
|362.846
|
|-
|22/13-comma
|520.845
|158.690
|362.535
|
|-
|5/3-comma
|520.500
|159.001
|361.499
|
|-
|23/14-comma
|520.179
|159.643
|360.536
|
|-
|18/11-comma
|520.091
|159.818
|360.274
|
|-
|13/8-comma
|519.938
|160.124
|359.814
|
|-
|ϕ-comma
|519.844
|160.311
|359.533
|
|-
|21/13-comma
|519.809
|160.383
|359.426
|
|-
|8/5-comma
|519.601
|160.797
|358.804
|
|-
|19/12-comma
|519.377
|161.246
|358.130
|
|-
|11/7-comma
|519.216
|161.567
|357.649
|
|-
|14/9-comma
|519.003
|161.995
|357.008
|
|-
|17/11-comma
|518.866
|162.267
|356.599
|
|-
|20/13-comma
|518.772
|162.456
|356.317
|
|-
|3/2-comma
|518.254
|163.492
|354.762
|
|-
|19/13-comma
|517.736
|164.528
|353.208
|
|-
|16/11-comma
|517.642
|164.717
|352.925
|
|-
|13/9-comma
|517.506
|164.989
|352.517
|
|-
|10/7-comma
|517.292
|165.417
|351.875
|
|-
|17/12-comma
|517.131
|165.737
|351.393
|
|-
|7/5-comma
|516.907
|166.186
|350.720
|
|-
|18/13-comma
|516.700
|166.600
|350.099
|
|-
|(ϕ+2)/(ϕ+1)-comma
|516.664
|166.328
|349.991
|
|-
|11/8-comma
|516.570
|166.860
|349.710
|
|-
|15/11-comma
|516.417
|167.164
|349.251
|
|-
|19/14-comma
|516.329
|167.341
|348.988
|
|-
|4/3-comma
|516.009
|167.983
|348.026
|
|-
|17/13-comma
|515.663
|168.674
|346.989
|
|-
|13/10-comma
|515.560
|168.881
|346.679
|
|-
|9/7-comma
|515.367
|169.266
|346.101
|
|-
|14/11-comma
|515.192
|169.616
|345.576
|
|-
|5/4-comma
|514.886
|170.228
|344.658
|
|-
|16/13-comma
|514.627
|170.746
|343.880
|
|-
|11/9-comma
|514.512
|170.977
|343.535
|
|-
|17/14-comma
|514.404
|171.191
|343.214
|
|-
|6/5-comma
|514.212
|171.576
|342.637
|
|-
|13/11-comma
|513.967
|172.065
|341.902
|
|-
|7/6-comma
|513.763
|172.474
|341.289
|
|-
|15/13-comma
|513.590
|173.811
|340.771
|
|-
|8/7-comma
|513.422
|173.115
|340.327
|
|-
|9/8-comma
|513.202
|173.596
|339.605
|
|-
|10/9-comma
|513.015
|173.971
|339.044
|
|-
|11/10-comma
|512.865
|174.270
|338.595
|
|-
|11/11-comma
|512.742
|174.515
|338.227
|
|-
|13/12-comma
|512.640
|174.719
|337.921
|
|-
|14/13-comma
|512.554
|174.892
|337.662
|
|-
|15/14-comma
|512.480
|175.040
|337.440
|
|-
|1-comma
|511.518
|176.965
|334.553
|
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 2-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|4-comma
|470.941
|258.178
|212.824
|
|-
|27/7-comma
|471.909
|256.181
|215.728
|
|-
|23/6-comma
|472.071
|255.858
|216.212
|
|-
|19/5-comma
|472.297
|255.407
|216.890
|
|-
|15/4-comma
|472.635
|254.769
|217.906
|
|-
|26/7-comma
|472.877
|254.243
|218.632
|
|-
|11/3-comma
|473.200
| 253.600
|216.600
|
|-
|(2+ϕ)-comma
|473.530
|252.940
|220.589
|
|-
|18/5-comma
|473.652
|252.696
|220.956
|
|-
|25/7-comma
|473.845
|252.309
|221.536
|
|-
|7/2-comma
|474.329
|251.341
|222.988
|
|-
|24/7-comma
|474.813
|250.373
|224.440
|
|-
|17/5-comma
|475.007
|249.986
|225.021
|
|-
|10/3-comma
|475.459
|249.083
|226.376
|
|-
|23/7-comma
|475.781
|248.437
|227.344
|
|-
|13/4-comma
|476.023
|247.953
|228.070
|
|-
|16/5-comma
|476.362
|247.258
|229.087
|
|-
|19/6-comma
|476.588
|246.824
|229.764
|
|-
|22/7-comma
|476.749
|246.501
|230.248
|
|-
|3-comma
|477.717
|244.565
|233.152
|
|-
|20/7-comma
|478.685
|242.629
|236.056
|
|-
|17/6-comma
|478.847
|242.307
|236.540
|
|-
|14/5-comma
|479.073
|241.855
|237.218
|
|-
|11/4-comma
|479.411
|241.177
|238.234
|
|-
|19/7-comma
|479.653
|240.693
|238.960
|
|-
|8/3-comma
|479.976
|240.048
|239.928
|
|-
|13/5-comma
|480.428
|239.145
|241.283
|
|-
|18/7-comma
|480.621
|238.757
|241.864
|
|-
|5/2-comma
|481.105
| 237.789
|243.316
|
|-
|17/7-comma
|481.589
|236.821
|244.768
|
|-
|12/5-comma
|481.783
|236.434
|245.349
|
|-
|(2ϕ+3)/(ϕ+1)-comma
|481.905
|236.190
|245.715
|
|-
|7/3-comma
|482.235
|235.531
|246.704
|
|-
|16/7-comma
|482.557
|234.115
|247.672
|
|-
|9/4-comma
|482.799
|234.401
|248.398
|
|-
|11/5-comma
|483.183
|233.276
|249.414
|
|-
|13/6-comma
|483.364
|233.272
|250.092
|
|-
|15/7-comma
|483.525
|232.051
|250.576
|
|-
|2-comma
|484.493
|231.014
|253.480
|
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 1-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
! third
!Comments
|-
|1-comma
|511.518
|176.965
| 334.553
|
|-
|13/14-comma
|510.555
|178.890
|331.666
|
|-
|12/13-comma
|510.481
|179.037
| 331.444
|
|-
|11/12-comma
|510.395
|179.210
|331.185
|
|-
|10/11-comma
| 510.293
| 179.414
|330.879
|
|-
|9/10-comma
| 510.170
|179.659
|330.511
|
|-
|8/9-comma
|510.021
|179.959
|330.062
|
|-
|7/8-comma
|509.834
|180.333
|329.501
|
|-
|6/7-comma
|509.593
|180.814
|328.779
|
|-
|11/13-comma
|509.445
|181.110
|328.335
|
|-
|5/6-comma
|509.272
|181.455
|327.817
|
|-
|9/11-comma
|509.068
|181.864
|327.204
|
|-
|4/5-comma
|508.823
|182.354
|326.469
|
|-
|11/14-comma
|508.630
|182.739
|325.892
|
|-
|7/9-comma
|508.523
|182.952
| 325.571
|
|-
|10/13-comma
|508.408
|183.183
|325.226
|
|-
|3/4-comma
|508.150
|183.701
|324.449
|
|-
|8/11-comma
|507.843
|184.687
|323.530
|
|-
|5/7-comma
|507.638
|184.633
|323.005
|
|-
|7/10-comma
|507.476
|184.952
|322.428
|
|-
|9/13-comma
|507.372
|185.255
|322.117
|
|-
|2/3-comma
|507.027
|185.946
|321.080
|
|-
|9/14-comma
|506.706
|186.588
|320.118
|
|-
|7/11-comma
|506.619
|186.763
|319.856
|
|-
|5/8-comma
|506.465
|187.069
|319.396
|
|-
|1/ϕ-comma
|506.372
|187.257
|319.115
|
|-
|8/13-comma
|506.336
|187.320
|319.008
|
|-
|3/5-comma
|506.129
|187.743
|318.386
|
|-
|7/12-comma
|505.904
|188.194
|317.712
|
|-
|4/7-comma
|505.744
|188.512
|317.231
|
|-
|5/9-comma
|505.530
|188.940
|316.590
|
|-
|6/11-comma
|505.394
|189.213
|316.181
|
|-
|7/13-comma
|505.300
|189.401
|315.899
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|504.781
|190.437
|314.344
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|6/13-comma
|504.263
|191.574
|312.790
|
|-
|5/11-comma
|504.169
|191.338
|312.507
|
|-
|4/9-comma
|504.033
|191.934
|312.099
|
|-
|3/7-comma
|503.819
|192.362
|311.457
|
|-
|5/12-comma
|503.659
|192.683
|310.976
|
|-
|2/5-comma
|503.434
|193.132
|310.302
|Almost exactly meantone
|-
|5/13-comma
|503.227
|193.546
|309.680
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|503.191
|193.618
| 309.573
|
|-
|3/8-comma
| 503.096
|193.805
|309.291
|
|-
|4/11-comma
|502.944
|194.112
|308.832
|
|-
|5/14-comma
|502.856
|194.287
|308.570
|
|-
|1/3-comma
|502.536
|194.928
|307.608
|
|-
|4/13-comma
|502.190
|195.619
|306.571
|
|-
|3/10-comma
|502.087
|195.174
|306.260
|
|-
|2/7-comma
|501.894
|196.211
|305.683
|
|-
|3/11-comma
|501.718
|196.561
|305.158
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|501.413
|197.174
|304.240
|
|-
|3/13-comma
|501.154
|197.692
|303.462
|
|-
|2/9-comma
|501.039
|197.922
|303.117
|
|-
|3/14-comma
|500.932
|198.136
|302.796
|
|-
|1/5-comma
|500.740
|198.521
|302.219
|
|-
|2/11-comma
|500.495
|199.011
|301.484
|
|-
|1/6-comma
|500.290
|199.419
|300.871
|
|-
|2/13-comma
|500.118
|199.765
|300.353
|
|-
|1/7-comma
|499.970
|200.061
|299.909
|
|-
|1/8-comma
| 499.729
|200.542
|299.187
|
|-
|1/9-comma
|499.542
|200.916
|298.626
|
|-
|1/10-comma
|499.392
|201.785
|298.177
|
|-
|1/11-comma
|499.270
|201.460
|297.810
|
|-
|1/12-comma
|499.168
|201.665
|297.503
|
|-
|1/13-comma
|499.081
|201.837
|297.244
|
|-
|1/14-comma
|499.007
|201.953
|297.022
|
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 2-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|2-comma
|484.493
|231.014
|253.480
|
|-
|13/7-comma
|485.461
|229.078
|256.384
|
|-
|11/6-comma
|485.623
|228.755
|256.868
|
|-
|9/5-comma
|485.848
|228.697
|257.545
|
|-
| 7/4-comma
|486.187
|227.626
|258.562
|
|-
|12/7-comma
|486.429
|227.142
|259.288
|
|-
|5/3-comma
|486.752
|226.496
|260.253
|
|-
|ϕ-comma
|487.081
|225.837
|261.244
|
|-
|8/5-comma
|487.204
|225.593
|261.611
|
|-
|11/7-comma
| 487.397
|225.206
|262.192
|
|-
|3/2-comma
|487.881
| 224.762
|263.644
|
|-
|10/7-comma
|488.365
|223.270
|265.096
|
|-
|7/5-comma
|488.559
|222.882
|265.676
|
|-
|4/3-comma
|489.010
|221.979
|267.031
|
|-
|9/7-comma
|489.333
|221.334
|267.999
|
|-
|5/4-comma
|489.575
|220.850
|268.725
|
|-
| 6/5-comma
|489.914
|220.172
|269.742
|
|-
|7/6-comma
|490.140
|219.720
|270.419
|
|-
|8/7-comma
|490.301
|219.398
|270.903
|
|-
|1-comma
|491.269
|217.538
| 273.807
|
|-
|6/7-comma
|492.237
|215.526
|276.711
|
|-
|5/6-comma
|492.398
|215.203
|277.195
|
|-
| 4/5-comma
| 492.624
|214.751
|277.873
|
|-
|3/4-comma
|492.963
|214.926
|278.889
|
|-
|5/7-comma
|493.205
|213.590
|279.615
|
|-
|2/3-comma
|493.528
|212.945
|280.583
|
|-
|3/5-comma
|493.979
|212.041
|281.938
|
|-
|4/7-comma
|494.173
|211.346
|282.519
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|494.657
|210.686
|283.971
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|495.141
|209.718
|285.423
|
|-
|2/5-comma
|495.335
|209.331
|286.004
|
|-
|1/(ϕ+1)-comma
|495.457
|209.086
|286.371
|
|-
|1/3-comma
|495.786
|208.573
|287.359
|
|-
|2/7-comma
|496.109
|207.782
|289.372
|
|-
|1/4-comma
|496.351
|207.293
|289.053
|
|-
|1/5-comma
|496.690
|206.620
|290.069
|
|-
|1/6-comma
|496.916
|206.169
|290.747
|
|-
|1/7-comma
|497.077
|205.846
|291.231
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from Pythagorean to -1-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|497.083
|205.835
|291.248
|
|-
| -1/13-comma
|497.009
|205.983
|291.026
|
|-
|<nowiki>-1/12-comma</nowiki>
|496.922
|206.155
|290.767
|
|-
|<nowiki>-1/11-comma</nowiki>
|496.820
|206.360
|290.460
|
|-
| -1/10-comma
|496.698
|206.605
|290.093
|
|-
| -1/9-comma
|496.548
|206.904
|289.644
|
|-
| -1/8-comma
|496.361
|207.278
|289.083
|
|-
| -1/7-comma
|496.120
|207.759
|288.361
|
|-
| -2/13-comma
|495.972
|208.055
|287.917
|
|-
| -1/6-comma
|495.800
|208.401
|287.399
|
|-
|<nowiki>-2/11-comma</nowiki>
|495.595
|208.809
|286.786
|
|-
| -1/5-comma
|495.350
|209.299
|286.051
|
|-
| -3/14-comma
|495.158
|209.684
|285.474
|
|-
| -2/9-comma
|495.051
|209.898
|285.153
|
|-
| -3/13-comma
|494.936
|210.128
|284.808
|
|-
| -1/4-comma
|494.677
|210.646
|284.030
|
|-
|<nowiki>-3/11-comma</nowiki>
|494.371
|211.259
|283.111
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|494.196
|211.609
|282.587
|
|-
| -3/10-comma
|494.003
|211.994
|282.010
|
|-
| -4/13-comma
|493.900
|212.799
|281.699
|
|-
| -1/3-comma
|493.554
|212.892
|280.662
|
|-
| -5/14-comma
|493.233
|213.537
|279.700
|
|-
|<nowiki>-4/11-comma</nowiki>
|493.146
|213.709
|279.437
|
|-
| -3/8-comma
|492.993
|214.014
|278.979
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|492,899
|214.203
|278.697
|
|-
| -5/13-comma
|492.863
|214.274
|278.590
|
|-
| -2/5-comma
|492.656
|214.688
|277.968
|
|-
|<nowiki>-5/12-comma</nowiki>
|492.431
|215.137
|277.294
|
|-
| -3/7-comma
|492.271
|215.458
|276.813
|
|-
| -4/9-comma
|492.057
|215.886
|276.171
|
|-
|<nowiki>-5/11-comma</nowiki>
|491.921
|216.158
|275.763
|
|-
| -6/13-comma
|491.827
|216.346
|275.480
|
|-
| -1/2-comma
|491.309
|217.383
|273.926
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161
|-
| -7/13-comma
|490.790
|218.419
|272.371
|
|-
|<nowiki>-6/11-comma</nowiki>
|490.696
|218.607
|272.089
|
|-
| -5/9-comma
|490.560
|218.880
|271.680
|
|-
| -4/7-comma
|490.346
|219.307
|271.039
|
|-
|<nowiki>-7/12-comma</nowiki>
|490.186
|219.629
|270.558
|
|-
| -3/5-comma
|489.961
|220.077
|269.884
|
|-
| -8/13-comma
|489.754
|220.492
|269.262
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|489.716
|220.563
|269.155
|
|-
| -5/8-comma
|489.625
|220.751
|268.874
|
|-
|<nowiki>-7/11-comma</nowiki>
|489.471
|221.057
|268.414
|
|-
| -9/14-comma
|489.384
|221.232
|268.152
|
|-
| -2/3-comma
|489.063
|221.874
|267.190
|
|-
| -9/13-comma
|488.718
|222.565
|266.153
|
|-
| -7/10-comma
|488.614
|222.772
|265.842
|
|-
| -5/7-comma
|488.422
|223.157
|265.265
|
|-
|<nowiki>-8/11-comma</nowiki>
|488.247
|223.507
|264.740
|
|-
| -3/4-comma
|487.940
|224.119
|263.821
|
|-
| -10/13-comma
|487.681
|224.637
|263.044
|
|-
| -7/9-comma
|487.566
|224.868
|262.698
|
|-
| -11/14-comma
|487.459
|225.081
|262.378
|
|-
| -4/5-comma
|487.267
|225.466
|261.801
|
|-
|<nowiki>-9/11-comma</nowiki>
|487.022
|225.957
|261.066
|
|-
| -5/6-comma
|486.818
|226.365
|260.453
|
|-
| -11/13-comma
|486.645
|226.710
|259.935
|
|-
| -6/7-comma
|486.497
|227.006
|259.491
|
|-
| -7/8-comma
|486.256
|227.487
|258.769
|
|-
| -8/9-comma
|486.069
|227.861
|258.208
|
|-
| -9/10-comma
|485.920
|228.161
|257.759
|
|-
|<nowiki>-10/11-comma</nowiki>
|485.797
|228.406
|257.391
|
|-
| -11/12-comma
|485.695
|228.610
|257.085
|
|-
| -12/13-comma
|485.609
|228.783
|256.826
|
|-
| -13/14-comma
|485.535
|228.931
|256.604
|
|-
|-
| -1-comma
|484.752
|230.855
|253.717
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from Pythagorean to -2-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|499.013
|201.974
|297.039
|
|-
| -1/6-comma
|499.174
|201.652
|297.523
|
|-
| -1/5-comma
|499.400
|201.200
|298.201
|
|-
| -1/4-comma
|499.739
|200.522
|299.217
|
|-
| -2/7-comma
|499.981
|200.038
|299.942
|
|-
| -1/3-comma
|500.303
|199.393
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|500.633
|198.734
|301.900
|
|-
| -2/5-comma
|500.755
|198.499
|302.266
|
|-
| -3/7-comma
|500.949
|198.102
|302.847
|
|-
| -1/2-comma
|501.433
|197.134
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|501.917
|196.166
|305.751
|
|-
| -3/5-comma
|502.111
|195.779
|306.332
|
|-
| -2/3-comma
|502.562
|194.876
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|502.885
|194.230
|308.655
|
|-
| -4/5-comma
|503.466
|193.069
|310.397
|
|-
| -5/6-comma
|503.692
|192.617
|311.075
|
|-
|-
| -6/7-comma
|503.853
|192.294
|311.556
|
|-
| -1-comma
|504.821
|190.352
|314.463
|
|-
| -8/7-comma
|505.789
|188.422
|317.367
|
|-
| -7/6-comma
|505.950
|188.100
|317.851
|
|-
| -6/5-comma
|506.176
|187.648
|318.528
|
|-
| -5/4-comma
|506.515
|186.970
|319.545
|
|-
| -9/7-comma
|506.757
|186.486
|320.271
|
|-
| -4/3-comma
|507.080
|185.841
|321.239
|
|-
| -7/5-comma
|507.531
|184.937
|322.594
|
|-
| -10/7-comma
|507.725
|184.550
|323.174
|
|-
| -3/2-comma
|508.209
|183.582
|324.626
|
|-
| -11/7-comma
|508.693
|182.614
|326.078
|
|-
| -8/5-comma
|508.886
|182.228
|326.659
|
|-
| -ϕ-comma
|509.009
|181.983
|327.026
|
|-
| -5/3-comma
|509.338
|181.324
|328.014
|
|-
| -12/7-comma
|509.661
|180.678
|328.982
|
|-
| -7/4-comma
|509.903
|180.194
|329.708
|
|-
| -9/5-comma
|510.242
|179.517
|330.725
|
|-
| -11/6-comma
|510.467
|179.065
|331.402
|
|-
| -13/7-comma
|510.629
|178.742
|331.886
|
|-
| -2-comma
|511.597
|176.807
|334.790
|}
===Beyond Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from -1-comma to -2-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
| -1-comma
|484.752
|230.855
|253.717
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|483.610
|232.780
|250.830
|
|-
| -14/13-comma
|483.536
|232.928
|250.608
|
|-
| -13/12-comma
|483.450
|233.101
|250.349
|
|-
| -12/11-comma
|483.348
|233.305
|250.043
|
|-
| -11/10-comma
|483.225
|233.550
|249.675
|
|-
| -10/9-comma
|483.075
|233.151
|249.226
|
|-
| -9/8-comma
|482.888
|234.234
|248.665
|
|-
| -8/7-comma
|482.648
|234.295
|247.943
|
|-
| -15/13-comma
|482.500
|235.001
|247.499
|
|-
| -7/6-comma
|482.327
|235.346
|246.981
|
|-
| -13/11-comma
|482.123
|235.755
|246.368
|
|-
| -6/5-comma
|481.878
|236.244
|245.633
|
|-
| -17/14-comma
|481.685
|236.629
|245.056
|
|-
| -11/9-comma
|481.578
|236.843
|244.735
|
|-
| -16/13-comma
|481.463
|237.926
|244.390
|
|-
| -5/4-comma
|481.204
|237.592
|243.612
|
|-
| -14/11-comma
|480.898
|238.204
|242.694
|
|-
| -9/7-comma
|480.723
|238.554
|242.169
|
|-
| -13/10-comma
|480.530
|238.939
|241.591
|
|-
| -17/13-comma
|480.427
|239.146
|241.280
|
|-
| -4/3-comma
|480.081
|239.837
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|479.761
|240.479
|239.282
|Everything from this point onwards has an oneirotonic scale.
|-
| -15/11-comma
|479.673
|240.634
|239.019
|
|-
| -11/8-comma
|479.520
|240.960
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|479.426
|241.148
|238.279
|
|-
| -18/13-comma
|479.390
|241.219
|238.171
|
|-
| -7/5-comma
|479.183
|241.634
|237.550
|
|-
| -17/12-comma
|478.959
|242.917
|236.876
|
|-
| -10/7-comma
|478.798
|242.403
|236.395
|
|-
| -13/9-comma
|478.584
|242.831
|235.753
|
|-
| -16/11-comma
|478.448
|243.103
|235.345
|
|-
| -19/13-comma
|478.354
|243.708
|235.062
|
|-
| -3/2-comma
|477.836
|244.328
|233.508
|
|-
| -20/13-comma
|477.318
|245.344
|231.953
|Close to [[93edo]] oneirotonic
|-
| -17/11-comma
|477.224
|245.553
|231.671
|Close to [[88edo]] oneirotonic
|-
| -14/9-comma
|477.087
|245.825
|231.262
|Close to [[83edo]] oneirotonic
|-
| -11/7-comma
|476.873
|246.747
|230.621
|Close to [[78edo]] oneirotonic
|-
| -19/12-comma
|476.713
|246.426
|230.140
|Close to [[73edo]] oneirotonic
|-
| -8/5-comma
|476.489
|247.023
|229.466
|Close to [[68edo]] oneirotonic
|-
| -21/13-comma
|476.281
|247.437
|228.844
|
|-
| -ϕ-comma
|476.246
|247.491
|228.737
|
|-
| -13/8-comma
|476.152
|247.696
|228.456
|Close to [[63edo]] oneirotonic
|-
| -18/11-comma
|475.999
|248.002
|227.996
|
|-
| -23/14-comma
|475.911
|248.823
|227.734
|Close to [[58edo]] oneirotonic
|-
| -5/3-comma
|475.590
|248.819
|226.771
|Close to [[53edo]] oneirotonic
|-
| -22/13-comma
|475.245
|249.510
|225.735
|
|-
| -17/10-comma
|475.141
|249.717
|225.424
|
|-
| -12/7-comma
|474.949
|250.105
|224.847
|Close to [[48edo]] oneirotonic
|-
| -19/11-comma
|474.774
|250.552
|224.322
|
|-
| -7/4-comma
|474.468
|251.064
|223.403
|Close to [[43edo]] oneirotonic
|-
| -23/13-comma
|474.209
|251.583
|222.626
|
|-
| -16/9-comma
|474.094
|251.823
|222.281
|
|-
| -25/14-comma
|473.987
|252.027
|221.960
|
|-
| -9/5-comma
|473.794
|252.412
|221.382
|Close to [[38edo]] oneirotonic
|-
| -20/11-comma
|473.549
|252.912
|220.648
|
|-
| -11/6-comma
|473.345
|253.610
|220.035
|
|-
| -24/13-comma
|473.172
|253.345
|219.517
|
|-
| -13/7-comma
|473.924
|253.951
|219.073
|
|-
| -15/8-comma
|472.784
|254.433
|218.351
|Close to [[33edo]] oneirotonic
|-
| -17/9-comma
|472.597
|254.807
|217.790
|
|-
| -19/10-comma
|472.447
|255.106
|217.341
|
|-
| -21/11-comma
|472.324
|255.351
|216.973
|
|-
| -23/12-comma
|472.222
|255.555
|216.667
|
|-
| -25/13-comma
|472.135
|255.728
|216.408
|
|-
| -27/14-comma
|472.052
|255.876
|216.186
|
|-
|-
| -2-comma
|471.100
|258.801
|213.299
|Close to [[28edo]] oneirotonic
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from -2 to -4-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
| -2-comma
|511.597
|176.807
|334.790
|
|-
| -15/7-comma
|512.565
|174.870
|337.694
|
|-
| -13/6-comma
|512.726
|174.548
|338.178
|
|-
| -11/5-comma
|512.952
|174.096
|338.856
|
|-
| -9/4-comma
|513.291
|173.419
|339.872
|
|-
| -16/7-comma
|513.533
|172.935
|340.598
|
|-
| -7/3-comma
|513.855
|172.289
|341.566
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|514.185
|171.630
|342.555
|
|-
| -12/5-comma
|514.307
|171.386
|342.921
|Close to [[7edo]]. Everything from this point onwards has a superdiatonic scale.
|-
| -17/7-comma
|514.501
|170.999
|343.502
|
|-
| -3/2-comma
|514.984
|170.031
|344.954
|
|-
| -18/7-comma
|515.469
|169.063
|346.406
|
|-
| -13/5-comma
|515.662
|168.675
|346.987
|
|-
| -8/3-comma
|516.114
|167.772
|348.342
|
|-
| -19/7-comma
|516.437
|167.167
|349.310
|
|-
| -14/5-comma
|517.017
|165.965
|351.052
|Very close to [[6ed6]]
|-
| -11/6-comma
|517.243
|165.513
|351.730
|
|-
|-
| -20/7-comma
|517.404
|165.191
|352.214
|
|-
| -3-comma
|518.373
|163.255
|355.118
|
|-
| -22/7-comma
|519.341
|161.389
|358.022
|
|-
| -19/6-comma
|519.502
|160.996
|358.501
|
|-
| -16/5-comma
|519.728
|160.544
|359.183
|
|-
| -13/4-comma
|520.067
|159.867
|360.200
|Close to [[30edo]] superdiatonic
|-
| -23/7-comma
|520.309
|159.383
|360.926
|
|-
| -10/3-comma
|520.631
|158.737
|361.894
|
|-
| -17/5-comma
|521.083
|157.834
|363.249
|
|-
| -24/7-comma
|521.277
|157.447
|363.830
|
|-
| -7/2-comma
|521.761
|156.479
|365.282
|Close to [[23edo]]
|-
| -25/7-comma
|522.245
|155.511
|366.734
|
|-
| -18/5-comma
|522.438
|155.124
|367.315
|
|-
| -(ϕ+2)-comma
|522.560
|154.879
|367.681
|
|-
| -11/3-comma
|522.890
|154.220
|368.670
|
|-
| -26/7-comma
|523.213
|153.575
|369.638
|Close to [[39edo]] superdiatonic
|-
| -15/4-comma
|523.455
|153.091
|370.364
|
|-
| -19/5-comma
|523.793
|152.433
|271.380
|
|-
| -23/6-comma
|524.020
|151.962
|372.058
|
|-
| -27/7-comma
|524.181
|151.639
|372.542
|
|-
| -4-comma
|525.149
|149.703
|375.446
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-18T05:32:53Z

Moremajorthanmajor:

{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -2 and 2 with a denominator 14 or smaller or the '''charisma''' ([[256/255]]) is being divided and n is a fraction between -4 and 4 with a denominator 7 or smaller. This range is almost the same as the range of m/n-comma Archytas temperaments.

==Cautions==
As tempering out either comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129 and 72:85:108 vs. 170:216:255), the mean minor tetrachord (root-whole tone-minor third-tempered fourth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with more complex thirds, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 or 85/48 and 16/9.

== The table==
=== Beyond historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 1-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|1-comma
|511.518
|176.965
|334.553
|
|-
|13/14-comma
|510.555
|178.890
|331.666
|
|-
|12/13-comma
|510.481
|179.037
|331.444
|
|-
|11/12-comma
|510.395
|179.210
|331.185
|
|-
|10/11-comma
|510.293
|179.414
|330.879
|
|-
|9/10-comma
|510.170
|179.659
|330.511
|
|-
|8/9-comma
|510.021
|179.959
|330.062
|
|-
|7/8-comma
|509.834
| 180.333
|329.501
|
|-
|6/7-comma
|509.593
|180.814
|328.779
|
|-
|11/13-comma
|509.445
|181.110
|328.335
|
|-
|5/6-comma
|509.272
|181.455
|327.817
|
|-
|9/11-comma
|509.068
|181.864
|327.204
|
|-
|4/5-comma
|508.823
|182.354
|326.469
|
|-
|11/14-comma
|508.630
|182.739
|325.892
|
|-
|7/9-comma
|508.523
|182.952
|325.571
|
|-
|10/13-comma
|508.408
|183.183
|325.226
|
|-
|3/4-comma
|508.150
|183.701
|324.449
|
|-
|8/11-comma
|507.843
|184.687
|323.530
|
|-
|5/7-comma
|507.638
|184.633
|323.005
|
|-
|7/10-comma
|507.476
|184.952
|322.428
|
|-
|9/13-comma
|507.372
|185.255
|322.117
|
|-
|2/3-comma
|507.027
|185.946
|321.080
|
|-
|9/14-comma
|506.706
|186.588
|320.118
|
|-
|7/11-comma
|506.619
|186.763
|319.856
|
|-
|5/8-comma
|506.465
|187.069
|319.396
|
|-
|1/ϕ-comma
|506.372
|187.257
|319.115
|
|-
|8/13-comma
|506.336
|187.320
|319.008
|
|-
|3/5-comma
|506.129
|187.743
|318.386
|
|-
|7/12-comma
|505.904
|188.194
|317.712
|
|-
|4/7-comma
|505.744
|188.512
|317.231
|
|-
|5/9-comma
|505.530
|188.940
|316.590
|
|-
|6/11-comma
|505.394
|189.213
|316.181
|
|-
|7/13-comma
|505.300
|189.401
|315.899
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|504.781
|190.437
|314.344
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|6/13-comma
|504.263
|191.574
|312.790
|
|-
|5/11-comma
|504.169
|191.338
|312.507
|
|-
|4/9-comma
|504.033
|191.934
|312.099
|
|-
|3/7-comma
|503.819
|192.362
|311.457
|
|-
|5/12-comma
|503.659
|192.683
|310.976
|
|-
|2/5-comma
|503.434
|193.132
|310.302
|Almost exactly meantone
|-
|5/13-comma
|503.227
|193.546
|309.680
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|503.191
|193.618
|309.573
|
|-
|3/8-comma
|503.096
|193.805
|309.291
|
|-
|4/11-comma
|502.944
|194.112
|308.832
|
|-
|5/14-comma
|502.856
|194.287
|308.570
|
|-
|1/3-comma
|502.536
|194.928
|307.608
|
|-
|4/13-comma
|502.190
|195.619
|306.571
|
|-
|3/10-comma
|502.087
|195.174
|306.260
|
|-
|2/7-comma
|501.894
|196.211
|305.683
|
|-
|3/11-comma
|501.718
|196.561
|305.158
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|501.413
|197.174
|304.240
|
|-
|3/13-comma
|501.154
|197.692
|303.462
|
|-
|2/9-comma
|501.039
|197.922
|303.117
|
|-
|3/14-comma
|500.932
|198.136
|302.796
|
|-
|1/5-comma
|500.740
|198.521
|302.219
|
|-
|2/11-comma
|500.495
|199.011
|301.484
|
|-
|1/6-comma
|500.290
|199.419
|300.871
|
|-
|2/13-comma
|500.118
|199.765
|300.353
|
|-
|1/7-comma
|499.970
|200.061
|299.909
|
|-
|1/8-comma
|499.729
|200.542
|299.187
|
|-
|1/9-comma
|499.542
|200.916
|298.626
|
|-
|1/10-comma
|499.392
|201.785
|298.177
|
|-
|1/11-comma
|499.270
|201.460
|297.810
|
|-
|1/12-comma
|499.168
|201.665
|297.503
|
|-
|1/13-comma
|499.081
|201.837
|297.244
|
|-
|1/14-comma
|499.007
|201.953
|297.022
|
|-
|1-comma
|511.518
|176.965
|334.553
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 2-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|2-comma
|484.493
|231.014
|253.480
|
|-
|13/7-comma
|485.461
|229.078
|256.384
|
|-
|11/6-comma
|485.623
|228.755
|256.868
|
|-
|9/5-comma
|485.848
|228.697
|257.545
|
|-
|7/4-comma
|486.187
|227.626
|258.562
|
|-
|12/7-comma
|486.429
|227.142
|259.288
|
|-
|5/3-comma
|486.752
| 226.496
|260.253
|
|-
|ϕ-comma
|487.081
|225.837
|261.244
|
|-
|8/5-comma
|487.204
|225.593
|261.611
|
|-
|11/7-comma
|487.397
|225.206
|262.192
|
|-
|3/2-comma
|487.881
|224.762
|263.644
|
|-
|10/7-comma
|488.365
|223.270
|265.096
|
|-
|7/5-comma
|488.559
|222.882
|265.676
|
|-
|4/3-comma
|489.010
|221.979
|267.031
|
|-
|9/7-comma
|489.333
|221.334
|267.999
|
|-
|5/4-comma
|489.575
|220.850
|268.725
|
|-
|6/5-comma
|489.914
|220.172
|269.742
|
|-
|7/6-comma
|490.140
|219.720
|270.419
|
|-
|8/7-comma
|490.301
|219.398
|270.903
|
|-
|1-comma
|491.269
|217.538
|273.807
|
|-
|6/7-comma
|492.237
|215.526
|276.711
|
|-
|5/6-comma
|492.398
|215.203
|277.195
|
|-
|4/5-comma
|492.624
|214.751
|277.873
|
|-
|3/4-comma
|492.963
|214.926
|278.889
|
|-
|5/7-comma
|493.205
|213.590
|279.615
|
|-
|2/3-comma
|493.528
|212.945
|280.583
|
|-
|3/5-comma
|493.979
|212.041
|281.938
|
|-
|4/7-comma
|494.173
|211.346
|282.519
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/2-comma
|494.657
| 210.686
|283.971
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|495.141
|209.718
|285.423
|
|-
|2/5-comma
|495.335
|209.331
|286.004
|
|-
|1/(ϕ+1)-comma
|495.457
|209.086
|286.371
|
|-
|1/3-comma
|495.786
|208.573
|287.359
|
|-
|2/7-comma
|496.109
|207.782
|289.372
|
|-
|1/4-comma
|496.351
|207.293
|289.053
|
|-
|1/5-comma
|496.690
|206.620
|290.069
|
|-
|1/6-comma
|496.916
|206.169
|290.747
|
|-
|1/7-comma
|497.077
|205.846
|291.231
|Almost exactly [[65edo]]
|-
|2-comma
|484.493
|231.014
|253.480
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Historically-defined mean tetrachord===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 1-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
! third
!Comments
|-
|1-comma
|511.518
|176.965
| 334.553
|
|-
|13/14-comma
|510.555
|178.890
|331.666
|
|-
|12/13-comma
|510.481
|179.037
| 331.444
|
|-
|11/12-comma
|510.395
|179.210
|331.185
|
|-
|10/11-comma
| 510.293
| 179.414
|330.879
|
|-
|9/10-comma
| 510.170
|179.659
|330.511
|
|-
|8/9-comma
|510.021
|179.959
|330.062
|
|-
|7/8-comma
|509.834
|180.333
|329.501
|
|-
|6/7-comma
|509.593
|180.814
|328.779
|
|-
|11/13-comma
|509.445
|181.110
|328.335
|
|-
|5/6-comma
|509.272
|181.455
|327.817
|
|-
|9/11-comma
|509.068
|181.864
|327.204
|
|-
|4/5-comma
|508.823
|182.354
|326.469
|
|-
|11/14-comma
|508.630
|182.739
|325.892
|
|-
|7/9-comma
|508.523
|182.952
| 325.571
|
|-
|10/13-comma
|508.408
|183.183
|325.226
|
|-
|3/4-comma
|508.150
|183.701
|324.449
|
|-
|8/11-comma
|507.843
|184.687
|323.530
|
|-
|5/7-comma
|507.638
|184.633
|323.005
|
|-
|7/10-comma
|507.476
|184.952
|322.428
|
|-
|9/13-comma
|507.372
|185.255
|322.117
|
|-
|2/3-comma
|507.027
|185.946
|321.080
|
|-
|9/14-comma
|506.706
|186.588
|320.118
|
|-
|7/11-comma
|506.619
|186.763
|319.856
|
|-
|5/8-comma
|506.465
|187.069
|319.396
|
|-
|1/ϕ-comma
|506.372
|187.257
|319.115
|
|-
|8/13-comma
|506.336
|187.320
|319.008
|
|-
|3/5-comma
|506.129
|187.743
|318.386
|
|-
|7/12-comma
|505.904
|188.194
|317.712
|
|-
|4/7-comma
|505.744
|188.512
|317.231
|
|-
|5/9-comma
|505.530
|188.940
|316.590
|
|-
|6/11-comma
|505.394
|189.213
|316.181
|
|-
|7/13-comma
|505.300
|189.401
|315.899
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|504.781
|190.437
|314.344
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|6/13-comma
|504.263
|191.574
|312.790
|
|-
|5/11-comma
|504.169
|191.338
|312.507
|
|-
|4/9-comma
|504.033
|191.934
|312.099
|
|-
|3/7-comma
|503.819
|192.362
|311.457
|
|-
|5/12-comma
|503.659
|192.683
|310.976
|
|-
|2/5-comma
|503.434
|193.132
|310.302
|Almost exactly meantone
|-
|5/13-comma
|503.227
|193.546
|309.680
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|503.191
|193.618
| 309.573
|
|-
|3/8-comma
| 503.096
|193.805
|309.291
|
|-
|4/11-comma
|502.944
|194.112
|308.832
|
|-
|5/14-comma
|502.856
|194.287
|308.570
|
|-
|1/3-comma
|502.536
|194.928
|307.608
|
|-
|4/13-comma
|502.190
|195.619
|306.571
|
|-
|3/10-comma
|502.087
|195.174
|306.260
|
|-
|2/7-comma
|501.894
|196.211
|305.683
|
|-
|3/11-comma
|501.718
|196.561
|305.158
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|501.413
|197.174
|304.240
|
|-
|3/13-comma
|501.154
|197.692
|303.462
|
|-
|2/9-comma
|501.039
|197.922
|303.117
|
|-
|3/14-comma
|500.932
|198.136
|302.796
|
|-
|1/5-comma
|500.740
|198.521
|302.219
|
|-
|2/11-comma
|500.495
|199.011
|301.484
|
|-
|1/6-comma
|500.290
|199.419
|300.871
|
|-
|2/13-comma
|500.118
|199.765
|300.353
|
|-
|1/7-comma
|499.970
|200.061
|299.909
|
|-
|1/8-comma
| 499.729
|200.542
|299.187
|
|-
|1/9-comma
|499.542
|200.916
|298.626
|
|-
|1/10-comma
|499.392
|201.785
|298.177
|
|-
|1/11-comma
|499.270
|201.460
|297.810
|
|-
|1/12-comma
|499.168
|201.665
|297.503
|
|-
|1/13-comma
|499.081
|201.837
|297.244
|
|-
|1/14-comma
|499.007
|201.953
|297.022
|
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}

