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<span style="display: block; text-align: right;">[[:de:19edo|Deutsch]] - [[19平均律|日本語]]</span>
{{interwiki
| de = 19-EDO
| en = 19edo
| es = 19 EDO
| ja = 19平均律
}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{ED intro}}


=Theory=
== 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 music, '''19 equal temperament''', called 19-TET, 19-[[EDO|EDO]], or 19-ET, is the scale derived by dividing the [[Octave|octave]] into 19 [[Equal|equal]]ly large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 [[cent|cents]]. It is the 8th [[prime_numbers|prime]] edo, following [[17edo|17edo]] and coming before [[23edo|23edo]].
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.


Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur_Dieu_ta_pitié|Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theorist Francisco de Salinas in effect proposed it. Salinas discussed [[1-3_Syntonic_Comma_Meantone|1/3-comma meantone]], in which the fifth is of size 694.786 cents; the fifth of 19-et is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19-et. In the nineteenth century 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://sonic-arts.org/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).
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]).


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


The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for [[Meantone_family|meantone]] temperament. It is also a suitable for [[Regular_Temperaments#magic|magic/muggles]] temperament, because five of its major thirds are equivalent to one of its ''twelfths.'' For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is [[31edo|31 equal temperament]]. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; [[41edo|41 equal temperament]] more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.
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.  


However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with [[Harmonic_Limit|5-limit]] music in a tolerable manner, and is the fifth (after 12) [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]]. It is less successful with [[7-limit|7-limit]] (but still better than 12-et), as it eliminates the distinction between a septimal minor third ([[7/6|7/6]]), and a septimal whole tone ([[8/7|8/7]]). 19-EDO 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 scales in 19-EDO offering a great abundance of septimal tetrads. The [[Graham_complexity|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.
=== Prime harmonics ===
{{Harmonics in equal|19|columns=12}}


Being a zeta integral tuning, the 13-limit is represented relatively well, and practically [[19edo|19-edo]] 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. The same cannot be said of 12edo, in which the 5th and 7th are - not only farther than they are in 19, but fairly sharp already.
=== 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.  


Another option would be to use a stretched octave; the [[The_Riemann_Zeta_Function_and_Tuning|zeta function]]-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them. 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 instance, if we're using [[93ed30|93ed30]] (a variant of 19edo in which 30:1 is just), 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.
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.


==As a means of extending harmony==
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.


Because 19 EDO allows for more blended, consonant harmonies than 12 EDO does, 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 12 EDO blend much better in 19 EDO.
=== 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 19 EDO 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 ton<span style="line-height: 15.6px;">l</span><span style="line-height: 1.5;">aity without sounding too alien.</span>
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.


The narrow whole tones and wide diatonic semitones of 19edo give the diatonic scale a somewhat duller quality, but has the opposite effect on the pentatonic scale, which becomes much more expressive owing to the 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.
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.


==Intervals and linear temperaments==
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.


* [[List_of_19et_rank_two_temperaments_by_badness|List of 19et rank two temperaments by badness]]
=== Adaptive tuning ===
* [[List_of_19et_rank_two_temperaments_by_complexity|List of 19et rank two temperaments by complexity]]
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&nbsp;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]].
* [[List_of_edo-distinct_19et_rank_two_temperaments|List of edo-distinct 19et rank two temperaments]]


Since 19 is prime, all rank two temperaments in 19edo have one period per octave. Therefore you can make a correspondence between intervals and the linear temperaments they generate.
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.  


{| class="wikitable"
Another option would be to use [[octave stretching]], which has similar benefits to adaptive use, but it also works for fixed-pitch Instruments. For more on that see the [[19edo#Octave stretch or compression|section on octave stretch]].
 
=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]]. As such, it does not contain any nontrivial subset edos, though it contains [[19ed4]].
 
