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| ja = 19平均律
| ja = 19平均律
}}
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{{High priority}}
{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{Wikipedia|19 equal temperament}}
{{ED intro}}
{{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 ==
== Theory ==
=== History ===
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.  
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.  
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.  


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


=== As an approximation of other temperaments ===
=== 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.
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.  


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.
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 ===
=== 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.
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.
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.
Line 32: Line 40:
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.
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 ===
=== Adaptive tuning ===
{{Harmonics in equal|19|columns=12}}
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]].
 
=== 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.  
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}}.
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 ===
=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]].
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. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].
[[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]].
=== 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 ==
== Intervals ==
{| class="wikitable right-1 right-2 center-5 center-8"
{| class="wikitable right-1 right-2"
|-
|-
! [[Degree]]
! [[Degree|#]]
! [[Cent]]s
! [[Cent]]s
! [[Interval region|Interval Region]]
! [[Interval category|Interval categories]]
! Approximated [[Just intonation|JI]] Intervals<ref group="note">{{sg|limit=2.3.5.7.13 subgroup}}</ref>
! Approximated ratios<ref group="note">As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament.</ref>
! [[Solfege]]
! colspan="2" | [[SKULO interval names|SKULO Interval]]
|-
|-
| 0
| 0
| 0.00
| 0.0
| Unison (prime)
| Unison (prime)
| [[1/1]]
| [[1/1]]
|-
| 1
| 63.2
| Augmented unison
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
|-
| 2
| 126.3
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 189.5
| Major second
| [[9/8]], [[10/9]]
|-
| 4
| 252.6
| Augmented second<br>Diminished third
| [[7/6]], [[8/7]], [[15/13]]
|-
| 5
| 315.8
| Minor third
| [[6/5]]
|-
| 6
| 378.9
| Major third
| [[5/4]], [[16/13]], [[56/45]]
|-
| 7
| 442.1
| Augmented third
| [[9/7]], [[13/10]], [[21/16]], [[32/25]]
|-
| 8
| 505.3
| Perfect fourth
| [[4/3]], [[75/56]]
|-
| 9
| 568.4
| Augmented fourth<br>(Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
|-
| 10
| 631.6
| Diminished fifth<br>(Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
|-
| 11
| 694.7
| Perfect fifth
| [[3/2]], [[112/75]]
|-
| 12
| 757.9
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]], [[32/21]]
|-
| 13
| 821.1
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
|-
| 14
| 884.2