====Tempering out [[256/255]]====

{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 2-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|2-comma
|484.493
|231.014
|253.480
|
|-
|13/7-comma
|485.461
|229.078
|256.384
|
|-
|11/6-comma
|485.623
|228.755
|256.868
|
|-
|9/5-comma
|485.848
|228.697
|257.545
|
|-
| 7/4-comma
|486.187
|227.626
|258.562
|
|-
|12/7-comma
|486.429
|227.142
|259.288
|
|-
|5/3-comma
|486.752
|226.496
|260.253
|
|-
|ϕ-comma
|487.081
|225.837
|261.244
|
|-
|8/5-comma
|487.204
|225.593
|261.611
|
|-
|11/7-comma
| 487.397
|225.206
|262.192
|
|-
|3/2-comma
|487.881
| 224.762
|263.644
|
|-
|10/7-comma
|488.365
|223.270
|265.096
|
|-
|7/5-comma
|488.559
|222.882
|265.676
|
|-
|4/3-comma
|489.010
|221.979
|267.031
|
|-
|9/7-comma
|489.333
|221.334
|267.999
|
|-
|5/4-comma
|489.575
|220.850
|268.725
|
|-
| 6/5-comma
|489.914
|220.172
|269.742
|
|-
|7/6-comma
|490.140
|219.720
|270.419
|
|-
|8/7-comma
|490.301
|219.398
|270.903
|
|-
|1-comma
|491.269
|217.538
| 273.807
|
|-
|6/7-comma
|492.237
|215.526
|276.711
|
|-
|5/6-comma
|492.398
|215.203
|277.195
|
|-
| 4/5-comma
| 492.624
|214.751
|277.873
|
|-
|3/4-comma
|492.963
|214.926
|278.889
|
|-
|5/7-comma
|493.205
|213.590
|279.615
|
|-
|2/3-comma
|493.528
|212.945
|280.583
|
|-
|3/5-comma
|493.979
|212.041
|281.938
|
|-
|4/7-comma
|494.173
|211.346
|282.519
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| 1/2-comma
|494.657
|210.686
|283.971
|Everything from this point onwards has a minor seventh between 85/48 and 16/9. This is the other canonical mean tetrachord tuning in universe.
|-
|3/7-comma
|495.141
|209.718
|285.423
|
|-
|2/5-comma
|495.335
|209.331
|286.004
|
|-
|1/(ϕ+1)-comma
|495.457
|209.086
|286.371
|
|-
|1/3-comma
|495.786
|208.573
|287.359
|
|-
|2/7-comma
|496.109
|207.782
|289.372
|
|-
|1/4-comma
|496.351
|207.293
|289.053
|
|-
|1/5-comma
|496.690
|206.620
|290.069
|
|-
|1/6-comma
|496.916
|206.169
|290.747
|
|-
|1/7-comma
|497.077
|205.846
|291.231
|Almost exactly [[65edo]]
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|}

===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from Pythagorean to -1-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|497.083
|205.835
|291.248
|
|-
| -1/13-comma
|497.009
|205.983
|291.026
|
|-
|<nowiki>-1/12-comma</nowiki>
|496.922
|206.155
|290.767
|
|-
|<nowiki>-1/11-comma</nowiki>
|496.820
|206.360
|290.460
|
|-
| -1/10-comma
|496.698
|206.605
|290.093
|
|-
| -1/9-comma
|496.548
|206.904
|289.644
|
|-
| -1/8-comma
|496.361
|207.278
|289.083
|
|-
| -1/7-comma
|496.120
|207.759
|288.361
|
|-
| -2/13-comma
|495.972
|208.055
|287.917
|
|-
| -1/6-comma
|495.800
|208.401
|287.399
|
|-
|<nowiki>-2/11-comma</nowiki>
|495.595
|208.809
|286.786
|
|-
| -1/5-comma
|495.350
|209.299
|286.051
|
|-
| -3/14-comma
|495.158
|209.684
|285.474
|
|-
| -2/9-comma
|495.051
|209.898
|285.153
|
|-
| -3/13-comma
|494.936
|210.128
|284.808
|
|-
| -1/4-comma
|494.677
|210.646
|284.030
|
|-
|<nowiki>-3/11-comma</nowiki>
|494.371
|211.259
|283.111
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|494.196
|211.609
|282.587
|
|-
| -3/10-comma
|494.003
|211.994
|282.010
|
|-
| -4/13-comma
|493.900
|212.799
|281.699
|
|-
| -1/3-comma
|493.554
|212.892
|280.662
|
|-
| -5/14-comma
|493.233
|213.537
|279.700
|
|-
|<nowiki>-4/11-comma</nowiki>
|493.146
|213.709
|279.437
|
|-
| -3/8-comma
|492.993
|214.014
|278.979
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|492,899
|214.203
|278.697
|
|-
| -5/13-comma
|492.863
|214.274
|278.590
|
|-
| -2/5-comma
|492.656
|214.688
|277.968
|
|-
|<nowiki>-5/12-comma</nowiki>
|492.431
|215.137
|277.294
|
|-
| -3/7-comma
|492.271
|215.458
|276.813
|
|-
| -4/9-comma
|492.057
|215.886
|276.171
|
|-
|<nowiki>-5/11-comma</nowiki>
|491.921
|216.158
|275.763
|
|-
| -6/13-comma
|491.827
|216.346
|275.480
|
|-
| -1/2-comma
|491.309
|217.383
|273.926
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161
|-
| -7/13-comma
|490.790
|218.419
|272.371
|
|-
|<nowiki>-6/11-comma</nowiki>
|490.696
|218.607
|272.089
|
|-
| -5/9-comma
|490.560
|218.880
|271.680
|
|-
| -4/7-comma
|490.346
|219.307
|271.039
|
|-
|<nowiki>-7/12-comma</nowiki>
|490.186
|219.629
|270.558
|
|-
| -3/5-comma
|489.961
|220.077
|269.884
|
|-
| -8/13-comma
|489.754
|220.492
|269.262
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|489.716
|220.563
|269.155
|
|-
| -5/8-comma
|489.625
|220.751
|268.874
|
|-
|<nowiki>-7/11-comma</nowiki>
|489.471
|221.057
|268.414
|
|-
| -9/14-comma
|489.384
|221.232
|268.152
|
|-
| -2/3-comma
|489.063
|221.874
|267.190
|
|-
| -9/13-comma
|488.718
|222.565
|266.153
|
|-
| -7/10-comma
|488.614
|222.772
|265.842
|
|-
| -5/7-comma
|488.422
|223.157
|265.265
|
|-
|<nowiki>-8/11-comma</nowiki>
|488.247
|223.507
|264.740
|
|-
| -3/4-comma
|487.940
|224.119
|263.821
|
|-
| -10/13-comma
|487.681
|224.637
|263.044
|
|-
| -7/9-comma
|487.566
|224.868
|262.698
|
|-
| -11/14-comma
|487.459
|225.081
|262.378
|
|-
| -4/5-comma
|487.267
|225.466
|261.801
|
|-
|<nowiki>-9/11-comma</nowiki>
|487.022
|225.957
|261.066
|
|-
| -5/6-comma
|486.818
|226.365
|260.453
|
|-
| -11/13-comma
|486.645
|226.710
|259.935
|
|-
| -6/7-comma
|486.497
|227.006
|259.491
|
|-
| -7/8-comma
|486.256
|227.487
|258.769
|
|-
| -8/9-comma
|486.069
|227.861
|258.208
|
|-
| -9/10-comma
|485.920
|228.161
|257.759
|
|-
|<nowiki>-10/11-comma</nowiki>
|485.797
|228.406
|257.391
|
|-
| -11/12-comma
|485.695
|228.610
|257.085
|
|-
| -12/13-comma
|485.609
|228.783
|256.826
|
|-
| -13/14-comma
|485.535
|228.931
|256.604
|
|-
|-
| -1-comma
|484.752
|230.855
|253.717
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from Pythagorean to -2-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 85/48 and 16/9
|-
| -1/7-comma
|499.013
|201.974
|297.039
|
|-
| -1/6-comma
|499.174
|201.652
|297.523
|
|-
| -1/5-comma
|499.400
|201.200
|298.201
|
|-
| -1/4-comma
|499.739
|200.522
|299.217
|
|-
| -2/7-comma
|499.981
|200.038
|299.942
|
|-
| -1/3-comma
|500.303
|199.393
|300.911
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|500.633
|198.734
|301.900
|
|-
| -2/5-comma
|500.755
|198.499
|302.266
|
|-
| -3/7-comma
|500.949
|198.102
|302.847
|
|-
| -1/2-comma
|501.433
|197.134
|304.299
|Everything from this point onwards does not have a minor seventh between 16/9 and 4096/2295
|-
| -4/7-comma
|501.917
|196.166
|305.751
|
|-
| -3/5-comma
|502.111
|195.779
|306.332
|
|-
| -2/3-comma
|502.562
|194.876
|307.687
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -5/7-comma
|502.885
|194.230
|308.655
|
|-
| -4/5-comma
|503.466
|193.069
|310.397
|
|-
| -5/6-comma
|503.692
|192.617
|311.075
|
|-
|-
| -6/7-comma
|503.853
|192.294
|311.556
|
|-
| -1-comma
|504.821
|190.352
|314.463
|
|-
| -8/7-comma
|505.789
|188.422
|317.367
|
|-
| -7/6-comma
|505.950
|188.100
|317.851
|
|-
| -6/5-comma
|506.176
|187.648
|318.528
|
|-
| -5/4-comma
|506.515
|186.970
|319.545
|
|-
| -9/7-comma
|506.757
|186.486
|320.271
|
|-
| -4/3-comma
|507.080
|185.841
|321.239
|
|-
| -7/5-comma
|507.531
|184.937
|322.594
|
|-
| -10/7-comma
|507.725
|184.550
|323.174
|
|-
| -3/2-comma
|508.209
|183.582
|324.626
|
|-
| -11/7-comma
|508.693
|182.614
|326.078
|
|-
| -8/5-comma
|508.886
|182.228
|326.659
|
|-
| -ϕ-comma
|509.009
|181.983
|327.026
|
|-
| -5/3-comma
|509.338
|181.324
|328.014
|
|-
| -12/7-comma
|509.661
|180.678
|328.982
|
|-
| -7/4-comma
|509.903
|180.194
|329.708
|
|-
| -9/5-comma
|510.242
|179.517
|330.725
|
|-
| -11/6-comma
|510.467
|179.065
|331.402
|
|-
| -13/7-comma
|510.629
|178.742
|331.886
|
|-
| -2-comma
|511.597
|176.807
|334.790
|}
===Beyond Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] and Reversed Archytas)===

====Tempering out [[129/128]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from -1-comma to -2-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
| -1-comma
|484.752
|230.855
|253.717
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -15/14-comma
|483.610
|232.780
|250.830
|
|-
| -14/13-comma
|483.536
|232.928
|250.608
|
|-
| -13/12-comma
|483.450
|233.101
|250.349
|
|-
| -12/11-comma
|483.348
|233.305
|250.043
|
|-
| -11/10-comma
|483.225
|233.550
|249.675
|
|-
| -10/9-comma
|483.075
|233.151
|249.226
|
|-
| -9/8-comma
|482.888
|234.234
|248.665
|
|-
| -8/7-comma
|482.648
|234.295
|247.943
|
|-
| -15/13-comma
|482.500
|235.001
|247.499
|
|-
| -7/6-comma
|482.327
|235.346
|246.981
|
|-
| -13/11-comma
|482.123
|235.755
|246.368
|
|-
| -6/5-comma
|481.878
|236.244
|245.633
|
|-
| -17/14-comma
|481.685
|236.629
|245.056
|
|-
| -11/9-comma
|481.578
|236.843
|244.735
|
|-
| -16/13-comma
|481.463
|237.926
|244.390
|
|-
| -5/4-comma
|481.204
|237.592
|243.612
|
|-
| -14/11-comma
|480.898
|238.204
|242.694
|
|-
| -9/7-comma
|480.723
|238.554
|242.169
|
|-
| -13/10-comma
|480.530
|238.939
|241.591
|
|-
| -17/13-comma
|480.427
|239.146
|241.280
|
|-
| -4/3-comma
|480.081
|239.837
|240.244
|Close to [[5edo]].
|-
| -19/14-comma
|479.761
|240.479
|239.282
|Everything from this point onwards has an oneirotonic scale.
|-
| -15/11-comma
|479.673
|240.634
|239.019
|
|-
| -11/8-comma
|479.520
|240.960
|238.560
|
|-
| -(ϕ+3)/(ϕ+1)-comma
|479.426
|241.148
|238.279
|
|-
| -18/13-comma
|479.390
|241.219
|238.171
|
|-
| -7/5-comma
|479.183
|241.634
|237.550
|
|-
| -17/12-comma
|478.959
|242.917
|236.876
|
|-
| -10/7-comma
|478.798
|242.403
|236.395
|
|-
| -13/9-comma
|478.584
|242.831
|235.753
|
|-
| -16/11-comma
|478.448
|243.103
|235.345
|
|-
| -19/13-comma
|478.354
|243.708
|235.062
|
|-
| -3/2-comma
|477.836
|244.328
|233.508
|
|-
| -20/13-comma
|477.318
|245.344
|231.953
|Close to [[93edo]] oneirotonic
|-
| -17/11-comma
|477.224
|245.553
|231.671
|Close to [[88edo]] oneirotonic
|-
| -14/9-comma
|477.087
|245.825
|231.262
|Close to [[83edo]] oneirotonic
|-
| -11/7-comma
|476.873
|246.747
|230.621
|Close to [[78edo]] oneirotonic
|-
| -19/12-comma
|476.713
|246.426
|230.140
|Close to [[73edo]] oneirotonic
|-
| -8/5-comma
|476.489
|247.023
|229.466
|Close to [[68edo]] oneirotonic
|-
| -21/13-comma
|476.281
|247.437
|228.844
|
|-
| -ϕ-comma
|476.246
|247.491
|228.737
|
|-
| -13/8-comma
|476.152
|247.696
|228.456
|Close to [[63edo]] oneirotonic
|-
| -18/11-comma
|475.999
|248.002
|227.996
|
|-
| -23/14-comma
|475.911
|248.823
|227.734
|Close to [[58edo]] oneirotonic
|-
| -5/3-comma
|475.590
|248.819
|226.771
|Close to [[53edo]] oneirotonic
|-
| -22/13-comma
|475.245
|249.510
|225.735
|
|-
| -17/10-comma
|475.141
|249.717
|225.424
|
|-
| -12/7-comma
|474.949
|250.105
|224.847
|Close to [[48edo]] oneirotonic
|-
| -19/11-comma
|474.774
|250.552
|224.322
|
|-
| -7/4-comma
|474.468
|251.064
|223.403
|Close to [[43edo]] oneirotonic
|-
| -23/13-comma
|474.209
|251.583
|222.626
|
|-
| -16/9-comma
|474.094
|251.823
|222.281
|
|-
| -25/14-comma
|473.987
|252.027
|221.960
|
|-
| -9/5-comma
|473.794
|252.412
|221.382
|Close to [[38edo]] oneirotonic
|-
| -20/11-comma
|473.549
|252.912
|220.648
|
|-
| -11/6-comma
|473.345
|253.610
|220.035
|
|-
| -24/13-comma
|473.172
|253.345
|219.517
|
|-
| -13/7-comma
|473.924
|253.951
|219.073
|
|-
| -15/8-comma
|472.784
|254.433
|218.351
|Close to [[33edo]] oneirotonic
|-
| -17/9-comma
|472.597
|254.807
|217.790
|
|-
| -19/10-comma
|472.447
|255.106
|217.341
|
|-
| -21/11-comma
|472.324
|255.351
|216.973
|
|-
| -23/12-comma
|472.222
|255.555
|216.667
|
|-
| -25/13-comma
|472.135
|255.728
|216.408
|
|-
| -27/14-comma
|472.052
|255.876
|216.186
|
|-
|-
| -2-comma
|471.100
|258.801
|213.299
|Close to [[28edo]] oneirotonic
|}

====Tempering out [[256/255]]====
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from -2 to -4-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
| -2-comma
|511.597
|176.807
|334.790
|
|-
| -15/7-comma
|512.565
|174.870
|337.694
|
|-
| -13/6-comma
|512.726
|174.548
|338.178
|
|-
| -11/5-comma
|512.952
|174.096
|338.856
|
|-
| -9/4-comma
|513.291
|173.419
|339.872
|
|-
| -16/7-comma
|513.533
|172.935
|340.598
|
|-
| -7/3-comma
|513.855
|172.289
|341.566
|
|-
| -(2ϕ+3)/(ϕ+1)-comma
|514.185
|171.630
|342.555
|
|-
| -12/5-comma
|514.307
|171.386
|342.921
|Close to [[7edo]]. Everything from this point onwards has a superdiatonic scale.
|-
| -17/7-comma
|514.501
|170.999
|343.502
|
|-
| -3/2-comma
|514.984
|170.031
|344.954
|
|-
| -18/7-comma
|515.469
|169.063
|346.406
|
|-
| -13/5-comma
|515.662
|168.675
|346.987
|
|-
| -8/3-comma
|516.114
|167.772
|348.342
|
|-
| -19/7-comma
|516.437
|167.167
|349.310
|
|-
| -14/5-comma
|517.017
|165.965
|351.052
|Very close to [[6ed6]]
|-
| -11/6-comma
|517.243
|165.513
|351.730
|
|-
|-
| -20/7-comma
|517.404
|165.191
|352.214
|
|-
| -3-comma
|518.373
|163.255
|355.118
|
|-
| -22/7-comma
|519.341
|161.389
|358.022
|
|-
| -19/6-comma
|519.502
|160.996
|358.501
|
|-
| -16/5-comma
|519.728
|160.544
|359.183
|
|-
| -13/4-comma
|520.067
|159.867
|360.200
|Close to [[30edo]] superdiatonic
|-
| -23/7-comma
|520.309
|159.383
|360.926
|
|-
| -10/3-comma
|520.631
|158.737
|361.894
|
|-
| -17/5-comma
|521.083
|157.834
|363.249
|
|-
| -24/7-comma
|521.277
|157.447
|363.830
|
|-
| -7/2-comma
|521.761
|156.479
|365.282
|Close to [[23edo]]
|-
| -25/7-comma
|522.245
|155.511
|366.734
|
|-
| -18/5-comma
|522.438
|155.124
|367.315
|
|-
| -(ϕ+2)-comma
|522.560
|154.879
|367.681
|
|-
| -11/3-comma
|522.890
|154.220
|368.670
|
|-
| -26/7-comma
|523.213
|153.575
|369.638
|Close to [[39edo]] superdiatonic
|-
| -15/4-comma
|523.455
|153.091
|370.364
|
|-
| -19/5-comma
|523.793
|152.433
|271.380
|
|-
| -23/6-comma
|524.020
|151.962
|372.058
|
|-
| -27/7-comma
|524.181
|151.639
|372.542
|
|-
| -4-comma
|525.149
|149.703
|375.446
|Close to [[16edo]]
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-16T19:41:45Z

Moremajorthanmajor: /* Tempering out 256/255 */

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-16T05:56:29Z

Moremajorthanmajor: /* Tempering out 129/128 */

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-16T02:46:50Z

Moremajorthanmajor:

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-15T22:57:44Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth, nineteenth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth, nineteenth (technically)
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis'', beside which the weakness of ''lenis'' is that its desired (sub)harmonics mostly form [[wolf interval]]<nowiki/>s. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean tetrachords|mean tetrachord]], which is primarily considered to temper out [[129/128]] or [[256/255]].

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-14T07:00:03Z

Moremajorthanmajor: /* Negative harmony theory-defined mean tetrachord (most often approached as Reversed meantone ) */

{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -1 and 1 with a denominator 14 or smaller. This range is almost the same as the range between [[61edo|61bedo]] and its complementary opposite.

== Cautions ==
As tempering out this comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129), the mean minor tetrachord (root-whole tone-minor third-tempered fourth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with a sharp third, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 and 16/9.

==The table==
===Historically-defined mean tetrachord===
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 1-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|1-comma
|511.518
|176.965
|334.553
|
|-
|13/14-comma
|510.555
|178.890
|331.666
|
|-
|12/13-comma
|510.481
|179.037
|331.444
|
|-
|11/12-comma
|510.395
|179.210
|331.185
|
|-
|10/11-comma
|510.293
|179.414
|330.879
|
|-
|9/10-comma
|510.170
|179.659
|330.511
|
|-
|8/9-comma
|510.021
| 179.959
|330.062
|
|-
|7/8-comma
| 509.834
|180.333
| 329.501
|
|-
|6/7-comma
|509.593
|180.814
|328.779
|
|-
|11/13-comma
|509.445
|181.110
|328.335
|
|-
|5/6-comma
|509.272
|181.455
|327.817
|
|-
|9/11-comma
|509.068
|181.864
| 327.204
|
|-
|4/5-comma
|508.823
|182.354
|326.469
|
|-
|11/14-comma
|508.630
|182.739
|325.892
|
|-
|7/9-comma
|508.523
|182.952
|325.571
|
|-
|10/13-comma
|508.408
|183.183
|325.226
|
|-
|3/4-comma
|508.150
|183.701
|324.449
|
|-
|8/11-comma
|507.843
|184.687
|323.530
|
|-
|5/7-comma
|507.638
|184.633
|323.005
|
|-
|7/10-comma
|507.476
|184.952
|322.428
|
|-
|9/13-comma
|507.372
|185.255
|322.117
|
|-
|2/3-comma
|507.027
|185.946
|321.080
|
|-
|9/14-comma
|506.706
|186.588
|320.118
|
|-
|7/11-comma
| 506.619
|186.763
|319.856
|
|-
|5/8-comma
|506.465
| 187.069
|319.396
|
|-
|1/ϕ-comma
|506.372
|187.257
|319.115
|
|-
|8/13-comma
|506.336
|187.320
|319.008
|
|-
|3/5-comma
|506.129
|187.743
|318.386
|
|-
|7/12-comma
|505.904
|188.194
|317.712
|
|-
|4/7-comma
|505.744
|188.512
|317.231
|
|-
|5/9-comma
|505.530
|188.940
|316.590
|
|-
|6/11-comma
|505.394
|189.213
|316.181
|
|-
|7/13-comma
|505.300
|189.401
|315.899
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|504.781
| 190.437
|314.344
|Close to [[88edo]] and [[Lucy tuning]]. Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|6/13-comma
|504.263
|191.574
|312.790
|
|-
|5/11-comma
|504.169
|191.338
|312.507
|
|-
|4/9-comma
|504.033
| 191.934
|312.099
|
|-
|3/7-comma
|503.819
|192.362
|311.457
|
|-
|5/12-comma
|503.659
|192.683
|310.976
|
|-
|2/5-comma
|503.434
|193.132
|310.302
|Almost exactly meantone
|-
|5/13-comma
|503.227
|193.546
|309.680
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|503.191
|193.618
|309.573
|
|-
|3/8-comma
| 503.096
|193.805
|309.291
|
|-
|4/11-comma
|502.944
|194.112
|308.832
|
|-
|5/14-comma
| 502.856
|194.287
|308.570
|
|-
|1/3-comma
| 502.536
|194.928
|307.608
|
|-
|4/13-comma
|502.190
|195.619
|306.571
|
|-
|3/10-comma
|502.087
|195.174
|306.260
|
|-
|2/7-comma
|501.894
|196.211
|305.683
|
|-
|3/11-comma
|501.718
|196.561
|305.158
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|501.413
|197.174
|304.240
|
|-
|3/13-comma
|501.154
|197.692
|303.462
|
|-
|2/9-comma
|501.039
|197.922
|303.117
|
|-
|3/14-comma
|500.932
|198.136
|302.796
|
|-
|1/5-comma
|500.740
|198.521
|302.219
|
|-
|2/11-comma
|500.495
|199.011
|301.484
|
|-
|1/6-comma
|500.290
|199.419
|300.871
|
|-
|2/13-comma
|500.118
|199.765
|300.353
|
|-
|1/7-comma
|499.970
|200.061
|299.909
|
|-
|1/8-comma
|499.729
|200.542
|299.187
|
|-
|1/9-comma
|499.542
|200.916
|298.626
|
|-
|1/10-comma
|499.392
|201.785
|298.177
|
|-
|1/11-comma
|499.270
|201.460
|297.810
|
|-
|1/12-comma
|499.168
|201.665
|297.503
|
|-
|1/13-comma
|499.081
|201.837
|297.244
|
|-
|1/14-comma
|499.007
|201.953
|297.022
|
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}
===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]])===
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from Pythagorean to -1-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|497.083
|205.835
|291.248
|
|-
| -1/13-comma
|497.009
|205.983
|291.026
|
|-
|<nowiki>-1/12-comma</nowiki>
|496.922
|206.155
|290.767
|
|-
|<nowiki>-1/11-comma</nowiki>
|496.820
|206.360
|290.460
|
|-
| -1/10-comma
|496.698
|206.605
|290.093
|
|-
| -1/9-comma
|496.548
|206.904
|289.644
|
|-
| -1/8-comma
|496.361
|207.278
|289.083
|
|-
| -1/7-comma
|496.120
|207.759
|288.361
|
|-
| -2/13-comma
|495.972
|208.055
|287.917
|
|-
| -1/6-comma
|495.800
|208.401
|287.399
|
|-
|<nowiki>-2/11-comma</nowiki>
|495.595
|208.809
|286.786
|
|-
| -1/5-comma
|495.350
|209.299
|286.051
|
|-
| -3/14-comma
|495.158
|209.684
|285.474
|
|-
| -2/9-comma
|495.051
|209.898
|285.153
|
|-
| -3/13-comma
|494.936
|210.128
|284.808
|
|-
| -1/4-comma
|494.677
|210.646
|284.030
|
|-
|<nowiki>-3/11-comma</nowiki>
|494.371
|211.259
|283.111
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|494.196
|211.609
|282.587
|
|-
| -3/10-comma
|494.003
|211.994
|282.010
|
|-
| -4/13-comma
|493.900
|212.799
|281.699
|
|-
| -1/3-comma
|493.554
|212.892
|280.662
|
|-
| -5/14-comma
|493.233
|213.537
|279.700
|
|-
|<nowiki>-4/11-comma</nowiki>
|493.146
|213.709
|279.437
|
|-
| -3/8-comma
|492.993
|214.014
|278.979
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|492,899
|214.203
|278.697
|
|-
| -5/13-comma
|492.863
|214.274
|278.590
|
|-
| -2/5-comma
|492.656
|214.688
|277.968
|
|-
|<nowiki>-5/12-comma</nowiki>
|492.431
|215.137
|277.294
|
|-
| -3/7-comma
|492.271
|215.458
|276.813
|
|-
| -4/9-comma
|492.057
|215.886
|276.171
|
|-
|<nowiki>-5/11-comma</nowiki>
|491.921
|216.158
|275.763
|
|-
| -6/13-comma
|491.827
|216.346
|275.480
|
|-
| -1/2-comma
|491.309
|217.383
|273.926
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161
|-
| -7/13-comma
|490.790
|218.419
|272.371
|
|-
|<nowiki>-6/11-comma</nowiki>
|490.696
|218.607
|272.089
|
|-
| -5/9-comma
|490.560
|218.880
|271.680
|
|-
| -4/7-comma
|490.346
|219.307
|271.039
|
|-
|<nowiki>-7/12-comma</nowiki>
|490.186
|219.629
|270.558
|
|-
| -3/5-comma
|489.961
|220.077
|269.884
|
|-
| -8/13-comma
|489.754
|220.492
|269.262
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|489.716
|220.563
|269.155
|
|-
| -5/8-comma
|489.625
|220.751
|268.874
|
|-
|<nowiki>-7/11-comma</nowiki>
|489.471
|221.057
|268.414
|
|-
| -9/14-comma
|489.384
|221.232
|268.152
|
|-
| -2/3-comma
|489.063
|221.874
|267.190
|
|-
| -9/13-comma
|488.718
|222.565
|266.153
|
|-
| -7/10-comma
|488.614
|222.772
|265.842
|
|-
| -5/7-comma
|488.422
|223.157
|265.265
|
|-
|<nowiki>-8/11-comma</nowiki>
|488.247
|223.507
|264.740
|
|-
| -3/4-comma
|487.940
|224.119
|263.821
|
|-
| -10/13-comma
|487.681
|224.637
|263.044
|
|-
| -7/9-comma
|487.566
|224.868
|262.698
|
|-
| -11/14-comma
|487.459
|225.081
|262.378
|
|-
| -4/5-comma
|487.267
|225.466
|261.801
|
|-
|<nowiki>-9/11-comma</nowiki>
|487.022
|225.957
|261.066
|
|-
| -5/6-comma
|486.818
|226.365
|260.453
|
|-
| -11/13-comma
|486.645
|226.710
|259.935
|
|-
| -6/7-comma
|486.497
|227.006
|259.491
|
|-
| -7/8-comma
|486.256
|227.487
|258.769
|
|-
| -8/9-comma
|486.069
|227.861
|258.208
|
|-
| -9/10-comma
|485.920
|228.161
|257.759
|
|-
|<nowiki>-10/11-comma</nowiki>
|485.797
|228.406
|257.391
|
|-
| -11/12-comma
|485.695
|228.610
|257.085
|
|-
| -12/13-comma
|485.609
|228.783
|256.826
|
|-
| -13/14-comma
|485.535
|228.931
|256.604
|
|-
|- 1-comma
| -1-comma
|484.752
|230.855
|253.717
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-14T06:51:21Z

Moremajorthanmajor: /* The table */

{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -1 and 1 with a denominator 14 or smaller. This range is almost the same as the range between [[61edo|61bedo]] and its complementary opposite.

== Cautions ==
As tempering out this comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129), the mean minor tetrachord (root-whole tone-minor third-tempered fourth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with a sharp third, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 and 16/9.

==The table==
===Historically-defined mean tetrachord===
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 1-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|1-comma
|511.518
|176.965
|334.553
|
|-
|13/14-comma
|510.555
|178.890
|331.666
|
|-
|12/13-comma
|510.481
|179.037
|331.444
|
|-
|11/12-comma
|510.395
|179.210
|331.185
|
|-
|10/11-comma
|510.293
|179.414
|330.879
|
|-
|9/10-comma
|510.170
|179.659
|330.511
|
|-
|8/9-comma
|510.021
| 179.959
|330.062
|
|-
|7/8-comma
| 509.834
|180.333
| 329.501
|
|-
|6/7-comma
|509.593
|180.814
|328.779
|
|-
|11/13-comma
|509.445
|181.110
|328.335
|
|-
|5/6-comma
|509.272
|181.455
|327.817
|
|-
|9/11-comma
|509.068
|181.864
| 327.204
|
|-
|4/5-comma
|508.823
|182.354
|326.469
|
|-
|11/14-comma
|508.630
|182.739
|325.892
|
|-
|7/9-comma
|508.523
|182.952
|325.571
|
|-
|10/13-comma
|508.408
|183.183
|325.226
|
|-
|3/4-comma
|508.150
|183.701
|324.449
|
|-
|8/11-comma
|507.843
|184.687
|323.530
|
|-
|5/7-comma
|507.638
|184.633
|323.005
|
|-
|7/10-comma
|507.476
|184.952
|322.428
|
|-
|9/13-comma
|507.372
|185.255
|322.117
|
|-
|2/3-comma
|507.027
|185.946
|321.080
|
|-
|9/14-comma
|506.706
|186.588
|320.118
|
|-
|7/11-comma
| 506.619
|186.763
|319.856
|
|-
|5/8-comma
|506.465
| 187.069
|319.396
|
|-
|1/ϕ-comma
|506.372
|187.257
|319.115
|
|-
|8/13-comma
|506.336
|187.320
|319.008
|
|-
|3/5-comma
|506.129
|187.743
|318.386
|
|-
|7/12-comma
|505.904
|188.194
|317.712
|
|-
|4/7-comma
|505.744
|188.512
|317.231
|
|-
|5/9-comma
|505.530
|188.940
|316.590
|
|-
|6/11-comma
|505.394
|189.213
|316.181
|
|-
|7/13-comma
|505.300
|189.401
|315.899
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|504.781
| 190.437
|314.344
|Close to [[88edo]] and [[Lucy tuning]]. Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|6/13-comma
|504.263
|191.574
|312.790
|
|-
|5/11-comma
|504.169
|191.338
|312.507
|
|-
|4/9-comma
|504.033
| 191.934
|312.099
|
|-
|3/7-comma
|503.819
|192.362
|311.457
|
|-
|5/12-comma
|503.659
|192.683
|310.976
|
|-
|2/5-comma
|503.434
|193.132
|310.302
|Almost exactly meantone
|-
|5/13-comma
|503.227
|193.546
|309.680
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|503.191
|193.618
|309.573
|
|-
|3/8-comma
| 503.096
|193.805
|309.291
|
|-
|4/11-comma
|502.944
|194.112
|308.832
|
|-
|5/14-comma
| 502.856
|194.287
|308.570
|
|-
|1/3-comma
| 502.536
|194.928
|307.608
|
|-
|4/13-comma
|502.190
|195.619
|306.571
|
|-
|3/10-comma
|502.087
|195.174
|306.260
|
|-
|2/7-comma
|501.894
|196.211
|305.683
|
|-
|3/11-comma
|501.718
|196.561
|305.158
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|501.413
|197.174
|304.240
|
|-
|3/13-comma
|501.154
|197.692
|303.462
|
|-
|2/9-comma
|501.039
|197.922
|303.117
|
|-
|3/14-comma
|500.932
|198.136
|302.796
|
|-
|1/5-comma
|500.740
|198.521
|302.219
|
|-
|2/11-comma
|500.495
|199.011
|301.484
|
|-
|1/6-comma
|500.290
|199.419
|300.871
|
|-
|2/13-comma
|500.118
|199.765
|300.353
|
|-
|1/7-comma
|499.970
|200.061
|299.909
|
|-
|1/8-comma
|499.729
|200.542
|299.187
|
|-
|1/9-comma
|499.542
|200.916
|298.626
|
|-
|1/10-comma
|499.392
|201.785
|298.177
|
|-
|1/11-comma
|499.270
|201.460
|297.810
|
|-
|1/12-comma
|499.168
|201.665
|297.503
|
|-
|1/13-comma
|499.081
|201.837
|297.244
|
|-
|1/14-comma
|499.007
|201.953
|297.022
|
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}
===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] )===
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from Pythagorean to -1-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|497.083
|205.835
|291.248
|
|-
| -1/13-comma
|497.009
|205.983
|291.026
|
|-
|<nowiki>-1/12-comma</nowiki>
|496.922
|206.155
|290.767
|
|-
|<nowiki>-1/11-comma</nowiki>
|496.820
|206.360
|290.460
|
|-
| -1/10-comma
|496.698
|206.605
|290.093
|
|-
| -1/9-comma
|496.548
|206.904
|289.644
|
|-
| -1/8-comma
|496.361
|207.278
|289.083
|
|-
| -1/7-comma
|496.120
|207.759
|288.361
|
|-
| -2/13-comma
|495.972
|208.055
|287.917
|
|-
| -1/6-comma
|495.800
|208.401
|287.399
|
|-
|<nowiki>-2/11-comma</nowiki>
|495.595
|208.809
|286.786
|
|-
| -1/5-comma
|495.350
|209.299
|286.051
|
|-
| -3/14-comma
|495.158
|209.684
|285.474
|
|-
| -2/9-comma
|495.051
|209.898
|285.153
|
|-
| -3/13-comma
|494.936
|210.128
|284.808
|
|-
| -1/4-comma
|494.677
|210.646
|284.030
|
|-
|<nowiki>-3/11-comma</nowiki>
|494.371
|211.259
|283.111
|Everything up to this point has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
| -2/7-comma
|494.196
|211.609
|282.587
|
|-
| -3/10-comma
|494.003
|211.994
|282.010
|
|-
| -4/13-comma
|493.900
|212.799
|281.699
|
|-
| -1/3-comma
|493.554
|212.892
|280.662
|
|-
| -5/14-comma
|493.233
|213.537
|279.700
|
|-
|<nowiki>-4/11-comma</nowiki>
|493.146
|213.709
|279.437
|
|-
| -3/8-comma
|492.993
|214.014
|278.979
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|492,899
|214.203
|278.697
|
|-
| -5/13-comma
|492.863
|214.274
|278.590
|
|-
| -2/5-comma
|492.656
|214.688
|277.968
|
|-
|<nowiki>-5/12-comma</nowiki>
|492.431
|215.137
|277.294
|
|-
| -3/7-comma
|492.271
|215.458
|276.813
|
|-
| -4/9-comma
|492.057
|215.886
|276.171
|
|-
|<nowiki>-5/11-comma</nowiki>
|491.921
|216.158
|275.763
|
|-
| -6/13-comma
|491.827
|216.346
|275.480
|
|-
| -1/2-comma
|491.309
|217.383
|273.926
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161
|-
| -7/13-comma
|490.790
|218.419
|272.371
|
|-
|<nowiki>-6/11-comma</nowiki>
|490.696
|218.607
|272.089
|
|-
| -5/9-comma
|490.560
|218.880
|271.680
|
|-
| -4/7-comma
|490.346
|219.307
|271.039
|
|-
|<nowiki>-7/12-comma</nowiki>
|490.186
|219.629
|270.558
|
|-
| -3/5-comma
|489.961
|220.077
|269.884
|
|-
| -8/13-comma
|489.754
|220.492
|269.262
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|489.716
|220.563
|269.155
|
|-
| -5/8-comma
|489.625
|220.751
|268.874
|
|-
|<nowiki>-7/11-comma</nowiki>
|489.471
|221.057
|268.414
|
|-
| -9/14-comma
|489.384
|221.232
|268.152
|
|-
| -2/3-comma
|489.063
|221.874
|267.190
|
|-
| -9/13-comma
|488.718
|222.565
|266.153
|
|-
| -7/10-comma
|488.614
|222.772
|265.842
|
|-
| -5/7-comma
|488.422
|223.157
|265.265
|
|-
|<nowiki>-8/11-comma</nowiki>
|488.247
|223.507
|264.740
|
|-
| -3/4-comma
|487.940
|224.119
|263.821
|
|-
| -10/13-comma
|487.681
|224.637
|263.044
|
|-
| -7/9-comma
|487.566
|224.868
|262.698
|
|-
| -11/14-comma
|487.459
|225.081
|262.378
|
|-
| -4/5-comma
|487.267
|225.466
|261.801
|
|-
|<nowiki>-9/11-comma</nowiki>
|487.022
|225.957
|261.066
|
|-
| -5/6-comma
|486.818
|226.365
|260.453
|
|-
| -11/13-comma
|486.645
|226.710
|259.935
|
|-
| -6/7-comma
|486.497
|227.006
|259.491
|
|-
| -7/8-comma
|486.256
|227.487
|258.769
|
|-
| -8/9-comma
|486.069
|227.861
|258.208
|
|-
| -9/10-comma
|485.920
|228.161
|257.759
|
|-
|<nowiki>-10/11-comma</nowiki>
|485.797
|228.406
|257.391
|
|-
| -11/12-comma
|485.695
|228.610
|257.085
|
|-
| -12/13-comma
|485.609
|228.783
|256.826
|
|-
| -13/14-comma
|485.535
|228.931
|256.604
|
|-
|- 1-comma
| -1-comma
|484.752
|230.855
|253.717
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-14T06:29:39Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Sol♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Do♯
|15
|Major seventh, fourteenth
|-
|4 fifths
|Fa♯
|5
|Major tenth, seventeenth
|-
|3 fifths
|Si
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Mi
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|La
|3
|Perfect twelfth
|-
|0
|Natural, ''naturalis''
|Re
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Sol
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Ut > Do
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa, originally ''supertripartiens''
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > Si♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Mi♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|La♭
|*11
|Diminished twelfth
|}
At the time the modal system was new, it was widespread, but not absolute, that only the true relations for the first three steps from the octave on the chain of fifths, and thus the 2.3.7.19.43 subgroup, were considered strictly in-bounds, thus it is that the modal system is considered to classify Re as natural. Major is considered as comparable to La as minor is to Sol, but La ''superparticularis'' and La ''superpartiens'' never saw as widespread usage as Fa ''superpartiens'' before the conversion of the latter to flats'','' Sol ''superparticularis'' and Sol ''superpartiens'' never seeing serious usage as they unnecessarily complicated notation. The paradox of this is that the true relations, only they and the tritone being considered to have distinct desired (sub)harmonics, generally do not have the same ones for ''fortis'' and ''lenis''. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean tetrachords|mean tetrachord]], which is primarily considered to temper out [[129/128]].

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-14T04:47:30Z

Moremajorthanmajor: /* Cautions */

{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -1 and 1 with a denominator 14 or smaller. This range is almost the same as the range between [[61edo|61bedo]] and its complementary opposite.

== Cautions ==
As tempering out this comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129), the mean minor tetrachord (root-whole tone-minor third-tempered fourth) is quoted as the lemma. This is similar to [[Pythagorean tuning]] itself or tempering out [[64/63]], [[352/351]], or [[513/512]]; but with a sharp third, and tempering out [[1053/1024]], but with thirds which sound as distinctly major and minor as the ideal [[5-limit]] thirds.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 and 16/9.