[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. 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 ==
{| class="wikitable center-1 right-2"
|-
|-
! | Degrees of 19edo
! [[Degree|#]]
! | Solfege
! [[Cent]]s
! | Diatonic Category
! Note
! | Dodecatonic category
! Approximated ratios<ref group="note">As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament</ref>
! | Cents
! [[Interval category]]
! colspan="2" | Ratios*
! | Generator for
|-
|-
| style="text-align:center;" | 0
| 0
| style="text-align:center;" | do
| 0.0
| style="text-align:center;" | P1
| D
| style="text-align:center;" | P1
| [[1/1]]
| style="text-align:center;" | 0
| Unison (prime)
| colspan="2" style="text-align:center;" | 1/1
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 1
| 1
| style="text-align:center;" | di
| 63.2
| style="text-align:center;" | A1, d2
| D♯
| style="text-align:center;" | A1, m2
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
| style="text-align:center;" | 63.1579
| Augmented unison
| colspan="2" style="text-align:center;" | 25/24, 21/20, 28/27, 26/25, 27/26
| style="text-align:center;" | Unicorn/rhinocerus
|-
|-
| style="text-align:center;" | 2
| 2
| style="text-align:center;" | ra
| 126.3
| style="text-align:center;" | m2
| E♭
| style="text-align:center;" | M2, m3
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
| style="text-align:center;" | 126.326
| Minor second
| colspan="2" style="text-align:center;" | 15/14, 16/15, 13/12, 14/13
| style="text-align:center;" | [[Negri|Negri]]
|-
|-
| style="text-align:center;" | 3
| 3
| style="text-align:center;" | re
| 189.5
| style="text-align:center;" | M2
| E
| style="text-align:center;" | M3
| [[9/8]], [[10/9]]
| style="text-align:center;" | 189.474
| Major second
| colspan="2" style="text-align:center;" | 9/8, 10/9
| style="text-align:center;" | Deutone (2-meantone) / spell
|-
|-
| style="text-align:center;" | 4
| 4
| style="text-align:center;" | ri/ma
| 252.6
| style="text-align:center;" | A2, d3
| E♯/F♭
| style="text-align:center;" | m4, a3
| [[7/6]], [[8/7]], [[15/13]]
| style="text-align:center;" | 252.632
| Augmented second/<br>Diminished third
| colspan="2" style="text-align:center;" | 7/6, 8/7, 15/13
| style="text-align:center;" | [[Godzilla|Godzilla]]
|-
|-
| style="text-align:center;" | 5
| 5
| style="text-align:center;" | me
| 315.8
| style="text-align:center;" | m3
| F
| style="text-align:center;" | M4, m5
| [[6/5]]
| style="text-align:center;" | 315.789
| Minor third
| colspan="2" style="text-align:center;" | 6/5, 25/21
| style="text-align:center;" | [[kleismic|Kleismic]] ([[Hanson|hanson]], [[Keemun|keemun]], [[catakleismic|catakleismic]])
|-
|-
| style="text-align:center;" | 6
| 6
| style="text-align:center;" | mi
| 378.9
| style="text-align:center;" | M3
| F♯
| style="text-align:center;" | M5
| [[5/4]], [[16/13]], [[56/45]]
| style="text-align:center;" | 378.947
| Major third
| colspan="2" style="text-align:center;" | 5/4, 16/13, 26/21
| style="text-align:center;" | [[Magic|Magic]]/charisma/glamour
|-
|-
| style="text-align:center;" | 7
| 7
| style="text-align:center;" | mo
| 442.1
| style="text-align:center;" | A3, d4
| F𝄪/G♭
| style="text-align:center;" | A5, d6
| [[9/7]], [[13/10]], [[21/16]], [[32/25]]
| style="text-align:center;" | 442.105
| Augmented third/<br>Diminished fourth
| colspan="2" style="text-align:center;" | 32/25, 9/7, 13/10
| style="text-align:center;" | [[Sensi|Sensi]]
|-
|-
| style="text-align:center;" | 8
| 8
| style="text-align:center;" | fa
| 505.3
| style="text-align:center;" | P4
| G
| style="text-align:center;" | P6
| [[4/3]], [[75/56]]
| style="text-align:center;" | 505.263
| Perfect fourth
| colspan="2" style="text-align:center;" | 4/3
| style="text-align:center;" | [[Meantone|Meantone]]/[[Flattone|flattone]]/[[meanennedecal|meanenneadecal]]/[[Meanpop|meanpop]]
|-
|-
| style="text-align:center;" | 9
| 9
| style="text-align:center;" | fi
| 568.4
| style="text-align:center;" | A4
| G♯
| style="text-align:center;" | A6, m7
| [[7/5]], [[18/13]], [[25/18]]
| style="text-align:center;" | 568.421
| Augmented fourth<br>(Small [[tritone]])
| colspan="2" style="text-align:center;" | 25/18, 7/5, 18/13
| style="text-align:center;" | [[Liese|Liese]]/[[Triton|triton]]/lisa
|-
|-
| style="text-align:center;" | 10
| 10
| style="text-align:center;" | se
| 631.6
| style="text-align:center;" | d5
| A♭
| style="text-align:center;" | M7, d8
| [[10/7]], [[13/9]], [[36/25]]
| style="text-align:center;" | 631.579
| Diminished fifth<br>(Large [[tritone]])
| colspan="2" style="text-align:center;" | 36/25, 10/7, 13/9
| style="text-align:center;" | Liese/triton/lisa
|-
|-
| style="text-align:center;" | 11
| 11
| style="text-align:center;" | sol
| 694.7
| style="text-align:center;" | P5
| A
| style="text-align:center;" | P8
| [[3/2]], [[112/75]]
| style="text-align:center;" | 694.737
| Perfect fifth
| colspan="2" style="text-align:center;" | 3/2
| style="text-align:center;" | Meantone
|-
|-
| style="text-align:center;" | 12
| 12
| style="text-align:center;" | lo
| 757.9
| style="text-align:center;" | A5, d6
| A♯/B𝄫
| style="text-align:center;" | A8, m9
| [[14/9]], [[20/13]], [[25/16]], [[32/21]]
| style="text-align:center;" | 757.895
| Augmented fifth/<br>Diminished sixth
| colspan="2" style="text-align:center;" | 25/16, 14/9, <span style="line-height: 1.5;">20/13</span>
| style="text-align:center;" | Sensi
|-
|-
| style="text-align:center;" | 13
| 13
| style="text-align:center;" | le
| 821.1
| style="text-align:center;" | m6
| B♭
| style="text-align:center;" | M9, m10
| [[8/5]], [[13/8]], [[45/28]]
| style="text-align:center;" | 821.053
| Minor sixth
| colspan="2" style="text-align:center;" | 8/5, <span style="line-height: 1.5;">13/8, 21/13</span>
| style="text-align:center;" | Magic
|-
|-
| style="text-align:center;" | 14
| 14
| style="text-align:center;" | la
| 884.2
| style="text-align:center;" | M6
| B
| style="text-align:center;" | M10
| [[5/3]]
| style="text-align:center;" | 884.210
| Major sixth
| colspan="2" style="text-align:center;" | 5/3, 42/25
| style="text-align:center;" | Kleismic (hanson, keemun, catakleismic)
|-
|-
| style="text-align:center;" | 15
| 15
| style="text-align:center;" | li/ta
| 947.4
| style="text-align:center;" | A6, d7
| B♯/C♭
| style="text-align:center;" | m11, A10
| [[7/4]], [[12/7]], [[26/15]]
| style="text-align:center;" | 947.368
| Augmented sixth<br>Diminished seventh
| colspan="2" style="text-align:center;" | 7/4, 12/7, 26/15
| style="text-align:center;" | Godzilla
|-
|-
| style="text-align:center;" | 16
| 16
| style="text-align:center;" | te
| 1010.5
| style="text-align:center;" | m7
| C
| style="text-align:center;" | M11, m12
| [[9/5]], [[16/9]]
| style="text-align:center;" | 1010.53
| Minor seventh
| colspan="2" style="text-align:center;" | 9/5, 16/9
| style="text-align:center;" | Deutone / spell
|-
|-
| style="text-align:center;" | 17
| 17
| style="text-align:center;" | ti
| 1073.7
| style="text-align:center;" | M7
| C♯
| style="text-align:center;" | M12
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
| style="text-align:center;" | 1073.68
| Major seventh
| colspan="2" style="text-align:center;" | 15/8, 13/7, <span style="line-height: 1.5;">28/15, 24/13</span>
| style="text-align:center;" | Negri
|-
|-
| style="text-align:center;" | 18
| 18
| style="text-align:center;" | da
| 1136.8
| style="text-align:center;" | A7, d8
| D♭
| style="text-align:center;" | A12, d13
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
| style="text-align:center;" | 1136.84
| Augmented seventh
| colspan="2" style="text-align:center;" | <span style="line-height: 1.5;">48/25, </span>40/21, 27/14, <span style="line-height: 1.5;">25/13, 52/27</span>
| style="text-align:center;" | Unicorn/rhinocerus
|-
|-
| style="text-align:center;" | 19
| 19
| style="text-align:center;" | do
| 1200.0
| style="text-align:center;" | P8
| D
| style="text-align:center;" | P13
| [[2/1]]
| style="text-align:center;" | 1200
| Octave
| style="text-align:center;" | 2/1
| style="text-align:center;" |
| style="text-align:center;" |
|}
|}
<nowiki>*</nowiki> based on treating 19-EDO as a 2.3.5.7.13 subgroup temperament; other approaches are possible.
<references group="note"/>


