Xenharmonic Wiki - User contributions [en]

Otonality and utonality

2026-08-03T18:58:10Z

FloraC: + an actual intro, more to come

{{Interwiki
| en = Otonality and utonality
| de =
| es =
| ja = OtonalityとUtonality
}}
{{Wikipedia|Otonality and Utonality}}

'''Otonality''' and '''utonality''' are properties of [[chord]]s that describe if it is simpler to treat them as part of the [[harmonic series]] or [[subharmonic series]].

== Introduction ==
Given a JI chord, how can we decide whether it is otonal or utonal? This might seem obvious at first, but it's actually surprisingly subtle. For example, the chord 10:12:15 is a 5-limit utonality (1/6:1/5:1/4), but it's also a 15-limit otonality, consisting of the 10th, 12th, and 15th harmonics of a fundamental. One reasonable definition is to say that a chord is otonal if its largest odd number is smaller than the largest odd number of its inverse, and utonal if the inverse has a smaller largest-odd-number. In other words, if inverting a chord increases its odd limit, it's otonal, and if it reduces it, it's utonal. That way 4:5:6 is otonal because it's simpler than its inverse, 10:12:15, and 10:12:15 is utonal because it is more simply expressed as 1/6:1/5:1/4. Because we're using odd limit and not integer limit, this definition is independent of the chord's voicing. Thus 4:5:6 is otonal even if voiced 3:4:5 or 2:3:5.

A chord's inverse can be visualized in a 2-D drawing of the harmonic lattice as a rotation by 180 degrees around 1/1.

== Precise definitions ==

To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by removing all factors of two from all numerators and denominators, followed by removing any duplicate ratios, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the all-odd voicing by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator).

For example, consider the chord {5/6, 5/3, 5/2, 25/16}. The all-odd voicing of this is {5/3, 5/1, 25/1}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {1, 3, 15}. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {1, 5, 15}. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it '''otonal'''; if the reverse is true, we call it '''utonal'''. If they are the same, as here, we may call it '''ambitonal'''. Examples of ambitonal chords include 8:9:12 = sus2 chord (inverse 6:8:9 = sus4 chord, with the same largest-odd-number) and 8:10:15 = maj7no5 (inverse 8:12:15 = maj7no3).

If a chord can be voiced as a "palindrome", it inverts to itself, and is ambitonal. Such a voicing makes the lowest interval the same as the highest, the next lowest the same as the next highest, etc. For example, the min7 chord can be voiced as 1-m3-P5-m7 = min 3rd, maj 3rd, min 3rd, therefore it must be ambitonal. Note that some ambitonal chords, such as the maj7no5, cannot be voiced as a palindrome.

=== Dyads vs. intervals ===
By this definition all [[monad]]s and [[dyad]]s are ambitonal. (Dyads and intervals are not the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)

Therefore take note that while [[43/32]] may be the "prime harmonic fourth" (in that it is rooted/of the form ''k'' / 2''n''), it is only because we are seeing it as an ''interval'' that it is so, because seeing it as a ''dyad'' would mean seeing it as 32:43:64 so that it isn't clear whether it is otonal or utonal as [[64/43]] is the "prime subharmonic fifth", so interpreting it as a dyad means that whether it is harmonic or subharmonic (or neither) depends on the voicing and/or inversion used.

Note that a dyad (consisting of ''two'' [[pitch class]]es) thus has ''two'' possible ''inversions'' (which is a distinct concept to [[octave complement]]s!). For further clarity, see the section directly below.

=== Telling inversion of an ''n''-ad ===
To determine the inversion of an (''n''+''d'')-note chord consisting of ''n'' pitches up to [[octave equivalence]] (that is, given an ''n''-ad), go through all the pitches from lowest to highest until every pitch class is accounted for; that representation will then tell you which inversion the ''n''-ad has.

Example: going through the pitches of the 5-note chord 5:8:10:16:20 lowest to highest, we find that 5:8 accounts for all higher pitches (in that all higher pitches are a whole number of octaves above one of those harmonics); therefore this chord is a ''dyad'' (''n''=2); in this case, as one of the integers in the ''interval'' is a power of 2, we can classify this inversion of the dyad as ''subharmonic''.

== Properties of types of chords ==

=== Otonal ===

* If we represent an otonal chord as a set of integers in the form A1:A2: ... :An, we may add any additional integers without affecting the chord's otonality.
* All chords with [[delta signature]]s that can be reduced (scaled by a positive real number) to +1+1, +1+1+1, +1+1+1+1, etc., are otonal.

=== Utonal ===

* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]{{dead link}}

=== Ambitonal ===

* [[Dyadic chord|Essentially tempered]] chords can be ambitonal, even though they do not have unique representations in the harmonic series.

== Ambitonal chord theorem ==

A chord can be represented as a set of integers whose gcd is 1. (If [[octave equivalence]] is assumed, we take the largest odd factors of all of these integers.) The inverse of this chord is the set of integers LCM(original chord)/x for each integer x in the original chord.

Assume a chord is ambitonal. Then its largest integer, max(chord), is equal to the largest integer of its inverse, which is LCM(chord)/min(chord). Therefore min(chord)*max(chord) = LCM(chord). Conversely, if a set of integers has gcd 1 and also satisfies this, then it is an ambitonal chord.

Thus, for any given odd number N (where N is not prime), all ambitonal chords with LCM N can easily be found by considering subsets of the factors of N. If a subset has at least three factors (as mentioned above, the statement always holds for two or fewer), has a GCD of 1, an LCM of N, and also satisfies min(subset)*max(subset) = N, then it is an ambitonal chord. These conditions are satisfied by any subset which includes 1 and N. There are usually other valid subsets as well.

For N = 15, the factors are 1, 3, 5 and 15, and the ambitonal chords are {1, 3, 5, 15}, {1, 3, 15} and {1, 5, 15}. These [[octave-reduce]] to {1/1, 3/2, 5/4, 15/8} = maj7 chord, {1/1, 3/2, 15/8} = maj7no3 chord, and {1/1, 5/4, 15/8} = maj7no5 chord.

For N = 45, the factors are 1, 3, 5, 9, 15 and 45. One ambitonal chord is {1, 3, 5, 9, 15, 45}, which octave-reduces to {1/1, 5/4, 3/2, 15/8, 9/4, 45/16} = 16:20:24:30:36:45 = maj9(#11) chord. Any note or notes can be dropped except the root and the 11th, and the chord will still be ambitonal. The only other chord is {3, 5, 9, 15} = {1/1, 5/4, 3/2, 5/3} = maj6 chord, or its [[Chord homonym|homonym]] the min7 chord. {3, 9, 15} is not ambitonal because the GCD isn't 1. {3, 5, 15} is not ambitonal because the LCM isn't 45.

== Scales ==

These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 1/1-9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 1/1-10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, Euler genera, the type of combination product sets where ''n'' = 2''k'', or tonality diamonds are necessarily ambitonal, whereas dwarf scales are always either otonal or ambitonal.

== Essentially tempered chords ==

This kind of reduction can also be used to analyze [[Dyadic_chord|essentially tempered chords]]. Consider for example the [[sinbadmic tetrad]], which is the 1001/1000-tempering of 1-11/10-13/10-10/7. The reduction of the JI version of this chord is {25, 35, 77, 91}; discarding the lowest number, 25, and reducing again gives {5, 11, 13}. This tells us the chord can be analyzed as an otonbal 1-11/10-13/10 chord plus a 10/7 addition requiring essential tempering.ed

[[Category:Otonality and utonality| ]] 
{{Todo|improve synopsis}}

2.3.5.7.11.13.19 subgroup

2026-08-03T18:48:15Z

FloraC: A little clarification & deduplication (I don't think we need to say it's close to the minor third twice)

The '''2.3.5.7.11.13.19 subgroup''' (a.k.a. ''yazalathana'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 19, but without 17. It is thus a subset of the [[19-limit]], or alternatively, it can be seen as the [[13-limit]] with an extra prime [[19/1|19]].

This subgroup is a [[rank and codimension|rank-7]] system, and can be modeled in a 6-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], [[13/1|13]] and [[19/1|19]] represented by each dimension. The prime [[2/1|2]] does not appear in typical lattices because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a seventh dimension is needed.

This subgroup is significant because 19 mirrors [[21/1|21]] in the 16::24 [[harmonic series segment]], and 21 is already present in the 13-limit; thus any fifth-bounded chord involving 21 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 19 in this subgroup. Because [[19/16]] is very close to the [[32/27|Pythagorean minor third]], it can bring a "pyth" flavor to the [[otonal]] chord. Meanwhile, primes [[17/1|17]] and [[23/1|23]] may be considered to clash with the fundamental, being close to a semitone and a tritone when [[octave reduction|octave reduced]], so people may wish to exclude them. Therefore the subgroup can be considered to complete the harmonic series up to the lower half of the fifth octave without the more difficult ones near the edges. The same reasons also give rise to the [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]].

== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
[[Edo]]s which represents the subgroup better ([[monotonic]] in the no-17 [[19-odd-limit]] and decreasing [[TE error]]): {{EDOs|'''27e''', 31, 34dh, 38df, 41f, '''41''', 50, '''53''', 58h, '''72''', 87, 94, 103h, 111, 121, '''130''', '''152f''', 190, 217, 224, '''270''', 552, 581, … }} and so on. Bold edos are records of [[Tenney–Euclidean temperament measures #TE simple badness|TE relative error]].

{{Note|[[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.}}

[[270edo]] is arguably one of the best equal temperaments for this subgroup, achieving a record of [[relative error]] that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions: [[2190edo|2190]], [[6079edo|6079]], [[8269edo|8269]] and [[8539edo|8539]]. The last two coincidentally differ by 270 and are prime edos.

=== Rank-2 temperaments ===
[[Cassandra|Cassandra (41 & 53)]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and equating together the [[Pythagorean comma|pythagorean]], [[Septimal comma|septimal]], and [[syntonic comma]] into one generic comma, that doubled approximates 33/32~1053/1024. It is well represented with [[41edo]] and [[53edo]], though [[94edo]] is more optimized.

For those searching higher-accuracy temperaments, [[cotoneum|cotoneum (41 & 217)]] keeps the chain of fifths and a pyth-septimal comma, but does not temper out the [[schisma]], instead equating it with the [[41-comma]]. Another similar temperament is [[gariwizmic|gariwizmic (94 & 270)]], which instead halves the octave and finds the [[aberschisma]] at +53 fifths -1/2 pythcomma.

[[newt|Newt (41 & 270)]] halves the fifth (tempering out [[2401/2400]]) and finds the [[aberschisma]] -41 hemififths away with much more efficiency. It is notable as one of the most efficient 13-limit and 2.3.5.7.11.13.19 temperaments, with errors down to tenths of a cent while being based on a [[Ploidacot/Dicot|dicot]] chain-of-fifths.

Other non-chain-of-fifths temperaments that are good candidates for the subgroup include [[vulture|vulture (53 & 217)]], [[satin|satin (94 & 217)]], and [[paramity|paramity (53 & 311)]].

=== Rank-3 temperaments ===
[[Cassaschismic]] is the union of all the rank-2 temperaments discussed above, relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma and a generic aberschisma, making it significant for notation systems based on a [[Ploidacot/Monocot|monocot]] chain of fifths. Other temperaments that achieve a similar level of accuracy and efficiency include [[lif]] and [[eir]].

All of these temperaments are very close to newt, and in fact, newt is the intersection of these three; Cassaschismic observes 2401/2400, lif observes 3025/3024, eir observes 4096/4095.

[[Category:Just intonation subgroups|#]]
[[Category:19-limit|#]]

19edo

2026-08-03T18:24:55Z

FloraC: /* Subsets and supersets */ restore certain info (I don't see why it should be removed)

{{interwiki
| de = 19-EDO
| en = 19edo
| es = 19 EDO
| ja = 19平均律
}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{ED intro}}

== History ==
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577, music theorist Francisco de Salinas proposed [[1/3-comma meantone]], in which the fifth is 694.786{{c}}; the fifth of 19edo is 694.737{{c}}, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

== Theory ==
19edo is the second edo, after [[12edo]], which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy (unless you count [[15edo]], which has a 18-[[cent]]-sharp fifth). Having an almost just minor third and perfect fifths and major thirds about 7 cents flat, it serves as a good tuning for [[meantone]]. Unlike 12edo, where [[enharmonic]] notes are conflated, 19edo distinguishes them, and differs from [[17edo]] in that its [[diatonic semitone]] is wider than the [[chromatic semitone]], rather than narrower. In fact, it is nearly identical to the enharmonic scale of [[1/3-comma meantone]], and can be considered a closed form thereof.

It is less successful in the [[7-limit]] as it conflates the septimal subminor third ([[7/6]]) with the septimal whole tone ([[8/7]]), but it is still better than 12edo overall.

=== Prime harmonics ===
{{Harmonics in equal|19|columns=12}}

=== As an approximation of other temperaments ===
Besides meantone, 19edo is also suitable for [[magic]]/[[muggles]] temperament, because five of its major thirds are equivalent to one of its twelfths. Its 7-step supermajor third can be used for [[sensi]], whose generator is a very sharp major third, two of which make an approximate 5/3 major sixth.

For all of these there are more optimal tunings: the fifth of 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, but in 19edo it is the same as magic. Finally, 19edo can be used as a tuning for sensi, though [[46edo]] provides a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19edo also has the advantage of being excellent for [[negri]], [[keemun]], [[godzilla]], muggles, and [[triton]]/[[liese]]. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their [[mos scale]]s in 19edo offering a great abundance of septimal tetrads. The [[Graham complexity]] of the [[4:5:6:7|7-odd-limit tetrad]] is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles, and 13 for sensi.

=== As a means of extending harmony ===
Because 19edo's 5-limit chords are more blended and concordant than those of 12edo, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non-diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

In addition, [[Joseph Yasser]] talks about the idea of a 12-tone supra-diatonic scale where the 7-tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra-diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of [[Bozuji tuning]], a 21st century tuning based on Gioseffo Zarlino's approach to just intonation. with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

=== Adaptive tuning ===
The no-11's 13-limit is represented relatively well and consistently. 19edo's negri, sensi and godzilla scales have many 13-limit chords. (You can think of the Sensi[8] [[3L 5s]] mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but Sensi[8] gives you additional ratios of 7 and 13.) Its diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3{{c}} off [[23/16]].

Practically 19edo can be used ''adaptively'' on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th, and 13th harmonics are all tuned flat. This is in contrast to 12edo, where this is not possible since the 5th and 7th harmonics are not only much farther from just than they are in 19edo, but fairly sharp already.

Another option would be to use [[octave stretching]], which has similar benefits to adaptive use, but it also works for fixed-pitch Instruments. For more on that see the [[19edo#Octave stretch or compression|section on octave stretch]].

=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]]. As such, it does not contain any nontrivial subset edos, though it contains [[19ed4]].

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best.

Large edos like [[190edo]], [[494edo]], [[665edo]], [[836edo]], [[1178edo]], and many others, contain 19edo and also exhibit high consistency in higher limits. These are also notable because they still temper out the [[enneadeca]], which preserves the precise [[6/5]] mapping 19edo has while improving upon it with microtemperaments.

See [[19th-octave temperaments]] for a detailed overview.

=== Miscellaneous properties ===
19edo has the flattest possible fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. Using 11\19 as the fifth, the sharpest possible mapping of [[5/4]] where [[10/9]] is no greater than [[9/8]] is 6\19, so the sharpest possible [[15/8]] is 17\19. Here [[16/15]] is a quarter of [[4/3]] (as in any [[negri]] tuning), so [[15/14]], [[14/13]], and [[13/12]] must all be equated with [[16/15]] to 2\19 for them to be monotone in size. If the fifth were any flatter, 5/4 and 15/8 would have to be flatter, and 16/15 would have to be sharper, so diamond monotone in the 15-odd-limit becomes impossible. 19edo is, in fact, diamond monotone in the 15-odd-limit, and even the [[17-odd-limit]] (see [[Monotonicity limits of small EDOs]]). The sharpest fifth where 15-odd-limit is possible is [[37edo#Miscellaneous properties|22\37]].

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! [[Degree|#]]
! [[Cent]]s
! Note
! Approximated ratios<ref group="note">As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament</ref>
! [[Interval category]]
|-
| 0
| 0.0
| D
| [[1/1]]
| Unison (prime)
|-
| 1
| 63.2
| D♯
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| Augmented unison
|-
| 2
| 126.3
| E♭
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| Minor second
|-
| 3
| 189.5
| E
| [[9/8]], [[10/9]]
| Major second
|-
| 4
| 252.6
| E♯/F♭
| [[7/6]], [[8/7]], [[15/13]]
| Augmented second/ Diminished third
|-
| 5
| 315.8
| F
| [[6/5]]
| Minor third
|-
| 6
| 378.9
| F♯
| [[5/4]], [[16/13]], [[56/45]]
| Major third
|-
| 7
| 442.1
| F𝄪/G♭
| [[9/7]], [[13/10]], [[21/16]], [[32/25]]
| Augmented third/ Diminished fourth
|-
| 8
| 505.3
| G
| [[4/3]], [[75/56]]
| Perfect fourth
|-
| 9
| 568.4
| G♯
| [[7/5]], [[18/13]], [[25/18]]
| Augmented fourth (Small [[tritone]])
|-
| 10
| 631.6
| A♭
| [[10/7]], [[13/9]], [[36/25]]
| Diminished fifth (Large [[tritone]])
|-
| 11
| 694.7
| A
| [[3/2]], [[112/75]]
| Perfect fifth
|-
| 12
| 757.9
| A♯/B𝄫
| [[14/9]], [[20/13]], [[25/16]], [[32/21]]
| Augmented fifth/ Diminished sixth
|-
| 13
| 821.1
| B♭
| [[8/5]], [[13/8]], [[45/28]]
| Minor sixth
|-
| 14
| 884.2
| B
| [[5/3]]
| Major sixth
|-
| 15
| 947.4
| B♯/C♭
| [[7/4]], [[12/7]], [[26/15]]
| Augmented sixth Diminished seventh
|-
| 16
| 1010.5
| C
| [[9/5]], [[16/9]]
| Minor seventh
|-
| 17
| 1073.7
| C♯
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| Major seventh
|-
| 18
| 1136.8
| D♭
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| Augmented seventh
|-
| 19
| 1200.0
| D
| [[2/1]]
| Octave
|}
<references group="note"/>

=== Proposed interval names and solfèges ===
{| class="wikitable right-1 right-2 center-3 center-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! [[Solfège]]
! colspan="2" | [[SKULO interval names]]
|-
| 0
| 0.0
| Do
| Unison
| P1
|-
| 1
| 63.2
| Di/Ro
| Super unison, subminor second
| S1, sm2
|-
| 2
| 126.3
| Ra
| Minor second
| m2
|-
| 3
| 189.5
| Re
| Major second
| M2
|-
| 4
| 252.6
| Ri/Ma
| Supermajor second, subminor third
| SM2, sm3
|-
| 5
| 315.8
| Me
| Minor third
| m3
|-
| 6
| 378.9
| Mi
| Major third
| M3
|-
| 7
| 442.1
| Mo/Fe
| Supermajor third, sub fourth
| SM3, s4
|-
| 8
| 505.3
| Fa
| Perfect fourth
| P4
|-
| 9
| 568.4
| Fi
| Augmented fourth
| A4
|-
| 10
| 631.6
| Se
| Diminished fifth
| d5
|-
| 11
| 694.7
| So
| Perfect fifth
| P5
|-
| 12
| 757.9
| Si/Lo
| Super fifth, subminor sixth
| S5, sm6
|-
| 13
| 821.1
| Le
| Minor sixth
| m6
|-
| 14
| 884.2
| La
| Major sixth
| M6
|-
| 15
| 947.4
| Li/Ta
| Supermajor sixth, subminor seventh
| SM6, sm7
|-
| 16
| 1010.5
| Te
| Minor seventh
| m7
|-
| 17
| 1073.7
| Ti
| Major seventh
| M7
|-
| 18
| 1136.8
| To/Da
| Supermajor seventh, sub octave
| SM7, s8
|-
| 19
| 1200.0
| Do
| Octave
| P8
|}

=== Interval quality and chord names in color notation ===
Using [[Kite's color notation]], qualities can be loosely associated with colors:

{| class="wikitable" style="text-align: center;"
|-
! Quality
! Color name
! Monzo format
! Examples
|-
| Diminished
| zo
| {{nowrap|(''a'', ''b'', 0, 1)}}
| 7/6, 7/4
|-
| rowspan="2" | Minor
| fourthward wa
| (''a'', ''b''), {{nowrap|''b'' < −1}}
| 32/27, 16/9
|-
| gu
| {{nowrap|(''a'', ''b'', −1)}}
| 6/5, 9/5
|-
| rowspan="2" | Major
| yo
| {{nowrap|(''a'', ''b'', 1)}}
| 5/4, 5/3
|-
| fifthward wa
| (''a'', ''b''), {{nowrap| ''b'' > 1 }}
| 9/8, 27/16
|-
| Augmented
| ru
| {{nowrap|(''a'', ''b'', 0, −1)}}
| 9/7, 12/7
|}

Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of B𝄫 would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A♯ might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished seconds, thirds, sixths and sevenths. Here are the zo, gu, yo and ru triads:

{| class="wikitable center-1 center-2 center-3 center-4"
|-
! Color of the third
! JI chord
! Edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo (7-over)
| 6:7:9
| 0–4–11
| C–E𝄫–G
| Cm(♭3) or Cmin(♭3) or C(d3)
| C subminor, C minor flat-three, C dim-three
|-
| gu (5-under)
| 10:12:15
| 0–5–11
| C–E♭–G
| Cm or Cmin
| C minor
|-
| yo (5-over)
| 4:5:6
| 0–6–11
| C–E–G
| C or Cmaj
| C, C major
|-
| ru (7-under)
| 14:18:21
| 0–7–11
| C–E♯–G
| C(♯3) or Cmaj(♯3) or C(A3)
| C supermajor, C major sharp-three, C aug-three
|-
| yo (5-over)
| 4:5:6:7
| 0–6–11–15
| C–E–G–B𝄫
| Ch7 or C,d7 or Cadd(d7)
| C harmonic 7, C (major) add dim-seven
|-
| gu (5-under)
| 1/(12:10:8:7) (1–6/5–3/2–12/7)
| 0–5–11–15
| C–E♭–G–A♯
| Cm♯6 or CmA6 or Cm(add(♯6)) or Cm(add(A6))
| C minor (add) sharp-six, C minor (add) aug-six
|}

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo conflates zo and ru ratios.

For a more complete list, see [[19edo chords #Ups and downs notation]] and [[Kite's ups and downs notation #Chords and chord progressions]].

== Notation ==
=== Standard notation ===
Standard 12edo notation can be used, whether it is staff notation (with five lines), letter [[chain-of-fifths notation]] (with standard accidentals), solfège, or sargam. Note that D# and Eb are two different notes.

Any 19edo note or interval can be [[enharmonic unison|respelled enharmonically]] by adding a double-diminished second to it or subtracting one from it. Adding a dd2 is equivalent to finding the 12edo equivalent with a higher degree, then diminishing it. For example, C# becomes Db, which is diminished to become Dbb.

{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 19edo
|-
! rowspan="2" | [[Degree|#]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic interval names]]
! Note names on D
|-
| 0
| 0.0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 63.2
| Augmented unison (A1) Diminished second (d2)
| D# Ebb
|-
| 2
| 126.3
| Doubly augmented unison (AA1) Minor second (m2)
| Dx Eb
|-
| 3
| 189.5
| '''Major second (M2)''' Doubly diminished third (dd3)
| '''E''' Fbb
|-
| 4
| 252.6
| Augmented second (A2) Diminished third (d3)
| E# Fb
|-
| 5
| 315.8
| Doubly augmented second (AA2) '''Minor third (m3)'''
| Ex '''F'''
|-
| 6
| 378.9
| '''Major third (M3)''' Doubly diminished fourth (dd4)
| '''F#''' Gbb
|-
| 7
| 442.1
| Augmented third (A3) Diminished fourth (d4)
| Fx Gb
|-
| 8
| 505.3
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 9
| 568.4
| Augmented fourth (A4) Doubly diminished fifth (dd5)
| G# Abb
|-
| 10
| 631.6
| Doubly augmented fourth (AA4) Diminished fifth (d5)
| Gx Ab
|-
| 11
| 694.7
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 12
| 757.9
| Augmented fifth (A5) Diminished sixth (d6)
| A# Bbb
|-
| 13
| 821.1
| Doubly augmented fifth (AA5) Minor sixth (m6)
| Ax Bb
|-
| 14
| 884.2
| '''Major sixth (M6)''' Doubly diminished seventh (dd7)
| '''B''' Cbb
|-
| 15
| 947.4
| Augmented sixth (A6) Diminished seventh (d7)
| B# Cb
|-
| 16
| 1010.5
| Doubly augmented sixth (AA6) '''Minor seventh (m7)'''
| Bx '''C'''
|-
| 17
| 1073.7
| Major seventh (M7) Doubly diminished octave (dd8)
| C# Dbb
|-
| 18
| 1136.8
| Augmented seventh (A7) Diminished octave (d8)
| Cx Db
|-
| 19
| 1200.0
| '''Perfect octave (P8)'''
| '''D'''
|}

In 19edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

{{Sharpness-sharp1}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[5edo #Sagittal notation|5]], [[12edo #Sagittal notation|12]], and [[26edo #Sagittal notation|26]], and is a subset of the notations for edos [[38edo #Sagittal notation|38]], [[57edo #Sagittal notation|57]], and [[76edo #Sagittal notation|76]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is identical to conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic notation of 19edo
|-
! [[Degree|#]]
! [[Cent]]s
! Interval names
|-
| 0
| 0.0
| P1
|-
| 1
| 63.2
| A1, m2
|-
| 2
| 126.3
| M2, m3
|-
| 3
| 189.5
| M3
|-
| 4
| 252.6
| m4, A3
|-
| 5
| 315.8
| M4, m5
|-
| 6
| 378.9
| M5
|-
| 7
| 442.1
| A5, d6
|-
| 8
| 505.3
| P6
|-
| 9
| 568.4
| A6, m7
|-
| 10
| 631.6
| M7, d8
|-
| 11
| 694.7
| P8
|-
| 12
| 757.9
| A8, m9
|-
| 13
| 821.1
| M9, m10
|-
| 14
| 884.2
| M10
|-
| 15
| 947.4
| m11, A10
|-
| 16
| 1010.5
| M11, m12
|-
| 17
| 1073.7
| M12
|-
| 18
| 1136.8
| A12, d13
|-
| 19
| 1200.0
| P13
|}

== Approximation to JI ==
[[File:19ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 19edo]]

=== Interval mappings ===
{{Q-odd-limit intervals|19}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -30 19 }}
| {{mapping| 19 30 }}
| +2.277
| 2.277
| 3.612
|-
| 2.3.5
| 81/80, 3125/3072
| {{mapping| 19 30 44 }}
| +2.578
| 1.911
| 3.025
|-
| 2.3.5.7
| 49/48, 81/80, 126/125
| {{mapping| 19 30 44 53 }}
| +3.848
| 2.755
| 4.362
|-
| 2.3.5.7.13
| 49/48, 65/64, 81/80, 91/90
| {{mapping| 19 30 44 53 70 }}
| +4.135
| 2.530
| 4.006
|-
| 2.3.5.7.13.23
| 49/48, 65/64, 70/69, 81/80, 91/90
| {{mapping| 19 30 44 53 70 86 }}
| +3.319
| 2.936
| 4.649
|}
* 19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit—''both'' 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next equal temperaments doing better in those subgroups are [[34edo|34]], [[31edo|31]], [[27edo|27e]], [[22edo|22]], and [[26edo|26]], respectively.
* 19et is best in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is [[53edo|53]].

=== Uniform maps ===
{{Uniform map|edo=19}}

=== Commas ===
19et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color notation/Temperament names|Color name]]
! Name
|-
| 3
| <abbr title="1162261467/1073741824">(20 digits)</abbr>
| {{monzo| -30 19 }}
| 137.14
| Trilawa
| [[19-comma]]
|-
| 5
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma
|-
| 5
| <abbr title="1594323/1562500">(14 digits)</abbr>
| {{monzo| -2 13 -8}}
| 34.91
| Laquadbigu
| [[Unicorn comma]]
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1224440064/1220703125">(20 digits)</abbr>
| {{monzo| 8 14 -13 }}
| 5.29
| Thegu
| [[Parakleisma]]
|-
| 5
| <abbr title="19073486328125/19042491875328">(28 digits)</abbr>
| {{monzo| -14 -19 19 }}
| 2.82
| Neyo
| [[Enneadeca]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
| 7.32
| Labirugu
| Cataharry comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Satrizo-agu
| Hemimage comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 13
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[343/338]]
| {{monzo| -1 0 0 3 0 -2 }}
| 25.42
| Thuthutrizo
|
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 23
| [[2187/2116]]
| {{monzo| -2 7 0 0 0 0 0 0 -2 }}
| 57.14
| Labitwethu
| Lipsett comma
|-
| 23
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -1 }}
| 24.91
| Twethuzoyo
| Small vicesimotertial eighth tone
|-
| 23
| 256/253
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| 20.41
| Twethulu
| 253rd subharmonic
|-
| 23
| [[161/160]]
| {{monzo| -5 0 -1 1 0 0 0 0 1 }}
| 10.79
| Twethozogu
| Major kirnbergisma
|-
| 23
| [[208/207]]
| {{monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[529/528]]
| {{monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
| 23
| [[576/575]]
| {{monzo| 6 2 -2 0 0 0 0 0 -1 }}
| 3.01
| Twethugugu
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Twethothuluzo
| Triaphonisma
|}
<references group="note" />

=== Linear temperaments ===
* [[List of 19et rank two temperaments by badness]]
* [[List of 19et rank two temperaments by complexity]]
* [[List of edo-distinct 19et rank two temperaments]]
* [[Syntonic–kleismic equivalence continuum]]

Since 19 is prime, all rank-2 temperaments in 19edo have one period per octave (i.e. are linear). Therefore you can make a correspondence between intervals and the linear temperaments they generate.

{| class="wikitable center-1 right-2 center-3"
|-
! Degree
! Cents
! Interval
! Mos scales
! Temperaments
|-
| 1
| 63.16
| A1, d2
|
| [[Unicorn]] / [[Rhinoceros]]
|-
| 2
| 126.32
| m2
| [[1L 8s]], [[9L 1s]]
| [[Negri]]
|-
| 3
| 189.47
| M2
| [[1L 5s]], [[6L 1s]], [[6L 7s]]
| [[Deutone]] [[Xenial]] / [[Sensamagic clan #Xenia|Xenia]] [[Spell]]
|-
| 4
| 252.63
| A2, d3
| [[1L 3s]], [[4L 1s]], [[5L 4s]], [[5L 9s]]
| [[Godzilla]] / [[Helayo]]
|-
| 5
| 315.79
| m3
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]]
| [[Cata]] / [[keemun]]
|-
| 6
| 378.95
| M3
| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]]
| [[Magic]] / [[muggles]]
|-
| 7
| 442.11
| A3, d4
| [[3L 2s]], [[3L 5s]], [[8L 3s]]
| [[Sensi]]
|-
| 8
| 505.26
| P4
| [[2L 3s]], [[5L 2s]], [[7L 5s]]
| [[Meantone]] / [[flattone]]
|-
| 9
| 568.42
| A4
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]]
| [[Liese]] [[Triton]] / [[pycnic]]
|}

== Octave stretch or compression ==
19edo is a promising option as a meantone tuning for pianos if split sharps are acceptable, since pianos are frequently tuned with stretched octaves due to the slight [[inharmonicity]] inherent in their strings. It also works well with harpsichords, since many have been, and are, built with split sharps.

Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-[[just]]. For example, if we are using [[49ed6]] or [[30edt]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57{{c}}, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-odd-limit [[tonality diamond]]. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. Another possible choice is [[ZPI|65zpi]].

