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38edo

2026-06-04T20:40:58Z

Overthink: /* Theory */ expand 38df paragraph

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]], [[25/22]], and their [[octave complement]]s, while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where primes [[7/1|7]] and [[13/1|13]] use their second-best approximations, and are mapped the same as in 19edo. The [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping of 19edo is preserved in 38df, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are mapped between steps of 19edo. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]]s of all [[19-odd-limit]] intervals in 38df align with their closest approximations in 38edo, except for [[7/4]], [[13/8]], and their octave complements [[8/7]] and [[16/13]], which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo. It tempers out [[49/48]], [[65/64]], [[81/80]], [[225/224]], etc. as in 19edo, as well as [[121/120]], [[289/288]], [[324/323]], [[361/360]], and many more.

The harmonic series from 1 to 20 is approximated within 38df by the step sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
| [[55/54]], [[45/44]], ''[[33/32]]''
| [[64/63]], ''[[36/35]]''
| [[56/55]]
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
| [[25/24]], [[34/33]]
| [[22/21]]
| [[28/27]], [[26/25]], [[27/26]]
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]'', [[21/20]]
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]'', [[35/32]]
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
| [[28/25]]
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[25/22]], [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]], [[32/25]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]], ''[[21/16]]''
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
| [[25/19]]
| [[21/16]], ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]], [[34/25]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
| [[25/18]]
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
| [[36/25]]
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[22/15]], [[16/11]], [[25/17]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
| [[38/25]]
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]], [[25/16]]
| ''[[26/17]]'', ''[[11/7]]''
| [[14/9]], [[20/13]], ''[[32/21]]''
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
| [[22/13]]
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[44/25]], [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
| [[25/14]]
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]'', [[64/35]]
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
| [[33/17]], [[48/25]]
|
| [[27/14]], [[25/13]], [[52/27]]
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
| ''[[64/33]]'', [[88/45]], [[108/55]]
| [[63/32]]
| [[55/28]]
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

38edo

2026-06-04T20:30:20Z

Overthink: /* Theory */ fix typo

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]], [[25/22]], and their [[octave complement]]s, while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are corrected. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]]s of all [[19-odd-limit]] intervals in 38df align with their closest approximations in 38edo, except for 7/4, 13/8, and their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo.

The harmonic series from 1 to 20 is approximated within 38df by the step sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
| [[55/54]], [[45/44]], ''[[33/32]]''
| [[64/63]], ''[[36/35]]''
| [[56/55]]
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
| [[25/24]], [[34/33]]
| [[22/21]]
| [[28/27]], [[26/25]], [[27/26]]
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]'', [[21/20]]
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]'', [[35/32]]
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
| [[28/25]]
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[25/22]], [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]], [[32/25]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]], ''[[21/16]]''
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
| [[25/19]]
| [[21/16]], ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]], [[34/25]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
| [[25/18]]
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
| [[36/25]]
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[22/15]], [[16/11]], [[25/17]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
| [[38/25]]
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]], [[25/16]]
| ''[[26/17]]'', ''[[11/7]]''
| [[14/9]], [[20/13]], ''[[32/21]]''
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
| [[22/13]]
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[44/25]], [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
| [[25/14]]
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]'', [[64/35]]
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
| [[33/17]], [[48/25]]
|
| [[27/14]], [[25/13]], [[52/27]]
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
| ''[[64/33]]'', [[88/45]], [[108/55]]
| [[63/32]]
| [[55/28]]
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

38edo

2026-06-04T20:18:38Z

Overthink: /* Intervals */ place this first

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]], [[25/22]], and their [[octave complement]]s, while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are corrected. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]]s of all [[19-odd-limit]] intervals in 38df align with their closest approximations in 38edo, except for 7/4, 13/8, and their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo.

The harmonic series from 1 to 20 is approximated within 38df by the step sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
| [[55/54]], [[45/44]], ''[[33/32]]''
| [[64/63]], ''[[36/35]]''
| [[56/55]]
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
| [[25/24]], [[34/33]]
| [[22/21]]
| [[28/27]], [[26/25]], [[27/26]]
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]'', [[21/20]]
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]'', [[35/32]]
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
| [[28/25]]
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[25/22]], [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]], [[32/25]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]], ''[[21/16]]''
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
| [[25/19]]
| [[21/16]], ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]], [[34/25]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
| [[25/18]]
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
| [[36/25]]
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[22/15]], [[16/11]], [[25/17]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
| [[38/25]]
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]], [[25/16]]
| ''[[26/17]]'', ''[[11/7]]''
| [[14/9]], [[20/13]], ''[[32/21]]''
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
| [[22/13]]
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[44/25]], [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
| [[25/14]]
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]'', [[64/35]]
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
| [[33/17]], [[48/25]]
|
| [[27/14]], [[25/13]], [[52/27]]
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
| ''[[64/33]]'', [[88/45]], [[108/55]]
| [[63/32]]
| [[55/28]]
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

38edo

2026-06-04T20:18:03Z

Overthink: /* Intervals */ add more ratios

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]], [[25/22]], and their [[octave complement]]s, while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are corrected. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]]s of all [[19-odd-limit]] intervals in 38df align with their closest approximations in 38edo, except for 7/4, 13/8, and their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo.