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


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


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


==Chord Names==
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 blue, green, yellow and red 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"
{| class="wikitable center-1 center-2 center-3 center-4"
|-
! Color of the third
! JI chord
! Edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo (7-over)
| 6:7:9
| 0–4–11
| C–E𝄫–G
| Cm(♭3) or Cmin(♭3) or C(d3)
| C subminor, C minor flat-three, C dim-three
|-
|-
! | color of the 3rd
| gu (5-under)
! | JI chord
| 10:12:15
! | notes as edosteps
| 0–5–11
! | notes of C chord
| C–E♭–G
! | written name
| Cm or Cmin
! | spoken name
| C minor
|-
|-
| style="text-align:center;" | zo
| yo (5-over)
| style="text-align:center;" | 6:7:9
| 4:5:6
| style="text-align:center;" | 0-4-11
| 0–6–11
| style="text-align:center;" | C Ebb G
| C–E–G
| style="text-align:center;" | C(b3) or C(d3)
| C or Cmaj
| style="text-align:center;" | C flat-three or C dim-three
| C, C major
|-
|-
| style="text-align:center;" | gu
| ru (7-under)
| style="text-align:center;" | 10:12:15
| 14:18:21
| style="text-align:center;" | 0-5-11
| 0–7–11
| style="text-align:center;" | C Eb G
| C–E♯–G
| style="text-align:center;" | Cm
| C(♯3) or Cmaj(♯3) or C(A3)
| style="text-align:center;" | C minor
| C supermajor, C major sharp-three, C aug-three
|-
|-
| style="text-align:center;" | yo
| yo (5-over)
| style="text-align:center;" | 4:5:6
| 4:5:6:7
| style="text-align:center;" | 0-6-11
| 0–6–11–15
| style="text-align:center;" | C E G
| C–E–G–B𝄫
| style="text-align:center;" | C
| Ch7 or C,d7 or Cadd(d7)
| style="text-align:center;" | C major or C
| C harmonic 7, C (major) add dim-seven
|-
|-
| style="text-align:center;" | ru
| gu (5-under)
| style="text-align:center;" | 14:18:27
| 1/(12:10:8:7)<br>(1–6/5–3/2–12/7)
| style="text-align:center;" | 0-7-11
| 0–5–11–15
| style="text-align:center;" | C E# G
| C–E♭–G–A♯
| style="text-align:center;" | C(#3) or C(A3)
| Cm♯6 or CmA6 or Cm(add(♯6)) or Cm(add(A6))
| style="text-align:center;" | C sharp-three or C aug-three
| C minor (add) sharp-six, C minor (add) aug-six
|}
|}


0-6-11-15 = C E G Bbb = C,bb7 or C,d7 = C double-flat-seven or C major dim-seven or C add dim-seven = 4:5:6:7
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.
 
0-5-11-15 = C Eb G A# is Cm,#6 or Cm,A6 = C minor sharp-six or C minor aug-six = 1/(4:5:6:7) = 1/1 - 6/5 - 3/2 - 12/7


The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate 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]].


For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].


==Selected just intervals by error==
== Notation ==
The following table shows how [[Just-24|some prominent just intervals]] are represented within 19edo (ordered by absolute error).
=== 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.


===Best direct mapping, even if inconsistent===
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"
{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 19edo
|-
|-
| | '''Interval, complement'''
! rowspan="2" | [[Degree|#]]
| | '''Error (abs., in [[cent|cents]])'''
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
! [[5L 2s|Diatonic interval names]]
| style="text-align:center;" | 0.148
! Note names<br>on D
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| 0
| style="text-align:center;" | 1.982
| 0.0
| '''Perfect unison (P1)'''
| '''D'''
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| 1
| style="text-align:center;" | 4.891
| 63.2
| Augmented unison (A1)<br>Diminished second (d2)
| D#<br>Ebb
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| 2
| style="text-align:center;" | 5.039
| 126.3
| Doubly augmented unison (AA1)<br>Minor second (m2)
| Dx<br>Eb
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| 3
| style="text-align:center;" | 6.873
| 189.5
| '''Major second (M2)'''<br>Doubly diminished third (dd3)
| '''E'''<br>Fbb
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| 4
| style="text-align:center;" | 7.021
| 252.6
| Augmented second (A2)<br>Diminished third (d3)
| E#<br>Fb
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| 5
| style="text-align:center;" | 7.070
| 315.8
| Doubly augmented second (AA2)<br>'''Minor third (m3)'''
| Ex<br>'''F'''
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| 6
| style="text-align:center;" | 7.218
| 378.9
| '''Major third (M3)'''<br>Doubly diminished fourth (dd4)
| '''F#'''<br>Gbb
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 7
| style="text-align:center;" | 7.366
| 442.1
| Augmented third (A3)<br>Diminished fourth (d4)
| Fx<br>Gb
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 8
| style="text-align:center;" | 12.109
| 505.3
| '''Perfect fourth (P4)'''
| '''G'''
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| 9
| style="text-align:center;" | 12.257
| 568.4
| Augmented fourth (A4)<br>Doubly diminished fifth (dd5)
| G#<br>Abb
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| 10
| style="text-align:center;" | 14.091
| 631.6
| Doubly augmented fourth (AA4)<br>Diminished fifth (d5)
| Gx<br>Ab
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 11
| style="text-align:center;" | 14.239
| 694.7
| '''Perfect fifth (P5)'''
| '''A'''
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| 12
| style="text-align:center;" | 14.436
| 757.9
| Augmented fifth (A5)<br>Diminished sixth (d6)
| A#<br>Bbb
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| 13
| style="text-align:center;" | 14.585
| 821.1
| Doubly augmented fifth (AA5)<br>Minor sixth (m6)
| Ax<br>Bb
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 14
| style="text-align:center;" | 17.103
| 884.2
| '''Major sixth (M6)'''<br>Doubly diminished seventh (dd7)
| '''B'''<br>Cbb
|-
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| 15
| style="text-align:center;" | 19.475
| 947.4
| Augmented sixth (A6)<br>Diminished seventh (d7)
| B#<br>Cb
|-
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| 16
| style="text-align:center;" | 21.457
| 1010.5
| Doubly augmented sixth (AA6)<br>'''Minor seventh (m7)'''
| Bx<br>'''C'''
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| 17
| style="text-align:center;" | 24.321
| 1073.7
| Major seventh (M7)<br>Doubly diminished octave (dd8)
| C#<br>Dbb
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| 18
| style="text-align:center;" | 24.469
| 1136.8
| Augmented seventh (A7)<br>Diminished octave (d8)
| Cx<br>Db
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| 19
| style="text-align:center;" | 24.597
| 1200.0
| '''Perfect octave (P8)'''
| '''D'''
|}
 
In 19edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.
 