==The table==
===Historically-defined mean tetrachord===
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from 1-comma to Pythagorean
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|1-comma
|511.518
|176.965
|334.553
|
|-
|13/14-comma
|510.555
|178.890
|331.666
|
|-
|12/13-comma
|510.481
|179.037
|331.444
|
|-
|11/12-comma
|510.395
|179.210
|331.185
|
|-
|10/11-comma
|510.293
|179.414
|330.879
|
|-
|9/10-comma
|510.170
|179.659
|330.511
|
|-
|8/9-comma
|510.021
| 179.959
|330.062
|
|-
|7/8-comma
| 509.834
|180.333
| 329.501
|
|-
|6/7-comma
|509.593
|180.814
|328.779
|
|-
|11/13-comma
|509.445
|181.110
|328.335
|
|-
|5/6-comma
|509.272
|181.455
|327.817
|
|-
|9/11-comma
|509.068
|181.864
| 327.204
|
|-
|4/5-comma
|508.823
|182.354
|326.469
|
|-
|11/14-comma
|508.630
|182.739
|325.892
|
|-
|7/9-comma
|508.523
|182.952
|325.571
|
|-
|10/13-comma
|508.408
|183.183
|325.226
|
|-
|3/4-comma
|508.150
|183.701
|324.449
|
|-
|8/11-comma
|507.843
|184.687
|323.530
|
|-
|5/7-comma
|507.638
|184.633
|323.005
|
|-
|7/10-comma
|507.476
|184.952
|322.428
|
|-
|9/13-comma
|507.372
|185.255
|322.117
|
|-
|2/3-comma
|507.027
|185.946
|321.080
|
|-
|9/14-comma
|506.706
|186.588
|320.118
|
|-
|7/11-comma
| 506.619
|186.763
|319.856
|
|-
|5/8-comma
|506.465
| 187.069
|319.396
|
|-
|1/ϕ-comma
|506.372
|187.257
|319.115
|
|-
|8/13-comma
|506.336
|187.320
|319.008
|
|-
|3/5-comma
|506.129
|187.743
|318.386
|
|-
|7/12-comma
|505.904
|188.194
|317.712
|
|-
|4/7-comma
|505.744
|188.512
|317.231
|
|-
|5/9-comma
|505.530
|188.940
|316.590
|
|-
|6/11-comma
|505.394
|189.213
|316.181
|
|-
|7/13-comma
|505.300
|189.401
|315.899
|Even closer to [[19edo]] than [[1/3-comma meantone]].
|-
|1/2-comma
|504.781
| 190.437
|314.344
|Close to [[88edo]] and [[Lucy tuning]]. Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|6/13-comma
|504.263
|191.574
|312.790
|
|-
|5/11-comma
|504.169
|191.338
|312.507
|
|-
|4/9-comma
|504.033
| 191.934
|312.099
|
|-
|3/7-comma
|503.819
|192.362
|311.457
|
|-
|5/12-comma
|503.659
|192.683
|310.976
|
|-
|2/5-comma
|503.434
|193.132
|310.302
|Almost exactly meantone
|-
|5/13-comma
|503.227
|193.546
|309.680
|Almost exactly [[31edo]]
|-
|1/(ϕ+1)-comma
|503.191
|193.618
|309.573
|
|-
|3/8-comma
| 503.096
|193.805
|309.291
|
|-
|4/11-comma
|502.944
|194.112
|308.832
|
|-
|5/14-comma
| 502.856
|194.287
|308.570
|
|-
|1/3-comma
| 502.536
|194.928
|307.608
|
|-
|4/13-comma
|502.190
|195.619
|306.571
|
|-
|3/10-comma
|502.087
|195.174
|306.260
|
|-
|2/7-comma
|501.894
|196.211
|305.683
|
|-
|3/11-comma
|501.718
|196.561
|305.158
|Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|1/4-comma
|501.413
|197.174
|304.240
|
|-
|3/13-comma
|501.154
|197.692
|303.462
|
|-
|2/9-comma
|501.039
|197.922
|303.117
|
|-
|3/14-comma
|500.932
|198.136
|302.796
|
|-
|1/5-comma
|500.740
|198.521
|302.219
|
|-
|2/11-comma
|500.495
|199.011
|301.484
|
|-
|1/6-comma
|500.290
|199.419
|300.871
|
|-
|2/13-comma
|500.118
|199.765
|300.353
|
|-
|1/7-comma
|499.970
|200.061
|299.909
|
|-
|1/8-comma
|499.729
|200.542
|299.187
|
|-
|1/9-comma
|499.542
|200.916
|298.626
|
|-
|1/10-comma
|499.392
|201.785
|298.177
|
|-
|1/11-comma
|499.270
|201.460
|297.810
|
|-
|1/12-comma
|499.168
|201.665
|297.503
|
|-
|1/13-comma
|499.081
|201.837
|297.244
|
|-
|1/14-comma
|499.007
|201.953
|297.022
|
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|}
===Negative harmony theory-defined mean tetrachord (most often approached as [[Reversed meantone]] )===
{| class="wikitable mw-collapsible"
|+Spectrum of mean tetrachord tunings from Pythagorean to -1-comma
!Mean tetrachord temperament
!g (cents)
!Tone
!third
!Comments
|-
|[[Pythagorean tuning]]
|498.045
|203.910
|294.135
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/14-comma
|497.083
|205.835
|291.248
|
|-
| -1/13-comma
|497.009
|205.983
|291.026
|
|-
|<nowiki>-1/12-comma</nowiki>
|496.922
|206.155
|290.767
|
|-
|<nowiki>-1/11-comma</nowiki>
|496.820
|206.360
|290.460
|
|-
| -1/10-comma
|496.698
|206.605
|290.093
|
|-
| -1/9-comma
|496.548
|206.904
|289.644
|
|-
| -1/8-comma
|496.361
|207.278
|289.083
|
|-
| -1/7-comma
|496.120
|207.759
|288.361
|
|-
| -2/13-comma
|495.972
|208.055
|287.917
|
|-
| -1/6-comma
|495.800
|208.401
|287.399
|
|-
|<nowiki>-2/11-comma</nowiki>
|495.595
|208.809
|286.786
|
|-
| -1/5-comma
|495.350
|209.299
|286.051
|
|-
| -3/14-comma
|495.158
|209.684
|285.474
|
|-
| -2/9-comma
|495.051
|209.898
|285.153
|
|-
| -3/13-comma
|494.936
|210.128
|284.808
|
|-
| -1/4-comma
|494.677
|210.646
|284.030
|
|-
|<nowiki>-3/11-comma</nowiki>
|494.371
|211.259
|283.111
|
|-
| -2/7-comma
|494.196
|211.609
|282.587
|
|-
| -3/10-comma
|494.003
|211.994
|282.010
|
|-
| -4/13-comma
|493.900
|212.799
|281.699
|
|-
| -1/3-comma
|493.554
|212.892
|280.662
|
|-
| -5/14-comma
|493.233
|213.537
|279.700
|
|-
|<nowiki>-4/11-comma</nowiki>
|493.146
|213.709
|279.437
|
|-
| -3/8-comma
|492.993
|214.014
|278.979
|
|-
|<nowiki>-1/(ϕ+1)-comma</nowiki>
|492,899
|214.203
|278.697
|
|-
| -5/13-comma
|492.863
|214.274
|278.590
|
|-
| -2/5-comma
|492.656
|214.688
|277.968
|
|-
|<nowiki>-5/12-comma</nowiki>
|492.431
|215.137
|277.294
|
|-
| -3/7-comma
|492.271
|215.458
|276.813
|
|-
| -4/9-comma
|492.057
|215.886
|276.171
|
|-
|<nowiki>-5/11-comma</nowiki>
|491.921
|216.158
|275.763
|
|-
| -6/13-comma
|491.827
|216.346
|275.480
|
|-
| -1/2-comma
|491.309
|217.383
|273.926
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161
|-
| -7/13-comma
|490.790
|218.419
|272.371
|
|-
|<nowiki>-6/11-comma</nowiki>
|490.696
|218.607
|272.089
|
|-
| -5/9-comma
|490.560
|218.880
|271.680
|
|-
| -4/7-comma
|490.346
|219.307
|271.039
|
|-
|<nowiki>-7/12-comma</nowiki>
|490.186
|219.629
|270.558
|
|-
| -3/5-comma
|489.961
|220.077
|269.884
|
|-
| -8/13-comma
|489.754
|220.492
|269.262
|
|-
|<nowiki>-1/ϕ-comma</nowiki>
|489.716
|220.563
|269.155
|
|-
| -5/8-comma
|489.625
|220.751
|268.874
|
|-
|<nowiki>-7/11-comma</nowiki>
|489.471
|221.057
|268.414
|
|-
| -9/14-comma
|489.384
|221.232
|268.152
|
|-
| -2/3-comma
|489.063
|221.874
|267.190
|
|-
| -9/13-comma
|488.718
|222.565
|266.153
|
|-
| -7/10-comma
|488.614
|222.772
|265.842
|
|-
| -5/7-comma
|488.422
|223.157
|265.265
|
|-
|<nowiki>-8/11-comma</nowiki>
|488.247
|223.507
|264.740
|
|-
| -3/4-comma
|487.940
|224.119
|263.821
|
|-
| -10/13-comma
|487.681
|224.637
|263.044
|
|-
| -7/9-comma
|487.566
|224.868
|262.698
|
|-
| -11/14-comma
|487.459
|225.081
|262.378
|
|-
| -4/5-comma
|487.267
|225.466
|261.801
|
|-
|<nowiki>-9/11-comma</nowiki>
|487.022
|225.957
|261.066
|
|-
| -5/6-comma
|486.818
|226.365
|260.453
|
|-
| -11/13-comma
|486.645
|226.710
|259.935
|
|-
| -6/7-comma
|486.497
|227.006
|259.491
|
|-
| -7/8-comma
|486.256
|227.487
|258.769
|
|-
| -8/9-comma
|486.069
|227.861
|258.208
|
|-
| -9/10-comma
|485.920
|228.161
|257.759
|
|-
|<nowiki>-10/11-comma</nowiki>
|485.797
|228.406
|257.391
|
|-
| -11/12-comma
|485.695
|228.610
|257.085
|
|-
| -12/13-comma
|485.609
|228.783
|256.826
|
|-
| -13/14-comma
|485.535
|228.931
|256.604
|
|-
|- 1-comma
| -1-comma
|484.752
|230.855
|253.717
|}

User:BudjarnLambeth/Table of n-comma meantone generators

2025-01-14T03:17:07Z

Moremajorthanmajor: /* Negative harmony theory-defined meantone (most often approached as superpyth) */

{{Editable user page}}

Here are all [[meantone]] tunings that can be written in the form "n-comma meantone", where the syntonic comma ([[81/80]]) is being divided and n is a fraction between -1 and 1 with a denominator 22 or smaller.

== Scope of table ==

=== Characteristics included ===
Some of the characteristics this table mentions for each temperament include:
* Whether it saw historical (pre-1950) use
* Whether it is close to (i.e. within 1/2 a degree of closing of) an [[edo]] smaller than 100
* Whether it is the closest on the table to the optimal [[CTE]] or [[POTE]] tuning of meantone or [[superpyth]] in any JI [[limit]]
* Whether it approximates a very simple n-[[Pythagorean comma]] meantone
* Whether it is about equally sharp of [[3/2]] as some other listed temperament is flat
* Whether it is close to exactly one [[just-noticeable difference]] away from 3/2

Occasional other comments may be included as well.

=== Characteristics omitted ===
Dozens of tunings on this table are significant to [[negative harmony temperaments|negative harmony temperament theory]], enough that labelling them all individually would clutter the table.

Every tuning on this table is close to some arbitrarily large edo, but labeling them beyond [[100edo]] would clutter the table.

=== Special cases included ===
A small number of additional temperaments are included. Not too many, to avoid clutter, just the bare minimum:
* {{EDOs|7, 12, 17 and 5}} edos (to delineate small [[MOS]] shapes and boundaries of [[diamond monotone]])
* any tunings listed under "[[historical temperaments]]" (e.g. 4/25-comma), ''but only the ones of the form "n-comma"''.

== Cautions ==
=== Preservation of meantone behavior ===
Temperaments that fall outside of the "[[Historical temperaments|historically-defined meantone]]" range will not possess most of the musical properties that meantone usually possesses, but they are included for completeness.

Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.

== The table ==

=== Flatter than flattest historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 1/1-comma to 1/2-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[1/1-comma meantone|1/1-comma]] ||680.449||Close to [[30edo]]
|-
|[[21/22-comma meantone|21/22-comma]]
|681.426
|Close to [[37edo]].
|-
|[[20/21-comma meantone|20/21-comma]]
|681.473
|
|-
|[[19/20-comma meantone|19/20-comma]]
|681.524
|
|-
|[[18/19-comma meantone|18/19-comma]]
|681.581
|
|-
|[[17/18-comma meantone|17/18-comma]]
|681.644
|
|-
|[[16/17-comma meantone|16/17-comma]]
|681.713
|
|-
|[[15/16-comma meantone|15/16-comma]] ||681.793|| Close to [[44edo]].
|-
|[[14/15-comma meantone|14/15-comma]] ||681.883||
|-
|[[13/14-comma meantone|13/14-comma]] ||681.985||
|-
|[[12/13-comma meantone|12/13-comma]] ||682.103||
|-
|[[11/12-comma meantone|11/12-comma]] ||682.241||
|-
|[[10/11-comma meantone|10/11-comma]] ||682.404||Close to [[51edo]].
|-
|[[19/21-comma meantone|19/21-comma]]
|682.497
|
|-
|[[9/10-comma meantone|9/10-comma]]||682.599||
|-
|[[17/19-comma meantone|17/19-comma]]
|682.713
|
|-
|[[8/9-comma meantone|8/9-comma]] ||682.838||Close to [[58edo]].
|-
|[[15/17-comma meantone|15/17-comma]]
|682.979
|
|-
|[[7/8-comma meantone|7/8-comma]] ||683.137||Close to [[65edo]].
|-
|[[13/15-comma meantone|13/15-comma]] ||683.316||Close to [[72edo]].
|-
|[[19/22-comma meantone|19/22-comma]]
|683.381
|
|-
|[[6/7-comma meantone|6/7-comma]] ||683.521||Close to [[79edo]].
|-
|[[17/20-comma meantone|17/20-comma]]
|683.675
|
|-
|[[11/13-comma meantone|11/13-comma]] ||683.757||Close to [[86edo]].
|-
|[[16/19-comma meantone|16/19-comma]]
|683.844
|
|-
|[[21/25-comma meantone|21/25-comma]]
|683.890
|Close to [[93edo]]
|-
|[[5/6-comma meantone|5/6-comma]] ||684.033|| Close to [[100edo]].
|-
|[[14/17-comma meantone|14/17-comma]]
|684.244
|
|-
|[[9/11-comma meantone|9/11-comma]] ||684.359||
|-
|[[13/16-comma meantone|13/16-comma]] ||684.481||
|-
|[[17/21-comma meantone|17/21-comma]]
|684.545
|
|-
|[[4/5-comma meantone|4/5-comma]] ||684.750||
|-
|[[15/19-comma meantone|15/19-comma]]
|684.976
|
|-
|[[11/14-comma meantone|11/14-comma]] ||685.057||
|-
|[[7/9-comma meantone|7/9-comma]] ||685.228||
|-
|[[17/22-comma meantone|17/22-comma]]
|685.337
|
|-
|[[10/13-comma meantone|10/13-comma]] ||685.412||
|-
|[[13/17-comma meantone|13/17-comma]]
|685.509
|
|-
|[[16/21-comma meantone|16/21-comma]]
|685.569
|Everything up to this point generates 9 and 16 tone MOS scales.
|-
|[[7edo]]||685.714||The largest MOS scale this can generate is 7 tone. '''Lower boundary of 5-limit diamond monotone.'''
|-
|[[3/4-comma meantone|3/4-comma]] ||685.825||Everything from this point onwards generates 12 and 19 tone MOS scales.
|-
|[[14/19-comma meantone|14/19-comma]]
|686.108
|
|-
|[[11/15-comma meantone|11/15-comma]] ||686.184||
|-
|[[19/26-comma meantone|19/26-comma]]
|686.239
|
|-
|[[8/11-comma meantone|8/11-comma]]||686.314||
|-
|[[13/18-comma meantone|13/18-comma]]
|686.423
|
|-
|[[5/7-comma meantone|5/7-comma]] ||686.593||
|-
|[[17/24-comma meantone|17/24-comma]]
|686.721
|
|-
|[[12/17-comma meantone|12/17-comma]]
|686.774
|
|-
|[[7/10-comma meantone|7/10-comma]] ||686.901||
|-
|[[9/13-comma meantone|9/13-comma]] ||687.066||
|-
|[[11/16-comma meantone|11/16-comma]] ||687.169||
|-
|[[13/19-comma meantone|13/19-comma]]
|687.240
|
|-
|[[15/22-comma meantone|15/22-comma]]
|687.292
|
|-
|[[17/25-comma meantone|17/25-comma]]
|687.331
|
|-
|[[19/28-comma]]
|687.361
|
|-
|[[2/3-comma meantone|2/3-comma]] ||687.617||Close to [[89edo]].
|-
|[[17/26-comma meantone|17/26-comma]]
|687.893
|Close to [[82edo]].
|-
|[[15/23-comma meantone|15/23-comma]]
|687.929
|
|-
|[[13/20-comma meantone|13/20-comma]]
|687.976
|
|-
|[[11/17-comma meantone|11/17-comma]]
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone|9/14-comma]] ||688.129||
|-
|[[7/11-comma meantone|7/11-comma]] ||688.269||Close to [[68edo]].
|-
|[[12/19-comma meantone|12/19-comma]]
|688.372
|
|-
|[[5/8-comma meantone|5/8-comma]] ||688.514||Close to [[61edo]] and [[43/32]].
|-
|[[13/21-comma meantone|13/21-comma]]
|688.641
|
|-
|[[1/φ-comma meantone|1/ϕ-comma]]
|688.663
|
|-
|[[8/13-comma meantone|8/13-comma]] ||688.720||
|-
|[[11/18-comma meantone|11/18-comma]]
|688.812
|Close to [[54edo]].
|-
|[[14/23-comma meantone|14/23-comma]]
|688.864
|
|-
|[[3/5-comma meantone|3/5-comma]] ||689.051||
|-
|[[13/22-comma meantone|13/22-comma]]
|689.247
|
|-
|[[10/17-comma meantone|10/17-comma]]
|689.304
|
|-
|[[7/12-comma meantone|7/12-comma]] ||689.410||Close to [[47edo]].
|-
|[[11/19-comma meantone|11/19-comma]]
|689.504
|
|-
|[[4/7-comma meantone|4/7-comma]] ||689.666||Close to [[87edo]].
|-
|[[9/16-comma meantone|9/16-comma]] ||689.858||
|-
|[[5/9-comma meantone|5/9-comma]] ||690.007||Close to [[40edo]].
|-
|[[11/20-comma meantone|11/20-comma]]
|690.127
|
|-
|[[6/11-comma meantone|6/11-comma]] ||690.224||
|-
|[[7/13-comma meantone|7/13-comma]] ||690.375||Close to [[73edo]].
|-
|[[8/15-comma meantone|8/15-comma]] ||690.485||
|-
|[[9/17-comma meantone|9/17-comma]]
|690.569
|
|-
|[[10/19-comma meantone|10/19-comma]]
|690.636
|
|-
|[[11/21-comma meantone|11/21-comma]]
|690.690
|Close to [[33edo]]
|-
|[[1/2-comma meantone|1/2-comma]] ||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not have a whole tone between 10/9 and 9/8.
|}

=== Historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 10/21-comma to 1/22-comma
!Temperament!!Generator (cents)!!Comments
|-
|[[10/21-comma meantone|10/21-comma]]
|691.714
|
|-
|[[9/19-comma meantone|9/19-comma]]
|691.768
|Close to [[85edo]].
|-
|[[8/17-comma meantone|8/17-comma]]
|691.834
|
|-
|[[7/15-comma meantone|7/15-comma]] ||691.919||
|-
|[[6/13-comma meantone|6/13-comma]] ||692.029||
|-
|[[5/11-comma meantone|5/11-comma]] ||692.179||
|-
|[[9/20-comma meantone|9/20-comma]]
|692.277
|Close to [[26edo]].
|-
|[[4/9-comma meantone|4/9-comma]] ||692.397||
|-
|[[7/16-comma meantone|7/16-comma]] ||692.546||
|-
|[[3/7-comma meantone|3/7-comma]] ||692.738||
|-
|[[8/19-comma meantone|8/19-comma]]
|692.899
|
|-
|[[5/12-comma meantone|5/12-comma]] ||692.994||Close to [[71edo]].
|-
|[[7/17-comma meantone|7/17-comma]] ||693.099||
|-
|[[9/22-comma meantone|9/22-comma]]
|693.157
|
|-
|[[2/5-comma meantone|2/5-comma]] ||693.352||Close to [[45edo]].
|-
|[[9/23-comma meantone|9/23-comma]]
|693.539
|
|-
|[[7/18-comma meantone|7/18-comma]] ||693.591||
|-
|[[5/13-comma meantone|5/13-comma]] ||693.683||
|-
|[[1/(φ+1)-comma meantone|1/(ϕ+1)-comma]]
|693.740
|Close to [[64edo]].
|-
|[[8/21-comma meantone|8/21-comma]]
|693.762
|
|-
|[[3/8-comma meantone|3/8-comma]] ||693.890||Close to [[83edo]].
|-
|[[7/19-comma meantone|7/19-comma]]
|694.032
|
|-
|[[4/11-comma meantone|4/11-comma]] ||694.134||Almost exactly 1/3-''Pythagorean'' comma meantone.
|-
|[[5/14-comma meantone|5/14-comma]] ||694.274||
|-
|[[6/17-comma meantone|6/17-comma]] ||694.365||
|-
|[[7/20-comma meantone|7/20-comma]]
|694.428
|
|-
|[[8/23-comma meantone|8/23-comma]]
|694.475
|
|-
|[[9/26-comma meantone|9/26-comma]]
|694.511
|
|-
|[[1/3-comma meantone|1/3-comma]] ||694.786||Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[9/28-comma meantone|9/28-comma]]
|695.042
|
|-
|[[8/25-comma meantone|8/25-comma]]
|695.073
|
|-
|[[7/22-comma meantone|7/22-comma]]
|695.112
|
|-
|[[6/19-comma meantone|6/19-comma]]
|695.164
|
|-
|[[5/16-comma meantone|5/16-comma]] ||695.234||
|-
|[[4/13-comma meantone|4/13-comma]] ||695.338||
|-
|[[3/10-comma meantone|3/10-comma]] ||695.503||Close to [[88edo]] and [[Lucy tuning]]. Historically significant (see [[historical temperaments]]).
|-
|[[5/17-comma meantone|5/17-comma]] ||695.630||Close to [[69edo]].
|-
|[[7/24-comma meantone|7/24-comma]]
|695.682
|
|-
|[[2/7-comma meantone|2/7-comma]] ||695.810||Historically significant (see [[historical temperaments]]).
|-
|[[5/18-comma meantone|5/18-comma]] ||695.981||Close to [[50edo]].
|-
|[[3/11-comma meantone|3/11-comma]] ||696.090||
|-
|[[7/26-comma meantone|7/26-comma]] ||696.165||Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/15-comma meantone|4/15-comma]] ||696.220||Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|[[5/19-comma meantone|5/19-comma]]
|696.295
|Close to [[81edo]].
|-
|[[Quarter-comma meantone|1/4-comma]] ||696.578||Close to [[7-limit|septimal]] and [[tridecimal]] meantone POTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[5/21-comma meantone|5/21-comma]]
|696.834
|Close to [[31edo]].
|-
|[[4/17-comma meantone|4/17-comma]] ||696.895||
|-
|[[3/13-comma meantone|3/13-comma]] ||696.992||Close to [[7-limit|septimal]] & [[tridecimal]] meantone [[CTE]] tunings. Close to [[undecimal]] meantone POTE tuning.
|-
|[[5/22-comma meantone|5/22-comma]]
|697.067
|
|-
|[[2/9-comma meantone|2/9-comma]] ||697.176||Close to [[5-limit]] and [[undecimal]] meantone CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone|3/14-comma]] ||697.346||Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/19-comma meantone|4/19-comma]]
|697.427
|
|-
|[[1/5-comma meantone|1/5-comma]] ||697.654||Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/21-comma meantone|4/21-comma]]
|697.859
|
|-
|[[3/16-comma meantone|3/16-comma]] ||697.923||
|-
|[[2/11-comma meantone|2/11-comma]] ||698.045||Close to [[55edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/17-comma meantone|3/17-comma]] ||698.159||
|-
|[[4/23-comma meantone|4/23-comma]]
|698.215
|
|-
|[[1/6-comma meantone|1/6-comma]] ||698.371||Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[4/25-comma meantone|4/25-comma]] ||698.514||Close to [[67edo]].
|-
|[[3/19-comma meantone|3/19-comma]]
|698.559
|
|-
|[[2/13-comma meantone|2/13-comma]] ||698.646|| Close to [[79edo]].
|-
|[[3/20-comma meantone|3/20-comma]]
|698.729
|
|-
|[[1/7-comma meantone|1/7-comma]] ||698.883||Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/22-comma meantone|3/22-comma]]
|699.022
|
|-
|[[2/15-comma meantone|2/15-comma]] ||699.088||
|-
|[[1/8-comma meantone|1/8-comma]] ||699.267||
|-
|[[2/17-comma meantone|2/17-comma]] ||699.425||
|-
|[[1/9-comma meantone|1/9-comma]] ||699.565||
|-
|[[2/19-comma meantone|2/19-comma]]
|699.691
|
|-
|[[1/10-comma meantone|1/10-comma]] ||699.804||
|-
|[[2/21-comma meantone|2/21-comma]]
|699.907
|
|-
|[[1/11-comma meantone|1/11-comma]] ||700.000||Everything up to this point generates 12 and 19 tone MOS scales.
|-
|[[12edo]]||700.000||The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|[[1/12-comma meantone|1/12-comma]] ||700.163||Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|[[1/13-comma meantone|1/13-comma]] ||700.301||
|-
|[[1/14-comma meantone|1/14-comma]] ||700.419||
|-
|[[1/15-comma meantone|1/15-comma]] ||700.521||
|-
|[[1/16-comma meantone|1/16-comma]] ||700.611||
|-
|[[1/17-comma meantone|1/17-comma]] ||700.690||
|-
|[[1/18-comma meantone|1/18-comma]] ||700.760||
|-
|[[1/19-comma meantone|1/19-comma]]
|700.823
|
|-
|[[1/20-comma meantone|1/20-comma]]
|700.879
|
|-
|[[1/21-comma meantone|1/21-comma]]
|700.931
|
|-
|[[1/22-comma meantone|1/22-comma]]
|700.977
|
|}

=== Negative harmony theory-defined meantone (most often approached as [[superpyth]]) ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 0/1-comma to -10/21-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[Pythagorean tuning]]
|701.955||Historically significant (see [[historical temperaments]].) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|-
|[[-1/22-comma meantone|-1/22-comma]]
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|705.538
| Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]]
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|706.563
| About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]]
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[-1/4-comma meantone|-1/4-comma]]
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|710.148
|
|-
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]]
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|712.196
|
|}

=== Sharper than sharpest negative harmonic-defined meantone ===
{| class="wikitable mw-collapsible"
|+ Spectrum of meantone tunings -1/2-comma to -1/1-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[-1/2-comma meantone|-1/2-comma]]
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|715.190
|
|-
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|715.538
|
|-
|[[-7/11-comma meantone|-7/11-comma]]
|715.641
|
|-
|[[-9/14-comma meantone|-9/14-comma]]
|715.780
|Close to [[57edo]].
|-
|[[-11/17-comma meantone|-11/17-comma]]
|715.871
|
|-
|[[-13/20-comma meantone|-13/20-comma]]
|715.934
|
|-
|[[-2/3-comma meantone|-2/3-comma]]
|716.293
|Close to [[62edo]].
|-
|[[-15/22 comma meantone|-15/22 comma]]
|716.618
|Close to [[67edo]].
|-
|[[-13/19 comma meantone|-13/19 comma]]
|716.669
|Close to [[72edo]].
|-
|[[-11/16-comma meantone|-11/16-comma]]
|716.741
|
|-
|[[-9/13-comma meantone|-9/13-comma]]
|716.844
|Close to [[77edo]].
|-
|[[-7/10-comma meantone|-7/10-comma]]
|717.009
|
|-
|[[-12/17-comma meantone|-12/17-comma]]
|717.136
|Close to [[82edo]].
|-
|[[-17/24-comma meantone|-17/24-comma]]
|717.188
|Close to [[87edo]].
|-
|[[-5/7-comma meantone|-5/7-comma]]
|717.317
|Close to [[92edo]].
|-
|[[-13/18-comma meantone|-13/18-comma]]
|717.487
|Close to [[97edo]].
|-
|[[-8/11-comma meantone|-8/11-comma]]
|717.596
|
|-
|[[-19/26-comma meantone|-19/26-comma]]
|717.671
|
|-
|[[-11/15-comma meantone|-11/15-comma]]
|717.726
|
|-
|[[-14/19-comma meantone|-14/19-comma]]
|717.802
|
|-
|[[-3/4-comma meantone|-3/4-comma]]
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|[[-21/26-comma meantone|-21/26-comma]]
|718.325
|
|-
|[[-16/21-comma meantone|-16/21-comma]]
|718.341
|
|-
|[[-13/17-comma meantone|-13/17-comma]]
|718.401
|
|-
|[[-10/13-comma meantone|-10/13-comma]]
|718.498
|
|-
|[[-17/22-comma meantone|-17/22-comma]]
|718.574
|
|-
|[[-7/9-comma meantone|-7/9-comma]]
|718.682
|
|-
|[[-11/14-comma meantone|-11/14-comma]]
|718.853
|
|-
|[[-15/19-comma meantone|-15/19-comma]]
|718.934
|
|-
|[[-4/5-comma meantone|-4/5-comma]]
|719.160
|
|-
|[[-17/21-comma meantone|-17/21-comma]]
|719.365
|
|-
|[[-13/16-comma meantone|-13/16-comma]]
|719.429
|
|-
|[[-9/11-comma meantone|-9/11-comma]]
|719.551
|
|-
|[[-14/17-comma meantone|-14/17-comma]]
|719.666
|
|-
|[[-5/6-comma meantone|-5/6-comma]]
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|[[5edo]]||720.000||The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|[[-21/25-comma meantone|-21/25-comma]]
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|[[-16/19-comma meantone|-16/19-comma]]
|720.066
|
|-
|[[-11/13-comma meantone|-11/13-comma]]
|720.153
|
|-
|[[-17/20-comma meantone|-17/20-comma]]
|720.235
|
|-
|[[-6/7-comma meantone|-6/7-comma]]
|720.399
|
|-
|[[-19/22-comma meantone|-19/22-comma]]
|720.529
|
|-
|[[-13/15-comma meantone|-13/15-comma]]
|720.594
|
|-
| -[[7/8-comma meantone|7/8-comma]]
|720.773
|
|-
|[[-15/17-comma meantone|-15/17-comma]]
|720.931
|
|-
|[[-8/9-comma meantone|-8/9-comma]]
|721.017
|
|-
|[[-17/19-comma meantone|-17/19-comma]]
|721.197
|
|-
|[[-9/10-comma meantone|-9/10-comma]]
|721.311
|
|-
|[[-19/21-comma meantone|-19/21-comma]]
|721.413
|
|-
|[[-10/11-comma meantone|-10/11-comma]]
|721.506
|
|-
|[[-11/12-comma meantone|-11/12-comma]]
|721.669
|
|-
|[[-12/13-comma meantone|-12/13-comma]]
|721.807
|
|-
|[[-13/14-comma meantone|-13/14-comma]]
|721.925
|
|-
|[[-14/15-comma meantone|-14/15-comma]]
|722.028
|
|-
|[[-15/16-comma meantone|-15/16-comma]]
|722.117
|
|-
|[[-16/17-comma meantone|-16/17-comma]]
|722.196
|
|-
|[[-17/18-comma meantone|-17/18-comma]]
|722.266
|
|-
|[[-18/19-comma meantone|-18/19-comma]]
|722.329
|
|-
|[[-19/20-comma meantone|-19/20-comma]]
|722.386
|
|-
|[[-20/21-comma meantone|-20/21-comma]]
|722.437
|
|-
|[[-21/22-comma meantone|-21/22-comma]]
|722.484
|
|-
|[[-1/1-comma meantone|-1/1-comma]]
|723.461
|Close to [[68edo]].
|-
|}

[[Category:Tables]]
[[Category:Meantone]]

User:BudjarnLambeth/Table of n-comma meantone generators

2025-01-14T02:54:27Z

Moremajorthanmajor: /* The table */

{{Editable user page}}

Here are all [[meantone]] tunings that can be written in the form "n-comma meantone", where the syntonic comma ([[81/80]]) is being divided and n is a fraction between -1 and 1 with a denominator 22 or smaller.

== Scope of table ==

=== Characteristics included ===
Some of the characteristics this table mentions for each temperament include:
* Whether it saw historical (pre-1950) use
* Whether it is close to (i.e. within 1/2 a degree of closing of) an [[edo]] smaller than 100
* Whether it is the closest on the table to the optimal [[CTE]] or [[POTE]] tuning of meantone or [[superpyth]] in any JI [[limit]]
* Whether it approximates a very simple n-[[Pythagorean comma]] meantone
* Whether it is about equally sharp of [[3/2]] as some other listed temperament is flat
* Whether it is close to exactly one [[just-noticeable difference]] away from 3/2

Occasional other comments may be included as well.

=== Characteristics omitted ===
Dozens of tunings on this table are significant to [[negative harmony temperaments|negative harmony temperament theory]], enough that labelling them all individually would clutter the table.

Every tuning on this table is close to some arbitrarily large edo, but labeling them beyond [[100edo]] would clutter the table.

=== Special cases included ===
A small number of additional temperaments are included. Not too many, to avoid clutter, just the bare minimum:
* {{EDOs|7, 12, 17 and 5}} edos (to delineate small [[MOS]] shapes and boundaries of [[diamond monotone]])
* any tunings listed under "[[historical temperaments]]" (e.g. 4/25-comma), ''but only the ones of the form "n-comma"''.

== Cautions ==
=== Preservation of meantone behavior ===
Temperaments that fall outside of the "[[Historical temperaments|historically-defined meantone]]" range will not possess most of the musical properties that meantone usually possesses, but they are included for completeness.

Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.

== The table ==

=== Flatter than flattest historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 1/1-comma to 1/2-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[1/1-comma meantone|1/1-comma]] ||680.449||Close to [[30edo]]
|-
|[[21/22-comma meantone|21/22-comma]]
|681.426
|Close to [[37edo]].
|-
|[[20/21-comma meantone|20/21-comma]]
|681.473
|
|-
|[[19/20-comma meantone|19/20-comma]]
|681.524
|
|-
|[[18/19-comma meantone|18/19-comma]]
|681.581
|
|-
|[[17/18-comma meantone|17/18-comma]]
|681.644
|
|-
|[[16/17-comma meantone|16/17-comma]]
|681.713
|
|-
|[[15/16-comma meantone|15/16-comma]] ||681.793|| Close to [[44edo]].
|-
|[[14/15-comma meantone|14/15-comma]] ||681.883||
|-
|[[13/14-comma meantone|13/14-comma]] ||681.985||
|-
|[[12/13-comma meantone|12/13-comma]] ||682.103||
|-
|[[11/12-comma meantone|11/12-comma]] ||682.241||
|-
|[[10/11-comma meantone|10/11-comma]] ||682.404||Close to [[51edo]].
|-
|[[19/21-comma meantone|19/21-comma]]
|682.497
|
|-
|[[9/10-comma meantone|9/10-comma]]||682.599||
|-
|[[17/19-comma meantone|17/19-comma]]
|682.713
|
|-
|[[8/9-comma meantone|8/9-comma]] ||682.838||Close to [[58edo]].
|-
|[[15/17-comma meantone|15/17-comma]]
|682.979
|
|-
|[[7/8-comma meantone|7/8-comma]] ||683.137||Close to [[65edo]].
|-
|[[13/15-comma meantone|13/15-comma]] ||683.316||Close to [[72edo]].
|-
|[[19/22-comma meantone|19/22-comma]]
|683.381
|
|-
|[[6/7-comma meantone|6/7-comma]] ||683.521||Close to [[79edo]].
|-
|[[17/20-comma meantone|17/20-comma]]
|683.675
|
|-
|[[11/13-comma meantone|11/13-comma]] ||683.757||Close to [[86edo]].
|-
|[[16/19-comma meantone|16/19-comma]]
|683.844
|
|-
|[[21/25-comma meantone|21/25-comma]]
|683.890
|Close to [[93edo]]
|-
|[[5/6-comma meantone|5/6-comma]] ||684.033|| Close to [[100edo]].
|-
|[[14/17-comma meantone|14/17-comma]]
|684.244
|
|-
|[[9/11-comma meantone|9/11-comma]] ||684.359||
|-
|[[13/16-comma meantone|13/16-comma]] ||684.481||
|-
|[[17/21-comma meantone|17/21-comma]]
|684.545
|
|-
|[[4/5-comma meantone|4/5-comma]] ||684.750||
|-
|[[15/19-comma meantone|15/19-comma]]
|684.976
|
|-
|[[11/14-comma meantone|11/14-comma]] ||685.057||
|-
|[[7/9-comma meantone|7/9-comma]] ||685.228||
|-
|[[17/22-comma meantone|17/22-comma]]
|685.337
|
|-
|[[10/13-comma meantone|10/13-comma]] ||685.412||
|-
|[[13/17-comma meantone|13/17-comma]]
|685.509
|
|-
|[[16/21-comma meantone|16/21-comma]]
|685.569
|Everything up to this point generates 9 and 16 tone MOS scales.
|-
|[[7edo]]||685.714||The largest MOS scale this can generate is 7 tone. '''Lower boundary of 5-limit diamond monotone.'''
|-
|[[3/4-comma meantone|3/4-comma]] ||685.825||Everything from this point onwards generates 12 and 19 tone MOS scales.
|-
|[[14/19-comma meantone|14/19-comma]]
|686.108
|
|-
|[[11/15-comma meantone|11/15-comma]] ||686.184||
|-
|[[19/26-comma meantone|19/26-comma]]
|686.239
|
|-
|[[8/11-comma meantone|8/11-comma]]||686.314||
|-
|[[13/18-comma meantone|13/18-comma]]
|686.423
|
|-
|[[5/7-comma meantone|5/7-comma]] ||686.593||
|-
|[[17/24-comma meantone|17/24-comma]]
|686.721
|
|-
|[[12/17-comma meantone|12/17-comma]]
|686.774
|
|-
|[[7/10-comma meantone|7/10-comma]] ||686.901||
|-
|[[9/13-comma meantone|9/13-comma]] ||687.066||
|-
|[[11/16-comma meantone|11/16-comma]] ||687.169||
|-
|[[13/19-comma meantone|13/19-comma]]
|687.240
|
|-
|[[15/22-comma meantone|15/22-comma]]
|687.292
|
|-
|[[17/25-comma meantone|17/25-comma]]
|687.331
|
|-
|[[19/28-comma]]
|687.361
|
|-
|[[2/3-comma meantone|2/3-comma]] ||687.617||Close to [[89edo]].
|-
|[[17/26-comma meantone|17/26-comma]]
|687.893
|Close to [[82edo]].
|-
|[[15/23-comma meantone|15/23-comma]]
|687.929
|
|-
|[[13/20-comma meantone|13/20-comma]]
|687.976
|
|-
|[[11/17-comma meantone|11/17-comma]]
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone|9/14-comma]] ||688.129||
|-
|[[7/11-comma meantone|7/11-comma]] ||688.269||Close to [[68edo]].
|-
|[[12/19-comma meantone|12/19-comma]]
|688.372
|
|-
|[[5/8-comma meantone|5/8-comma]] ||688.514||Close to [[61edo]] and [[43/32]].
|-
|[[13/21-comma meantone|13/21-comma]]
|688.641
|
|-
|[[1/φ-comma meantone|1/ϕ-comma]]
|688.663
|
|-
|[[8/13-comma meantone|8/13-comma]] ||688.720||
|-
|[[11/18-comma meantone|11/18-comma]]
|688.812
|Close to [[54edo]].
|-
|[[14/23-comma meantone|14/23-comma]]
|688.864
|
|-
|[[3/5-comma meantone|3/5-comma]] ||689.051||
|-
|[[13/22-comma meantone|13/22-comma]]
|689.247
|
|-
|[[10/17-comma meantone|10/17-comma]]
|689.304
|
|-
|[[7/12-comma meantone|7/12-comma]] ||689.410||Close to [[47edo]].
|-
|[[11/19-comma meantone|11/19-comma]]
|689.504
|
|-
|[[4/7-comma meantone|4/7-comma]] ||689.666||Close to [[87edo]].
|-
|[[9/16-comma meantone|9/16-comma]] ||689.858||
|-
|[[5/9-comma meantone|5/9-comma]] ||690.007||Close to [[40edo]].
|-
|[[11/20-comma meantone|11/20-comma]]
|690.127
|
|-
|[[6/11-comma meantone|6/11-comma]] ||690.224||
|-
|[[7/13-comma meantone|7/13-comma]] ||690.375||Close to [[73edo]].
|-
|[[8/15-comma meantone|8/15-comma]] ||690.485||
|-
|[[9/17-comma meantone|9/17-comma]]
|690.569
|
|-
|[[10/19-comma meantone|10/19-comma]]
|690.636
|
|-
|[[11/21-comma meantone|11/21-comma]]
|690.690
|Close to [[33edo]]
|-
|[[1/2-comma meantone|1/2-comma]] ||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not have a whole tone between 10/9 and 9/8.
|}

=== Historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 10/21-comma to 1/22-comma
!Temperament!!Generator (cents)!!Comments
|-
|[[10/21-comma meantone|10/21-comma]]
|691.714
|
|-
|[[9/19-comma meantone|9/19-comma]]
|691.768
|Close to [[85edo]].
|-
|[[8/17-comma meantone|8/17-comma]]
|691.834
|
|-
|[[7/15-comma meantone|7/15-comma]] ||691.919||
|-
|[[6/13-comma meantone|6/13-comma]] ||692.029||
|-
|[[5/11-comma meantone|5/11-comma]] ||692.179||
|-
|[[9/20-comma meantone|9/20-comma]]
|692.277
|Close to [[26edo]].
|-
|[[4/9-comma meantone|4/9-comma]] ||692.397||
|-
|[[7/16-comma meantone|7/16-comma]] ||692.546||
|-
|[[3/7-comma meantone|3/7-comma]] ||692.738||
|-
|[[8/19-comma meantone|8/19-comma]]
|692.899
|
|-
|[[5/12-comma meantone|5/12-comma]] ||692.994||Close to [[71edo]].
|-
|[[7/17-comma meantone|7/17-comma]] ||693.099||
|-
|[[9/22-comma meantone|9/22-comma]]
|693.157
|
|-
|[[2/5-comma meantone|2/5-comma]] ||693.352||Close to [[45edo]].
|-
|[[9/23-comma meantone|9/23-comma]]
|693.539
|
|-
|[[7/18-comma meantone|7/18-comma]] ||693.591||
|-
|[[5/13-comma meantone|5/13-comma]] ||693.683||
|-
|[[1/(φ+1)-comma meantone|1/(ϕ+1)-comma]]
|693.740
|Close to [[64edo]].
|-
|[[8/21-comma meantone|8/21-comma]]
|693.762
|
|-
|[[3/8-comma meantone|3/8-comma]] ||693.890||Close to [[83edo]].
|-
|[[7/19-comma meantone|7/19-comma]]
|694.032
|
|-
|[[4/11-comma meantone|4/11-comma]] ||694.134||Almost exactly 1/3-''Pythagorean'' comma meantone.
|-
|[[5/14-comma meantone|5/14-comma]] ||694.274||
|-
|[[6/17-comma meantone|6/17-comma]] ||694.365||
|-
|[[7/20-comma meantone|7/20-comma]]
|694.428
|
|-
|[[8/23-comma meantone|8/23-comma]]
|694.475
|
|-
|[[9/26-comma meantone|9/26-comma]]
|694.511
|
|-
|[[1/3-comma meantone|1/3-comma]] ||694.786||Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[9/28-comma meantone|9/28-comma]]
|695.042
|
|-
|[[8/25-comma meantone|8/25-comma]]
|695.073
|
|-
|[[7/22-comma meantone|7/22-comma]]
|695.112
|
|-
|[[6/19-comma meantone|6/19-comma]]
|695.164
|
|-
|[[5/16-comma meantone|5/16-comma]] ||695.234||
|-
|[[4/13-comma meantone|4/13-comma]] ||695.338||
|-
|[[3/10-comma meantone|3/10-comma]] ||695.503||Close to [[88edo]] and [[Lucy tuning]]. Historically significant (see [[historical temperaments]]).
|-
|[[5/17-comma meantone|5/17-comma]] ||695.630||Close to [[69edo]].
|-
|[[7/24-comma meantone|7/24-comma]]
|695.682
|
|-
|[[2/7-comma meantone|2/7-comma]] ||695.810||Historically significant (see [[historical temperaments]]).
|-
|[[5/18-comma meantone|5/18-comma]] ||695.981||Close to [[50edo]].
|-
|[[3/11-comma meantone|3/11-comma]] ||696.090||
|-
|[[7/26-comma meantone|7/26-comma]] ||696.165||Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/15-comma meantone|4/15-comma]] ||696.220||Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|[[5/19-comma meantone|5/19-comma]]
|696.295
|Close to [[81edo]].
|-
|[[Quarter-comma meantone|1/4-comma]] ||696.578||Close to [[7-limit|septimal]] and [[tridecimal]] meantone POTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[5/21-comma meantone|5/21-comma]]
|696.834
|Close to [[31edo]].
|-
|[[4/17-comma meantone|4/17-comma]] ||696.895||
|-
|[[3/13-comma meantone|3/13-comma]] ||696.992||Close to [[7-limit|septimal]] & [[tridecimal]] meantone [[CTE]] tunings. Close to [[undecimal]] meantone POTE tuning.
|-
|[[5/22-comma meantone|5/22-comma]]
|697.067
|
|-
|[[2/9-comma meantone|2/9-comma]] ||697.176||Close to [[5-limit]] and [[undecimal]] meantone CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone|3/14-comma]] ||697.346||Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/19-comma meantone|4/19-comma]]
|697.427
|
|-
|[[1/5-comma meantone|1/5-comma]] ||697.654||Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/21-comma meantone|4/21-comma]]
|697.859
|
|-
|[[3/16-comma meantone|3/16-comma]] ||697.923||
|-
|[[2/11-comma meantone|2/11-comma]] ||698.045||Close to [[55edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/17-comma meantone|3/17-comma]] ||698.159||
|-
|[[4/23-comma meantone|4/23-comma]]
|698.215
|
|-
|[[1/6-comma meantone|1/6-comma]] ||698.371||Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[4/25-comma meantone|4/25-comma]] ||698.514||Close to [[67edo]].
|-
|[[3/19-comma meantone|3/19-comma]]
|698.559
|
|-
|[[2/13-comma meantone|2/13-comma]] ||698.646|| Close to [[79edo]].
|-
|[[3/20-comma meantone|3/20-comma]]
|698.729
|
|-
|[[1/7-comma meantone|1/7-comma]] ||698.883||Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/22-comma meantone|3/22-comma]]
|699.022
|
|-
|[[2/15-comma meantone|2/15-comma]] ||699.088||
|-
|[[1/8-comma meantone|1/8-comma]] ||699.267||
|-
|[[2/17-comma meantone|2/17-comma]] ||699.425||
|-
|[[1/9-comma meantone|1/9-comma]] ||699.565||
|-
|[[2/19-comma meantone|2/19-comma]]
|699.691
|
|-
|[[1/10-comma meantone|1/10-comma]] ||699.804||
|-
|[[2/21-comma meantone|2/21-comma]]
|699.907
|
|-
|[[1/11-comma meantone|1/11-comma]] ||700.000||Everything up to this point generates 12 and 19 tone MOS scales.
|-
|[[12edo]]||700.000||The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|[[1/12-comma meantone|1/12-comma]] ||700.163||Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|[[1/13-comma meantone|1/13-comma]] ||700.301||
|-
|[[1/14-comma meantone|1/14-comma]] ||700.419||
|-
|[[1/15-comma meantone|1/15-comma]] ||700.521||
|-
|[[1/16-comma meantone|1/16-comma]] ||700.611||
|-
|[[1/17-comma meantone|1/17-comma]] ||700.690||
|-
|[[1/18-comma meantone|1/18-comma]] ||700.760||
|-
|[[1/19-comma meantone|1/19-comma]]
|700.823
|
|-
|[[1/20-comma meantone|1/20-comma]]
|700.879
|
|-
|[[1/21-comma meantone|1/21-comma]]
|700.931
|
|-
|[[1/22-comma meantone|1/22-comma]]
|700.977
|
|}

=== Negative harmony theory-defined meantone (most often approached as [[superpyth]]) ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 0/1-comma to -10/21-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[Pythagorean tuning]]
|701.955||Historically significant (see [[historical temperaments]].) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|-
|[[-1/22-comma meantone|-1/22-comma]]
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|705.538
| Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]]
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|706.563
| About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]]
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[Negative Quarter-comma meantone|Negative Quarter-comma]]
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|710.148
|
|-
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]]
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|712.196
|
|}

=== Sharper than sharpest negative harmonic-defined meantone ===
{| class="wikitable mw-collapsible"
|+ Spectrum of meantone tunings -1/2-comma to -1/1-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[-1/2-comma meantone|-1/2-comma]]
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|715.190
|
|-
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|715.538
|
|-
|[[-7/11-comma meantone|-7/11-comma]]
|715.641
|
|-
|[[-9/14-comma meantone|-9/14-comma]]
|715.780
|Close to [[57edo]].
|-
|[[-11/17-comma meantone|-11/17-comma]]
|715.871
|
|-
|[[-13/20-comma meantone|-13/20-comma]]
|715.934
|
|-
|[[-2/3-comma meantone|-2/3-comma]]
|716.293
|Close to [[62edo]].
|-
|[[-15/22 comma meantone|-15/22 comma]]
|716.618
|Close to [[67edo]].
|-
|[[-13/19 comma meantone|-13/19 comma]]
|716.669
|Close to [[72edo]].
|-
|[[-11/16-comma meantone|-11/16-comma]]
|716.741
|
|-
|[[-9/13-comma meantone|-9/13-comma]]
|716.844
|Close to [[77edo]].
|-
|[[-7/10-comma meantone|-7/10-comma]]
|717.009
|
|-
|[[-12/17-comma meantone|-12/17-comma]]
|717.136
|Close to [[82edo]].
|-
|[[-17/24-comma meantone|-17/24-comma]]
|717.188
|Close to [[87edo]].
|-
|[[-5/7-comma meantone|-5/7-comma]]
|717.317
|Close to [[92edo]].
|-
|[[-13/18-comma meantone|-13/18-comma]]
|717.487
|Close to [[97edo]].
|-
|[[-8/11-comma meantone|-8/11-comma]]
|717.596
|
|-
|[[-19/26-comma meantone|-19/26-comma]]
|717.671
|
|-
|[[-11/15-comma meantone|-11/15-comma]]
|717.726
|
|-
|[[-14/19-comma meantone|-14/19-comma]]
|717.802
|
|-
|[[-3/4-comma meantone|-3/4-comma]]
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|[[-21/26-comma meantone|-21/26-comma]]
|718.325
|
|-
|[[-16/21-comma meantone|-16/21-comma]]
|718.341
|
|-
|[[-13/17-comma meantone|-13/17-comma]]
|718.401
|
|-
|[[-10/13-comma meantone|-10/13-comma]]
|718.498
|
|-
|[[-17/22-comma meantone|-17/22-comma]]
|718.574
|
|-
|[[-7/9-comma meantone|-7/9-comma]]
|718.682
|
|-
|[[-11/14-comma meantone|-11/14-comma]]
|718.853
|
|-
|[[-15/19-comma meantone|-15/19-comma]]
|718.934
|
|-
|[[-4/5-comma meantone|-4/5-comma]]
|719.160
|
|-
|[[-17/21-comma meantone|-17/21-comma]]
|719.365
|
|-
|[[-13/16-comma meantone|-13/16-comma]]
|719.429
|
|-
|[[-9/11-comma meantone|-9/11-comma]]
|719.551
|
|-
|[[-14/17-comma meantone|-14/17-comma]]
|719.666
|
|-
|[[-5/6-comma meantone|-5/6-comma]]
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|[[5edo]]||720.000||The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|[[-21/25-comma meantone|-21/25-comma]]
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|[[-16/19-comma meantone|-16/19-comma]]
|720.066
|
|-
|[[-11/13-comma meantone|-11/13-comma]]
|720.153
|
|-
|[[-17/20-comma meantone|-17/20-comma]]
|720.235
|
|-
|[[-6/7-comma meantone|-6/7-comma]]
|720.399
|
|-
|[[-19/22-comma meantone|-19/22-comma]]
|720.529
|
|-
|[[-13/15-comma meantone|-13/15-comma]]
|720.594
|
|-
| -[[7/8-comma meantone|7/8-comma]]
|720.773
|
|-
|[[-15/17-comma meantone|-15/17-comma]]
|720.931
|
|-
|[[-8/9-comma meantone|-8/9-comma]]
|721.017
|
|-
|[[-17/19-comma meantone|-17/19-comma]]
|721.197
|
|-
|[[-9/10-comma meantone|-9/10-comma]]
|721.311
|
|-
|[[-19/21-comma meantone|-19/21-comma]]
|721.413
|
|-
|[[-10/11-comma meantone|-10/11-comma]]
|721.506
|
|-
|[[-11/12-comma meantone|-11/12-comma]]
|721.669
|
|-
|[[-12/13-comma meantone|-12/13-comma]]
|721.807
|
|-
|[[-13/14-comma meantone|-13/14-comma]]
|721.925
|
|-
|[[-14/15-comma meantone|-14/15-comma]]
|722.028
|
|-
|[[-15/16-comma meantone|-15/16-comma]]
|722.117
|
|-
|[[-16/17-comma meantone|-16/17-comma]]
|722.196
|
|-
|[[-17/18-comma meantone|-17/18-comma]]
|722.266
|
|-
|[[-18/19-comma meantone|-18/19-comma]]
|722.329
|
|-
|[[-19/20-comma meantone|-19/20-comma]]
|722.386
|
|-
|[[-20/21-comma meantone|-20/21-comma]]
|722.437
|
|-
|[[-21/22-comma meantone|-21/22-comma]]
|722.484
|
|-
|[[-1/1-comma meantone|-1/1-comma]]
|723.461
|Close to [[68edo]].
|-
|}

[[Category:Tables]]
[[Category:Meantone]]

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-13T21:15:44Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Fa♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Si
|15
|Major seventh, fourteenth
|-
|4 fifths
|Mi
|5
|Major tenth, seventeenth
|-
|3 fifths
|La
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Re
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|Sol
|3
|Perfect twelfth
|-
|0
|Natural, ''naturalis''
|Ut > Do
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Fa, originally ''superparticularis''
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Fa ''superbipartiens'' > Si♭
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa ''supertripartiens'' > Mi♭
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > La♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Re♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|Sol♭
|*11
|Diminished twelfth
|}
Major is considered as comparable to Sol as minor is to Fa, but Sol ''superparticularis'' and Sol ''superpartiens'' never saw as widespread usage as Fa ''superparticularis'' and Fa ''superpartiens'' before the conversion of the latter to flats. At that time, it was also widespread, but not absolute, that only the true relations, at least for the first three steps from the octave on the chain of fifths, and thus the 2.3.(5).7.(13).(17).19.43 subgroup, were considered within the bounds of the modal system. The paradox of this is that the true relations generally do not have the same desired (sub)harmonics for ''fortis'' and ''lenis''. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean tetrachords|mean tetrachord]], which is primarily considered to temper out [[129/128]].

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-13T20:56:11Z

Moremajorthanmajor: Moremajorthanmajor moved page User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n comma mean tetrachords to User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean tetrachords

{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -1 and 1 with a denominator 14 or smaller. This range is almost the same as the range between [[61edo|61bedo]] and its complementary opposite.

== Cautions ==
As tempering out this comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129), the mean minor tetrachord (root-whole tone-minor third-tempered fourth) is quoted as the lemma.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 and 16/9.
{| class="wikitable"
|+Spectrum of mean tetrachord tunings
!Mean tetrachord temperament
!Intervals (cents)
!Comments
|-
|Whole-comma
|176.965-334.553-511.518
|
|-
|8/9-comma
|179.959-330.062-510.021
|
|-
|7/8-comma
|180.333-329.501-509.834
|
|-
|6/7-comma
|180.814-328.779-509.593
|
|-
|5/6-comma
|181.455-327.817-509.272
|
|-
|4/5-comma
|182.354-326.469-508.823
|
|-
|7/9-comma
|182.952-325.571-508.523
|
|-
|3/4-comma
|183.701-324.449-508.150
|
|-
|5/7-comma
|184.633-323.005-507.638
|
|-
|2/3-comma
|185.946-321.080-507.027
|
|-
|5/8-comma
|187.069-319.396-506.465
|
|-
|3/5-comma
|187.743-318.386-506.129
|
|-
|4/7-comma
|188.512-317.231-505.744
|
|-
|5/9-comma
|188.940-316.590-505.530
|
|-
|1/2-comma
|190.437-314.344-504.781
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|4/9-comma
|191.934-312.099-504.033
|
|-
|3/7-comma
|192.362-311.457-503.819
|
|-
|2/5-comma
|193.132-310.302-503.434
|Almost exactly meantone
|-
|3/8-comma
|193.805-309.291-503.096
|
|-
|1/3-comma
|194.928-307.608-502.536
|
|-
|2/7-comma
|196.211-305.683-501.894
|
|-
|1/4-comma
|197.174-304.240-501.413
|
|-
|2/9-comma
|197.922-303.117-501.039
|
|-
|1/5-comma
|198.521-302.219-500.740
|
|-
|1/6-comma
|199.419-300.871-500.290
|
|-
|1/7-comma
|200.061-299.909-499.970
|
|-
|1/8-comma
|200.542-299.187-499.729
|
|-
|1/9-comma
|200.916-298.626-499.542
|
|-
|[[Pythagorean tuning]]
|203.910-294.135-498.045
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/9-comma
|206.904-289.644-496.548
|
|-
| -1/8-comma
|207.278-289.083-496.361
|
|-
| -1/7-comma
|207.759-288.361-496,120
|
|-
| -1/6-comma
|208.401-287.399-495.800
|
|-
| -1/5-comma
|209.299-286.051-495.350
|
|-
| -2/9-comma
|209.898-285.153-495.051
|
|-
| -1/4-comma
|210.646-284.030-494.677
|
|-
| -2/7-comma
|211.609-282.587-494.196
|
|-
| -1/3-comma
|212.892-280.662-493.554
|
|-
| -3/8-comma
|214.014-278.979-492.993
|
|-
| -2/5-comma
|214.688-277.968-492.656
|
|-
| -3/7-comma
|215.458-276.813-492.271
|
|-
| -4/9-comma
|215.886-276.171-492.057
|
|-
| -1/2-comma
|217.383-273.926-491.309
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161
|-
| -5/9-comma
|218.880-271.680-490.560
|
|-
| -4/7-comma
|219.307-271.039-490.346
|
|-
| -3/5-comma
|220.077-269.884-489.961
|
|-
| -5/8-comma
|220.751-268.874-489,625
|
|-
| -2/3-comma
|221.874-267.190-489.063
|
|-
| -5/7-comma
|223.157-265.265-488.422
|
|-
| -3/4-comma
|224.119-263.821–487.940
|
|-
| -7/9-comma
|224.868-262.698-487.566
|
|-
| -4/5-comma
|225.466-261.801-487.267
|
|-
| -5/6-comma
|226.365-260.453-486.818
|
|-
| -6/7-comma
|227.006-259.491-486.497
|
|-
| -7/8-comma
|227.487-258.769-486.256
|
|-
| -8/9-comma
|227.861-258.208-486.069
|
|-
|Negative Whole-comma
|230.855-253.717-484.752
|
|}

User:BudjarnLambeth/Table of n-comma meantone generators

2025-01-13T20:34:59Z

Moremajorthanmajor: /* The table */

{{Editable user page}}

Here are all [[meantone]] tunings that can be written in the form "n-comma meantone", where the syntonic comma ([[81/80]]) is being divided and n is a fraction between -1 and 1 with a denominator 22 or smaller.

== Scope of table ==

=== Characteristics included ===
Some of the characteristics this table mentions for each temperament include:
* Whether it saw historical (pre-1950) use
* Whether it is close to (i.e. within 1/2 a degree of closing of) an [[edo]] smaller than 100
* Whether it is the closest on the table to the optimal [[CTE]] or [[POTE]] tuning of meantone or [[superpyth]] in any JI [[limit]]
* Whether it approximates a very simple n-[[Pythagorean comma]] meantone
* Whether it is about equally sharp of [[3/2]] as some other listed temperament is flat
* Whether it is close to exactly one [[just-noticeable difference]] away from 3/2

Occasional other comments may be included as well.

=== Characteristics omitted ===
Dozens of tunings on this table are significant to [[negative harmony temperaments|negative harmony temperament theory]], enough that labelling them all individually would clutter the table.

Every tuning on this table is close to some arbitrarily large edo, but labeling them beyond [[100edo]] would clutter the table.

=== Special cases included ===
A small number of additional temperaments are included. Not too many, to avoid clutter, just the bare minimum:
* {{EDOs|7, 12, 17 and 5}} edos (to delineate small [[MOS]] shapes and boundaries of [[diamond monotone]])
* any tunings listed under "[[historical temperaments]]" (e.g. 4/25-comma), ''but only the ones of the form "n-comma"''.

== Cautions ==
=== Preservation of meantone behavior ===
Temperaments that fall outside of the "[[Historical temperaments|historically-defined meantone]]" range will not possess most of the musical properties that meantone usually possesses, but they are included for completeness.

Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.

== The table ==

=== Flatter than flattest historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 1/1-comma to 1/2-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[1/1-comma meantone|1/1-comma]] ||680.449||Close to [[30edo]]
|-
|[[21/22-comma meantone|21/22-comma]]
|681.426
|Close to [[37edo]].
|-
|[[20/21-comma meantone|20/21-comma]]
|681.473
|
|-
|[[19/20-comma meantone|19/20-comma]]
|681.524
|
|-
|[[18/19-comma meantone|18/19-comma]]
|681.581
|
|-
|[[17/18-comma meantone|17/18-comma]]
|681.644
|
|-
|[[16/17-comma meantone|16/17-comma]]
|681.713
|
|-
|[[15/16-comma meantone|15/16-comma]] ||681.793|| Close to [[44edo]].
|-
|[[14/15-comma meantone|14/15-comma]] ||681.883||
|-
|[[13/14-comma meantone|13/14-comma]] ||681.985||
|-
|[[12/13-comma meantone|12/13-comma]] ||682.103||
|-
|[[11/12-comma meantone|11/12-comma]] ||682.241||
|-
|[[10/11-comma meantone|10/11-comma]] ||682.404||Close to [[51edo]].
|-
|[[19/21-comma meantone|19/21-comma]]
|682.497
|
|-
|[[9/10-comma meantone|9/10-comma]]||682.599||
|-
|[[17/19-comma meantone|17/19-comma]]
|682.713
|
|-
|[[8/9-comma meantone|8/9-comma]] ||682.838||Close to [[58edo]].
|-
|[[15/17-comma meantone|15/17-comma]]
|682.979
|
|-
|[[7/8-comma meantone|7/8-comma]] ||683.137||Close to [[65edo]].
|-
|[[13/15-comma meantone|13/15-comma]] ||683.316||Close to [[72edo]].
|-
|[[19/22-comma meantone|19/22-comma]]
|683.381
|
|-
|[[6/7-comma meantone|6/7-comma]] ||683.521||Close to [[79edo]].
|-
|[[17/20-comma meantone|17/20-comma]]
|683.675
|
|-
|[[11/13-comma meantone|11/13-comma]] ||683.757||Close to [[86edo]].
|-
|[[16/19-comma meantone|16/19-comma]]
|683.844
|
|-
|[[21/25-comma meantone|21/25-comma]]
|683.890
|Close to [[93edo]]
|-
|[[5/6-comma meantone|5/6-comma]] ||684.033|| Close to [[100edo]].
|-
|[[14/17-comma meantone|14/17-comma]]
|684.244
|
|-
|[[9/11-comma meantone|9/11-comma]] ||684.359||
|-
|[[13/16-comma meantone|13/16-comma]] ||684.481||
|-
|[[17/21-comma meantone|17/21-comma]]
|684.545
|
|-
|[[4/5-comma meantone|4/5-comma]] ||684.750||
|-
|[[15/19-comma meantone|15/19-comma]]
|684.976
|
|-
|[[11/14-comma meantone|11/14-comma]] ||685.057||
|-
|[[7/9-comma meantone|7/9-comma]] ||685.228||
|-
|[[17/22-comma meantone|17/22-comma]]
|685.337
|
|-
|[[10/13-comma meantone|10/13-comma]] ||685.412||
|-
|[[13/17-comma meantone|13/17-comma]]
|685.509
|
|-
|[[16/21-comma meantone|16/21-comma]]
|685.569
|Everything up to this point generates 9 and 16 tone MOS scales.
|-
|[[7edo]]||685.714||The largest MOS scale this can generate is 7 tone. '''Lower boundary of 5-limit diamond monotone.'''
|-
|[[3/4-comma meantone|3/4-comma]] ||685.825||Everything from this point onwards generates 12 and 19 tone MOS scales.
|-
|[[14/19-comma meantone|14/19-comma]]
|686.108
|
|-
|[[11/15-comma meantone|11/15-comma]] ||686.184||
|-
|[[19/26-comma meantone|19/26-comma]]
|686.239
|
|-
|[[8/11-comma meantone|8/11-comma]]||686.314||
|-
|[[13/18-comma meantone|13/18-comma]]
|686.423
|
|-
|[[5/7-comma meantone|5/7-comma]] ||686.593||
|-
|[[17/24-comma meantone|17/24-comma]]
|686.721
|
|-
|[[12/17-comma meantone|12/17-comma]]
|686.774
|
|-
|[[7/10-comma meantone|7/10-comma]] ||686.901||
|-
|[[9/13-comma meantone|9/13-comma]] ||687.066||
|-
|[[11/16-comma meantone|11/16-comma]] ||687.169||
|-
|[[13/19-comma meantone|13/19-comma]]
|687.240
|
|-
|[[15/22-comma meantone|15/22-comma]]
|687.292
|
|-
|[[17/25-comma meantone|17/25-comma]]
|687.331
|
|-
|[[19/28-comma]]
|687.361
|
|-
|[[2/3-comma meantone|2/3-comma]] ||687.617||Close to [[89edo]].
|-
|[[17/26-comma meantone|17/26-comma]]
|687.893
|Close to [[82edo]].
|-
|[[15/23-comma meantone|15/23-comma]]
|687.929
|
|-
|[[13/20-comma meantone|13/20-comma]]
|687.976
|
|-
|[[11/17-comma meantone|11/17-comma]]
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone|9/14-comma]] ||688.129||
|-
|[[7/11-comma meantone|7/11-comma]] ||688.269||Close to [[68edo]].
|-
|[[12/19-comma meantone|12/19-comma]]
|688.372
|
|-
|[[5/8-comma meantone|5/8-comma]] ||688.514||Close to [[61edo]] and [[43/32]].
|-
|[[13/21-comma meantone|13/21-comma]]
|688.641
|
|-
|[[Golden ratio-comma meantone|Golden ratio-comma]]
|688.663
|
|-
|[[8/13-comma meantone|8/13-comma]] ||688.720||
|-
|[[11/18-comma meantone|11/18-comma]]
|688.812
|Close to [[54edo]].
|-
|[[14/23-comma meantone|14/23-comma]]
|688.864
|
|-
|[[3/5-comma meantone|3/5-comma]] ||689.051||
|-
|[[13/22-comma meantone|13/22-comma]]
|689.247
|
|-
|[[10/17-comma meantone|10/17-comma]]
|689.304
|
|-
|[[7/12-comma meantone|7/12-comma]] ||689.410||Close to [[47edo]].
|-
|[[11/19-comma meantone|11/19-comma]]
|689.504
|
|-
|[[4/7-comma meantone|4/7-comma]] ||689.666||Close to [[87edo]].
|-
|[[9/16-comma meantone|9/16-comma]] ||689.858||
|-
|[[5/9-comma meantone|5/9-comma]] ||690.007||Close to [[40edo]].
|-
|[[11/20-comma meantone|11/20-comma]]
|690.127
|
|-
|[[6/11-comma meantone|6/11-comma]] ||690.224||
|-
|[[7/13-comma meantone|7/13-comma]] ||690.375||Close to [[73edo]].
|-
|[[8/15-comma meantone|8/15-comma]] ||690.485||
|-
|[[9/17-comma meantone|9/17-comma]]
|690.569
|
|-
|[[10/19-comma meantone|10/19-comma]]
|690.636
|
|-
|[[11/21-comma meantone|11/21-comma]]
|690.690
|Close to [[33edo]]
|-
|[[1/2-comma meantone|1/2-comma]] ||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not have a whole tone between 10/9 and 9/8.
|}

=== Historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 10/21-comma to 1/22-comma
!Temperament!!Generator (cents)!!Comments
|-
|[[10/21-comma meantone|10/21-comma]]
|691.714
|
|-
|[[9/19-comma meantone|9/19-comma]]
|691.768
|Close to [[85edo]].
|-
|[[8/17-comma meantone|8/17-comma]]
|691.834
|
|-
|[[7/15-comma meantone|7/15-comma]] ||691.919||
|-
|[[6/13-comma meantone|6/13-comma]] ||692.029||
|-
|[[5/11-comma meantone|5/11-comma]] ||692.179||
|-
|[[9/20-comma meantone|9/20-comma]]
|692.277
|Close to [[26edo]].
|-
|[[4/9-comma meantone|4/9-comma]] ||692.397||
|-
|[[7/16-comma meantone|7/16-comma]] ||692.546||
|-
|[[3/7-comma meantone|3/7-comma]] ||692.738||
|-
|[[8/19-comma meantone|8/19-comma]]
|692.899
|
|-
|[[5/12-comma meantone|5/12-comma]] ||692.994||Close to [[71edo]].
|-
|[[7/17-comma meantone|7/17-comma]] ||693.099||
|-
|[[9/22-comma meantone|9/22-comma]]
|693.157
|
|-
|[[2/5-comma meantone|2/5-comma]] ||693.352||Close to [[45edo]].
|-
|[[9/23-comma meantone|9/23-comma]]
|693.539
|
|-
|[[7/18-comma meantone|7/18-comma]] ||693.591||
|-
|[[5/13-comma meantone|5/13-comma]] ||693.683||
|-
|[[Split golden ratio-comma meantone|Split golden ratio-comma]]
|693.740
|Close to [[64edo]].
|-
|[[8/21-comma meantone|8/21-comma]]
|693.762
|
|-
|[[3/8-comma meantone|3/8-comma]] ||693.890||Close to [[83edo]].
|-
|[[7/19-comma meantone|7/19-comma]]
|694.032
|
|-
|[[4/11-comma meantone|4/11-comma]] ||694.134||Almost exactly 1/3-''Pythagorean'' comma meantone.
|-
|[[5/14-comma meantone|5/14-comma]] ||694.274||
|-
|[[6/17-comma meantone|6/17-comma]] ||694.365||
|-
|[[7/20-comma meantone|7/20-comma]]
|694.428
|
|-
|[[8/23-comma meantone|8/23-comma]]
|694.475
|
|-
|[[9/26-comma meantone|9/26-comma]]
|694.511
|
|-
|[[1/3-comma meantone|1/3-comma]] ||694.786||Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[9/28-comma meantone|9/28-comma]]
|695.042
|
|-
|[[8/25-comma meantone|8/25-comma]]
|695.073
|
|-
|[[7/22-comma meantone|7/22-comma]]
|695.112
|
|-
|[[6/19-comma meantone|6/19-comma]]
|695.164
|
|-
|[[5/16-comma meantone|5/16-comma]] ||695.234||
|-
|[[4/13-comma meantone|4/13-comma]] ||695.338||
|-
|[[3/10-comma meantone|3/10-comma]] ||695.503||Close to [[88edo]] and [[Lucy tuning]]. Historically significant (see [[historical temperaments]]).
|-
|[[5/17-comma meantone|5/17-comma]] ||695.630||Close to [[69edo]].
|-
|[[7/24-comma meantone|7/24-comma]]
|695.682
|
|-
|[[2/7-comma meantone|2/7-comma]] ||695.810||Historically significant (see [[historical temperaments]]).
|-
|[[5/18-comma meantone|5/18-comma]] ||695.981||Close to [[50edo]].
|-
|[[3/11-comma meantone|3/11-comma]] ||696.090||
|-
|[[7/26-comma meantone|7/26-comma]] ||696.165||Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/15-comma meantone|4/15-comma]] ||696.220||Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|[[5/19-comma meantone|5/19-comma]]
|696.295
|Close to [[81edo]].
|-
|[[Quarter-comma meantone|1/4-comma]] ||696.578||Close to [[7-limit|septimal]] and [[tridecimal]] meantone POTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[5/21-comma meantone|5/21-comma]]
|696.834
|Close to [[31edo]].
|-
|[[4/17-comma meantone|4/17-comma]] ||696.895||
|-
|[[3/13-comma meantone|3/13-comma]] ||696.992||Close to [[7-limit|septimal]] & [[tridecimal]] meantone [[CTE]] tunings. Close to [[undecimal]] meantone POTE tuning.
|-
|[[5/22-comma meantone|5/22-comma]]
|697.067
|
|-
|[[2/9-comma meantone|2/9-comma]] ||697.176||Close to [[5-limit]] and [[undecimal]] meantone CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone|3/14-comma]] ||697.346||Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/19-comma meantone|4/19-comma]]
|697.427
|
|-
|[[1/5-comma meantone|1/5-comma]] ||697.654||Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/21-comma meantone|4/21-comma]]
|697.859
|
|-
|[[3/16-comma meantone|3/16-comma]] ||697.923||
|-
|[[2/11-comma meantone|2/11-comma]] ||698.045||Close to [[55edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/17-comma meantone|3/17-comma]] ||698.159||
|-
|[[4/23-comma meantone|4/23-comma]]
|698.215
|
|-
|[[1/6-comma meantone|1/6-comma]] ||698.371||Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[4/25-comma meantone|4/25-comma]] ||698.514||Close to [[67edo]].
|-
|[[3/19-comma meantone|3/19-comma]]
|698.559
|
|-
|[[2/13-comma meantone|2/13-comma]] ||698.646|| Close to [[79edo]].
|-
|[[3/20-comma meantone|3/20-comma]]
|698.729
|
|-
|[[1/7-comma meantone|1/7-comma]] ||698.883||Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/22-comma meantone|3/22-comma]]
|699.022
|
|-
|[[2/15-comma meantone|2/15-comma]] ||699.088||
|-
|[[1/8-comma meantone|1/8-comma]] ||699.267||
|-
|[[2/17-comma meantone|2/17-comma]] ||699.425||
|-
|[[1/9-comma meantone|1/9-comma]] ||699.565||
|-
|[[2/19-comma meantone|2/19-comma]]
|699.691
|
|-
|[[1/10-comma meantone|1/10-comma]] ||699.804||
|-
|[[2/21-comma meantone|2/21-comma]]
|699.907
|
|-
|[[1/11-comma meantone|1/11-comma]] ||700.000||Everything up to this point generates 12 and 19 tone MOS scales.
|-
|[[12edo]]||700.000||The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|[[1/12-comma meantone|1/12-comma]] ||700.163||Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|[[1/13-comma meantone|1/13-comma]] ||700.301||
|-
|[[1/14-comma meantone|1/14-comma]] ||700.419||
|-
|[[1/15-comma meantone|1/15-comma]] ||700.521||
|-
|[[1/16-comma meantone|1/16-comma]] ||700.611||
|-
|[[1/17-comma meantone|1/17-comma]] ||700.690||
|-
|[[1/18-comma meantone|1/18-comma]] ||700.760||
|-
|[[1/19-comma meantone|1/19-comma]]
|700.823
|
|-
|[[1/20-comma meantone|1/20-comma]]
|700.879
|
|-
|[[1/21-comma meantone|1/21-comma]]
|700.931
|
|-
|[[1/22-comma meantone|1/22-comma]]
|700.977
|
|}

=== Negative harmony theory-defined meantone (most often approached as [[superpyth]]) ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 0/1-comma to -10/21-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[Pythagorean tuning]]
|701.955||Historically significant (see [[historical temperaments]].) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|-
|[[-1/22-comma meantone|-1/22-comma]]
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|705.538
| Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]]
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|706.563
| About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]]
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[Negative Quarter-comma meantone|Negative Quarter-comma]]
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|710.148
|
|-
|[[Negative split golden ratio-comma meantone|Negative split golden ratio-comma]]
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]]
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|712.196
|
|}

=== Sharper than sharpest negative harmonic-defined meantone ===
{| class="wikitable mw-collapsible"
|+ Spectrum of meantone tunings -1/2-comma to -1/1-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[-1/2-comma meantone|-1/2-comma]]
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|715.190
|
|-
|[[Negative golden ratio-comma meantone|Negative golden ratio-comma]]
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|715.538
|
|-
|[[-7/11-comma meantone|-7/11-comma]]
|715.641
|
|-
|[[-9/14-comma meantone|-9/14-comma]]
|715.780
|Close to [[57edo]].
|-
|[[-11/17-comma meantone|-11/17-comma]]
|715.871
|
|-
|[[-13/20-comma meantone|-13/20-comma]]
|715.934
|
|-
|[[-2/3-comma meantone|-2/3-comma]]
|716.293
|Close to [[62edo]].
|-
|[[-15/22 comma meantone|-15/22 comma]]
|716.618
|Close to [[67edo]].
|-
|[[-13/19 comma meantone|-13/19 comma]]
|716.669
|Close to [[72edo]].
|-
|[[-11/16-comma meantone|-11/16-comma]]
|716.741
|
|-
|[[-9/13-comma meantone|-9/13-comma]]
|716.844
|Close to [[77edo]].
|-
|[[-7/10-comma meantone|-7/10-comma]]
|717.009
|
|-
|[[-12/17-comma meantone|-12/17-comma]]
|717.136
|Close to [[82edo]].
|-
|[[-17/24-comma meantone|-17/24-comma]]
|717.188
|Close to [[87edo]].
|-
|[[-5/7-comma meantone|-5/7-comma]]
|717.317
|Close to [[92edo]].
|-
|[[-13/18-comma meantone|-13/18-comma]]
|717.487
|Close to [[97edo]].
|-
|[[-8/11-comma meantone|-8/11-comma]]
|717.596
|
|-
|[[-19/26-comma meantone|-19/26-comma]]
|717.671
|
|-
|[[-11/15-comma meantone|-11/15-comma]]
|717.726
|
|-
|[[-14/19-comma meantone|-14/19-comma]]
|717.802
|
|-
|[[-3/4-comma meantone|-3/4-comma]]
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|[[-21/26-comma meantone|-21/26-comma]]
|718.325
|
|-
|[[-16/21-comma meantone|-16/21-comma]]
|718.341
|
|-
|[[-13/17-comma meantone|-13/17-comma]]
|718.401
|
|-
|[[-10/13-comma meantone|-10/13-comma]]
|718.498
|
|-
|[[-17/22-comma meantone|-17/22-comma]]
|718.574
|
|-
|[[-7/9-comma meantone|-7/9-comma]]
|718.682
|
|-
|[[-11/14-comma meantone|-11/14-comma]]
|718.853
|
|-
|[[-15/19-comma meantone|-15/19-comma]]
|718.934
|
|-
|[[-4/5-comma meantone|-4/5-comma]]
|719.160
|
|-
|[[-17/21-comma meantone|-17/21-comma]]
|719.365
|
|-
|[[-13/16-comma meantone|-13/16-comma]]
|719.429
|
|-
|[[-9/11-comma meantone|-9/11-comma]]
|719.551
|
|-
|[[-14/17-comma meantone|-14/17-comma]]
|719.666
|
|-
|[[-5/6-comma meantone|-5/6-comma]]
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|[[5edo]]||720.000||The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|[[-21/25-comma meantone|-21/25-comma]]
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|[[-16/19-comma meantone|-16/19-comma]]
|720.066
|
|-
|[[-11/13-comma meantone|-11/13-comma]]
|720.153
|
|-
|[[-17/20-comma meantone|-17/20-comma]]
|720.235
|
|-
|[[-6/7-comma meantone|-6/7-comma]]
|720.399
|
|-
|[[-19/22-comma meantone|-19/22-comma]]
|720.529
|
|-
|[[-13/15-comma meantone|-13/15-comma]]
|720.594
|
|-
| -[[7/8-comma meantone|7/8-comma]]
|720.773
|
|-
|[[-15/17-comma meantone|-15/17-comma]]
|720.931
|
|-
|[[-8/9-comma meantone|-8/9-comma]]
|721.017
|
|-
|[[-17/19-comma meantone|-17/19-comma]]
|721.197
|
|-
|[[-9/10-comma meantone|-9/10-comma]]
|721.311
|
|-
|[[-19/21-comma meantone|-19/21-comma]]
|721.413
|
|-
|[[-10/11-comma meantone|-10/11-comma]]
|721.506
|
|-
|[[-11/12-comma meantone|-11/12-comma]]
|721.669
|
|-
|[[-12/13-comma meantone|-12/13-comma]]
|721.807
|
|-
|[[-13/14-comma meantone|-13/14-comma]]
|721.925
|
|-
|[[-14/15-comma meantone|-14/15-comma]]
|722.028
|
|-
|[[-15/16-comma meantone|-15/16-comma]]
|722.117
|
|-
|[[-16/17-comma meantone|-16/17-comma]]
|722.196
|
|-
|[[-17/18-comma meantone|-17/18-comma]]
|722.266
|
|-
|[[-18/19-comma meantone|-18/19-comma]]
|722.329
|
|-
|[[-19/20-comma meantone|-19/20-comma]]
|722.386
|
|-
|[[-20/21-comma meantone|-20/21-comma]]
|722.437
|
|-
|[[-21/22-comma meantone|-21/22-comma]]
|722.484
|
|-
|[[-1/1-comma meantone|-1/1-comma]]
|723.461
|Close to [[68edo]].
|-
|}

[[Category:Tables]]
[[Category:Meantone]]

User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n-comma mean minor triads

2025-01-11T07:16:48Z

Moremajorthanmajor: Created page with "{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma (129/128) i..."