The harmonic series from 1 to 20 is approximated within 38df by the step sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
| [[55/54]], [[45/44]], ''[[33/32]]''
| [[64/63]], ''[[36/35]]''
| [[56/55]]
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
| [[25/24]], [[34/33]]
| [[22/21]]
| [[26/25]], [[27/26]], [[28/27]]
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]'', [[21/20]]
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]'', [[35/32]]
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
| [[28/25]]
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[25/22]], [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]], [[32/25]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]], ''[[21/16]]''
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
| [[25/19]]
| [[21/16]], ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]], [[34/25]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
| [[25/18]]
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
| [[36/25]]
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[22/15]], [[16/11]], [[25/17]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
| [[38/25]]
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]], [[25/16]]
| ''[[26/17]]'', ''[[11/7]]''
| [[14/9]], [[20/13]], ''[[32/21]]''
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
| [[22/13]]
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[44/25]], [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
| [[25/14]]
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]'', [[64/35]]
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
| [[33/17]], [[48/25]]
|
| [[27/14]], [[25/13]], [[52/27]]
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
| ''[[64/33]]'', [[88/45]], [[108/55]]
| [[63/32]]
| [[55/28]]
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

Talk:39edo

2026-06-04T20:02:10Z

Overthink: /* 39 isn't a dual-7 edo */ alright

== 39dfgijk ==

If you don't want the second row of odd harmonics (or prime harmonics if switched to that), you should also get rid of "39edo can be usefully mapped onto the val 39dfgijk", since the argument about higher harmonics being too inaccurate would make this val not so useful. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 03:04, 2 April 2026 (UTC)

: Octave compression certainly makes the higher harmonics more accurate, though one needs to be careful about intervals with many powers of 2 (and also 11, since it loses accuracy at that level of compression). A second row won't do too much harm, so I guess adding it back is fine. [[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:54, 2 April 2026 (UTC)

: Also, when you added the second table you added an extra line between the templates, which makes them more spaced apart than they should be. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:57, 2 April 2026 (UTC)

== 39 isn't a dual-7 edo ==
39d is clearly the best val up to the 11-limit and 39 patent should not be put in the interval table as a competing column (39df might be considered as the 13-limit mapping tho that's besides the point), for mostly the same reason 44d should not as I showed in Talk: 44edo.

To be clear, the question of a dual-prime edo concerns whether two mappings are nearly equally valid. If one mapping is considerably more accurate, it is hard for one to hear the other mapping as a valid approximation to the same set of intervals, since their presence in the same tuning system means the difference in quality is highlighted thru contrast. As such, for many edo articles we present a main mapping most useful for composition. This mapping is discussed at length in the theory section and put in the interval table. The distinction of a main mapping and various ancillary mappings is a consistent feature of edo articles on this wiki.

The ancillary mappings can also be used, and may be interesting for various reasons. I think they deserve to be discussed briefly in the theory section. However, we can't afford to put whatever we think is potentially or marginally useful in the interval table, cuz human readers have limited attention resource and wish to spend it on the best things. A less valid mapping in the interval table means divided attention and less efficiency of presenting information.

For example, in 145edo, there is this short sentence discussing the utility of a less accurate mapping: "The 145c val provides a tuning for magic which is nearly identical to the POTE tuning." But the main mapping is discussed in the rest of the article.

The reasons that 39d commends itself as the main mapping are mostly the same as that for 44. Specifically:
* The sharp 3, 5, 11 justifies the sharp 7. The interactions of 7 with 3, 5, 9, 11, and 15 all favor the sharp mapping. Iow 7 itself is the only inconsistently mapped interval in the 11-limit 15-odd-limit. While this is also true for 34edo, which is treated as dual-7, 39edo differs from 34edo in that the other primes and especially the 5 are very sharp, which brings us to …
* With the flat 7, the 7/5 will have 93% error and the 15/14 will have ''112%'' error, whereas with the sharp 7, the maximum error comes from 7 itself, only 51%.
* TE error for 39d: 2.43 cents; 39dee: 3.13 cents; 39: 3.79 cents. Note that 39dee has a lower error than 39, so if 39dee isn't reasonable to consider, neither is 39 logically.

The only difference here is that the flat-7 mapping is a patent val. On that account one might argue that the mapping is of some special importance. I think the value of patentness has been overstated in the community at large. What we mean by a patent val is really using the closest approximation for the basis elements, but basis elements can change. For example, many ppl consider 5/3 and/or 7/6 to be as important in composition as 5/4 and 7/4, and one can generate the 7-limit with 2, 3, 5/3, and 7/6. In this basis, the patent val for 39edo isn't the same as the one found for 2, 3, 5, and 7. In fact it's the sharp-7 mapping. That reveals the lack of unique significance of patent vals in practice (and in math, as every GPV is demonstrably patent in some way); as such the importance of a mapping solely from being a patent val in this specific case is baseless from a broader perspective.