== Scales ==
=== MOS scales ===
{{Main|List of MOS scales in {{PAGENAME}}}}

==== Octave-equivalent mosses ====
* [[Meantone]] pentic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[Meantone]] diatonic, [[5L 2s]] (gen = 11\19): 3 3 2 3 3 3 2
* [[Meantone]] chromatic, [[7L 5s]] (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
* [[Semaphore]][5], [[4L 1s]] (gen = 4\19): 4 4 3 4 4
* [[Semaphore]][9], [[5L 4s]] (gen = 4\19): 3 1 3 1 3 3 1 3 1
* [[Semaphore]][14], [[5L 9s]] (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
* [[Sensi]][5], [[2L 3s]] (gen = 7\19): 5 2 5 2 5
* [[Sensi]][8], [[3L 5s]] (gen = 7\19): 2 3 2 2 3 2 2 3
* [[Sensi]][11], [[8L 3s]] (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
* [[Negri]][9], [[1L 8s]] (gen = 2\19): 2 2 2 2 3 2 2 2 2
* [[Negri]][10], [[9L 1s]] (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
* [[Kleismic]][7], [[4L 3s]] (gen = 5\19): 1 4 1 4 1 4 4
* [[Kleismic]][11], [[4L 7s]] (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
* [[Kleismic]][15], [[4L 11s]] (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
* [[Magic]][7], [[3L 4s]] (gen = 6\19): 5 1 5 1 5 1 1
* [[Magic]][10], [[3L 7s]] (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
* [[Magic]][13], [[3L 10s]] (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
* [[Magic]][16], [[3L 13s]] (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
* [[Liese]][17], [[2L 15s]] (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

=== Other scales ===
{{Main|19edo modes}}

* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2 (ascending), 3 2 3 3 2 3 3 (descending)
* Meantone harmonic major: 3 3 2 3 2 4 2
* Chromatic octave species – meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* Chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* Chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* Enharmonic pentatonic: 2 6 3 2 6
* Enharmonic pentatonic: 6 2 3 6 2
* Enharmonic octave species: 1 1 6 3 1 1 6
* Enharmonic octave species: 6 1 1 3 6 1 1
* Enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
* [[Pinetone #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
* [[Pinetone #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
* [[Pinetone #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Semiquartal]] 3|5 b2: 1 3 3 1 3 1 3 3 1
* [[5-odd-limit]] tonality diamond: 5 1 2 3 2 1 5
* [[7-odd-limit]] tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
* [[9-odd-limit]] tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3

== Instruments ==
[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
[[File:Bass19.jpg|alt=19edo 5 string Bass 34"-37" scale length|512x384px|thumb|none|19edo bass conversion by Ron Sword]]

== Music ==
{{Main| 19edo/Music }}
{{Catrel| 19edo tracks }}

; [http://micro.soonlabel.com/19-ET/ XA 19-ET Index]
; A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]{{dead link}}

== See also ==
* [[19edo modes]]
* [[19edo chords]]
* [[Strictly proper 19edo scales]]
* [[How to tune a 19edo guitar by ear]]
* [[Primer for 19edo]]
* [[Mason Green's New Common Practice Notation]]
* [[Extraclassical tonality]]
* [[Lumatone mapping for 19edo]]

== Further reading ==
* [[Darreg, Ivor]]. ''[http://www.tonalsoft.com/sonic-arts/darreg/case.htm A Case for Nineteen]''. 1982.
* Darreg, Ivor. ''[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties]''{{dead link}}. (Unknown date of publication).
* Howe, Hubert S., Jr. [http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications]. c. 2004.
* [[Sethares, William A]]. [http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar]. 1991.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave]''. 2010.
* Yasser, Joseph. ''[https://www.worldcat.org/fr/title/726192994 Theory of Evolving Tonality]''. 1932.

== External links ==
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on the [[Tonalsoft Encyclopedia]]
* [http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
* [http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]

=== References ===
* Bucht, Saku and Huovinen, Erkki, ''Perceived consonance of harmonic intervals in 19-tone equal temperament'', CIM04_proceedings.
* Levy, Kenneth J., ''Costeley's Chromatic Chanson'', Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

[[Category:19-tone scales]]
[[Category:Godzilla]]
[[Category:Golden meantone]]
[[Category:Kleismic]]
[[Category:Meantone]]
[[Category:Magic]]
[[Category:Negri]]
[[Category:Semaphore]]
[[Category:Sensi]]
[[Category:Teentuning]]
[[Category:Historical]]

Amity comma

2026-08-03T18:21:00Z

FloraC: Undo (vague)

{{Infobox Interval
| Ratio = 1600000/1594323
| Name = amity comma, amiton
| Color name = sy51, Saquinyo comma
| Comma = yes
}}
The '''amity comma''' or '''amiton''' ({{monzo|legend=1| 9 -13 5 }}, [[ratio]]: 1600000/1594323) is a [[small comma|small]] [[5-limit]] [[comma]] of 6.154 [[cent]]s, the amount by which five [[10/9|minor whole tones (10/9)]] exceed the [[27/16|Pythagorean major sixth (27/16)]]. It belongs to the [[syntonic–chromatic equivalence continuum]] and is equal to the difference between an [[apotome]] and a stack of five [[syntonic comma]]s ((2187/2048)/(81/80)5), or in terms of classic chromatic semitone, between a classic chromatic semitone and a stack of three syntonic commas ((25/24)/(81/80)3).

== Temperaments ==
Tempering it out leads to the [[amity family]] of temperaments.

== Etymology ==
The corresponding temperament was discovered first and named by [[Gene Ward Smith]] as ''acute minor third'' or ''amt'' in 2001–2002<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2064.html Yahoo! Tuning Group | ''Kleismic & co'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3481.html Yahoo! Tuning Group | ''32 best 5-limit linear temperaments redux'']</ref>. The temperament was renamed to ''amity'', and the comma was at one point dubbed ''amitisma'', both by Gene Ward Smith in late 2002, though it was ''amity comma'' that stuck<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5025.html Yahoo! Tuning Group | ''5-limit comma names'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5080.html#5114 Yahoo! Tuning Group | ''Ultimate 5-limit comma list'']</ref>.

In 2025, [[User: VectorGraphics|Vector]] and [[Lériendil]] proposed ''amiton'' by analogy with [[graviton]], as both amity and gravity are on the syntonic–chromatic equivalence continuum.

== See also ==
* [[Small comma]]

== Notes ==

[[Category:Amity|#]] 
[[Category:Commas named for their regular temperament properties]]

3/2

2026-08-03T18:20:10Z

FloraC: /* As a dyad */ migrate certain info from 1:2:3

{{interwiki
| de =
| en =
| es =
| ja =3/2
| ko =
| ro = 3/2 (ro)
}}
{{Infobox interval
| Name = just perfect fifth
| Color name = w5, wa 5th
| Sound = jid_3_2_pluck_adu_dr220.mp3
}}
{{Wikipedia|Perfect fifth}}

'''3/2''', the '''just perfect fifth''', is a very [[consonance|consonant]] interval, due to the numerator and denominator of its ratio being very small numbers, with only the [[octave]] and the [[3/1|tritave]] having smaller numbers. As such, it is very important in western music and many musical traditions, and approximating it is key in systems like [[12edo]] and other [[edo]]s.

For harmonic [[timbre]]s, the loudest harmonics are usually the second and third ones (2/1 and 3/1). 3/2 is the interval between these two harmonics (which incidentally makes 3/2 [[superparticular]]). Thus 3/2 is easy to tune by ear, and it is easy to hear if it is mistuned.

== Usage ==
Variations of the perfect fifth (whether [[just]] or tempered) appear in most [[approaches to musical tuning|music of the world]]. [[Historical temperaments|Historically]], European music treated the perfect fifth as consonant long before it treated the major third—specifically [[5/4]]—as consonant. In the present day, the dominant tuning [[12edo]] approximates 3/2 very accurately.

A [[chain of fifths|chain of just perfect fifths]] generates [[Pythagorean tuning]]. The chain continues indefinitely and theoretically never returns to the starting note. A chain that ends at seven notes generates the historically important Pythagorean [[5L 2s|diatonic]] scale. This scale is also the 7 natural notes of all "pyth-spine" notations, in which all uninflected notes are Pythagorean, such as [[HEJI]], [[Sagittal notation|Sagittal]], [[ups and downs notation|ups and downs]], [[FJS]] and [[color notation]].

Music using unusual intervals can be very disorienting. The presence of perfect fifths can provide a "ground" that make it less so. Some composers deliberately use tunings that lack fifths, to make their music sound more [[xenharmonic]].

=== In regular temperament theory ===
Because 3/2 is a very simple and concordant interval, it is still recognizable even when heavily tempered. Often it is tempered so that an octave-reduced stack of fourths or fifths approximates some other interval. Some examples:

[[Meantone]] temperament flattens the fifth from just (to around 695–700 cents) such that the major third generated by stacking four fifths is closer to (or even identical to) 5/4. The minor third generated by stacking three fourths is closer to 6/5.

[[Superpyth]] temperaments ''sharpen'' the fifth from just so that the major third is closer to 9/7 and the minor third is closer to 7/6. Thus the minor seventh 16/9 approximates 7/4 instead of 9/5.

[[Schismic]] temperament adjusts the fifth such that the ''diminished fourth'' generated by stacking eight fourths approximates 5/4. As this is already a close approximation, the tuning of the fifth can be varied around its just tuning, but is most accurately flattened by a tiny amount. Thus a triad with 5/4 is written as {{nowrap|{{dash|C, F♭, G}}}} (unless the notation has accidentals for [[81/80]], e.g. {{nowrap|{{dash|C, vE, G}}}}).

* Garibaldi temperament is an extension of schismic that sharpens the fifth so that the small interval between the major third and diminished fourth can also be used to create simple 7-limit intervals.

== Approximations by edos ==
12edo approximates 3/2 to within only 2{{c}}. [[29edo]], [[41edo]], and [[53edo]] are even more accurate. In regards to [[telicity]], while 12edo is a 2-strong 3-2 telic system, 53edo is notably a 3-strong 3-2 telic system.

The following edos (up to 200) approximate 3/2 to within both 7{{c}} and 7%. Errors are unsigned so that the table can be sorted by them. The arrow column indicates a sharp (↑) or flat (↓) fifth.

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | Deg\edo
! Absolute error ([[cent|¢]])
! Relative error (%)
! ↕
! class="unsortable" | Equally accurate multiples
|-
| [[12edo|12]] || 7\12 || 1.955 || 1.955 || ↓ || [[24edo|14\24]], [[36edo|21\36]]
|-
| [[17edo|17]] || 10\17 || 3.927 || 5.564 || ↑ ||
|-
| [[29edo|29]] || 17\29 || 1.493 || 3.609 || ↑ ||
|-
| [[41edo|41]] || 24\41 || 0.484 || 1.654 || ↑ || [[82edo|48\82]], [[123edo|72\123]], [[164edo|96\164]]
|-
| [[53edo|53]] || 31\53 || 0.068 || 0.301 || ↓ || [[106edo|62\106]], [[159edo|93\159]]
|-
| [[65edo|65]] || 38\65 || 0.416 || 2.256 || ↓ || [[130edo|76\130]], [[195edo|114\195]]
|-
| [[70edo|70]] || 41\70 || 0.902 || 5.262 || ↑ ||
|-
| [[77edo|77]] || 45\77 || 0.656 || 4.211 || ↓ ||
|-
| [[89edo|89]] || 52\89 || 0.831 || 6.166 || ↓ ||
|-
| [[94edo|94]] || 55\94 || 0.173 || 1.352 || ↑ || [[188edo|110\188]]
|-
| [[111edo|111]] || 65\111 || 0.748 || 6.916 || ↑ ||
|-
| [[118edo|118]] || 69\118 || 0.260 || 2.557 || ↓ ||
|-
| [[135edo|135]] || 79\135 || 0.267 || 3.006 || ↑ ||
|-
| [[142edo|142]] || 83\142 || 0.547 || 6.467 || ↓ ||
|-
| [[147edo|147]] || 86\147 || 0.086 || 1.051 || ↑ ||
|-
| [[171edo|171]] || 100\171 || 0.200 || 2.859 || ↓ ||
|-
| [[176edo|176]] || 103\176 || 0.318 || 4.660 || ↑ ||
|-
| [[183edo|183]] || 107\183 || 0.316 || 4.814 || ↓ ||
|-
| [[200edo|200]] || 117\200 || 0.045 || 0.750 || ↑ ||
|}

Edos can be classified by their approximation of 3/2 as:
* '''Superflat''' edos have fifths narrower than {{nowrap| 4\7 {{=}} ~686{{c}} }}
* '''Perfect''' edos have fifths of exactly 4\7
* '''Diatonic''' edos have fifths between 4\7 and {{nowrap| 3\5 {{=}} 720{{c}} }}
* '''Pentatonic''' have fifths of exactly 3\5
* '''Supersharp''' edos have fifths wider than 3\5

{| class="wikitable sortable"
|+ style="font-size: 105%;" | Comparison of the fifths of edos 5 to 31
|-
! Edo
! Degree
! Cents
! Edo category
! Error (¢)
|-
| [[5edo]]
| 3\5
| 720.000
| Pentatonic edo
| +18.045
|-
| [[7edo]]
| 4\7
| 685.714
| Perfect edo
| −16.241
|-
| [[8edo]]
| 5\8
| 750.000
| Supersharp edo
| +48.045
|-
| [[9edo]]
| 5\9
| 666.667
| Superflat edo
| −35.288
|-
| [[10edo]]
| 6\10
| 720.000
| Pentatonic edo
| +18.045
|-
| [[11edo]]
| 6\11
| 654.545
| Superflat edo
| −47.41
|-
| [[12edo]]
| 7\12
| 700.000
| Diatonic edo
| −1.955
|-
| [[13edo]]
| 8\13
| 738.462
| Supersharp edo
| +36.507
|-
| [[14edo]]
| 8\14
| 685.714
| Perfect edo
| −16.241
|-
| [[15edo]]
| 9\15
| 720.000
| Pentatonic edo
| +18.045
|-
| [[16edo]]
| 9\16
| 675.000
| Superflat edo
| −26.955
|-
| [[17edo]]
| 10\17
| 705.882
| Diatonic edo
| +3.927
|-
| [[18edo]]
| 11\18
| 733.333
| Supersharp edo
| +31.378
|-
| [[19edo]]
| 11\19
| 694.737
| Diatonic edo
| −7.218
|-
| [[20edo]]
| 12\20
| 720.000
| Pentatonic edo
| +18.045
|-
| [[21edo]]
| 12\21
| 685.714
| Perfect edo
| −16.241
|-
| [[22edo]]
| 13\22
| 709.091
| Diatonic edo
| +7.136
|-
| [[23edo]]
| 13\23
| 678.261
| Superflat edo
| −23.694
|-
| [[24edo]]
| 14\24
| 700.000
| Diatonic edo
| −1.955
|-
| [[25edo]]
| 15\25
| 720.000
| Pentatonic edo
| +18.045
|-
| [[26edo]]
| 15\26
| 692.308
| Diatonic edo
| −9.647
|-
| [[27edo]]
| 16\27
| 711.111
| Diatonic edo
| +9.156
|-
| [[28edo]]
| 16\28
| 685.714
| Perfect edo
| −16.241
|-
| [[29edo]]
| 17\29
| 703.448
| Diatonic edo
| +1.493
|-
| [[30edo]]
| 18\30
| 720.000
| Pentatonic edo
| +18.045
|-
| [[31edo]]
| 18\31
| 696.774
| Diatonic edo
| −5.181
|}

== As a dyad ==
{{Infobox Chord
| 2:3
| ColorName=5
| debug=1
}}
'''2:3''' is a 3-limit [[dyad]], known as the ''five chord'' (as in C5 not V) or ''power chord''. It is used to create an open, stable, and powerful sound in contemporary western music. It is also indispensable in certain musical genres such as [[African music #Equiheptatonic tunings|mbira music]] and late medieval music. In the latter, when voiced as '''1:2:3''', it is known as the ''trine'', a very common closing chord.

=== Notable voicings ===
{| class="wikitable"
|+
! Voices
! [[EFR]]
! [[Kite's thoughts on hi-lo notation|Hi-lo name]]
! Special properties
|-
| rowspan="3" | 2 voices
| 1:3
| hi5
| AOV ([[Odd limit #Proposed extensions|all-odd voicing]])
|-
| 2:3
| basic
| CAOV (condensed AOV)
|-
| 3:4
| lo5
| 1st inversion
|-
| rowspan="3" |3 voices
| 1:2:3
| hi5add8
| The trine
|-
| 2:3:4
| add8
|
|-
| 3:4:6
| addlo5
| 2:3:4 melodically inverted
|}
{{Clear}}

== See also ==
* [[4/3]] – its [[octave complement]]
* [[Fifth complement]]
* [[Edf]] – tunings which equally divide 3/2
* [[Gallery of just intervals]]
* {{OEIS|A060528}} – sequence of edos with increasingly better approximations of 3/2 (and by extension 4/3)
* {{OEIS|A005664}} – denominators of the convergents to log2(3)
* {{OEIS|A206788}} – denominators of the semiconvergents to log2(3)

[[Category:Fifth]]
[[Category:Taxicab-2 intervals]]

1:2:3

2026-08-03T18:18:53Z

FloraC: Voicing of 2:3

#redirect [[3/2 #As a dyad]]

Quartonic family

2026-08-03T06:13:35Z

FloraC: Switch to Sintel's badness, WE & CWE tunings (2/2)

{{Technical data page}}
The '''quartonic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quartonic comma]] ({{monzo|legend=1| 3 -18 11 }}, [[ratio]]: 390 625 000 / 387 420 489).

== Quartonic ==
The name ''quartonic'' refers to the [[quartertone]], since a somewhat flattened quartertone (representing the triptolemaic chromatic semitone, i.e. [[porcupine comma]]) is the [[generator]] of this temperament. The [[ploidacot]] for the temperament is omega-hendecacot, with eleven generator steps giving the [[4/3|perfect fourth]]. It is a member of the [[schismic–Mercator equivalence continuum]], with equivalence number ''n'' = 11/2.

[[Subgroup]]: 2.3.5

[[Comma list]]: 390625000/387420489

{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
: mapping generators: ~2, ~250/243

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9709{{c}}, ~250/243 = 45.2318{{c}}
: [[error map]]: {{val| -0.029 +0.437 -0.574 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 45.2349{{c}}
: error map: {{val| 0.000 +0.461 -0.541 }}

{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}

[[Badness]] (Sintel): 2.75

=== Overview to extensions ===
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].
* [[10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[390625/388962]] or [[4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.

== Septimal quartonic ==
Septimal quartonic tempers out the [[orwellisma]], the [[octagar comma]], as well as the [[ragisma]], and may be described as the {{nowrap| 26 & 27 }} temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 4000/3969

{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1424{{c}}, ~36/35 = 45.1064{{c}}
: [[error map]]: {{val| -0.858 +0.160 -0.801 +3.069 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 45.1881{{c}}
: error map: {{val| 0.000 +0.976 +0.301 +5.234 }}

{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}

[[Badness]] (Sintel): 1.08

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 2200/2187

Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}

Optimal tunings:
* WE: ~2 = 1198.8787{{c}}, ~36/35 = 44.9994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.0863{{c}}

{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}

Badness (Sintel): 1.13

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 540/539

Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}

Optimal tunings:
* WE: ~2 = 1199.2991{{c}}, ~36/35 = 45.0539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.1049{{c}}

{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d }}

Badness (Sintel): 0.987

=== Quarto ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 864/847

Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}

Optimal tunings:
* WE: ~2 = 1198.3094{{c}}, ~36/35 = 45.0688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2329{{c}}

{{Optimal ET sequence|legend=0| 26, 27e, 53e, 80ee }}

Badness (Sintel): 1.38

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 245/242

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}

Optimal tunings:
* WE: ~2 = 1198.9315{{c}}, ~36/35 = 45.1650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2640{{c}}

{{Optimal ET sequence|legend=0| 26, 27e, 53e }}

Badness (Sintel): 1.14

=== Quartz ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 4000/3969

Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}

Optimal tunings:
* WE: ~2 = 1200.5082{{c}}, ~33/32 = 45.4047{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}

{{Optimal ET sequence|legend=0| 26, 53 }}

Badness (Sintel): 1.76

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 275/273, 385/384

Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}

Optimal tunings:
* WE: ~2 = 1200.5899{{c}}, ~33/32 = 45.4096{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}

{{Optimal ET sequence|legend=0| 26, 53 }}

Badness (Sintel): 1.19

=== Biquartonic ===
Subgroup: 2.3.5.7.11

Comma list: 1728/1715, 2420/2401, 2560/2541

Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
: mapping generators: ~99/70, ~36/35

Optimal tunings:
* WE: ~99/70 = 599.5389{{c}}, ~36/35 = 45.0947{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~36/35 = 45.1692{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 186de, 266dde }}

Badness (Sintel): 2.01

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 640/637

Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}

Optimal tunings:
* WE: ~55/39 = 599.6870{{c}}, ~40/39 = 45.1293{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~40/39 = 45.1789{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}

Badness (Sintel): 1.65

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539

Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}

Optimal tunings:
* WE: ~17/12 = 599.7706{{c}}, ~40/39 = 45.1427{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~40/39 = 45.1790{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}

Badness (Sintel): 1.43

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539, 400/399

Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}

Optimal tunings:
* WE: ~17/12 = 599.7848{{c}}, ~39/38 = 45.1288{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~39/38 = 45.1634{{c}}

{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}

Badness (Sintel): 1.30

=== Yarm ===
Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1728/1715, 4000/3969

Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}

Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0363{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0617{{c}}

{{Optimal ET sequence|legend=0| 79, 80, 239dd }}

Badness (Sintel): 3.30

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 640/637, 1331/1323

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}

Optimal tunings:
* WE: ~2 = 1199.5798{{c}}, ~100/99 = 15.0551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0681{{c}}

{{Optimal ET sequence|legend=0| 79, 80 }}

Badness (Sintel): 2.55

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 325/324, 561/560, 640/637, 850/847

Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}

Optimal tunings:
* WE: ~2 = 1199.6800{{c}}, ~100/99 = 15.0624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0705{{c}}

{{Optimal ET sequence|legend=0| 79, 80 }}

Badness (Sintel): 2.38

== Yarman I ==
Yarman I is the first interpretation of [[Ozan Yarman]]'s proposed tuning for the Turkish makam.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 244140625/243045684

{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}
: mapping generators: ~2, ~126/125

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8726{{c}}, ~126/125 = 15.0651{{c}}
: [[error map]]: {{val| -0.127 +0.641 -0.213 -0.523 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~126/125 = 15.0689{{c}}
: error map: {{val| 0.000 +0.771 -0.036 -0.374 }}

{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 637, 876b }}

[[Badness]] (Sintel): 4.89

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 | 0 -33 -54 -95 -43 }}

Optimal tunings:
* WE: ~2 = 1199.8339{{c}}, ~100/99 = 15.0637{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0685{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 876be }}

Badness (Sintel): 1.63

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 1001/1000, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 4 | 0 -33 -54 -95 -43 -24 }}

Optimal tunings:
* WE: ~2 = 1200.1080{{c}}, ~100/99 = 15.0766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0737{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f }}

Badness (Sintel): 1.69

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 595/594, 1001/1000, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 4 4 | 0 -33 -54 -95 -43 -24 7 }}

Optimal tunings:
* WE: ~2 = 1200.0222{{c}}, ~100/99 = 15.0718{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0713{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f }}

Badness (Sintel): 1.58

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 361/360, 364/363, 595/594, 969/968, 1001/1000

Mapping: {{mapping| 1 2 3 4 4 4 4 5 | 0 -33 -54 -95 -43 -24 7 -60 }}

Optimal tunings:
* WE: ~2 = 1200.0698{{c}}, ~100/99 = 15.0722{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0706{{c}}

{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 398fh }}

Badness (Sintel): 1.41

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 325/324, 361/360, 364/363, 460/459, 507/506, 529/528, 760/759

Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 | 0 -33 -54 -95 -43 -24 7 -60 -38 }}

Optimal tunings:
* WE: ~2 = 1200.0946{{c}}, ~100/99 = 15.0732{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0710{{c}}

{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 398fh }}

Badness (Sintel): 1.27

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 325/324, 361/360, 364/363, 406/405, 460/459, 494/493, 507/506, 529/528

Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 6 | 0 -33 -54 -95 -43 -24 7 -60 -38 -91 }}

Optimal tunings:
* WE: ~2 = 1200.0995{{c}}, ~100/99 = 15.0726{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0703{{c}}

{{Optimal ET sequence|legend=0| 79dhj, 80, 159, 239, 398fh }}

Badness (Sintel): 1.19

== Yarman II ==
Yarman II is the other interpretation of [[Ozan Yarman]]'s proposed tuning for the Turkish makam.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5359375/5308416, 390625000/387420489

{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}
: mapping generators: ~2, ~875/864

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4734{{c}}, ~875/864 = 15.1122{{c}}
: [[error map]]: {{val| +0.473 +0.291 -0.950 -0.701 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/864 = 15.1048{{c}}
: error map: {{val| 0.000 -0.412 -1.971 -2.121 }}

{{Optimal ET sequence|legend=1| 79, 159, 397cd, 556cd, 715ccdd }}

[[Badness]] (Sintel): 16.6

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 78121827/77948684

Mapping: {{mapping| 1 2 3 2 4 | 0 -33 -54 64 -43 }}

Optimal tunings:
* WE: ~2 = 1200.4357{{c}}, ~100/99 = 15.1126{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1054{{c}}

{{Optimal ET sequence|legend=0| 79, 159, 397cd, 556cd }}

Badness (Sintel): 4.74

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 85683/85184

Mapping: {{mapping| 1 2 3 2 4 4 | 0 -33 -54 64 -43 -24 }}

Optimal tunings:
* WE: ~2 = 1200.5028{{c}}, ~100/99 = 15.1135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1052{{c}}

{{Optimal ET sequence|legend=0| 79, 159, 397cdff, 556cdff }}

Badness (Sintel): 2.82

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 385/384, 1575/1573, 4928/4913

Mapping: {{mapping| 1 2 3 2 4 4 4 | 0 -33 -54 64 -43 -24 7 }}

Optimal tunings:
* WE: ~2 = 1200.3840{{c}}, ~100/99 = 15.1086{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1025{{c}}

{{Optimal ET sequence|legend=0| 79, 159 }}

Badness (Sintel): 2.60

== Tertiseptisix ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 390625000/387420489

{{Mapping|legend=1| 1 -31 -51 -32 | 0 44 72 47 }}
: mapping generators: ~2, ~1458/875

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0156{{c}}, ~1458/875 = 888.7028{{c}}
: [[error map]]: {{val| +0.016 +0.485 -0.507 -0.293 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1458/875 = 888.6915{{c}}
: error map: {{val| 0.000 +0.472 -0.523 -0.324 }}

{{Optimal ET sequence|legend=1| 27, …, 212, 239, 451 }}

[[Badness]] (Sintel): 3.95

== Triquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 117649/116640, 250047/250000

{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}
: mapping generators: ~63/50, ~250/243

[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0521{{c}}, ~250/243 = 45.2373{{c}}
: [[error map]]: {{val| +0.156 +0.747 -0.117 -1.628 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~250/243 = 45.2204{{c}}
: error map: {{val| 0.000 +0.621 -0.281 -1.911 }}

{{Optimal ET sequence|legend=1| 27, 105cd, 132d, 159, 345d, 506d }}

[[Badness]] (Sintel): 4.30

== Quartiquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 390625/388962, 4802000/4782969

{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}
: mapping generators: ~25/21, ~250/243

[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.9955{{c}}, ~250/243 = 45.2382{{c}}
: [[error map]]: {{val| -0.018 +0.389 -0.655 +0.152 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~250/243 = 45.2400{{c}}
: error map: {{val| 0.000 +0.405 -0.634 +0.174 }}

{{Optimal ET sequence|legend=1| 80, 132d, 212, 292, 504 }}

[[Badness]] (Sintel): 5.04

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 14641/14580

Mapping: {{mapping| 4 8 12 15 17 | 0 -11 -18 -25 -21 }}

Optimal tunings:
* WE: ~25/21 = 300.0140{{c}}, ~77/75 = 45.2398{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~77/75 = 45.2342{{c}}

{{Optimal ET sequence|legend=0| 80, 132de, 212, 292, 504e }}

Badness (Sintel): 2.06

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1001/1000, 1375/1372, 10648/10647

Mapping: {{mapping| 4 8 12 15 17 16 | 0 -11 -18 -25 -21 -8 }}

Optimal tunings:
* WE: ~25/21 = 300.0805{{c}}, ~40/39 = 45.2812{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~40/39 = 45.2517{{c}}

{{Optimal ET sequence|legend=0| 80, 132de, 212 }}

Badness (Sintel): 1.86

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 561/560, 1001/1000, 10648/10647

Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}

Optimal tunings:
* WE: ~25/21 = 300.1108{{c}}, ~40/39 = 45.2990{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~40/39 = 45.2604{{c}}

{{Optimal ET sequence|legend=0| 80, 132deg, 212g }}

Badness (Sintel): 1.59

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 325/324, 361/360, 561/560, 1001/1000, 1331/1330

Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}

Optimal tunings:
* WE: ~25/21 = 300.1212{{c}}, ~39/38 = 45.3015{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~39/38 = 45.2590{{c}}

{{Optimal ET sequence|legend=0| 80, 132degh, 212gh }}

Badness (Sintel): 1.36

== Quintiquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807, 390625000/387420489

{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}
: mapping generators: ~35721/31250, ~250/243

[[Optimal tuning]]s:
* [[WE]]: ~35721/31250 = 239.9950{{c}}, ~250/243 = 45.2520{{c}}
: [[error map]]: {{val| -0.025 +0.223 -0.925 +0.792 }}
* [[CWE]]: ~35721/31250 = 240.0000{{c}}, ~250/243 = 45.2546{{c}}
: error map: {{val| 0.000 +0.244 -0.897 +0.826 }}

{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d }}

[[Badness]] (Sintel): 9.04

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 390625000/387420489

Mapping: {{mapping| 5 10 15 18 19 | 0 -11 -18 -21 -9 }}

Optimal tunings:
* WE: ~8019/7000 = 239.9613{{c}}, ~250/243 = 45.2270{{c}}
* CWE: ~8019/7000 = 240.0000{{c}}, ~250/243 = 45.2460{{c}}

{{Optimal ET sequence|legend=0| 80, 185c, 265, 345 }}

Badness (Sintel): 3.42

[[Category:Temperament families]]
[[Category:Quartonic family| ]] 
[[Category:Rank 2]]

Ozan Yarman

2026-08-03T05:39:02Z

FloraC: Fix typo; + YouTube and SoundCloud links

'''Ozan Yarman''' is a Turkish composer and musician trained in [[historical temperaments|Western classical music]] and with an interest in microtonality, especially in connection with [[Middle-Eastern music|traditional Turkish music]]. He received his doctorate in musicology in 2008, and became an associate professor of musicology and music theory in 2011.

== External links ==
* [http://www.ozanyarman.com/welcome.html Official website]
* [https://www.youtube.com/user/DrOzanYarman YouTube channel]
* [https://soundcloud.com/dr-oz SoundCloud profile]

{{DEFAULTSORT:Yarman, Ozan}}
[[Category:People]]
[[Category:Composers]]
[[Category:Musicians]]

{{Stub}}

Quartonic family

2026-08-03T05:32:41Z

FloraC: Switch to Sintel's badness, WE & CWE tunings (1/)

{{Technical data page}}
The '''quartonic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quartonic comma]] ({{monzo|legend=1| 3 -18 11 }}, [[ratio]]: 390 625 000 / 387 420 489).

== Quartonic ==
The name ''quartonic'' refers to the [[quartertone]], since a somewhat flattened quartertone (representing the triptolemaic chromatic semitone, i.e. [[porcupine comma]]) is the [[generator]] of this temperament. The [[ploidacot]] for the temperament is omega-hendecacot, with eleven generator steps giving the [[4/3|perfect fourth]]. It is a member of the [[schismic–Mercator equivalence continuum]], with equivalence number ''n'' = 11/2.