{{Editable user page}}Here are all mean tetrachord tunings that can be written in the form "m/n-comma mean tetrachord", where the '''43-limit Johnston''' comma ([[129/128]]) is being divided and n is a fraction between -1 and 1 with a denominator 14 or smaller. This range is almost the same as the range between [[61edo|61bedo]] and its complementary opposite.

== Cautions ==
As tempering out this comma renders minor the simpler triad than major (36:43:54 vs. 86:108:129), the mean minor tetrachord (root-whole tone-minor third-tempered fourth) is quoted as the lemma.

As this comma is considered to most importantly distinguish the harmonic and perfect fourths, the entire spectrum until [[Pythagorean tuning]] is fictionally significant, though the desired minor seventh falls between 43/24 and 16/9.
{| class="wikitable"
|+Spectrum of mean tetrachord tunings
!Mean tetrachord temperament
!Intervals (cents)
!Comments
|-
|Whole-comma
|176.965-334.553-511.518
|
|-
|8/9-comma
|179.959-330.062-510.021
|
|-
|7/8-comma
|180.333-329.501-509.834
|
|-
|6/7-comma
|180.814-328.779-509.593
|
|-
|5/6-comma
|181.455-327.817-509.272
|
|-
|4/5-comma
|182.354-326.469-508.823
|
|-
|7/9-comma
|182.952-325.571-508.523
|
|-
|3/4-comma
|183.701-324.449-508.150
|
|-
|5/7-comma
|184.633-323.005-507.638
|
|-
|2/3-comma
|185.946-321.080-507.027
|
|-
|5/8-comma
|187.069-319.396-506.465
|
|-
|3/5-comma
|187.743-318.386-506.129
|
|-
|4/7-comma
|188.512-317.231-505.744
|
|-
|5/9-comma
|188.940-316.590-505.530
|
|-
|1/2-comma
|190.437-314.344-504.781
|Everything from this point onwards has a minor seventh between 43/24 and 16/9. This is the canonical mean tetrachord tuning in universe
|-
|4/9-comma
|191.934-312.099-504.033
|
|-
|3/7-comma
|192.362-311.457-503.819
|
|-
|2/5-comma
|193.132-310.302-503.434
|Almost exactly meantone
|-
|3/8-comma
|193.805-309.291-503.096
|
|-
|1/3-comma
|194.928-307.608-502.536
|
|-
|2/7-comma
|196.211-305.683-501.894
|
|-
|1/4-comma
|197.174-304.240-501.413
|
|-
|2/9-comma
|197.922-303.117-501.039
|
|-
|1/5-comma
|198.521-302.219-500.740
|
|-
|1/6-comma
|199.419-300.871-500.290
|
|-
|1/7-comma
|200.061-299.909-499.970
|
|-
|1/8-comma
|200.542-299.187-499.729
|
|-
|1/9-comma
|200.916-298.626-499.542
|
|-
|[[Pythagorean tuning]]
|203.910-294.135-498.045
|Everything from this point onwards does not have a minor seventh between 43/24 and 16/9
|-
| -1/9-comma
|206.904-289.644-496.548
|
|-
| -1/8-comma
|207.278-289.083-496.361
|
|-
| -1/7-comma
|207.759-288.361-496,120
|
|-
| -1/6-comma
|208.401-287.399-495.800
|
|-
| -1/5-comma
|209.299-286.051-495.350
|
|-
| -2/9-comma
|209.898-285.153-495.051
|
|-
| -1/4-comma
|210.646-284.030-494.677
|
|-
| -2/7-comma
|211.609-282.587-494.196
|
|-
| -1/3-comma
|212.892-280.662-493.554
|
|-
| -3/8-comma
|214.014-278.979-492.993
|
|-
| -2/5-comma
|214.688-277.968-492.656
|
|-
| -3/7-comma
|215.458-276.813-492.271
|
|-
| -4/9-comma
|215.886-276.171-492.057
|
|-
| -1/2-comma
|217.383-273.926-491.309
|Everything from this point onwards does not have a minor seventh between 16/9 and 2048/1161
|-
| -5/9-comma
|218.880-271.680-490.560
|
|-
| -4/7-comma
|219.307-271.039-490.346
|
|-
| -3/5-comma
|220.077-269.884-489.961
|
|-
| -5/8-comma
|220.751-268.874-489,625
|
|-
| -2/3-comma
|221.874-267.190-489.063
|
|-
| -5/7-comma
|223.157-265.265-488.422
|
|-
| -3/4-comma
|224.119-263.821–487.940
|
|-
| -7/9-comma
|224.868-262.698-487.566
|
|-
| -4/5-comma
|225.466-261.801-487.267
|
|-
| -5/6-comma
|226.365-260.453-486.818
|
|-
| -6/7-comma
|227.006-259.491-486.497
|
|-
| -7/8-comma
|227.487-258.769-486.256
|
|-
| -8/9-comma
|227.861-258.208-486.069
|
|-
|Negative Whole-comma
|230.855-253.717-484.752
|
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-11T00:44:18Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the chain of fifths to give a sense of finality to the last piece of a sequence. The chain of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Fa♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Si
|15
|Major seventh, fourteenth
|-
|4 fifths
|Mi
|5
|Major tenth, seventeenth
|-
|3 fifths
|La
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Re
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|Sol
|3
|Perfect twelfth
|-
|0
|Natural, ''naturalis''
|Ut > Do
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Fa, originally ''superparticularis''
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Fa ''superbipartiens'' > Si♭
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa ''supertripartiens'' > Mi♭
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > La♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Re♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|Sol♭
|*11
|Diminished twelfth
|}
Major is considered as comparable to Sol as minor is to Fa, but Sol ''superparticularis'' and Sol ''superpartiens'' never saw as widespread usage as Fa ''superparticularis'' and Fa ''superpartiens'' before the conversion of the latter to flats. At that time, it was also widespread, but not absolute, that only the true relations, and thus the 2.3.5.7.13.17.19.43 subgroup, were considered within the bounds of the modal system. The paradox of this is that the true relations generally do not have the same desired (sub)harmonics for ''fortis'' and ''lenis''. To solve this problem, theorists quickly created the [[User:Moremajorthanmajor/United Kingdom of Musical Instruments/List of m/n comma mean tetrachords|mean tetrachord]], which is primarily considered to temper out [[129/128]].

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-09T23:25:17Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the circle of fifths to give a sense of finality to the last piece of a sequence. The circle of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Fa♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Si
|15
|Major seventh, fourteenth
|-
|4 fifths
|Mi
|5
|Major tenth, seventeenth
|-
|3 fifths
|La
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Re
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|Sol
|3
|Perfect twelfth
|-
|0
|Natural, ''naturalis''
|Ut > Do
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Fa, originally ''superparticularis''
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Fa ''superbipartiens'' > Si♭
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weakest, ''lenissimus''
|Fa ''supertripartiens'' > Mi♭
|19 (technically)
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > La♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Re♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|Sol♭
|*11
|Diminished twelfth
|}
Major is considered as comparable to Sol as minor is to Fa, but Sol ''superparticularis'' and Sol ''superpartiens'' never saw as widespread usage as Fa ''superparticularis'' and Fa ''superpartiens'' before the conversion of the latter to flats. At that time, it was also widespread, but not absolute, that only the true relations were considered within the bounds of the modal system.

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-09T22:08:26Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various translations, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the circle of fifths to give a sense of finality to the last piece of a sequence. The circle of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="4" |Strongest, ''fortissimus''
|Fa♯
|*11
|Augmented eleventh, eighteenth (technically)
|-
|5 fifths
|Si
|15
|Major seventh, fourteenth
|-
|4 fifths
|Mi
|5
|Major tenth, seventeenth
|-
|3 fifths
|La
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Stronger, ''fortior''
|Re
|9
|Major ninth, sixteenth
|-
|1 fifth
|Strong, ''fortis''
|Sol
|3
|Perfect twelfth
|-
|0
|Natural, ''naturalis''
|Ut > Do
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
|Weak, ''lenis''
|Fa, originally ''superparticularis''
|43 (technically)
|Perfect eleventh, eighteenth
|-
|2 fourths
|Weaker, ''lenior''
|Fa ''superbipartiens'' > Si♭
|7
|Minor seventh, fourteenth
|-
|3 fourths
| rowspan="4" |Weak, ''lenissimus''
|Fa ''supertripartiens'' > Mi♭
|19 (technically)
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > La♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''superquinquipartiens'' > Re♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|Sol♭
|*11
|Diminished twelfth
|}
Major is considered as comparable to Sol as minor is to Fa, but Sol ''superparticularis'' and Sol ''superpartiens'' never saw as widespread usage as Fa ''superparticularis'' and Fa ''superpartiens'' before the conversion of the latter to flats. At that time, it was also widespread, but not absolute, that only the true relations were considered within the bounds of the modal system.

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-09T07:29:50Z

Moremajorthanmajor:

Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legitimate proposals of how to talk about any musical practice in the real world.

The '''musical system of the modern United Kingdom of Musical Instruments''' fundamentally obeys the concepts of conventional human musical systems to the whole depth of musical history. That is, it is ideally based on [[Just Intonation]] and thus normally prioritizes the perfect consonances of the [[3-limit]], followed closely by the remaining true relations of the ideally consonant thirds and sixths and the commonly dissonant steps/seconds and sevenths. False relations are normally more important for how they are averted or masked than for compositions which proceed into them.

Where the instruments differ is in their underlying system(s) of functionality though their systems of functionality with the most native documentation are also originally Eurasian and North African. The globally most popular system of functionality is that which fully crystallized in Medieval Western Europe. There is no single standard name for this system, which is derived from the real-world music history of very late medieval and later pre-classical theorists, who used terms like ''musica mensurata'' ("measured music") or ''cantus mensurabilis'' ("measurable song") to refer to the rhythmically defined polyphonic music of their age, as opposed to ''musica plana'' or ''musica choralis'', i.e., Gregorian plainchant which is happening alongside this system in-universe. The most common terms for this system have changed across its history from the medieval ''chordon conjugans'' (“conjugating chord”) to the modern “conjugable tone” and its various transitions, as opposed to the presumed “non-conjugating” octaves underlying both Gregorian plainchant and the rhythmically defined polyphonic music. The main defining feature of compositions in this tradition are the progressions from one “chord” to another by changing the balance of perfect fourths and perfect fifths in the frame interval of the simple gamut which would traditionally signal the start of a new piece of a sequence. The traditional goal of these “chord progressions” would be a “chord” within one step of the octave on the circle of fifths to give a sense of finality to the last piece of a sequence. The circle of fifths is often grouped into the three parts of the “regular conjugation”.
{| class="wikitable"
|+
!Distance from octave
!Class
!Name
!Desired (sub)harmonic
!Regular conjugation
|-
|6 fifths
| rowspan="6" |Strong, ''fortis''
|Fa♯
|*11
|Augmented eleventh
|-
|5 fifths
|Si
|15
|Major seventh, fourteenth
|-
|4 fifths
|Mi
|5
|Major tenth, seventeenth
|-
|3 fifths
|La
|27 (technically)
|Major sixth, thirteenth
|-
|2 fifths
|Re
|9
|Major ninth, sixteenth
|-
|1 fifth
|Sol
|3
|Perfect twelfth
|-
|0
|Natural, ''naturalis''
|Ut > Do
|(2)
|Perfect octave, fifteenth
|-
|1 fourth
| rowspan="6" |Weak, ''lenis''
|Fa, originally ''superparticularis''
|1/3
|Perfect eleventh, eighteenth
|-
|2 fourths
|Fa ''superbipartiens'' > Si♭
|7
|Minor seventh, fourteenth
|-
|3 fourths
|Fa ''supertripartiens'' > Mi♭
|19
|Minor tenth, seventeenth
|-
|4 fourths
|Fa ''superquadripartiens'' > La♭
|1/5 > 13
|Minor sixth, thirteenth
|-
|5 fourths
|Fa ''supequinquipartiens'' > Re♭
|17
|Minor ninth, sixteenth
|-
|6 fourths
|Sol♭
|*11
|Diminished twelfth
|}

User:Moremajorthanmajor/United Kingdom of Musical Instruments

2025-01-08T21:42:29Z

Moremajorthanmajor: Created page with "Notice: Even though this topic is from a fictional world based on real types of musical instruments which have human lives, no terminology given here is to confused with legit..."

User:Moremajorthanmajor/5L 3s (minor ninth-equivalent)

2024-12-30T02:20:23Z

Moremajorthanmajor: /* See also */

The minor ninth of a diatonic scale has a '''5L 3s''' [[MOS]] structure with generators ranging from 2\5 (two degrees of 5ed8\7 = 548.6¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5ed8\7, s becomes so small it disappears (and all that remains are the five equal L's).

Any edIX of an interval up to 8\7 with an interval between 450¢ and 548.6¢ has a 5L 3s scale. [[13edIX]] is the smallest edIX with a (non-degenerate) 5L 3s scale and thus is the most commonly used 5L 3s tuning.
==Standing assumptions==
The [[TAMNAMS]] system is used in this article to name 5L 3s<minor ninth> intervals and step size ratios and step ratio ranges.

The notation used in this article is G Mixolydian Mediant (LLsLLsLs) = GABCQDEFG, unless specified otherwise. We denote raising and lowering by a chroma (L − s) by # and f "flat (F molle)".

The chain of perfect 4ths becomes: ... Ef Af Qf Ff Bf Df G C E A Q F B D ...

Thus the [[13edIX]] gamut is as follows:

'''G/F#''' G#/Af '''A''' A#/Bf '''B/Cf''' '''C/B#''' C#/Qf '''Q''' Q#/Df '''D/Ef E/D#''' E#/Ff '''F/Gf G'''

The 18edIX gamut is notated as follows:

'''G''' F#/Af G# '''A''' Bf A#/Cf '''B''' '''C''' B#/Qf C# '''Q''' Df Q#/Ef '''D E''' D#/Ff E#/Gf '''F G'''

The 21edIX gamut:

'''G''' G# Af '''A''' A# Bf '''B''' B#/Cf '''C''' C# Qf '''Q''' Q# Df '''D''' D#/Ef '''E''' E# Ff '''F''' F#/Gf '''G'''
==Names==
The author suggests the name '''Neapolitan'''-'''oneirotonic'''.
==Intervals==
The table of Neapolitan-oneirotonic intervals below takes the fourth as the generator. Given the size of the fourth generator ''g'', any oneirotonic interval can easily be found by noting what multiple of ''g'' it is, and multiplying the size by the number ''k'' of generators it takes to reach the interval and reducing if necessary (so you can use "''k''*''g'' % ''x''" for search engines, for plugged-in values of ''k'' and ''g''). For example, since the major third is reached by six fourth generators, 18edIX's major third is 6*494.12 mod 1270.59 = 2964.71 mod 1270.59 = 423.53¢.
{| class="wikitable center-all"
|-
!
!Notation (1/1 = G)
!name
!In L's and s's
!# generators up
!Notation of 2/1 inverse
!name
!In L's and s's
|-
| colspan="8" |The 8-note MOS has the following intervals (from some root):
|-
|0
|G
|perfect unison
|0L + 0s
|0
|G
|“perfect” minor 9th
|5L + 3s
|-
|1
|C
|natural 4th
|2L + 1s
| -1
|Df
|minor 6th
|3L + 2s
|-
|2
|E
|major 7th
|4L + 2s
| -2
|Bf
|minor 3rd
|1L + 1s
|-
|3
|A
|major 2nd
|1L + 0s
| -3
|Ff
|diminished octave
|4L + 3s
|-
|4
|Q
|perfect 5th
|3L + 1s
| -4
|Qf
|diminished 5th
|2L + 2s
|-
|5
|F
|perfect octave
|5L + 2s
| -5
|Af
|minor 2nd
|0L + 1s
|-
|6
|B
|major 3rd
|2L + 0s
| -6
|Ef
|minor 7th
|3L + 3s
|-
|7
|D
|major 6th
|4L + 1s
| -7
|Cf
|diminished 4th
|1L + 2s
|-
| colspan="8" |The chromatic 13-note MOS (either [[5L 8s (minor ninth equivalent)|5L 8s]], [[8L 5s (minor ninth equivalent)|8L 5s]], or [[13edIX]]) also has the following intervals (from some root):
|-
|8
|G#
|augmented unison
|1L - 1s
| -8
|Gf
|diminished 9th
|4L + 4s
|-
|9
|C#
|augmented 4th
|3L + 0s
| -9
|Dff
|diminished 6th
|2L + 3s
|-
|10
|E#
|augmented 7th
|5L + 1s
| -10
|Bff
|diminished 3rd
|0L + 2s
|-
|11
|A#
|augmented 2nd
|2L - 1s
| -11
|Fff
|doubly diminished octave
|3L + 4s
|-
|12
|Q#
|augmented 5th
|4L + 0s
| -12
|Qff
|doubly diminished 5th
|1L + 3s
|}
==Tuning ranges==
===Simple tunings===
Table of intervals in the simplest Neapolitan-oneirotonic tunings:
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in 18edIX (hard)
!Size in 21edIX (soft)
! class="unsortable" |Note name on G
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\13, 0.00
|0\18, 0.00
|0\21, 0.00
|G
|0
|-
|minor 2nd
|1\13, 100.00
|1\18, 70.59
|2\21, 126.32
|Af
| -5
|-
|major 2nd
|2\13, 200.00
|3\18, 211.76
|3\21, 189.47
|A
| +3
|- bgcolor="#eaeaff"
|minor 3rd
|3\13, 300.00
|4\18, 282.35
|5\21, 315.79
|Bf
| -2
|- bgcolor="#eaeaff"
|major 3rd
| rowspan="2" |4\13, 400.00
|6\18, 423.53
|6\21, 378.95
|B
| +6
|-
|diminished 4th
|5\18, 352.94
|7\21, 442.105
|Cf
| -7
|-
|natural 4th
|5\13, 500.00
|7\18, 494.12
|8\21, 505.24
|C
| +1
|-
|augmented 4th
| rowspan="2" |6\13, 600.00
|9\18, 635.29
|9\21, 568.42
|C#
| +9
|- bgcolor="#eaeaff"
|diminished 5th
|8\18, 564.71
|10\21, 631.58
|Qf
| -4
|- bgcolor="#eaeaff"
|perfect 5th
|7\13, 700.00
|10\18, 705.88
|11\31, 694.74
|Q
| +4
|-
|minor 6th
|8\13, 800.00
|11\18, 776.47
|13\21, 821.05
|Df
| -1
|-
|major 6th
| rowspan="2" |9\13, 900.00
|13\18, 917.65
|14\21, 884.21
|D
| +7
|- bgcolor="#eaeaff"
|minor 7th
|12\18, 847.06
|15\21, 947.37
|Ef
| -6
|- bgcolor="#eaeaff"
|major 7th
|10\13, 1000.00
|14\18, 988.24
|16\21, 1017.53
|E
| +2
|-
|diminished octave
|11\13, 1100.00
|15\18, 1052.82
|18\21, 1136.84
|Ff
| -3
|-
|perfect octave
|12\13, 1200.00
|17\18, 1200.00
|19\21, 1200.00
|F
| +5
|}
===Hypohard===
[[Hypohard]] Neapolitan-oneirotonic tunings (with generator between 5\13 and 7\18) have step ratios between 2/1 and 3/1.

Hypohard Neapolitan-oneirotonic can be considered " superpythagorean Neapolitan-oneirotonic". This is because these tunings share the following features with [[Superpyth|superpythagorean]] diatonic tunings:
**The large step is near the Pythagorean 9/8 whole tone, somewhere between as in [[12edo]] and as in [[17edo]].
**The major 3rd (made of two large steps) is a near-[[Pythagorean]] to [[Neogothic]] major third.
EDIXs that are in the hypohard range include [[13edIX]], 18edIX, and 31edIX.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hypohard Neapolitan-oneiro tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
![[13edIX]] (basic)
!18edIX (hard)
!31edIX (semihard)
|-
|generator (g)
|5\13, 500.00
|7\18, 494.12
|12\31, 496.55
|-
|L (3g - minor 9th)
|2\13, 200.00
|3\18, 211.765
|5\31, 206.87
|-
|s (-5g + 2 minor 9ths)
|1\13, 100.00
|1\18, 70.59
|2\31, 82.76
|}
====Intervals====
Sortable table of major and minor intervals in hypohard Neapolitan-oneiro tunings:
{| class="wikitable right-2 right-3 right-4 sortable"
|-
! class="unsortable" |Degree
!Size in 13edIX (basic)
!Size in 18edIX (hard)
!Size in 31edIX (semihard)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios<ref>The ratio interpretations that are not valid for 18edo are italicized.</ref>
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\13, 0.00
|0\18, 0.00
|0\31, 0.00
|G
|1/1
|0
|-
|minor 2nd
|1\13, 100.00
|1\18, 70.59
|2\31, 82.76
|Af
|21/20, ''22/21''
| -5
|-
|major 2nd
|2\13, 200.00
|3\18, 211.76
|5\31, 206.87
|A
|9/8
| +3
|- bgcolor="#eaeaff"
|minor 3rd
|3\13, 300.00
|4\18, 282.35
|7\31, 289.66
|Bf
|13/11, 33/28
| -2
|- bgcolor="#eaeaff"
|major 3rd
| rowspan="2" |4\13, 400.00
|6\18, 423.53
|10\31, 413.79
|B
|14/11, 33/26
| +6
|-
|diminished 4th
|5\18, 352.94
|9\31, 372.41
|Cf
|''5/4, 11/9''
| -7
|-
|natural 4th
|5\13, 500.00
|7\18, 494.12
|12\31, 496.55
|C
|4/3
| +1
|- bgcolor="#eaeaff"
|augmented 4th
| rowspan="2" |6\13, 600.00
|9\18, 635.29
|15\31, 620.69
|C#
|''10/7, 18/13, 11/8''
| +9
|- bgcolor="#eaeaff"
|diminished 5th
|8\18, 564.71
|14\31, 579.31
|Qf
|''7/5, 13/9'', ''16/11''
| -4
|-
|perfect 5th
|7\13, 700.00
|10\18, 705.88
|17\31, 703.45
|Q
|3/2
| +4
|-
|minor 6th
|8\13, 800.00
|11\18, 776.47
|19\31, 786.21
|Df
|52/33, 11/7
| -1
|- bgcolor="#eaeaff"
|major 6th
| rowspan="2" |9\13, 900.00
|13\18, 917.65
|22\31, 910.34
|D
|56/33, 22/17
| +7
|- bgcolor="#eaeaff"
|minor 7th
|12\18, 847.06
|21\31, 868.97
|Ef
|5/3, 18/11
| -6
|-
|major 7th
|10\13, 1000.00
|14\18, 988.24
|24\31, 993.13
|E
|16/9
| +2
|-
|diminished octave
|11\13, 1100.00
|15\18, 1052.82
|26\31, 1075.86
|Ff
|11/6, 13/7, 15/8
| -3
|-
|perfect octave
|12\13, 1200.00
|17\18, 1200.00
|29\31, 1200.00
|F
|2/1
| +5
|}<references />
===Hyposoft===
[[Hyposoft]] Neapolitan-oneirotonic tunings (with generator between 8\21 and 5\13) have step ratios between 3/2 and 2/1. The 8\21-to-5\13 range of Neapolitan-oneirotonic tunings can be considered "meantone Neapolitan-oneirotonic”. This is because these tunings share the following features with meantone diatonic tunings:
*The large step is between near the meantone and near the Pythagorean 9/8 whole tone, somewhere between as in [[19edo]] and as in [[17edo|12edo]].
*The major 3rd (made of two large steps) is a near-[[Just intonation|just]] to near-[[Pythagorean]] major third.
The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various hyposoft Neapolitan-oneiro tunings (13edIX not shown).
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!21edIX (soft)
!34edIX (semisoft)
|-
|generator (g)
|8\21, 505.26
|13\34, 503.23
|-
|L (3g - minor 9th)
|3\21, 189.47
|5\34, 193.55
|-
|s (-5g + 2 minor 9ths)
|2\21, 126.32
|3\34, 116.19
|}
====Intervals====
Sortable table of major and minor intervals in hyposoft tunings (13edIX not shown):
{| class="wikitable right-2 right-3 sortable"
|-
! class="unsortable" |Degree
!Size in 21edIX (soft)
!Size in 34edIX (semisoft)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\21, 0.00
|0\34, 0.00
|G
|1/1
|0
|-
|minor 2nd
|2\21, 126.32
|3\34, 116.19
|Af
|16/15
| -5
|-
|major 2nd
|3\21, 189.47
|5\34, 193.55
|A
|10/9, 9/8
| +3
|- bgcolor="#eaeaff"
|minor 3rd
|5\21, 315.79
|8\34, 309.68
|Bf
|6/5
| -2
|- bgcolor="#eaeaff"
|major 3rd
|6\21, 378.95
|10\34, 387.10
|B
|5/4
| +6
|-
|diminished 4th
|7\21, 442.105
|11\34, 425.81
|Cf
|9/7
| -7
|-
|natural 4th
|8\21, 505.24
|13\34, 503.23
|C
|4/3
| +1
|- bgcolor="#eaeaff"
|augmented 4th
|9\21, 568.42
|15\34, 580.645
|C#
|7/5
| +9
|- bgcolor="#eaeaff"
|diminished 5th
|10\21, 631.58
|16\34, 619.355
|Qf
|10/6
| -4
|-
|perfect 5th
|11\31, 694.74
|18\34, 696.77
|Q
|3/2
| +4
|-
|minor 6th
|13\21, 821.05
|21\34, 812.90
|Df
|8/5
| -1
|- bgcolor="#eaeaff"
|major 6th
|14\21, 884.21
|23\34, 890.32
|D
|5/3
| +7
|- bgcolor="#eaeaff"
|minor 7th
|15\21, 947.37
|24\34, 929.03
|Ef
|12/7
| -6
|-
|major 7th
|16\21, 1017.53
|26\34, 1006.45
|E
|9/5, 16/9
| +2
|-
|diminished octave
|18\21, 1136.84
|29\34, 1122.58
|Ff
|27/14, 48/25
| -3
|-
|perfect octave
|19\21, 1200.00
|31\34, 1200.00
|F
|2/1
| +5
|}
===Parasoft to ultrasoft tunings===
The range of Neapolitan-oneirotonic tunings of step ratio between 6/5 and 3/2 (thus in the [[parasoft]] to [[ultrasoft]] range) may be of interest because it is closely related to [[Meantone family|flattone]] temperament.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various tunings in this range.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!29edIX (supersoft)
!37edIX
|-
|generator (g)
|11\29, 507.69
|14\37, 509.09
|-
|L (3g - minor 9th)
|4\29, 184.615
|5\37, 181.82
|-
|s (-5g + 2 minor 9ths)
|3\29, 138.46
|4\37, 145.455
|}
====Intervals====
The intervals of the extended generator chain (-15 to +15 generators) are as follows in various softer-than-soft Neapolitan-oneirotonic tunings.
{| class="wikitable right-2 right-3 sortable"
|-
! class="unsortable" |Degree
!Size in 29edIX (supersoft)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\29, 0.00
|G
|1/1
|0
|- bgcolor="#eaeaff"
|chroma
|1\29, 46.15
|G#
|[[33/32]], [[49/48]], [[36/35]], [[25/24]]
| +8
|-
|diminished 2nd
|2\29, 92.31
|Aff
|[[21/20]], [[22/21]], [[26/25]]
| -13
|-
|minor 2nd
|3\29, 138.46
|Af
|[[12/11]], [[13/12]], [[14/13]], [[16/15]]
| -5
|-
|major 2nd
|4\29, 184.615
|A
|[[9/8]], [[10/9]], [[11/10]]
| +3
|-
|augmented 2nd
|5\29, 230.77
|A#
|[[8/7]], [[15/13]]
| +11
|- bgcolor="#eaeaff"
|diminished 3rd
|6\29, 276.92
|Bff
|[[7/6]], [[13/11]], [[33/28]]
| -10
|- bgcolor="#eaeaff"
|minor 3rd
|7\29, 323.08
|Bf
|[[135/112]], [[6/5]]
| -2
|- bgcolor="#eaeaff"
|major 3rd
|8\29, 369.23
|B
|[[5/4]], [[11/9]], [[16/13]]
| +6
|- bgcolor="#eaeaff"
|augmented 3rd
|9\29, 415.385
|B#
|[[9/7]], [[14/11]], [[33/26]]
| +14
|-
|diminished 4th
|10\29, 461.54
|Cf
|[[21/16]], [[13/10]]
| -7
|-
|natural 4th
|11\29, 507.69
|C
|[[75/56]], [[4/3]]
| +1
|-
|augmented 4th
|12\29, 553.85
|C#
|[[11/8]], [[18/13]]
| +9
|-
|doubly augmented 4th, doubly diminished 5th
|13\29, 600.00
|Cx, Qff
|[[7/5]], [[10/7]]
| -12
|- bgcolor="#eaeaff"
|diminished 5th
|14\29, 646.15
|Qf
|[[16/11]], [[13/9]]
| -4
|- bgcolor="#eaeaff"
|perfect 5th
|15\29, 692.31
|Q
|[[112/75]], [[3/2]]
| +4
|- bgcolor="#eaeaff"
|augmented 5th
|16\29, 738.46
|Q#
|[[32/21]], [[20/13]]
| +12
|- bgcolor="#eaeaff"
|diminished 6th
|17\29, 784.615
|Dff
|[[11/7]], [[14/9]]
| -9
|-
|minor 6th
|18\29, 830.77
|Df
|[[13/8]], [[8/5]]
| -1
|-
|major 6th
|19\29, 876.92
|D
|[[5/3]], [[224/135]]
| +7
|-
|augmented 6th
|20\29, 923.08
|D#
|[[12/7]], [[22/13]]
| -14
|-
|minor 7th
|21\29, 969.23
|Ef
|[[7/4]], [[26/15]]
| -6
|- bgcolor="#eaeaff"
|major 7th
|22\29, 1015.385
|E
|[[9/5]], [[16/9]], [[20/11]]
| +2
|- bgcolor="#eaeaff"
|augmented 7th
|23\29, 1061.54
|E#
|[[11/6]], [[13/7]], [[15/8]], [[24/13]]
| +10
|- bgcolor="#eaeaff"
|doubly augmented 7th, doubly diminished octave
|24\29, 1107.69
|Ex, Fff
|[[21/11]], [[25/13]], [[40/21]]
|<nowiki>-11</nowiki>
|- bgcolor="#eaeaff"
|diminished octave
|25\29, 1153.85
|Ff
|[[64/33]], [[96/49]], [[35/18]], [[48/25]]
| -3
|-
|perfect octave
|26\29, 1200.00
|F
|2/1
| +5
|-
|augmented octave
|27\29, 1246.15
|F#
|33/16, 49/24, 72/35, 25/12
| +13
|-
|doubly augmented octave, diminished 9th
|28\29, 1292.31
|Fx, Gf
|21/10, 44/21, 52/25
| -8
|}
===Parahard===
23edIX Neapolitan-oneiro combines the sound of the 32/15 minor ninth and the [[8/7]] whole tone. This is because 23edIX Neapolitan-oneirotonic has a large step of 218.2¢, 22edIX's superpythagorean major second, which is both a warped Pythagorean [[9/8]] whole tone and a warped [[8/7]] septimal whole tone.
====Intervals====
The intervals of the extended generator chain (-12 to +12 generators) are as follows in various Neapolitan-oneirotonic tunings close to 23edIX.
{| class="wikitable right-2 right-3 sortable"
|-
! class="unsortable" |Degree
!Size in 23edIX
(superhard)
! class="unsortable" |Note name on G
! class="unsortable" |Approximate ratios (23edIX)
!#Gens up
|- bgcolor="#eaeaff"
|unison
|0\23, 0.00
|G
|1/1
|0
|- bgcolor="#eaeaff"
|chroma
|3\23, 163.63
|G#
|12/11, 11/10, 10/9
| +8
|-
|minor 2nd
|1\23, 54.545
|Af
|[[36/35]], [[34/33]], [[33/32]], [[32/31]]
| -5
|-
|major 2nd
|4\23, 218.18
|A
|[[9/8]], [[17/15]], [[8/7]]
| +3
|-
|aug. 2nd
|7\23, 381.82
|A#
|5/4
| +11
|- bgcolor="#eaeaff"
|dim. 3rd
|2\23, 109.09
|Bf
|16/15
| -10
|- bgcolor="#eaeaff"
|minor 3rd
|5\23, 272.73
|B
|7/6
| -2
|- bgcolor="#eaeaff"
|major 3rd
|8\23, 436.36
|B#
|9/7, 14/11
| +6
|-
|dim. 4th
|6\23, 327.27
|Cf
|6/5
| -7
|-
|nat. 4th
|9\23, 490.91
|C
|4/3
| +1
|-
|aug. 4th
|12\23, 654.545
|C#
|[[16/11]], [[22/15]]
| +9
|- bgcolor="#eaeaff"
|double dim. 5th
|7\23, 381.82
|Qff
|5/4
| -12
|- bgcolor="#eaeaff"
|dim. 5th
|10\23, 545.455
|Qf
|[[15/11]], [[11/8]]
| -4
|- bgcolor="#eaeaff"
|perf. 5th
|13\23, 709.09
|Q
|3/2
| +4
|- bgcolor="#eaeaff"
|aug. 5th
|16\23, 872.73
|Q#
|5/3
| +12
|-
|dim. 6th
|11\23, 600.00
|Dff
|[[7/5]], [[24/17]], [[17/12]], [[10/7]]
| -9
|-
|minor 6th
|14\23, 763.64
|Df
|14/9, 11/7
| -1
|-
|major 6th
|17\23, 927.27
|D
|12/7
| +7
|-
|- bgcolor="#eaeaff"
|minor 7th
|15\23, 818.18
|Ef
|8/5
| -6
|- bgcolor="#eaeaff"
|major 7th
|18\23, 981.82
|E
|[[7/4]], [[30/17]], [[16/9]]
| +2
|- bgcolor="#eaeaff"
|aug. 7th
|21\23, 1145.455
|E#
|[[31/16]], [[64/33]], [[33/17]], [[35/18]]
| +10
|-
|dim. octave
|19\23, 1036.36
|Ff
|11/6, 20/11, 9/5
| -11
|-
|perf. octave
|22\23, 1200.00
|F
|2/1
| -3
|-
|aug. octave
|25\23, 1363.64
|F#
|24/11, 11/5, 20/9
| +5
|-
|- bgcolor="#eaeaff"
|dim. ninth
|20\23, 1090.91
|J@
|15/8
| -8
|}
===Ultrahard===
[[Archytas clan#Ultrapyth|Ultrapythagorean]] Neapolitan-oneirotonic is a rank-2 temperament in the [[Step ratio|pseudopaucitonic]] range. It represents the [[harmonic entropy]] minimum of the Neapolitan-oneirotonic spectrum where [[7/4]] is the major seventh.

In the broad sense, Ultrapyth can be viewed as any tuning that divides a 16/7 into 2 equal parts. 23edIX, 28edIX and 33edIX can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edIX and true Buzzard in terms of harmonies. 38edIX & 43edIX are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but 48edIX is where it really comes into its own in terms of harmonies, providing not only excellent 7:8:9 melodies, but also [[5/4]] and [[The_Archipelago|archipelago]] harmonies, as by shifting one whole tone done a comma, it shifts from [[The Archipelago|archipelago]] to septimal harmonies.

Beyond that, it's a question of which intervals you want to favor. 53edIX has an essentially perfect [[7/4]], 58edIX also gives three essentially perfect chains of third-comma meantone, while 63edIX has a double chain of essentially perfect quarter-comma meantone and gives about as low overall error as 83edIX does for the basic 4:6:7 triad. But beyond 83edIX, general accuracy drops off rapidly and you might as well be playing equal pentatonic.

The sizes of the generator, large step and small step of Neapolitan-oneirotonic are as follows in various ultrapyth tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
!38edIX
!53edIX
!63edIX
!Optimal ([[POTE|PNTE]]) Ultrapyth tuning
!JI intervals represented (2.3.5.7.13 subgroup)
|-
|generator (g)
|15\38, 486.49
|21\53, 484.615
|25\63, 483.87
|484.07
|4/3
|-
|L (3g - minor 9th)
|7/38, 227.03
|10/53, 230.77
|12/63, 232.26
|231.51
|8/7
|-
|s (-5g + 2 minor 9ths)
|1/38, 32.43
|1/53, 23.08
|1/63, 19.355
|21.05
|50/49 81/80 91/90
|}
====Intervals====
Sortable table of intervals in the Neapolitan-Dylathian mode and their Ultrapyth interpretations:
{| class="wikitable right-2 right-3 right-4 right-5 sortable"
|-
!Degree
!Size in 38edIX
!Size in 53edIX
!Size in 63edIX
!Size in PNTE tuning
!Note name on G
! class="unsortable" |Approximate ratios
!#Gens up
|-
|1
|0\38, 0.00
|0\53, 0.00
|0\63, 0.00
|0.00
|G
|1/1
|0
|-
|2
|7/38, 227.03
|10/53, 230.77
|12/63, 232.26
|231.51
|A
|8/7
| +3
|-
|3
|14\38, 454.05
|20\53, 461.54
|24\63, 464.52
|463.03
|B
|13/10, 21/16
| +6
|-
|4
|15\38, 486.49
|21\53, 484.615
|25\63, 483.87
|484.07
|C
|4/3
| +1
|-
|5
|22\38, 713.51
|31\53, 715.385
|37\63, 716.13
|715.59
|Q
|3/2
| +4
|-
|6
|29\38, 940.54
|41\53, 946.15
|49\63, 948.39
|947.10
|D
|26/15
| +7
|-
|7
|30\38, 972.97
|42\53, 969.23
|50\63, 967.74
|968.15
|E
|7/4
| +2
|-
|8
|37\38, 1200.00
|52\53, 1200.00
|62\63, 1200.00
|1199.66
|F
|2/1
| +5
|}
==Modes==
Neapolitan-Oneirotonic modes are named after cities in the Dreamlands.
{| class="wikitable"
|-
|'''Mode'''
|[[Modal UDP Notation|'''UDP''']]
|'''Name'''
|-
| |LLsLLsLs
|<nowiki>7|0</nowiki>
| |Neapolitan-Dylathian (də-LA(H)TH-iən)
|-
| |LLsLsLLs
|<nowiki>6|1</nowiki>
| |Neapolitan-Illarnekian (ill-ar-NEK-iən)
|-
| |LsLLsLLs
|<nowiki>5|2</nowiki>
| |Neapolitan-Celephaïsian (kel-ə-FAY-zhən)
|-
| |LsLLsLsL
|<nowiki>4|3</nowiki>
| |Neapolitan-Ultharian (ul-THA(I)R-iən)
|-
| |LsLsLLsL
|<nowiki>3|4</nowiki>
| |Neapolitan-Mnarian (mə-NA(I)R-iən)
|-
| |sLLsLLsL
|<nowiki>2|5</nowiki>
| |Neapolitan-Kadathian (kə-DA(H)TH-iən)
|-
| |sLLsLsLL
|<nowiki>1|6</nowiki>
| |Neapolitan-Hlanithian (lə-NITH-iən)
|-
| |sLsLLsLL
|<nowiki>0|7</nowiki>
| |Neapolitan-Sarnathian (sar-NA(H)TH-iən), can be shortened to "Sarn"
|}
==Scale tree==
{| class="wikitable center-all"
! colspan="3" |Normalized Generator
!Cents
!L
!s
!L/s
!Comments
|-
| colspan="3" |3\8||514.286||1||1||1.000||
|-
| colspan="3" |17\45||510.000||6||5||1.200||
|-
| colspan="3" |14\37||509.091||5||4||1.250||
|-
| colspan="3" |25\66||508.475||9||7||1.286||
|-
| colspan="3" |11\29||507.692||4||3||1.333||
|-
| colspan="3" |19\50||506.667||7||5||1.400||
|-
| colspan="3" |8\21||505.263||3||2||1.500||L/s = 3/2
|-
| colspan="3" |29\76||504.348||11||7||1.571||
|-
| colspan="3" |21\55||504.000||8||5||1.600||
|-
| colspan="3" |34\89||503.704||13||8||1.625||Golden Neapolitan-oneirotonic
|-
| colspan="3" |13\34||503.226||5||3||1.667||
|-
| colspan="3" |31\81||502.703||12||7||1.714||
|-
| colspan="3" |18\47||502.326||7||4||1.750||
|-
| colspan="3" |23\60||501.818||9||5||1.800||
|-
| colspan="3" |28\73
|501.493
|11
|6
|1.833
|
|-
| colspan="3" |33\86
|501.265
|13
|7
|1.857
|
|-
| colspan="3" |38\99
|501.099
|15
|8
|1.875
|
|-
| colspan="3" |43\112
|500.971
|17
|9
|1.889
|
|-
| colspan="3" |5\13||500||2||1||2.000||Basic Neapolitan-oneirotonic
(generators smaller than this are proper)
|-
| colspan="3" |42\109
|499.010
|17
|8
|2.125
|
|-
| colspan="3" |37\96
|498.876
|15
|7
|2.143
|
|-
| colspan="3" |32\83
|498.701
|13
|6
|2.167
|
|-
| colspan="3" |27\70
|498.462
|11
|5
|2.200
|
|-
| colspan="3" |22\57||498.113||9||4||2.250||
|-
| colspan="3" |17\44||497.561||7||3||2.333||
|-
| colspan="3" |29\75||497.143||12||5||2.400||
|-
| colspan="3" |12\31||496.552||5||2||2.500||
|-
| colspan="3" |31\80||496.000||13||5||2.600||
|-
| colspan="3" |19\49||495.652||8||3||2.667||
|-
| colspan="3" |26\67||495.238||11||4||2.750||
|-
| colspan="3" |7\18||494.118||3||1||3.000||L/s = 3/1
|-
| colspan="3" |30\77
|493.151
|13
|4
|3.250
|
|-
| colspan="3" |23\59||492.857||10||3||3.333||
|-
| colspan="3" |16\41||492.308||7||2||3.500||
|-
| colspan="3" |25\64||491.803||11||3||3.667||
|-
| colspan="3" |9\23||490.909||4||1||4.000||
|-
| colspan="3" |20\51||489.796||9||2||4.500||
|-
| colspan="3" |11\28||488.889||5||1||5.000||
|-
| colspan="3" |24\61
|488.136
|11
|2
|5.500
|
|-
| colspan="3" |13\33||487.500||6||1||6.000||
|-
| colspan="3" |2\5||480.000||1||0||→ inf||
|}

== See also ==
[[5L 3s (45/22-equivalent)]] - undecimal small diesis tuning

[[5L 3s (33/16-equivalent)]] - harmonic subminor ninth tuning

[[5L 3s (56/27-equivalent)]] - Archytas diatonic minor ninth tuning

[[5L 3s (25/12-equivalent)]] - classical chromatic minor ninth tuning

[[5L 3s (44/21-equivalent)]], [[5L 3s (208/99-equivalent)]] - Neogothic undecimal diatonic minor ninth tuning

[[5L 3s (21/10-equivalent)]] - septimal chromatic minor ninth tuning

[[5L 3s (32/15-equivalent)]] - classical diatonic minor ninth tuning

[[5L 3s (891/416-equivalent)]], [[5L 3s (189/88-equivalent)]] - Neogothic chromatic minor ninth tuning

[[5L 3s (15/7-equivalent)]] - septimal diatonic minor ninth tuning

[[5L 3s (243/112-equivalent)]] - Archytas chromatic minor ninth tuning

[[5L 3s (11/5-equivalent)]] - undecimal neutral ninth tuning

User:Moremajorthanmajor/TAMNAMS Extension

2024-12-30T00:18:46Z

Moremajorthanmajor: /* Non-octave extensions (proposed) */

This is a system for describing and naming mos scales beyond the set of named TAMNAMS mosses. Both [[User:Frostburn]] ([[User:Frostburn/TAMNAMS Extension]]) and I have similar systems for how to name mos descendants. However, this page describes several more systems that apply to non-octave mosses.