—[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:51, 29 May 2026 (UTC)

: Overall, I agree that 39d feels more natural to use. However, the wiki is supposed to present info from a neutral point of view rather than pushing a perspective. Not everyone agrees that more accurate necessarily means "better", and that patent vals are completely arbitrary. People often think of the octave as the equivalence interval, so they want to keep it pure. The pure-octave patent val with prime harmonics as basis entries feels like the most natural mapping to use for many people, even if it is less accurate overall. The patent val isn't completely uninteresting, supporting structures like immunity and triforce. 39edo is a medium-sized edo, and someone who uses it very much may not be focused on accuracy.

: Overall, I think 39 and 39d should have about equal coverage, with structures in both presented. The page definitely should explain how 39d improves accuracy of many intervals, and it should be up to the reader to decide which perspective they agree with, and which mapping to use. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 17:08, 30 May 2026 (UTC)

:: I agree with Overthink on this front. At the very least, the patent val should be given in the interval table, and it should probably be discussed in other parts of the page, but I recognize 39d as a legitimate interpretation as well. Most people looking for an edo to use don't really care about TE either. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 10:12, 3 June 2026 (UTC)

::: > I agree with Overthink on this front. […]

::: This is not a poll. Address my points and back your own points with reasoning.

::: > Most people looking for an edo to use don't really care about TE either.

::: You don't have to "care about TE" to see why 39d is clearly the best val. I showed three aspects for it, only one of which deals with TE errors, so my other two aspects still stand. TE errors here are but a math model to quantify the general observation that 39d is the most accurate mapping, considerably more so than the second place, and that 39 patent isn't necessarily the second place.

::: —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:05, 3 June 2026 (UTC)

:: That sounds very sensible — put my vote in for that as well. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 18:36, 30 May 2026 (UTC)

:: > Not everyone agrees that more accurate necessarily means "better", and that patent vals are completely arbitrary.

:: That's why I need these ppl to hear me out.

:: > The patent val isn't completely uninteresting, supporting structures like immunity and triforce.

:: I think it's less interesting than 39c. I think Leri has made a case for 39d vs 39c, so dual-5 dual-7, which is more acceptable than dual-7 alone. I should add that the prime 17 in 39c doesn't really work given it tunes 17/16 < 18/17, and that 39d might benefit more from a 19 than 13 tho I don't really mind 39df. I also think 2.3.25.35.11 is too sparse a subgroup to be interesting. Overall I'm willing to take the compromise of two columns, one based on 39d and the other based on 39c, with the patent val deducible from them.

:: —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 20:30, 30 May 2026 (UTC)

:: So, as I'm editing the table for a 39c column, I've realized 39c isn't significantly better than 39. In light of this, I propose two tables. First for 39d as we had before (plus primes 13 and 19 via 39df), and a second table for a comparison on various other mappings: 39 in 2.25.35.11, 39c, 39, and 39d. This will cater to more demands than ever. Those who don't think of 39 as a dual-prime system get their cleaner table, and those who do get their table too.

:: —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:28, 3 June 2026 (UTC)

::: Seems like an okay solution, worth a try. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 20:02, 4 June 2026 (UTC)

: I have no opinion on 39et in particular, but I will agree that "patentness" is a completely arbitrary quality and the procedure of rounding is merely heuristic to get something that usually works. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 19:47, 30 May 2026 (UTC)

Diaschismic family

2026-06-02T18:39:02Z

Overthink: /* Pajara */ subgroup extensions

{{Technical data page}}
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].

== Diaschismic ==
{{Main| Diaschismic }}

The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.

This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.

[[Subgroup]]: 2.3.5

[[Comma list]]: 2048/2025

{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
: mapping generators: ~45/32, ~3

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4107{{c}}, ~3/2 = 704.2059{{c}}
: [[error map]]: {{val| -1.179 +1.072 +1.150 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}
: error map: {{val| 0.000 +3.003 +3.769 }}

[[Tuning ranges]]:
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}

[[Badness]] (Sintel): 0.467

=== Overview to extensions ===
==== 7-limit extensions ====
To get the 7-limit extensions, we add another comma:
* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
* Pajara adds [[50/49]] or [[64/63]] and is a popular and well-known choice.
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
* Keen adds [[875/864]].

Those all keep the same half-octave period and fifth generator.

Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.

==== Subgroup extensions ====
Since the diaschisma factors into ([[256/255]])2([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.

== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}

A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.

Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2048/2025

{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4449{{c}}, ~3/2 = 703.0299{{c}}
: [[error map]]: {{val| -1.110 -0.035 +3.740 -1.391 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}

[[Badness]] (Sintel): 0.959

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

Comma list: 126/125, 176/175, 896/891

Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}

Optimal tunings:
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}

Badness (Sintel): 0.828

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

Comma list: 126/125, 176/175, 196/195, 364/363

Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}

Optimal tunings:
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]

{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}

Badness (Sintel): 0.782

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

Comma list: 126/125, 136/135, 176/175, 196/195, 256/255

Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}

Optimal tunings:
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}

Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]

{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}

Badness (Sintel): 0.837

=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
Na"Naa' is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23. It is yet to be found why it got this strange name.