{{Sharpness-sharp1}}
 
=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[5edo #Sagittal notation|5]], [[12edo #Sagittal notation|12]], and [[26edo #Sagittal notation|26]], and is a subset of the notations for edos [[38edo #Sagittal notation|38]], [[57edo #Sagittal notation|57]], and [[76edo #Sagittal notation|76]].
 
==== Evo flavor ====
{{Sagittal chart|Evo}}
 
Because it includes no Sagittal symbols, this Evo Sagittal notation is identical to conventional notation.
 
==== Revo flavor ====
{{Sagittal chart}}
 
=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic notation of 19edo
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
! [[Degree|#]]
| style="text-align:center;" | 26.580
! [[Cent]]s
! Interval names
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| 0
| style="text-align:center;" | 31.470
| 0.0
| P1
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| 1
| style="text-align:center;" | 31.539
| 63.2
|}
| A1, m2
 
===Patent val mapping===
 
{| class="wikitable"
|-
|-
| | '''Interval, complement'''
| 2
| | '''Error (abs., in [[cent|cents]])'''
| 126.3
| M2, m3
|-
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| 3
| style="text-align:center;" | 0.148
| 189.5
| M3
|-
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| 4
| style="text-align:center;" | 1.982
| 252.6
| m4, A3
|-
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| 5
| style="text-align:center;" | 4.891
| 315.8
| M4, m5
|-
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| 6
| style="text-align:center;" | 5.039
| 378.9
| M5
|-
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| 7
| style="text-align:center;" | 6.873
| 442.1
| A5, d6
|-
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| 8
| style="text-align:center;" | 7.021
| 505.3
| P6
|-
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| 9
| style="text-align:center;" | 7.070
| 568.4
| A6, m7
|-
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| 10
| style="text-align:center;" | 7.218
| 631.6
| M7, d8
|-
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| 11
| style="text-align:center;" | 7.366
| 694.7
| P8
|-
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| 12
| style="text-align:center;" | 12.109
| 757.9
| A8, m9
|-
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| 13
| style="text-align:center;" | 12.257
| 821.1
| M9, m10
|-
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| 14
| style="text-align:center;" | 14.091
| 884.2
| M10
|-
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| 15
| style="text-align:center;" | 14.239
| 947.4
| m11, A10
|-
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| 16
| style="text-align:center;" | 14.436
| 1010.5
| M11, m12
|-
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| 17
| style="text-align:center;" | 14.585
| 1073.7
| M12
|-
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| 18
| style="text-align:center;" | 17.103
| 1136.8
| A12, d13
|-
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| 19
| style="text-align:center;" | 19.475
| 1200.0
| P13
|}
 
== Approximation to JI ==
[[File:19ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 19edo]]
 
=== Interval mappings ===
{{Q-odd-limit intervals|19}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
! rowspan="2" | [[Subgroup]]
| style="text-align:center;" | 21.457
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
! [[TE error|Absolute]] (¢)
| style="text-align:center;" | 24.321
! [[TE simple badness|Relative]] (%)
|-
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| 2.3
| style="text-align:center;" | 24.469
| {{monzo| -30 19 }}
| {{mapping| 19 30 }}
| +2.277
| 2.277
| 3.612
|-
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| 2.3.5
| style="text-align:center;" | 31.539
| 81/80, 3125/3072
| {{mapping| 19 30 44 }}
| +2.578
| 1.911
| 3.025
|-
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| 2.3.5.7
| style="text-align:center;" | 31.688
| 49/48, 81/80, 126/125
| {{mapping| 19 30 44 53 }}
| +3.848
| 2.755
| 4.362
|-
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| 2.3.5.7.13
| style="text-align:center;" | 36.578
| 49/48, 65/64, 81/80, 91/90
| {{mapping| 19 30 44 53 70 }}
| +4.135
| 2.530
| 4.006
|-
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| 2.3.5.7.13.23
| style="text-align:center;" | 38.561
| 49/48, 65/64, 70/69, 81/80, 91/90
| {{mapping| 19 30 44 53 70 86 }}
| +3.319
| 2.936
| 4.649
|}
|}
* 19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit&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 best in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is [[53edo|53]].


[[File:19ed2-001.svg|alt=alt : Your browser has no SVG support.]]
=== Uniform maps ===
{{Uniform map|edo=19}}


[[:File:19ed2-001.svg|19ed2-001.svg]]
=== Commas ===
19et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.)


==Commas==
{| class="commatable wikitable center-all left-3 right-4 left-6"
19 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes the [[val|val]] &lt; 19 30 44 53 66 70 |.)
|-
 