The schemes proposed here are '''not meant to be a definitive naming scheme'''. Rather, it's meant to be a starting point for a naming scheme discussion. Some parts of this page also serves as a sandbox.

The scope of this TAMNAMS extension is as follows:

#Systematically name mosses beyond the named range by how they're related to TAMNAMS-named mosses. The most common way of doing this is by considering what mosses descend from a TAMNAMS-named mos.
##Secondarily, propose unique names, or provide suggestions for possible names, for certain mosses in case they're worth having distinct names. Some of these names are old names that have been around long enough to be memorable.
###Catalog any names that had already existed or have been proposed elsewhere on the wiki.
#Systematically name mosses regardless of the equave. Such names should be as general as possible. Names for mosses with no more than 10 notes are prioritized.
#Propose names for 3/2 (fifth) and 3/1 (tritave) equivalent mosses, or provide suggestions for possible name ideas. Names for mosses with no more than 10 notes are prioritized.

==Naming mos descendants==
To name mosses that have more than 10 notes, rather than giving mosses unique names, names are based on how they're related to another (named) mos and, optionally, what step ratio is needed for the parent to produce that mos.
{| class="wikitable"
! colspan="12" |Base names
|-
! colspan="2" |Parent mos
! colspan="3" |Child (1st descendant)
! colspan="3" |Grandchild (2nd descendant)
! colspan="3" |Great-grandchild (3rd descendant)
!''k''th descendant
|-
| colspan="2" |''(mos-name)''
| colspan="3" |''(step-ratio)-''chromatic ''(mos-name)''
''(step-ratio)-''chro ''(mos-name)''

''(step-ratio)-(mos-prefix)''enharmonic
| colspan="3" |''(step-ratio)''-enharmonic ''(mos-name)''
''(step-ratio)''-enhar ''(mos-name)''

''(step-ratio)-(mos-prefix)''enharmonic
| colspan="3" |''(step-ratio)''-subchromatic ''(mos-name)''
''(step-ratio)''-subchro ''(mos-name)''

''(step-ratio)-(mos-prefix)''subchromatic
|''(k''th'') (mos-name)'' descendant
''(k''th'')-(mos-prefix)''descendant
|-
! colspan="12" |Step ratio prefixes (optional)
|-
! colspan="2" |Parent mos
! colspan="3" |Child (1st descendant)
! colspan="3" |Grandchild (2nd descendant)
! colspan="3" |Great-grandchild (3rd descendant)
!''k''th descendant
|-
!Mos
!L:s range
!Mos
!L:s range
!Prefix
!Mos
!L:s range
!Prefix
!Mos
!L:s range
!Prefix
!Prefixes not applicable
|-
| rowspan="8" |xL ys
| rowspan="8" |1:1 to 1:0
| rowspan="4" |(x+y)L xs
| rowspan="4" |1:1 to 2:1
(general soft range)
| rowspan="4" |s-
| rowspan="2" |(x+y)L (2x+y)s
| rowspan="2" |1:1 to 3:2
(soft)
| rowspan="2" |s-
|(x+y)L (3x+2y)s
|1:1 to 4:3
(ultrasoft)
|us-
| rowspan="8" |
|-
|(3x+2y)L (x+y)s
|4:3 to 3:2
(parasoft)
|ps-
|-
| rowspan="2" |(2x+y)L (x+y)s
| rowspan="2" |3:2 to 2:1
(hyposoft)
| rowspan="2" |os-
|(3x+2y)L (2x+y)s
|3:2 to 5:3
(quasisoft)
|qs-
|-
|(2x+y)L (3x+2y)s
|5:3 to 2:1
(minisoft)
|ms-
|-
| rowspan="4" |xL (x+y)s
| rowspan="4" |2:1 to 1:0
(general hard range)
| rowspan="4" |h-
| rowspan="2" |(2x+y)L xs
| rowspan="2" |2:1 to 3:1
(hypohard)
| rowspan="2" |oh-
|(2x+y)L (3x+y)s
|2:1 to 5:2
(minihard)
|mh-
|-
|(3x+y)L (2x+y)s
|5:2 to 3:1
(quasihard)
|qh-
|-
| rowspan="2" |xL (2x+y)s
| rowspan="2" |3:1 to 1:0
(hard)
| rowspan="2" |h-
|(3x+y)L xs
|3:1 to 4:1
(parahard)
|ph-
|-
|xL (3x+y)s
|4:1 to 1:0
(ultrahard)
|uh-
|}
Mos descendant names have two main forms: a multi-part name, where the base name (''chromatic'', ''enharmonic'', ''subchromatic'', and ''descendant'') and mos name are separate words, and a one-part name, formed by prefixing the mos's prefix to the base names. The latter is recommended for mosses with no more than three periods, as the only 4 and 5-period mosses named by TAMNAMS are tetrawood and pentawood respectively. If a step ratio is specified for the former, it may be written out fully instead of prefixed to the base word.

The term ''k''th descendant can be used to refer to any mos that descends from another mos, regardless of how many generations apart the two are. To find the number of generations ''n'' separating the two mosses, use the following algorithm:
#Let z and w be the number of large and small steps of the parent mos to be found. Assign to z and w the values x and y respectively. Let n = 0, where n is the number of generations away from zL ws.
#Let m1 be equal to max(z, w) and m2 be equal to min(z, w).
#Assign to z the value m2 and w the value m1-m2. Increment n by 1.
#If the sum of z and w is no more than 10, then the parent mos is zL ws and is n generations from the mos descendant xL ys. If not, repeat the process starting at step 2.
As diatonic (5L 2s) doesn't have a prefix, the terms ''chromatic'', ''enharmonic'', and ''subchromatic'' by themselves (and with no other context suggesting a non-diatonic mos) refer to 1st (child), 2nd (grandchild), and 3rd (great-grandchild) diatonic descendants. For consistency, mos descendant names apply to mosses whose child mosses exceed 10 notes. Since all mosses ultimately descend from some nL ns mos, every possible descendant up to 5 periods will be related to a named mos.
{| class="wikitable center-all"
|+Mosses whose grandchildren have more than 10 notes (1st and 2nd descendants only)
|-
! colspan="2" |6-note mosses
! colspan="2" |Chromatic mosses
! colspan="2" |Enharmonic mosses
|-
!Pattern!!Name
!Patterns
!Names
!Patterns
!Names
|-
|[[1L 5s]]
|antimachinoid
|1L 6s, 6L 1s
|onyx, arch(a)eotonic
|1A 7B, 6A 7B
|antipine, pine; atetarquintal, tetarquintal
|-
|[[2L 4s]]
|malic
|2L 6s, 6L 2s
|subaric, ekic
|2A 8B, 6A 8B
|jaric, taric; ekchromatic
|-
|[[3L 3s]]
|triwood: augmented
|3L 6s, 6L 3s
|tcherepnin, hyrulic
|3A 9B, 6A 9B
|sergic, ivanic; hyruchromatic
|-
|[[4L 2s]]
|citric
|4L 6s, 6L 4s
|lime, lemon
|4A 10B, 6A 10B
|limechromatic, lemchromatic
|-
|[[5L 1s]]||machinoid
|5L 6s, 6L 5s
|xeimtonic, antixeimtonic
|5A 11B, 6A 11B
|xeimchromattic, axeimchromatic
|-
! colspan="2" |7-note mosses
! colspan="2" |Chromatic mosses
! colspan="2" |Enharmonic mosses
|-
!Pattern!!Name
!Patterns
!Names
!Patterns
!Names
|-
|[[1L 6s]]
|onyx
|1L 7s, 7L 1s
|antipine, pine
|1A 8B, 7A 8B
|antisubneutralic, subneutralic; pinechromatic
|-
|[[2L 5s]]
|antidiatonic
|2L 7s, 7L 2s
|balzano, superdiatonic
|2A 9B, 7A 9B
|joanatonic, ultradiatonic; armochromatic
|-
|[[3L 4s]]
|mosh
|3L 7s, 7L 3s
|sephiroid, dicoid
|3A 10B, 7A 10B
|magitonic, luachoid; dicochromatic
|-
|[[4L 3s]]||smitonic
|4L 7s, 7L 4s
|kleistonic, suprasmitonic
|4A 11B, 7A 11B
|kleichromatic, suprasmichromatic
|-
|[[5L 2s]]||diatonic
|5L 7s, 7L 5s
|chromatic
|5A 12B, 7A 12B
|enharmonic
|-
|[[6L 1s]]||arch(a)eotonic
|6L 7s, 7L 6s
|antitetarquintal, tetarquintal
|6A 13B, 7A 13B
|atetarquinchromatic, tetarquinchromatic
|-
! colspan="2" |8-note mosses
! colspan="2" |Chromatic mosses
! colspan="2" |Enharmonic mosses
|-
!Pattern!!Name
!Patterns
!Names
!Patterns
!Names
|-
|[[1L 7s]]
|antipine
|1L 8s, 8L 1s
|antisubneutralic, subneutralic
|1A 9B, 8A 9B
|antisinatonic, sinatonic; bluchromatic
|-
|[[2L 6s]]
|subaric
|2L 8s, 8L 2s
|jaric, taric
|2A 10B, 8A 10B
|rujaric, talaric; tarachromatic
|-
|[[3L 5s]]||checkertonic
|3L 8s, 8L 3s
|squaroid, sensoid
|3A 11B, 8A 11B
|squarochromatic, sensochromatic
|-
|[[4L 4s]]||tetrawood; diminished
|4L 8s, 8L 4s
|chromatic diminished
|4A 12B, 8A 12B
|enharmonic diminished
|-
|[[5L 3s]]||oneirotonic
|5L 8s, 8L 5s
|antipetroid, petroid
|5A 13B, 8A 13B
|apetrochromatic, petrochromatic
|-
|[[6L 2s]]||ekic
|6L 8s, 8L 6s
|ekchromatic
|6A 14B, 8A 14B
|ekenharmonic
|-
|[[7L 1s]]||pine
|7L 8s, 8L 7s
|pinechromatic
|7A 15B, 8A 15B
|pinenharmonic
|-
! colspan="2" |9-note mosses
! colspan="2" |Chromatic mosses
! colspan="2" |Enharmonic mosses
|-
!Pattern!!Name
!Patterns
!Names
!Patterns
!Names
|-
|[[1L 8s]]
|antisubneutralic
|1L 9s, 9L 1s
|antisinatonic, sinatonic
|1A 10B, 9A 10B
|tenorite, miratonic; sinachromatic
|-
|[[2L 7s]]
|balzano
|2L 9s, 9L 2s
|joanatonic, ultradiatonic
|2A 11B, 9A 11B
|litonic, hendecoid; ultrachromatic
|-
|[[3L 6s]]||tcherepnin
|3L 9s, 9L 3s
|sergic, ivanic
|3A 12B, 9A 12B
|sergichromatic, ivanichromatic
|-
|[[4L 5s]]||gramitonic
|4L 9s, 9L 4s
|huxloga, orwelloid
|4A 13B, 9A 13B
|huxlochromatic, orwellchromatic
|-
|[[5L 4s]]||semiquartal
|5L 9s, 9L 5s
|chtonchromatic
|5A 14B, 9A 14B
|chtonenharmonic
|-
|[[6L 3s]]||hyrulic
|6L 9s, 9L 6s
|hyruchromatic
|6A 15B, 9A 15B
|hyrenharmonic
|-
|[[7L 2s]]||superdiatonic
|7L 9s, 9L 7s
|armochromatic
|7A 16B, 9A 16B
|armenharmonic
|-
|[[8L 1s]]||subneutralic
|8L 9s, 9L 8s
|bluchromatic
|8A 17B, 9A 17B
|bluenharmonic
|-
! colspan="2" |10-note mosses
! colspan="2" |Chromatic mosses
! colspan="2" |Enharmonic mosses
|-
!Pattern!!Name
!Patterns
!Names
!Patterns
!Names
|-
|[[1L 9s]]||antisinatonic
|1L 10s, 10L 1s
|tenorite, miratonic
|1A 11B, 10A 11B
|helenite, ripploid; miracloid, antimiracloid
|-
|[[2L 8s]]||jaric
|2L 10s, 10L 2s
|rujaric, talaric
|2A 12B, 10A 12B
|rujachromatic, talachromatic
|-
|[[3L 7s]]||sephiroid
|3L 10s, 10L 3s
|magitonic, luachoid
|3A 13B, 10A 13B
|magichromatic, luachromatic
|-
|[[4L 6s]]||lime
|4L 10s, 10L 4s
|limechromatic
|4A 14B, 10A 14B
|limenharmonic
|-
|[[5L 5s]]||pentawood; blackwood
|5L 10s, 10L 5s
|chromatic blackwood
|5A 15B, 10A 15B
|enharmonic blackwood
|-
|[[6L 4s]]||lemon
|6L 10s, 10L 6s
|lemchromatic
|6A 16B, 10A 16B
|lemenharmonic
|-
|[[7L 3s]]||dicoid
|7L 10s, 10L 7s
|dicochromatic
|7A 17B, 10A 17B
|dicoenharmonic
|-
|[[8L 2s]]||taric
|8L 10s, 10L 8s
|tarachromatic
|8A 18B, 10A 18B
|tarenharmonic
|-
|[[9L 1s]]||sinatonic
|9L 10s, 10L 9s
|sinachromatic
|9A 19B, 10A 19B
|sinenharmonic
|}
==Names for mosses beyond 10 notes==
This section outlines proposed names and naming suggestions for mosses beyond 10 notes.

===Extended ''k''-wood names===
To name mos descendants with more than 5 periods, the names for wood mosses are extended to hexawood, heptawood, octawood, enneawood, and decawood. (This is not too different from Frostburn's proposal.) Names for descendants for these scales follow the same scheme as with other TAMNAMS-named mosses.
{| class="wikitable"
|+Names for wood scales up to 10 periods
!Mos
!Name
!Prefix
!Abbrev.
|-
|6L 6s
|hexawood
|hexwd-
|hxw
|-
|7L 7s
|heptawood
|hepwd-
|hpw
|-
|8L 8s
|octawood
|octwd-
|ocw
|-
|9L 9s
|enneawood
|ennwd-
|enw
|-
|10L 10s
|decawood
|dekwd-
|dkw
|-
|11L 11s
|11-wood
|11-wud-
|11wd
|-
|12L 12s
|12-wood
|12-wud
|12wd
|-
|etc...
|
|
|
|}
===Specific names for mosses beyond 10 notes (proposed)===
These names are intended for notable mosses outside the named range for which its mos descendant name would be insufficient.
{| class="wikitable"
! colspan="4" |11-note mosses
|-
!Mos
!Suggested name(s)
!Proposed by
!Reasoning
|-
|1L 10s
|tanzanite or tenorite
|[[User:Ganaram inukshuk]]
|More naming puns ('''ten'''zanite or '''ten'''orite)
|-
|2L 9s
|joanatonic
|
|Restoration of an old name that applied to its parent scale
|-
|3L 8s
|squaroid
|
|Restoration of an old name
|-
| rowspan="3" |4L 7s
|p-chromatic smitonic
soft-chromatic smitonic

soft smichromatic
|
|TAMNAMS descendant mos naming schemes
|-
|kleistonic
|
|Restoration of an old name
|-
|angelic or ecclesial
|[[User:Eliora]]
|
|-
|5L 6s
|xeimtonic
|
|Restoration of an old name
|-
|6L 5s
|
|
|
|-
| rowspan="3" |7L 4s
|suprasmitonic
|
|Restoration of an old name
|-
|demonic or infernal
|[[User:Eliora]]
|Described as being "furthest removed from typical xen approaches of RTT or JI."
|-
|daemotonic
|[[User:Ganaram inukshuk]]
|Alternative for name described above.
|-
|8L 3s
|sentonic or sensoid
|
|Modification or restoration of an old name that applied to its parent scale
|-
| rowspan="2" |9L 2s
|villatonic
|[[User:Ganaram inukshuk]]
|Indirectly references a'''vila''' and '''casa'''blanca (Spanish for "white house", and a villa is a type of house) temperaments
|-
|ultradiatonic, superarmotonic
|[[User:CompactStar]]
|In reference to diatonic and armotonic
|-
|10L 1s
|miratonic or miraculoid
|[[User:Ganaram inukshuk]]
|Modification or restoration of an old name (miraculoid); reference miracle temperament
|-
! colspan="4" |12-note mosses
|-
!Mos
!Suggested name(s)
!Proposed by
!Reasoning
|-
|1L 11s
|helenite
|[[User:Ganaram inukshuk]]
|In reference to the "ele" substring found in the word "eleven"
|-
|2L 10s
|rujaric
|[[User:Ganaram inukshuk]]
|Named based off of injera and shrutar temperaments
|-
|3L 9s
|sergic
|[[User:Ganaram inukshuk]]
|Named after one of Alexander Nikolayevich Tcherepnin's sons
|-
|4L 8s
|
|
|
|-
|5L 7s
|p-chromatic
|
|Restoration of an old name
|-
|6L 6s
|hexawood
|
|Extension of -wood scales; coincidentally references hexe temperament
|-
|7L 5s
|m-chromatic
|
|Restoration of an old name
|-
|8L 4s
|
|
|
|-
|9L 3s
|ivanic
|[[User:Ganaram inukshuk]]
|Named after one of Alexander Nikolayevich Tcherepnin's sons
|-
|10L 2s
|talaric
|[[User:Ganaram inukshuk]]
|Names based off of srutal/pajara temepraments
|-
|11L 1s
|ripploid
|[[User:Ganaram inukshuk]]
|Restoration of an old name
|-
! colspan="4" |13-note mosses
|-
!Mos
!Suggested name(s)
!Proposed by
!Reasoning
|-
|1L 12s
|zircon
|[[User:Ganaram inukshuk]]
|Zircon is used as a birthstone for December
|-
|2L 11s
|litonic
|[[User:Ganaram inukshuk]]
|Portmanteau of liese, triton, and tritonic temperaments
|-
|3L 10s
|magitonic or mystic
|[[User:Ganaram inukshuk]]
|In reference to magic temperament
|-
|4L 9s
|huxloga
|[[User:Ganaram inukshuk]]
|Portmanteau of huxley, lovecraft, and gariberttet temperaments
|-
|5L 8s
|
|
|
|-
|6L 7s
|
|
|
|-
|7L 6s
|tetarquintal
|[[User:Ganaram inukshuk]]
|Indirect reference to tetracot temperament, which divides the perfect 5th (3/2) into four
|-
|8L 5s
|petroid
|
|Restoration of an old name
|-
|9L 4s
|orwelloid
|
|Restoration of an old name that applied to its parent scale
|-
|10L 3s
|luachoid
|
|Already proposed name
|-
| rowspan="2" |11L 2s
|maioquartal
|[[User:Ganaram inukshuk]]
|In reference to the "major fourths" scale used by Tcherepnin
|-
|hendecoid
|[[User:Eliora]]
|From Greek "eleven", references how "its generator is so close to 11/8 as to be called nothing but that".
|-
|12L 1s
|quasidozenal
|[[User:Ganaram inukshuk]]
|Meant to invoke the phrase "almost twelve"
|-
! colspan="4" |14-note mosses
|-
!Mos
!Suggested name(s)
!Proposed by
!Reasoning
|-
|11L 3s
|ketradektriatoh
|[[User:Osmiorisbendi‎]]
|Already established name
|-
|13L 1s
|trollic
|[[User:Godtone]]
|Refers to 12L 1s, but refers to 13L 1s as a troll move
|-
! colspan="4" |15-note mosses
|-
!Mos
!Suggested name(s)
!Proposed by
!Reasoning
|-
|14L 1s
|sextiliquartal
|[[User:Eliora]]
|Already proposed name, references temperaments that divide 4/3 into 6 pieces
|}
{| class="wikitable"
|+
! colspan="5" |Other higher note count mosses
|-
!Note count
!Mos
!Suggested name(s)
!Proposed by
!Reasoning
|-
|17
|2L 15s
|liesic
|[[User:Frostburn]]
|Frostburn's extension scheme stops here, so this name is suggested
|-
|21
|10L 11s
|miracloid
|[[User:Eliora]]
|In reference to miracle temperament
|-
| rowspan="3" |22
|3L 19s
|zheligowskic
|[[User:Frostburn]]
|In reference to Lucjan Żeligowski leading fights against the town of Giedraičiai.
|-
|19L 3s
|giedraitic
|[[User:Frostburn]]
|Named after the basic magic layout of [[Kite Giedraitis]]' [[Kite guitar|guitar]].
|-
|21L 1s
|escapist
|[[User:Eliora]]
|References escapade temperament, which is supported by both 21edo and 22edo, covering the entire range.
|-
|23
|22L 1s
|quartismoid
|[[User:Eliora]]
|Five generators of roughly 33/32 quartertone are equal to 7/6 in the harmonic entropy minimum; also, the extreme ranges of 22edo and 23edo both support this mos.
|-
|25
|4L 21s
|moulinoid
|[[User:Eliora]]
|In reference to moulin temperament
|}

==Non-octave extensions (proposed)==
Since the perfect 5th and tritave (or perfect 12th) are the two most common non-octave equivalence intervals for which there are scales described, mosses for these two intervals should be the most likely to receive TAMNAMS-like names. For mosses with any other equivalence interval, describing nested mos structures, or in situations where the notion of an equivalence interval is unimportant, equave-agnostic names are proposed.

===Equave-agnostic names (proposed)===
This is a proposed scheme to name mosses regardless of the equivalence interval, These names are meant for nonoctave mosses and nested mos patterns such as with a mos cradle. These names are not final and are open to better suggestions.
{| class="wikitable"
! colspan="5" |4-note mosses (new names only)
|-
!Mos
!Name
!Multi-period?
!Prefix
!Abbrev.
|-
|2L 2s
|double trivial
|Yes (2)
|2triv-
|2trv
|-
! colspan="5" |6-note mosses
|-
!Mos
!Name
!Multi-period?
!Prefix
!Abbrev.
|-
|1L 5s
|anhexic
|No
|ahex-
|ahx
|-
|2L 4s
|double antrial
|Yes (2)
|2atri-
|2tri
|-
|3L 3s
|triple trivial
|Yes (3)
|3triv-
|3trv
|-
|4L 2s
|double trial
|Yes (2)
|2tri-
|2tri
|-
|5L 1s
|hexic
|No
|hex-
|hx
|-
! colspan="5" |7-note mosses
|-
!Mos
!Name
!Multi-period?
!Prefix
!Abbrev.
|-
|1L 6s
|ansaptic
|No
|ansap-
|asp
|-
|2L 5s
|anheptic
|No
|anhep-
|ahp
|-
|3L 4s
|anseptenic
|No
|ansep-
|asep
|-
|4L 3s
|septenic
|No
|sep-
|sep
|-
|5L 2s
|heptic
|No
|hep-
|hp
|-
|6L 1s
|saptic
|No
|sap-
|sp
|-
! colspan="5" |8-note mosses
|-
!Mos
!Name
!Multi-period?
!Prefix
!Abbrev.
|-
|1L 7s
|anashtaic
|No
|anasht-
|aasht
|-
|2L 6s
|double antetric
|Yes (2)
|2atetra-
|2att
|-
|3L 5s
|anoctic
|No
|anoct-
|aoct
|-
|4L 4s
|quadruple trivial
|Yes (4)
|4triv-
|4trv
|-
|5L 3s
|octic
|No
|oct-
|oct
|-
|6L 2s
|double tetric
|Yes (2)
|2tetra-
|2tt
|-
|7L 1s
|ashtaic
|No
|asht-
|asht
|-
! colspan="5" |9-note mosses
|-
!Mos
!Name
!Multi-period?
!Prefix
!Abbrev.
|-
|1L 8s
|annavic
|No
|annav-
|anv
|-
|2L 7s
|anennaic
|No
|anenn-
|aenn
|-
|3L 6s
|triple antrial
|Yes (3)
|3atri-
|3atri
|-
|4L 5s
|annovemic
|No
|annov-
|anv
|-
|5L 4s
|novemic
|No
|nov-
|nv
|-
|6L 3s
|triple trial
|Yes (3)
|3tri-
|3tri
|-
|7L 2s
|ennaic
|No
|enn-
|enn
|-
|8L 1s
|navic
|No
|nav-
|nv
|-
! colspan="5" |10-note mosses
|-
!Mos
!Name
!Multi-period?
!Prefix
!Abbrev.
|-
|1L 9s
|andashic
|No
|andash-
|adsh
|-
|2L 8s
|double pedal
|Yes (2)
|2ped-
|2ped
|-
|3L 7s
|andeckic
|No
|andeck-
|adek
|-
|4L 6s
|double pentic
|Yes (2)
|2pent-
|2pt
|-
|5L 5s
|quintuple trivial
|Yes (5)
|5triv-
|5trv
|-
|6L 4s
|double anpentic
|Yes (2)
|2apent-
|2apt
|-
|7L 3s
|deckic
|No
|deck-
|dek
|-
|8L 2s
|double manual
|Yes (2)
|2manu-
|2manu
|-
|9L 1s
|dashic
|No
|dash-
|dsh
|}
Names for these mosses are meant to be as general as possible, starting with established names that are already equave-agnostic: trivial, (an)trial, (an)tetric, (an)pentic, and pedal/manual. Mosses are named in pairs of xL ys and yL xs, where the mos with more small steps than large steps is given the an- prefix, short for anti-; this rule doesn't apply to pentic (2L 3s) and anpentic (3L 2s), where the former is the familiar pentatonic scale.

As there is only one pair of 6-note single-period mosses, 5L 1s and 1L 5s, the pair is named '''hexic'''.

With 7-note mosses, there are three pairs of mosses, whose names are based on three languages: Greek, Latin, and Sanskrit. The pair 5L 2s and 2L 5s are given the Greek-based name of '''heptic''', as 5L 2s is the familiar diatonic scale. The next pair, 3L 4s and 4L 3s, are given the Latin-based name of '''septenic'''. The last pair, 1L 6s and 6L 1s, are given the Sanskrit-based name of '''saptic''.'''''

This pattern is continued for all successive sequences of mosses for each successive note count: 1L ns and nL 1s are given a Sanskrit-based name, the next single-period pair after that are given a Greek-based name, and the next single-period pair after that are given a Latin-based name. The two 8-note pairs are named '''ashtaic''' (7L 1s and 1L 7s) and '''octic''' (5L 3s and 3L 5s) respectively. The three 9-note pairs are named '''navic''' (8L 1s and 1L 8s), '''ennaic''' (7L 2s and 2L 7s), and '''novemic''' (4L 5s and 5L 4s). Finally the two 10-note pairs are named '''dashic''' (9L 1s and 1L 9s) and '''dekic''' (7L 3s and 3L 7s).

11-note mosses require naming five pairs, so this naming scheme stops at 10-note mosses.

Since the equivalence interval can be anything, names for multi-period mosses are named as a smaller mos repeated (double, triple, quadruple, etc) some number of times. The prefix and abbreviation of the base mos is preceded by the number of duplications. For example, 2L 2s is double trivial, its prefix is 2triv-, and its abbreviation is 2trv.

===Non-octave twins of diatonic ===
{| class="wikitable"
|+
! colspan="5" |2-note mos
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|1L 1s <32/27, 6/5, 7/6, etc.><ref name=":1">Partial detemperament of diminished temperament</ref>
|dorianic[2], aeolianic[2], phrygianic[2], locrianic[2]
|dor-, aeol-, phryg-, locri-,
|dor, aeol, phryg, locri
|
|-
! colspan="5" |3-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|1L 2s <81/64, 5/4, 9/7 (21/16), etc.>*
|Second magitonic or mystic antrial
| colspan="2" rowspan="2" |use compound
| rowspan="2" |Linear combination of undivided major third and equal multiples of whole tone, by analogy with magic temperament having a major third generator
|-
|2L 1s <81/64, 5/4, 9/7 (21/16), etc.>*
|Second magitonic or mystic trial
|-
|2L 1s <4/3>
|ionianic[4]
|ion-
|ion
|“Perfect” fourth is the characteristic interval of Ionian (major) mode
|-
! colspan="5" |4-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|1L 3s<729/512, 1024/729, 25/18 (45/32), 36/25 (64/45), 7/5 (49/36), 10/7 (72/49), 3/2, etc.><ref>Partial detemperament of subaric temperament</ref>*
|(hard) lydianic antetric
| colspan="2" |use compound
|Linear combination of Lydian tetrachord and undivided tritone, analogous to 3:1 step ratio of hard mosses
|-
|2L 2s <729/512, 1024/729, 25/18 (45/32), 64/45, 7/5, 10/7 (72/49), 4/3, etc.><ref name=":1" />
|locrianic
|locri-
|locri-
|Locrian tritone is a diminished fifth
|-
|3L 1s<1024/729, 25/18 (45/32), 36/25 (64/45), 7/5 (49/36), 10/7 (72/49), etc.><ref>Partial detemperament of hedgehog temperament</ref>*
|(hard) lydianic tetric
| colspan="2" |use compound
|Linear combination of Lydian tetrachord and undivided tritone, analogous to 3:1 step ratio of hard mosses
|-
|3L 1s <3/2>
|phrygianic[4], (hard) lydianic tetric
|phryg-
|phryg
|“Perfect” fifth is the characteristic interval of Phrygian mode
Lydian pentachord is analogous to major scale

Mason Green proposes angelic specifically for the instance of this mos within 12L 4s <5/1>

antineptunian (prefix anept-, abbrev. anep) is by analogy with 1L 3s neptunian
|-
! colspan="5" |5-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|1L 4s <augmented fifth>*
|indopedal
| colspan="2" |use compound
|Linear combination of Hindu pentachord and undivided augmented fifth
|-
|2L 3s <diminished sixth>*
|Micro-pentic
| colspan="2" |use compound
|
|-
|3L 2s <128/81, 8/5, 14/9 (32/21), etc.>
|aeolianic, phrygianic[5]
|aeol-, phryg-
|aeol, phryg
|Commonly invoked as Aeolian (natural minor) hexachord
Phrygian hexachord is analogous to major scale
|-
|4L 1s <augmented fifth>*
|indomanual
| colspan="2" |use compound
|Linear combination of Hindu pentachord and undivided augmented fifth
|-
|4L 1s <27/16, 5/3, 12/7, etc.>
|dorianic[5]
|dor-
|dor
|Major sixth is the characteristic interval of Dorian mode
|-
! colspan="5" |6-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|3L 3s <diminished seventh><ref name=":1" />
|
|
|
|
|-
|1L 5s <15/8, 27/14 (63/32), 9/5, 7/4, 27/16, 5/3, 12/7, etc.>*
|Neapolitan-antimachinoid
| colspan="2" |use compound
|Linear combination of Neapolitan hexachord and undivided augmented sixth
|-
|4L 2s <9/5, 7/4, etc.><ref name=":0" />
|mixolydianic, dorianic[6]
|mixo-, dor-
|mixo, dor
|Minor seventh is the characteristic interval of Mixolydian mode
Dorian heptachord is analogous to major scale
|-
|5L 1s <16/9, 9/5, 7/4, 27/16, 5/3, 12/7, etc.>*
|Neapolitan-machinoid
| colspan="2" |use compound
|Linear combination of Neapolitan hexachord and undivided augmented sixth
|-
|5L 1s <243/128, 15/8, 27/14 (63/32), etc.>
|ionianic[6], lydianic[6]
|ion-, lyd-
|ion, lyd
|Commonly invoked as Ionian (major) heptachord
|-
! colspan="5" |8-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|4L 4s<diminished ninth>*<ref name=":1" /><ref name=":0" />
|diminished
|dimi-
|dimi-
|coincidentally references scale closing at a diminished ninth
|-
|5L 3s <512/243, 32/15, 56/27, etc.>
|Neapolitan-oneirotonic
| colspan="2" |use compound
|Neapolitan 6/9 scales appear just above oneirotonic proper
|-
|6L 2s <20/9, 16/7, etc.><ref name=":0">Major tempered variants</ref>
|napolitonic
|nap-
|nap-
|Translates minor triad to Neapolitan sixth
|-
! colspan="5" |9-note mosses (Dominant Seventh Scales)
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|6L 3s <12/5, 7/3, etc.><ref name=":0" />
|mahuric
|mahu-
|mahu
|Regularisation of Maqam Mahur scale
|-
|7L 2s <81/16, 5/2, 18/7 (21/8), etc.>
|armodecadic
|ardec-
|ard(e)
|In reference to Terra Rubra temperament, makes affix via translation (Terra = Du aarde, Ar ‘ard)
|-
! colspan="5" |10-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|7L 3s <8/3>
|choralic (Major)
| rowspan="2" |chor-
| rowspan="2" |chor
| rowspan="2" |More transparent of two given names, references how it puts triads in four parts
|-
|8L 2s <2048/729, 45/16, 72/25 (128/45), 14/5 (49/18), 20/7 (144/49), 3/1, etc.><ref name=":0" />
|choralic (Lydian)*
|-
! colspan="5" |11-note mosses (Dominant Ninth Scales)
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|7L 4s <729/256, 2048/729, 25/9 (45/16), 72/25 (128/45), 14/5 (49/18), 20/7 (144/49), 8/3, etc.,><ref name=":0" />
|Obikhodic (Locrian)*
| rowspan="2" |obi-
| rowspan="2" |obi
| rowspan="2" |In reference to Russian Orthodox Obikhod chants
|-
|8L 3s <3/1>
|Obikhodic (Phrygian)
|-
! colspan="5" |12-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|7L 5s <3/1, 729/256, 2048/729, 25/9 (45/16), 72/25 (128/45), 14/5 (49/18), 20/7 (144/49), 8/3, 16/5, 28/9 (64/21), etc.*>*
|
|
|
|Canonical macrochromatic scale
|-
|8L 4s <256/81, 16/5, 28/9 (64/21), etc.><ref name=":0" />
|m-chromatic bastonic
|
|
|bastonic is actually the macro-tetrawood temperament in a thirteenth (one of the '''four''' “Spanish” suits is '''wood'''en batons, or ''bastos'' in Spanish)
|-
|9L 3s <27/8, 10/3, 24/7, etc.><ref name=":0" />
|ivanimajiangic
|
|
|majiangic is actually the macro-tcherepnin temperament in a thirteenth (mahjong tiles have 3 1-9 suits)
|-
! colspan="5" |13-note mosses (Dominant Eleventh Scales)
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|8L 5s <diminished fourteenth>
|
|
|
|
|-
|9L 4s <32/9, 18/5, 7/2, etc.>
|shōsūshoid
|shō-
|shō
|References Japanese mahjong rules
|-
|10L 3s <243/64, 15/4, 27/7 (63/16), etc.>
|
|
|
|
|-
! colspan="5" |15-note mosses (Dominant Thirteenth Scales)
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|10L 5s <1024/243, 64/15, 112/27, etc.><ref name=":0" />
|
|
|
|
|-
|11L 4s <9/2, 40/9, 32/7, etc.>
|
|
|
|
|-
! colspan="5" |16-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|11L 5s <128/27, 24/5, 14/3, etc.>
|
|
|
|
|-
|12L 4s <81/16, 5/1, 36/7 (21/4), etc.><ref name=":0" />
|quasiangelic
|qang-
|qang
|In reference to Mason Green’s angelic generating 12L 4s <5/1>
|-
! colspan="5" |17-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|12L 5s <16/3>
|
|
|
|Canonical heptadecatonic macroenharmonic scale
|-
|13L 4s <729/128, 4096/729, 50/9 (45/8), 144/25 (256/45), 28/5 (49/9), 40/7 (288/49), 6/1, etc.>
|subsextal[17]
|sub6-
|sub6-
|In reference to its period being under the sixth harmonic
|-
! colspan="5" |18-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|12L 6s <729/128, 4096/729, 50/9 (45/8), 256/45, 28/5, 40/7 (288/49), 16/3, etc.><ref name=":0" />
|subsextal[18]
|sub6-
|sub6-
|In reference to its period being under the sixth harmonic
|-
|13L 5s <6/1>
|daseianic
|asper-
|asp-
|In reference to daseian notation
|-
! colspan="5" |19-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|12L 7s <diminished twentieth><ref name=":0" />
|
|
|
|Canonical enneadecatonic macroenharmonic scale
|-
|13L 6s <512/81, 32/5, 56/9 (128/21), etc.>
|
|
|
|
|-
|14L 5s <27/4, 20/3, 48/7, etc.>
|
|
|
|
|-
! colspan="5" |20-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|13L 7s <diminished twenty-first>
|Guidotonic (diminished)
| rowspan="3" |guido-
| rowspan="3" |guid-
| rowspan="3" |In reference to the Guidonian hand
|-
|14L 6s <36/5, 7/1, etc.><ref name=":0" />
|Guidotonic (dominant)
|-
|15L 5s <243/32, 15/2, 54/7 (63/8), etc.><ref name=":0" />
|Guidotonic (major)
|-
! colspan="5" |22-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|15L 7s <2048/243, 128/15, 224/27, etc.>
|
|
|
|
|-
|16L 6s <80/9, 64/7, etc.><ref name=":0" />
|
|
|
|
|-
|17L 5s <augmented twenty-third>*
|
|
|
|Canonical macroprotofractalic macrosubchromatic scale
|-
! colspan="5" |23-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|16L 7s <256/27, 48/5, 28/3, etc.>
|
|
|
|
|-
|17L 6s <81/8, 10/1, 72/7 (21/2), etc.>
|archangelic
|
|
|17L 6s <10/1> is the canonical decimal pitch scale
|-
! colspan="5" |24-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|17L 7s <32/3>
|tressettine
|
|
|In reference to Tressette scoring counting the whole deck as 10⅔ points
|-
|18L 6s <8192/729, 100/9 (45/4), 288/25 (512/45), 56/5 (98/9), 80/7 (576/49), 12/1, etc.><ref name=":0" />
|hendecoidal[24]
|hendec-
|hendec-
|From Greek "eleven", references 18L 6s <11/1>.
|-
! colspan="5" |25-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|19L 6s <729/64, 8192/729, 100/9 (45/4), 288/25 (512/45), 56/5 (98/9), 80/7 (576/49), 32/3, etc.>
|hendecoidal[25]
|hendec-
|hendec-
|From Greek "eleven", references 19L 6s <11/1>.
|-
|18L 7s <12/1>
|violic
|viol-
|vio
|In reference to the viol family commonly having French music for it notated in clefs a third above or below the grand staff
|-
! colspan="5" |26-note mosses
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|18L 8s <64/5, 112/9 (256/21), etc.><ref name=":0" />
|Petrushkatonic (minor)
| rowspan="2" |petrushka-
| rowspan="2" |petru-
| rowspan="2" |In reference to the “27th chord” which appears in Stravinsky’s ''Petrushka''
|-
|19L 7s <27/2, 40/3, 96/7, etc.>
|Petrushkatonic (major)
|}

=== Secondary names for tritone-equivalent mosses ===
{| class="wikitable"
|+
|-
! colspan="5" |5-note mosses <tritone>
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|2L 3s<7/5, 10/7, 11/8*><ref>Partial detemperament of lime temperaments </ref>
|(soft) lydianic pentic
| colspan="2" rowspan="2" |use compound
| rowspan="2" |Linear combination of Lydian tetrachord and equal multiples of sesquitone, analogous to 3:2 step ratio of soft mosses
|-
|3L 2s<7/5, 10/7, 11/8*><ref>Partial detemperament of lemon temperaments</ref>
|(soft) lydianic anpentic
|}

===Names for 3/2-equivalent mosses===
Names are based on information that is available on their respective pages. Otherwise, possible ideas are given. Only mosses with 10 or fewer notes are prioritized for names.
{| class="wikitable"
! colspan="5" |4-note mosses <3/2>
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|1L 3s
|neptunian
|nept-
|nep
|Name proposed by CompactStar, analogous to uranian
|-
! colspan="5" |5-note mosses <3/2>
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|2L 3s
|saturnian
|sat-
|sat
|Name proposed by CompactStar, analogous to uranian
|-
|3L 2s
|uranian
|ura-
|ura
|Already-existing name
|}

===Names for 3/1-equivalent mosses===
Names are based on information that is is available on their respective pages. Otherwise, possible ideas are given. Only mosses with 10 or fewer notes are prioritized for names.
{| class="wikitable"
! colspan="5" |7-note mosses <3/1>
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|3L 4s
|
|
|
|In reference to electromagnetism, 3L 4s <3/1> could be named "magnetic"
|-
|4L 3s
|electric
|elec-
|ele
|Name proposed by CompactStar
|-
! colspan="5" |9-note mosses <3/1>
|-
!Mos
!Name (if given)
!Prefix
!Abbrev.
!Reasoning or ideas
|-
|4L 5s
|lambdatonic
|lam-
|lam
|"Lambda" already refers to tritave-equivalent 4L 5s
|}

====Reasoning for names====
The overall motivation for these names is to give names to closely related mosses and refer to individual mosses as some member of a broader family, rather than name individual mosses. Various terms have been used to similarly describe child mosses, but not under a temperament-agnostic viewpoint.
{| class="wikitable"
|-
!Source of terms
!Grandparent (2nd predecessor)
!Parent (1st predecessor)
!Mos
!Child (1st descendant)
!Grandchild (2nd descendant)
!Great-grandchild (3rd descendant)
!''k''th descendant
|-
|From [[Diatonic, Chromatic, Enharmonic, Subchromatic]]
|n/a
|n/a
|diatonic
|chromatic
|enharmonic
|subchromatic
|n/a
|-
| rowspan="2" |From [[Chromatic pairs]]
| rowspan="2" |sub-haplotonic
(not called this on page)
| rowspan="2" |haplotonic
| rowspan="2" |albitonic
|chromatic
|mega-chromatic
|
| rowspan="2" |n/a
|-
|mega-albitonic
|chromatic
|mega-chromatic
|-
|Terminology used for this page
|n/a
|n/a
|mos
|chromatic mos
|enharmonic mos
|subchromatic mos
|''k''th descendant
|}The format of adding a mos's prefix to the terms descendant, chromatic, enharmonic, and subchromatic is best applied to mosses that have no more than three periods. With mosses that descend directly from nL ns mosses especially (4L 4s and above), this is to keep names from being too complicated (eg, ''chromatic (number)-wood'' instead of ''(number)-woodchromatic'').

