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


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


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


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


== Notation ==
== Notation ==
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| 23
| 23
| [[70/69]]
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -}}
| {{monzo| 1 -1 1 1 0 0 0 0 -1 }}
| 24.91
| 24.91
| Twethuzoyo
| Twethuzoyo
Line 1,037: Line 1,059:
| M2
| M2
| [[1L&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]]
| [[1L&nbsp;5s]], [[6L&nbsp;1s]], [[6L&nbsp;7s]]
| [[Deutone]]<br>[[Spell]]
| [[Deutone]] <br>[[Xenial]] / [[Sensamagic clan #Xenia|Xenia]] <br>[[Spell]]
|-
|-
| 4
| 4
Line 1,043: Line 1,065:
| A2, d3
| A2, d3
| [[1L&nbsp;3s]], [[4L&nbsp;1s]], <br>[[5L&nbsp;4s]], [[5L&nbsp;9s]]
| [[1L&nbsp;3s]], [[4L&nbsp;1s]], <br>[[5L&nbsp;4s]], [[5L&nbsp;9s]]
| [[Godzilla]]
| [[Godzilla]] / [[Helayo]]
|-
|-
| 5
| 5
Line 1,073: Line 1,095:
| A4
| A4
| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]], <br>[[2L&nbsp;15s]]
| [[2L&nbsp;3s]], [[2L&nbsp;5s]], [[2L&nbsp;7s]], <br>[[2L&nbsp;9s]], [[2L&nbsp;11s]], [[2L&nbsp;13s]], <br>[[2L&nbsp;15s]]
| [[Liese]] / [[pycnic]]<br>[[Triton]]
| [[Liese]] <br>[[Triton]] / [[pycnic]]
|}
|}


Line 1,086: Line 1,108:


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


=== Other scales ===
=== Other scales ===
{{Main|19edo modes}}
{{Main|19edo modes}}
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2
* Meantone melodic minor: 3 2 3 3 3 3 2 (ascending), 3 2 3 3 2 3 3 (descending)
* Meantone harmonic major: 3 3 2 3 2 4 2
* Meantone harmonic major: 3 3 2 3 2 4 2
* chromatic octave species - Meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* Chromatic octave species – meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* Chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* Chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* enharmonic pentatonic: 2 6 3 2 6
* Enharmonic pentatonic: 2 6 3 2 6
* enharmonic pentatonic: 6 2 3 6 2
* Enharmonic pentatonic: 6 2 3 6 2
* enharmonic octave species: 1 1 6 3 1 1 6
* Enharmonic octave species: 1 1 6 3 1 1 6
* enharmonic octave species: 6 1 1 3 6 1 1
* Enharmonic octave species: 6 1 1 3 6 1 1
* enharmonic octave species: 1 6 1 3 1 6 1
* Enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone#Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
* [[Pinetone #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
* [[Pinetone#Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
* [[Pinetone #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
* [[Pinetone#Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
* [[Pinetone #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
* [[Pinetone#Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Pinetone #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Antipental blues]]: 4 4 1 2 4 4
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