[[Subgroup]]: 2.3.5

[[Comma list]]: 390625000/387420489

{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
: mapping generators: ~2, ~250/243

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9709{{c}}, ~250/243 = 45.2318{{c}}
: [[error map]]: {{val| -0.029 +0.437 -0.574 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 45.2349{{c}}
: error map: {{val| 0.000 +0.461 -0.541 }}

{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}

[[Badness]] (Sintel): 2.75

=== Overview to extensions ===
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].
* [[10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[390625/388962]] or [[4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.

== Septimal quartonic ==
Septimal quartonic tempers out the [[orwellisma]], the [[octagar comma]], as well as the [[ragisma]], and may be described as the {{nowrap| 26 & 27 }} temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 4000/3969

{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1424{{c}}, ~36/35 = 45.1064{{c}}
: [[error map]]: {{val| -0.858 +0.160 -0.801 +3.069 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 45.1881{{c}}
: error map: {{val| 0.000 +0.976 +0.301 +5.234 }}

{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}

[[Badness]] (Sintel): 1.08

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 2200/2187

Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}

Optimal tunings:
* WE: ~2 = 1198.8787{{c}}, ~36/35 = 44.9994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.0863{{c}}

{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}

Badness (Sintel): 1.13

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 540/539

Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}

Optimal tunings:
* WE: ~2 = 1199.2991{{c}}, ~36/35 = 45.0539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.1049{{c}}

{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d }}

Badness (Sintel): 0.987

=== Quarto ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 864/847

Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}

Optimal tunings:
* WE: ~2 = 1198.3094{{c}}, ~36/35 = 45.0688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2329{{c}}

{{Optimal ET sequence|legend=0| 26, 27e, 53e, 80ee }}

Badness (Sintel): 1.38

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 245/242

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}

Optimal tunings:
* WE: ~2 = 1198.9315{{c}}, ~36/35 = 45.1650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2640{{c}}

{{Optimal ET sequence|legend=0| 26, 27e, 53e }}

Badness (Sintel): 1.14

=== Quartz ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 4000/3969

Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}

Optimal tunings:
* WE: ~2 = 1200.5082{{c}}, ~33/32 = 45.4047{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}

{{Optimal ET sequence|legend=0| 26, 53 }}

Badness (Sintel): 1.76

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 275/273, 385/384

Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}

Optimal tunings:
* WE: ~2 = 1200.5899{{c}}, ~33/32 = 45.4096{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}

{{Optimal ET sequence|legend=0| 26, 53 }}

Badness (Sintel): 1.19

=== Biquartonic ===
Subgroup: 2.3.5.7.11

Comma list: 1728/1715, 2420/2401, 2560/2541

Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
: mapping generators: ~99/70, ~36/35

Optimal tunings:
* WE: ~99/70 = 599.5389{{c}}, ~36/35 = 45.0947{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~36/35 = 45.1692{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 186de, 266dde }}

Badness (Sintel): 2.01

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 640/637

Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}

Optimal tunings:
* WE: ~55/39 = 599.6870{{c}}, ~40/39 = 45.1293{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~40/39 = 45.1789{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}

Badness (Sintel): 1.65

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539

Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}

Optimal tunings:
* WE: ~17/12 = 599.7706{{c}}, ~40/39 = 45.1427{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~40/39 = 45.1790{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}

Badness (Sintel): 1.43

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539, 400/399

Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}

Optimal tunings:
* WE: ~17/12 = 599.7848{{c}}, ~39/38 = 45.1288{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~39/38 = 45.1634{{c}}

{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}

Badness (Sintel): 1.30

=== Yarm ===
Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1728/1715, 4000/3969

Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}

Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0363{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0617{{c}}

{{Optimal ET sequence|legend=0| 79, 80, 239dd }}

Badness (Sintel): 3.30

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 640/637, 1331/1323

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}

Optimal tunings:
* WE: ~2 = 1199.5798{{c}}, ~100/99 = 15.0551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0681{{c}}

{{Optimal ET sequence|legend=0| 79, 80 }}

Badness (Sintel): 2.55

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 325/324, 561/560, 640/637, 850/847

Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}

Optimal tunings:
* WE: ~2 = 1199.6800{{c}}, ~100/99 = 15.0624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0705{{c}}

{{Optimal ET sequence|legend=0| 79, 80 }}

Badness (Sintel): 2.38

== Yarman I ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 244140625/243045684

{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~126/125 = 15.0714{{c}}

{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 398, 637 }}

[[Badness]] (Smith): 0.193315

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 | 0 -33 -54 -95 -43 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0724{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398, 637, 1035bd }}

Badness (Smith): 0.049170

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 1001/1000, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 4 | 0 -33 -54 -95 -43 -24 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0707{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f, 637ff }}

Badness (Smith): 0.040929

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 595/594, 1001/1000, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 4 4 | 0 -33 -54 -95 -43 -24 7 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0706{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f, 637ff }}

Badness (Smith): 0.031015

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 361/360, 364/363, 595/594, 969/968, 1001/1000

Mapping: {{mapping| 1 2 3 4 4 4 4 5 | 0 -33 -54 -95 -43 -24 7 -60 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0683{{c}}

{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 637ffh }}

Badness (Smith): 0.023193

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 325/324, 361/360, 364/363, 460/459, 507/506, 529/528, 760/759

Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 | 0 -33 -54 -95 -43 -24 7 -60 -38 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0676{{c}}

{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 637ffhi }}

Badness (Smith): 0.017682

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 325/324, 361/360, 364/363, 406/405, 460/459, 494/493, 507/506, 529/528

Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 6 | 0 -33 -54 -95 -43 -24 7 -60 -38 -91 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0667{{c}}

{{Optimal ET sequence|legend=0| 79dhj, 80, 159, 239 }}

Badness (Smith): 0.014289

== Yarman II ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5359375/5308416, 390625000/387420489

{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~875/864 = 15.0995{{c}}

{{Optimal ET sequence|legend=1| 79, 159 }}

[[Badness]] (Smith): 0.655487

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 78121827/77948684

Mapping: {{mapping| 1 2 3 2 4 | 0 -33 -54 64 -43 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0982{{c}}

{{Optimal ET sequence|legend=0| 79, 159 }}

Badness (Smith): 0.143477

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 85683/85184

Mapping: {{mapping| 1 2 3 2 4 4 | 0 -33 -54 64 -43 -24 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0952{{c}}

{{Optimal ET sequence|legend=0| 79, 159 }}

Badness (Smith): 0.068150

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 385/384, 1575/1573, 4928/4913

Mapping: {{mapping| 1 2 3 2 4 4 4 | 0 -33 -54 64 -43 -24 7 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0950{{c}}

{{Optimal ET sequence|legend=0| 79, 159 }}

Badness (Smith): 0.051019

== Tertiseptisix ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 390625000/387420489

{{Mapping|legend=1| 1 13 21 15 | 0 -44 -72 -47 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~875/729 = 311.308{{c}}

{{Optimal ET sequence|legend=1| 27, 131bccd, 158cd, 185c, 212, 239, 451 }}

[[Badness]] (Smith): 0.155952

== Triquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 117649/116640, 250047/250000

{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}

[[Optimal tuning]] ([[CTE]]): ~63/50 = 400.0000{{c}}, ~250/243 = 45.2083{{c}}

{{Optimal ET sequence|legend=1| 27, 105cd, 132d, 159, 186, 345d }}

[[Badness]] (Smith): 0.170062

== Quartiquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 390625/388962, 4802000/4782969

{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}

[[Optimal tuning]] ([[CTE]]): ~25/21 = 300.0000{{c}}, ~250/243 = 45.2411{{c}}

{{Optimal ET sequence|legend=1| 80, 132d, 212, 292, 504 }}

[[Badness]] (Smith): 0.199116

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 14641/14580

Mapping: {{mapping| 4 8 12 15 17 | 0 -11 -18 -25 -21 }}

Optimal tuning (CTE): ~25/21 = 300.0000{{c}}, ~77/75 = 45.2303{{c}}

{{Optimal ET sequence|legend=0| 80, 132de, 212, 292, 504e }}

Badness (Smith): 0.062450

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1001/1000, 1375/1372, 10648/10647

Mapping: {{mapping| 4 8 12 15 17 16 | 0 -11 -18 -25 -21 -8 }}

Optimal tuning (CTE): ~25/21 = 300.0000{{c}}, ~40/39 = 45.2243{{c}}

{{Optimal ET sequence|legend=0| 80, 132de, 212 }}

Badness (Smith): 0.045028

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 561/560, 1001/1000, 10648/10647

Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}

Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~40/39 = 45.218{{c}}

{{Optimal ET sequence|legend=0| 52cdeg, 80, 132deg, 212g }}

Badness (Smith): 0.0312

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 325/324, 361/360, 561/560, 1001/1000, 1331/1330

Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}

Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~39/38 = 45.210{{c}}

{{Optimal ET sequence|legend=0| 52cdegh, 80, 132degh, 212gh }}

Badness (Smith): 0.0224

== Quintiquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807, 390625000/387420489

{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}

[[Optimal tuning]] ([[CTE]]): ~35721/31250 = 240.0000{{c}}, ~250/243 = 45.2563{{c}}

{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d, 875cd }}

[[Badness]] (Smith): 0.357387

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 390625000/387420489

Mapping: {{mapping| 5 10 15 18 19 | 0 -11 -18 -21 -9 }}

Optimal tuning (CTE): ~8019/7000 = 240.0000{{c}}, ~250/243 = 45.2624{{c}}

{{Optimal ET sequence|legend=0| 80, 185c, 265, 610de, 875cde }}

Badness (Smith): 0.103496

[[Category:Temperament families]]
[[Category:Quartonic family| ]] 
[[Category:Rank 2]]

User:FloraC/Temperament extension issues

2026-08-03T05:01:54Z

FloraC:

{{Editable user page|Please add ideas and name proposals. }}

A list of issues with temperament extensions.

== Open issues ==
=== Rank-2 temperaments ===
; Alphatrident
: 7-limit → 11-limit
: 11-limit alphatrident should be decanonicalized. The tuning is not very representative of 7-limit alphatrident. Specifically, alphatrident has three 11-limit extensions: 176 & 229, 229 & 282, and 282 & 335d. The first and last are supported by 53 which means they are simpler, but they are not representative of the 7-limit optimum. The middle one is representative of the 7-limit optimum, but is very complex. Therefore, the situation is similar to slendric, where none of the extensions should be canon.
: Proposed solution:

; Diminished
: 7-limit → 11-limit
: 11-limit diminished should be decanonicalized. It flips the optimal tuning from flat to sharp of just.
: Proposed solution:

; Gammic
: 7-limit → 11-limit
: 11-limit gammic should be decanonicalized. It's a low-accuracy extension.
: Proposed solution:

; Grendel
: 11-limit → 13-limit
: 13-limit grendel should be decanonicalized. The tuning is not very representative of 11-limit grendel. Specifically, grendel has three 13-limit extensions: 121 & 152f, 152f & 183, and 183 & 214. The first and last are supported by 31 which means they are simpler, but they are not representative of the 11-limit optimum. The middle one is representative of the 11-limit optimum, but somewhat more complex than the last. Therefore, the situation is similar to slendric, where none of the extensions should be canon.
: Proposed solution:

; Hemiwürschmidt
: 7-limit → 11-limit
: 11-limit hemiwürschmidt should be decanonicalized. The 7-limit optimum is between 99 and 130. The 11-limit extension shifts it to somewhere between 130 and 161, and is a significant drop in accuracy. Hemiwürschmidt is also described as a 2.3.5.7.23-subgroup temperament, which is more natural.
: Proposed solution:

; Keemun
: 11-limit → 13-limit
: 13-limit keemun should be decanonicalized. Kema, which maps prime 13 to +14 steps, is the right extension to use, as in all kleismic temperaments. Its edo join is 15 & 19, while the current canonical 13-limit extension is 15f & 19, and 15 is better than 15f. Kema is additionally supported by patent 34, while the current canonical extension uses the much less accurate 34f. The mapping of prime 13 to -5 generators is inaccurate and in the opposite direction compared to primes 3, 5, and 7, making other intervals of 13 more complex.
: Proposed solution:

; Lunatic
: 7-limit → 11-limit
: 11-limit lunatic should be decanonicalized. Lunatic has at least three extensions: 205 & 323, 323 & 441, and 441 & 559. Perhaps, the canonicity of 441 & 559 could be argued on the basis that the 7-limit optimum is very close to 441; however, compared to the good efficiency and portability of the 7-limit temperament, this 11-limit extension is pretty mid.
: Proposed solution:

; Neptune
: 7-limit → 11-limit
: 11-limit neptune should be decanonicalized. It's a low-accuracy extension. Neptune is also described as a 2.3.5.7.17-subgroup temperament, which is more natural. However, the 2.3.5.7.17.31-subgroup extension is a significant drop in accuracy, so that one should at least be dual-named, like ''triceneptune''.
: Proposed solution:

; Pajara
: 11-limit → 13-limit
: 13-limit pajara should be decanonicalized. It's way too inaccurate. Pajara should be a no-13 17-limit temperament.
: Proposed solution:

; Parakleismic
: 7-limit → 11-limit
: 11-limit parakleismic should be decanonicalized. Extending parakleismic to the 11- and 13-limit is complicated, mainly cuz both 99edo and 118edo have ambiguous interpretations, but paralytic (99e & 118) is the "main" extension here, since septimal parakleismic is really an interpolation between 99 and 118, with 217 giving a good middle ground. Note that 217 is more accurate than both 99 and 118 in the 7-limit, showing that the optimum isn't very near 99. Paralytic uses the 19e val, which is reasonable, and the other extensions zigzag around this segment: 80 & 99e, called paradigmic, 118 & 137d, which is our "undecimal parakleismic" tho the optimum is sharp of 87\118, making it 99 & 118, and finally 61de & 80, called parkleismic, tho the optimum is flat of 59\80, making it 80 & 99.
: Paralytic has four notable 13-limit extensions: 99ef & 118, 99e & 118, 99ef & 118f, and 99e & 118f. Note that 99e & 118 and 99ef & 118f are the more accurate ones, supported by 217, and the other two are less accurate, but use the simpler 19e-form in the 13-limit. The 13-odd-limit Graham complexities of 99e & 118 and 99ef & 118f are 157 and 142, respectively, showing that 99e & 118 isn't simpler than 99ef & 118f; thus 13-limit paralytic needs a distinct name too.
: Paradigmic has two notable 13-limit extensions: 80 & 99e and 80 & 99ef, and like above, 80 & 99e is the simpler 19-form whereas 80 & 99ef is more accurate. 80 & 99ef also supports 13-limit parapyth, which is an extra point. It follows that 13-limit paradigmic needs a distinct name too.
: Proposed solution:

; Secund
: 11-limit → 13-limit
: 13-limit secund should be decanonicalized, for the other extension is obviously way better.
: Proposed solution: delete 13-limit secund. Secundly → 13-limit secund. Status: pending community reactions.

; Tertiaseptal
: 7-limit → 11-limit
: 11-limit tertiaseptal should be decanonicalized. Tertiaseptal has three 11-limit extensions: 109 & 140, 140 & 171, and 171 & 202. Perhaps, the canonicity of 171 & 202 could be argued on the basis that the 7-limit optimum is very close to 171; however, compared to the absolute supremacy of the 7-limit temperament, this 11-limit extension is plain bad. ''Tertia'' and ''tertiaseptia'' are bad-tasted names so we might wanna reconsider these names too.
: Proposed solution:

; Wesley
: 5-limit → 7-limit
: 7-limit wesley should be decanonicalized. The tuning is not very representative of 5-limit wesley. Roman (26 & 29) is most representative, even tho it's a little more complex.
: Proposed solution:

; Würschmidt
: 5-limit → 7-limit
: 7-limit würschmidt should be decanonicalized. Almost all the notable supporting equal temperaments are dual-7: 34, 65, 96, and 127, and that speaks something about it. Würschmidt is also described as a 2.3.5.23-subgroup temperament, which is more natural.
: Proposed solution:

; Schismatic family
: Various dubious extensions. See [[Talk: Schismatic family #Extensions]]. Pending public reactions.

; Tetracot family
: Various dubious extensions to the 17-limit. See [[Talk: Tetracot family #17- and 19-limit extensions]]. Pending public reactions.

=== Rank-3 temperaments ===
; Agni
: 11-limit → 13-limit
: 13-limit agni should be decanonicalized. There's no reason to favor 31 & 72 over 41 & 72, even tho the latter has a little higher badness. Proposing to rename it to pavaka, after another name of Agni.
: Proposed solution: 13-limit agni → pavaka. Status: pending community reactions.

; Artemis
: 11-limit → 13-limit
: 13-limit artemis should be decanonicalized, as diana is a very competitive extension. This is a series of awkward cases related to orwell, which doesn't have a simple and accurate 13-limit extension, but since the more accurate one, 31 & 53, was dubbed tridecimal orwell over the simpler 22f & 31, these rank-3 temperaments should not be utterly at odds with that. Related temperaments include guanyin and zeus, both go with the 31 & 53 path.
: Proposed solution:

; Big brother
: 11-limit → 13-limit
: 13-limit big brother should be decanonicalized. See the case with artemis.
: Proposed solution:

; Freya
: 11-limit → 13-limit
: 13-limit freya should be decanonicalized. 11-limit freya has three extensions: 31 & 41 & 270, 31 & 72 & 270, and 41 & 72 & 270. 31 & 41 & 270 is a pretty neutral choice, but is also the least efficient out of all three.
: Proposed solution:

== Closed issues ==
=== Rank-2 temperaments ===
; Augmented/augene
: 5-limit → 7-limit
: 7-limit augene should be canonicalized while 13-limit augene should be decanonicalized. Tempering out 126/125 is natural for diaschismic, augmented and diminished, and since both diaschismic and diminished have canonical 7-limit extensions, so should augmented (even tho septimal diaschismic is essentially grandfathered in). Furthermore, the 11-limit extension should remain as canon, for the 11-odd-limit diamond monotone range is the same as 9-odd-limit. The name ''augene'' may remain as an alias for this temperament, but the 13-limit extension should eventually have its name changed cuz that one is much worse.
: Proposed solution: 7-, 11-, and 13-limit augene → augmented (augene). Status: changed.

; Dicot
: 5-limit → 7-limit
: 7-limit dicot should be decanonicalized. It has been clear that it's difficult to discuss these as a single temperament. The 7-limit extension is in fact the best extension of 2.3.7-subgroup {54/49} (in the same way beep is the best extension of bug), known as ''mujannabic'', so it may inherit that name.
: Proposed solution: 7- and 11-limit dicot → mujannabic. Status: changed.

; Ennealimmal
: 7-limit → 11-limit → 13-limit
: 11-limit ennealimmal should be decanonicalized. Further, 13-limit ennealimmal should be decanonicalized with respect to the 11-limit to make way for the other extension to be canon. Compared to the absolute supremacy of the 7-limit temperament, the 11-limit is mid, and the 13-limit is just utterly wrong. This should be obvious from the badness values and the optimal GPV sequences, but also the Graham complexities: 225 for 99e & 171e vs 189 for 99ef & 171ef.
: The names of the ennealimmal extensions demonstrate an absolute devoidness of creativity. The same morpheme combination ''ennealimma'' is re-used again and again, making all the extension names almost undistinguishable. Therefore I believe in rectifying these names we should inject new morphemes into the scene, while keeping the ''ennea-'' part to remind users of their relations. 11-limit ennealimmal (99e & 171e) tempers out the symbiotic comma, so it can be called ''enneabiotic''. Further, since 13-limit 99e & 171e is the worse extension, it can be ''enneabio''. The other high-accuracy 11-limit extension, 99 & 171, tempers out the olympic comma, so it can be called ''ennealympic''. Ennealimnic and ennealiminal were named earlier so I think they can stay.
: ''Semiennealimmal'' is extremely misleading. It tempers out the pine comma so it can be ''ennealimmapine''. This name is a little long but the temp itself isn't very notable anyway.
: Proposed solution: 11-limit ennealimmal, 13-limit ennealimmalis → enneabiotic; 13-, 17-, 19-limit ennealimmal → enneabio; ennealimmia → ennealympic; semiennealimmal → ennealimmapine. Status: changed.

; Hendecatonic
: 7-limit → 11-limit
: 11-limit hendecatonic should be decanonicalized. It's not better than 22 & 99e, known as ''cohendecatonic''.
: Proposed solution: 11-, 13- and 17-limit hendecatonic → hendecaton. Status: changed.

; Ketchup
: 17-limit → 19-limit
: 19-limit ketchup should be decanonicalized. Ketchup should be a no-19 23-limit temperament.
: Proposed solution: replace 19- and 23-limit ketchup with no-19 23-limit. Status: changed.

; Leapday
: 17-limit → 19-limit
: 19-limit leapday should be decanonicalized. Leapday should be a no-19 23-limit temperament.
: Proposed solution: replace 19- and 23-limit leapday and leapling with no-19 23-limit leapday. Status: changed.

; Magus
: 7-limit → 5-limit
: 5-limit magus should be decanonicalized. The 5-limit temperament's optimum is between 43edo and 46edo, which corresponds to the tuning range of amigo (43 & 46), not magus (46 & 49).
: Proposed solution: 5-limit magus → amigo. Status: suspended.

; Misty
: 7-limit → 11-limit
: 11-limit misty should be decanonicalized. Misty is more naturally a 2.3.5.7.17.19-subgroup temperament. Besides, 99 & 111, 99ef & 111, and 87 & 99 are all competitive extensions. Proposing these extensions as murky (87 & 99ef), smoky (87 & 99), hazy (99 & 111), and ashy (99ef & 111).
: Proposed solution: 11-, 13-, 17-, and 19-limit misty → murky, delete mystic (87 & 99e). Status: changed.

; Nessafof
: 7-limit → 11-limit
: 11-limit nessafof should be decanonicalized. Extending nessafof to the 11-limit is complicated since both 84edo and 99edo have ambiguous interpretations (see the case with parakleismic). Currently we have catalogued the two low-complexity 15-form extensions, which makes sense. One is being called ''nessa'' so the other may simply be ''fof''.
: Proposed solution: 11-limit nessafof → fof. Status: changed.

=== Rank-3 temperaments ===
; Aphrodite
: 11-limit → 13-limit
: 13-limit aphrodite should be decanonicalized. It's the worst of the four catalogued extensions. Eros is perhaps the best extension, and based on that, it would make sense to name the other extensions after the Erotes (Eros, Anteros, Himeros, Pothos). However, the other extensions are named after the equivalent deities of Aphrodite in other mythologies, so if we follow that pattern, the temperament in question may be named astarte, after Phoenician goddess Astarte.
: Proposed solution: 13-limit aphrodite → astarte. Status: changed.

; Jubilismic/jubilee
: 7-limit → 11-limit
: Jubilee should be canonicalized. This is obvious. We should spare the name ''jubilee'' for the 2.5.7-subgroup temp, mirroring ''semaphore''.
: Proposed solution: 11-limit jubilee → jubilismic; 2.5.7-subgroup jubilic → jubilee. Status: changed/suspended.

; Laka
: 13-limit → 17-limit
: 17-limit laka should be decanonicalized. Laka should be a no-17 19-limit temperament.
: Proposed solution: delete 17-limit laka. Status: changed.

; Marvel
: 11-limit → 13-limit
: 13-limit marvel should be decanonicalized. Hecate is a very competitive extension, and based on that one could consider naming the temperament in question after Greek deities such as Helios, Hebe, Hermes, or Hephaestus.
: Proposed solution: 13-limit marvel → helios. Status: changed.

; Starling
: 7-limit → 11-limit
: 11-limit starling should be decanonicalized. Thrush is a better extension.
: Proposed solution: undecimal starling → starnova. Status: changed.

Wesley family

2026-08-03T05:00:07Z

FloraC: Switch to Sintel's badness, WE & CWE tunings

{{Technical data page}}
The '''wesley family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] 78125/73728, the [[wesley comma]]. The wesley comma is unchanged in [[Wesley Woolhouse]]'s 7/26-comma meantone – it is an [[eigenmonzo|eigenmonzo (i.e. unchanged-interval)]], which [[Gene Ward Smith]] has talked about<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12793.html Yahoo! Tuning Group | ''26et and meantone projections'']</ref>.

== Wesley ==
The generator of wesley is one half of a [[~]][[5/2]], around 786 [[cent]]s, seven of which after [[octave reduction]] reach the [[3/2|perfect fifth]]. The [[ploidacot]] signature of this temperament is delta-heptacot. A good tuning for the generator may be found between [[26edo|17\26]] and [[29edo|19\29]].

[[Subgroup]]: 2.3.5

[[Comma list]]: 78125/73728

{{Mapping|legend=1| 1 -3 1 | 0 7 2 }}
: mapping generators: ~2, ~192/125

[[Optimal tuning]]s:
* [[WE]] ~2 = 1202.9225{{c}}, ~192/125 = 787.4042{{c}}
: [[error map]]: {{val| +2.922 +1.107 -8.583 }}
* [[CWE]] ~2 = 1200.0000{{c}}, ~192/125 = 785.7808{{c}}
: error map: {{val| 0.000 -1.490 -14.752 }}

{{Optimal ET sequence|legend=1| 3, …, 23, 26, 29, 55c }}

[[Badness]] (Sintel): 5.81

== Septimal wesley ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 405/392, 875/864

{{Mapping|legend=1| 1 -3 1 -7 | 0 7 2 15 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.6434{{c}}, ~14/9 = 786.8631{{c}}
: [[error map]]: {{val| +3.634 -4.844 -8.944 +8.617 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 784.7754{{c}}
: error map: {{val| 0.000 -8.527 -16.763 +2.805 }}

{{Optimal ET sequence|legend=1| 3d, …, 23d, 26 }}

[[Badness]] (Sintel): 2.41

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 875/864

Mapping: {{mapping| 1 -3 1 -7 -7 | 0 7 2 15 16 }}

Optimal tunings:
* WE: ~2 = 1203.6405{{c}}, ~11/7 = 786.6101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 784.5242{{c}}

{{Optimal ET sequence|legend=0| 3de, …, 23de, 26 }}

Badness (Sintel): 1.62

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 99/98, 325/324

Mapping: {{mapping| 1 -3 1 -7 -7 -12 | 0 7 2 15 16 24 }}

Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~11/7 = 786.5973{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 784.5923{{c}}

{{Optimal ET sequence|legend=0| 3def, 23deff, 26 }}

Badness (Sintel): 1.59

== Snipes ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 6125/5832

{{Mapping|legend=1| 1 -3 1 -9 | 0 7 2 18 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.9268{{c}}, ~54/35 = 787.7496{{c}}
: [[error map]]: {{val| +1.927 +6.511 -8.888 -6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/35 = 786.6189{{c}}
: error map: {{val| 0.000 +4.377 -13.076 -9.686 }}

{{Optimal ET sequence|legend=1| 3d, …, 26d, 29 }}

[[Badness]] (Sintel): 2.98

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 225/224, 245/242

Mapping: {{mapping| 1 -3 1 -9 -9 | 0 7 2 18 19 }}

Optimal tunings:
* WE: ~2 = 1201.9201{{c}}, ~11/7 = 787.7690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 786.6402{{c}}

{{Optimal ET sequence|legend=0| 3de, …, 26de, 29 }}

Badness (Sintel): 1.79

== Roman ==
Roman tempers out 525/512, the [[avicennma]]. The 7-limit version has also been proposed as ''crusher''.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 525/512, 3125/3024

{{Mapping|legend=1| 1 -3 1 10 | 0 7 2 -11 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.0258{{c}}, ~63/40 = 787.4289{{c}}
: [[error map]]: {{val| +3.026 +0.970 -8.430 -0.286 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~63/40 = 785.4717{{c}}
: error map: {{val| 0.000 -3.653 -15.370 -9.0145 }}

{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}

[[Badness]] (Sintel): 2.87

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 525/512

Mapping: {{mapping| 1 -3 1 10 10 | 0 7 2 -11 -10 }}

Optimal tunings:
* WE: ~2 = 1202.7762{{c}}, ~11/7 = 787.3465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 785.5130{{c}}

{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55c }}

Badness (Sintel): 1.75

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 100/99, 105/104, 245/242

Mapping: {{mapping| 1 -3 1 10 10 5 | 0 7 2 -11 -10 -2 }}

Optimal tunings:
* WE: ~2 = 1202.8327{{c}}, ~11/7 = 787.3820{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 785.5085{{c}}

{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55cf }}

Badness (Sintel): 1.24

== Dubbla ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49, 78125/73728

{{Mapping|legend=1| 2 1 4 5 | 0 7 2 2 }}
: mapping generators: ~7/5, ~192/175

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.8055{{c}}, ~192/175 = 185.9882{{c}}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~192/175 = 185.8808{{c}}

{{Optimal ET sequence|legend=1| 6, 20b, 26, 58c }}

[[Badness]] (Sintel): 4.60

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 125/121, 1344/1331

Mapping: {{mapping| 2 1 4 5 6 | 0 7 2 2 3 }}

Optimal tunings:
* WE: ~7/5 = 600.3702{{c}}, ~11/10 = 185.9936{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~11/10 = 185.9408{{c}}

{{Optimal ET sequence|legend=0| 6, 20b, 26 }}

Badness (Sintel): 2.51

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 105/104, 125/121, 144/143

Mapping: {{mapping| 2 1 4 5 6 4 | 0 7 2 2 3 11 }}

Optimal tunings:
* WE: ~7/5 = 600.3563{{c}}, ~11/10 = 185.8121{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~11/10 = 185.7623{{c}}

{{Optimal ET sequence|legend=0| 6f, 20bff, 26 }}

Badness (Sintel): 2.05

== References ==

[[Category:Temperament families]]
[[Category:Wesley family| ]] 
[[Category:Rank 2]]

5120/5103

2026-08-03T04:30:18Z

FloraC: /* Temperaments */

{{Infobox Interval
| Name = aberschisma
| Color name = sry1, saruyo 1sn, sryM, saruyoma
| Comma = yes
}}
'''5120/5103''', the '''aberschisma''' (formerly ''hemifamity comma''), is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 5.76 [[cent]]s. It is the difference between [[7/5]] (small septimal tritone) and [[1024/729]] (Pythagorean diminished fifth), or between [[10/7]] (large septimal tritone) and [[729/512]] (Pythagorean augmented fourth). It is also the difference between [[15/14]] and [[2187/2048]], between [[21/20]] and [[256/243]], between [[80/63]] and [[81/64]], etc., hence, between the septimal comma of [[64/63]] and the syntonic comma of [[81/80]], as well as between [[225/224]], the marvel comma, and [[32805/32768]], the schisma.

== Temperaments ==
[[Tempering out]] this comma in the 7-limit leads to the rank-3 [[aberschismic]] temperament, which splits the septimal quartertone [[36/35]] into two equal parts, each representing 81/80~64/63. Rank-2 temperaments that temper it out include [[hemififths]], [[amity]], and [[garibaldi]] to name three. [[Edo]]s that temper it out include {{EDOs| 12, 29, 34, 41, 46, 53, 58, 87, 94, 99, 111, 140, 145, 152, 239, 292, 391 }}, etc.

See [[Aberschismic family]] for the [[family]] of rank-3 temperaments where it is tempered out. See [[Aberschismic temperaments]] for a collection of rank-2 temperaments where it is tempered out.

== Sagittal notation ==
In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented by the sagittal {{sagittal | !( }} and is called the '''7/5 kleisma''', or '''7/5k''' for short, because the simplest interval it notates is 7/5, as for example in B–F{{nbhsp}}{{sagittal | !( }}. The upward version is called '''5/7k''' or '''7/5k up''' and is represented by {{sagittal| |( }}.

== Etymology ==
The name ''aberschisma'' was coined by [[Tristan Bay]] in 2024 in reference to [[groundfault]]'s [[aberrismic theory]], since 5120/5103 is the difference between two of the most common aberrismas in just intonation (81/80 and 64/63). By 2026, this is the most widely used name for the comma.

The formerly official name ''hemifamity'' was given by [[Gene Ward Smith]] in 2005 as a contraction of ''hemififths'' and ''amity''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12900.html Yahoo! Tuning Group | ''Seven limit comma names from pairs of temperament names'']</ref>.

