User:Hkm/19edo: Difference between revisions

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Created page with "{{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. C..."
 
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{{interwiki
A6d5
 
A4A2{{interwiki
| de = 19-EDO
| de = 19-EDO
| en = 19edo
| en = 19edo
| es = 19 EDO
| es = 19 EDO
| ja = 19平均律
| ja = 19平均律
}}
}}{{Harmonics in equal|19|columns=12}}
{{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|{{frac|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 ==
=== As an approximation of other temperaments ===
19edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for [[meantone]]. It is also suitable for [[magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its twelfths. 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. 19edo's 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, though [[46edo]] is 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 is in fact the second edo, after [[12edo]] which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy (unless you count [[15edo]]), and is the fifth [[zeta integral edo]], after 12edo. It is less successful in the [[7-limit]] (but still better than 12edo), as it conflates the septimal subminor third ([[7/6]]) with the septimal whole tone ([[8/7]]). 19edo also has the advantage of being excellent for [[negri]], [[keemun]], [[godzilla]], magic/muggles, and [[triton]]/[[liese]], and fairly decent for sensi. 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 a 7-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 consonant 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.
 
=== Prime harmonics ===
{{Harmonics in equal|19|columns=12}}
 
=== Adaptive tuning and octave stretch ===
Being a zeta integral 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]]; the closest [[the Riemann zeta function and tuning #Optimal octave stretch|local zeta peak]] to 19 occurs at 18.9481, which makes the octaves 1203.29{{c}}, and a step size of between 63.2–63.4{{c}} would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos 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 [[30ed3]] (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-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. The most extreme of these options would be [[11edf]], which has octaves stretched by 12.47{{c}}.
 
=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].
 
[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].
 