Various people have suggested the use of p- and m- as prefixes to refer to specific chromatic mosses, as well as the use of f- and s- for enharmonic mosses. Generalizing the pattern to 3rd mos descendants shows the letters diverging from one another, notably where m- is no longer next to p- and f- and s- are no longer along the extremes. Rather than using these letters, as well as being temperament-agnostic, prefixes based on step ratios are used instead. However, temperament-based prefixes may be used specifically for diatonic descendants as alternatives to the prefixes based on step ratios.
{| class="wikitable"
|+Prefixes for diatonic descendants
! rowspan="2" |Diatonic scale
! colspan="3" |Chromatic mosses
! colspan="3" |Enharmonic mosses
! colspan="3" |Subchromatic mosses
|-
!Steps
!Temp-based prefix
!Ratio-based prefix
!Steps
!Temp-based prefix
!Ratio-based prefix
!Steps
!Temp-based prefix
!Ratio-based prefix
|-
| rowspan="8" |[[5L 2s]]
| rowspan="4" |[[7L 5s]]
| rowspan="4" |m- (from meantone)
| rowspan="4" |s-
| rowspan="2" |[[7L 12s]]
| rowspan="2" |f- (from flattone)
| rowspan="2" |s-
|[[7L 19s]]
|t- (from tridecimal)
|us-
|-
|[[19L 7s]]
|f- (from flattone)
|ps-
|-
| rowspan="2" |[[12L 7s]]
| rowspan="2" |m- (from meantone)
| rowspan="2" |os-
|[[19L 12s]]
|m- (from meanpop)
|qs-
|-
|[[12L 19s]]
|h- (from huygens)
|ms-
|-
| rowspan="4" |[[5L 7s]]
| rowspan="4" |p- (from pythagorean)
| rowspan="4" |h-
| rowspan="2" |[[12L 5s]]
| rowspan="2" |p- (from pythagorean)
| rowspan="2" |oh-
|[[12L 17s]]
|p- (from pythagorean)
|mh-
|-
|[[17L 12s]]
|g- (from gentle)
|qh-
|-
| rowspan="2" |[[5L 12s]]
| rowspan="2" |s- (from superpyth)
| rowspan="2" |h-
|[[17L 5s]]
|s- (from superpyth)
|ph-
|-
|[[5L 17s]]
|u- (from ultrapyth)
|uh-
|}

<references />

User:Moremajorthanmajor/3L 1s (perfect fifth-equivalent)

2024-12-29T20:43:54Z

Moremajorthanmajor: /* Notation */

'''3L 1s<perfect fifth>''' is constructed by repeating the fifth-spanning pattern LLLs of the ordinary diatonic mos ([[5L 2s]]) at the equave of 3/2. The so-called "Super Ultra Hyper Mega Meta Lydian" is one mode of this mos.

The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating [[3L 1s]]. The name of the period interval is called the '''sesquitave''' (by analogy to the [[tritave]]). The generator range is 171.4 to 240 cents, placing it near the [[9/8|diatonic major second]], usually representing a major second of some type. The dark (chroma-negative) generator is, however, its fifth complement (480 to 514.3 cents).

In the fifth-repeating version of the diatonic scale, each tone has a 3/2 perfect fifth above it. The scale has two major chords and two minor chords.

'''Angel''' is a proposed name for this mos. [[Basic]] Angel is in [[7edf]], which is a very good fifth-based equal tuning similar to [[12edo]].

==Notation==

There are 6 main ways to notate the angel scale. One method uses a simple sesquitave (fifth) repeating notation consisting of 4 naturals (eg. Do Re Mi Fa, Fa Sol La Si, Sol La Si Do). Given that 1-5/4-5/3 is fifth-equivalent to a tone cluster of 1-10/9-5/4, it may be more convenient to notate angel scales as repeating at the double, triple, quadruple, quintuple or sextuple sesquitave (major ninth, thirteenth, seventeenth i. e. ~pentave or twenty-first or augmented twenty-fifth), however it does make navigating the [[Generator|genchain]] harder. This way, 5/3 is its own pitch class, distinct from 10/9. Notating this way produces a major ninth which is the Aeolian mode of Napoli[6L 2s], a major thirteenth which is the Dorian mode of Bijou[9L 3s], an ~pentave which is the Mixolydian mode of Hextone[12L 4s], a major twenty-first which is the Ionian mode of Guidotonic[15L 5s] or an augmented twenty-fifth which is the Lydian mode of Subdozenal[18L 6s]. Since there are exactly 8 naturals in double sesquitave notation, 12 in triple sesquitave notation, 16 in quadruple sesquitave notation, 20 in quintuple sesquitave notation and 24 in sextuple sesquitave notation, letters A-H (FGABHCDEF), dozenal or hex digits (0123456789XE0 or D1234567FGACD with flats written C molle or 0123456789ABCDEF0 or G123456789ABCDEFG with flats written F molle), the Guidonian names with F as the lowest ut or letters except I and O may be used.

{| class="wikitable"

|+

Cents

! Notation

!Supersoft

!Soft

!Semisoft

!Basic

!Semihard

!Hard

!Superhard

|-

!Diatonic
!~15edf

!~11edf

!~18edf

!~7edf

!~17edf

!~10edf

!~13edf

|-

|Do#, Fa#, Sol#
|1\15, 46.154

|1\11, 63.158

|2\18, 77.419

| rowspan="2" | 1\7, 100

|3\17, 124.138

|2\10, 141.176

|3\13, 163.636

|-

|Reb, Solb, Lab
|3\15, 138.462

|2\11. 126.316

|3\18, 116.129

|2\17, 82.759

|1\10, 70.588

|1\13, 54.545

|-

|'''Re, Sol, La'''
|'''4\15,''' '''184.615'''

|'''3\11,''' '''189.474'''

|'''5\18,''' '''193.548'''

|'''2\7,''' '''200'''

|'''5\17,''' '''206.897'''

|'''3\10,''' '''211.765'''

|'''4\13,''' '''218.182'''

|-

|Re#, Sol#, La#
|5\15, 230.769

|4\11, 252.632

|7\18, 270.968

| rowspan="2" | 3\7, 300

|8\17, 331.034

|5\10, 352.941

|7\13, 381.818

|-

|Mib, Lab, Sib
|7\15, 323.077

|5\11, 315.789

|8\18, 309.677

|7\17, 289.655

|4\10, 282.353

|5\13, 272.727

|-

|Mi, La, Si
|8\15, 369.231

|6\11, 378.947

|10\18, 387.097

|4\7, 400

|10\17, 413.793

|6\10, 423.529

|8\13, 436.364

|-

|Mi#, La#, Si#
|9\15, 415.385

| rowspan="2" | 7\11, 442.105

|12\18, 464.516

|5\7, 500

|13\17, 537.069

|8\10, 564.706

|11\13, 600

|-

|Fab, Sibb, Dob
|10\15, 461.538

|11\18, 425.806

|4\7, 400

|9\17, 372.414

|5\10, 352.941

|6\13, 327.273

|-

|'''Fa, Sib, Do'''
|'''11\15,''' '''507.692'''

|'''8\11,''' '''505.263'''

|'''13\18,''' '''503.226'''

|'''5\7, 500'''

|'''12\17,''' '''496.552'''

|'''7\10,''' '''494.118'''

|'''9\13,''' '''490.909'''

|-

|Fa#, Si, Do#
|12\15, 553.846

|9\11, 568.421

|15\18, 580.645

|6\7, 600

|15\17, 620.690

|9\10, 635.294

|12\13, 654.545

|-

|Fax, Si#, Dox
|13\15, 600

| rowspan="2" | 10\11, 631.579

|17\18, 658.064

|7\7, 700

|18\17, 744.828

|11\10, 776.471

|15\13, 818.182

|-

|Dob, Fab, Solb
|14\15, 646.154
|16\18, 619.355
|6\7, 600
|14\17, 579.310
|8\10, 564.706
|10\13, 545.455

|-

!Do, Fa, Sol
!'''15\15,''' '''692.308'''

!'''11\11,''' '''694.737'''

!'''18\18,''' '''696.774'''

!7\7, 700

!'''17\17,''' '''703.448'''

!'''10\10,''' '''705.882'''

!'''13\13,''' '''709.091'''

|}

{| class="wikitable"

|+

Cents
!Notation
!Supersoft

! Soft

!Semisoft

!Basic

!Semihard

! Hard

! Superhard

|-

!Napoli
! ~15edf

! ~11edf

!~18edf

!~7edf

!~17edf

!~10edf

!~13edf

|-

|F#
|1\15, 46.154

|1\11, 63.158

| 2\18, 77.419

| rowspan="2" |1\7, 100

|3\17, 124.138

| 2\10, 141.176

|3\13, 163.636

|-

| Gb, Ge
|3\15, 138.462

| 2\11. 126.316

|3\18, 116.129

|2\17, 82.759

|1\10, 70.588

|1\13, 54.545

|-

|'''G'''
|'''4\15,''' '''184.615'''

|'''3\11,''' '''189.474'''

|'''5\18,''' '''193.548'''

|'''2\7,''' '''200'''

|'''5\17,''' '''206.897'''

|'''3\10,''' '''211.765'''

|'''4\13,''' '''218.182'''

|-

|G#
|5\15, 230.769

|4\11, 252.632

|7\18, 270.968

| rowspan="2" |3\7, 300

| 8\17, 331.034

|5\10, 352.941

|7\13, 381.818

|-

|Ab, Æ
|7\15, 323.077

|5\11, 315.789

|8\18, 309.677

|7\17, 289.655

|4\10, 282.353

|5\13, 272.727

|-

|A
| 8\15, 369.231

|6\11, 378.947

|10\18, 387.097

| 4\7, 400

|10\17, 413.793

|6\10, 423.529

|8\13, 436.364

|-

|A#
| 9\15, 415.385

| rowspan="2" |7\11, 442.105

|12\18, 464.516

|5\7, 500

|13\17, 537.069

|8\10, 564.706

|11\13, 600

|-

|Bbb, Bee
|10\15, 461.538

|11\18, 425.806

|4\7, 400

|9\17, 372.414

| 5\10, 352.941

|6\13, 327.273

|-

|'''Bb, Be'''
|'''11\15,''' '''507.692'''

|'''8\11,''' '''505.263'''

|'''13\18,''' '''503.226'''

|'''5\7, 500'''

|'''12\17,''' '''496.552'''

|'''7\10,''' '''494.118'''

|'''9\13,''' '''490.909'''

|-

|B
|12\15, 553.846

|9\11, 568.421

|15\18, 580.645

|6\7, 600

| 15\17, 620.690

|9\10, 635.294

|12\13, 654.545

|-
| B#
| 13\15, 600

| rowspan="2" |10\11, 631.579

|17\18, 658.064

|7\7, 700

|18\17, 744.828

|11\10, 776.471

|15\13, 818.182

|-
|Hb, He
|14\15, 646.154
| 16\18, 619.355
|6\7, 600
|14\17, 579.310
|8\10, 564.706
|10\13, 545.455

|-

! H
!'''15\15,''' '''692.308'''

!'''11\11,''' '''694.737'''

!'''18\18,''' '''696.774'''

! 7\7, 700

!'''17\17,''' '''703.448'''

!'''10\10,''' '''705.882'''

!'''13\13,''' '''709.091'''

|-

|Η#
|16\15, 738.462

|12\11, 757.895

|20\18, 774.194

| rowspan="2" |8\8, 800

|20\17, 827.586

|12\10, 847.059

|16\13, 872.727

|-

|Cb, Ce
|18\15, 830.769

|13\11, 821.053

|21\18, 812.903

|19\17, 786.207

|11\10, 776.471

|14\13, 763.63

|-

|'''C'''
|'''19\15,''' '''876.923'''

|'''14\11,''' '''884.211'''

|'''23\18,''' '''890.323'''

|'''9\5,''' '''900'''

|'''22\17,''' '''910.345'''

|'''13\10,''' '''917.647'''

|'''17\13,''' '''927.273'''

|-

|C#
|20\15, 923.077

|15\11, 947.368

|25\18, 967.742

| rowspan="2" |10\7, 1000

|25\17, 1034.483

|15\10, 1058.824

|20\13, 1090.909

|-

| Db, De
|22\15, 1015.385

|16\11, 1010.526

|26\18, 1006.452

|24\17, 993.103

|14\10, 988.235

|18\13, 981.818

|-

|D
|23\15, 1061.538

|17\11, 1073.684

|28\18, 1083.871

|11\7, 1100

|27\17, 1117.241

|16\10, 1129.412

|21\9, 1145.455

|-

|D#
|24\15, 1107.923

| rowspan="2" |18\11, 1136.842

|30\18, 1161.29

|12\7, 1200

|30\17, 1241.379

|18\10, 1270.588

|24\13, 1309.091

|-

|Ebb, Ëe
|25\15, 1153.846

|29\18, 1122.581

|11\7, 1100

|26\17, 1075.862

|15\10, 1058.824

| 19\13, 1036.364

|-

|'''Eb, Ë'''
|'''26\15,''' '''1200'''

|'''19\11,''' '''1200'''

|'''31\18,''' '''1200'''

|'''12\7, 1200'''

|'''29\17,''' '''1200'''

|'''17\10,''' '''1200'''

|'''22\13,''' '''1200'''

|-

|E
|27\15, 1246.154

|20\11, 1263.158

|33\18, 1277.419

|13\7, 1300

|32\17, 1324.138

|19\10, 1341.176

|25\13, 1363.636

|-

|E#
|28\15, 1292.308

| rowspan="2" |21\11, 1326.318

|35\18, 1354.834

|14\7, 1400

|35\17, 1448.275

| 21\10, 1482.353

|28\13, 1527.273

|-

| Fb, Fe
|29\15, 1338.462

|34\18, 1316.129

|13\7, 1300

|31\17, 1282.759

|18\10, 1270.588

|23\13, 1254.545

|-

!F
!30\15, 1384.615

!22\11, 1389.473

!36\18, 1393.548

!14\7, 1400

!34\17, 1406.897

!20\10, 1411.765

!26\13, 1418.182
|}

{| class="wikitable"
|+Cents
! Notation
!Supersoft
!Soft
!Semisoft
!Basic
!Semihard
!Hard
! Superhard
|-
!Bijou
!~15edf
!~11edf
!~18edf
!~7edf
!~17edf
!~10edf
!~13edf
|-
|0#, D#
|1\15, 46.154
|1\11, 63.158
|2\18, 77.419
| rowspan="2" |1\7, 100
|3\17, 124.138
|2\10, 141.176
|3\13, 163.636
|-
|1b, 1c
|3\15, 138.462
| 2\11. 126.316
|3\18, 116.129
|2\17, 82.759
|1\10, 70.588
|1\13, 54.545
|-
|'''1'''
|'''4\15,''' '''184.615'''
|'''3\11,''' '''189.474'''
|'''5\18,''' '''193.548'''
|'''2\7,''' '''200'''
|'''5\17,''' '''206.897'''
|'''3\10,''' '''211.765'''
|'''4\13,''' '''218.182'''
|-
|1#
|5\15, 230.769
|4\11, 252.632
|7\18, 270.968
| rowspan="2" |3\7, 300
|8\17, 331.034
|5\10, 352.941
|7\13, 381.818
|-
|2b, 2c
|7\15, 323.077
|5\11, 315.789
| 8\18, 309.677
| 7\17, 289.655
|4\10, 282.353
|5\13, 272.727
|-
|2
|8\15, 369.231
|6\11, 378.947
|10\18, 387.097
|4\7, 400
|10\17, 413.793
|6\10, 423.529
|8\13, 436.364
|-
|2#
| 9\15, 415.385
| rowspan="2" |7\11, 442.105
|12\18, 464.516
|5\7, 500
|13\17, 537.069
|8\10, 564.706
|11\13, 600
|-
|3b, 3c
| 10\15, 461.538
| 11\18, 425.806
|4\7, 400
|9\17, 372.414
|5\10, 352.941
|6\13, 327.273
|-
|'''3'''
|'''11\15,''' '''507.692'''
|'''8\11,''' '''505.263'''
|'''13\18,''' '''503.226'''
|'''5\7, 500'''
|'''12\17,''' '''496.552'''
|'''7\10,''' '''494.118'''
|'''9\13,''' '''490.909'''
|-
|3#
|12\15, 553.846
|9\11, 568.421
|15\18, 580.645
|6\7, 600
|15\17, 620.690
|9\10, 635.294
|12\13, 654.545
|-
|3x
|13\15, 600
| rowspan="2" |10\11, 631.579
|17\18, 658.064
|7\7, 700
|18\17, 744.828
|11\10, 776.471
|15\13, 818.182
|-
|4b, 4c
|14\15, 646.154
|16\18, 619.355
|6\7, 600
|14\17, 579.310
|8\10, 564.706
|10\13, 545.455
|-
!4
!'''15\15,''' '''692.308'''
!'''11\11,''' '''694.737'''
!'''18\18,''' '''696.774'''
!7\7, 700
!'''17\17,''' '''703.448'''
!'''10\10,''' '''705.882'''
!'''13\13,''' '''709.091'''
|-
|4#
| 16\15, 738.462
|12\11, 757.895
|20\18, 774.194
| rowspan="2" |8\8, 800
|20\17, 827.586
|12\10, 847.059
| 16\13, 872.727
|-
|5b, 5c
|18\15, 830.769
|13\11, 821.053
|21\18, 812.903
|19\17, 786.207
|11\10, 776.471
|14\13, 763.63
|-
|'''5'''
|'''19\15,''' '''876.923'''
|'''14\11,''' '''884.211'''
|'''23\18,''' '''890.323'''
|'''9\5,''' '''900'''
|'''22\17,''' '''910.345'''
|'''13\10,''' '''917.647'''
|'''17\13,''' '''927.273'''
|-
|5#
|20\15, 923.077
|15\11, 947.368
|25\18, 967.742
| rowspan="2" |10\7, 1000
|25\17, 1034.483
|15\10, 1058.824
|20\13, 1090.909
|-
|6b, 6c
|22\15, 1015.385
|16\11, 1010.526
|26\18, 1006.452
|24\17, 993.103
|14\10, 988.235
|18\13, 981.818
|-
|6
|23\15, 1061.538
|17\11, 1073.684
| 28\18, 1083.871
|11\7, 1100
|27\17, 1117.241
|16\10, 1129.412
|21\9, 1145.455
|-
|6#
|24\15, 1107.923
| rowspan="2" |18\11, 1136.842
|30\18, 1161.290
|12\7, 1200
|30\17, 1241.379
|18\10, 1270.588
|24\13, 1309.091
|-
| 7b, 7c
|25\15, 1153.846
|29\18, 1122.581
|11\7, 1100
|26\17, 1075.862
|15\10, 1058.824
|19\13, 1036.364
|-
|'''7'''
|'''26\15,''' '''1200'''
|'''19\11,''' '''1200'''
|'''31\18,''' '''1200'''
|'''12\7, 1200'''
|'''29\17,''' '''1200'''
|'''17\10,''' '''1200'''
|'''22\13,''' '''1200'''
|-
|7#
|27\15, 1246.154
|20\11, 1263.158
|33\18, 1277.419
|13\7, 1300
|32\17, 1324.138
|19\10, 1341.176
|25\13, 1363.636
|-
|7x
|28\15, 1292.308
| rowspan="2" |21\11, 1326.318
|35\18, 1354.834
|14\7, 1400
|35\17, 1448.275
|21\10, 1482.353
|28\13, 1527.273
|-
|8b, Fc
|29\15, 1338.462
|34\18, 1316.129
|13\7, 1300
|31\17, 1282.759
|18\10, 1270.588
|23\13, 1254.545
|-
!8, F
!30\15, 1384.615
!22\11, 1389.473
!36\18, 1393.548
!14\7, 1400
!34\17, 1406.897
!20\10, 1411.765
!26\13, 1418.182
|-
|8#, F#
|31\15, 1430.769
|23\11, 1452.632
|38\18, 1470.968
| rowspan="2" |15\7, 1500
|37\17, 1531.034
|22\10, 1552.941
|29\13, 1581.818
|-
|9b, Gc
|33\15, 1523.077
|24\11, 1515.789
|39\18, 1509.677
|36\17, 1489.655
|21\10, 1482.759
|27\13, 1472.273
|-
|'''9, G'''
|'''34\15,''' '''1569.231'''
|'''25\11,''' '''1578.947'''
|'''41\18,''' '''1587.097'''
|'''16\7,''' '''1600'''
|'''39\17,''' '''1613.793'''
|'''23\10,''' '''1623.529'''
|'''30\13,''' '''1636.364'''
|-
|9#, G#
|35\15, 1615.385
|26\11, 1642.105
|43\18, 1664.516
| rowspan="2" |17\7, 1700
|42\17, 1737.069
|25\10, 1764.706
|33\13, 1800
|-
|Xb, Ac
|37\15, 1707.692
|27\11, 1705.263
|44\18, 1703.226
|41\17, 1696.552
|24\10, 1694.118
|31\13, 1690.909
|-
|X, A
|38\15, 1753.846
|28\11, 1768.421
|46\18, 1780.645
|18\7, 1800
|44\17, 1820.690
|26\10, 1835.294
|34\13, 1854.545
|-
|X#, A#
|39\15, 1800
| rowspan="2" |29\11, 1831.579
|48\18, 1858.064
|19\7, 1900
|47\17, 1944.828
|28\10, 1976.471
|37\13, 2018.182
|-
|Ebb, Ccc
|40\15, 1846.154
|47\18, 1819.355
|18\7, 1800
|43\17, 1779.310
|25\10, 1764.706
|32\13, 1745.545
|-
|'''Eb, Cc'''
|'''41\15,''' '''1892.308'''
|'''30\11,''' '''1894.737'''
|'''49\18,''' '''1896.774'''
|'''19\7, 1900'''
|'''46\17,''' '''1903.448'''
|'''27\10,''' '''1905.882'''
|'''35\13,''' '''1909.091'''
|-
|E, C
|42\15, 1938.462
|31\11, 1957.895
|51\18, 1974.194
|20\7, 2000
|49\17, 2027.586
|29\10, 2047.059
|38\13, 2072.727
|-
|Ex, Cx
|43\15, 1984.615
| rowspan="2" |32\11, 2021.053
|53\18, 2051.612
|21\7, 2100
|52\17, 2151.725
|31\10, 2188.235
|41\13, 2236.364
|-
|0b, Dc
|44\15, 2030.769
|52\18, 2012.903
|20\7, 2000
|48\17, 1986.207
|28\10, 1976.471
|36\13, 1963.636
|-
! 0, D
!45\15, 2076.923
!33\11, 2084.211
!54\18, 2090.323
!21\7, 2100
!51\17, 2110.345
!30\10, 2117.647
!39\13, 2127.273
|}

{| class="wikitable"
|+Cents
!Notation
!Supersoft
!Soft
!Semisoft
! Basic
!Semihard
!Hard
!Superhard
|-
!Hextone
!~15edf
!~11edf
!~18edf
!~7edf
!~17edf
!~10edf
!~13edf
|-
|0#, G#
|1\15, 46.154
|1\11, 63.158
|2\18, 77.419
| rowspan="2" |1\7, 100
|3\17, 124.138
|2\10, 141.176
|3\13, 163.636
|-
| 1f
|3\15, 138.462
|2\11. 126.316
|3\18, 116.129
|2\17, 82.759
|1\10, 70.588
|1\13, 54.545
|-
|'''1'''
|'''4\15,''' '''184.615'''
|'''3\11,''' '''189.474'''
|'''5\18,''' '''193.548'''
|'''2\7,''' '''200'''
|'''5\17,''' '''206.897'''
|'''3\10,''' '''211.765'''
|'''4\13,''' '''218.182'''
|-
|1#
|5\15, 230.769
|4\11, 252.632
|7\18, 270.968
| rowspan="2" |3\7, 300
|8\17, 331.034
|5\10, 352.941
|7\13, 381.818
|-
|2f
|7\15, 323.077
|5\11, 315.789
|8\18, 309.677
|7\17, 289.655
|4\10, 282.353
|5\13, 272.727
|-
|2
|8\15, 369.231
|6\11, 378.947
|10\18, 387.097
| 4\7, 400
|10\17, 413.793
|6\10, 423.529
|8\13, 436.364
|-
|2#
|9\15, 415.385
| rowspan="2" |7\11, 442.105
|12\18, 464.516
|5\7, 500
|13\17, 537.069
|8\10, 564.706
|11\13, 600
|-
|3f
| 10\15, 461.538
|11\18, 425.806
|4\7, 400
|9\17, 372.414
|5\10, 352.941
|6\13, 327.273
|-
|'''3'''
|'''11\15,''' '''507.692'''
|'''8\11,''' '''505.263'''
|'''13\18,''' '''503.226'''
|'''5\7, 500'''
|'''12\17,''' '''496.552'''
|'''7\10,''' '''494.118'''
|'''9\13,''' '''490.909'''
|-
|3#
|12\15, 553.846
|9\11, 568.421
|15\18, 580.645
|6\7, 600
|15\17, 620.690
|9\10, 635.294
|12\13, 654.545
|-
| 3x
|13\15, 600
| rowspan="2" | 10\11, 631.579
|17\18, 658.064
|7\7, 700
|18\17, 744.828
|11\10, 776.471
|15\13, 818.182
|-
|4f
| 14\15, 646.154
|16\18, 619.355
|6\7, 600
|14\17, 579.310
|8\10, 564.706
|10\13, 545.455
|-
!4
!'''15\15,''' '''692.308'''
!'''11\11,''' '''694.737'''
!'''18\18,''' '''696.774'''
!7\7, 700
!'''17\17,''' '''703.448'''
!'''10\10,''' '''705.882'''
!'''13\13,''' '''709.091'''
|-
| 4#
|16\15, 738.462
|12\11, 757.895
|20\18, 774.194
| rowspan="2" |8\8, 800
|20\17, 827.586
|12\10, 847.059
|16\13, 872.727
|-
|5
|18\15, 830.769
|13\11, 821.053
|21\18, 812.903
|19\17, 786.207
| 11\10, 776.471
|14\13, 763.63
|-
|'''5'''
|'''19\15,''' '''876.923'''
|'''14\11,''' '''884.211'''
|'''23\18,''' '''890.323'''
|'''9\5,''' '''900'''
|'''22\17,''' '''910.345'''
|'''13\10,''' '''917.647'''
|'''17\13,''' '''927.273'''
|-
|5#
|20\15, 923.077
|15\11, 947.368
| 25\18, 967.742
| rowspan="2" |10\7, 1000
|25\17, 1034.483
|15\10, 1058.824
|20\13, 1090.909
|-
|6f
|22\15, 1015.385
|16\11, 1010.526
|26\18, 1006.452
|24\17, 993.103
|14\10, 988.235
|18\13, 981.818
|-
|6
|23\15, 1061.538
|17\11, 1073.684
|28\18, 1083.871
|11\7, 1100
|27\17, 1117.241
|16\10, 1129.412
|21\9, 1145.455
|-
|6#
|24\15, 1107.923
| rowspan="2" |18\11, 1136.842
|30\18, 1161.290
|12\7, 1200
|30\17, 1241.379
|18\10, 1270.588
|24\13, 1309.091
|-
| 7f
|25\15, 1153.846
|29\18, 1122.581
|11\7, 1100
|26\17, 1075.862
|15\10, 1058.824
|19\13, 1036.364
|-
|'''7'''
|'''26\15,''' '''1200'''
|'''19\11,''' '''1200'''
|'''31\18,''' '''1200'''
|'''12\7, 1200'''
|'''29\17,''' '''1200'''
|'''17\10,''' '''1200'''
|'''22\13,''' '''1200'''
|-
|7#
|27\15, 1246.154
|20\11, 1263.158
|33\18, 1277.419
|13\7, 1300
|32\17, 1324.138
|19\10, 1341.176
|25\13, 1363.636
|-
|7x
|28\15, 1292.308
| rowspan="2" |21\11, 1326.318
|35\18, 1354.834
|14\7, 1400
|35\17, 1448.275
|21\10, 1482.353
|28\13, 1527.273
|-
|8f
|29\15, 1338.462
| 34\18, 1316.129
|13\7, 1300
|31\17, 1282.759
|18\10, 1270.588
|23\13, 1254.545
|-
! 8
!30\15, 1384.615
!22\11, 1389.473
!36\18, 1393.548
!14\7, 1400
!34\17, 1406.897
!20\10, 1411.765
!26\13, 1418.182
|-
|8#
|31\15, 1430.769
|23\11, 1452.632
| 38\18, 1470.968
| rowspan="2" |15\7, 1500
|37\17, 1531.034
|22\10, 1552.941
|29\13, 1581.818
|-
|9f
|33\15, 1523.077
|24\11, 1515.789
|39\18, 1509.677
| 36\17, 1489.655
|21\10, 1482.759
|27\13, 1472.273
|-
|9
|'''34\15,''' '''1569.231'''
|'''25\11,''' '''1578.947'''
|'''41\18,''' '''1587.097'''
|'''16\7,''' '''1600'''
|'''39\17,''' '''1613.793'''
|'''23\10,''' '''1623.529'''
|'''30\13,''' '''1636.364'''
|-
|9#
|35\15, 1615.385
|26\11, 1642.105
|43\18, 1664.516
| rowspan="2" |17\7, 1700
|42\17, 1737.069
|25\10, 1764.706
|33\13, 1800
|-
|Af
| 37\15, 1707.692
| 27\11, 1705.263
|44\18, 1703.226
|41\17, 1696.552
|24\10, 1694.118
|31\13, 1690.909
|-
|A
| 38\15, 1753.846
|28\11, 1768.421
|46\18, 1780.645
|18\7, 1800
|44\17, 1820.690
|26\10, 1835.294
|34\13, 1854.545
|-
|A#
| 39\15, 1800
| rowspan="2" |29\11, 1831.579
| 48\18, 1858.064
|19\7, 1900
|47\17, 1944.828
|28\10, 1976.471
|37\13, 2018.182
|-
|Ax
|40\15, 1846.154
|47\18, 1819.355
|18\7, 1800
|43\17, 1779.310
|25\10, 1764.706
|32\13, 1745.545
|-
|'''Bf'''
|'''41\15,''' '''1892.308'''
|'''30\11,''' '''1894.737'''
|'''49\18,''' '''1896.774'''
|'''19\7, 1900'''
|'''46\17,''' '''1903.448'''
|'''27\10,''' '''1905.882'''
|'''35\13,''' '''1909.091'''
|-
|B
|42\15, 1938.462
|31\11, 1957.895
|51\18, 1974.194
|20\7, 2000
|49\17, 2027.586
| 29\10, 2047.059
|38\13, 2072.727
|-
|B#
|43\15, 1984.615
| rowspan="2" |32\11, 2021.053
|53\18, 2051.612
|21\7, 2100
|52\17, 2151.725
|31\10, 2188.235
|41\13, 2236.364
|-
|Cf
|44\15, 2030.769
|52\18, 2012.903
|20\7, 2000
|48\17, 1986.207
|28\10, 1976.471
|36\13, 1963.636
|-
!C
!45\15, 2076.923
!33\11, 2084.211
!54\18, 2090.323
!21\7, 2100
!51\17, 2110.345
!30\10, 2117.647
!39\13, 2127.273
|-
|C#
|46\15, 2123.077
|34\11, 2147.368
|56\15, 2167.742
| rowspan="2" |22\7, 2200
|54\17, 2234.483
|32\10, 2258.824
|42\13, 2090.909
|-
|Df
|48\15, 2215.385
|35\11, 2210.526
|57\15, 2206.452
|53\17, 2193.103
|31\10, 2188.235
|40\13, 2181.818
|-
|'''D'''
|'''49\15, 2261.538'''
|'''36\11, 1073.684'''
|'''59\18, 2283.871'''
|'''23\7, 2300'''
|'''56\17, 2317.241'''
|'''33\10, 2329.412'''
|'''43\13,''' '''2345.455'''
|-
|D#
|50\15, 2307.692
|37\11, 2336.842
|61\18, 2361.290
| rowspan="2" |24\7, 2400
|59\17, 2441.379
|35\10, 2470.588
|46\13, 2509.091
|-
|Ef
|52\15, 2400
|38\11, 2400
|62\18, 2400
|58\17, 2400
|34\10, 2400
| 44\13, 2400
|-
|E
|53\15, 2446.154
| 39\11, 2463.158
|64\18, 2477,419
|25\7, 2500
|61\17, 2524.138
|36\10, 2541.176
|47\13, 2563.636
|-
|E#
|54\15, 2492.308
| rowspan="2" |40\11, 2526.316
|66\18, 2554.838
|26\7, 2600
|64\17, 2648.275
|38\10, 2682.353
|50\13, 2727.273
|-
|Fff
| 55\15, 2538.462
| 65\18, 2516.129
|25\7, 2500
|60\17, 2482.759
|35\10, 2470.588
|45\13, 2454.545
|-
|'''Ff'''
|'''56\15, 2584.615'''
|'''41\11, 2589.474'''
|'''67\18, 2593.548'''
|'''26\7, 2600'''
|'''63\17, 2606.897'''
|'''37\10, 2611.765'''
|'''48\13,''' '''2618.182'''
|-
|F
|57\15, 2630.769
|42\11, 2652.632
|69\18, 2670.968
|27\7, 2700
|66\17, 2731.034
|39\10, 2752.941
|51\13, 2781.818
|-
| F#
| rowspan="2" |58\15, 2676.923
|43\11, 2715.789
|71\18, 2748.387
| 28\7, 2800
|69\17, 2855.172
|41\10, 2894.118
|54\13, 2945.455
|-
|0ff, Gff
|42\11, 2652.632
|68\18, 2632.258
|26\7, 2600
|62\17, 2565.517
|36\10, 2541.176
|46\13, 2509.091
|-
|0f, Gf
|59\15, 2723.077
|43\11, 2715.789
|70\18, 2709.677
|27\7, 2700
|65\17, 2689.552
|38\10, 2682.353
|49\13, 2672.273
|-
!0, G
!60\15, 2769.231
!44\11, 2778.947
!72\18, 2787.097
!28\7, 2800
!68\17, 2813.793
!40\10, 2823.529
!52\13, 2836.364
|}