In 2025, [[Flora Canou|FloraC]] et al. gave this comma the name ''argent comma'' due to the fifth size from the [[argent tuning]] being essentially optimal for tuning the full 7-limit and 2.3.7/5-subgroup temperament that temper only it out. While not gaining community-wide support over ''aberschisma'', the name has since been reused for the 2.3.7/5-subgroup temperament in the form ''argentic''.

== References ==

[[Category:Aberschismic]]

Family of temperaments

2026-08-03T04:29:44Z

FloraC: Redirected page to Temperament families and clans

#redirect [[Temperament families and clans]]

Vishnu comma

2026-08-03T04:25:18Z

FloraC: /* Temperaments */ link on *family of temperaments*

{{Infobox Interval
| Ratio = 6115295232/6103515625
| Name = vishnu comma, vishnuzma, semisuper comma
| Color name = sg144, sasepbigu 4th
| Comma = yes
}}
The '''vishnu comma''', '''vishnuzma''' or '''semisuper comma''' ([[ratio]]: 6 115 295 232 / 6 103 515 625, [[monzo]]: {{monzo| 23 6 -14 }}), is an [[unnoticeable comma|unnoticeable]] [[5-limit]] [[comma]] measuring about 3.34 [[cent]]s. It is the amount by which a stack of seven [[25/24|classical chromatic semitones]] falls short of a [[4/3|just perfect fourth]], in other words the interval (4/3)/(25/24)7. Equivalently, it is the amount by which a stack of fourteen [[5/4|classical major thirds]] misses the [[729/512|Pythagorean augmented fourth]].

== Temperaments ==
[[Tempering out]] this comma leads to the 5-limit [[vishnu]] temperament.

See [[Vishnu family]] for the [[family of temperaments]] where it is tempered out.

== Etymology ==
This comma was first named by [[Gene Ward Smith]] in 2001 in terms of the corresponding temperament, ''semisuper'', for one possible generator of the temperament was a "semisuper fourth"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2121.html Yahoo! Tuning Group | ''Shismic & co'']</ref>. ''Semisuper comma'' appeared shortly after<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_5025.html Yahoo! Tuning Group | ''5-limit comma names'']</ref>. It is not clear yet how and why the later names ''vishnu'' (for the temperament) and ''vishnuzma'' (for the comma) were added.

== References ==

[[Category:Vishnu]]
[[Category:Commas with unknown etymology]]

Quartonic comma

2026-08-03T04:24:06Z

FloraC: Cleanup

{{Infobox Interval
| Name = Quartonic comma
| Monzo = 3 -18 11
| Comma = yes
}}
The '''quartonic comma''' ({{monzo|legend=1| 3 -18 11 }}, [[ratio]]: 390 625 000 / 387 420 489) is a [[small comma|small]] [[5-limit]] [[comma]] of about 14.3 [[cent]]s.

== Temperaments ==
[[Tempering out]] this comma in the 5-limit defines the [[quartonic]] temperament.

See [[Quartonic family]] for the [[family of temperaments]] where it is tempered out.

{{todo|expand}}
[[Category:Commas named for their regular temperament properties]]

256/243

2026-08-03T04:24:01Z

FloraC: /* Temperaments */ link on *family of temperaments*

{{interwiki
| de = 256/243
| en = 256/243
| es =
| ja =
}}
{{Infobox Interval
| Name = Pythagorean limma, Pythagorean diatonic semitone, blackwood comma
| Color name = sw2, sawa 2nd
| Sound = jid_256_243_pluck_adu_dr220.mp3
| Comma = yes
}}
{{Wikipedia| Semitone #Pythagorean tuning }}

'''256/243''', the '''Pythagorean limma''' or '''Pythagorean diatonic semitone''', is the [[diatonic semitone]] in [[Pythagorean tuning]]. In other words, it is the [[3-limit]] minor second. It factors as 28/35, and is about 90.2 [[cent]]s in size. It can be generated by stacking five [[4/3]] just perfect fourths and [[octave reduction|octave-reducing]] the resulting interval, or equivalently by decreasing 4/3 by two [[9/8]] major seconds. Unlike the situation in [[meantone]] tunings, it is smaller, not larger, than the corresponding [[chromatic semitone]], which is the Pythagorean augmented unison of [[2187/2048]].

== Approximation ==
This interval is well approximated by any tuning generated with accurate octaves and fifths. For example, [[53edo|4\53]] is a very good approximation.

== Temperaments ==
When this ratio is taken as a comma to be tempered in the [[5-limit]], it produces the [[blackwood]] temperament, and it may be called the '''blackwood comma''', named after [[Easley Blackwood Jr]]. Edos tempering it out include [[5edo]], [[10edo]], [[15edo]], [[20edo]], [[25edo]] and [[30edo]].

See [[Blackwood family]] for the [[family of temperaments]] where it is tempered out.

== Notation ==
In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the limma is represented by the distances between B and C, as well as between E and F.

The scale is structured with the following step pattern:
* A to B: [[9/8|whole tone]]
* B to C: [[256/243|limma]]
* C to D: [[9/8|whole tone]]
* D to E: [[9/8|whole tone]]
* E to F: [[256/243|limma]]
* F to G: [[9/8|whole tone]]
* G to A: [[9/8|whole tone]]
This pattern highlights the placement of the limma intervals between the note pairs above, distinguishing them from the [[9/8|whole tone]] that occur between the other note pairs.

== See also ==
* [[243/128]] – its [[octave complement]]
* [[729/512]] – its [[fifth complement]]
* [[16/15]] – the classic (5-limit) diatonic semitone
* [[Gallery of just intervals]]
* [[Medium comma]]
* [[Pythagorean tuning]]

[[Category:Second]]
[[Category:Semitone]]
[[Category:Blackwood]]
[[Category:Commas named after composers]]

2.3.5.7.11.13.19 subgroup

2026-08-03T04:12:54Z

FloraC: Explain the significance

The '''2.3.5.7.11.13.19 subgroup''' (a.k.a. ''yazalathana'' in [[color notation]]) consists of [[just intonation]] [[interval]]s such that the highest [[prime factor]] in all [[ratio]]s is 19, but without 17. It is thus a subset of the [[19-limit]], or alternatively, it can be seen as the [[13-limit]] with an extra prime [[19/1|19]].

This subgroup is a [[rank and codimension|rank-7]] system, and can be modeled in a 6-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], [[13/1|13]] and [[19/1|19]] represented by each dimension. The prime [[2/1|2]] does not appear in typical lattices because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a seventh dimension is needed.

This subgroup is significant because 19 mirrors [[21/1|21]] in the 16::24 [[harmonic series segment]], and 21 is already present in the 13-limit; thus any fifth-bounded chord involving 21 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 19 in this subgroup. Meanwhile, primes [[17/1|17]] and [[23/1|23]] may be considered to clash with the fundamental, being a semitone and a tritone when [[octave reduction|octave reduced]], so people may wish to exclude them. Therefore, the subgroup can be considered to complete the harmonic series up to the lower half of the fifth octave without the difficult ones near the edges. The same reasons will also give rise to the [[2.3.5.7.11.13.19.29 subgroup]] as an [[expansion]].

== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
[[Edo]]s which represents the subgroup better ([[monotonic]] in the no-17 [[19-odd-limit]] and decreasing [[TE error]]): {{EDOs|'''27e''', 31, 34dh, 38df, 41f, '''41''', 50, '''53''', 58h, '''72''', 87, 94, 103h, 111, 121, '''130''', '''152f''', 190, 217, 224, '''270''', 552, 581, … }} and so on. Bold edos are records of [[Tenney–Euclidean temperament measures #TE simple badness|TE relative error]].

{{Note|[[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.}}

[[270edo]] is arguably one of the best equal temperaments for this subgroup, achieving a record of [[relative error]] that no other equal temperament of its grain comes close to achieving. The next best ones are in the thousands of divisions: [[2190edo|2190]], [[6079edo|6079]], [[8269edo|8269]] and [[8539edo|8539]]. The last two coincidentally differ by 270 and are prime edos.

=== Rank-2 temperaments ===
[[Cassandra|Cassandra (41 & 53)]] provides a very intuitive approximation to this subgroup using the [[chain of fifths]], naturally mapping 19/16 to the minor third, and equating together the [[Pythagorean comma|pythagorean]], [[Septimal comma|septimal]], and [[syntonic comma]] into one generic comma, that doubled approximates 33/32~1053/1024. It is well represented with [[41edo]] and [[53edo]], though [[94edo]] is more optimized.

For those searching higher-accuracy temperaments, [[cotoneum|cotoneum (41 & 217)]] keeps the chain of fifths and a pyth-septimal comma, but does not temper out the [[schisma]], instead equating it with the [[41-comma]]. [[newt|Newt (41 & 270)]] halves the fifth (tempering out [[2401/2400]]) and finds the [[aberschisma]] -41 hemififths away with much more efficiency. Another similar temperament is [[gariwizmic|gariwizmic (94 & 270)]], which instead of halving the fifth, halves the octave and finds the [[aberschisma]] at +53 fifths -1/2 pythcomma.

Other non-chain-of-fifths temperaments that are good candidates for the subgroup include [[vulture|vulture (53 & 217)]], [[satin|satin (94 & 217)]], and [[paramity|paramity (53 & 311)]].

=== Rank-3 temperaments ===
[[Cassaschismic]] is the union of all the rank-2 temperaments discussed above, relates several [[formal comma]]s in this subgroup to reduce them to essentially a generic comma and a generic aberschisma, making it significant for notation systems based on the diatonic chain of fifths. Other temperaments that achieve a similar level of accuracy include [[lif]] and [[eir]].

[[Category:Just intonation subgroups|#]]
[[Category:19-limit|#]]

Root

2026-08-03T04:09:27Z

FloraC: - typo

:''This page is about the root of a chord. For the ''n''-th root of an interval, see [[Radical interval]]. For the tonic of a scale, occasionally called its 'root', see [[Tonic]].''
{{Wikipedia|Root (chord)}}
The '''root''' is a specific note that names and characterizes a given chord. Chords are often spoken about in terms of their root, their quality, and their extensions. The root of the chord often appears in the bass, although this is not always the case. When the chord is voiced such that the root is the lowest note, it is said to be in ''root position''.

== In tertian harmony ==
If the chord can be voiced as a stack of thirds, then the root is typically the lowest note in the stack.

== In other types of harmony ==
If a chord is not composed of thirds, the root can be ambiguous. The chord may still be given a root, although there is no established procedure for doing so.

== See also ==
* [[Rooted interval]]
* [[Uprooted interval]]

{{Stub}}
{{Todo| add examples }}

Olympian

2026-08-03T03:45:35Z

FloraC: Redirected page to Olympic clan#Olympian

#redirect [[Olympic clan #Olympian]]

[[Category:Rank-3 temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Olympic clan]]

2.3.7.11 subgroup

2026-08-03T03:43:08Z

FloraC: /* Regular temperaments */ expand

The '''2.3.7.11 subgroup''' ('''zala''' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where 2, 3, 7, and 11 are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 7, and 11. This is an infinite set and still infinite even if we restrict consideration to a single octave. Some examples within the [[octave]] include [[3/2]], [[7/4]], [[9/7]], [[21/16]], [[11/9]], [[22/21]], and so on.

The 2.3.7.11 subgroup is a retraction of the [[11-limit]], obtained by removing prime 5. Its simplest expansion is the 2.3.7.11.13 subgroup, which adds prime 13. It can also be retracted to the [[2.3.7 subgroup]] by removing prime 11.

A notable subset of the 2.3.7.11 subgroup is the {1, 3, 7, 9, 11} [[tonality diamond]], comprising all intervals in which 1, 3, 7, 9, and 11 are the only allowable odd numbers, once all powers of 2 are removed, either for the intervals of the scale or the ratios between successive or simultaneously sounding notes of the composition. The complete list of intervals in this tonality diamond within the octave is [[1/1]], [[12/11]], [[9/8]], [[8/7]], [[7/6]], [[11/9]], [[14/11]], [[9/7]], [[4/3]], [[11/8]], [[16/11]], [[3/2]], [[14/9]], [[11/7]], [[18/11]], [[12/7]], [[7/4]], [[16/9]], [[11/6]], and [[2/1]].

When [[octave equivalence]] is assumed, an interval can be taken as representing that interval in every possible voicing. This leaves primes 3, 7, and 11, which can be represented in a 3-dimensional [[lattice diagram]], each prime represented by a different dimension, such that each point on the lattice represents a different [[interval class]].

== Scales ==
* Genus(3*7*11) 12/11, 14/11, 21/16, 16/11, 3/2, 7/4, 21/11, 2
* Genus(3*7^2*11) 49/48, 49/44, 7/6, 14/11, 4/3, 16/11, 49/33, 49/32, 56/33, 7/4, 64/33, 2
* Ptolemy's Intense Chromatic at 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1 (disjunct form)
* A conjunct Rast: 1/1 9/8 27/22 4/3 3/2 11/8 16/9 2/1 (note: actually 2.3.11 subgroup)
* Qutb al-Din al-Shirazi's version of Hijaz: 1/1-12/11-14/11-4/3 (12:11-7:6-22:21), itself a permutation of Ptolemy's Intense Chromatic
** [[Margo Schulter]] notes: "A modern form of Maqam Hijaz based on this tuning might be 1/1-12/11-14/11-4/3-3/2-18/11-16/9-2/1 ascending, and 1/1-12/11-14/11-4/3-3/2-128/81-16/9-2/1 descending (with the minor sixth maybe a bit smaller, say 11/7 or the like)."
* A septimal flavor of Sazkar, which has a minor third above the 1/1 ascending but a tone descending: thus ascending 1/1-7/6-11/9-4/3-3/2-27/16-11/6-2/1, and descending 1/1-9/8-11/9-4/3-3/2-27/16-11/6-2/1
** Margo Schulter adds: "I should caution that an Arab Rast, of which Sazkar is an offshoot, might usually have a Zalzalian third more like 27/22 or 16/13 rather than 11/9, so this may be more of a new tuning than a traditional Arab Sazkar."
* The 1-3-7-9-11 [[Combination product set|2(5 Dekany]] within a 12-tone [[Periodic_scale#Constant Structure|constant structure]] (see [http://anaphoria.com/dekanyconstantstructures.pdf link]): 1*3, 9*11, 3*9, 1*7, 3*7*11, 7*9, 3*11, 1*9, 3*9*11, 7*11, 3*7, 1*11.
* (with prime 13 added) A 12f, {352/351, 364/363} 2.3.7.11.13 elf transversal: 28/27-9/8-13/11-9/7-4/3-11/8-3/2-14/9-22/13-16/9-27/14-2

== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
The 2.3.7.11 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''5''', 9, 10, 12, 14, '''17''', 31, '''41''', 58, 63, '''72''', 94, 118, 130, '''135''', 342, … }}

=== Rank-2 temperaments ===
[[Supra]], which extends [[archy]], provides a simple yet high-damage approximation to the subgroup. It is generated by a perfect fifth, tuned a little sharp so that two make [[8/7]][[~]][[9/8]] and six make [[16/11]], tempering out [[64/63]] and [[99/98]]. Alternatively, [[suhajira]] can be considered an extension of archy that adds neutral intervals, with the perfect fifth split into two neutral third generators, each representing [[11/9]]~[[27/22]], thus tempering out 64/63 and [[243/242]].

[[Skwares]] takes the 11/9~27/22 neutral third, adds an octave to it and splits it in halves for ~[[11/7]], tempering out 99/98 and 243/242.

[[Radon]], which adds prime 11 to [[slendric]] by tempering out [[896/891]], provides a more complex entry, well represented by 41edo and 46edo. A similar temperament at this level is [[hemif]], which tempers out 243/242 and 896/891 and can be tuned to 58edo. In both cases, the diatonic major third represents [[14/11]].

On the high-accuracy side, [[gary]] is an important temperament that finds 7 and 11 far into the [[chain of fifths]].

=== Rank-3 temperaments ===
[[Parapyth]] equates [[28/27]] with [[33/32]] and uses this interval as a spacer added to the chain of fifths to finds intervals of 7 and 11.

[[Symbiotian]], which makes 33/32 and 28/27 sum to the [[Pythagorean apotome]], is an efficient high-accuracy counterpart of parapyth.

[[Olympian]] equates 33/32 with a stack of two 64/63's. It is very accurate yet still easy to notate on the staff, using the septimal comma as spacer added to the chain of fifths to finds intervals of 7 and 11.

[[Category:Just intonation subgroups|#]]
[[Category:Rank-4 temperaments|#]]
[[Category:11-limit|#]]
[[Category:Lists of scales|#]]

36/1

2026-08-03T02:42:13Z

FloraC: Correction & improve linking

{{Mathematical interest}}
{{Infobox interval
| Name = 36th harmonic
}}
'''36/1''', the 36th harmonic, is the [[harmonic]] after [[35/1]] and before [[37/1]]. It is equal to a [[9/8|Pythagorean major second]] plus five [[octave]]s (9/8 × 25), or alternatively the [[6/1|6th harmonic]] stacked twice (62). It is a [[highly composite harmonic]], meaning it has dozens of possible makeups.

35/1

2026-08-03T02:37:14Z

FloraC: Improve formatting; use more consistent terms; + links to previous and next harmonics

{{Mathematical interest}}
{{Infobox interval
| Name = 35th harmonic
}}
'''35/1''', the 35th harmonic, is the [[harmonic]] after [[34/1]] and before [[36/1]]. It can be compounded in multiple ways: seven [[5/1|quintuples]] (7 × 5), five [[7/1|septuples]] (5 × 7), or a [[35/32|septimal neutral second]] plus five octaves (35/32 × 25).

34/1

2026-08-03T02:35:08Z

FloraC: + link

{{Mathematical interest}}
{{Infobox interval
| Name = 34th harmonic
}}
'''34/1''', the 34th harmonic, is the [[harmonic]] after [[33/1]] and before [[35/1]]. It is equal to a [[17/1|17th harmonic]] plus an [[octave]], or a large septendecimal semitone ([[17/16]]) plus five octaves.

33/1

2026-08-03T02:34:47Z

FloraC: Correction

{{Mathematical interest}}
{{Infobox interval
| Name = 33rd harmonic
| Color name = quincolo unison, c^51o1
}}
'''33/1''', the 33rd harmonic, is the [[harmonic]] after [[32/1]] and before [[34/1]]. It is equal to a stack consisting of a [[3/1|3rd harmonic]] and an [[11/1|11th harmonic]], or the undecimal quartertone ([[33/32]]) plus five [[octave]]s.

34/1

2026-08-03T02:33:03Z

FloraC: Correction. + links to previous and next harmonics

{{Mathematical interest}}
{{Infobox interval
| Name = 34th harmonic
}}
'''34/1''', the 34th harmonic, is the harmonic after [[33/1]] and before [[35/1]]. It is equal to a [[17/1|17th harmonic]] plus an [[octave]], or a large septendecimal semitone ([[17/16]]) plus five octaves.

17/8

2026-08-03T02:32:37Z

FloraC: Improve linking

{{Infobox Interval
| Name = septendecimal minor ninth, septendecimal augmented octave
| Color name = 17o9, iso 9th
}}
'''17/8''' is the distance between the 17th and the 8th harmonic. It spans the size of 1304.95541 [[cent]]s. It is essentially a large septendecimal semitone ([[17/16]]) but one [[octave]] higher.

This interval is a near-perfect approximation of the minor ninth in [[12edo]], with an error of only ~5 cents, stemming from 12edo's near-perfect approximation of the [[2.3.17.19 subgroup]].

Quintiquart

2026-08-02T15:55:13Z

FloraC: Recategorize

#REDIRECT [[Quartonic family #Quintiquart]]

[[Category:Rank-2 temperaments]]
[[Category:Quartonic family]]
[[Category:Canopic clan]]

Quartiquart

2026-08-02T15:54:13Z

FloraC: Recategorize

#REDIRECT [[Quartonic family #Quartiquart]]

[[Category:Rank-2 temperaments]]
[[Category:Quartonic family]]
[[Category:Dimcomp temperaments]]
[[Category:Canousmic temperaments]]

Triquart

2026-08-02T15:53:04Z

FloraC: Recategorize

#REDIRECT [[Quartonic family #Triquart]]

[[Category:Rank-2 temperaments]]
[[Category:Quartonic family]]
[[Category:Landscape microtemperaments]]

Tertiseptisix

2026-08-02T15:51:48Z

FloraC: Recategorize

#REDIRECT [[Quartonic family #Tertiseptisix]]

[[Category:Rank-2 temperaments]]
[[Category:Quartonic family]]
[[Category:Breedsmic temperaments]]

Yarman II

2026-08-02T15:50:33Z

FloraC: Update & + categories

#redirect [[Quartonic family #Yarman II]]

[[Category:Rank-2 temperaments]]
[[Category:Quartonic family]]

Yarman I

2026-08-02T15:50:02Z

FloraC: Update & + categories

#redirect [[Quartonic family #Yarman I]]

[[Category:Rank-2 temperaments]]
[[Category:Quartonic family]]
[[Category:Hemimage temperaments]]

Category:Quartonic family

2026-08-02T15:47:57Z

FloraC: + category

{{Cat main}}

[[Category:Quartonic]]
[[Category:Temperament families]]

Quartonic

2026-08-02T15:47:38Z

FloraC: Recategorize

#redirect [[Quartonic family #Quartonic]]

[[Category:Quartonic| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Quartonic family]]
[[Category:Orwellismic temperaments]]
[[Category:Octagar temperaments]]

Quartonic family

2026-08-02T15:46:52Z

FloraC: Units; misc. cleanup; - redundant category

{{Technical data page}}
The '''quartonic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quartonic comma]] ({{monzo|legend=1| 3 -18 11 }}, [[ratio]]: 390 625 000 / 387 420 489).

== Quartonic ==
The name ''quartonic'' means quarter-tone, which is the [[generator]] of this temperament.

[[Subgroup]]: 2.3.5

[[Comma list]]: 390625000/387420489

{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}

[[Optimal tuning]] ([[CTE]]): 2 = 1200.0000{{c}}, ~250/243 = 45.2368{{c}}

{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}

[[Badness]] (Smith): 0.117250

=== Overview to extensions ===
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].
* [[Hemimage comma|10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[Landscape comma|250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[Dimcomp comma|390625/388962]] or [[canousma|4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[Mirkwai comma|16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.

== Septimal quartonic ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 4000/3969

{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~36/35 = 45.2652{{c}}

{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}

[[Badness]] (Smith): 0.042632

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 2200/2187

Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1674{{c}}

{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}

Badness (Smith): 0.034031

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 540/539

Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1632{{c}}

{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d, 186d }}

Badness (Smith): 0.023875

=== Quarto ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 864/847

Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.4022{{c}}

{{Optimal ET sequence|legend=0| 26, 53e, 132ee }}

Badness (Smith): 0.041786

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 245/242

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.3857{{c}}

{{Optimal ET sequence|legend=0| 26, 53e, 132ee }}

Badness (Smith): 0.027692

=== Quartz ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 4000/3969

Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3313{{c}}

{{Optimal ET sequence|legend=0| 26, 53 }}

Badness (Smith): 0.053285

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 275/273, 385/384

Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3168{{c}}

{{Optimal ET sequence|legend=0| 26, 53 }}

Badness (Smith): 0.028818

=== Biquartonic ===
Subgroup: 2.3.5.7.11

Comma list: 1728/1715, 2420/2401, 2560/2541

Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}

Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~36/35 = 45.2678{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}

Badness (Smith): 0.060737

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 640/637

Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}

Optimal tuning (CTE): ~55/39 = 600.0000{{c}}, ~40/39 = 45.2544{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}

Badness (Smith): 0.039891

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539

Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}

Optimal tuning (CTE): ~17/12 = 600.0000{{c}}, ~34/33 = 45.2397{{c}}

{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}

Badness (Smith): 0.028112

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539, 400/399

Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}

Optimal tuning (CTE): ~17/12 = 600.000{{c}}, ~39/38 = 45.222{{c}}

{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}

Badness (Smith): 0.0213

=== Yarm ===
Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1728/1715, 4000/3969

Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0880{{c}}

{{Optimal ET sequence|legend=0| 79, 80, 159d }}

Badness (Smith): 0.099950

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 640/637, 1331/1323

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0842{{c}}

{{Optimal ET sequence|legend=0| 79, 80, 159d }}

Badness (Smith): 0.061645

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 325/324, 561/560, 640/637, 850/847

Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0840{{c}}

{{Optimal ET sequence|legend=0| 79, 80, 159d }}

Badness (Smith): 0.046718

== Yarman I ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 244140625/243045684

{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~126/125 = 15.0714{{c}}

{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 398, 637 }}

[[Badness]] (Smith): 0.193315

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 | 0 -33 -54 -95 -43 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0724{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398, 637, 1035bd }}

Badness (Smith): 0.049170

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 1001/1000, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 4 | 0 -33 -54 -95 -43 -24 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0707{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f, 637ff }}

Badness (Smith): 0.040929

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 595/594, 1001/1000, 10976/10935

Mapping: {{mapping| 1 2 3 4 4 4 4 | 0 -33 -54 -95 -43 -24 7 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0706{{c}}

{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f, 637ff }}

Badness (Smith): 0.031015

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 361/360, 364/363, 595/594, 969/968, 1001/1000

Mapping: {{mapping| 1 2 3 4 4 4 4 5 | 0 -33 -54 -95 -43 -24 7 -60 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0683{{c}}

{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 637ffh }}

Badness (Smith): 0.023193

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 325/324, 361/360, 364/363, 460/459, 507/506, 529/528, 760/759

Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 | 0 -33 -54 -95 -43 -24 7 -60 -38 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0676{{c}}

{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 637ffhi }}

Badness (Smith): 0.017682

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 325/324, 361/360, 364/363, 406/405, 460/459, 494/493, 507/506, 529/528

Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 6 | 0 -33 -54 -95 -43 -24 7 -60 -38 -91 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0667{{c}}

{{Optimal ET sequence|legend=0| 79dhj, 80, 159, 239 }}

Badness (Smith): 0.014289

== Yarman II ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5359375/5308416, 390625000/387420489

{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~875/864 = 15.0995{{c}}

{{Optimal ET sequence|legend=1| 79, 159 }}

[[Badness]] (Smith): 0.655487

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 78121827/77948684

Mapping: {{mapping| 1 2 3 2 4 | 0 -33 -54 64 -43 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0982{{c}}

{{Optimal ET sequence|legend=0| 79, 159 }}

Badness (Smith): 0.143477

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 85683/85184

Mapping: {{mapping| 1 2 3 2 4 4 | 0 -33 -54 64 -43 -24 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0952{{c}}

{{Optimal ET sequence|legend=0| 79, 159 }}

Badness (Smith): 0.068150

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 385/384, 1575/1573, 4928/4913

Mapping: {{mapping| 1 2 3 2 4 4 4 | 0 -33 -54 64 -43 -24 7 }}

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0950{{c}}

{{Optimal ET sequence|legend=0| 79, 159 }}

Badness (Smith): 0.051019

== Tertiseptisix ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 390625000/387420489

{{Mapping|legend=1| 1 13 21 15 | 0 -44 -72 -47 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~875/729 = 311.308{{c}}

{{Optimal ET sequence|legend=1| 27, 131bccd, 158cd, 185c, 212, 239, 451 }}

[[Badness]] (Smith): 0.155952

== Triquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 117649/116640, 250047/250000

{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}

[[Optimal tuning]] ([[CTE]]): ~63/50 = 400.0000{{c}}, ~250/243 = 45.2083{{c}}

{{Optimal ET sequence|legend=1| 27, 105cd, 132d, 159, 186, 345d }}

[[Badness]] (Smith): 0.170062

== Quartiquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 390625/388962, 4802000/4782969

{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}

[[Optimal tuning]] ([[CTE]]): ~25/21 = 300.0000{{c}}, ~250/243 = 45.2411{{c}}

{{Optimal ET sequence|legend=1| 80, 132d, 212, 292, 504 }}

[[Badness]] (Smith): 0.199116

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 14641/14580

Mapping: {{mapping| 4 8 12 15 17 | 0 -11 -18 -25 -21 }}

Optimal tuning (CTE): ~25/21 = 300.0000{{c}}, ~77/75 = 45.2303{{c}}

{{Optimal ET sequence|legend=0| 80, 132de, 212, 292, 504e }}

Badness (Smith): 0.062450

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1001/1000, 1375/1372, 10648/10647

Mapping: {{mapping| 4 8 12 15 17 16 | 0 -11 -18 -25 -21 -8 }}

Optimal tuning (CTE): ~25/21 = 300.0000{{c}}, ~40/39 = 45.2243{{c}}

{{Optimal ET sequence|legend=0| 80, 132de, 212 }}

Badness (Smith): 0.045028

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 561/560, 1001/1000, 10648/10647

Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}

Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~40/39 = 45.218{{c}}

{{Optimal ET sequence|legend=0| 52cdeg, 80, 132deg, 212g }}

Badness (Smith): 0.0312

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 325/324, 361/360, 561/560, 1001/1000, 1331/1330

Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}

Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~39/38 = 45.210{{c}}

{{Optimal ET sequence|legend=0| 52cdegh, 80, 132degh, 212gh }}

Badness (Smith): 0.0224

== Quintiquart ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807, 390625000/387420489

{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}

[[Optimal tuning]] ([[CTE]]): ~35721/31250 = 240.0000{{c}}, ~250/243 = 45.2563{{c}}

{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d, 875cd }}

[[Badness]] (Smith): 0.357387

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 390625000/387420489

Mapping: {{mapping| 5 10 15 18 19 | 0 -11 -18 -21 -9 }}

Optimal tuning (CTE): ~8019/7000 = 240.0000{{c}}, ~250/243 = 45.2624{{c}}

{{Optimal ET sequence|legend=0| 80, 185c, 265, 610de, 875cde }}

Badness (Smith): 0.103496

[[Category:Temperament families]]
[[Category:Quartonic family| ]] 
[[Category:Rank 2]]

Dubbla

2026-08-02T15:36:58Z

FloraC: Recategorize

#REDIRECT [[Wesley family #Dubbla]]

[[Category:Rank-2 temperaments]]
[[Category:Wesley family]]
[[Category:Jubilismic clan]]

Roman

2026-08-02T15:35:36Z

FloraC: Recategorize

#redirect [[Wesley family #Roman]]

[[Category:Rank-2 temperaments]]
[[Category:Wesley family]]
[[Category:Avicennmic temperaments]]
[[Category:Octagar temperaments]]

Snipes

2026-08-02T15:33:27Z

FloraC: Recategorize

#REDIRECT [[Wesley family #Snipes]]

[[Category:Rank-2 temperaments]]
[[Category:Wesley family]]
[[Category:Marvel temperaments]]

Greenwoodmic temperaments

2026-08-02T15:32:30Z

FloraC: + link to wesley

{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] '''greenwoodmic temperaments''', which [[tempering out|temper out]] [[405/392]], the [[greenwoodma]].

Temperaments discussed elsewhere are
* ''[[Schism]]'' (+64/63) → [[Archytas clan #Schism|Archytas clan]]
* ''[[Superpelog]]'' (+49/48) → [[Semaphoresmic clan #Superpelog|Semaphoresmic clan]]
* ''[[Injera]]'' (+50/49 or 81/80) → [[Meantone family #Injera|Meantone family]]
* ''[[August]]'' (+36/35) → [[Augmented family #August|Augmented family]]
* ''[[Sidi]]'' (+25/24) → [[Dicot family #Sidi|Dicot family]]
* ''[[Wesley]]'' (+875/864) → [[Wesley family #Septimal wesley|Wesley family]]
* ''[[Greenwood]]'' (+1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]

Considered below are secund and semishallowtone.