! [[Harmonic limit|Prime<br>limit]]
{| class="wikitable"
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color notation/Temperament names|Color name]]
! Name
|-
| 3
| <abbr title="1162261467/1073741824">(20 digits)</abbr>
| {{monzo| -30 19 }}
| 137.14
| Trilawa
| [[19-comma]]
|-
| 5
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma
|-
| 5
| <abbr title="1594323/1562500">(14 digits)</abbr>
| {{monzo| -2 13 -8}}
| 34.91
| Laquadbigu
| [[Unicorn comma]]
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1224440064/1220703125">(20 digits)</abbr>
| {{monzo| 8 14 -13 }}
| 5.29
| Thegu
| [[Parakleisma]]
|-
| 5
| <abbr title="19073486328125/19042491875328">(28 digits)</abbr>
| {{monzo| -14 -19 19 }}
| 2.82
| Neyo
| [[Enneadeca]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
|-
! | Comma
| 7
! | Monzo
| [[126/125]]
! | Value (Cents)
| {{monzo| 1 2 -3 1 }}
! | Name(s)
| 13.79
| Zotrigu
| Starling comma
|-
|-
| style="text-align:center;" | 16875/16384
| 7
| |<nowiki> | -14 3 4 </nowiki>&gt;
| [[225/224]]
| style="text-align:right;" | 51.12
| {{monzo| -5 2 2 -1 }}
| style="text-align:center;" | Negri Comma, <span style="line-height: 1.5;">Double Augmentation Diesis</span>
| 7.71
| Ruyoyo
| Marvel comma
|-
|-
| style="text-align:center;" | 3125/3072
| 7
| |<nowiki> | -10 -1 5 </nowiki>&gt;
| [[19683/19600]]
| style="text-align:right;" | 29.61
| {{monzo| -4 9 -2 -2 }}
| style="text-align:center;" | Small Diesis, <span style="line-height: 1.5;">Magic Comma</span>
| 7.32
| Labirugu
| Cataharry comma
|-
|-
| style="text-align:center;" | [[81/80|81/80]]
| 7
| |<nowiki> | -4 4 -1 </nowiki>&gt;
| [[10976/10935]]
| style="text-align:right;" | 21.51
| {{monzo| 5 -7 -1 3 }}
| style="text-align:center;" | [[Syntonic_Comma|Syntonic Comma]], <span style="line-height: 1.5;">Didymos Comma, Meantone Comma</span>
| 6.48
| Satrizo-agu
| Hemimage comma
|-
|-
| style="text-align:center;" | 78732/78125
| 7
| |<nowiki> | 2 9 -7 </nowiki>&gt;
| [[3136/3125]]
| style="text-align:right;" | 13.40
| {{monzo| 6 0 -5 2 }}
| style="text-align:center;" | Medium Semicomma, <span style="line-height: 1.5;">Sensipent Comma</span>
| 6.08
| Zozoquingu
| Hemimean comma
|-
|-
| style="text-align:center;" | 15625/15552
| 7
| |<nowiki> | -6 -5 6 </nowiki>&gt;
| <abbr title="703125/702464">(12 digits)</abbr>
| style="text-align:right;" | 8.11
| {{monzo| -11 2 7 -3 }}
| style="text-align:center;" | Kleisma, <span style="line-height: 1.5;">Semicomma Majeur</span>
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
|-
| style="text-align:center;" |  
| 7
| |<nowiki> | 8 14 -13 </nowiki>&gt;
| [[4375/4374]]
| style="text-align:right;" | 5.29
| {{monzo| -1 -7 4 1 }}
| style="text-align:center;" | Parakleisma
| 0.40
| Zoquadyo
| Ragisma
|-
|-
| style="text-align:center;" |  
| 11
| |<nowiki> | -14 -19 19 </nowiki>&gt;
| [[45/44]]
| style="text-align:right;" | 2.82
| {{monzo| -2 2 1 0 -1 }}
| style="text-align:center;" | Enneadeca, <span style="line-height: 1.5;">19-Tone-Comma</span>
| 38.91
| Luyo
| Undecimal fifth tone
|-
|-
| style="text-align:center;" | 1029/1000
| 11
| |<nowiki> | -3 1 -3 3 </nowiki>&gt;
| [[56/55]]
| style="text-align:right;" | 49.49
| {{monzo| 3 0 -1 1 -1 }}
| style="text-align:center;" | Keega
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
|-
| style="text-align:center;" | 525/512
| 11
| |<nowiki> | -9 1 2 1 </nowiki>&gt;
| [[100/99]]
| style="text-align:right;" | 43.41
| {{monzo| 2 -2 2 0 -1 }}
| style="text-align:center;" | Avicenna, <span style="line-height: 1.5;">Avicenna's Enharmonic Diesis</span>
| 17.40
| Luyoyo
| Ptolemisma
|-
|-
| style="text-align:center;" | [[49/48|49/48]]
| 11
| |<nowiki> | -4 -1 0 2 </nowiki>&gt;
| [[896/891]]
| style="text-align:right;" | 35.70
| {{monzo| 7 -4 0 1 -1 }}
| style="text-align:center;" | Slendro Diesis
| 9.69
| Saluzo
| Pentacircle comma
|-
|-
| style="text-align:center;" | 686/675
| 11
| |<nowiki> | 1 -3 -2 3 </nowiki>&gt;
| [[65536/65219]]
| style="text-align:right;" | 27.99
| {{monzo| 16 0 0 -2 -3 }}
| style="text-align:center;" | Senga
| 8.39
| Satrilu-aruru
| Orgonisma
|-
|-
| style="text-align:center;" | 875/864
| 11
| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
| [[385/384]]
| style="text-align:right;" | 21.90
| {{monzo| -7 -1 1 1 1 }}
| style="text-align:center;" | Keema
| 4.50
| Lozoyo
| Keenanisma
|-
|-
| style="text-align:center;" | [[245/243|245/243]]
| 11
| |<nowiki> | 0 -5 1 2 </nowiki>&gt;
| [[540/539]]
| style="text-align:right;" | 14.19
| {{monzo| 2 3 1 -2 -1 }}
| style="text-align:center;" | Sensamagic
| 3.21
| Lururuyo
| Swetisma
|-
|-
| style="text-align:center;" | [[126/125|126/125]]
| 13
| |<nowiki> | 1 2 -3 1 </nowiki>&gt;
| [[39/38]]
| style="text-align:right;" | 13.79
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| style="text-align:center;" | Septimal Semicomma, <span style="line-height: 1.5;">Starling Comma</span>
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
|-
| style="text-align:center;" | [[225/224|225/224]]
| 13
| |<nowiki> | -5 2 2 -1 </nowiki>&gt;
| [[65/64]]
| style="text-align:right;" | 7.71
| {{monzo| -6 0 1 0 0 1 }}
| style="text-align:center;" | Septimal Kleisma, <span style="line-height: 1.5;">Marvel Comma</span>
| 26.84
| Thoyo
| Wilsorma
|-
|-
| style="text-align:center;" | 19683/19600
| 13
| |<nowiki> | -4 9 -2 -2 </nowiki>&gt;
| [[343/338]]
| style="text-align:right;" | 7.32
| {{monzo| -1 0 0 3 0 -2 }}
| style="text-align:center;" | Cataharry
| 25.42
| Thuthutrizo
|  
|-
|-
| style="text-align:center;" | 10976/10935
| 13
| |<nowiki> | 5 -7 -1 3 </nowiki>&gt;
| [[91/90]]
| style="text-align:right;" | 6.48
| {{monzo| -1 -2 -1 1 0 1 }}
| style="text-align:center;" | Hemimage
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
|-
| style="text-align:center;" | 3136/3125
| 13
| |<nowiki> | 6 0 -5 2 </nowiki>&gt;
| [[676/675]]
| style="text-align:right;" | 6.08
| {{monzo| 2 -3 -2 0 0 2 }}
| style="text-align:center;" | Hemimean
| 2.56
| Bithogu
| Island comma
|-
|-
| style="text-align:center;" |  
| 13
| |<nowiki> | -11 2 7 -3 </nowiki>&gt;
| [[1001/1000]]
| style="text-align:right;" | 1.63
| {{monzo| -3 0 -3 1 1 1 }}
| style="text-align:center;" | Meter
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
|-
| style="text-align:center;" | 4375/4374
| 23
| |<nowiki> | -1 -7 4 1 </nowiki>&gt;
| [[2187/2116]]
| style="text-align:right;" | 0.40
| {{monzo| -2 7 0 0 0 0 0 0 -2 }}
| style="text-align:center;" | Ragisma
| 57.14
| Labitwethu
| Lipsett comma
|-
|-
| style="text-align:center;" | [[100/99|100/99]]
| 23
| |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
| [[70/69]]
| style="text-align:right;" | 17.40
| {{monzo| 1 -1 1 1 0 0 0 0 -1 }}
| style="text-align:center;" | Ptolemisma
| 24.91
| Twethuzoyo
| Small vicesimotertial eighth tone
|-
|-
| style="text-align:center;" | 896/891
| 23
| |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
| 256/253
| style="text-align:right;" | 9.69
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| style="text-align:center;" | Pentacircle
| 20.41
| Twethulu
| 253rd subharmonic
|-
|-
| style="text-align:center;" | 65536/65219
| 23
| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
| [[161/160]]
| style="text-align:right;" | 8.39
| {{monzo| -5 0 -1 1 0 0 0 0 1 }}
| style="text-align:center;" | Orgonisma
| 10.79
| Twethozogu
| Major kirnbergisma
|-
|-
| style="text-align:center;" | [[385/384|385/384]]
| 23
| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| [[208/207]]
| style="text-align:right;" | 4.50
| {{monzo| 4 -2 0 0 0 1 0 0 -1 }}
| style="text-align:center;" | Keenanisma
| 8.34
| Twethutho
| Vicetone comma
|-
|-
| style="text-align:center;" | 540/539
| 23
| |<nowiki> | 2 3 1 -2 -1 </nowiki>&gt;
| [[529/528]]
| style="text-align:right;" | 3.21
| {{monzo| -4 -1 0 0 -1 0 0 0 2 }}
| style="text-align:center;" | Swetisma
| 3.28
| Bitwetho-alu
| Preziosisma
|-
|-
| style="text-align:center;" | [[91/90|91/90]]
| 23
| |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| [[576/575]]
| style="text-align:right;" | 19.13
| {{monzo| 6 2 -2 0 0 0 0 0 -1 }}
| style="text-align:center;" | Superleap
| 3.01
| Twethugugu
| Worcester comma
|-
|-
| style="text-align:center;" | 676/675
| 23
| |<nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
| [[1288/1287]]
| style="text-align:right;" | 2.56
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| style="text-align:center;" | Parizeksma
| 1.34
| Twethothuluzo
| Triaphonisma
|}
|}
<references group="note" />