| Major sixth
| [[5/3]]
|-
| 15
| 947.4
| Augmented sixth<br>Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
|-
| 16
| 1010.5
| Minor seventh
| [[9/5]], [[16/9]]
|-
| 17
| 1073.7
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
|-
| 18
| 1136.8
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
|-
| 19
| 1200.0
| Octave
| [[2/1]]
|}
<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
| Do
| unison
| Unison
| P1
| P1
|-
|-
| 1
| 1
| 63.16
| 63.2
| Augmented unison
| [[25/24]], [[26/25]], [[28/27]]
| Di/Ro
| Di/Ro
| super unison, subminor second
| Super unison, subminor second
| S1, sm2
| S1, sm2
|-
|-
| 2
| 2
| 126.32
| 126.3
| Ra
| Minor second
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| Ra
| minor second
| m2
| m2
|-
|-
| 3
| 3
| 189.47
| 189.5
| Re
| Major second
| Major second
| [[9/8]], [[10/9]]
| Re
| major second
| M2
| M2
|-
|-
| 4
| 4
| 252.63
| 252.6
| Augmented second<br />Diminished third
| [[7/6]], [[8/7]], [[15/13]]
| Ri/Ma
| Ri/Ma
| supermajor second, subminor third
| Supermajor second, subminor third
| SM2, sm3
| SM2, sm3
|-
|-
| 5
| 5
| 315.79
| 315.8
| Me
| Minor third
| Minor third
| [[6/5]]
| Me
| minor third
| m3
| m3
|-
|-
| 6
| 6
| 378.95
| 378.9
| Mi
| Major third
| Major third
| [[5/4]], [[16/13]], [[56/45]]
| Mi
| major third
| M3
| M3
|-
|-
| 7
| 7
| 442.11
| 442.1
| Augmented third
| [[9/7]], [[13/10]], [[32/25]]
| Mo/Fe
| Mo/Fe
| supermajor third, sub fourth
| Supermajor third, sub fourth
| SM3, s4
| SM3, s4
|-
|-
| 8
| 8
| 505.26
| 505.3
| Fa
| Perfect fourth
| Perfect fourth
| [[4/3]], [[75/56]]
| Fa
| perfect fourth
| P4
| P4
|-
|-
| 9
| 9
| 568.42
| 568.4
| Augmented fourth<br />(Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
| Fi
| Fi
| augmented fourth
| Augmented fourth
| A4
| A4
|-
|-
| 10
| 10
| 631.58
| 631.6
| Diminished fifth<br />(Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
| Se
| Se
| diminished fifth
| Diminished fifth
| d5
| d5
|-
|-
| 11
| 11
| 694.74
| 694.7
| So
| Perfect fifth
| Perfect fifth
| [[3/2]], [[112/75]]
| So
| perfect fifth
| P5
| P5
|-
|-
| 12
| 12
| 757.89
| 757.9
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]]
| Si/Lo
| Si/Lo
| super fifth, subminor sixth
| Super fifth, subminor sixth
| S5, sm6
| S5, sm6
|-
|-
| 13
| 13
| 821.05
| 821.1
| Le
| Minor sixth
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
| Le
| minor sixth
| m6
| m6
|-
|-
| 14
| 14
| 884.21
| 884.2
| La
| Major sixth
| Major sixth
| [[5/3]]
| La
| major sixth
| M6
| M6
|-
|-
| 15
| 15
| 947.37
| 947.4
| Augmented sixth<br />Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
| Li/Ta
| Li/Ta
| supermajor sixth, subminor seventh
| Supermajor sixth, subminor seventh
| SM6, sm7
| SM6, sm7
|-
|-
| 16
| 16
| 1010.53
| 1010.5
| Te
| Minor seventh
| Minor seventh
| [[9/5]], [[16/9]]
| Te
| minor seventh
| m7
| m7
|-
|-
| 17
| 17
| 1073.68
| 1073.7
| Ti
| Major seventh
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| Ti
| major seventh
| M7
| M7
|-
|-
| 18
| 18
| 1136.84
| 1136.8
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]]
| To/Da
| To/Da
| supermajor seventh, sub octave
| Supermajor seventh, sub octave
| SM7, s8
| SM7, s8
|-
|-
| 19
| 19
| 1200.00
| 1200.0
| Do
| Octave
| Octave
| [[2/1]]
| Do
| octave
| P8
| P8
|}
|}