== Secund ==
Secund tempers out the greendwoodma, the [[avicennma]], and the [[orwellisma]]. It may be described as the {{nowrap| 9 & 26 }} temperament, with a [[ploidacot]] signature of pentacot. It divides the [[3/2|perfect fifth]] into five [[~]][[16/15]] generators, two for [[7/6]] and three for [[9/7]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 405/392, 525/512

{{Mapping|legend=1| 1 1 3 2 | 0 5 -6 7 }}
: mapping generators: ~2, ~16/15

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.6786{{c}}, ~16/15 = 138.3805{{c}}
: [[error map]]: {{val| +3.679 -6.374 -5.561 +7.195 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 138.0648{{c}}
: error map: {{val| 0.000 -11.631 -14.702 -2.372 }}

{{Optimal ET sequence|legend=1| 9, 17, 26, 61bc, 87bcc }}

[[Badness]] (Sintel): 2.27

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 385/384

Mapping: {{mapping| 1 1 3 2 3 | 0 5 -6 7 4 }}

Optimal tunings:
* WE: ~2 = 1202.9415{{c}}, ~12/11 = 138.2377{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.9970{{c}}

{{Optimal ET sequence|legend=0| 9, 17, 26, 61bc }}

Badness (Sintel): 1.41

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 385/384

Mapping: {{mapping| 1 1 3 2 3 2 | 0 5 -6 7 4 15 }}

Optimal tunings:
* WE: ~2 = 1202.4465{{c}}, ~12/11 = 137.2165{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 137.0309{{c}}

{{Optimal ET sequence|legend=0| 9, 35 }}

Badness (Sintel): 2.05

==== Secundly ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 99/98

Mapping: {{mapping| 1 1 3 2 3 3 | 0 5 -6 7 4 6 }}

Optimal tunings:
* WE: ~2 = 1203.0195{{c}}, ~13/12 = 138.2642{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 138.0317{{c}}

{{Optimal ET sequence|legend=0| 9, 17, 26, 61bcf }}

Badness (Sintel): 1.08

== Semishallowtone ==
{{See also| Shallowtone }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 405/392, 35721/32768

{{Mapping|legend=1| 2 0 36 15 | 0 1 -10 -3 }}
: mapping generators: ~189/128, ~3

[[Optimal tuning]]s:
* [[WE]]: ~189/128 = 603.1653{{c}}, ~3/2 = 686.6365{{c}} (~64/63 = 83.4712{{c}})
: [[error map]]: {{val| +6.331 -8.988 -2.034 -0.248 }}
* [[CWE]]: ~189/128 = 600.0000{{c}}, ~3/2 = 682.6498{{c}} (~64/63 = 82.6498{{c}})
: error map: {{val| 0.000 -19.305 -12.811 -16.775 }}

{{Optimal ET sequence|legend=1| 14c, 44bd, 58bcd }}

[[Badness]] (Sintel): 8.42

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 35721/32768

Mapping: {{mapping| 2 0 36 15 32 | 0 1 -10 -3 -8 }}

Optimal tunings:
* WE: ~189/128 = 603.0536{{c}}, ~3/2 = 686.6175{{c}} (~64/63 = 83.5640{{c}})
* CWE: ~189/128 = 600.0000{{c}}, ~3/2 = 682.7238{{c}} (~64/63 = 82.7238{{c}})

{{Optimal ET sequence|legend=0| 14c, 44bd, 58bcde, 72bbccddee }}

Badness (Sintel): 4.15

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 35721/32768

Mapping: {{mapping| 2 0 36 15 32 -2 | 0 1 -10 -3 -8 3 }}

Optimal tunings:
* WE: ~91/64 = 601.8872{{c}}, ~3/2 = 684.4774{{c}} (~64/63 = 82.5902{{c}})
* CWE: ~91/64 = 600.0000{{c}}, ~3/2 = 682.1676{{c}} (~64/63 = 82.1676{{c}})

{{Optimal ET sequence|legend=0| 14c, 30b }}

Badness (Sintel): 4.38

[[Category:Temperament collections]]
[[Category:Greenwoodmic temperaments| ]] 
[[Category:Rank 2]]

Keemic temperaments

2026-08-02T15:29:19Z

FloraC: + link to wesley

{{Technical data page}}
This is a collection of [[rank-2 temperament|linear]] [[regular temperament|temperaments]] that [[tempering out|temper out]] the [[keema]] ({{monzo|legend=1| -5 -3 3 1 }}, [[ratio]]: 875/864), with [[S-expression]] S5/S6. Its fundamental equivalence entails that [[6/5]] is sharpened so that it stacks three times to reach [[7/4]], and the interval between 6/5 and [[5/4]] is compressed so that [[7/6]]–6/5–5/4–[[9/7]] are set equidistant from each other. As the canonical extension of rank-3 [[keemic]] to the [[11-limit]] tempers out the commas [[100/99]] and [[385/384]] (whereby ([[6/5]])2 is identified with [[16/11]]), this provides a clean way to extend the various keemic temperaments to the 11-limit as well.

Full [[7-limit]] keemic temperaments discussed elsewhere are:
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* ''[[Mujannabic]]'' (+25/24) → [[Dicot family #Dicot|Dicot family]]
* [[Porcupine]] (+64/63) → [[Porcupine family #Septimal porcupine|Porcupine family]]
* [[Monkey]] (+5120/5103) → [[Tetracot family #Monkey|Tetracot family]]
* [[Magic]] (+225/224) → [[Magic family #Septimal magic|Magic family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* ''[[Wesley]]'' (+405/392) → [[Wesley family #Septimal wesley|Wesley family]]
* ''[[Doublewide]]'' (+50/49) → [[Jubilismic clan #Doublewide|Jubilismic clan]]
* [[Superkleismic]] (+1029/1024) → [[Gamelismic clan #Superkleismic|Gamelismic clan]]
* ''[[Fifives]]'' (+83349/81920) → [[Fifive family #Fifives|Fifive family]]
* ''[[Sycamore]]'' (+686/675) → [[Sycamore family #Septimal sycamore|Sycamore family]]

Discussed below are quasitemp, chromo, barbad, pentadecal, undeka, hyperkleismic, and sevond, in the order of increasing [[TE logflat badness]].

== Quasitemp ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasitemp]].''

Quasitemp tempers out [[2401/2400]] in addition to 875/864 and may be described as the {{nowrap| 37 & 41 }} temperament. It is characterized by equating the interval between the pental and septimal thirds ([[36/35]]) with the classical chromatic semitone ([[25/24]]), and by tempering together the septimal dieses of [[49/48]] and [[50/49]]. In that sense, it is opposed to [[orwellismic temperaments]], in particular [[myna]], where the distance between the pental and septimal thirds is the same as the septimal dieses and different from the classical chromatic semitone.

Quasitemp can also be thought of as a [[strong extension]] of the 2.5/3.7/3-subgroup temperament called [[gariberttet]], which is defined by tempering out [[3125/3087]]. In gariberttet, three generators reach [[5/3]] and five reach [[7/3]], so that the generator itself has the interpretation of [[25/21]]. This implies that 3:5:7 and 5:6:7 chords are reached rather quickly. Quasitemp tempering out 875/864 entails that [[8/7]] is found after 9 generators, from which the mappings of 3 and 5 follow.

Note that the generator is given as 25/21's octave complement, 42/25, in the data that follow, since a stack of 14 such generators octave-reduced is the perfect fifth, whence the temperament's [[ploidacot]] is iota-14-cot. This generator is equated to [[22/13]] for the 13-limit extension, tempering out [[275/273]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 2401/2400

{{Mapping|legend=1| 1 -9 -6 -4 | 0 14 11 9 }}
: mapping generators: ~2, ~42/25

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9237{{c}}, ~42/25 = 907.9887{{c}}
: [[error map]]: {{val| +0.924 +1.573 -3.981 -0.623 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~42/25 = 907.3471{{c}}
: error map: {{val| 0.000 +0.905 -5.495 -2.702 }}

{{Optimal ET sequence|legend=1| 4, …, 37, 41 }}

[[Badness]] (Sintel): 1.53

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1375/1372

Mapping: {{mapping| 1 -9 -6 -4 8 | 0 14 11 9 -6 }}

Optimal tunings:
* WE: ~2 = 1199.9585{{c}}, ~42/25 = 907.4221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.4521{{c}}

{{Optimal ET sequence|legend=0| 4, 37, 41, 119 }}

Badness (Sintel): 1.43

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 275/273, 385/384

Mapping: {{mapping| 1 -9 -6 -4 8 9 | 0 14 11 9 -6 -7 }}

Optimal tunings:
* WE: ~2 = 1199.4376{{c}}, ~22/13 = 907.1175{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 907.5314{{c}}

{{Optimal ET sequence|legend=0| 4, 37, 41, 78, 119f }}

Badness (Sintel): 1.36

=== Quato ===
Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 625/616

Mapping: {{mapping| 1 -9 -6 -4 -23 | 0 14 11 9 35 }}

Optimal tunings:
* WE: ~2 = 1201.2729{{c}}, ~42/25 = 908.1116{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.2109{{c}}

{{Optimal ET sequence|legend=0| 41, 127cd, 168cd }}

Badness (Sintel): 1.36

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 243/242, 275/273, 325/324

Mapping: {{mapping| 1 -9 -6 -4 -23 -22 | 0 14 11 9 35 34 }}

Optimal tunings:
* WE: ~2 = 1201.4078{{c}}, ~42/25 = 908.1362{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~42/25 = 907.1370{{c}}

{{Optimal ET sequence|legend=0| 41, 86ce }}

Badness (Sintel): 1.24

== Chromo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Chromo]].''

Chromo represents the [[13edf]] chain as a rank-2 temperament, with [[6/5]] and [[5/4]] mapped to 6 and 7 steps, respectively. Since the difference of those two intervals is abbreviated considerably from just, keemic provides the most meaningful 7-limit extension (setting [[7/6]], 6/5, 5/4, [[9/7]] equidistant) so that the temperament then approximates the [[4:5:6:7]] tetrad with 0:7:13:18 generator steps.

Note that if one allows a more complex mapping for prime 7 and wants a larger prime limit, one may prefer [[escapade]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 2430/2401

{{Mapping|legend=1| 1 1 2 2 | 0 13 7 18 }}
: mapping generators: ~2, ~36/35

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4060{{c}}, ~36/35 = 53.8791{{c}}
: [[error map]]: {{val| +1.406 -0.121 -6.348 +3.810 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 53.9055{{c}}
: error map: {{val| 0.000 -1.183 -8.975 +1.474 }}

{{Optimal ET sequence|legend=1| 22, 45, 67c }}

[[Badness]] (Sintel): 2.30

== Barbad ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 16875/16807

{{Mapping|legend=1| 1 -10 -5 -10 | 0 19 12 21 }}
: mapping generators: ~2, ~98/75

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.0462{{c}}, ~75/49 = 732.3071{{c}}
: [[error map]]: {{val| +1.046 +1.418 -3.859 -0.838 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/49 = 731.7183{{c}}
: error map: {{val| 0.000 +0.692 -5.694 -2.742 }}

{{Optimal ET sequence|legend=0| 18, 23d, 41 }}

[[Badness]] (Sintel): 2.80

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 245/242, 540/539, 625/616

Mapping: {{mapping| 1 -10 -5 -10 -13 | 0 19 12 21 27 }}

Optimal tunings:
* WE: ~2 = 1200.8513{{c}}, ~75/49 = 732.1519{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~75/49 = 731.6740{{c}}

{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}

Badness (Sintel): 1.66

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 245/242, 275/273

Mapping: {{mapping| 1 -10 -5 -10 -13 -3 | 0 19 12 21 27 11 }}

Optimal tunings:
* WE: ~2 = 1199.7960{{c}}, ~20/13 = 731.6053{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 731.7208{{c}}

{{Optimal ET sequence|legend=0| 18e, 23de, 41 }}

Badness (Sintel): 1.62

== Pentadecal ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Pentadecal]].''

Named by [[Xenllium]] in 2021, pentadecal tempers out the 15-5/3-comma ({{monzo| -11 -15 15 }}) in the 5-limit. This temperament can be described as {{nowrap| 15 & 60 }} temperament, tempering out the [[cloudy comma]], 16807/16384 and the [[keema]], 875/864 in the 7-limit.

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 16807/16384

{{Mapping|legend=1| 15 0 11 42 | 0 1 1 0 }}
: mapping generators: ~21/20, ~3

[[Optimal tuning]]s:
* [[WE]]: ~21/20 = 80.1141{{c}}, ~3/2 = 700.2213{{c}} (~126/125 = 19.8053{{c}})
: [[error map]]: {{val| +1.711 +0.977 -2.127 -4.035 }}
* [[CWE]]: ~21/20 = 80.0000{{c}}, ~3/2 = 701.2357{{c}} (~126/125 = 19.7643{{c}})
: error map: {{val| 0.000 -0.719 -5.078 -8.826 }}

{{Optimal ET sequence|legend=1| 15, 45, 60 }}

[[Badness]] (Sintel): 2.91

==== 2.3.5.7.13 subgroup ====
Subgroup: 2.3.5.7.13

Comma list: 105/104, 325/324, 15625/15379

Mapping: {{mapping| 15 0 11 42 52 8 | 0 1 1 0 0 2 }}

Optimal tunings:
* WE: ~21/20 = 80.1133{{c}}, ~3/2 = 700.1871{{c}} (~91/90 = 20.8325{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 700.2086{{c}} (~91/90 = 19.7914{{c}})

{{Optimal ET sequence|legend=0| 15, 30bcff, 45f, 60 }}

Badness (Sintel): 1.80

=== Quindecal ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 875/864

Mapping: {{mapping| 15 0 11 42 28 | 0 1 1 0 1 }}

Optimal tunings:
* WE: ~21/20 = 80.1322{{c}}, ~3/2 = 701.4751{{c}} (~126/125 = 19.7148{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 701.5453{{c}} (~126/125 = 18.4547{{c}})

{{Optimal ET sequence|legend=0| 15, 45e, 60e }}

Badness (Sintel): 1.47

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 275/273, 325/324

Mapping: {{mapping| 15 0 11 42 28 8 | 0 1 1 0 1 2 }}

Optimal tunings:
* WE: ~21/20 = 80.1359{{c}}, ~3/2 = 700.2567{{c}} (~91/90 = 20.9661{{c}})
* CWE: ~21/20 = 80.0000{{c}}, ~3/2 = 700.2955{{c}} (~91/90 = 19.7045{{c}})

{{Optimal ET sequence|legend=0| 15, 30bceff, 45ef, 60e }}

Badness (Sintel): 1.34

== Undeka ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Undeka]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 3200/3087

{{Mapping|legend=1| 11 0 8 31 | 0 1 1 0 }}
: mapping generators: ~21/20, ~3

[[Optimal tuning]]s:
* [[WE]]: ~21/20 = 108.9318{{c}}, ~3/2 = 707.7579{{c}}
: [[error map]]: {{val| -1.750 +4.053 -8.852 +8.059 }}
* [[CWE]]: ~21/20 = 109.0909{{c}}, ~3/2 = 707.7526{{c}}
: error map: {{val| 0.000 +5.798 -5.834 +12.992 }}

{{Optimal ET sequence|legend=1| 11b, 22 }}

[[Badness]] (Sintel): 3.59

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 352/343, 385/384

Mapping: {{mapping| 11 0 8 31 38 | 0 1 1 0 0 }}

Optimal tunings:
* WE: ~21/20 = 109.0321{{c}}, ~3/2 = 706.3870{{c}}
* CWE: ~21/20 = 109.0909{{c}}, ~3/2 = 706.4785{{c}}

{{Optimal ET sequence|legend=0| 11c, 22 }}

Badness (Sintel): 2.27

=== 2.3.5.7.11.17 subgroup ===
Subgroup: 2.3.5.7.11.17

Comma list: 85/84, 100/99, 121/119, 385/384

Mapping: {{mapping| 11 0 8 31 38 45 | 0 1 1 0 0 0 }}

Optimal tunings:
* WE: ~17/16 = 109.0232{{c}}, ~3/2 = 706.5074{{c}}
* CWE: ~17/16 = 109.0909{{c}}, ~3/2 = 706.6786{{c}}

{{Optimal ET sequence|legend=0| 11c, 22 }}

Badness (Sintel): 1.82

== Hyperkleismic ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 51200/50421

{{Mapping|legend=1| 1 -3 -2 2 | 0 17 16 3 }}
: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0290{{c}}, ~6/5 = 323.7882{{c}}
: [[error map]]: {{val| +0.029 +2.358 -5.759 +2.597 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 323.7816{{c}}
: error map: {{val| 0.000 +2.332 -5.808 +2.519 }}

{{Optimal ET sequence|legend=1| 26, 37, 63 }}

[[Badness]] (Sintel): 3.99

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 2420/2401

Mapping: {{mapping| 1 -3 -2 2 4 | 0 17 16 3 -2}}

Optimal tunings:
* WE: ~2 = 1199.9010{{c}}, ~6/5 = 323.7691{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7931{{c}}

{{Optimal ET sequence|legend=0| 26, 37, 63 }}

Badness (Sintel): 2.16

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 275/273, 385/384

Mapping: {{mapping| 1 -3 -2 2 4 1 | 0 17 16 3 -2 10 }}

Optimal tunings:
* WE: ~2 = 1200.0524{{c}}, ~6/5 = 323.8039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 323.7912{{c}}

{{Optimal ET sequence|legend=0| 26, 37, 63 }}

Badness (Sintel): 1.48

== Sevond ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Sevond (5-limit)]].''

10/9 is tempered to be exactly 1\7. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 327680/321489

{{Mapping|legend=1| 7 0 -6 53 | 0 1 2 -3 }}
: mapping generators: ~10/9, ~3

[[Optimal tuning]]s:
* [[WE]]: ~10/9 = 171.4007{{c}}, ~3/2 = 705.4982{{c}}
: [[error map]]: {{val| -0.195 +3.348 -4.112 -0.499 }}
* [[CWE]]: ~10/9 = 171.4286{{c}}, ~3/2 = 705.6057{{c}}
: error map: {{val| 0.000 +3.651 -3.674 +0.071 }}

{{Optimal ET sequence|legend=1| 7, …, 56, 63, 119 }}

[[Badness]] (Sintel): 5.23

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 6655/6561

Mapping: {{mapping| 7 0 -6 53 2 | 0 1 2 -3 2 }}

Optimal tunings:
* WE: ~11/10 = 171.3859{{c}}, ~3/2 = 705.3421{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.4973{{c}}

{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}

Badness (Sintel): 2.33

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 169/168, 352/351, 385/384

Mapping: {{mapping| 7 0 -6 53 2 37 | 0 1 2 -3 2 -1 }}

Optimal tunings:
* WE: ~11/10 = 171.4163{{c}}, ~3/2 = 705.2930{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~3/2 = 705.3402{{c}}

{{Optimal ET sequence|legend=0| 7, 56, 63, 119 }}

Badness (Sintel): 1.70

[[Category:Temperament collections]]
[[Category:Keemic temperaments| ]] 
[[Category:Rank 2]]

Wesley

2026-08-02T15:26:45Z

FloraC: Recategorize

#REDIRECT [[Wesley family #Wesley]]

[[Category:Wesley| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Wesley family]]
[[Category:Greenwoodmic temperaments]]
[[Category:Keemic temperaments]]

Wesley family

2026-08-02T15:25:13Z

FloraC: Units; misc. cleanup; - redundant category

{{Technical data page}}
The '''wesley family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] 78125/73728, the [[wesley comma]]. The wesley comma is unchanged in [[Wesley Woolhouse]]'s 7/26-comma meantone – it is an [[eigenmonzo|eigenmonzo (i.e. unchanged-interval)]], which [[Gene Ward Smith]] has talked about<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12793.html Yahoo! Tuning Group | ''26et and meantone projections'']</ref>.

== Wesley ==
[[Subgroup]]: 2.3.5

[[Comma list]]: 78125/73728

{{Mapping|legend=1| 1 4 3 | 0 -7 -2 }}
: mapping generators: ~2, ~125/96

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~125/96 = 414.509{{c}}

{{Optimal ET sequence|legend=1| 3, …, 23, 26, 29, 55c }}

[[Badness]] (Smith): 0.247718

== Septimal wesley ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 405/392, 875/864

{{Mapping|legend=1| 1 4 3 8 | 0 -7 -2 -15 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 415.519{{c}}

{{Optimal ET sequence|legend=1| 3d, …, 23d, 26 }}

[[Badness]] (Smith): 0.095344

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 875/864

Mapping: {{mapping| 1 4 3 8 9 | 0 -7 -2 -15 -16 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 415.769{{c}}

{{Optimal ET sequence|legend=0| 3de, …, 23de, 26 }}

Badness (Smith): 0.049066

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 78/77, 99/98, 325/324

Mapping: {{mapping| 1 4 3 8 9 12 | 0 -7 -2 -15 -16 -24 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~9/7 = 415.645{{c}}

{{Optimal ET sequence|legend=0| 3def, 23deff, 26 }}

Badness (Smith): 0.038402

== Snipes ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 6125/5832

{{Mapping|legend=1| 1 4 3 9 | 0 -7 -2 -18 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/27 = 413.513{{c}}

{{Optimal ET sequence|legend=1| 3d, …, 26d, 29 }}

[[Badness]] (Smith): 0.117943

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 225/224, 245/242

Mapping: {{mapping| 1 4 3 9 10 | 0 -7 -2 -18 -19 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 413.490{{c}}

{{Optimal ET sequence|legend=0| 3de, …, 26de, 29 }}

Badness (Smith): 0.054296

== Roman ==
{{See also| Avicennmic temperaments }}

7-limit Roman is also known as "crusher".

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 525/512, 3125/3024

{{Mapping|legend=1| 1 4 3 -1 | 0 -7 -2 11 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~63/50 = 414.552{{c}}

{{Optimal ET sequence|legend=1| 3, 23, 26, 29, 55c }}

[[Badness]] (Smith): 0.113386

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 525/512

Mapping: {{mapping| 1 4 3 -1 0 | 0 -7 -2 11 10 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 414.471{{c}}

{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55c }}

Badness (Smith): 0.052841

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 100/99, 105/104, 245/242

Mapping: {{mapping| 1 4 3 -1 0 3 | 0 -7 -2 11 10 2 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~14/11 = 414.472{{c}}

{{Optimal ET sequence|legend=0| 3, 23, 26, 29, 55cf }}

Badness (Smith): 0.030043

== Dubbla ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49, 78125/73728

{{Mapping|legend=1| 2 1 4 5 | 0 7 2 2 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~192/175 = 185.738{{c}}

{{Optimal ET sequence|legend=1| 6, 20b, 26, 58c }}

[[Badness]] (Smith): 0.181726

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 125/121, 1344/1331

Mapping: {{mapping| 2 1 4 5 6 | 0 7 2 2 3 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 185.879{{c}}

{{Optimal ET sequence|legend=0| 6, 20b, 26 }}

Badness (Smith): 0.075842

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 105/104, 125/121, 144/143

Mapping: {{mapping| 2 1 4 5 6 4 | 0 7 2 2 3 11 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/10 = 185.702{{c}}

{{Optimal ET sequence|legend=0| 6f, 20bff, 26 }}

Badness (Smith): 0.049603

== References ==

[[Category:Temperament families]]
[[Category:Wesley family| ]] 
[[Category:Rank 2]]

Hemimage temperaments

2026-08-02T15:11:19Z

FloraC:

{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).

Temperaments discussed elsewhere are:
* [[Quasisuper]] (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]
* ''[[Cotoneum]]'' (+33554432/33480783) → [[Garischismic clan #Cotoneum|Garischismic clan]]
* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]]
* ''[[Subfourth]]'' (+65536/64827) → [[Buzzardsmic clan #Subfourth|Buzzardsmic clan]]
* [[Magic]] (+225/224 or 245/243) → [[Magic family #Magic|Magic family]]
* ''[[Echidna]]'' (+1728/1715 or 2048/2025) → [[Diaschismic family #Echidna|Diaschismic family]]
* ''[[Pluto]]'' (+4000/3969) → [[Octagar temperaments #Pluto|Octagar temperaments]]
* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]
* ''[[Hendecatonic (temperament)|Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]
* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]
* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat|Amity family]]
* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]
* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]

Considered below are bisupermajor, bicommatic, degrees, squarschmidt, and leapmonth, in the order of increasing [[badness]].

== Bisupermajor ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 65625/65536

{{Mapping|legend=1| 2 1 6 1 | 0 8 -5 17 }}
: mapping generators: ~1225/864, ~192/175

[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.0294{{c}}, ~192/175 = 162.8141{{c}}
: [[error map]]: {{val| +0.059 +0.587 -0.208 -0.957 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~192/175 = 162.8082{{c}}
: error map: {{val| 0.000 +0.510 -0.355 -1.087 }}

{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}

[[Badness]] (Sintel): 1.66

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 3388/3375, 9801/9800

Mapping: {{mapping| 2 1 6 1 8 | 0 8 -5 17 -4 }}

Optimal tunings:
* WE: ~99/70 = 600.1224{{c}}, ~11/10 = 162.8065{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}

{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}

Badness (Sintel): 1.06

== Bicommatic ==
Used to be known simply as the ''commatic'' temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 50421/50000

{{Mapping|legend=1| 2 3 4 5 | 0 5 19 18 }}
: mapping generators: ~567/400, ~81/80

[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.0497{{c}}, ~81/80 = 20.3790{{c}}
: [[error map]]: {{val| +0.099 +0.089 +1.085 -1.756 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~81/80 = 20.3837{{c}}
: error map: {{val| 0.000 -0.037 +0.976 -1.920 }}

{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}

[[Badness]] (Sintel): 2.13

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

Mapping: {{mapping| 2 3 4 5 6 | 0 5 19 18 27 }}

Optimal tunings:
* WE: ~99/70 = 600.0401{{c}}, ~81/80 = 20.3913{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}

{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}

Badness (Sintel): 1.01

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 729/728, 1001/1000

Mapping: {{mapping| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}

Optimal tunings:
* WE: ~99/70 = 599.8514{{c}}, ~66/65 = 20.4215{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}

{{Optimal ET sequence|legend=0| 58, 118, 176f }}

Badness (Sintel): 1.09

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 170/169, 196/195, 289/288, 352/351, 561/560

Mapping: {{mapping| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}

Optimal tunings:
* WE: ~17/12 = 600.0257{{c}}, ~66/65 = 20.3789{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}

{{Optimal ET sequence|legend=0| 58, 118 }}

Badness (Sintel): 1.14

== Degrees ==
{{About|the regular temperament|scale degrees|degree}}
{{See also| 20th-octave temperaments }}

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.

An obvious extension to the 23-limit exists by mapping [[23/20]] to 4\20 (1\5), [[69/56]] to 6\20 (3\10), and [[23/18]] to 7\20. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by mapping [[29/22]] to 2\5, we get a uniquely elegant extension to the 29-limit which tempers out [[726/725]], which is the difference between [[33/25]] and [[29/22]], as well as [[784/783]] ({{S|28}}) and [[841/840]] ({{S|29}}). An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.

By equating [[37/28]] with 2\5 and more accurately [[85/74]] with 1\5 and [[44/37]] with 1\4 (among many other equivalences), we get an extension for prime [[37/1|37]] agreeing with many [[semiconvergent]]s, tempering out [[481/480]]. By mapping [[60/41]] and [[41/28]] to 11\20 or equivalently [[56/41]] and [[41/30]] to 9\20 and by mapping [[44/41]] to 1\10 (among many other equivalences), there is a very efficient extension for prime [[41/1|41]] tempering out [[451/450]].

The 80-note generator chain is ideal, so [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 390625/388962

{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}
: mapping generators: ~28/27, ~3

[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})
: [[error map]]: {{val| -0.157 +0.812 -0.647 -0.220 }}
* [[CWE]]: ~28/27 = 60.0000{{c}}, ~3/2 = 702.9324{{c}} (~126/125 = 17.0676{{c}})
: error map: {{val| 0.000 +0.977 -0.449 -0.029 }}

{{Optimal ET sequence|legend=1| 60, 80, 140, 640b, 780b }}

[[Badness]] (Sintel): 2.69

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1375/1372, 2200/2187

Mapping: {{mapping| 20 0 -17 -39 -26 | 0 1 2 3 3 }}

Optimal tunings:
* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}

Badness (Sintel): 1.55

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

Mapping: {{mapping| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}

Optimal tunings:
* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.35

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000

Mapping: {{mapping| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}

Optimal tunings:
* WE: ~28/27 = 60.0058{{c}}, ~3/2 = 703.0364{{c}} (~100/99 = 17.0335{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0061{{c}} (~100/99 = 16.9939{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.17

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475

Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 | 0 1 2 3 3 0 1 0 }}

Optimal tunings:
* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.27

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399

Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}

Optimal tunings:
* WE: ~28/27 = 59.9990{{c}}, ~3/2 = 703.1804{{c}} (~100/99 = 16.8074{{c}})
* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1870{{c}} (~100/99 = 16.8130{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.21

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405

Mapping: {{mapping| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}

Optimal tunings:
* WE: ~29/28 = 59.9990{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})
* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.13

== Squarschmidt ==
: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''

Squarschimidt may be described as {{nowrap| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 29360128/29296875

{{Mapping|legend=1| 1 -8 1 -20 | 0 29 4 69 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9006{{c}}, ~1125/896 = 396.6104{{c}}
: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1125/896 = 396.6417{{c}}
: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}

{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}

[[Badness]] (Sintel): 3.36

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 5632/5625, 10976/10935

Mapping: {{mapping| 1 -8 1 -20 -21 | 0 29 4 69 74 }}

Optimal tunings:
* WE: ~2 = 1199.9005{{c}}, ~44/35 = 396.6107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}

{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}

Badness (Sintel): 1.26

== Leapmonth ==
Leapmonth may be described as the {{nowrap| 63 & 80 }} temperament, generated by a [[3/2|perfect fifth]] and being a strong extension of [[leapfrog]]. It was named by [[Flora Canou]] in 2025 following the pattern demonstrated by ''leapday'' and ''leapweek'', the two simpler extensions of leapfrog.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 51200/50421

{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8005{{c}}, ~3/2 = 704.2543{{c}}
: [[error map]]: {{val| -1.200 +1.100 -0.659 +2.186 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}
: error map: {{val| 0.000 +2.977 +1.093 +5.150 }}

{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}

[[Badness]] (Sintel): 4.79

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 1331/1323

Mapping: {{mapping| 1 0 -58 -21 -14 | 0 1 38 15 11 }}

Optimal tunings:
* WE: ~2 = 1198.8679{{c}}, ~3/2 = 704.2911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}

{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}

Badness (Sintel): 1.88

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 352/351, 364/363, 540/539

Mapping: {{mapping| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}

Optimal tunings:
* WE: ~2 = 1199.1781{{c}}, ~3/2 = 704.4551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}}

{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}

Badness (Sintel): 1.53

[[Category:Temperament collections]]
[[Category:Hemimage temperaments| ]] 
[[Category:Rank 2]]

Harp

2026-08-02T09:13:17Z

FloraC: Improve sectioning. No need to start with an "introduction" section

{{Wikipedia}}

'''Harps''' are instruments that have multiple strings perpendicular to the sound board, plucked by the player's hands. Each string corresponds to one pitch, which in the case of lever and {{W|pedal harp}}s, may be modified by the lever or pedal mechanism to get more than one pitch. In the case of the pedal harp, the strings are tuned to a diatonic scale, and depressing a pedal (of which one is available for each note of the diatonic scale), depressing a pedal rotates a set of discs with pins set to stop the corresponding strings (in all octaves) so as to raise their pitch by a chromatic semitone (including allowance for the resulting slight increase in string tension). The double-acting pedal harp extends this concept to have two sets of string-stopping discs, actuated by successive further depressions of the corresponding pedal, which can be locked at either of its depressed positions; the strings are tuned to all flats, and the first depressed position of a pedal raises the pitch of the strings to achieve the corresponding natural notes, while the second depressed position of the pedal raises the pitch further to achieve the corresponding sharp notes. The modern double-acting harp also has two lowest C and D strings and one highest G string to which this mechanism does not apply, so these strings are tuned to whichever pitches are required for the music being played.