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


[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
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.


=Compositions=
{| class="wikitable center-1 right-2 center-3"
'''[http://micro.soonlabel.com/19-ET/ XA 19-ET Index]'''
|-
! Degree
! Cents
! Interval
! Mos scales
! Temperaments
|-
| 1
| 63.16
| A1, d2
|
| [[Unicorn]] / [[Rhinoceros]]
|-
| 2
| 126.32
| m2
| [[1L&nbsp;8s]], [[9L&nbsp;1s]]
| [[Negri]]
|-
| 3
| 189.47
| M2
| [[1L&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]]
| [[Deutone]] <br>[[Xenial]] / [[Sensamagic clan #Xenia|Xenia]] <br>[[Spell]]
|-
| 4
| 252.63
| A2, d3
| [[1L&nbsp;3s]], [[4L&nbsp;1s]], <br>[[5L&nbsp;4s]], [[5L&nbsp;9s]]
| [[Godzilla]] / [[Helayo]]
|-
| 5
| 315.79
| m3
| [[3L&nbsp;1s]], [[4L&nbsp;3s]], <br>[[4L&nbsp;7s]], [[4L&nbsp;11s]]
| [[Cata]] / [[keemun]]
|-
| 6
| 378.95
| M3
| [[3L&nbsp;1s]], [[3L&nbsp;4s]], [[3L&nbsp;7s]], <br>[[3L&nbsp;10s]], [[3L&nbsp;13s]]
| [[Magic]] / [[muggles]]
|-
| 7
| 442.11
| A3, d4
| [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[8L&nbsp;3s]]
| [[Sensi]]
|-
| 8
| 505.26
| P4
| [[2L&nbsp;3s]], [[5L&nbsp;2s]], [[7L&nbsp;5s]]
| [[Meantone]] / [[flattone]]
|-
| 9
| 568.42
| A4
| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]], <br>[[2L&nbsp;15s]]
| [[Liese]] <br>[[Triton]] / [[pycnic]]
|}


<span style=""><span style=""><span style=""><span style="right: auto;"><span style="">[https://soundcloud.com/aaron-krister-johnson/the-juggler The Juggler]</span></span></span></span> </span>by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]
== 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.


[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=145852&songID=1008195 Foum] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Barton/Foum.mp3 play]</span></span></span></span></span> by [[Jacob_Barton|Jacob Barton]]
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]].


[http://music.columbia.edu/%7Echris/sand.html Sand] by [[Christopher_Bailey|Christopher Bailey]]
== Scales ==
=== MOS scales ===
{{Main|List of MOS scales in {{PAGENAME}}}}


<span style=""><span style=""><span style=""><span style="right: auto;"><span style="">[http://works.music.columbia.edu/%7Echris/19mix1.mp3 Walking Down the Hillside at Cortona, and Seeing its Towers Rise Before Me]</span></span></span></span> </span>by Christopher Bailey
==== Octave-equivalent mosses ====
* [[Meantone]] pentic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[Meantone]] diatonic, [[5L 2s]] (gen = 11\19): 3 3 2 3 3 3 2
* [[Meantone]] chromatic, [[7L 5s]] (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
* [[Semaphore]][5], [[4L 1s]] (gen = 4\19): 4 4 3 4 4
* [[Semaphore]][9], [[5L 4s]] (gen = 4\19): 3 1 3 1 3 3 1 3 1
* [[Semaphore]][14], [[5L 9s]] (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
* [[Sensi]][5], [[2L 3s]] (gen = 7\19): 5 2 5 2 5
* [[Sensi]][8], [[3L 5s]] (gen = 7\19): 2 3 2 2 3 2 2 3
* [[Sensi]][11], [[8L 3s]] (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
* [[Negri]][9], [[1L 8s]] (gen = 2\19): 2 2 2 2 3 2 2 2 2
* [[Negri]][10], [[9L 1s]] (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
* [[Kleismic]][7], [[4L 3s]] (gen = 5\19): 1 4 1 4 1 4 4
* [[Kleismic]][11], [[4L 7s]] (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
* [[Kleismic]][15], [[4L 11s]] (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
* [[Magic]][7], [[3L 4s]] (gen = 6\19): 5 1 5 1 5 1 1
* [[Magic]][10], [[3L 7s]] (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
* [[Magic]][13], [[3L 10s]] (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
* [[Magic]][16], [[3L 13s]] (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
* [[Liese]][17], [[2L 15s]] (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1