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255

Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}

Optimal tunings:
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}

{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}

Badness (Sintel): 0.882

== Pajara ==
{{Main| Pajara }}

Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49, 64/63

{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
: [[error map]]: {{val| -2.303 +1.432 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]

{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}

[[Badness]] (Sintel): 0.507

=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17

Comma list: 50/49, 64/63, 85/84

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

Optimal tunings:
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}

{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}

Badness (Sintel): 0.438

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

Comma list: 50/49, 64/63, 99/98

Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}

Optimal tunings:
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]

{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}

Badness (Sintel): 0.673

==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17

Comma list: 50/49, 64/63, 85/84, 99/98

Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}

Optimal tunings:
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}

{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}

Badness (Sintel): 0.645

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

Comma list: 50/49, 64/63, 65/63, 99/98

Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}

Optimal tunings:
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}

{{Optimal ET sequence|legend=0| 10e, 12, 22 }}

Badness (Sintel): 1.14

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

Comma list: 50/49, 52/51, 64/63, 65/63, 99/98

Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}

Optimal tunings:
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}

{{Optimal ET sequence|legend=0| 10e, 12, 22 }}

Badness (Sintel): 1.06

==== Pajarina ====
Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 78/77, 99/98

Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}

Optimal tunings:
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}

{{Optimal ET sequence|legend=0| 12f, 22, 34d }}

Badness (Sintel): 0.923

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

Comma list: 50/49, 64/63, 78/77, 85/84, 99/98

Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}

Optimal tunings:
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}

{{Optimal ET sequence|legend=0| 12f, 22, 34d }}

Badness (Sintel): 0.936

==== Pajarita ====
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 64/63, 66/65

Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}

Optimal tunings:
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}

{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}

Badness (Sintel): 0.937

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

Comma list: 40/39, 50/49, 64/63, 66/65, 85/84

Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}

Optimal tunings:
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}

{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}

Badness (Sintel): 0.968

=== Pajarous ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 64/63

Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}

Optimal tunings:
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]

{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}

Badness (Sintel): 0.937

==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}

Optimal tunings:
* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}

{{Optimal ET sequence|legend=0| 10, 12e, 22 }}

Badness (Sintel): 0.766

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

Comma list: 50/49, 55/54, 64/63, 65/63

Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}

Optimal tunings:
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}

{{Optimal ET sequence|legend=0| 10, 22 }}

Badness (Sintel): 1.04

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

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}

Optimal tunings:
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}

{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}

Badness (Sintel): 0.930

==== Pajaro ====
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 55/54, 64/63

Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}

Optimal tunings:
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}

{{Optimal ET sequence|legend=0| 10, 22f, 32f }}

Badness (Sintel): 1.13

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

Comma list: 40/39, 50/49, 55/54, 64/63, 85/84

Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}

Optimal tunings:
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}

{{Optimal ET sequence|legend=0| 10, 22f, 32f }}

Badness (Sintel): 1.01

=== Pajaric ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 56/55

Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}

Optimal tunings:
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}

{{Optimal ET sequence|legend=0| 10, 12, 22e }}

Badness (Sintel): 0.787

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

Comma list: 40/39, 45/44, 50/49, 56/55

Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}

Optimal tunings:
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}

{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}

Badness (Sintel): 0.845

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

Comma list: 34/33, 40/39, 45/44, 50/49, 56/55

Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}

Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}

{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}

Badness (Sintel): 0.896

=== Hemipaj ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 121/120

Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}

: mapping generators: ~2, ~16/11

Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}

{{Optimal ET sequence|legend=0| 2, 20, 22 }}

Badness (Sintel): 1.29

=== Hemifourths ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 243/242

Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
: mapping generators: ~2, ~55/32

Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}

{{Optimal ET sequence|legend=0| 10, 24d, 34d }}

Badness (Sintel): 1.62

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

Comma list: 50/49, 64/63, 78/77, 144/143

Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}

Optimal tunings:
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}

{{Optimal ET sequence|legend=0| 10, 24d, 34d }}

Badness (Sintel): 1.19

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

Comma list: 50/49, 64/63, 78/77, 85/84, 144/143

Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}

Optimal tunings:
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}

{{Optimal ET sequence|legend=0| 10, 24d, 34d }}

Badness (Sintel): 1.11

== Srutal ==
{{See also| Srutal vs diaschismic }}

Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 4375/4374

{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4046{{c}}, ~3/2 = 704.1150{{c}}
: [[error map]]: {{val| -1.191 +0.969 +1.289 +0.044 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7646{{c}}
: error map: {{val| 0.000 +2.810 +4.157 +2.643 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}

[[Badness]] (Sintel): 2.32

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

Comma list: 176/175, 896/891, 1331/1323

Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}

Optimal tunings:
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}

Badness (Sintel): 1.17

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

Comma list: 169/168, 176/175, 325/324, 364/363

Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}

Optimal tunings:
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]

{{Optimal ET sequence|legend=0| 34d, 46, 80 }}

Badness (Sintel): 1.04

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

Comma list: 136/135, 169/168, 176/175, 221/220, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}

Optimal tunings:
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}

Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]