{| class="wikitable"
|+Cents
!Notation
!Supersoft
!Soft
! Semisoft
! Basic
!Semihard
!Hard
!Superhard
|-
!Guidotonic
!~15edf
!~11edf
!~18edf
!~7edf
!~17edf
!~10edf
!~13edf
|-
|F ut#
|1\15, 46.154
|1\11, 63.158
|2\18, 77.419
| rowspan="2" |1\7, 100
|3\17, 124.138
|2\10, 141.176
|3\13, 163.636
|-
|G reb
|3\15, 138.462
|2\11. 126.316
|3\18, 116.129
|2\17, 82.759
|1\10, 70.588
|1\13, 54.545
|-
|'''G re'''
|'''4\15,''' '''184.615'''
|'''3\11,''' '''189.474'''
|'''5\18,''' '''193.548'''
|'''2\7,''' '''200'''
|'''5\17,''' '''206.897'''
|'''3\10,''' '''211.765'''
|'''4\13,''' '''218.182'''
|-
|G re#
|5\15, 230.769
|4\11, 252.632
|7\18, 270.968
| rowspan="2" |3\7, 300
|8\17, 331.034
|5\10, 352.941
|7\13, 381.818
|-
|A mib
|7\15, 323.077
|5\11, 315.789
|8\18, 309.677
|7\17, 289.655
|4\10, 282.353
|5\13, 272.727
|-
|A mi
|8\15, 369.231
| 6\11, 378.947
|10\18, 387.097
|4\7, 400
|10\17, 413.793
|6\10, 423.529
|8\13, 436.364
|-
| A mi#
|9\15, 415.385
| rowspan="2" |7\11, 442.105
|12\18, 464.516
|5\7, 500
|13\17, 537.069
|8\10, 564.706
|11\13, 600
|-
|B fa utb
|10\15, 461.538
|11\18, 425.806
|4\7, 400
|9\17, 372.414
|5\10, 352.941
|6\13, 327.273
|-
|'''B fa ut'''
|'''11\15,''' '''507.692'''
|'''8\11,''' '''505.263'''
|'''13\18,''' '''503.226'''
|'''5\7, 500'''
|'''12\17,''' '''496.552'''
|'''7\10,''' '''494.118'''
|'''9\13,''' '''490.909'''
|-
|B fa ut#
|12\15, 553.846
|9\11, 568.421
|15\18, 580.645
|6\7, 600
|15\17, 620.690
|9\10, 635.294
|12\13, 654.545
|-
|B fa utx
| 13\15, 600
| rowspan="2" |10\11, 631.579
|17\18, 658.064
|7\7, 700
|18\17, 744.828
|11\10, 776.471
|15\13, 818.182
|-
|C sol re utb
| 14\15, 646.154
|16\18, 619.355
|6\7, 600
|14\17, 579.310
|8\10, 564.706
|10\13, 545.455
|-
!C sol re ut
!'''15\15,''' '''692.308'''
!'''11\11,''' '''694.737'''
!'''18\18,''' '''696.774'''
!7\7, 700
!'''17\17,''' '''703.448'''
!'''10\10,''' '''705.882'''
!'''13\13,''' '''709.091'''
|-
|C sol re ut#
|16\15, 738.462
|12\11, 757.895
|20\18, 774.194
| rowspan="2" |8\8, 800
|20\17, 827.586
|12\10, 847.059
|16\13, 872.727
|-
|D la mi reb
|18\15, 830.769
|13\11, 821.053
|21\18, 812.903
|19\17, 786.207
|11\10, 776.471
|14\13, 763.63
|-
|'''D la mi re'''
|'''19\15,''' '''876.923'''
|'''14\11,''' '''884.211'''
|'''23\18,''' '''890.323'''
|'''9\5,''' '''900'''
|'''22\17,''' '''910.345'''
|'''13\10,''' '''917.647'''
|'''17\13,''' '''927.273'''
|-
|D la mi re#
|20\15, 923.077
| rowspan="2" |15\11, 947.368
|25\18, 967.742
|10\7, 1000
|25\17, 1034.483
|15\10, 1058.824
|20\13, 1090.909
|-
|E fa utb
|21\15, 969.231
|24\18, 929.032
| 9\5, 900
|21\17, 868.966
|12\10, 847.059
|15\13, 818.182
|-
|E fa ut
| 22\15, 1015.385
|16\11, 1010.526
|26\18, 1006.452
|10\7, 1000
|24\17, 993.103
|14\10, 988.235
|18\13, 981.818
|-
|E si mi re
|23\15, 1061.538
|17\11, 1073.684
|28\18, 1083.871
|11\7, 1100
|27\17, 1117.241
|16\10, 1129.412
|21\9, 1145.455
|-
| E si mi re#
|24\15, 1107.923
| rowspan="2" |18\11, 1136.842
|30\18, 1161.29
|12\7, 1200
|30\17, 1241.379
| 18\10, 1270.588
|24\13, 1309.091
|-
|F sol fa ut reb
|25\15, 1153.846
|29\18, 1122.581
|11\7, 1100
|26\17, 1075.862
|15\10, 1058.824
|19\13, 1036.364
|-
|'''F sol fa ut re'''
|'''26\15,''' '''1200'''
|'''19\11,''' '''1200'''
|'''31\18,''' '''1200'''
|'''12\7, 1200'''
|'''29\17,''' '''1200'''
|'''17\10,''' '''1200'''
|'''22\13,''' '''1200'''
|-
|F sol fa ut re#
|27\15, 1246.154
|20\11, 1263.158
|33\18, 1277.419
|13\7, 1300
|32\17, 1324.138
| 19\10, 1341.176
| 25\13, 1363.636
|-
|F sol fa ut rex
|28\15, 1292.308
| rowspan="2" |21\11, 1326.318
|35\18, 1354.834
| 14\7, 1400
|35\17, 1448.275
|21\10, 1482.353
|28\13, 1527.273
|-
|G la sol re mib
| 29\15, 1338.462
|34\18, 1316.129
| 13\7, 1300
|31\17, 1282.759
|18\10, 1270.588
|23\13, 1254.545
|-
!G la sol re mi
!30\15, 1384.615
!22\11, 1389.473
!36\18, 1393.548
!14\7, 1400
!34\17, 1406.897
!20\10, 1411.765
!26\13, 1418.182
|-
|G la sol re mi#
|31\15, 1430.769
|23\11, 1452.632
|38\18, 1470.968
| rowspan="2" |15\7, 1500
|37\17, 1531.034
|22\10, 1552.941
|29\13, 1581.818
|-
|A si la mi fab
|33\15, 1523.077
| 24\11, 1515.789
|39\18, 1509.677
|36\17, 1489.655
|21\10, 1482.759
| 27\13, 1472.273
|-
|'''A si la mi fa'''
|'''34\15,''' '''1569.231'''
|'''25\11,''' '''1578.947'''
|'''41\18,''' '''1587.097'''
|'''16\7,''' '''1600'''
|'''39\17,''' '''1613.793'''
|'''23\10,''' '''1623.529'''
|'''30\13,''' '''1636.364'''
|-
|A si la mi fa#
| 35\15, 1615.385
| rowspan="2" |26\11, 1642.105
|43\18, 1664.516
|17\7, 1700
|42\17, 1737.069
| 25\10, 1764.706
|33\13, 1800
|-
|B sol fa utb
|36\61, 1661.538
|42\18, 1625.806
|16\7, 1600
|38\29, 1572.414
|22\10, 1552.941
|28\13, 1527.273
|-
|B sol fa ut
|37\15, 1707.692
|27\11, 1705.263
| 44\18, 1703.226
| 17\7, 1700
|41\17, 1696.552
|24\10, 1694.118
|31\13, 1690.909
|-
|B si
|38\15, 1753.846
| 28\11, 1768.421
|46\18, 1780.645
|18\7, 1800
|44\17, 1820.690
|26\10, 1835.294
|34\13, 1854.545
|-
|B si
|39\15, 1800
| rowspan="2" |29\11, 1831.579
|48\18, 1858.064
|19\7, 1900
|47\17, 1944.828
|28\10, 1976.471
|37\13, 2018.182
|-
|C la sol re utb
|40\15, 1846.154
|47\18, 1819.355
| 18\7, 1800
| 43\17, 1779.310
|25\10, 1764.706
|32\13, 1745.545
|-
|'''C la sol re ut'''
|'''41\15,''' '''1892.308'''
|'''30\11,''' '''1894.737'''
|'''49\18,''' '''1896.774'''
|'''19\7, 1900'''
|'''46\17,''' '''1903.448'''
|'''27\10,''' '''1905.882'''
|'''35\13,''' '''1909.091'''
|-
|C la sol re ut#
|42\15, 1938.462
|31\11, 1957.895
|51\18, 1974.194
|20\7, 2000
|49\17, 2027.586
| 29\10, 2047.059
|38\13, 2072.727
|-
|C la sol re utx
| rowspan="2" |43\15, 1984.615
|32\11, 2021.053
|53\18, 2051.612
|21\7, 2100
|52\17, 2151.725
|31\10, 2188.235
|41\13, 2236.364
|-
|D fa la mi reb
|31\11, 1957.895
|50\18, 1935.484
|19\7, 1900
|45\17, 1862.069
|26\10, 1835.294
|33\13, 1800
|-
|D fa la mi re
|44\15, 2030.769
|32\11, 2021.053
|52\18, 2012.903
|20\7, 2000
|48\17, 1986.207
|28\10, 1976.471
|36\13, 1963.636
|-
!D si la mi re
!45\15, 2076.923
!33\11, 2084.211
!54\18, 2090.323
!21\7, 2100
! 51\17, 2110.345
!30\10, 2117.647
!39\13, 2127.273
|-
|D si la mi re#
|46\15, 2123.077
| rowspan="2" |34\11, 2147.368
|56\18, 2167.742
|22\7, 2200
|54\17, 2234.483
| 32\10, 2258.824
|42\13, 2090.909
|-
|E fab
|47\26, 2169.231
|55\16, 2129.032
|21\7, 2100
|50\17, 2068.966
|29\10, 2047.059
|37\13, 2018.182
|-
|E fa
|48\15, 2215.385
|35\11, 2210.526
|57\18, 2206.452
|23\7, 2300
|53\17, 2193.103
|31\10, 2188.235
|40\13, 2181.818
|-
|E si mi
|49\15, 2261.538
|36\11, 1073.684
|59\18, 2283.871
|24\7, 2400
|56\17, 2317.241
|33\10, 2329.412
|43\13, 2345.455
|-
|E si mi#
|50\15, 2307.692
| rowspan="2" |37\11, 2336.842
|61\18, 2361.290
| rowspan="2" |23\7, 2300
| 59\17, 2441.379
|35\10, 2470.588
|46\13, 2509.091
|-
|F sol fa utb
|51\15, 2353.846
|60\18, 2322.581
|55\17, 2275.862
|32\10, 2258.824
|41\13, 2236.364
|-
|F sol fa ut
|52\15, 2400
|38\11, 2400
|62\18, 2400
|24\7, 2400
|58\17, 2400
|34\10, 2400
|44\13, 2400
|-
|F sol fa ut#
|53\15, 2446.154
|39\11, 2463.158
|64\18, 2477,419
| rowspan="2" |25\7, 2500
|61\17, 2524.138
|36\10, 2541.176
|47\13, 2563.636
|-
|G la sol reb
|55\15, 2538.462
|40\11, 2526.316
|65\18, 2516.129
|60\17, 2482.759
|35\10, 2470.588
|45\13, 2454.545
|-
|'''G la sol re'''
|'''56\15, 2584.615'''
|'''41\11, 2589.474'''
|'''67\18, 2593.548'''
|'''26\7, 2600'''
|'''63\17, 2606.897'''
|'''37\10, 2611.765'''
|'''48\13,''' '''2618.182'''
|-
|G la sol re#
|57\15, 2630.769
|42\11, 2652.632
|69\18, 2670.968
| rowspan="2" |27\7, 2700
|66\17, 2731.034
|39\10, 2752.941
|51\13, 2781.818
|-
|A si la mib
|59\15, 2723.077
|43\11, 2715.789
|70\18, 2709.677
|65\17, 2689.552
|38\10, 2682.353
|49\13, 2672.273
|-
!A si la mi
!60\15, 2769.231
!44\11, 2778.947
!72\18, 2787.097
!28\7, 2800
!68\17, 2813.793
!40\10, 2823.529
!52\13, 2836.364
|-
|A si la mi#
|61\15, 2815.385
| rowspan="2" |45\11, 2842.105
| 74\18, 2864.516
|29\7, 2900
|71\17, 2937.069
|42\10, 2964.706
|55\13, 3000
|-
|B fab
|62\15, 2861.538
|73\18, 2825.806
| 28\7, 2800
|67\17, 2772.414
|39\10, 2752.941
|50\13, 2727.273
|-
|B fa
|63\15, 2907.692
|46\11, 2905.263
|75\18, 2903.226
|29\7, 2900
|70\17, 2896.552
|41\10, 2894.118
|53\13, 2890.909
|-
|'''B si'''
|'''64\15, 2953.846'''
|'''47\11, 2968.421'''
|'''77\18, 2980.645'''
|'''30\7, 3000'''
|'''73\17, 3020.690'''
|'''43\10, 3035.294'''
|'''56\13, 3054.545'''
|-
|B si#
|65\15, 3000
|48\11, 3031.579
|79\18, 3058.064
| rowspan="2" |31\7, 3100
|76\17, 3144.828
|45\10, 3176.471
|59\13, 3218.182
|-
|C solb
|67\15, 3092.308
|49\11, 3094.737
|80\18, 3096.774
|75\17, 3103.448
|44\10, 3105.882
|57\13, 3109.091
|-
|C sol
|68\15, 3138.462
|50\11, 3157.895
| 82\18, 3174.194
|32\7, 3200
|78\17, 3227.586
| 46\10, 3247.059
|60\13, 3272.273
|-
|C sol#
| 69\15, 3184.615
| rowspan="2" |51\11, 3221.053
|84\18, 3251.612
|33\7, 3300
|81\17, 3351.725
|48\10, 3388.235
|63\13, 3436.364
|-
|D labb
|70\15, 3230.769
|83\18, 3212.903
|32\7, 3200
|77\17, 3186.207
|45\10, 3176.471
|58\13, 3163.636
|-
|'''D lab'''
|'''71\15,''' '''3276.923'''
|'''52\11,''' '''3284.211'''
|'''85\18,''' '''3290.323'''
|'''33\7, 3300'''
|'''80\17,''' '''3310.345'''
|'''47\10,''' '''3317.647'''
|'''61\13,''' '''3327.{{Overline|27}}'''
|-
|D la
|72\15, 3323.077
|53\11, 3347.368
|87\18, 3367.742
|34\7, 3400
|83\17, 3434.583
|49\10, 3458.824
|64\13, 3490.909
|-
|D la#
|73\15, 3369.231
| rowspan="2" |54\11, 3410.625
|89\18, 3445.162
|35\7, 3500
|86\17, 3558.621
|51\10, 3600
|67\13, 3654.545
|-
|F utb
|74\15, 3415.385
|88\18, 3406.452
|34\7, 3400
|82\17, 3393.103
|48\10, 3388.235
|62\13, 3381.818
|-
!F ut
!75\15, 3461.538
!55\11, 3473.684
!90\18, 3483.871
!35\7, 3500
!85\17, 3517.241
!50\10, 3529.412
!65\13, 3545.455
|}

{| class="wikitable"
|+Cents
!Notation
!Supersoft
!Soft
! Semisoft
!Basic
!Semihard
!Hard
!Superhard
|-
!Subdozenal
!~15edf
!~11edf
!~18edf
!~7edf
!~17edf
!~10edf
!~13edf
|-
|F#
|1\15, 46.154
|1\11, 63.158
|2\18, 77.419
| rowspan="2" |1\7, 100
|3\17, 124.138
|2\10, 141.176
|3\13, 163.636
|-
|Gb, Ge
|3\15, 138.462
|2\11. 126.316
|3\18, 116.129
|2\17, 82.759
|1\10, 70.588
|1\13, 54.545
|-
|'''G'''
|'''4\15,''' '''184.615'''
|'''3\11,''' '''189.474'''
|'''5\18,''' '''193.548'''
|'''2\7,''' '''200'''
|'''5\17,''' '''206.897'''
|'''3\10,''' '''211.765'''
|'''4\13,''' '''218.182'''
|-
|G#
|5\15, 230.769
|4\11, 252.632
|7\18, 270.968
| rowspan="2" |3\7, 300
|8\17, 331.034
|5\10, 352.941
|7\13, 381.818
|-
|Hb, He
|7\15, 323.077
|5\11, 315.789
|8\18, 309.677
|7\17, 289.655
|4\10, 282.353
|5\13, 272.727
|-
|H
|8\15, 369.231
|6\11, 378.947
|10\18, 387.097
|4\7, 400
|10\17, 413.793
|6\10, 423.529
|8\13, 436.364
|-
|H#
|9\15, 415.385
| rowspan="2" |7\11, 442.105
|12\18, 464.516
|5\7, 500
|13\17, 537.069
|8\10, 564.706
|11\13, 600
|-
|Jbb, Jee
|10\15, 461.538
|11\18, 425.806
|4\7, 400
|9\17, 372.414
|5\10, 352.941
|6\13, 327.273
|-
|'''Jb, Je'''
|'''11\15,''' '''507.692'''
|'''8\11,''' '''505.263'''
|'''13\18,''' '''503.226'''
|'''5\7, 500'''
|'''12\17,''' '''496.552'''
|'''7\10,''' '''494.118'''
|'''9\13,''' '''490.909'''
|-
|J
|12\15, 553.846
|9\11, 568.421
|15\18, 580.645
|6\7, 600
|15\17, 620.690
|9\10, 635.294
|12\13, 654.545
|-
|J#
|13\15, 600
| rowspan="2" |10\11, 631.579
|17\18, 658.064
|7\7, 700
|18\17, 744.828
|11\10, 776.471
|15\13, 818.182
|-
|Kb, Ke
|14\15, 646.154
|16\18, 619.355
|6\7, 600
|14\17, 579.310
|8\10, 564.706
|10\13, 545.455
|-
!K
!'''15\15,''' '''692.308'''
!'''11\11,''' '''694.737'''
!'''18\18,''' '''696.774'''
!7\7, 700
!'''17\17,''' '''703.448'''
!'''10\10,''' '''705.882'''
!'''13\13,''' '''709.091'''
|-
|K#
|16\15, 738.462
|12\11, 757.895
|20\18, 774.194
| rowspan="2" |8\8, 800
|20\17, 827.586
|12\10, 847.059
|16\13, 872.727
|-
|Lb, Le
|18\15, 830.769
|13\11, 821.053
|21\18, 812.903
|19\17, 786.207
|11\10, 776.471
|14\13, 763.63
|-
|'''L'''
|'''19\15,''' '''876.923'''
|'''14\11,''' '''884.211'''
|'''23\18,''' '''890.323'''
|'''9\5,''' '''900'''
|'''22\17,''' '''910.345'''
|'''13\10,''' '''917.647'''
|'''17\13,''' '''927.273'''
|-
|L#
|20\15, 923.077
| rowspan="2" |15\11, 947.368
|25\18, 967.742
|10\7, 1000
|25\17, 1034.483
|15\10, 1058.824
|20\13, 1090.909
|-
|Mbb, Mee
|21\15, 969.231
|24\18, 929.032
|9\5, 900
|21\17, 868.966
|12\10, 847.059
|15\13, 818.182
|-
|Mb, Me
|22\15, 1015.385
|16\11, 1010.526
|26\18, 1006.452
|10\7, 1000
|24\17, 993.103
|14\10, 988.235
|18\13, 981.818
|-
|M
|23\15, 1061.538
|17\11, 1073.684
|28\18, 1083.871
|11\7, 1100
|27\17, 1117.241
|16\10, 1129.412
|21\9, 1145.455
|-
|M#
|24\15, 1107.923
| rowspan="2" |18\11, 1136.842
|30\18, 1161.29
|12\7, 1200
|30\17, 1241.379
|18\10, 1270.588
|24\13, 1309.091
|-
|Nbb, Nee
|25\15, 1153.846
|29\18, 1122.581
|11\7, 1100
|26\17, 1075.862
|15\10, 1058.824
|19\13, 1036.364
|-
|'''Nb, Ne'''
|'''26\15,''' '''1200'''
|'''19\11,''' '''1200'''
|'''31\18,''' '''1200'''
|'''12\7, 1200'''
|'''29\17,''' '''1200'''
|'''17\10,''' '''1200'''
|'''22\13,''' '''1200'''
|-
|N
|27\15, 1246.154
|20\11, 1263.158
|33\18, 1277.419
|13\7, 1300
|32\17, 1324.138
|19\10, 1341.176
|25\13, 1363.636
|-
|N#
|28\15, 1292.308
| rowspan="2" |21\11, 1326.318
|35\18, 1354.834
|14\7, 1400
|35\17, 1448.275
|21\10, 1482.353
|28\13, 1527.273
|-
|Pb, Pe
|29\15, 1338.462
|34\18, 1316.129
|13\7, 1300
|31\17, 1282.759
|18\10, 1270.588
|23\13, 1254.545
|-
!P
!30\15, 1384.615
!22\11, 1389.473
!36\18, 1393.548
!14\7, 1400
!34\17, 1406.897
!20\10, 1411.765
!26\13, 1418.182
|-
|P#
|31\15, 1430.769
|23\11, 1452.632
|38\18, 1470.968
| rowspan="2" |15\7, 1500
|37\17, 1531.034
|22\10, 1552.941
|29\13, 1581.818
|-
|Qb, Qe
|33\15, 1523.077
|24\11, 1515.789
|39\18, 1509.677
|36\17, 1489.655
|21\10, 1482.759
|27\13, 1472.273
|-
|'''Q'''
|'''34\15,''' '''1569.231'''
|'''25\11,''' '''1578.947'''
|'''41\18,''' '''1587.097'''
|'''16\7,''' '''1600'''
|'''39\17,''' '''1613.793'''
|'''23\10,''' '''1623.529'''
|'''30\13,''' '''1636.364'''
|-
|Q#
|35\15, 1615.385
| rowspan="2" |26\11, 1642.105
|43\18, 1664.516
|17\7, 1700
|42\17, 1737.069
|25\10, 1764.706
|33\13, 1800
|-
|Rb, Re
|36\61, 1661.538
|42\18, 1625.806
|16\7, 1600
|38\29, 1572.414
|22\10, 1552.941
|28\13, 1527.273
|-
|R
|37\15, 1707.692
|27\11, 1705.263
|44\18, 1703.226
|17\7, 1700
|41\17, 1696.552
|24\10, 1694.118
|31\13, 1690.909
|-
|R#
|38\15, 1753.846
|28\11, 1768.421
|46\18, 1780.645
|18\7, 1800
|44\17, 1820.690
|26\10, 1835.294
|34\13, 1854.545
|-
|R#
|39\15, 1800
| rowspan="2" |29\11, 1831.579
|48\18, 1858.064
|19\7, 1900
|47\17, 1944.828
|28\10, 1976.471
|37\13, 2018.182
|-
|Sb, Se
|40\15, 1846.154
|47\18, 1819.355
|18\7, 1800
|43\17, 1779.310
|25\10, 1764.706
|32\13, 1745.545
|-
|'''S'''
|'''41\15,''' '''1892.308'''
|'''30\11,''' '''1894.737'''
|'''49\18,''' '''1896.774'''
|'''19\7, 1900'''
|'''46\17,''' '''1903.448'''
|'''27\10,''' '''1905.882'''
|'''35\13,''' '''1909.091'''
|-
|S#
|42\15, 1938.462
|31\11, 1957.895
|51\18, 1974.194
|20\7, 2000
|49\17, 2027.586
|29\10, 2047.059
|38\13, 2072.727
|-
|Sx
|43\15, 1984.615
| rowspan="2" |32\11, 2021.053
|53\18, 2051.612
|21\7, 2100
|52\17, 2151.725
|31\10, 2188.235
|41\13, 2236.364
|-
|Tb, Te
|44\15, 2030.769
|52\18, 2012.903
|20\7, 2000
|48\17, 1986.207
|28\10, 1976.471
|36\13, 1963.636
|-
!T
!45\15, 2076.923
!33\11, 2084.211
!54\18, 2090.323
!21\7, 2100
!51\17, 2110.345
!30\10, 2117.647
!39\13, 2127.273
|-
|T#
|46\15, 2123.077
| rowspan="2" |34\11, 2147.368
|56\18, 2167.742
|22\7, 2200
|54\17, 2234.483
|32\10, 2258.824
|42\13, 2090.909
|-
|Ub, Üe
|47\26, 2169.231
|55\16, 2129.032
|21\7, 2100
|50\17, 2068.966
|29\10, 2047.059
|37\13, 2018.182
|-
|Ub, Ü
|48\15, 2215.385
|35\11, 2210.526
|57\18, 2206.452
|23\7, 2300
|53\17, 2193.103
|31\10, 2188.235
|40\13, 2181.818
|-
|U
|49\15, 2261.538
|36\11, 1073.684
|59\18, 2283.871
|24\7, 2400
|56\17, 2317.241
|33\10, 2329.412
|43\13, 2345.455
|-
|U#
|50\15, 2307.692
| rowspan="2" |37\11, 2336.842
|61\18, 2361.290
| rowspan="2" |23\7, 2300
|59\17, 2441.379
|35\10, 2470.588
|46\13, 2509.091
|-
|Vb, Ve
|51\15, 2353.846
|60\18, 2322.581
|55\17, 2275.862
|32\10, 2258.824
|41\13, 2236.364
|-
|V
|52\15, 2400
|38\11, 2400
|62\18, 2400
|24\7, 2400
|58\17, 2400
|34\10, 2400
|44\13, 2400
|-
|V#
|53\15, 2446.154
|39\11, 2463.158
|64\18, 2477,419
| rowspan="2" |25\7, 2500
|61\17, 2524.138
|36\10, 2541.176
|47\13, 2563.636
|-
|Wb, We
|55\15, 2538.462
|40\11, 2526.316
|65\18, 2516.129
|60\17, 2482.759
|35\10, 2470.588
|45\13, 2454.545
|-
|'''Wb'''
|'''56\15, 2584.615'''
|'''41\11, 2589.474'''
|'''67\18, 2593.548'''
|'''26\7, 2600'''
|'''63\17, 2606.897'''
|'''37\10, 2611.765'''
|'''48\13,''' '''2618.182'''
|-
|W#
|57\15, 2630.769
|42\11, 2652.632
|69\18, 2670.968
| rowspan="2" |27\7, 2700
|66\17, 2731.034
|39\10, 2752.941
|51\13, 2781.818
|-
|Xb, Xe
|59\15, 2723.077
|43\11, 2715.789
|70\18, 2709.677
|65\17, 2689.552
|38\10, 2682.353
|49\13, 2672.273
|-
!X
!60\15, 2769.231
!44\11, 2778.947
!72\18, 2787.097
!28\7, 2800
!68\17, 2813.793
!40\10, 2823.529
!52\13, 2836.364
|-
|X#
|61\15, 2815.385
| rowspan="2" |45\11, 2842.105
|74\18, 2864.516
|29\7, 2900
|71\17, 2937.069
|42\10, 2964.706
|55\13, 3000
|-
|Ybb, Yee
|62\15, 2861.538
|73\18, 2825.806
|28\7, 2800
|67\17, 2772.414
|39\10, 2752.941
|50\13, 2727.273
|-
|Yb, Ye
|63\15, 2907.692
|46\11, 2905.263
|75\18, 2903.226
|29\7, 2900
|70\17, 2896.552
|41\10, 2894.118
|53\13, 2890.909
|-
|'''Y'''
|'''64\15, 2953.846'''
|'''47\11, 2968.421'''
|'''77\18, 2980.645'''
|'''30\7, 3000'''
|'''73\17, 3020.690'''
|'''43\10, 3035.294'''
|'''56\13, 3054.545'''
|-
|Y#
|65\15, 3000
|48\11, 3031.579
|79\18, 3058.064
| rowspan="2" |31\7, 3100
|76\17, 3144.828
|45\10, 3176.471
|59\13, 3218.182
|-
|Zb. Ze
|67\15, 3092.308
|49\11, 3094.737
|80\18, 3096.774
|75\17, 3103.448
|44\10, 3105.882
|57\13, 3109.091
|-
|Z
|68\15, 3138.462
|50\11, 3157.895
|82\18, 3174.194
|32\7, 3200
|78\17, 3227.586
|46\10, 3247.059
|60\13, 3272.273
|-
|Z#
|69\15, 3184.615
| rowspan="2" |51\11, 3221.053
|84\18, 3251.612
|33\7, 3300
|81\17, 3351.725
|48\10, 3388.235
|63\13, 3436.364
|-
|Ab, Æ
|70\15, 3230.769
|83\18, 3212.903
|32\7, 3200
|77\17, 3186.207
|45\10, 3176.471
|58\13, 3163.636
|-
|'''A'''
|'''71\15,''' '''3276.923'''
|'''52\11,''' '''3284.211'''
|'''85\18,''' '''3290.323'''
|'''33\7, 3300'''
|'''80\17,''' '''3310.345'''
|'''47\10,''' '''3317.647'''
|'''61\13,''' '''3327.{{Overline|27}}'''
|-
|A#
|72\15, 3323.077
|53\11, 3347.368
|87\18, 3367.742
|34\7, 3400
|83\17, 3434.583
|49\10, 3458.824
|64\13, 3490.909
|-
|Ax
|73\15, 3369.231
| rowspan="2" |54\11, 3410.625
|89\18, 3445.162
|35\7, 3500
|86\17, 3558.621
|51\10, 3600
|67\13, 3654.545
|-
|Bb, Be
|74\15, 3415.385
|88\18, 3406.452
|34\7, 3400
|82\17, 3393.103
|48\10, 3388.235
|62\13, 3381.818
|-
!B
!75\15, 3461.538
!55\11, 3473.684
!90\18, 3483.871
!35\7, 3500
!85\17, 3517.241
!50\10, 3529.412
!65\13, 3545.455
|-
|B#
|76\15, 3507.692
|56\11, 3536.842
|92\18, 3561.290
| rowspan="2" |36\7, 3600
|88\17, 3641.379
|52\10, 3670.588
|68\13, 3709.091
|-
|Cb, Ce
|78\15, 3600
|57\11, 3600
|93\18, 3600
|87\17, 3600
|51\10, 3600
|66\13, 3600
|-
|'''C'''
|'''79\15,''' '''3646.154'''
|'''58\11,''' '''3663.158'''
|'''95\18,''' '''3677.419'''
|'''37\7,''' '''3700'''
|'''90\17,''' '''3724.138'''
|'''53\10,''' '''3741.176'''
|'''69\13,''' '''3763.636'''
|-
|C#
|80\15, 3692.308
|59\11, 3726.316
|97\18, 3755.838
| rowspan="2" |38\7, 3800
|93\17, 3848.275
|55\10, 3882.353
|72\13, 3927.273
|-
|Db, De
|82\15, 3784.615
|60\11, 3789.474
|98\18, 3793.548
|92\17, 3806.897
|54\10, 3811.765
|70\13, 3818.182
|-
|D
|83\15, 3830.769
|61\11, 3852.632
|100\18, 3870.968
|39\7, 3900
|95\17, 3931.03$
|56\10, 3952.941
|73\13, 3981.818
|-
|D#
|84\15, 3876.923
| rowspan="2" |62\11, 3915.789
|102\18, 3948.387
|40\7, 4000
|98\17, 4055.172
|58\10, 4094.118
|76\13, 4145.455
|-
|Ebb, Ëe
|85\15, 3923.077
|101\18, 3909.677
|39\7, 3900
|94\17, 3889.552
|55\10, 3882.353
|71\13, 3872.727
|-
|'''Eb, Ë'''
|'''86\15,''' '''3969.231'''
|'''63\11,''' '''3978.947'''
|'''103\18,''' '''3987.097'''
|'''40\7, 4000'''
|'''97\17,''' '''4013.793'''
|'''57\10,''' '''4023.529'''
|'''74\13,''' '''4036.364'''
|-
|E
|87\15, 4015.385
|64\11, 4042.105
|105\18, 4064.516
|41\7, 4100
|100\17, 4137.931
|59\10, 4164.706
|77\13, 4200
|-
|E#
|88\15, 4061.583
| rowspan="2" |65\11, 4105.263
|107\18, 4141.956
|42\7, 4200
|103\17, 4262.069
|61\10, 4305.882
|80\13, 4363.636
|-
|Fb, Fe
|89\15, 4107.692
|106\18, 4103.226
|41\7, 4100
|99\17, 4096.552
|58\10, 4094.118
|75\13, 4090.909
|-
!F
!90\15, 4153.846
!66\11, 4168.421
!108\18, 4180.645
!42\7, 4200
!102\17, 4220.690
!60\10, 4235.294
!78\13, 4254.545
|}
==Intervals==
{| class="wikitable"
!Generators
!Sesquitave notation
!Interval category name
!Generators
!Notation of 3/2 inverse
!Interval category name
|-
| colspan="6" |The 4-note MOS has the following intervals (from some root):
|-
|0
|Do, Fa, Sol
|perfect unison
|0
|Do, Fa, Sol
|sesquitave (just fifth)
|-
|1
|Fa, Sib, Do
|perfect fourth
| -1
|Re, Sol, La
|perfect second
|-
|2
|Mib, Lab, Sib
|minor third
| -2
|Mi, La, Si
|major third
|-
|3
|Reb, Solb, Lab
|diminished second
| -3
|Fa#, Si, Do#
|augmented fourth
|-
| colspan="6" |The chromatic 7-note MOS also has the following intervals (from some root):
|-
|4
|Dob, Fab, Solb
|diminished sesquitave
| -4
|Do#, Fa#, Sol#
|augmented unison (chroma)
|-
|5
|Fab, Sibb, Dob
|diminished fourth
| -5
|Re#, Sol#, La#
|augmented second
|-
|6
|Mibb, Labb, Sibb
|diminished third
| -6
|Mi#, La#, Si#
|augmented third
|}

==Genchain==

The generator chain for this scale is as follows:
{| class="wikitable"
|Mibb
Labb

Sibb
|Fab
Sibb

Dob
|Dob
Fab

Solb
|Reb
Solb

Lab
|Mib
Lab

Sib
|Fa
Sib

Do
|Do
Fa

Sol
|Re
Sol

La
|Mi
La

Si
|Fa#
Si

Do#
|Do#
Fa#

Sol#
|Re#
Sol#

La#
|Mi#
La#

Si#
|-
|d3
|d4
|d5
|d2
|m3
|P4
|P1
|P2
|M3
|A4
|A1
|A2
|A3
|}

==Modes==

The mode names are based on the species of fifth:
{| class="wikitable"
!Mode
!Scale
![[Modal UDP Notation|UDP]]
! colspan="3" |Interval type
|-
!name
!pattern
!notation
!2nd
!3rd
!4th
|-
|Lydian
|LLLs
|<nowiki>3|0</nowiki>
|P
|M
|A
|-
|Major
|LLsL
|<nowiki>2|1</nowiki>
|P
|M
|P
|-
|Minor
|LsLL
|<nowiki>1|2</nowiki>
|P
|m
|P
|-
|Phrygian
|sLLL
|<nowiki>0|3</nowiki>
|d
|m
|P
|}

==Temperaments==

The most basic rank-2 temperament interpretation of angel is '''Napoli'''. The name "Napoli" comes from the “Neapolitan” sixth triad spelled <code>root-(p-2g)-(2p-3g)</code> (p = 3/2, g = the whole tone) which serves as its minor triad approximating 5:6:8 in pental interpretations or 18:21:28 in septimal ones. Basic ~7edf fits both interpretations.
==='''Napoli-Meantone (Hex meantone)'''===

[[Subgroup]]: 3/2.6/5.8/5 (5.2.3)

[[Comma]] list: [[81/80]]

[[POL2]] generator: ~9/8 = 192.6406¢

[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]

[[Optimal ET sequence]]: *[[28ed5]], [[44ed5]], [[72ed5]] ≈ [[7edf]], [[11edf]], [[18edf]]
==='''Napoli-Archy (Hex Archytas)'''===

[[Subgroup]]: 3/2.7/6.14/9 (36/7.2.3)

[[Comma]] list: [[64/63]]

[[POL2]] generator: ~8/7 = 218.6371¢

[[Mapping]]: [{{val|1 1 2}}, {{val|0 -2 -3}}]

[[Optimal ET sequence]]: *[[28ed36/7]], [[40ed36/7]], [[52ed36/7]] ≈ [[7edf]], [[10edf]], [[13edf]]
===Scale tree===

The spectrum looks like this:
{| class="wikitable"
!Generator

(bright)
!Cents
!L
!s
!L/s
!Comments
|-
|1\4
|171.429
|1
|1
|1.000
|Equalised
|-
|6\23
|180.000
|6
|5
|1.200
|
|-
|5\19
|181.818
|5
|4
|1.250
|
|-
|14\53
|182.609
|14
|11
|1.273
|
|-
|9\34
|183.051
|9
|7
|1.286
|
|-
|4\15
|184.615
|4
|3
|1.333
|
|-
|11\41
|185.915
|11
|8
|1.375
|
|-
|7\26
|186.667
|7
|5
|1.400
|
|-
|10\37
|187.5
|10
|7
|1.429
|
|-
|13\48
|187.952
|13
|9
|1.444
|
|-
|16\59
|188.253
|16
|11
|1.455
|
|-
|3\11
|189.474
|3
|2
|1.500
|Napoli-Meantone starts here
|-
|14\51
|190.909
|14
|9
|1.556
|
|-
|11\40
|191.304
|11
|7
|1.571
|
|-
|8\29
|192.000
|8
|5
|1.600
|
|-
|5\18
|193.548
|5
|3
|1.667
|
|-
|12\43
|194.595
|12
|7
|1.714
|
|-
|7\25
|195.348
|7
|4
|1.750
|
|-
|9\32
|196.364
|9
|5
|1.800
|
|-
|11\39
|197.015
|11
|6
|1.833
|
|-
|13\46
|197.468
|13
|7
|1.857
|
|-
|15\53
|197.802
|15
|8
|1.875
|
|-
|17\60
|198.058
|17
|9
|1.889
|
|-
|19\67
|198.261
|19
|10
|1.900
|
|-
|21\74
|198.425
|21
|11
|1.909
|
|-
|23\81
|198.561
|23
|12
|1.917
|
|-
|25\88
|198.675
|25
|13
|1.923
|
|-
|27\95
|198.773
|27
|14
|1.929
|
|-
|29\102
|198.857
|29
|15
|1.933
|
|-
|31\109
|198.930
|31
|16
|1.9375
|
|-
|33\116
|198.995
|33
|17
|1.941
|
|-
|35\123
|199.009
|35
|18
|1.944
|
|-
|2\7
|200
|2
|1
|2.000
|Napoli-Meantone ends, Napoli-Pythagorean begins
|-
|17\59
|201.980
|17
|8
|2.125
|
|-
|15\52
|202.247
|15
|7
|2.143
|
|-
|13\45
|202.597
|13
|6
|2.167
|
|-
|11\38
|203.077
|11
|5
|2.200
|
|-
|9\31
|203.774
|9
|4
|2.250
|
|-
|7\24
|204.878
|7
|3
|2.333
|
|-
|12\41
|205.714
|12
|5
|2.400
|
|-
|5\17
|206.897
|5
|2
|2.500
|Napoli-Neogothic heartland is from here…
|-
|18\61
|207.693
|18
|7
|2.571
|
|-
|13\44
|208.000
|13
|5
|2.600
|
|-
|8\27
|208.696
|8
|3
|2.667
|…to here
|-
|11\37
|209.524
|11
|4
|2.750
|
|-
|14\47
|210.000
|14
|5
|2.800
|
|-
|3\10
|211.765
|3
|1
|3.000
|Napoli-Pythagorean ends, Napoli-Archy begins
|-
|22\73
|212.903
|22
|7
|3.143
|
|-
|19\63
|213.084
|19
|6
|3.167
|
|-
|16\53
|213.333
|16
|5
|3.200
|
|-
|13\43
|213.699
|13
|4
|3.250
|
|-
|10\33
|214.286
|10
|3
|3.333
|
|-
|7\23
|215.385
|7
|2
|3.500
|
|-
|11\36
|216.393
|11
|3
|3.667
|
|-
|15\49
|216.867
|15
|4
|3.750
|
|-
|19\62
|217.143
|19
|5
|3.800
|
|-
|4\13
|218.182
|4
|1
|4.000
|
|-
|13\42
|219.718
|13
|3
|4.333
|
|-
|9\29
|220.408
|9
|2
|4.500
|
|-
|14\45
|221.053
|14
|3
|4.667
|
|-
|5\16
|222.222
|5
|1
|5.000
|Napoli-Archy ends
|-
|11\35
|223.728
|11
|2
|5.500
|
|-
|17\54
|224.176
|17
|3
|5.667
|
|-
|6\19
|225.000
|6
|1
|6.000
|
|-
|1\3
|240.000
|1
|0
|→ inf
|Paucitonic
|}

==See also==
[[3L 1s (3/2-equivalent)]] - idealized tuning

[[6L 2s (20/9-equivalent)]] - Neapolitan 1/2-comma meantone

[[6L 2s (88/39-equivalent)]] - Neapolitan gentle temperament

[[6L 2s (16/7-equivalent)]] - Neapolitan 1/2-comma archy

[[9L 3s (10/3-equivalent)]] - Bijou 1/3-comma meantone

[[9L 3s (44/13-equivalent)]] - Bijou gentle temperament

[[9L 3s (24/7-equivalent)]] - Bijou 1/3-comma archy

[[12L 4s (5/1-equivalent)]] - Hex meantone

[[12L 4s (56/11-equivalent)]] - Hextone gentle temperament

[[12L 4s (36/7-equivalent)]] - Hextone 1/4-comma archy

[[15L 5s (15/2-equivalent)]] - Guidotonic major 1/5-comma meantone

[[15L 5s (84/11-equivalent)]] - Guidotonic major gentle temperament

[[15L 5s (54/7-equivalent)]] - Guidotonic major 1/5-comma archy

[[18L 6s (11/1-equivalent)]] - Subdozenal harmonic tuning

[[18L 6s (56/5-equivalent)]] - Subdozenal low septimal (meantone) tuning

[[18L 6s (512/45-equivalent)]] - Subdozenal 1/6-comma meantone

[[18L 6s (80/7-equivalent)]] - Subdozenal high septimal tuning

[[18L 6s (128/11-equivalent)]] - Subdozenal subharmonic tuning

[[18L 6s (11/1-equivalent)|18L 6s (12/1-equivalent)]] - Warped Pythagorean tuning