<span style=""><span style=""><span style=""><span style="right: auto;"><span style="">[http://www.emcollective.org/chris/items/ditty/ditty_fast.mp3 Ditty]</span></span></span></span> </span>by Christopher Bailey
=== Other scales ===
{{Main|19edo modes}}


''[http://micro.soonlabel.com/gene_ward_smith/misc/Guillaume%20Costeley%20-%20Seigneur%20Dieu%20ta%20piti%e9.mp3 Seigneur Dieu ta pitié]'' by Guillaume Costeley
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2 (ascending), 3 2 3 3 2 3 3 (descending)
* Meantone harmonic major: 3 3 2 3 2 4 2
* Chromatic octave species – meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* Chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* Chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* Enharmonic pentatonic: 2 6 3 2 6
* Enharmonic pentatonic: 6 2 3 6 2
* Enharmonic octave species: 1 1 6 3 1 1 6
* Enharmonic octave species: 6 1 1 3 6 1 1
* Enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
* [[Pinetone #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
* [[Pinetone #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
* [[Pinetone #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Semiquartal]] 3|5 b2: 1 3 3 1 3 1 3 3 1
* [[5-odd-limit]] tonality diamond: 5 1 2 3 2 1 5
* [[7-odd-limit]] tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
* [[9-odd-limit]] tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3


''[http://micro.soonlabel.com/0-hosted-albums/ivor/02%20Prelude%202%20for%2019%20tone%20guitar.mp3 Prelude 2 for 19 tone guitar]'' by [[Ivor_Darreg|Ivor Darreg]]
== Instruments ==
 
[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
[http://sethares.engr.wisc.edu/mp3s/sympathetic.html Sympathetic Metaphor] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/SympatheticMetaphor.mp3 play]</span></span></span></span></span> by [[William_Sethares|William Sethares]] [http://www.webcitation.org/5zFdC6u4V Permalink]
[[File:Bass19.jpg|alt=19edo 5 string Bass 34"-37" scale length|512x384px|thumb|none|19edo bass conversion by Ron Sword]]
 
[http://sethares.engr.wisc.edu/mp3s/truthonabus.html Truth on a bus] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Truth_on_a_Bus.mp3 play]</span></span></span></span></span> by William Sethares [http://www.webcitation.org/5zFdSw1g4 Permalink]
 
<span style=""><span style=""><span style=""><span style="right: auto;"><span style="">[http://www.h-pi.com/mp3/Rondo19ET.mp3 Rondo in 19ET]</span></span></span></span> </span>by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]
 
<span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Starrett/CitifiedNotions19tet.mp3 Citified Notions]''</span></span></span></span></span> and
 
''[http://archive.org/download/LimpOffToSchool/LimpOffToSchool19tet.mp3 Limp Off to School]'' by [[John_Starrett|John Starrett]].
 
<span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/betelgeuse.mp3 The Light Of My Betelgeuse]''</span></span></span></span></span> by [[Mykhaylo_Khramov|Mykhaylo Khramov]]
 
''[http://archive.org/download/ElementalsFor2PianosIn19EqualTemperament/01.Undines.mp3 Undines]'', ''[http://archive.org/download/ElementalsFor2PianosIn19EqualTemperament/02.Sylphs.mp3 Sylphs]'', ''[http://archive.org/download/ElementalsFor2PianosIn19EqualTemperament/03.Gnomes.mp3 Gnomes]'', and ''[http://archive.org/download/ElementalsFor2PianosIn19EqualTemperament/04.Salamanders.mp3 Salamanders]'' by Jon Lyle Smith
 
[http://archive.org/details/AnotherAireForLute Another Aire For Lute] by Jon Lyle Smith
 
A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]
 
Fanfare in 19-note Equal Tuning by Easley Blackwood
 
[http://www.uvnitr.cz/flaoyg/flao_yg/zvire.mp3 Zvíře] by [http://www.uvnitr.cz/flaoyg/flao_yg/zvire.html Milan Guštar]
 
[http://www.ziaspace.com/ZIA/sections/music.html 19tet downloadable mp3s by ZIA, Elaine Walker and D.D.T.]
 
<span style=""><span style=""><span style="right: auto;"><span style="">''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/14%20-%2014.%2019%20octave.mp3 Comets Over Flatland 14]''</span></span></span></span> by [[Randy_Winchester|Randy Winchester]]
 
[http://chrisvaisvil.com/?p=865 Forgetting Even Her Beauty blog] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://micro.soonlabel.com/19-ET/daily20110516-19-forgetting_even_her_beauty.mp3 play Forgetting Even Her Beauty]''</span></span></span></span> </span>by [[Chris_Vaisvil|Chris Vaisvil]]
 
[http://chrisvaisvil.com/?p=855 19 Black Hawks for Osama blog] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://www.youtube.com/watch?v=gGO0R381HQs play video for 19 Black Hawks for Osama]'' </span></span></span></span> </span>by Chris Vaisvil
 
[http://chrisvaisvil.com/?p=823 Summer Song blog] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://micro.soonlabel.com/19-ET/daily20110501-19-summer_song_mix3.mp3 play Summer Song]''</span></span></span></span> </span>by Trevor (The TwoRegs) and Norm Harris and Chris Vaisvil
 
[http://chrisvaisvil.com/?p=820 19 ImprovFridays blog] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://www.youtube.com/watch?v=-KaRXFoYwT4 play video of performance of 19 ImprovFridays]'' </span></span></span></span> </span>by Chris Vaisvil
 
[http://chrisvaisvil.com/?p=800 The World has Changed blog] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://micro.soonlabel.com/19-ET/daily20110429-19edo-the_world_has_changed3.mp3 play The World has Changed]''</span></span></span></span> </span>by Chris Vaisvil
 
[http://chrisvaisvil.com/?p=883 jjj] <span style=""><span style=""><span style=""><span style="right: auto;"><span style="">''[http://micro.soonlabel.com/19-ET/daily20110523-19take2.mp3 play]''</span></span></span></span></span> by Chris Vaisvil
 
[http://soundcloud.com/omega9/chip-chamber Now listen! Pitch!] [http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Now%20listen%21%20Pitch%21.mp3 play] by Omega9
 
[https://www.youtube.com/watch?v=mlvme3ukPLE Cordas (19-edo version)] [http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Omega9%20-%20Cordas.mp3 play] by Omega9
 