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


{| class="wikitable" style="text-align: center;"
{| class="wikitable" style="text-align: center;"
|-
|-
! Quality
! Quality
! [[Color name|Color Name]]
! Color name
! Monzo Format
! Monzo format
! Examples
! Examples
|-
|-
| diminished
| Diminished
| zo
| zo
| (a, b, 0, 1)
| {{nowrap|(''a'', ''b'', 0, 1)}}
| 7/6, 7/4
| 7/6, 7/4
|-
|-
| rowspan="2" | minor
| rowspan="2" | Minor
| fourthward wa
| fourthward wa
| (a, b), b &lt; -1
| (''a'',&nbsp;''b''), {{nowrap|''b'' < −1}}
| 32/27, 16/9
| 32/27, 16/9
|-
|-
| gu
| gu
| (a, b, -1)
| {{nowrap|(''a'', ''b'', −1)}}
| 6/5, 9/5
| 6/5, 9/5
|-
|-
| rowspan="2" | major
| rowspan="2" | Major
| yo
| yo
| (a, b, 1)
| {{nowrap|(''a'', ''b'', 1)}}
| 5/4, 5/3
| 5/4, 5/3
|-
|-
| fifthward wa
| fifthward wa
| (a, b), b &gt; 1
| (''a'',&nbsp;''b''), {{nowrap| ''b'' > 1 }}
| 9/8, 27/16
| 9/8, 27/16
|-
|-
| augmented
| Augmented
| ru
| ru
| (a, b, 0, -1)
| {{nowrap|(''a'', ''b'', 0, −1)}}
| 9/7, 12/7
| 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.
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 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads:
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"
{| class="wikitable center-1 center-2 center-3 center-4"
|-
|-
! [[Kite's color notation|Color of the 3rd]]
! Color of the third
! JI Chord
! JI chord
! Edosteps
! Edosteps
! Notes of C Chord
! Notes of C chord
! Written Name
! Written name
! Spoken Name
! Spoken name
|-
|-
| zo (7-over)
| zo (7-over)
| 6:7:9
| 6:7:9
| 0–4–11
| 0–4–11
| C–E&#x1D12B;–G
| C–E𝄫–G
| Cm(&#x266D;3) or Cmin(&#x266D;3) or C(d3)
| Cm(♭3) or Cmin(♭3) or C(d3)
| C subminor, C minor flat-three, C dim-three
| C subminor, C minor flat-three, C dim-three
|-
|-
Line 280: Line 357:
| 10:12:15
| 10:12:15
| 0–5–11
| 0–5–11
| C–E&#x266D;–G
| C–E♭–G
| Cm or Cmin
| Cm or Cmin
| C minor
| C minor
Line 294: Line 371:
| 14:18:21
| 14:18:21
| 0–7–11
| 0–7–11
| C–E&#x266F;–G
| C–E♯–G
| C(&#x266F;3) or Cmaj(&#x266F;3) or C(A3)
| C(♯3) or Cmaj(♯3) or C(A3)
| C supermajor, C major sharp-three, C aug-three
| C supermajor, C major sharp-three, C aug-three
|-
|-
Line 301: Line 378:
| 4:5:6:7
| 4:5:6:7
| 0–6–11–15
| 0–6–11–15
| C–E–G–B&#x1D12B;
| C–E–G–B𝄫
| Ch7 or C,d7 or Cadd(d7)
| Ch7 or C,d7 or Cadd(d7)
| C harmonic 7, C (major) add dim-seven
| C harmonic 7, C (major) add dim-seven
|-
|-
| gu (5-under)
| gu (5-under)
| 12:10:8:7 or 1:6/5:3/2:12/7
| 1/(12:10:8:7)<br>(1–6/5–3/2–12/7)
| 0–5–11–15
| 0–5–11–15
| C–E&#x266D;–G–A&#x266F;
| C–E♭–G–A♯
| Cm&#x266F;6 or CmA6 or Cm(add(&#x266F;6)) or Cm(add(A6))
| Cm♯6 or CmA6 or Cm(add(♯6)) or Cm(add(A6))
| C minor (add) sharp-six, C minor (add) aug-six
| C minor (add) sharp-six, C minor (add) aug-six
|}
|}
Line 315: Line 392:
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.
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]].
For a more complete list, see [[19edo chords #Ups and downs notation]] and [[Kite's ups and downs notation #Chords and chord progressions]].