== Intervals ==
{| class="wikitable right-1 right-2 center-5 center-8"
{| class="wikitable right-1 right-2 center-5 center-8"
|-
|-
! [[Degree]]
! [[Degree]]
! [[Cent]]s
! [[Cent]]s
! [[Interval region|Interval Region]]
! [[Interval region|Interval]]
! Approximated [[Just intonation|JI]] Intervals<ref group="note">{{sg|limit=2.3.5.7.13 subgroup}}</ref>
! Approximated [[Just intonation|JI]]<ref group="note">{{sg|limit=2.3.5.7.13 subgroup}}</ref>
! [[Solfege]]
! [[Solfege]]
! colspan="2" | [[SKULO interval names|SKULO Interval]]
|-
|-
| 0
| 0
| 0.00
| 0.00
| Unison (prime)
| P1
| [[1/1]]
| [[1/1]]
| Do
| Do
| unison
| P1
|-
|-
| 1
| 1
| 63.16
| 63.16
| Augmented unison
| A1
| [[25/24]], [[26/25]], [[28/27]]
| [[25/24]], [[26/25]], [[28/27]]
| Di/Ro
| Di/Ro
| super unison, subminor second
| S1, sm2
|-
|-
| 2
| 2
| 126.32
| 126.32
| Minor second
| m2
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| Ra
| Ra
| minor second
| m2
|-
|-
| 3
| 3
| 189.47
| 189.47
| Major second
| M2
| [[9/8]], [[10/9]]
| [[9/8]], [[10/9]]
| Re
| Re
| major second
| M2
|-
|-
| 4
| 4
| 252.63
| 252.63
| Augmented second<br />Diminished third
| A2/d3
| [[7/6]], [[8/7]], [[15/13]]
| [[7/6]], [[8/7]], [[15/13]]
| Ri/Ma
| Ri/Ma
| supermajor second, subminor third
| SM2, sm3
|-
|-
| 5
| 5
| 315.79
| 315.79
| Minor third
| m3
| [[6/5]]
| [[6/5]]
| Me
| Me
| minor third
| m3
|-
|-
| 6
| 6
| 378.95
| 378.95
| Major third
| M3
| [[5/4]], [[16/13]], [[56/45]]
| [[5/4]], [[16/13]], [[56/45]]
| Mi
| Mi
| major third
| M3
|-
|-
| 7
| 7
| 442.11
| 442.11
| Augmented third
| A3/d4
| [[9/7]], [[13/10]], [[32/25]]
| [[9/7]], [[13/10]], [[32/25]]
| Mo/Fe
| Mo/Fe
| supermajor third, sub fourth
| SM3, s4
|-
|-
| 8
| 8
| 505.26
| 505.26
| Perfect fourth
| P4
| [[4/3]], [[75/56]]
| [[4/3]], [[75/56]]
| Fa
| Fa
| perfect fourth
| P4
|-
|-
| 9
| 9
| 568.42
| 568.42
| Augmented fourth<br />(Small [[tritone]])
| A4
| [[7/5]], [[18/13]], [[25/18]]
| [[7/5]], [[18/13]], [[25/18]]
| Fi
| Fi
| augmented fourth
| A4
|-
|-
| 10
| 10
| 631.58
| 631.58
| Diminished fifth<br />(Large [[tritone]])
| d5
| [[10/7]], [[13/9]], [[36/25]]
| [[10/7]], [[13/9]], [[36/25]]
| Se
| Se
| diminished fifth
| d5
|-
|-
| 11
| 11
| 694.74
| 694.74
| Perfect fifth
| P5
| [[3/2]], [[112/75]]
| [[3/2]], [[112/75]]
| So
| So
| perfect fifth
| P5
|-
|-
| 12
| 12
| 757.89
| 757.89
| Augmented fifth
| A5
| [[14/9]], [[20/13]], [[25/16]]
| [[14/9]], [[20/13]], [[25/16]]
| Si/Lo
| Si/Lo
| super fifth, subminor sixth
| S5, sm6
|-
|-
| 13
| 13
| 821.05
| 821.05
| Minor sixth
| m6
| [[8/5]], [[13/8]], [[45/28]]
| [[8/5]], [[13/8]], [[45/28]]
| Le
| Le
| minor sixth
| m6
|-
|-
| 14
| 14
| 884.21
| 884.21
| Major sixth
| M6
| [[5/3]]
| [[5/3]]
| La
| La
| major sixth
| M6
|-
|-
| 15
| 15
| 947.37
| 947.37
| Augmented sixth<br />Diminished seventh
| A6/d7
| [[7/4]], [[12/7]], [[26/15]]
| [[7/4]], [[12/7]], [[26/15]]
| Li/Ta
| Li/Ta
| supermajor sixth, subminor seventh
| SM6, sm7
|-
|-
| 16
| 16
| 1010.53
| 1010.53
| Minor seventh
| m7
| [[9/5]], [[16/9]]
| [[9/5]], [[16/9]]
| Te
| Te
| minor seventh
| m7
|-
|-
| 17
| 17
| 1073.68
| 1073.68
| Major seventh
| M7
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| Ti
| Ti
| major seventh
| M7
|-
|-
| 18
| 18
| 1136.84
| 1136.84
| Augmented seventh
| A7/d8
| [[25/13]], [[27/14]], [[48/25]]
| [[25/13]], [[27/14]], [[48/25]]
| To/Da
| To/Da
| supermajor seventh, sub octave
| SM7, s8
|-
|-
| 19
| 19
| 1200.00
| 1200.00
| Octave
| P8
| [[2/1]]
| [[2/1]]
| Do
| Do
| octave
| P8
|}
=== Interval quality and chord names in color notation ===
Using [[color notation]], qualities can be loosely associated with colors:
{| class="wikitable" style="text-align: center;"
|-
! Quality
! [[Color name|Color Name]]
! Monzo Format
! Examples
|-
| diminished
| zo
| (a, b, 0, 1)
| 7/6, 7/4
|-
| rowspan="2" | minor
| fourthward wa
| (a, b), b &lt; -1
| 32/27, 16/9
|-
| gu
| (a, b, -1)
| 6/5, 9/5
|-
| rowspan="2" | major
| yo
| (a, b, 1)
| 5/4, 5/3
|-
| fifthward wa
| (a, b), b &gt; 1
| 9/8, 27/16
|-
| augmented
| ru
| (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&#x1D12B; would have double-flats on B and E, and flats on C, D, F, G, and A.  Thinking of rewriting this key as A&#x266F; 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 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads:
{| class="wikitable center-1 center-2 center-3 center-4"
|-
! [[Kite's color notation|Color of the 3rd]]
! JI Chord
! Edosteps
! Notes of C Chord
! Written Name
! Spoken Name
|-
| zo (7-over)
| 6:7:9
| 0–4–11
| C–E&#x1D12B;–G
| Cm(&#x266D;3) or Cmin(&#x266D;3) or C(d3)
| C subminor, C minor flat-three, C dim-three
|-
| gu (5-under)
| 10:12:15
| 0–5–11
| C–E&#x266D;–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&#x266F;–G
| C(&#x266F;3) or Cmaj(&#x266F;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&#x1D12B;
| Ch7 or C,d7 or Cadd(d7)
| C harmonic 7, C (major) add dim-seven
|-
| gu (5-under)
| 12:10:8:7 or 1:6/5:3/2:12/7
| 0–5–11–15
| C–E&#x266D;–G–A&#x266F;
| Cm&#x266F;6 or CmA6 or Cm(add(&#x266F;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 Chord Names]] and [[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), solfege, 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 2nd 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<br />on D
|-
| 0
| 0.00
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 63.16
| Augmented unison (A1)<br />Diminished second (d2)
| D#<br />Ebb
|-
| 2
| 126.32
| Doubly augmented unison (AA1)<br />Minor second (m2)
| Dx<br />Eb
|-
| 3
| 189.47
| '''Major second (M2)'''<br />Doubly diminished third (dd3)
| '''E'''<br />Fbb
|-
| 4
| 252.63
| Augmented second (A2)<br />Diminished third (d3)
| E#<br />Fb
|-
| 5
| 315.79
| Doubly augmented second (AA2)<br />'''Minor third (m3)'''
| Ex<br />'''F'''
|-
| 6
| 378.95
| '''Major third (M3)'''<br />Doubly diminished fourth (dd4)
| '''F#'''<br />Gbb
|-
| 7
| 442.11
| Augmented third (A3)<br />Diminished fourth (d4)
| Fx<br />Gb
|-
| 8
| 505.26
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 9
| 568.42
| Augmented fourth (A4)<br />Doubly diminished fifth (dd5)
| G#<br />Abb
|-
| 10
| 631.58
| Doubly augmented fourth (AA4)<br />Diminished fifth (d5)
| Gx<br />Ab
|-
| 11
| 694.74
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 12
| 757.89
| Augmented fifth (A5)<br />Diminished sixth (d6)
| A#<br />Bbb
|-
| 13
| 821.05
| Doubly augmented fifth (AA5)<br />Minor sixth (m6)
| Ax<br />Bb
|-
| 14
| 884.21
| '''Major sixth (M6)'''<br />Doubly diminished seventh (dd7)
| '''B'''<br />Cbb
|-
| 15
| 947.37
| Augmented sixth (A6)<br />Diminished seventh (d7)
| B#<br />Cb
|-
| 16
| 1010.53
| Doubly augmented sixth (AA6)<br />'''Minor seventh (m7)'''
| Bx<br />'''C'''
|-
| 17
| 1073.68
| Major seventh (M7)<br />Doubly diminished octave (dd8)
| C#<br />Dbb
|-
| 18
| 1136.84
| Augmented seventh (A7)<br />Diminished octave (d8)
| Cx<br />Db
|-
| 19
| 1200.00
| '''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 (&#x266F;) and flats (&#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.