''[http://www.youtube.com/watch?v=DSOGF4gDYu8 A Piece in 19edo]'' by Omega9
 
[http://soundcloud.com/omega9/a-piece-in-19edo-version-3 A Piece in 19edo (ver.3)] [http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Omega9%20-%20A%20Piece%20in%2019edo.mp3 play] by Omega9
 
<span style=""><span style="right: auto;"><span style="">''[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/WTC2-24/19-edo-A-Prelude.mp3 Bach’s Prelude number 24 from Well Tempered Clavier, Book II]''</span></span></span>
 
<span style=""><span style="right: auto;"><span style="">''[http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/WTC2-24/19-edo-B-Fugue.mp3 Bach’s Fugue number 24 from Well Tempered Clavier, Book II]''</span></span></span>
 
rendered by Claudi Meneghin
 
''[http://micro.soonlabel.com/gene_ward_smith/Others/Roncevaux/_Movi_Nove__by_Roncevaux_on_SoundCloud___Hear_the_world_s_sounds.mp3 Movi-Nove]'' by [https://soundcloud.com/lois-lancaster/movi-nove Roncevaux (Löis Lancaster)]
 
[https://soundcloud.com/rewarrp/psychedelic-delt Psychedelic Delt] by Rewarrp


[https://soundcloud.com/camtaylor-1/17-out-of-19edo-bright-objects Bright Objects] - 17-tone fifths chain in 19EDO by Cam Taylor
== Music ==
{{Main| 19edo/Music }}
{{Catrel| 19edo tracks }}


[https://soundcloud.com/cam-taylor-2-1/17-out-of-19edo-wurly-minor-thirds-minor-sevens Minor Thirds, Minor Sevens] - 17-tone fifths chain in 19EDO by Cam Taylor
; [http://micro.soonlabel.com/19-ET/ XA 19-ET Index]
; A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]{{dead link}}


[https://soundcloud.com/camtaylor-1/wurly-minors-and-enharmonic-tetrachords-17-out-of-19edo Wurly Minors and Enharmonic Tetrachords] - 17-tone fifths chain in 19EDO by Cam Taylor
== See also ==
* [[19edo modes]]
* [[19edo chords]]
* [[Strictly proper 19edo scales]]
* [[How to tune a 19edo guitar by ear]]
* [[Primer for 19edo]]
* [[Mason Green's New Common Practice Notation]]
* [[Extraclassical tonality]]
* [[Lumatone mapping for 19edo]]


=See also=
== Further reading ==
*[[19edo_Modes|19edo Modes]]
* [[Darreg, Ivor]]. ''[http://www.tonalsoft.com/sonic-arts/darreg/case.htm A Case for Nineteen]''. 1982.
*[[Strictly_proper_19edo_scales|Strictly proper 19edo scales]]
* Darreg, Ivor. ''[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties]''{{dead link}}. (Unknown date of publication).
*[[How_to_tune_a_19edo_guitar_by_ear|How to tune a 19edo guitar by ear]]
* Howe, Hubert S., Jr. [http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications]. c. 2004.
*[[Chord_progressions_in_19edo-family_scales|Chord progressions in 19edo-family scales]]
* [[Sethares, William A]]. [http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar]. 1991.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave]''. 2010.
* Yasser, Joseph. ''[https://www.worldcat.org/fr/title/726192994 Theory of Evolving Tonality]''. 1932.


==Articles==
== External links ==
*[http://sonic-arts.org/darreg/case.htm A Case For Nineteen] by [[Ivor_Darreg|Ivor Darreg]] [http://www.webcitation.org/5xZzBtDGF Permalink]
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on the [[Tonalsoft Encyclopedia]]
*[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties] by Ivor Darreg
* [http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
*[http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications] by Hubert S. Howe Jr. [http://www.webcitation.org/5xbMKVaqa Permalink]
* [http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
*[http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar] by William A. Sethares [http://www.webcitation.org/5xeCbEPZ0 Permalink]
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]  
*[http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Bailey, Morrison, Pearson and Parncutt [http://www.webcitation.org/5xeDFQDvn Permalink]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]
*[http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone - Encyclopedia of Microtonal Music Theory] [http://tonalsoft.com/enc/number/19edo.aspx Permalink]
*[http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
*[http://www.ronsword.com/books.html Enneadecaphonic Scales for Guitar] by [[Ron_Sword|Ron Sword]]


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


[[Category:19-tone]]
[[Category:19-tone scales]]
[[Category:19edo]]
[[Category:Godzilla]]
[[Category:edo]]
[[Category:Golden meantone]]
[[Category:golden]]
[[Category:Kleismic]]
[[Category:intervals]]
[[Category:Meantone]]
[[Category:listen]]
[[Category:Magic]]
[[Category:meantone]]
[[Category:Negri]]
[[Category:prime_edo]]
[[Category:Semaphore]]
[[Category:teentuning]]
[[Category:Sensi]]
[[Category:todo:unify_precision]]
[[Category:Teentuning]]
[[Category:zeta]]
[[Category:Historical]]

Latest revision as of 03:13, 6 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 Note Approximated ratios[note 1] Interval category
0 0.0 D 1/1 Unison (prime)
1 63.2 D♯ 25/24, 26/25, 27/26, 28/27 Augmented unison
2 126.3 E♭ 13/12, 14/13, 15/14, 16/15 Minor second
3 189.5 E 9/8, 10/9 Major second
4 252.6 E♯/F♭ 7/6, 8/7, 15/13 Augmented second/
Diminished third
5 315.8 F 6/5 Minor third
6 378.9 F♯ 5/4, 16/13, 56/45 Major third
7 442.1 F𝄪/G♭ 9/7, 13/10, 21/16, 32/25 Augmented third/
Diminished fourth
8 505.3 G 4/3, 75/56 Perfect fourth
9 568.4 G♯ 7/5, 18/13, 25/18 Augmented fourth
(Small tritone)
10 631.6 A♭ 10/7, 13/9, 36/25 Diminished fifth
(Large tritone)
11 694.7 A 3/2, 112/75 Perfect fifth
12 757.9 A♯/B𝄫 14/9, 20/13, 25/16, 32/21 Augmented fifth/
Diminished sixth
13 821.1 B♭ 8/5, 13/8, 45/28 Minor sixth
14 884.2 B 5/3 Major sixth
15 947.4 B♯/C♭ 7/4, 12/7, 26/15 Augmented sixth
Diminished seventh
16 1010.5 C 9/5, 16/9 Minor seventh
17 1073.7 C♯ 13/7, 15/8, 24/13, 28/15 Major seventh
18 1136.8 D♭ 25/13, 27/14, 48/25, 52/27 Augmented seventh
19 1200.0 D 2/1 Octave
  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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