== Notation ==
== Notation ==
=== Standard 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.
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 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.
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"
{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 19edo
|+ style="font-size: 105%;" | Notation of 19edo
|-
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Degree|#]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard Notation]]
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
|-
! [[5L 2s|Diatonic Interval Names]]
! [[5L 2s|Diatonic interval names]]
! Note Names<br />on D
! Note names<br>on D
|-
|-
| 0
| 0
| 0.00
| 0.0
| '''Perfect unison (P1)'''
| '''Perfect unison (P1)'''
| '''D'''
| '''D'''
|-
|-
| 1
| 1
| 63.16
| 63.2
| Augmented unison (A1)<br />Diminished second (d2)
| Augmented unison (A1)<br>Diminished second (d2)
| D#<br />Ebb
| D#<br>Ebb
|-
|-
| 2
| 2
| 126.32
| 126.3
| Doubly augmented unison (AA1)<br />Minor second (m2)
| Doubly augmented unison (AA1)<br>Minor second (m2)
| Dx<br />Eb
| Dx<br>Eb
|-
|-
| 3
| 3
| 189.47
| 189.5
| '''Major second (M2)'''<br />Doubly diminished third (dd3)
| '''Major second (M2)'''<br>Doubly diminished third (dd3)
| '''E'''<br />Fbb
| '''E'''<br>Fbb
|-
|-
| 4
| 4
| 252.63
| 252.6
| Augmented second (A2)<br />Diminished third (d3)
| Augmented second (A2)<br>Diminished third (d3)
| E#<br />Fb
| E#<br>Fb
|-
|-
| 5
| 5
| 315.79
| 315.8
| Doubly augmented second (AA2)<br />'''Minor third (m3)'''
| Doubly augmented second (AA2)<br>'''Minor third (m3)'''
| Ex<br />'''F'''
| Ex<br>'''F'''
|-
|-
| 6
| 6
| 378.95
| 378.9
| '''Major third (M3)'''<br />Doubly diminished fourth (dd4)
| '''Major third (M3)'''<br>Doubly diminished fourth (dd4)
| '''F#'''<br />Gbb
| '''F#'''<br>Gbb
|-
|-
| 7
| 7
| 442.11
| 442.1
| Augmented third (A3)<br />Diminished fourth (d4)
| Augmented third (A3)<br>Diminished fourth (d4)
| Fx<br />Gb
| Fx<br>Gb
|-
|-
| 8
| 8
| 505.26
| 505.3
| '''Perfect fourth (P4)'''
| '''Perfect fourth (P4)'''
| '''G'''
| '''G'''
|-
|-
| 9
| 9
| 568.42
| 568.4
| Augmented fourth (A4)<br />Doubly diminished fifth (dd5)
| Augmented fourth (A4)<br>Doubly diminished fifth (dd5)
| G#<br />Abb
| G#<br>Abb
|-
|-
| 10
| 10
| 631.58
| 631.6
| Doubly augmented fourth (AA4)<br />Diminished fifth (d5)
| Doubly augmented fourth (AA4)<br>Diminished fifth (d5)
| Gx<br />Ab
| Gx<br>Ab
|-
|-
| 11
| 11
| 694.74
| 694.7
| '''Perfect fifth (P5)'''
| '''Perfect fifth (P5)'''
| '''A'''
| '''A'''
|-
|-
| 12
| 12
| 757.89
| 757.9
| Augmented fifth (A5)<br />Diminished sixth (d6)
| Augmented fifth (A5)<br>Diminished sixth (d6)
| A#<br />Bbb
| A#<br>Bbb
|-
|-
| 13
| 13
| 821.05
| 821.1
| Doubly augmented fifth (AA5)<br />Minor sixth (m6)
| Doubly augmented fifth (AA5)<br>Minor sixth (m6)
| Ax<br />Bb
| Ax<br>Bb
|-
|-
| 14
| 14
| 884.21
| 884.2
| '''Major sixth (M6)'''<br />Doubly diminished seventh (dd7)
| '''Major sixth (M6)'''<br>Doubly diminished seventh (dd7)
| '''B'''<br />Cbb
| '''B'''<br>Cbb
|-
|-
| 15
| 15
| 947.37
| 947.4
| Augmented sixth (A6)<br />Diminished seventh (d7)
| Augmented sixth (A6)<br>Diminished seventh (d7)
| B#<br />Cb
| B#<br>Cb
|-
|-
| 16
| 16
| 1010.53
| 1010.5
| Doubly augmented sixth (AA6)<br />'''Minor seventh (m7)'''
| Doubly augmented sixth (AA6)<br>'''Minor seventh (m7)'''
| Bx<br />'''C'''
| Bx<br>'''C'''
|-
|-
| 17
| 17
| 1073.68
| 1073.7
| Major seventh (M7)<br />Doubly diminished octave (dd8)
| Major seventh (M7)<br>Doubly diminished octave (dd8)
| C#<br />Dbb
| C#<br>Dbb
|-
|-
| 18
| 18
| 1136.84
| 1136.8
| Augmented seventh (A7)<br />Diminished octave (d8)
| Augmented seventh (A7)<br>Diminished octave (d8)
| Cx<br />Db
| Cx<br>Db
|-
|-
| 19
| 19
| 1200.00
| 1200.0
| '''Perfect octave (P8)'''
| '''Perfect octave (P8)'''
| '''D'''
| '''D'''
Line 436: Line 513:
In 19edo:
In 19edo:
* [[Ups and downs notation]] is identical to standard notation;
* [[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.
* 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}}
{{Sharpness-sharp1}}