Unfortunately, the above design in its standard implementation is wedded to [[12edo]] – the value of the chromatic semitone is fixed, and a standard pedal harp has no provision for adjusting the position of the string stops, nor for partial action of the string stops   attempting to move a pedal to an intermediate position does not produce an in-between note, but instead produces a buzz or a fearsome rattle (depending upon how hard the string was plucked) as the string hits the pins on the incompletely turned disc. The only microtonal tuning systems that can make efficient use of pedal harps to get in-between notes are multiples of 12edo, with multiple harps tuned to different rings of 12edo, as done by [[Ivan Wyschnegradsky]] with pianos. Although no recording is known of actual harp use in this way, [[Norokusi]] has written a composition in [[24edo]], rendered by soft synthesizers, that uses this technique (see below); in principle, it could be transcribed and then played on a full set of acoustic instruments, including two harps with the orchestra. Thus, microtonality on most harps (unless using multiple harps) is limited to whatever can be achieved by retuning the available 7 strings per octave. See the music section below for examples of obtaining microtones by [[#Retuned individual strings|retuning individual strings]] and [[#Retuned multiple harps|retuning two or more harps offset by one step within a multiple of 12edo]].

Here is a demonstration of harp string buzz:

; [https://www.youtube.com/@daniellekuntz/ Danielle Kuntz]
* [https://www.youtube.com/shorts/jjAOcndfwzM ''How to play pedal buzzes on the harp''] (2023)

== A solution that does not yet exist ==
An instrument called a "fluid harp" exists that has sliders to enable continuously-variable pitch variation on each string, as demonstrated in [https://www.youtube.com/watch?v=SzhlYaevFR8 ''Fluid Harp – Adjustable Microtonal Harp''] (2017), but it is actually a zither and produces a very different tonal quality from a harp; as such, it belongs to a page about zithers, and is not further covered here. In principle, such a mechanism – or better yet, a pedal-actuated slider mechanism – could be applied to an actual harp, but so far, none have been built.

== Lever harp ==
Lever harps have one lever for each string to tighten and/or stop the string to raise the pitch by a chromatic semitone (one disadvantage being that the levers do not have enough range to go all the way from a flat to a sharp). On a subset of lever harps, it is possible to achieve in-between notes as well as portamento and vibrato by moving levers to intermediate positions; on a subset of these harps (including the [https://www.harp-e.com/en-us Harp-E]), the levers have enough friction to stay in an intermediate position after being moved there. Since the levers operate independently, changing all of the notes in a pitch class requires changing the positions of all of the corresponding levers, and this will require too much time to achieve in a rapidly modulating passage.

=== Demonstration ===
; [https://www.youtube.com/@HarpistKT Kristan Toczko]
* [https://www.youtube.com/shorts/pelElN6Wr68 ''Levers AND bends!''] (2024)

== Manual string stopping ==
Another possible solution is to wear a metal cylinder on one finger and use it to stop pairs of adjacent strings. This enables any amount of pitch bend (potentially while plucking these and nearby strings with other fingers on the same hand), and can be used on any harp, although it only applies to one pair of strings at a time for each metal tube (conceivably, one could wear a metal tube on one finger on each hand to do this with two pairs of strings). See the music section below for examples of [[#Manual string stopping|manual string stopping]].

== More strings ==
The {{w|cross-strung harp}} or chromatic double harp foregoes changing the pitch of strings to get accidentals and instead has more strings — usually one row of 7 strings per octave tuned to a diatonic scale of the natural notes and one row of 5 strings per octave tuned to a pentatonic scale of the sharp/flat notes. (A few have been made that instead have twin whole-tone scales.) Since the strings are only tuned using the normal tuning mechanism (unless one stops the strings with a metal tube as above), it is in principle possible to tune the strings to sound in a 12 notes per octave subset of tuning systems other than 12edo, including complete tuning systems having less than 12 notes per octave.

The {{w|triple harp}} (including the Welsh version) has even more strings, with 2 outer rows of 7 strings each normally tuned to the natural notes and a middle row normally tuned to the flat/sharp notes as indicated above. Again, since the strings are only tuned using the normal tuning mechanism (unless one stops the strings with a metal tube as above), it is in principle possible to tune the strings to sound in a 19 notes per octave subset of tuning systems other than 12edo, as well as complete tuning systems of up to 19 notes per octave, including [[19edo]].

Unfortunately, no music or other demonstration appears to be available at this time for these types of harps.

== Music ==
Despite the general unfriendliness of the predominant harp design towards microtonality, some microtonal music has been composed, played (or rendered as a proof of concept), and recorded. For some of this music, the method of obtaining microtones was not described; this music is in [[#Unspecified method of obtaining microtones]] at the end; if the method is discovered, it should be moved to one of the other sections (or a new section created, as appropriate).

=== Manual string stopping ===
; [https://calyane.com/en/bio/ Caroline Lizotte]
* [https://www.youtube.com/watch?v=ZHxUnzVWHdQ ''KHEÍRŌN opus 53, Nocturne pour harpe – Caroline Lizotte (Juliette Duguay, harpiste)''] (2026) (uses the metal tube on finger technique)

=== Retuned individual strings ===
; {{W|Alain Bancquart}}
* [https://www.youtube.com/watch?v=BSDAz4qS3H0 ''Ma manière de chat''] (1979) (from video comments: some strings are tuned down by a quarter-tone)

; [[Nick, The NRG]]
* [https://www.youtube.com/watch?v=5itQAT_Ns2c ''Microtonal Semi-improvisation on Harpsicle/Folk Harp!''] (2019) (the harp is tuned to 1/1, 9/8, 5/4, 4/3, 3/2, 14/9, 7/4)

=== Retuned multiple harps ===
; [[Norokusi]]
* [https://www.youtube.com/watch?v=IJIKCXEah4Y ''Serenade for Harp and Strings''] (2021) (both examples using dual harps tuned apart by 1\24, rendered by soft synthesizer, but in principle playable on acoustic instruments)
** [https://www.youtube.com/watch?v=pxLgv0HEG-0 arrangement for harp and orchestra] (2026)

=== Unspecified method of obtaining microtones ===
; {{W|Alain Bancquart}}
* [https://www.youtube.com/watch?v=fyJ-SuQ2SxY ''Étrennes''] (1992), from [https://www.discogs.com/release/21693193-Catherine-Michel-Catherine-Michel-En-Récital-Anthologie-De-La-Musique-Pour-Harpe Catherine Michel En Récital] (by analogy with ''Ma manière de chat'', may be using retuning of individual strings, but not confirmed)

[[Category:Harp| ]]

Xenharmonic Wiki:Manual of Style/Temperament cataloging

2026-08-02T09:00:03Z

FloraC: Amendments

This is a '''guide for where to put and/or find temperament data'''.

{{Note| You can use <code>XW:TEMPCAT</code> as a shortcut to this page. }}
{{Note| As this guide cannot account for all edge cases, editors are advised to use their best judgement on where to put and/or find temperament data. Additionally, nothing about this guide is set in stone; this guide can (and should!) be updated to fit current developments regarding temperament pages. }}

== General rules ==
# Avoid duplication. A temperament should be catalogued exactly once on the wiki.
# Equal temperaments must be discussed in individual [[edo]] pages. The rest of this guide only concerns multirank temperaments.

== Families, clans, and collections ==
{{Main| Temperament families and clans }}

A family has a head that is a full-prime-limit temperament. A clan has a head that is a subgroup temperament. Family and clan pages should both be titled after the full-prime-limit temperament. Rarely, a clan may be referred to by the name of the subgroup temperament + family. For example:
* Archytas family is the rank-3 family of 2.3.5.7 {64/63};
* Archytas clan is the rank-2 clan of 2.3.7 {64/63}.
** ''Archy family'' is an alias of ''archytas clan''.

Temperament families and clans should only be created when the head temperament is notable by itself, has multiple branches to higher limits, and for the sake of navigability, is within rank-2 to rank-4. The head temperament should be one of the 7-limit tri-prime subgroups (2.3.5, 2.3.7, 2.5.7, and 3.5.7) for rank-2, the 11-limit analog for rank-3, or the 13-limit analog for rank-4.

Besides families and clans, there are pages that collect temperaments of a certain comma on this wiki. These pages are for temperaments one rank lower than its corresponding family, and are used because some temperaments are more closely associated with the comma than any other or otherwise do not belong to any notable family or clan.

Temperament families, clans, and collections are the bread and butter for documenting individual temperaments on this wiki. Temperament data should be placed there whenever applicable. The specific hierarchy of priority is explained as follows.

If a temperament belongs to an existing family or clan, or one that is supposed to exist according to the rules above, consider putting it there. If a temperament belongs to multiple families or clans, determine if it is a [[strong extension]] of the head of each and put it where it is a strong extension of the head. A strong extension is an extension where the generators are not split; it is generally the preferred place for the temperament since the relation between the temperament in question and the head is most clear. And if a temperament is zero or multiple strong extensions at the same time, it may be placed in the family or clan of the simpler subgroup or the simpler head, depending on which is more relevant.

Otherwise, it should be placed in a collection page of the simplest comma tempered out in the temperament.

For example:
* Mothra is found in Gamelismic clan, ''not'' Meantone family, since it is a strong extension of slendric, the head of the gamelismic clan, and only a weak extension of meantone;
* Dominant is found in Meantone family, since 2.3.5 is simpler than 2.3.7, but schism is found in Archytas clan, since the archytas comma is more "structural" to the temperament.

However, no-2 or no-3 clans should be treated just like collections, since the relation between the members and the head is often not obvious there.

For example:
* Sensi is found in Sensipent family, ''not'' Sensamagic clan;
* Miracle is found in Gamelismic clan, ''not'' Quince clan.

== Other pages ==
=== Very high/low accuracy temperaments ===
The criteria of inclusion for these pages are given in the introductions. Currently no [[badness]] constraint is in effect but in future there might be. The head of each entry should only be temperaments of the 7-limit tri-prime subgroups for rank-2, the 11-limit analog for rank-3, or the 13-limit analog for rank-4. Extensions may be put there if they meet the criteria above.

=== Equivalence continua ===
Equivalence continua only hold temperament data of the particular subgroup they are defined in. An equivalence continuum for an equal temperament should only be created if the equal temperament is special in some way, e.g., it is a record 5-limit temperament. Any rank-2 temperament of a 7-limit tri-prime subgroup which does not fit in a family or clan should be placed in an equivalence continuum, when applicable.

=== Miscellaneous 5-limit/7-limit temperaments and catalog of rank-4 temperaments ===
The pages Miscellaneous 5-limit temperaments and Miscellaneous 7-limit temperaments hold temperament data of the particular limit given in the title, for rank-2 and rank-3 temperaments, respectively.

Any 5-limit rank-2 temperament which does not fit in a family, nor an equivalence continuum, should go to Miscellaneous 5-limit temperaments. Any 7-limit rank-3 temperament which does not fit in a family should go to Miscellaneous 7-limit temperaments. Any 11-limit rank-4 temperament as well as their extensions which does not fit in a family should go to Catalog of rank-4 temperaments.

=== Subgroup temperaments ===
Subgroup temperament pages include Subgroup temperaments and the various No-''p'''s subgroup temperaments. Any subgroup temperament should go here unless they fit any of the above categories.

=== Fractional-octave temperaments and comma pages ===
Unless necessary, putting temperament data in these pages should generally be avoided. Fractional-octave temperament pages have poor discoverability, whereas temperament data in comma pages tend to be intrusive for average readers, and both have poor interoperability as they unnecessarily isolate certain temperaments from the rest.

[[Category:Xenharmonic Wiki guidelines]]

Fynn's comma

2026-08-02T08:55:17Z

FloraC: Do not name/catalog equal temps. Refer to them by *2.7-subgroup 26et*, etc.

{{Infobox interval
| Monzo = 73 0 0 -26
| Name = Fynn's comma, 26-7-comma
| Comma = true
}}
'''Fynn's comma''' ({{monzo|legend=1| 73 0 0 -26 }}, [[ratio]]: 9 444 732 965 739 290 427 392 / 9 387 480 337 647 754 305 649), or systematically, the '''26-7-comma''', is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 10.5 [[cent]]s. It is the amount by which twenty-one [[2/1|octaves]] exceed twenty-six [[7/4|harmonic sevenths]], or the amount by which twenty-one [[8/7|septimal major seconds]] exceed five octaves. It explains the high accuracy of [[26edo]]'s approximate [[harmonic]] [[7/1|7]].

== Temperaments ==
[[Tempering out]] this comma splits the octave into 26 equal parts and maps 7/4 to 21\26, For edos ''N'' up to 1456, it is tempered out if and only if 26 divides ''N''.

== Trivia ==
This interval is the difference between the [[dilemma]], the difference between thirteen 7/4's and 10 octaves, and the [[antidilemma]], difference between thirteen 7/4's and 11 octaves. Since 26edo has an accurate 7/4 not shared with any lower edo, [[13edo]] has almost 50% relative error on it.

== See also ==
* [[26th-octave temperaments]]

[[Category:Commas someone named after themselves]]

26th-octave temperaments

2026-08-02T08:36:39Z

FloraC: Do not name/catalog equal temps. Refer to them by *2.7-subgroup 26et*, etc.

{{Technical data page}}
{{Infobox fractional-octave|26}}
All temperaments on this page have a period that is 1/26th of an octave. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo| 73 0 0 -26 }}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple edos up to 1456 (such as [[26edo|26-]], [[130edo|130-]], [[286edo|286-]] or [[546edo|546edo]]).

Temperaments discussed elsewhere include [[varunismic temperaments #Bosonic|bosonic]]. Considered below is iron.

== Iron ==
Iron may be described as the {{nowrap| 130 & 494 }} temperament. It was named by [[Eliora]] in 2023 after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

{{Mapping|legend=1| 26 0 184 73 | 0 1 -3 0 }}
: mapping generators: ~17280/16807, ~3/2

[[Optimal tuning]]s:
* [[WE]]: ~17280/16807 = 46.1521{{c}}, ~3/2 = 701.9475{{c}}
: [[error map]]: {{val| -0.047 -0.054 -0.039 +0.274 }}
* [[CWE]]: ~17280/16807 = 46.1538{{c}}, ~3/2 = 701.9779{{c}}
: error map: {{val| 0.000 +0.023 +0.060 +0.405 }}

{{Optimal ET sequence|legend=1| 104, 130, 364, 494, 1118, 2730d, 3848dd }}

[[Badness]] (Sintel): 2.61

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: {{mapping| 26 0 184 73 296 | 0 1 -3 0 -5 }}

Optimal tunings:
* WE: ~77/75 = 46.1519{{c}}, ~3/2 = 701.9621{{c}}
* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 701.9967{{c}}

{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1118, 1612d, 2106d }}

Badness (Sintel): 1.14

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: {{mapping| 26 0 184 73 296 55 | 0 1 -3 0 -5 1 }}

Optimal tunings:
* WE: ~77/75 = 46.1525{{c}}, ~3/2 = 701.9820{{c}}
* CWE: ~77/75 = 46.1538{{c}}, ~3/2 = 702.0051{{c}}

{{Optimal ET sequence|legend=0| 104e, 130, 364, 494, 1612d, 2106d }}

Badness (Sintel): 0.711

{{Navbox fractional-octave}}

[[Category:26edo]]

Sensipent family

2026-08-02T08:34:41Z

FloraC: /* Bison */ update

{{Technical data page}}

[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.

== Sensipent ==
{{Main| Sensipent }}

The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125. Two generators make 5/3, seven make harmonic 6 and nine make harmonic 10. Its [[ploidacot]] is beta-heptacot ([[pergen]] (P8, ccP5/7)) and its color name is Sepguti.

[[Subgroup]]: 2.3.5

[[Comma list]]: 78732/78125

{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
: mapping generators: ~2, ~162/125

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
: error map: {{val| 0.000 -0.600 +1.143 }}

{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}

[[Badness]] (Sintel): 0.826

=== Overview to extensions ===
The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.

Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.

Temperaments discussed elsewhere include:
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)

Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.

=== Sensible ===
{{See also| Sensipent #Sensible interval table }}

Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])3. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[S-expression|S31⋅S322]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])2 with [[16/15]] as well as that a natural extension to prime 31 exists through {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.

The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.

Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])2. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].

Subgroup: 2.3.5.11

Comma list: 8019/8000, 16384/16335

Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}

: mapping generators: ~2, ~128/99

Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}

Badness (Sintel): 0.728

==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17

Comma list: 256/255, 1089/1088, 1377/1375

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}

: mapping generators: ~2, ~22/17

Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

Badness (Sintel): 0.639

==== 2.3.5.11.17.23 subgroup ====
Subgroup: 2.3.5.11.17.23

Comma list: 256/255, 576/575, 1089/1088, 1377/1375

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}

Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

Badness (Sintel): 0.555

==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31

Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}

Optimal tunings:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

Badness (Sintel): 0.490

== Sensi ==
{{Main| Sensi }}

Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.

=== Septimal sensi ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 245/243

{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[Tuning ranges of regular temperaments|Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
* 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
* 7-odd-limit [[diamond tradeoff]]: ~9/7 = [442.179, 445.628]
* 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]

[[Algebraic generator]]: The real root of ''x''5 + ''x''4 - 4''x''2 + ''x'' - 1, at 443.3783 cents.

{{Optimal ET sequence|legend=1| 19, 27, 46 }}

[[Badness]] (Sintel): 0.648

==== 2.3.5.7.13 subgroup (sensation) ====
Subgroup: 2.3.5.7.13

Comma list: 91/90, 126/125, 169/168

Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}

Optimal tunings:
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}

{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}

Badness (Sintel): 0.484

=== Sensor ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/243, 385/384

Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}

Optimal tunings:
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}

{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}

Badness (Sintel): 1.25

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 385/384

Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}

Optimal tunings:
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}

{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}

Badness (Sintel): 1.06

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168, 256/255

Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}

Optimal tunings:
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}

{{Optimal ET sequence|legend=0| 19, 27, 46 }}

Badness (Sintel): 1.17

=== Sensus ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 245/243

Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}

Optimal tunings:
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}

{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}

Badness (Sintel): 0.975

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 352/351

Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}

Optimal tunings:
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}

{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}

Badness (Sintel): 0.859

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 136/135, 154/153, 169/168

Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}

Optimal tunings:
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}

{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}

Badness (Sintel): 0.827

=== Sensis ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 245/243

Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}

Optimal tunings:
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}

{{Optimal ET sequence|legend=0| 8d, 19, 27e }}

Badness (Sintel): 0.948

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90, 100/99

Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}

Optimal tunings:
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}

{{Optimal ET sequence|legend=0| 8d, 19, 27e }}

Badness (Sintel): 0.827

=== Sensa ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 77/75, 99/98

Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}

Optimal tunings:
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}

{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}

Badness (Sintel): 1.22

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 77/75, 143/140

Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}

Optimal tunings:
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}

{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}

Badness (Sintel): 0.961

=== Bisensi ===
Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).

Subgroup: 2.3.5.7.11

Comma list: 121/120, 126/125, 245/243

Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}

: mapping generators: ~99/70, ~9/7

Optimal tunings:
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})

{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}

Badness (Sintel): 1.38

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 126/125, 169/168

Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}

Optimal tunings:
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})

{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}

Badness (Sintel): 1.09

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 126/125, 154/153, 169/168

Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}

Optimal tunings:
* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})

{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}

Badness (Sintel): 0.960

=== Hemisensi ===
Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-14-cot (pergen (P8, ccP5/14)).

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 245/242

Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}

: mapping generators: ~2, ~25/22

Optimal tunings:
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}

{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}

Badness (Sintel): 1.61

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 243/242

Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}

Optimal tunings:
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}

{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}

Badness (Sintel): 1.36

== Sensei ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 78732/78125

{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6422{{c}}, ~162/125 = 442.9920{{c}}
: [[error map]]: {{val| +0.642 -1.653 -0.028 +1.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}

{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}

[[Badness]] (Sintel): 1.50

== Warrior ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 78732/78125

{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2419{{c}}, ~162/125 = 443.0087{{c}}
: [[error map]]: {{val| -0.758 -0.136 +1.523 +0.516 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}

{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}

[[Badness]] (Sintel): 2.99

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323, 5120/5103

Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}

Optimal tunings:
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}

{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}

Badness (Sintel): 1.53

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 1331/1323

Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}

Optimal tunings:
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}

{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}

Badness (Sintel): 1.19

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 442/441, 715/714

Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}

Optimal tunings:
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}

{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}

Badness (Sintel): 0.922

== Bison ==
Bison has a 1/2-octave period and the generator can be taken as [[~]][[162/125]] or its semi-octave complement, ~[[35/32]]. Its [[ploidacot]] is diploid delta-heptacot ([[pergen]] (P8/2, ccP5/7)).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 6144/6125, 78732/78125

{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
: mapping generators: ~567/400, ~162/125

[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 599.9413{{c}}, ~162/125 = 443.0320{{c}} (~35/32 = 156.9093{{c}})
: [[error map]]: {{val| -0.117 -0.613 +1.092 +0.091 }}
* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})
: error map: {{val| 0.000 -0.446 +1.341 +0.446 }}

{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}

[[Badness]] (Sintel): 1.78

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 8019/8000

Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}

Optimal tunings:
* WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})

{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}

Badness (Sintel): 1.23

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 10985/10976

Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}

Optimal tunings:
* WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})

{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}

Badness (Sintel): 0.971

== Subpental ==
Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c4P4/14)).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 19683/19600

{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
: mapping generators: ~2, ~45/28

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9261{{c}}, ~45/28 = 821.4823{{c}}
: [[error map]]: {{val| -0.074 -0.611 +1.107 -0.052 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}

{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}

[[Badness]] (Sintel): 1.37

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 8019/8000

Mapping: {{mapping| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}

Optimal tunings:
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}

{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}

Badness (Sintel): 1.50

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 3136/3125

Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}

Optimal tunings:
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}

{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}

Badness (Sintel): 0.989

== Heinz ==
Heinz splits the sensipent generator ~324/125 in three. Its ploidacot is theta-21-cot (pergen (P8, c9P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 78732/78125

{{Mapping|legend=1| 1 -8 -10 6 | 0 21 27 -7 }}
: mapping generators: ~2, ~48/35

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
: [[error map]]: {{val| +0.425 -0.758 +1.061 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}

{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}

[[Badness]] (Sintel): 2.92

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 78732/78125

Mapping: {{mapping| 1 -8 -10 6 3 | 0 21 27 -7 1}}

: mapping generators: ~2, ~11/8

Optimal tunings:
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}

{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}

Badness (Sintel): 1.40

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440, 847/845

Mapping: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}

Optimal tunings:
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}

{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}

Badness (Sintel): 1.07

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440, 847/845

Mapping: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}

Optimal tunings:
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}

{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}

Badness (Sintel): 0.941

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968

Mapping: {{mapping| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}

Optimal tunings:
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}

{{Optimal ET sequence|legend=0| 46, 103h, 149h }}

Badness (Sintel): 1.16

== Trisensory ==
Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 78732/78125

{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
: mapping generators: ~63/50, ~36/35

[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.8117{{c}}, ~36/35 = 43.1270{{c}}
: [[error map]]: {{val| -0.565 -0.819 +0.700 +2.176 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}

{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}

[[Badness]] (Sintel): 2.27

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 78732/78125

Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}

Optimal tunings:
* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}

{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}

Badness (Sintel): 1.93

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 9295/9261

Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}

: mapping generators: ~49/39, ~36/35

Optimal tunings:
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}

{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}

Badness (Sintel): 1.44

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 351/350, 442/441, 540/539, 715/714

Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}

Optimal tunings:
* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}

{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}

Badness (Sintel): 1.23

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475

Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}

Optimal tunings:
* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}

{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}

Badness (Sintel): 1.12

== Other subgroup extensions ==
=== Sensipent (2.3.5.31 subgroup) ===
The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.

For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].

Subgroup: 2.3.5.31

Comma list: 961/960, 2511/2500

Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}

Optimal tunings:
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}

{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}

Badness (Sintel): 0.243

=== Sendai ===
{{See also| Sensipent #Sendai interval table }}

Sendai is an accurate extension of sensipent with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).

Subgroup: 2.3.5.23.29.31

Comma list: 465/464, 576/575, 621/620, 900/899

Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}

Optimal tunings:
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}

{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}

Badness (Sintel): 0.283

[[Category:Temperament families]]
[[Category:Sensipent family| ]] 
[[Category:Rank 2]]

10th-octave temperaments

2026-08-02T08:31:38Z

FloraC: Do not name/catalog equal temps. Refer to them by *2.13-subgroup 10et*, etc.

{{Technical data page}}
{{Infobox fractional-octave|10}}
[[10edo]] is notable for having close approximations of [[15/14]] to one step, the associated comma being the [[linus comma]] and [[13/8]] to 7 steps. 10th-octave temperaments naturally occur between any equal divisions of the octave whose greatest common divisor is 10.

Temperaments discussed elsewhere include: [[Quintosec family #Decoid|decoid]], [[Quintile family #Decile|decile]], [[Metric microtemperaments #Decimetra|decimetra]], [[Stearnsmic clan #Decistearn|decistearn]], [[Vishnu family #Decavish|decavish]], and [[Kalismic temperaments #Linus|linus]] (the 15/14 relationship mentioned above).

== Neon ==
: ''For extensions, see [[Ragismic microtemperaments #Deca]], [[Landscape microtemperaments #Zinc]], and [[20th-octave temperaments #Calcium]].''

Neon tempers out the [[neon comma]], {{monzo| 21 60 -50 }} in the 5-limit, equating [[3125/2916]] with one step of [[10edo]]. It was named by [[Eliora]] in 2023 after the 10th element.

Extending neon with [[ragisma]], or equivalently with the linus comma, produces [[deca]].

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 21 60 -50 }}

{{Mapping|legend=1| 10 4 9 | 0 5 6 }}
: mapping generators: ~3125/2916, {{monzo| 10 29 -24 }}

[[Optimal tuning]]s:
* [[WE]]: ~3125/2916 = 119.9999{{c}}, {{monzo| 10 29 -24 }} = 284.3890{{c}}
: [[error map]]: {{val| -0.001 -0.011 +0.019 }}
* [[CWE]]: ~3125/2916 = 120.0000{{c}}, {{monzo| 10 29 -24 }} = 284.3891{{c}}
: error map: {{val| 0.000 -0.010 +0.021 }}

{{Optimal ET sequence|legend=1| 80, 190, 270, 460, 730, 1730, 2460, 3190, 5650 }}

[[Badness]] (Sintel): 4.85

{{Navbox fractional-octave}}

[[Category:10edo]]

Gallery of just intervals

2026-08-01T11:35:37Z

FloraC: Updates

The following is a '''gallery of many just intervals''' that contributors to this wiki have found notable. It is of course not meant to be comprehensive, and is not the result of a systematic search.

Do not be intimidated by all the common names: there is no need to memorise the names. For pretty much all use cases, is perfectly acceptable - preferred, even - to just refer to an interval by its [[ratio]] (e.g. 3/2, 5/4).

== Introduction ==
In [[just intonation]], a musical [[interval]] is specified as a ratio of two frequencies. When two (or more) pitches are sounded that are in simple proportions to one another, there is a "fusing" quality to the sound which is often described as pleasing; hence the interest in tuning the pitches of musical systems according to such proportions. There is much debate as to what "[[consonance]]" means in a musical system, but in just intonation, it is generally assumed that lower numbers in frequency ratios lead to greater consonance.

In the actual performance of a piece of music, the number of factors involved are enormous, and it is not often helpful to reduce a musical experience to a one-dimensional description of "consonance versus dissonance." Hence the need for this gallery, to give life to conversation about what an interval means beyond the numerical description: "5/3" or "21/16" or what have you.

What follows is a gallery of just intervals in ascending order from [[1/1]] to [[2/1]] and beyond. No such list could possibly be complete (as there are infinite possible ratios), so please add intervals of interest as you see fit. Any rational interval is welcome, as long as the wiki author has some interest in it.

This page lists links to dedicated pages for each interval, under the heading "frequency ratio".

Said dedicated pages may include: descriptions of common usage, technical notes, poetry, links, reservations, complaints, chords or compositions that feature it, edos that approximate it, intervals that are functionally (or emotionally) related to it, nicknames, love letters, fan art, etc. Readers are encouraged to add such things to the interval pages. If your contribution is unconventional, add your name or pseudonym so future readers know where the idea originated.