{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}

Badness (Sintel): 0.947

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

Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}

Optimal tunings:
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}

{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}

Badness (Sintel): 1.04

==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}

Optimal tunings:
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}

{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}

Badness (Sintel): 0.971

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

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}

Optimal tunings:
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}

{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}

Badness (Sintel): 1.10

===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}

Optimal tunings:
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}

{{Optimal ET sequence|legend=0| 46, 80, 126 }}

Badness (Sintel): 1.44

== Keen ==
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 2048/2025

{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
: [[error map]]: {{val| -0.679 +4.536 -3.033 -2.591 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}
: error map: {{val| 0.000 +5.574 -1.373 -1.061 }}

{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}

[[Badness]] (Sintel): 2.13

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

Comma list: 100/99, 385/384, 1232/1215

Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}

Optimal tunings:
* WE: ~45/32 = 599.6286{{c}}, ~3/2 = 707.1712{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5984{{c}}

{{Optimal ET sequence|legend=0| 22, 56, 78 }}

Badness (Sintel): 1.50

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

Comma list: 100/99, 105/104, 144/143, 1078/1053

Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}

Optimal tunings:
* WE: ~45/32 = 599.3498{{c}}, ~3/2 = 706.4009{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.1309{{c}}

{{Optimal ET sequence|legend=0| 22f, 34, 56f }}

Badness (Sintel): 1.85

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

Comma list: 100/99, 105/104, 119/117, 144/143, 154/153

Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}

Optimal tunings:
* WE: ~17/12 = 599.4053{{c}}, ~3/2 = 706.4544{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.1243{{c}}

{{Optimal ET sequence|legend=0| 22f, 34, 56f }}

Badness (Sintel): 1.54

==== Keenic ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 352/351, 385/384

Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}

Optimal tunings:
* WE: ~45/32 = 599.8547{{c}}, ~3/2 = 707.0858{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.2596{{c}}

{{Optimal ET sequence|legend=0| 22, 34, 56 }}

Badness (Sintel): 1.67

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

Comma list: 91/90, 100/99, 136/135, 154/153, 256/255

Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}

Optimal tunings:
* WE: ~17/12 = 599.8338{{c}}, ~3/2 = 707.0558{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.2537{{c}}

{{Optimal ET sequence|legend=0| 22, 34, 56 }}

Badness (Sintel): 1.37

== Bidia ==
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 68 }} temperament; its ploidacot is tetraploid monocot. Scales of bidia [[cluster temperament|cluster]] around [[12edo]], with a small residue left behind when three semitones exceed the quarter-octave period. This residue represents [[64/63]], and somewhat peculiarly, [[81/80]] is represented by ''two'' of these intervals.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 3136/3125

{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}
: mapping generators: ~25/21, ~3

[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.6887{{c}}, ~3/2 = 704.6318{{c}}
: [[error map]]: {{val| -1.245 +1.432 +0.064 +0.854 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 705.5070{{c}}
: error map: {{val| 0.000 +3.552 +2.672 +3.639 }}

{{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}

[[Badness]] (Sintel): 1.43

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

Comma list: 176/175, 896/891, 1375/1372

Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}

Optimal tunings:
* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.2170{{c}}

{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}

Badness (Sintel): 1.33

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

Comma list: 176/175, 325/324, 640/637, 896/891

Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}

Optimal tunings:
* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3241{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}

Badness (Sintel): 1.70

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

Comma list: 136/135, 176/175, 256/255, 325/324, 640/637

Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}

Optimal tunings:
* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3496{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}

Badness (Sintel): 1.46

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

Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637

Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}

Optimal tunings:
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}

Badness (Sintel): 1.25

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

Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637

Mapping: {{mapping| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}

Optimal tunings:
* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}

Badness (Sintel): 1.24

== Shrutar ==
Shrutar adds 245/243 to the commas, and also tempers out [[6144/6125]]. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either ~36/35 or ~35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14(1/7), making 7's just.

By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14(1/7) generator can again be used as tunings.

Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 2048/2025

{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
: mapping generators: ~45/32, ~35/24

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.5401{{c}}, ~35/24 = 652.3108{{c}}
: [[error map]]: {{val| -0.920 +2.207 +0.304 -1.730 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~35/24 = 652.7736{{c}}
: error map: {{val| 0.000 +3.592 +2.592 +0.589 }}

{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}

[[Badness]] (Sintel): 1.20

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

Comma list: 121/120, 176/175, 245/243

Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}

Optimal tunings:
* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}

{{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}

Badness (Sintel): 0.876

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

Comma list: 121/120, 176/175, 196/195, 245/243

Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}

Optimal tunings:
* WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}

{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}

Badness (Sintel): 1.16

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

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195

Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}

Optimal tunings:
* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}

{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}

Badness (Sintel): 0.953

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

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342

Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}

Optimal tunings:
* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}

{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}

Badness (Sintel): 1.07

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

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342

Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}

Optimal tunings:
* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}

{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}

Badness (Sintel): 1.03

== Shru ==
Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 392/375, 1323/1280

{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
: mapping generators: ~45/32, ~10/7