=== Sagittal notation ===
=== 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]].
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 ====
==== Evo flavor ====
<imagemap>
{{Sagittal chart|Evo}}
File:19-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 327 0 487 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:19-EDO_Evo_Sagittal.svg]]
</imagemap>


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


==== Revo flavor ====
==== Revo flavor ====
<imagemap>
{{Sagittal chart}}
File:19-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 367 0 527 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:19-EDO_Revo_Sagittal.svg]]
</imagemap>


=== Dodecatonic notation ===
=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic Notation of 19edo
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic notation of 19edo
|-
|-
! [[Degree]]
! [[Degree|#]]
! [[Cent]]s
! [[Cent]]s
! Interval Names
! Interval names
|-
|-
| 0
| 0
| 0.00
| 0.0
| P1
| P1
|-
|-
| 1
| 1
| 63.16
| 63.2
| A1, m2
| A1, m2
|-
|-
| 2
| 2
| 126.32
| 126.3
| M2, m3
| M2, m3
|-
|-
| 3
| 3
| 189.47
| 189.5
| M3
| M3
|-
|-
| 4
| 4
| 252.63
| 252.6
| m4, A3
| m4, A3
|-
|-
| 5
| 5
| 315.79
| 315.8
| M4, m5
| M4, m5
|-
|-
| 6
| 6
| 378.95
| 378.9
| M5
| M5
|-
|-
| 7
| 7
| 442.11
| 442.1
| A5, d6
| A5, d6
|-
|-
| 8
| 8
| 505.26
| 505.3
| P6
| P6
|-
|-
| 9
| 9
| 568.42
| 568.4
| A6, m7
| A6, m7
|-
|-
| 10
| 10
| 631.58
| 631.6
| M7, d8
| M7, d8
|-
|-
| 11
| 11
| 694.74
| 694.7
| P8
| P8
|-
|-
| 12
| 12
| 757.89
| 757.9
| A8, m9
| A8, m9
|-
|-
| 13
| 13
| 821.05
| 821.1
| M9, m10
| M9, m10
|-
|-
| 14
| 14
| 884.21
| 884.2
| M10
| M10
|-
|-
| 15
| 15
| 947.37
| 947.4
| m11, A10
| m11, A10
|-
|-
| 16
| 16
| 1010.53
| 1010.5
| M11, m12
| M11, m12
|-
|-
| 17
| 17
| 1073.68
| 1073.7
| M12
| M12
|-
|-
| 18
| 18
| 1136.84
| 1136.8
| A12, d13
| A12, d13
|-
|-
| 19
| 19
| 1200.00
| 1200.0
| P13
| P13
|}
|}
Line 557: Line 622:
=== Interval mappings ===
=== Interval mappings ===
{{Q-odd-limit intervals|19}}
{{Q-odd-limit intervals|19}}
=== Zeta peak index ===
{{ZPI
| zpi = 65
| steps = 18.9480867166984
| step size = 63.3309324546460
| tempered height = 5.980169
| pure height = 5.214351
| integral = 1.313799
| gap = 16.699651
| octave = 1203.28771663827
| consistent = 10
| distinct = 7
}}