== Gallery of just intervals ==
See also [[list of superparticular intervals]] and [http://www.huygens-fokker.org/docs/intervals.html List of intervals (Huygens-Fokker foundation)]

{| class="wikitable sortable right-3"
|-
! Frequency Ratio
! Cents Value (7 sig. dig.)
! [[Sagittal notation|Sagittal notation]]
! [[Color notation|Color Name]]
! [[Functional Just System|FJS Name]]
! class="unsortable" | Some common names
|-
| [[1/1]]
| 0.000000
| <ref group=note>The [[Sagittal]] column shows a pitch-class-sensitive [[Sagittal_notation#Athenian|Athenian]] [[Sagittal_notation#Evo|Evo]] notation with 1/1 = C. A white notehead {{sagittal|nhhf|size=300%}} indicates exact notation while black {{sagittal|nhbl|size=300%}} indicates approximation (typically within 2 ¢).</ref>&numsp;{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| w1, wa unison
| P1&numsp;&numsp;<ref group=note>If any [[FJS]] names are missing, please [https://misotanni.github.io/fjs/en/calc.html add] them.</ref>
| unity, perfect prime, Tonic, unison
|-
| [[32805/32768]]
| 1.953721
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| Ly-2, Layoma
| d-25
| schisma ({{Monzo|-15, 8, 1}})
|-
| [[256/255]]
| 6.775876
| {{sagittal| |( |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 17ug1, Suguma
| P185
| charisma, charic comma, septendecimal kleisma
|-
| [[225/224]]
| 7.711523
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ryy-2, Ruyoyoma
| d-2257
| septimal kleisma, marvel comma ({{Monzo|-5, 2, 2, -1}})
|-
| [[126/125]]
| 13.79477
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zg32, Zotriguma
| d27125
| septimal semicomma, starling comma ({{Monzo|1, 2, -3, 1}})
|-
| [[100/99]]
| 17.39948
| {{sagittal| ~|( |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1uyy1, Luyoyoma
| A12511
| Ptolemy's comma, ptolemisma
|-
| [[99/98]]
| 17.57613
| {{sagittal| !( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| 1orr-2, Loruruma
| m-21149
| mothwellsma
|-
| [[2048/2025]]
| 19.55257
| {{sagittal| //| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| sgg2, Saguguma
| d225
| diaschisma ({{Monzo|11, -4, -2}})
|-
| [[81/80]]
| 21.50629
| {{sagittal| /| |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| g1, Guma
| P15
| meantone comma, syntonic comma, Didymus comma
|-
| [[531441/524288]]
| 23.46001
| {{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| LLw-2, Lalawama
| d-2
| Pythagorean comma, ditonic comma ({{Monzo|-19, 12}})
|-
| [[66/65]]
| 26.43157
| {{sagittal| |) |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| 3u1og1, Thuloguma
| P11165
| winmeanma
|-
| [[65/64]]
| 26.84138
| {{sagittal| |) |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| 3oy1, Thoyoma
| P165
| wilsorma, 13th-partial chroma
|-
| [[64/63]]
| 27.26409
| {{sagittal| |) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| r1, Ruma
| P17
| septimal comma, Archytas' comma
|-
| [[3125/3072]]
| 29.61357
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| Ly5-2, Laquinyoma
| dd-23125
| magic comma, small diesis ({{Monzo|-10, -1, 5}})
|-
| [[50/49]]
| 34.97562
| {{sagittal| )|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| rryy-2, Biruyoma
| d-22549
| small septimal diesis, small septimal sixth-tone, septimal tritonic diesis, jubilisma
|-
| [[49/48]]
| 35.69681
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zz2, Zozoma
| m249
| large septimal diesis, large septimal sixth-tone, slendro diesis, semaphoresma
|-
| [[45/44]]
| 38.90577
| {{sagittal|(|(|size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1uy1, Luyoma
| A1511
| undecimal 1/5 tone
|-
| [[128/125]]
| 41.05886
| {{sagittal| (|\ |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| g32, Triguma
| d2125
| diesis, minor diesis, augmented comma, enharmonic comma ({{Monzo|7, 0, -3}})
|-
| [[525/512]]
| 43.40834
| {{sagittal| //| |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Lzyy1, Lazoyoyoma
| A1175
| Avicenna's enharmonic diesis, avicennma ({{Monzo|-9, 1, 2, 1}})
|-
| [[1053/1024]]
| 48.34767
| {{sagittal| (!/ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| L3o1, Lathoma
| P113
| tridecimal quartertone, tridecimal flattone comma
|-
| [[36/35]]
| 48.77038
| {{sagittal| /|) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| rg1, Ruguma
| P135
| septimal quarter tone, double comma
|-
| [[250/243]]
| 49.16614
| {{sagittal| (!/ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| y31, Triyoma
| A1125
| porcupine comma ({{Monzo|1, -5, 3}})
|-
| [[59049/57344]]
| 50.72410
| {{sagittal| |) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| Lr-2, Laruma
| d-27
| Harrison's comma ({{Monzo|-13, 10, 0, -1}})
|-
| [[100/97]]
| 52.73202
| {{sagittal| (!) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| 97uyy1, ninety-suyoyo unison
| A12597
| shrutar quarter tone
|-
| [[33/32]]
| 53.27294
| {{sagittal| /|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1o1, ilo unison
| P111
| undecimal quarter tone, undecimal diesis, al-Farabi's 1/4-tone, [[octave-reduced]] 33rd [[harmonic]]
|-
| [[648/625]]
| 62.56515
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| g42, quadgu 2nd
| d2625
| diminished comma, major diesis ({{Monzo|8, 4, -4}})
|-
| [[28/27]]
| 62.96090
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| z2, zo 2nd
| m27
| septimal third-tone, subminor second, septimal minor second
|-
| [[27/26]]
| 65.33734
| {{sagittal| (|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 3u1, thu unison
| A113
| small tridecimal third tone
|-
| [[26/25]]
| 67.90023
| {{sagittal| \! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3ogg2, thogugu 2nd
| d21325
| large tridecimal third tone
|-
| [[25/24]]
| 70.67243
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| yy1, yoyo unison
| A125
| chroma, classic chromatic semitone, Zarlinian semitone
|-
| [[24/23]]
| 73.68065
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23u1, twethu unison
| m223
| lesser vicesimotertial semitone
|-
|[[23/22]]
|76.95641
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23o1u2, twetholu 2nd
|A12311
|greater vicesimotertial semitone
|-
| [[68/65]]
| 78.11403
| {{sagittal| )!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17o3ug2, sothugu 2nd
| m21765
| valentine semitone
|-
| [[22/21]]
| 80.53704
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1or1, loru unison
| P1117
| undecimal minor semitone
|-
| [[64/61]]
| 83.11520
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 61u2, siwu 2nd
| m261
| harry minor semitone
|-
| [[21/20]]
| 84.46719
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zg2, zogu 2nd
| m275
| septimal chromatic semitone, aberschismic diatonic semitone, minor diatonic semitone
|-
| [[20/19]]
| 88.80070
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 19uy1, nuyo unison
| A1519
| small undevicesimal semitone, undevicesimal chroma, Eratosthenes' semitone
|-
| [[256/243]]
| 90.22500
| {{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| sw2, sawa 2nd
| m2
| Pythagorean limma, Pythagorean diatonic semitone, Pythagorean minor second ({{monzo| 8 -5 }})
|-
| [[135/128]]
| 92.17872
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Ly1, layo unison
| A15
| major limma ({{Monzo|-7, 3, 1}})
|-
| [[19/18]]
| 93.60301
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19o2, ino 2nd
| m219
| large undevicesimal semitone, undevicesimal limma, Boethius' semitone
|-
| [[18/17]]
| 98.95459
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 17u1, su unison
| A117
| small septendecimal semitone, Arabic lute index finger
|-
| [[17/16]]
| 104.9554
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17o2, iso 2nd
| m217
| large septendecimal semitone, octave-reduced 17th harmonic
|-
| [[16/15]]
| 111.7313
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| g2, gu 2nd
| m25
| classic diatonic semitone, classic minor second, octave-reduced 15th subharmonic
|-
| [[2187/2048]]
| 113.6850
| {{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Lw1, lawa unison
| A1
| Pythagorean apotome, Pythagorean chromatic semitone, Pythagorean augmented unison ({{monzo| -11 7 }})
|-
| [[77/72]]
| 116.2338
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1oz2, lozo 2nd
| m277
| undecimal secor
|-
| [[15/14]]
| 119.4428
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| ry1, ruyo unison
| A157
| septimal diatonic semitone, aberschismic chromatic semitone, major chromatic semitone
|-
| [[14/13]]
| 128.2982
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3uz2, thuzo 2nd
| M2713
| tridecimal 2/3-tone, trienthird, tridecimal supraminor second
|-
| [[27/25]]
| 133.2376
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| gg2, gugu 2nd
| m225
| large limma
|-
| [[13/12]]
| 138.5727
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3o2, tho 2nd
| m213
| tridecimal neutral second, tridecimal 2/3-tone
|-
| [[243/224]]
| 140.9491
| {{sagittal| |) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| Lr1, laru unison
| A17
| septimal subtone ({{Monzo|-5, 5, 0, -1}})
|-
| [[88/81]]
| 143.4980
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1o2, ilo 2nd
| m211
| undecimal subtone
|-
| [[25/23]]
| 144.3531
| {{sagittal| (!) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23uyy1, twethuyoyo unison
| M25,523
| small vicesimotertial neutral second
|-
| [[49/45]]
| 147.4281
| {{sagittal| (! |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| zzg3, zozogu 3rd
| d3495
| swetismic neutral second
|-
| [[12/11]]
| 150.6371
| {{sagittal| \!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1u2, lu 2nd
| M211
| small undecimal neutral second, 3/4-tone
|-
| [[35/32]]
| 155.1396
| {{sagittal| \!) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| zy2, zoyo 2nd
| M235
| septimal neutral second
|-
|[[23/21]]
|157.4934
| {{sagittal| //| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23or2, twethoru 2nd
|A1237
|large vicesimotertial neutral second
|-
| [[78/71]]
| 162.7861
| {{sagittal| \\! |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 71u3o2, seventy-wutho 2nd
| m21371
| porcupine neutral second
|-
| [[11/10]]
| 165.0042
| {{sagittal| (!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1og2, logu 2nd
| m2115
| undecimal submajor second, large undecimal neutral second, 4/5-tone, Ptolemy's second
|-
| [[54/49]]
| 168.2132
| {{sagittal| /|\ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| rr1, ruru unison
| A149
| Zalzal's mujannab
|-
| [[21/19]]
| 173.2679
| {{sagittal| (! |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19uz2, nuzo 2nd
| M2719
| large neutral undevicesimal second
|-
| [[10/9]]
| 182.4037
| {{sagittal| \! |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| y2, yo 2nd
| M25
| classic (whole) tone, minor (whole) tone
|-
| [[49/44]]
| 186.3339
| {{sagittal| |( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1uzz3, luzozo 3rd
| m34911
| mothwellsmic major second
|-
| [[39/35]]
| 187.3430
| {{sagittal| ~!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3org2, thorugu 2nd
| m21335
| animist major second
|-
| [[6272/5625]]
| 188.4870
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| szzg44, sabizogugu 4th
| ddd449625
| double marvelous second, double marvelous (whole) tone
|-
| [[19/17]]
| 192.5576
| {{sagittal| )!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19o17u2, nosu 2nd
| M21917
| quasi-meantone
|-
| [[28/25]]
| 196.1985
| {{sagittal| ~|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| zgg3, zogugu 3rd
| d3725
| middle major second
|-
| [[55/49]]
| 199.9798
| {{sagittal| !) |size=300%}}{{sagittal| x |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 1orry1, loruruyo unison
| A15549
| werckismic major second
|-
| [[64/57]]
| 200.5320
| {{sagittal| !( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19u2, inu 2nd
| M219
| quasi-tempered whole tone, octave-reduced 57th subharmonic
|-
| [[9/8]]
| 203.9100
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| w2, wa 2nd
| M2
| Pythagorean (whole) tone, major (whole) tone, octave-reduced 9th harmonic
|-
| [[26/23]]
| 212.2533
| {{sagittal| (| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 23u3o2, twethutho 2nd
| d31323
| vicesimotertial whole tone
|-
| [[17/15]]
| 216.6867
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 17og3, sogu 3rd
| d3175
| septendecimal whole tone, septendecimal eventone
|-
| [[25/22]]
| 221.3095
| {{sagittal| ~|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1uyy2, luyoyo 2nd
| A22511
| ptolemismic whole tone, undecimal acute whole tone
|-
| [[729/640]]
| 225.4163
| {{sagittal| /| |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| Lg2, lagu 2nd
| M25
| acute whole tone
|-
| [[57/50]]
| 226.8406
| {{sagittal| /|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19ogg3, nogugu 3rd
| d31925
| devichromic supermajor second
|-
| [[8/7]]
| 231.1741
| {{sagittal| |) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| r2, ru 2nd
| M27
| supermajor second, septimal whole tone, octave-reduced 7th subharmonic
|-
| [[63/55]]
| 235.1043
| {{sagittal| (!) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1uzg3, luzogu 3rd
| m3755
| werckismic supermajor second
|-
| [[55/48]]
| 235.6767
| {{sagittal| (| |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1oy2, loyo 2nd
| M255
| keenanismic supermajor second
|-
| [[224/195]]
| 240.0295
| {{sagittal| (|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3uzg3, thuzogu 3rd
| m3765
| quasi-tempered 1/5-octave, "5EDO"-esque tone
|-
| [[23/20]]
| 241.9606
| {{sagittal| (|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23og3, twethogu 3rd
| M2235
| vicesimotertial inframinor third, vicesimotertial ultramajor second, vicesimotertial semifourth
|-
| [[15/13]]
| 247.7411
| {{sagittal| //| |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3uy2, thuyo 2nd
| A2513
| tridecimal semifourth, tridecimal ultramajor second, tridecimal inframinor third
|-
| [[97/84]]
| 249.1145
| {{sagittal| //| |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 97or2, ninety-soru 2nd
| M2977
| homothetic semifourth
|-
| [[81/70]]
| 252.2680
| {{sagittal| /|) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| rg2, rugu 2nd
| M235
| septimal semi-augmented second, septimal ultramajor second
|-
| [[22/19]]
| 253.8049
| {{sagittal| /|) |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 19u1o2, nulo 2nd
| M21119
| undevicesimal semifourth, minimal minor third, godzilla semifourth
|-
| [[64/55]]
| 262.3683
| {{sagittal| (! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1ug3, lugu 3rd
| m355
| keenanismic subminor third, octave-reduced 55th subharmonic
|-
| [[7/6]]
| 266.8709
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| z3, zo 3rd
| m37
| subminor third, septimal minor third
|-
| [[90/77]]
| 270.0799
| {{sagittal| \!) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1ury2, luruyo 2nd
| A2577
| swetismic subminor third
|-
| [[62/53]]
| 271.5310
| {{sagittal| \! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 53u31o2, fithu-thiwo 2nd
| M23153
| orwell subminor third
|-
| [[75/64]]
| 274.5824
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| yy2, yoyo 2nd
| A225
| classic augmented second
|-
| [[27/23]]
| 277.5907
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| twethu 2nd, 23u2
| m323
| vicesimotertial augmented second
|-
| [[20/17]]
| 281.3583
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17uy2, suyo 2nd
| A2517
| septendecimal augmented second, septendecimal minor third, diatismic minor third
|-
| [[13/11]]
| 289.2097
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3o1u3, tholu 3rd
| m31311
| tridecimal minor third
|-
| [[45/38]]
| 292.7107
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19uy2, nuyo 2nd
| A2519
| Eratosthenes' minor third
|-
| [[32/27]]
| 294.1350
| {{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| w3, wa 3rd
| m3
| Pythagorean minor third, octave-reduced 27th subharmonic
|-
| [[19/16]]
| 297.5130
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19o3, ino 3rd
| m319
| otonal minor third, octave-reduced 19th harmonic
|-
| [[289/243]]
| 300.1358
| {{sagittal| (| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17oo4, soso 4th
| dd417, 17
| semitonismic quasi-tempered minor third
|-
| [[25/21]]
| 301.8465
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| ryy2, ruyoyo 2nd
| A2257
| quasi-tempered minor third
|-
| [[61/51]]
| 309.9744
| {{sagittal| )!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDbl|size=400%}}&numsp;&numsp;
| 61o17u2, siwosu 2nd
| A26117
| myna third
|-
| [[6/5]]
| 315.6413
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| g3, gu 3rd
| m35
| classic minor third, just minor third
|-
| [[77/64]]
| 320.1438
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1oz3, lozo 3rd
| m377
| keenanismic minor third, octave-reduced 77th harmonic
|-
| [[135/112]]
| 323.3528
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| ry2, ruyo 2nd
| A257
| large septimal minor third, marvelous minor third ({{Monzo|-4, 3, 1, -1}})
|-
| [[35/29]]
| 325.5624
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 29uzy3, twenuzoyo 3rd
| M33529
| doublewide minor third
|-
| [[23/19]]
| 330.7613
| {{sagittal| )|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 23o19u3, twethonu 3rd
| A22319
| vicesimotertial supraminor third
|-
| [[17/14]]
| 336.1295
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 17or3, soru 3rd
| m3175
| septendecimal supraminor third
|-
| [[175/144]]
| 337.5433
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| zyy3, zoyoyo 3rd
| M35,5,7
| keemic minor third
|-
| [[73/60]]
| 339.5208
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 73og3, seventy-thogu 3rd
| m3735
| amity supraminor third
|-
| [[28/23]]
| 340.5516
| {{sagittal| \\! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23uz3, twethuzo 3rd
| d4723
| vicesimotertial neutral third
|-
| [[39/32]]
| 342.4827
| {{sagittal| (!/ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3o3, tho 3rd
| m313
| lesser tridecimal neutral third, octave-reduced 39th harmonic
|-
| [[128/105]]
| 342.9054
| {{sagittal| /|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| rg3, rugu 3rd
| m335
| quasi-tempered 2/7-octave, 105th subharmonic
|-
| [[625/512]]
| 345.2549
| {{sagittal| |) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| Ly42, laquadyo 2nd
| AA2625
| 5-limit neutral third ({{Monzo|-9, 0, 4}})
|-
| [[11/9]]
| 347.4079
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1o3, ilo 3rd
| m311
| undecimal neutral third
|-
| [[60/49]]
| 350.6169
| {{sagittal| (| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| rry2, ruruyo 2nd
| A2549
| smaller septimal neutral third, (purple 3rd)
|-
| [[49/40]]
| 351.3381
| {{sagittal| (! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| zzg4, zozogu 4th
| d4495
| larger septimal neutral third, (purple 3rd)
|-
| [[27/22]]
| 354.5471
| {{sagittal| \!/ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1u3, lu 3rd
| M311
| rastmic neutral third
|-
| [[16/13]]
| 359.4723
| {{sagittal| (|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 3u3, thu 3rd
| M313
| greater tridecimal neutral third, octave reduced 13subharmonic
|-
|[[21/17]]
|365.8255
| {{sagittal| \\! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
|17uz3, suzo 3rd
|M3717
|septendecimal submajor third
|-
|[[26/21]]
|369.7468
| {{sagittal| (!( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
|thoru 3rd, 3or3
|m3137
|tridecimal submajor third
|-
|[[51/41]]
|377.8480
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
|41u17o4, fowuso 4th
|d41741
|
|-
|[[56/45]]
|378.6022
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
|zg4, zogu 4th
|d475
|narrow perde segah, marvelous major third
|-
| [[71/57]]
| 380.2285
| {{sagittal| !) |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 71o19u3, seventy-wonu 3rd
| M37119
| witchcraft major third
|-
| [[76/61]]
| 380.6282
| {{sagittal| b |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 61u19o4, siwuno 4th
| d41961
| magic major third
|-
| [[96/77]]
| 381.8112
| {{sagittal| !) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1ur3, luru 3rd
| M377
| undecimal perde segah, keenanismic major third
|-
| [[5/4]]
| 386.3137
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| y3, yo 3rd
| M35
| classic major third, just major third, octave-reduced 5th harmonic
|-
| [[44/35]]
| 396.1783
| {{sagittal| )!( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 1org3, lorugu 3rd
| n3115,7
| fwiwismic major third
|-
| [[161/128]]
| 397.1003
| {{sagittal| )!( |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 23oz4, twethozo 4th
| M3161
| just/Pythagorean major third meantone, octave-reduced 161th harmonic
|-
| [[34/27]]
| 399.0904
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 17o4, iso 4th
| d417
| septendecimal quasi-tempered major third
|-
| [[63/50]]
| 400.1085
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| zgg4, zogugu 4th
| d4725
| quasi-tempered major third
|-
| [[24/19]]
| 404.4420
| {{sagittal| !( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19u3, inu 3rd
| M319
| Boethius' major third
|-
| [[81/64]]
| 407.8200
| {{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| Lw3, lawa 3rd
| M3
| Pythagorean major third, octave-reduced 81st harmonic
|-
| [[19/15]]
| 409.2443
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19og4, nogu 4th
| d4195
| Eratosthenes' major third
|-
| [[33/26]]
| 412.7453
| {{sagittal| |( |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 3u1o3, thulo 3rd
| M31113
| tridecimal major third
|-
| [[80/63]]
| 413.5778
| {{sagittal| |( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ry3, ruyo 3rd
| M357
| 5/7-kleismic major third
|-
| [[14/11]]
| 417.5080
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1uz4, luzo 4th
| P4711
| undecimal major third, undecimal diminished fourth
|-
| [[23/18]]
| 424.3643
| {{sagittal| ~|( |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 23o4, twetho 4th
| M323
| vicesimotertial diminished fourth
|-
| [[32/25]]
| 427.3726
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| gg4, gugu 4th
| d425
| classic diminished fourth
|-
| [[41/32]]
| 429.0624
| {{sagittal| /| |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 41o3, fowo 3rd
| M341
| octave-reduced 41st harmonic, quadracesimoprimal supermajor third
|-
| [[77/60]]
| 431.8751
| {{sagittal| /|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1ozg4, lozogu 4th
| d4775
| swetismic supermajor third
|-
| [[9/7]]
| 435.0841
| {{sagittal| |) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| r3, ru 3rd
| M37
| supermajor third, septimal major third
|-
| [[31/24]]
| 443.0806
| {{sagittal| \!/ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 31o3, thiwo 3rd
| M331
| sensi supermajor third
|-
| [[22/17]]
| 446.3625
| {{sagittal| (|\ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 17u1o3, sulo 3rd
| M31117
| septendecimal supermajor third
|-
| [[35/27]]
| 449.2746
| {{sagittal| \!) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| zy4, zoyo 4th
| P435
| septimal semidiminished fourth
|-
| [[83/64]]
| 450.0473
| {{sagittal| \!) |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 83o4
| P483
| octave-reduced 83rd harmonic, octacesimotertial harmonic semisixth
|-
| [[13/10]]
| 454.2139
| {{sagittal| \\! |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 3og4, thogu 4th
| d4135
| tridecimal semisixth, Barbados third, tridecimal 9/4 tone
|-
| [[30/23]]
| 459.9944
| {{sagittal| (!( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23uy3, twethuyo 3rd
| P4523
| vicesimotertial ultramajor third
|-
| [[64/49]]
| 462.3482
| {{sagittal| /|\ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| rr3, ruru 3rd
| M349
| septatonic major third
|-
| [[17/13]]
| 464.4277
| {{sagittal| (! |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 17o3u4, sothu 4th
| P41713
| septendecimal sub-fourth
|-
| [[21/16]]
| 470.7809
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| z4, zo 4th
| P47
| sub-fourth, narrow fourth, 8ve-reduced 21st harmonic
|-
| [[25/19]]
| 475.1144
| {{sagittal| \!) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19uyy3, nuyoyo 3rd
| A35,519
| undevicesimal augmented third, undevicesimal grave fourth
|-
| [[95/72]]
| 479.9167
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 19oy4, noyo 4th
| P45,19
| undevicesimal quasi-tempered 2/5-octave, undevicesimal "[[5edo|5EDO]]"-esque fourth
|-
| [[128/97]]
| 480.1046
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 97u4
| P497
| nonacesimoseptimal subharmonic fourth
|-
| [[33/25]]
| 480.6455
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1ogg4, logugu 4th
| d41125
| ptolemismic fourth, "5EDO"-esque fourth
|-
| [[160/121]]
| 483.6778
| {{sagittal| ~!( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 1uuy4, luluyo 4th
| A4511,11
| narrow biyatismic fourth
|-
| [[324/245]]
| 483.8545
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| rrg3, rurugu 3rd
| M35,7,7
| sensamagic fourth
|-
| [[85/64]]
| 491.2691
| {{sagittal| !( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17oy4, soyo 4th
| P485
| charismic fourth, septendecimal harmonic fourth
|-
| [[117/88]]
| 493.1197
| {{sagittal| !( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3o1u4, tholu 4th
| P41311
| major-minthmic fourth ({{monzo|-3 2 0 0 -1 1}})
|-
| [[4/3]]
| 498.0450
| {{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| w4, wa 4th
| P4
| just perfect fourth, octave-reduced 3rd subharmonic, diatessaron
|-
| [[171/128]]
| 501.4230
| {{sagittal| |( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19o4, ino 4th
| P419
| undevicesimal harmonic fourth
|-
| [[75/56]]
| 505.7565
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ryy3, ruyoyo 3rd
| A3257
| marvelous fourth
|-
| [[121/90]]
| 512.4122
| {{sagittal| ~|( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 1oog4, lologu 4th
| d411,115
| wide biyatismic fourth
|-
| [[35/26]]
| 514.6120
| {{sagittal| ~|( |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3uzy4, thuzoyo 4th
| A45,713
| animist fourth
|-
| [[128/95]]
| 516.1733
| {{sagittal| ~|( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19ug4, nugu 4th
| P45,19
| undevicesimal subharmonic fourth
|-
| [[27/20]]
| 519.5513
| {{sagittal| /| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| g4, gu 4th
| P45
| acute fourth, wolf fourth
|-
| [[875/648]]
| 519.9470
| {{sagittal| /| |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| zy34, zotriyo 4th
| A45,5,5,7
| maviloid generator superfourth
|-
| [[23/17]]
| 523.3189
| {{sagittal| # |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 23o17u4, twethosu 4th
| A32317
| vicesimotertial acute fourth
|-
| [[19/14]]
| 528.6871
| {{sagittal| (| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19or4, noru 4th
| P4197
| hendrix fourth
|-
| [[34/25]]
| 532.3280
| {{sagittal| (|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17ogg5, sogugu 5th
| dd5175,5
| vengeance superfourth
|-
| [[49/36]]
| 533.7418
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zz5, zozo 5th
| d549
| Arabic lute acute fourth
|-
| [[15/11]]
| 536.9508
| {{sagittal| (|( |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1uy4, luyo 4th
| A4511
| sub-augmented fourth, undecimal augmented fourth
|-
| [[26/19]]
| 543.0146
| {{sagittal| //| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19u3o4, nutho 4th
| P41319
| undevicesimal super-fourth
|-
| [[48/35]]
| 546.8154
| {{sagittal| /|) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| rg4, rugu 4th
| P435
| septimal super-fourth
|-
| [[11/8]]
| 551.3179
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1o4, ilo 4th
| P411
| super-fourth, paramajor fourth, undecimal semi-augmented fourth, octave-reduced 11th harmonic, Alphorn-Fa
|-
| [[18/13]]
| 563.3823
| {{sagittal| (|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 3u4, thu 4th
| A413
| tridecimal augmented fourth
|-
| [[25/18]]
| 568.7174
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| yy4, yoyo 4th
| A425
| classic augmented fourth, pental augmented fourth ({{Monzo|-1 -2 2}})
|-
| [[32/23]]
| 571.7257
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 23u4, twethu 4th
| d523
| vicesimotertial narrow tritone, octave-reduced 23rd subharmonic
|-
| [[88/63]]
| 578.5820
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1or4, loru 4th
| P4117
| pentacircle diminished fifth
|-
| [[7/5]]
| 582.5122
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zg5, zogu 5th
| d575
| augmented fourth, septimal tritone, Huygen's tritone
|-
| [[108/77]]
| 585.7212
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1ur4, luru 4th
| A477
| swetismic augmented fourth ({{Monzo|2, 3, 0, -1, -1}})
|-
| [[1024/729]]
| 588.2700
| {{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| sw5, sawa 5th
| d5
| Pytharogean diminished fifth
|-
| [[45/32]]
| 590.2237
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| y4, yo 4th
| A45
| smaller pental tritone, diatonic tritone ({{Monzo|-5 2 1}})
|-
| [[128/91]]
| 590.6464
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3ur4, thuru 4th
| A491
| smaller huntmic tritone, 91st subharmonic
|-
| [[38/27]]
| 591.6480
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| ino 5th, 19o5
| d519
| smaller undevicesimal tritone
|-
| [[55/39]]
| 595.1490
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 3u1oy4, thuloyo 4th
| A45513
| smaller gassormic tritone
|-
| [[24/17]]
| 596.9996
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17u4, su 4th
| A417
| smaller septendecimal tritone
|-
| [[181/128]]
| 599.8151
| {{sagittal| )!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFbl|size=400%}}&numsp;&numsp;
| 181o4
| A4181
| octave-reduced 181st harmonic, otonal quasi-tempered tritone
|-
| [[99/70]]
| 600.0883
| {{sagittal| )!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1org4, lorugu 4th
| P41135
| homothetic quasi-tempered tritone
|-
| [[256/181]]
| 600.1849
| {{sagittal| )|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 181u5
| d5181
| octave-reduced 181st subharmonic, utonal quasi-tempered tritone
|-
| [[17/12]]
| 603.0004
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17o5, iso 5th
| d517
| larger septendecimal tritone
|-
| [[78/55]]
| 604.8510
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3o1ug5, tholugu 5th
| d51355
| larger gassormic tritone
|-
| [[27/19]]
| 608.3520
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| inu 4th, 19u4
| A419
| larger undevicesimal tritone
|-
| [[91/64]]
| 609.3536
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3oz5, thozo 5th
| d591
| larger huntmic tritone, octave-reduced 91st harmonic
|-
| [[64/45]]
| 609.7763
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| g5, gu 5th
| d55
| larger pental tritone, diatonic tritone ({{Monzo|6 -2 -1}})
|-
| [[729/512]]
| 611.7300
| {{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| Lw4, lawa 4th
| A4
| Pythagorean tritone
|-
| [[10/7]]
| 617.4878
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ry4, ruyo 4th
| A457
| diminished fifth, Euler's tritone, superaugmented fourth
|-
| [[23/16]]
| 628.2743
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23o5, twetho 5th
| A423
| vicesimotertial superaugmented fourth, octave-reduced 23rd harmonic
|-
| [[36/25]]
| 631.2826
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| gg5, gugu 5th
| d525
| pental diminished fifth, classic diminshed fifth ({{Monzo|2 2 -2}})
|-
| [[13/9]]
| 636.6177
| {{sagittal| (!/ |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 3o5, tho 5th
| d513
| tridecimal diminished fifth
|-
| [[16/11]]
| 648.6821
| {{sagittal| \!/ |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1u5, lu 5th
| P511
| sub-fifth, paraminor fifth, octave-reduced 11th subharmonic
|-
| [[35/24]]
| 653.1846
| {{sagittal| \!) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zy5, zoyo 5th
| P535
| septimal sub-fifth
|-
| [[19/13]]
| 656.9854
| {{sagittal| \\! |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19o3u5, nothu 5th
| P51913
| undevicesimal sub-fifth
|-
| [[22/15]]
| 663.0492
| {{sagittal| (!( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1og5, logu 5th
| d5115
| undecimal diminished fifth, semidiminished fifth
|-
| [[72/49]]
| 666.2589
| {{sagittal| /|\ |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| rr4, ruru 4th
| A449
| septimal catafifth
|-
| [[25/17]]
| 667.6720
| {{sagittal| (!) |size=300%}}{{sagittal| x |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 17uyy4, suyoyo 4th
| AA45,517
| vengeance subfifth
|-
| [[81/55]]
| 670.1883
| {{sagittal| (! |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1ug5, lugu 5th
| P555
| undecimal catafifth
|-
| [[28/19]]
| 671.3129
| {{sagittal| (! |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19uz5, nuzo 5th
| P5719
| hendrix fifth
|-
| [[34/23]]
| 676.6811
| {{sagittal| bb |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
| 23u17o5, twethuso 5th
| d61723
| vicesimotertial grave fifth
|-
| [[262144/177147]]
| 678.4950
| {{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| sasawa 6th, ssw6
| d6
| Pythagorean wolf fifth
|-
| [[40/27]]
| 680.4487
| {{sagittal| \!|size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| y5, yo 5th
| P55
| grave fifth
|-
| [[95/64]]
| 683.8267
| {{sagittal| ~!( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19oy5, noyo 5th
| P55,19
| undevicesimal harmonic fifth
|-
| [[52/35]]
| 685.3880
| {{sagittal| ~!( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3org5, thorugu 5th
| d51335
| animist fifth
|-
| [[180/121]]
| 687.5878
| {{sagittal| ~!( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1uuy5, luluyo 5th
| A5511,11
| narrow biyatismic fifth
|-
| [[112/75]]
| 694.2435
| {{sagittal| ~|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zgg6, zogugu 6th
| d6725
| marvelous fifth
|-
| [[121/81]]
| 694.8159
| {{sagittal| !( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1oo5, lolo 5th
| d511,11
| rastmic fifth, Alpharabian narrow fifth
|-
| [[323/216]]
| 696.6034
| {{sagittal| /| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
| 19o17o6, noso 6th
| d617,19
| undevicesimal meantone fifth
|-
| [[256/171]]
| 698.5770
| {{sagittal| !( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19u5, inu 5th
| P519
| undevicesimal subharmonic fifth
|-
| [[16384/10935]]
| 700.0013
| {{sagittal| /| |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| sg6, sagu 6th
| d65
| Kirnberger's fifth ("[[12edo]]"-esque fifth)
|-
| [[3/2]]
| 701.9550
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| w5, wa 5th
| P5
| [[just perfect fifth]], octave-reduced 3rd harmonic, diapente
|-
| [[182/121]]
| 706.7177
| {{sagittal| |) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 3o1uuz6, tholuluzo 6th
| m67,1311,11
| minor-minthmic fifth ({{monzo| 1 0 0 1 -2 1}})
|-
| [[176/117]]
| 706.8803
| {{sagittal| |( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 3u1o5, thulo 5th
| P51113
| major-minthmic fifth ({{monzo| 4 -2 0 0 1 -1 }})
|-
| [[128/85]]
| 708.7309
| {{sagittal| |( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17ug5, sugu 5th
| P585
| charismic fifth, septendecimal subharmonic fifth
|-
| [[245/162]]
| 716.1455
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zzy6, zozoyo 6th
| m65,7,7
| sensamagic fifth
|-
| [[121/80]]
| 716.3222
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 1oog5, lologu 5th
| d51215
| wide biyatismic fifth
|-
| [[50/33]]
| 719.3545
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1uyy5, luyoyo 5th
| A52511
| ptolemismic fifth, "5EDO"-esque fifth
|-
| [[97/64]]
| 719.8954
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 97o5
| P597
| nonacesimoseptimal harmonic fifth
|-
| [[144/95]]
| 720.0833
| {{sagittal| ~|( |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 19ug5, nugu 5th
| P55,19
| undevicesimal quasi-tempered 3/5-octave, undevicesimal "5EDO"-esque fifth
|-
| [[38/25]]
| 724.8856
| {{sagittal| /|) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 19ogg6, nogugu 6th
| d6195,5
| undevicesimal diminished sixth, undevicesimal acute fifth
|-
| [[32/21]]
| 729.2191
| {{sagittal| |) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| r5, ru 5th
| P57
| super-fifth, wide fifth, octave-reduced 21st subharmonic
|-
| [[26/17]]
| 735.5723
| {{sagittal| (| |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 17u3o5, sutho 5th
| P51317
| septendecimal super-fifth
|-
| [[49/32]]
| 737.6518
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zz6, zozo 6th
| m649
| superduper fifth, octave-reduced 49th harmonic
|-
| [[23/15]]
| 740.0056
| {{sagittal| (|( |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 23og6, twethogu 6th
| P5235
| vicesimotertial ultraminor sixth
|-
| [[20/13]]
| 745.7861
| {{sagittal| //| |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 3uy5, thuyo 5th
| A5513
| tridecimal semitenth, Barbados sixth, ratwolf wolf fifth
|-
| [[128/83]]
| 749.9527
| {{sagittal| /|) |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 83u5
| P583
| 83rd subharmonic, octacesimotertial subharmonic semitenth
|-
| [[17/11]]
| 753.6375
| {{sagittal| (!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 17o1u6, solu 6th
| m61711
| septendecimal subminor sixth
|-
| [[14/9]]
| 764.9159
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| z6, zo 6th
| m67
| subminor sixth, septimal minor sixth
|-
| [[25/16]]
| 772.6274
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| yy5, yoyo 5th
| A525
| pental augmented fifth, classic augmented fifth, otonal minor sixth, octave-reduced 25th harmonic
|-
| [[36/23]]
| 775.6357
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 23u5, twethu 5th
| m623
| vicesimotertial augmented fifth
|-
| [[11/7]]
| 782.4920
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 1or5, loru 5th
| P5117
| undecimal subminor sixth, undecimal augmented fifth
|-
| [[63/40]]
| 786.4222
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| zg6, zogu 6th
| m675
| narrow minor sixth
|-
| [[52/33]]
| 787.2547
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
| 3o1u6, tholu 6th
| m61311
| tridecimal minor sixth
|-
| [[30/19]]
| 790.7557
| {{sagittal| !) |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 19uy5, nuyo 5th
| A5519
| small undevicesimal minor sixth, Eratosthenes' minor sixth
|-
| [[128/81]]
| 792.1800
| {{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| sw6, sawa 6th
| m6
| Pythagorean minor sixth, octave-reduced 81st subharmonic
|-
| [[19/12]]
| 795.5580
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 19o6, ino 6th
| m619
| large undevicesimal minor sixth, Boethius' minor sixth
|-
| [[27/17]]
| 800.9096
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| 17u5, su 5th
| A517
| septendecimal quasi-tempered minor sixth
|-
|[[35/22]]
|803.8217
| {{sagittal| )|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|1uzy6, luzoyo 6th
|n65,711
|fwiwismic minor sixth
|-
|[[8/5]]
|813.6863
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|g6, gu 6th
|m65
|classic minor sixth, just minor sixth, octave-reduced 5th subharmonic
|-
|[[21/13]]
|830.2532
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|3uz6, thuzo 6th
|M6713
|tridecimal supraminor sixth
|-
|[[34/21]]
|834.1745
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|17or6, soru 6th
|m6177
|septendecimal supraminor sixth
|-
|[[13/8]]
|840.5277
| {{sagittal| (!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|3o6, tho 6th
|m613
|lesser tridecimal neutral sixth, octave-reduced 13th harmonic
|-
|[[80/49]]
|848.6619
| {{sagittal| (| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
|rry5, ruruyo aug 5th
|A5549
|(purple 6th)
|-
|[[49/30]]
|849.3832
| {{sagittal| (! |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|zzg7, zozogu 7th
|d7495
|(purple 6th)
|-
|[[18/11]]
|852.5921
| {{sagittal| \!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|1u6, lu 6th
|M611
|undecimal neutral sixth
|-
|[[105/64]]
|857.0946
| {{sagittal| \!) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|zy6, zoyo 6th
|M635
|quasi-tempered 5/7-octave, octave-reduced 105th harmonic
|-
|[[23/14]]
|859.4484
| {{sagittal| //| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
|23or6, twethoru 6th
|A5237
|vicesimotertial neutral sixth
|-
|[[28/17]]
|863.8705
| {{sagittal| \\! |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
|17uz6, suzo 6th
|M6717
|septendecimal submajor sixth
|-
|[[38/23]]
|869.2387
| {{sagittal| )!( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|23u19o6, twethuno 6th
|d71923
|vicesimotertial submajor sixth
|-
|[[5/3]]
|884.3587
| {{sagittal| \! |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|y6, yo 6th
|M65
|classic major sixth, just major sixth
|-
|[[42/25]]
|898.1535
| {{sagittal| ~|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|zgg7, zogugu 7th
|d7725
|quasi-tempered major sixth
|-
|[[486/289]]
|899.8642
| {{sagittal| (! |size=300%}}{{sagittal| x |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
|17uu5, susu 5th
|AA517, 17
|semitonismic quasi-tempered major sixth
|-
|[[32/19]]
|902.4870
| {{sagittal| !( |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|19u6, inu 6th
|M619
|utonal major sixth, octave-reduced 19th subharmonic
|-
|[[27/16]]
|905.8650
| {{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|w6, wa 6th
|M6
|Pythagorean major sixth, octave-reduced 27th harmonic
|-
| [[76/45]]
| 907.2893
| {{sagittal| |) |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| 19og7, nogu 7th
| d7195
| Eratosthenes' major sixth
|-
|[[22/13]]
|910.7908
| {{sagittal| |( |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|3u1o6, thulo 6th
|M61113
| tridecimal major sixth
|-
|[[17/10]]
|918.6417
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|17og7, sogu 7th
|d7175
|septendecimal diminished seventh, septendecimal major sixth, diatismic major sixth
|-
|[[46/27]]
|922.4093
| {{sagittal| ~|( |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|twetho 7th, 23o7
|M623
|vicesimotertial diminished seventh, vicesiomtertial supermajor sixth
|-
| [[12/7]]
|933.1291
| {{sagittal| |) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|r6, ru 6th
| M67
|supermajor sixth, septimal major sixth
|-
|[[55/32]]
|937.6317
| {{sagittal| (| |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|1oy6, loyo 6th
|M655
|keenanismic supermajor sixth, octave-reduced 55th harmonic
|-
|[[19/11]]
|946.1951
| {{sagittal| \!) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|19o1u7, nolu 7th
|m71911
|undevicesimal semitwelfth, maximal major sixth
|-
|[[140/81]]
|947.3196
| {{sagittal| \!) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|zy7, zoyo 7th
|m735
|septimal semidiminished seventh, septimal inframinor seventh
|-
| [[97/56]]
|951.0695
| {{sagittal| //| |size=300%}}{{sagittal|ntAbl|size=400%}}&numsp;&numsp;
|97or6, ninety-soru 6th
|M6977
|homothetic semitwelth
|-
|[[26/15]]
|952.2589
| {{sagittal| \\! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3og7, thogu 7th
|d7135
| tridecimal semitwelfth
|-
|[[40/23]]
|958.0394
| {{sagittal| (!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|23uy6, twethuyo 6th
|m7523
|vicesimotertial ultramajor sixth, vicesimotertial inframinor seventh, vicesimotertial semitwelfth
|-
|[[7/4]]
|968.8259
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|z7, zo 7th
|m77
|subminor seventh, septimal minor seventh, harmonic seventh, natural seventh, octave-reduced 7th harmonic
|-
|[[225/128]]
|976.5374
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|Lyy6, layoyo 6th
|A625
|marvel five-limit harmonic seventh, octave-reduced 225th harmonic ({{Monzo|-7, 2, 2}})
|-
|[[44/25]]
|978.6905
| {{sagittal| ~!( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|1ogg7, logugu 7th
|d7115,5
|ptolemismic minor seventh, undecimal grave minor seventh
|-
|[[30/17]]
|983.3133
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|17uy6, suyo 6th
|A6517
|septendecimal minor seventh
|-
|[[23/13]]
|987.7467
| {{sagittal| (! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|23o3u7, twethothu 7th
|A62313
|vicesimotertial minor seventh
|-
|[[71/40]]
|993.3828
|{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|71oy7
|m7715
| harmonic/just minor seventh meantone
|-
| [[16/9]]
|996.0900
|{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|w7, wa 7th
|m7
|Pythagorean minor seventh, small minor seventh, octave-reduced 9th subharmonic
|-
| [[57/32]]
| 999.4680
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| 19o7, ino 7th
| m719
| quasi-tempered minor seventh, octave-reduced 57th harmonic
|-
|[[25/14]]
|1003.802
| {{sagittal| ~!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ryy6, ruyoyo 6th
|A6257
|middle minor seventh
|-
|[[34/19]]
| 1007.442
| {{sagittal| )|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|19u17o7, nuso 7th
|m71719
|quasi-meantone minor seventh
|-
| [[9/5]]
|1017.596
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|g7, gu 7th
|m75
|classic minor seventh, large minor seventh
|-
|[[38/21]]
|1026.732
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|19or7, noru 7th
|m7197
|minor neutral undevicesimal seventh
|-
|[[29/16]]
|1029.577
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|29o7, tweno 7th
|m729
|vicesimononal supraminor seventh, octave-reduced 29th harmonic
|-
|[[20/11]]
|1034.996
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|1uy7, luyo 7th
|M7511
|undecimal supraminor seventh, small undecimal neutral seventh
|-
|[[42/23]]
|1042.507
| {{sagittal| \\! |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23uz7, twethuzo 7th
|d8723
|small vicesimotertial neutral seventh
|-
|[[64/35]]
|1044.860
| {{sagittal| /|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|rg7, rugu 7th
|m735
|septimal neutral seventh
|-
|[[11/6]]
|1049.363
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|1o7, ilo 7th
| m711
|undecimal neutral seventh, 21/4-tone
|-
|[[46/25]]
|1055.647
| {{sagittal| (|) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|twethogugu octave, 23ogg8
|m7235,5
|large vicesimotertial neutral seventh
|-
|[[24/13]]
|1061.427
| {{sagittal| (|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3u7, thu 7th
|M713
|tridecimal neutral seventh
|-
|[[50/27]]
|1066.762
| {{sagittal| \\! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|yoyo 7th, yy7
|M75,5
|grave major seventh
|-
|[[13/7]]
|1071.702
| {{sagittal| (!( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3or7, thoru 7th
|m7137
|tridecimal submajor seventh, 16/3-tone
|-
| [[28/15]]
| 1080.557
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| zg8, zogu octave
| d875
| septimal grave major seventh, aberschismic diminished octave
|-
|[[15/8]]
|1088.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|y7, yo 7th
|M75
| classic major seventh, just major seventh, octave-reduced 15th harmonic
|-
|[[32/17]]
|1095.045
| {{sagittal| ~!( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|17u7, su 7th
|M717
| septendecimal major seventh (FJS), septendecimal diminished octave (HEJI), [[octave-reduced]] 17th [[subharmonic]]
|-
|[[17/9]]
|1101.045
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|17o8, iso octave
|d817
| septendecimal major seventh (HEJI), septendecimal diminished octave (FJS)
|-
|[[36/19]]
|1106.397
| {{sagittal| !( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|19u7, inu 7th
|M719
|undevicesimal major seventh, Boethius' major seventh
|-
| [[243/128]]
| 1109.775
| {{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|Lw7, lawa 7th
|M7
|Pythagorean major seventh, [[octave-reduced]] 243rd harmonic
|-
|[[19/10]]
|1111.199
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|19og8, nogu 8ve
|d8195
|undevicesimal diminished octave, Eratosthenes' major seventh
|-
| [[40/21]]
| 1115.533
| {{sagittal| |( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ry7, ruyo 7th
| M757
| septimal acute major seventh, aberschismic major seventh
|-
|[[61/32]]
|1116.885
| {{sagittal| |( |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
|61o7, siwo 7th
|M761
|octave-reduced 61st harmonic
|-
|[[21/11]]
|1119.463
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|1uz8, luzo 8ve
|P8711
|large undecimal diminished octave, undecimal major seventh
|-
|[[44/23]]
|1123.044
| {{sagittal| (|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|23u1o7, twethulo 7th
|d81123
|small vicesimotertial major seventh
|-
|[[23/12]]
|1126.319
| {{sagittal| ~|( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|23o8, twetho 8ve
|M723
|large vicesimotertial major seventh
|-
|[[48/25]]
|1129.328
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|gg8, gugu octave
|d825
|classic diminished octave
|-
|[[25/13]]
|1132.100
| {{sagittal| /| |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|3uyy7, thuyoyo 7th
|A75,513
|lesser tridecimal diminished octave
|-
| [[27/14]]
| 1137.039
| {{sagittal| |) |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| r7, ru 7th
| M77
| supermajor seventh, septimal major seventh
|-
|[[31/16]]
|1145.036
| {{sagittal| \!/ |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
|31o7, thiwo 7th
|M731
|tricesimoprimal ultramajor seventh (FJS), tricesimoprimal semidiminished octave (HEJI), octave-reduced 31st harmonic
|-
|[[64/33]]
|1146.727
| {{sagittal| \!/ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|1u8, lu octave
| P811
|undecimal semidiminished octave, octave-reduced 33rd subharmonic
|-
|[[35/18]]
|1151.240
| {{sagittal| \!) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
|zy8, zoyo octave
|P835
|septimal semidiminished octave
|-
|[[96/49]]
|1164.303
| {{sagittal| /|\ |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
|rr7, ruru 7th
|M749
|septimal ultramajor seventh
|-
|[[49/25]]
| 1165.024
| {{sagittal| )!( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
|zzgg9, bizogu 9th
|d94925
|BP eighth
|-
| [[63/32]]
| 1172.736
| {{sagittal| !) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| z8, zo octave
| P87
| septimal suboctave, octave-reduced 63rd harmonic
|-
| [[160/81]]
| 1178.494
| {{sagittal| \! |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| y8, yo octave
| P85
| syntonic suboctave
|-
| [[255/128]]
| 1193.224
| {{sagittal| !( |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 17oy8, soyo octave
| P885
| charismic suboctave, octave-reduced 255th harmonic
|-
|[[2/1]]
| 1200.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
|w8, wa octave
|P8
|[[Octave|octave]], [[Wikipedia:Diapason|diapason]]
|}
Notes<references group=note/>