{{Sharpness-sharp1}}
{{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 ====
<imagemap>
<imagemap>
File:19-EDO_Evo_Sagittal.svg
File:19-EDO_Evo_Sagittal.svg
Line 460: Line 157:
default [[File:19-EDO_Revo_Sagittal.svg]]
default [[File:19-EDO_Revo_Sagittal.svg]]
</imagemap>
</imagemap>
=== 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
| 0.00
| P1
|-
| 1
| 63.16
| A1, m2
|-
| 2
| 126.32
| M2, m3
|-
| 3
| 189.47
| M3
|-
| 4
| 252.63
| m4, A3
|-
| 5
| 315.79
| M4, m5
|-
| 6
| 378.95
| M5
|-
| 7
| 442.11
| A5, d6
|-
| 8
| 505.26
| P6
|-
| 9
| 568.42
| A6, m7
|-
| 10
| 631.58
| M7, d8
|-
| 11
| 694.74
| P8
|-
| 12
| 757.89
| A8, m9
|-
| 13
| 821.05
| M9, m10
|-
| 14
| 884.21
| M10
|-
| 15
| 947.37
| m11, A10
|-
| 16
| 1010.53
| M11, m12
|-
| 17
| 1073.68
| M12
|-
| 18
| 1136.84
| A12, d13
|-
| 19
| 1200.00
| P13
|}