== Regular temperament properties ==
== Regular temperament properties ==
Line 587: Line 638:
| {{monzo| -30 19 }}
| {{monzo| -30 19 }}
| {{mapping| 19 30 }}
| {{mapping| 19 30 }}
| +2.28
| +2.277
| 2.28
| 2.277
| 3.61
| 3.612
|-
|-
| 2.3.5
| 2.3.5
| 81/80, 3125/3072
| 81/80, 3125/3072
| {{mapping| 19 30 44 }}
| {{mapping| 19 30 44 }}
| +2.58
| +2.578
| 1.91
| 1.911
| 3.02
| 3.025
|-
|-
| 2.3.5.7
| 2.3.5.7
| 49/48, 81/80, 126/125
| 49/48, 81/80, 126/125
| {{mapping| 19 30 44 53 }}
| {{mapping| 19 30 44 53 }}
| +3.85
| +3.848
| 2.76
| 2.755
| 4.35
| 4.362
|-
|-
| 2.3.5.7.13
| 2.3.5.7.13
| 49/48, 65/64, 81/80, 91/90
| 49/48, 65/64, 81/80, 91/90
| {{mapping| 19 30 44 53 70 }}
| {{mapping| 19 30 44 53 70 }}
| +4.14
| +4.135
| 2.53
| 2.530
| 3.99
| 4.006
|-
|-
| 2.3.5.7.13.23
| 2.3.5.7.13.23
| 49/48, 65/64, 70/69, 81/80, 91/90
| 49/48, 65/64, 70/69, 81/80, 91/90
| {{mapping| 19 30 44 53 70 86 }}
| {{mapping| 19 30 44 53 70 86 }}
| +3.32
| +3.319
| 2.93
| 2.936
| 4.64
| 4.649
|}
|}
* 19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit&mdash;''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 lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit&mdash;''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.  
Line 905: Line 956:
| 23
| 23
| [[70/69]]
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -}}
| {{monzo| 1 -1 1 1 0 0 0 0 -1 }}
| 24.91
| 24.91
| Twethuzoyo
| Twethuzoyo
Line 952: Line 1,003:
| Triaphonisma
| Triaphonisma
|}
|}
<references group="note" />


=== Linear temperaments ===
=== Linear temperaments ===
Line 985: Line 1,037:
| M2
| M2
| [[1L&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]]
| [[1L&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]]
| [[Deutone]]<br>[[Spell]]
| [[Deutone]] <br>[[Xenial]] / [[Sensamagic clan #Xenia|Xenia]] <br>[[Spell]]
|-
|-
| 4
| 4
Line 991: Line 1,043:
| A2, d3
| A2, d3
| [[1L&nbsp;3s]], [[4L&nbsp;1s]], <br>[[5L&nbsp;4s]], [[5L&nbsp;9s]]
| [[1L&nbsp;3s]], [[4L&nbsp;1s]], <br>[[5L&nbsp;4s]], [[5L&nbsp;9s]]
| [[Godzilla]]
| [[Godzilla]] / [[Helayo]]
|-
|-
| 5
| 5
Line 1,021: Line 1,073:
| A4
| A4
| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]], <br>[[2L&nbsp;15s]]
| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]], <br>[[2L&nbsp;15s]]
| [[Liese]] / [[pycnic]]<br>[[Triton]]
| [[Liese]] <br>[[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 ==
== Scales ==
Line 1,029: Line 1,086:


==== Octave-equivalent mosses ====
==== Octave-equivalent mosses ====
* [[meantone]] pentatonic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[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]] 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
* [[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]][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]][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
* [[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]][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]][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
* [[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]][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
* [[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]][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]][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
* [[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]][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]][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]][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
* [[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
* [[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 ===
=== Other scales ===
{{Main|19edo modes}}
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 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
* 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 – 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 (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
* 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
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* enharmonic pentatonic: 2 6 3 2 6
* Enharmonic pentatonic: 2 6 3 2 6
* enharmonic pentatonic: 6 2 3 6 2
* Enharmonic pentatonic: 6 2 3 6 2
* enharmonic octave species: 1 1 6 3 1 1 6
* Enharmonic octave species: 1 1 6 3 1 1 6
* enharmonic octave species: 6 1 1 3 6 1 1
* Enharmonic octave species: 6 1 1 3 6 1 1
* enharmonic octave species: 1 6 1 3 1 6 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 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 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 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
* [[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
* [[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
* [[Antipental blues]]: 4 4 1 2 4 4
Line 1,108: Line 1,167:
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]  
* [[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]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]
=== Notes ===
<references group="note" />


=== References ===
=== References ===

Latest revision as of 00:42, 2 June 2026

← 18edo 19edo 20edo →
Prime factorization 19 (prime)
Step size 63.1579 ¢ 
Fifth 11\19 (694.737 ¢)
Semitones (A1:m2) 1:2 (63.16 ¢ : 126.3 ¢)
Consistency limit 9
Distinct consistency limit 5
English Wikipedia has an article on:

19 equal divisions of the octave (abbreviated 19edo or 19ed2), also called 19-tone equal temperament (19tet) or 19 equal temperament (19et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 19 equal parts of about 63.2 ¢ each. Each step represents a frequency ratio of 21/19, or the 19th root of 2.