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}

{{Optimal ET sequence|legend=1| 2, 22d, 24 }}

[[Badness]] (Sintel): 3.99

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

Comma list: 56/55, 77/75, 1323/1280

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

Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}

{{Optimal ET sequence|legend=0| 2, 22d, 24 }}

Badness (Sintel): 2.10

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

Comma list: 56/55, 77/75, 105/104, 507/500

Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}

Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}

{{Optimal ET sequence|legend=0| 2, 24 }}

Badness (Sintel): 2.12

== Sruti ==
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 19683/19600

{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
: mapping generators: ~45/32, ~140/81

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.2764{{c}}, ~140/81 = 950.7284{{c}}
: [[error map]]: {{val| -1.447 -0.498 +2.813 +1.497 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}
: error map: {{val| 0.000 +1.690 +6.395 +4.869 }}

{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}

[[Badness]] (Sintel): 2.97

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

Comma list: 176/175, 243/242, 896/891

Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}

Optimal tunings:
* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}

{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}

Badness (Sintel): 1.37

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

Comma list: 144/143, 176/175, 351/350, 676/675

Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}

Optimal tunings:
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}

{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}

Badness (Sintel): 0.983

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

Comma list: 136/135, 144/143, 170/169, 176/175, 221/220

Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}

Optimal tunings:
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}

{{Optimal ET sequence|legend=0| 24, 34d, 58 }}

Badness (Sintel): 1.05

== Anguirus ==
As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 49/48, 2048/2025

{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
: mapping generators: ~45/32, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2758{{c}}, ~7/4 = 953.4593{{c}}
: [[error map]]: {{val| +0.552 +4.964 +2.883 -14.264 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}
: error map: {{val| 0.000 +4.083 +1.611 -15.807 }}

{{Optimal ET sequence|legend=1| 10, 24, 34 }}

[[Badness]] (Sintel): 1.97

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

Comma list: 49/48, 56/55, 243/242

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

Optimal tunings:
* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}

{{Optimal ET sequence|legend=0| 10, 24, 34 }}

Badness (Sintel): 1.63

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

Comma list: 49/48, 56/55, 91/90, 243/242

Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}

Optimal tunings:
* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}

{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}

Badness (Sintel): 1.27

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

Comma list: 49/48, 56/55, 91/90, 119/117, 154/153

Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}

Optimal tunings:
* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}

{{Optimal ET sequence|legend=0| 10, 24, 34 }}

Badness (Sintel): 1.10

== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.

The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.

Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 2048/2025

{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
: mapping generators: ~45/32, ~9/7

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}

{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}

[[Badness]] (Sintel): 1.47

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

Comma list: 176/175, 540/539, 896/891

Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}

Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}

Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7

{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}

Badness (Sintel): 0.859

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

Comma list: 176/175, 351/350, 364/363, 540/539

Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}

Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}

{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}

Badness (Sintel): 0.978

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

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}

Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}

{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}

Badness (Sintel): 1.03

== Echidnic ==
Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 686/675, 1029/1024

{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
: mapping generators: ~45/32, ~8/7

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}

{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}

[[Badness]] (Sintel): 1.83

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

Comma list: 385/384, 441/440, 686/675

Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}

Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}

{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}

Badness (Sintel): 1.49

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

Comma list: 91/90, 169/168, 385/384, 441/440

Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}

Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}

{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}

Badness (Sintel): 1.19

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

Comma list: 91/90, 136/135, 154/153, 169/168, 256/255

Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}

Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}

{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}

Badness (Sintel): 0.983

; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)

== Quadrasruta ==
Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 2401/2400

{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
: mapping generators: ~45/32, ~21/16

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4443{{c}}, ~21/16 = 475.7746{{c}}
: [[error map]]: {{val| -1.111 +1.143 +1.377 -0.595 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}
: error map: {{val| 0.000 +3.003 +3.771 +2.456 }}

{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}

[[Badness]] (Sintel): 1.86

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

Comma list: 176/175, 896/891, 2401/2400

Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}

Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}

{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}

Badness (Sintel): 1.62

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

Comma list: 176/175, 196/195, 512/507, 676/675

Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}

Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}

{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}

Badness (Sintel): 1.18

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

Comma list: 136/135, 170/169, 176/175, 196/195, 256/255

Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}

Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}

{{Optimal ET sequence|legend=0| 10e, 58, 126f }}

Badness (Sintel): 1.21

=== Quadrafourths ===
Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 2048/2025

Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}

Optimal tunings:
* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}

{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}

Badness (Sintel): 1.62

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

Comma list: 144/143, 196/195, 243/242, 676/675

Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}

Optimal tunings:
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}

{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}

Badness (Sintel): 1.11

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

Comma list: 136/135, 144/143, 170/169, 196/195, 221/220

Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}

Optimal tunings:
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}

{{Optimal ET sequence|legend=0| 10, 48c, 58 }}

Badness (Sintel): 1.13

== Subgroup extensions ==
=== Srutal archagall (2.3.5.17) ===
{{See also | Fiventeen }}

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
: mapping generators: ~17/12, ~3

Optimal tunings:
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}

{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}

Badness (Sintel): 0.212

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

Subgroup temperaments

2026-06-02T17:45:59Z

Overthink: /* Sburb */ cent value

{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.

For temperaments that omit various prime harmonics, see:
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].