=== Intervals larger than an octave ===
{| class="wikitable sortable right-3"
|-
! Frequency Ratio
! Cents Value (7 sig. dig.)
! [[Sagittal notation|Sagittal notation]]
! colspan="2" |[[Color notation|Color Name]]
! FJS Name
! Some common names
|-
| [[27/13]]
| 1265.337
| <ref group=note>The [[Sagittal]] column shows a pitch-class-sensitive [[Sagittal_notation#Athenian|Athenian]] [[Sagittal_notation#Evo|Evo]] notation with 1/1 = C. A white notehead {{sagittal|nhhf|size=300%}} indicates exact notation while black {{sagittal|nhbl|size=300%}} indicates approximation (typically within 2 ¢).</ref>&hairsp;&hairsp;{{sagittal| (|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 3u8
| thu8ve&numsp;&numsp;&numsp;&numsp;&numsp;<ref group=note>In the color names, "co" or "c" stands for compound, {{w|Interval (music) #Compound intervals|a conventional music term}}.</ref>
| A813&numsp;<ref group=note>If any [[FJS]] names are missing, please [https://misotanni.github.io/fjs/en/calc.html add] them.</ref>
| Luxembourg eighth
|-
| [[23/11]]
| 1276.956
| {{sagittal| (!( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| 23o1u9
| twetholu 9th
| A82311
| vicesimotertial minor ninth
|-
| [[19/9]]
| 1293.603
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 19o9
| ino 9th
| m919
| undevicesimal minor ninth
|-
| [[17/8]]
| 1304.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 17o9
| iso 9th
| m917
| septendecimal minor ninth
|-
| [[32/15]]
| 1311.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| g9
| gu 9th
| m95
| classic minor ninth
|-
| [[15/7]]
| 1319.443
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| ry8
| ruyo 8ve
| A857
| septimal minor ninth
|-
| [[13/6]]
| 1338.573
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 3o9
| tho 9th
| m913
| tridecimal minor ninth, tridecimal neutral ninth
|-
| [[11/5]]
| 1365.004
| {{sagittal| (!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1o9
| ilo 9th
| m9115
| neutral ninth
|-
| [[20/9]]
| 1382.404
| {{sagittal| \! |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| y9
| yo 9th
| M95
| Ptolemaic ninth
|-
| [[9/4]]
| 1403.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| w9
| wa 9th
| M9
| major ninth
|-
| [[25/11]]
| 1421.309
| {{sagittal| ~|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| 1uyy9
| luyoyo 9th
| A92511
| undecimal major ninth
|-
| [[16/7]]
| 1431.174
| {{sagittal| |) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| r9
| ru 9th
| M97
| septimal major ninth
|-
| [[7/3]]
| 1466.871
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| z10
| zo 10th
| m107
| septimal minor tenth
|-
| [[19/8]]
| 1497.513
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| 19o3
| ino 10th
|
| undevicesimal minor tenth
|-
| [[12/5]]
| 1515.641
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| g10
| gu 10th
| m105
| classic minor tenth
|-
| [[17/7]]
| 1536.130
| {{sagittal| //| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEbl|size=400%}}&numsp;&numsp;
| 17or10
| soru 10th
| m10177
| septendecimal minor tenth
|-
| [[27/11]]
| 1554.547
| {{sagittal| \!/ |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c1u3
| colu 3rd
| M1011
| rastmic neutral tenth
|-
| [[5/2]]
| 1586.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| y10
| yo 10th
| M105
| classic major tenth
|-
| [[18/7]]
| 1635.084
| {{sagittal| |) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| r10
| ru 10th
|
| septimal supermajor tenth
|-
| [[13/5]]
| 1654.214
| {{sagittal| \\! |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 3og11
| thogu 11th
| d11135
| tridecimal eleventh
|-
| [[21/8]]
| 1670.781
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| z11
| zo 11th
|
| septimal subeleventh
|-
| [[8/3]]
| 1698.045
| {{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| w11, cw4
| wa 11th, cowa 4th
| P11
| perfect eleventh
|-
| [[19/7]]
| 1728.687
| {{sagittal| (| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 19or11
| noru 11th
|
| undevicesimal supereleventh
|-
| [[11/4]]
| 1751.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 1o11, c1o4
| ilo 11th, colo 4th
| P1111
| undecimal semi-augmented eleventh
|-
| [[14/5]]
| 1782.512
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zg12, czg5
| zogu 12th, cozogu 5th
| d1275
| short septimal navatone
|-
| [[17/6]]
| 1803.000
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| 17o12
| iso 12th
| d1217
| septendecimal navatone
|-
| [[20/7]]
| 1817.488
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ry11
| ruyo 11th
| A1157
| tall septimal navatone
|-
| [[23/8]]
| 1828.274
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| 23o12
| twetho 12th
| A1123
| trivigintal navatone
|-
| [[35/12]]
| 1853.185
| {{sagittal| \!) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| zy12, czy5
| zoyo 12th, cozoyo 5th
|
| septimal semi-diminished twelfth
|-
| [[3/1]]
| 1901.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| w12, cw5
| wa 12th, cowa 5th
| P12
| 3rd harmonic, perfect twelfth, tritave
|-
| [[28/9]]
| 1964.916
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| z13, cz6
| zo 13th, cozo 6th
|
| septimal subminor thirteenth
|-
| [[19/6]]
| 1995.558
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 19o13, c19o6
| ino 13th, cono 6th
|
| undevicesimal minor thirteenth
|-
| [[16/5]]
| 2013.686
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| g13, cg6
| gu 13th, cogu 6th
| m135
| classic minor thirteenth
|-
| [[13/4]]
| 2040.528
| {{sagittal| (!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| 3o13, c3o6
| tho 13th, cotho 6th
| m1313
| tridecimal neutral thirteenth
|-
| [[23/7]]
| 2059.448
| {{sagittal| //| |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c23or13, c23or6
| cotwethoru 6th
|
| trivigintal neutral/submajor thirteenth
|-
| [[10/3]]
| 2084.359
| {{sagittal| \! |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| y13, cy6
| yo 13th, coyo 6th
| M135
| just/classic(al)/ptolemaic major thirteenth
|-
| [[17/5]]
| 2118.642
| {{sagittal| (|( |size=300%}}{{sagittal| bb |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c17og7
| cosogu 7th
| d14175
| septendecimal major thirteenth
|-
| [[24/7]]
| 2133.129
| {{sagittal| |) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| r13, cr6
| ru 13th, coru 6th
|
| septimal supermajor thirteenth
|-
| [[7/2]]
| 2168.826
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cz7
| cozo 7th
| m147
| harmonic fourteenth
|-
| [[32/9]]
| 2196.090
| {{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cw7
| cowa 7th
|
| 3-limit minor fourteenth
|-
| [[18/5]]
| 2217.596
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cg7
| cogu 7th
| m145
| 5-limit minor fourteenth
|-
| [[11/3]]
| 2249.363
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c1o7
| colo 7th
| m1411
| undecimal neutral fourteenth
|-
| [[15/4]]
| 2288.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cy7
| coyo 7th
| M145
| just/classic(al)/ptolemaic major fourteenth
|-
| [[19/5]]
| 2311.199
| {{sagittal| |) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c19og8
| nogu double 8ve
|
| undevicesimal major fourteenth
|-
| [[35/9]]
| 2351.230
| {{sagittal| \!) |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| czy8
| zoyo double 8ve
|
| septimal semi-augmented fourteenth
|-
| [[4/1]]
| 2400.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
| cw8, ccw1
| wa double 8ve
| P15
| 4th harmonic, two octaves, just double octave
|-
| [[81/20]]
| 2421.506
| {{sagittal| /| |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| cg8, ccg1
| gu double 8ve
| P155
| syntonic double octave
|-
| [[25/6]]
| 2470.672
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| cyy8, ccyy1
| yoyo double 8ve
|
| classic chromatic minor sixteenth
|-
| [[17/4]]
| 2504.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c17o9, cc17o2
| coso 9th, cocoso 2nd
| m1617
| septendecimal minor sixteenth
|-
| [[64/15]]
| 2511.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| cg9, ccg2
| cogu 9th, cocogu 2nd
| m165
| classic minor sixteenth
|-
| [[13/3]]
| 2538.573
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3o9, cc3o2
| cotho 9th, cocotho 2nd
| m1613
| tridecimal minor sixteenth
|-
| [[22/5]]
| 2565.004
| {{sagittal| (!( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c1og9, cc1og2
| coco-logu 2nd
|
| large undecimal neutral sixteenth
|-
| [[9/2]]
| 2603.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| cw9, ccw2
| cowa 9th, cocowa 2nd
| M16
| classic large major sixteenth
|-
| [[23/5]]
| 2641.961
| {{sagittal| (|( |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| cc23og3
| coco-twethogu 3rd
|
| trivigintal supermajor sixteenth
|-
| [[14/3]]
| 2666.871
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccz3
| cocozo 3rd
| m177
| septimal subminor seventeenth
|-
| [[19/4]]
| 2696.513
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| cc19o3
| cocono 3rd
|
| undevicesimal minor seventeenth
|-
| [[24/5]]
| 2715.641
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccg3
| cocogu 3rd
|
| classic minor seventeenth
|-
| [[44/9]]
| 2747.408
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| cc1o3
| cocolo 3rd
|
| undecimal neutral seventeenth
|-
| [[5/1]]
| 2786.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccy3
| cocoyo 3rd
| M175
| 5th harmonic, pentave
|-
| [[36/7]]
| 2835.084
| {{sagittal| |) |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| ccr3
| cocoru 3rd
|
| septimal supermajor seventeenth
|-
| [[21/4]]
| 2870.781
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccz4
| cocozo 4th
|
| septimal subeighteenth
|-
| [[16/3]]
| 2898.045
| {{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccw4
| cocowa 4th
| P18
| perfect eighteenth
|-
| [[27/5]]
| 2919.551
| {{sagittal| /| |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccg4
| cocogu 4th
|
| 5-limit supereighteenth
|-
| [[11/2]]
| 2951.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| cc1o4
| cocolo 4th
| P1811
| undecimal semi-augmented eighteenth
|-
| [[28/5]]
| 2982.512
| {{sagittal| !( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cczg5
| coco-zogu 5th
| d1975
| short septimal panchadashatone
|-
| [[45/8]]
| 2990.224
| {{sagittal| \! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccy4
| cocoyo 4th
| A185
| quinquadragintimal panchadashatone
|-
| [[17/3]]
| 3003.000
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cc17o5
| cocoso 5th
| d1917
| septendecimal panchadasatone
|-
| [[40/7]]
| 3017.488
| {{sagittal| |( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| ccry4
| coco-ruyo 4th
| A1857
| tall septimal panchadashatone
|-
| [[23/4]]
| 3028.274
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| cc23o5
| coco-twetho 5th
|
| vicesimotertial panchadashatone
|-
| [[35/6]]
| 3053.185
| {{sagittal| \!) |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cczy5
| coco-zoyo 5th
|
| septimal semi-diminished nineteenth
|-
| [[6/1]]
| 3101.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| ccw5
| cocowa 5th
| P19
| 6th harmonic
|-
| [[55/9]]
| 3133.722
| {{sagittal| (| |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| cc1og5
| coco-logu 5th
|
| undecimal superninteenth
|-
| [[25/4]]
| 3172.627
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| ccyy5
| coco-yoyo 5th
|
| 5-limit subminor twentieth
|-
| [[19/3]]
| 3195.558
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| cc19o6
| cocono 6th
|
| undevicesimal minor twentieth
|-
| [[32/5]]
| 3213.686
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ccg6
| cocogu 6th
|
| classic minor twentieth
|-
| [[13/2]]
| 3240.528
| {{sagittal| (!/ |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| cc3o6
| cocotho 6th
| m2013
| tridecimal neutral twentieth
|-
| [[20/3]]
| 3284.359
| {{sagittal| \! |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ccy6
| cocoyo 6th
|
| classic major twentieth
|-
| [[48/7]]
| 3333.129
| {{sagittal| |) |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| ccr6
| cocoru 6th
|
| septimal supermajor twentieth
|-
| [[7/1]]
| 3368.826
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ccz7
| cocozo 7th
| m217
| 7th harmonic
|-
| [[36/5]]
| 3417.596
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ccg7
| cocogu 7th
|
| 5-limit minor twentyfirst
|-
| [[22/3]]
| 3449.363
| {{sagittal| /|\ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| cc1o7
| cocolo 7th
|
| undecimal neutral twentyfirst
|-
| [[15/2]]
| 3488.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| ccy7
| cocoyo 7th
| M215
| just/classic(al)/ptolemaic major twentyfirst
|-
| [[23/3]]
| 3526.319
| {{sagittal| ~|( |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c323o1
| twetho triple 8ve
|
| vicesimotertial supermajor twentyfirst
|-
| [[31/4]]
| 3545.036
| {{sagittal| \!/ |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| c331o1
| thiwo triple 8ve
|
| 31-limit semi-augmented twentyfirst
|-
| [[8/1]]
| 3600.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c3w1
| wa triple 8ve
| P22
| 8th harmonic, three octaves
|-
| [[33/4]]
| 3653.273
| {{sagittal| /|\ |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c31o1
| ilo triple 8ve
|
| undecimal semi-diminished twentythird
|-
| [[25/3]]
| 3670.672
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c3yy1
| yoyo triple 8ve
|
| classic chromatic twentythird
|-
| [[17/2]]
| 3704.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c317o2
| tricoso 2nd
|
| septendecimal minor twentythird
|-
| [[128/15]]
| 3711.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3g2
| tricogu 2nd
| m235
| classic minor twenty-third
|-
| [[26/3]]
| 3738.573
| {{sagittal| (!/ |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c33o2
| tricotho 2nd
|
| tridecimal neutral twentythird
|-
| [[35/4]]
| 3755.140
| {{sagittal| \!) |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3zy2
| trico-zoyo 2nd
|
| septimal neutral twentythird
|-
| [[9/1]]
| 3803.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c3w2
| tricowa 2nd
| M23
| 9th harmonic
|-
| [[10/1]]
| 3986.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c3y3
| tricoyo 3rd
| M245
| 10th harmonic, decade
|-
| [[11/1]]
| 4151.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c31o4
| tricolo 4th
| P2511
| 11th harmonic
|-
| [[12/1]]
| 4301.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c3w5
| tricowa 5th
| P26
| 12th harmonic
|-
| [[13/1]]
| 4440.528
| {{sagittal| /|) |size=300%}}{{sagittal|ntGbl|size=400%}}&numsp;&numsp;
| c33o6
| tricotho 6th
| m2713
| 13th harmonic
|-
| [[14/1]]
| 4568.826
| {{sagittal| !) |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c3z7
| tricozo 7th
| m287
| 14th harmonic
|-
| [[15/1]]
| 4688.269
| {{sagittal| \! |size=300%}}{{sagittal|ntBhf|size=400%}}&numsp;&numsp;
| c3y7
| tricoyo 7th
| M285
| 15th harmonic
|-
| [[16/1]]
| 4800.000
| {{sagittal|ntChf|size=400%}}&numsp;&numsp;
| c4w1
| wa quadruple 8ve
| P29
| 16th harmonic, four octaves
|-
| [[17/1]]
| 4904.955
| {{sagittal| ~|( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c417o2
| quadcoso 2nd
| m3017
| 17th harmonic
|-
| [[256/15]]
| 4911.731
| {{sagittal| /| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c4g2
| quadcogu 2nd
| m305
| classic minor thirtieth
|-
| [[18/1]]
| 5004.910
| {{sagittal|ntDhf|size=400%}}&numsp;&numsp;
| c4w2
| quadcowa 2nd
| M30
| 18th harmonic
|-
| [[19/1]]
| 5097.513
| {{sagittal| |( |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c419o3
| quadcono 3rd
| m3119
| 19th harmonic
|-
| [[20/1]]
| 5186.314
| {{sagittal| \! |size=300%}}{{sagittal|ntEhf|size=400%}}&numsp;&numsp;
| c4y3
| quadcoyo 3rd
| M315
| 20th harmonic
|-
| [[21/1]]
| 5270.781
| {{sagittal| !) |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c4z4
| quadcozo 4th
| P327
| 21st harmonic
|-
| [[22/1]]
| 5351.318
| {{sagittal| /|\ |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c41o4
| quadcolo 4th
| P3211
| 22nd harmonic
|-
| [[23/1]]
| 5428.274
| {{sagittal| ~|( |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntFhf|size=400%}}&numsp;&numsp;
| c423o5
| quadco-twetho 5th
| A3223
| 23rd harmonic
|-
| [[24/1]]
| 5501.955
| {{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c4w5
| quadcowa 5th
| P33
| 24th harmonic
|-
| [[25/1]]
| 5572.617
| {{sagittal| \\! |size=300%}}{{sagittal| # |size=300%}}{{sagittal|ntGhf|size=400%}}&numsp;&numsp;
| c4yy5
| quadco-yoyo 5th
| A335,5
| 25th harmonic
|-
| [[27/1]]
| 5705.865
| {{sagittal|ntAhf|size=400%}}&numsp;&numsp;
| c4w6
| quadcowa 6th
| M34
| 27th harmonic
|-
| [[29/1]]
| 5829.577
| {{sagittal| (| |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntBbl|size=400%}}&numsp;&numsp;
| c429o7
| quadco-tweno 7th
| m3529
| 29th harmonic
|-
| [[31/1]]
| 5945.036
| {{sagittal| \!/ |size=300%}}{{sagittal|ntCbl|size=400%}}&numsp;&numsp;
| c431o7
| quadco-thiwo 7th
| M3531
| 31st harmonic
|-
| [[49/1]]
| 6737.652
| {{sagittal| \!/ |size=300%}}{{sagittal| b |size=300%}}{{sagittal|ntAhf|size=400%}}&numsp;&numsp;
|c5zz6
|quinco-zozo 6th
| M417,7
| 49th harmonic
|}
Notes<references group=note />

== See also ==
* [[List of superparticular intervals]]
* [[:Category:Lists of intervals]] - Other galleries of intervals with different inclusion criteria to this one

== Links ==
* [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum] by [[Margo Schulter]] [https://www.webcitation.org/5xeoz4zmC Permalink]
* [http://www.huygens-fokker.org/docs/intervals.html Manuel Op de Coul interval list]
* [http://www.kylegann.com/Octave.html Anantomy of an Octave] by [[Kyle Gann]]
* [http://www.tallkite.com/AlternativeTunings.html Alternative Tunings: Theory, Notation and Practice] by [[Kite Giedraitis]]
* [https://misotanni.github.io/fjs/en/calc.html FJS interval calculators]
* [http://www.huygens-fokker.org/docs/intervals.html Stichting Huygens-Fokker: List of intervals]

[[Category:Lists]]
[[Category:Lists of intervals| ]]
[[Category:Just intonation]]