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 ¢; the fifth of 19edo is 694.737 ¢, 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 50 equal temperament (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

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)

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 scales in 19edo offering a great abundance of septimal tetrads. The Graham complexity of the 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 ¢ 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 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. See undevigintone. 57edo effectively corrects the harmonic 7 to just, although it is 76edo that fits the best. See meanmag.

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 22\37.

Intervals

# Cents Interval categories Approximated ratios[note 1]
0 0.0 Unison (prime) 1/1
1 63.2 Augmented unison 25/24, 26/25, 27/26, 28/27
2 126.3 Minor second 13/12, 14/13, 15/14, 16/15
3 189.5 Major second 9/8, 10/9
4 252.6 Augmented second
Diminished third 		7/6, 8/7, 15/13
5 315.8 Minor third 6/5
6 378.9 Major third 5/4, 16/13, 56/45
7 442.1 Augmented third 9/7, 13/10, 21/16, 32/25
8 505.3 Perfect fourth 4/3, 75/56
9 568.4 Augmented fourth
(Small tritone) 		7/5, 18/13, 25/18
10 631.6 Diminished fifth
(Large tritone) 		10/7, 13/9, 36/25
11 694.7 Perfect fifth 3/2, 112/75
12 757.9 Augmented fifth 14/9, 20/13, 25/16, 32/21
13 821.1 Minor sixth 8/5, 13/8, 45/28
14 884.2 Major sixth 5/3
15 947.4 Augmented sixth
Diminished seventh 		7/4, 12/7, 26/15
16 1010.5 Minor seventh 9/5, 16/9
17 1073.7 Major seventh 13/7, 15/8, 24/13, 28/15
18 1136.8 Augmented seventh 25/13, 27/14, 48/25, 52/27
19 1200.0 Octave 2/1
  1. ↑ As a 2.3.5.7.13-subgroup temperament.

Proposed interval names and solfèges

Table of proposed interval names and solfèges
# Cents Solfège 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:

Quality Color name Monzo format Examples
Diminished zo (a, b, 0, 1) 7/6, 7/4
Minor fourthward wa (a, b), b < −1 32/27, 16/9
gu (a, b, −1) 6/5, 9/5
Major yo (a, b, 1) 5/4, 5/3
fifthward wa (a, b), b > 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𝄫 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:

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 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.

Notation of 19edo
# Cents Standard notation
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 () and sagittal flat () respectively.
Step offset −2 −1 0 +1 +2
Symbol  
 		 
 		 
 		 
 		 

Sagittal notation

This notation uses the same sagittal sequence as edos 5, 12, and 26, and is a subset of the notations for edos 38, 57, and 76.

Evo flavor

19-EDO_Evo_Sagittal.svg

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

Revo flavor

19-EDO_Revo_Sagittal.svg

Dodecatonic notation

Dodecatonic notation of 19edo
# Cents 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

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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

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-30 19⟩ [⟨19 30]] +2.277 2.277 3.612
2.3.5 81/80, 3125/3072 [⟨19 30 44]] +2.578 1.911 3.025
2.3.5.7 49/48, 81/80, 126/125 [⟨19 30 44 53]] +3.848 2.755 4.362
2.3.5.7.13 49/48, 65/64, 81/80, 91/90 [⟨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 [⟨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 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 1] 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 -1⟩ 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
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

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
Xenial / 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 ¢, 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 65zpi.

Scales

MOS scales

Main article: List of MOS scales in 19edo

Octave-equivalent mosses

Other scales

Main article: 19edo modes

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

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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