== Approximation to JI ==
== Approximation to JI ==
Line 559: Line 167:
{{ZPI
{{ZPI
| zpi = 65
| zpi = 65
| steps = 18.9480867166984
| steps = 18.948
| step size = 63.3309324546460
| step size = 63.330
| tempered height = 5.980169
| tempered height = 5.980
| pure height = 5.214351
| pure height = 5.214
| integral = 1.313799
| integral = 1.313
| gap = 16.699651
| gap = 16.699
| octave = 1203.28771663827
| octave = 1203.287
| consistent = 10
| consistent = 10
| distinct = 7
| distinct = 7

Revision as of 16:03, 11 April 2025

A6d5

A4A2

Approximation of prime harmonics in 19edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
Error Absolute (¢) +0.0 -7.2 -7.4 -21.5 +17.1 -19.5 +21.4 +18.3 +3.3 -19.1 -8.2 +1.3
Relative (%) +0.0 -11.4 -11.7 -34.0 +27.1 -30.8 +33.8 +28.9 +5.2 -30.2 -13.0 +2.0
Steps
(reduced) 		19
(0) 		30
(11) 		44
(6) 		53
(15) 		66
(9) 		70
(13) 		78
(2) 		81
(5) 		86
(10) 		92
(16) 		94
(18) 		99
(4)
Degree Cents Interval Approximated JI[note 1] Solfege
0 0.00 P1 1/1 Do
1 63.16 A1 25/24, 26/25, 28/27 Di/Ro
2 126.32 m2 13/12, 14/13, 15/14, 16/15 Ra
3 189.47 M2 9/8, 10/9 Re
4 252.63 A2/d3 7/6, 8/7, 15/13 Ri/Ma
5 315.79 m3 6/5 Me
6 378.95 M3 5/4, 16/13, 56/45 Mi
7 442.11 A3/d4 9/7, 13/10, 32/25 Mo/Fe
8 505.26 P4 4/3, 75/56 Fa
9 568.42 A4 7/5, 18/13, 25/18 Fi
10 631.58 d5 10/7, 13/9, 36/25 Se
11 694.74 P5 3/2, 112/75 So
12 757.89 A5 14/9, 20/13, 25/16 Si/Lo
13 821.05 m6 8/5, 13/8, 45/28 Le
14 884.21 M6 5/3 La
15 947.37 A6/d7 7/4, 12/7, 26/15 Li/Ta
16 1010.53 m7 9/5, 16/9 Te
17 1073.68 M7 13/7, 15/8, 24/13, 28/15 Ti
18 1136.84 A7/d8 25/13, 27/14, 48/25 To/Da
19 1200.00 P8 2/1 Do
Step offset −2 −1 0 +1 +2
Symbol  
 		 
 		 
 		 
 		 
Sagittal notationPeriodic table of EDOs with sagittal notation

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation

Approximation to JI

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Selected 19-limit intervals approximated in 19edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 19edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 19edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
5/3, 6/5 0.148 0.2
13/7, 14/13 1.982 3.1
15/13, 26/15 4.891 7.7
13/9, 18/13 5.039 8.0
15/14, 28/15 6.873 10.9
9/7, 14/9 7.021 11.1
9/5, 10/9 7.070 11.2
3/2, 4/3 7.218 11.4
5/4, 8/5 7.366 11.7
13/10, 20/13 12.109 19.2
13/12, 24/13 12.257 19.4
7/5, 10/7 14.091 22.3
7/6, 12/7 14.239 22.5
9/8, 16/9 14.436 22.9
15/8, 16/15 14.585 23.1
11/8, 16/11 17.103 27.1
13/8, 16/13 19.475 30.8
7/4, 8/7 21.457 34.0
11/6, 12/11 24.321 38.5
11/10, 20/11 24.469 38.7
11/7, 14/11 24.597 38.9
13/11, 22/13 26.580 42.1
15/11, 22/15 31.470 49.8
11/9, 18/11 31.539 49.9
15-odd-limit intervals in 19edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
5/3, 6/5 0.148 0.2
13/7, 14/13 1.982 3.1
15/13, 26/15 4.891 7.7
13/9, 18/13 5.039 8.0
15/14, 28/15 6.873 10.9
9/7, 14/9 7.021 11.1
9/5, 10/9 7.070 11.2
3/2, 4/3 7.218 11.4
5/4, 8/5 7.366 11.7
13/10, 20/13 12.109 19.2
13/12, 24/13 12.257 19.4
7/5, 10/7 14.091 22.3
7/6, 12/7 14.239 22.5
9/8, 16/9 14.436 22.9
15/8, 16/15 14.585 23.1
11/8, 16/11 17.103 27.1
13/8, 16/13 19.475 30.8
7/4, 8/7 21.457 34.0
11/6, 12/11 24.321 38.5
11/10, 20/11 24.469 38.7
11/9, 18/11 31.539 49.9
15/11, 22/15 31.688 50.2
13/11, 22/13 36.578 57.9
11/7, 14/11 38.561 61.1

Zeta peak index

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
65zpi 18.948 63.33 5.98 5.214 1.313 16.699 1203.287 3.287 10 7

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-30 19⟩ [⟨19 30]] +2.28 2.28 3.61
2.3.5 81/80, 3125/3072 [⟨19 30 44]] +2.58 1.91 3.02
2.3.5.7 49/48, 81/80, 126/125 [⟨19 30 44 53]] +3.85 2.76 4.35
2.3.5.7.13 49/48, 65/64, 81/80, 91/90 [⟨19 30 44 53 70]] +4.14 2.53 3.99
2.3.5.7.13.23 49/48, 65/64, 70/69, 81/80, 91/90 [⟨19 30 44 53 70 86]] +3.32 2.93 4.64
  • 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 34, 31, 27e, 22, and 26, respectively.
  • 19et is best in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is 53.

Uniform maps

13-limit uniform maps between 18.8 and 19.2
Min. size Max. size Wart notation Map
18.7816 18.9337 19e ⟨19 30 44 53 65 70]
18.9337 19.0518 19 ⟨19 30 44 53 66 70]
19.0518 19.0571 19f ⟨19 30 44 53 66 71]
19.0571 19.1651 19df ⟨19 30 44 54 66 71]
19.1651 19.2228 19cdf ⟨19 30 45 54 66 71]

Commas

19et tempers out the following commas. (Note: This assumes the val ⟨19 30 44 53 66 70].)

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

Linear temperaments

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.

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
Spell
4 252.63 A2, d3 1L 3s, 4L 1s,
5L 4s, 5L 9s 		Godzilla
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 / pycnic
Triton

Scales

MOS scales

Main article: List of MOS scales in Hkm/19edo

Octave-equivalent mosses

Other scales

Instruments

19 note per octave Ibanez conversion by Brad Smith (Indianapolis)
19edo 5 string Bass 34"-37" scale length
19edo bass conversion by Ron Sword

Music

Main article: 19edo/Music
See also: Category:19edo tracks
XA 19-ET Index
A number of compositions that were perfomed at the midwestmicrofest concert in 2007[dead link]

See also

Further reading

External links

Notes

  1. ↑ Based on treating 19edo as a 2.3.5.7.13 subgroup temperament; other approaches are also possible.
  2. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

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.
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