= Composite subgroup temperaments =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.165

==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

: [[gencom]]: [2 9/8; 225/224 325/324 640/637]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.3.25 subgroup ==

=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.

Subgroup: 2.3.25

Edo join: 17 & 12

Comma list: [[2048/2025]]

{{Mapping|legend=2| 1 1 7| 0 1 -4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136

==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''

Edo join: 17 & 12

Comma list: 2048/2025, 576/575, 82/81

{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264

===== Sburb =====
This temperament sets the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.

Subgroup: 2.3.7.23.25.41.59

Edo join: 17 & 12

Comma list: 64/63, 225/224, 162/161, 82/81, 177/175

{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387

== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

[[Badness]]: 0.00263

=== Joan ===
{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.7.13.17 subgroup ==

=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4. It can be formed by doubling both generator and period of [[gizzard]].

[[Subgroup]]: 2.9.7.13.17

[[Comma list]]: 729/728, 442/441, 833/832

{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836

== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

{{Mapping|legend=2|1 1 2|0 3 2}}

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

=== Genius ===

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

{{Mapping|legend=2| 1 2 4 | 0 3 1 }}

: sval mapping generators: ~2, ~21/16

{{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }}

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

=== Tetrameantone ===
{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

=== Tetramagic ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

{{Mapping|legend=2| 1 0 1 | 0 5 1 }}

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

{{Todo|inline=1|cleanup}}

<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)
⟨2399.3973, 193.8643]

TE Step Tunings (cents)
⟨25.21211, 47.81337]

TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]

TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]

Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave

Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩

TE Generator Tunings (cents)
⟨2399.4195, 193.8654]

TE Step Tunings (cents)
⟨25.23883, 47.79592]

TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]

TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave

Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)
⟨2400.1097, 193.9498]

TE Step Tunings (cents)
⟨24.18752, 48.52491]

TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave

Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>

==== 4.6.5.7.11.13 ====

<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩

TE Generator Tunings (cents)
⟨2401.2305, 193.5378]

TE Step Tunings (cents)
⟨42.79107, 35.98524]

TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>

==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)
⟨2400.4701, 193.4599]

TE Step Tunings (cents)
⟨43.39350, 35.55764]

TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>

==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩

TE Generator Tunings (cents)
⟨2399.9219, 193.3952]

TE Step Tunings (cents)
⟨44.14256, 35.03670]

TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>

==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

TE Generator Tunings (cents)
⟨2399.3286, 193.5316]

TE Step Tunings (cents)
⟨37.31613, 39.63311]

TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>

== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

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

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

== 8.9.7 subgroup ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10-3

= Fractional subgroup temperaments =
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

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

: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====
{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

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

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====
{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====
{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===
{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

{{Mapping|legend=2| 1 0 0 | 0 2 1 }}

{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}
: <nowiki/>* wart for 5/3
: † wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

[[Subgroup]]: 2.5/3.13/9

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}

{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])3 = [[13/10]] and ([[15/14]])3 = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.

[[Subgroup]]: 2.5.7/3.11/3.13

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})

{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} using subgroup TE 
: <nowiki/>* wart for 7/3

Badness (Sintel): 0.329

==== Laz ====
{{See also | Chromatic pairs #Laz }}

Laz is related to [[avalokita]] as well as to [[winston]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}

{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}

Kryptonite is related to [[krypton]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12

{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}

Kiribati is related to [[nakika]] as well as to [[octacot]].

[[Subgroup]]: 2.9/5.7/3.11/9

[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})

{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20

{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents

=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.

[[Subgroup]]: 2.7/3.11

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

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

{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents

== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.

[[Subgroup]]: 2.5.9/7

[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})

{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5

{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents

==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[50/49]]

{{Mapping|legend=2| 2 3 1 | 0 1 0 }}

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}

{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830

{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
 based on subgroup TE 

Badness (Sintel): 0.119

==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}

Edson is related to [[pele]] and [[andromeda]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3

{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

{{Optimal ET sequence|legend=1| 12, 17, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

==== Haumea ====
{{See also| Chromatic pairs #Haumea }}

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}

{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: 364/363, 441/440, 1001/1000

{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

[[Subgroup]]: 2.7/5.9/5

[[Comma list]]: [[250047/250000]]

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

{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461

{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents

=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].

[[Subgroup]]: 2.7/5.11/5.13/5

[[Comma list]]: [[847/845]], [[1001/1000]]

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

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents

==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

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

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95t, 124t }}
: t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents

== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

[[Subgroup]]: 2.11/5.13/5

[[Comma list]]: [[2200/2197]]

{{Mapping|legend=2| 1 0 1| 0 3 1 }}

{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012

{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents

==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168, 540/539, 729/728

{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents

=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].

Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].

[[Subgroup]]: 2.3.7.11/5

[[Comma list]]: 1029/1024, 4000/3993

{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742

{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 1029/1024, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918

{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17

[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19

[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
: <nowiki />* wart for 11/7
: † wart for 13/7

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====
{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

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

{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

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

[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5

=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]

[[Subgroup]]: 2.3.49/5

[[Comma list]]: 2401/2400

{{Mapping|legend=2| 1 1 3 | 0 2 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

[[Tp tuning #T2 tuning|RMS error]]: ?

== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

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

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})

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

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

<nowiki/>* wart for 19/7

[[Badness]] (Sintel): 0.082

=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].

[[Subgroup]]: 2.17/7.19/7

[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})

{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}

{{Optimal ET sequence|legend=1| 7, 18, 25 }}

[[Badness]] (Sintel): 0.005

==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

Subgroup: 2.17/7.19/7.23/7

Comma list: [[323/322]], [[392/391]]

Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}

Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}

{{Optimal ET sequence|legend=0| 7, 18, 25 }}

Badness (Sintel): 0.029

==== 2.25/7.17/7.19/7.23/7 subgroup ====

Subgroup: 2.25/7.17/7.19/7.23/7

Comma list: [[323/322]], [[392/391]], [[476/475]]

Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}

Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}

{{Optimal ET sequence|legend=0| 7, 18, 25 }}

Badness (Sintel): 0.053

== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}

=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: [[10976/10935]]

{{Mapping|legend=2| 1 2 3 | 0 3 1 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909

[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

=== Halftone ===
{{Main| Halftone }}

[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: 9604/9375

{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2

[[Comma list]]: 1232/1215, 27783/27500

{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2

[[Comma list]]: 275/273, 1232/1215, 1323/1300

{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2

=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4

[[Comma list]]: 245/243

{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}

: sval mapping generators: ~3/2, ~9/7

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728

[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]

==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 100/99, 245/243

{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771

[[Optimal ET sequence]]: [[8edf]], [[13edf]]

==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 245/243, 441/440

{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939

[[Optimal ET sequence]]: [[8edf]], [[11edf]]

== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''

[[Subgroup]]: 3/2.5/4.11/8

[[Comma list]]: 121/120

{{Mapping|legend=2| 1 1 1 | 0 2 -1 }}]

: [[gencom]]: [3/2 12/11; 121/120]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29

{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}

== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 87808/85293

{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}

: sval mapping generators: ~3/2, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2

=== Doubleton ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 1352/1323

{{Mapping|legend=2| 2 0 3 | 0 1 1 }}

: sval mapping generators: ~26/21, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2

== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11

[[Comma list]]: {{monzo| 11 1 -5 }}

{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}

: [[gencom]]: [5/2 125/88; 341796875/329832448]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2

= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments

[[Category:Subgroup temperaments| ]] 
[[Category:Temperament collections]]
{{Todo| review | cleanup }}

Kryptonite19

2026-06-02T04:21:18Z

Overthink: info

<pre>
! kryptonite19.scl
!
Kryptonite[19] 2.5.7/3.11/3.13/3 subgroup MOS in 65\589 tuning
19
!
124.27844
132.42784
256.70628
264.85569
273.00509
397.28353
405.43294
529.71138
537.86078
662.13922
670.28862
794.56706
802.71647
926.99491
935.14431
1059.42275
1067.57216
1191.85059
2/1
</pre>

Pattern: [[9L 10s]]

Mode ([[UDP]]): 11|7

[[Category:19-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Kryptonite]]
[[Category:MOS scales]]
[[Category:13-limit]]
[[Category:589edo]]
[[Category:Pages with Scala files]]

5L 5s

2026-06-02T04:18:21Z

Overthink: standardize layout

{{Infobox MOS
| Name = pentawood
| Periods = 5
| nLargeSteps = 5
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 0
| Pattern = LsLsLsLsLs
}}
{{MOS intro}}

There is only one significant [[harmonic entropy]] minimum with this MOS pattern: [[limmic temperaments#5-limit_.28blackwood.29|blackwood]], in which intervals of the prime numbers [[3/1|3]] and [[7/1|7]] are all represented using steps of [[5edo|5edo]], and the generator reaches intervals of [[5/1|5]], such as [[6/5]], [[5/4]], and [[7/5]].

In addition to the true MOS form (LsLsLsLsLs and sLsLsLsLsL), there are 6 near-MOS forms, which are also single-alteration [[MODMOS]]ses – LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss – in which the period and its multiples (intervals of 2, 4, 6, and 8 mossteps) have more than two varieties. These forms are proper if the bright generator is less than 160¢.

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
{{MOS intervals}}

=== Modes ===
{{MOS mode degrees}}

=== Scale tree ===
{{MOS tuning spectrum
| 6/5 = [[Qintosec]] ↑
| 7/5 = [[Warlock]]
| 13/8 = Unnamed golden tuning (148.328{{c}})
| 7/4 = [[Quinkee]]
| 2/1 = [[Blackwood]] (optimal around here)
| 9/4 = [[Trisedodge]]
| 13/5 = Unnamed golden tuning (173.666{{c}})
| 6/1 = [[Cloudtone]] ↓
}}

[[Category:Pentawood| ]] 
[[Category:10-tone scales]]

41L 12s

2026-06-02T04:15:13Z

Overthink: Cotoneum in scales and scale tree; grammar

{{Infobox MOS}}{{MOS intro}} Its [[chroma-positive]] generator is a almost-perfect fifth of no more than 31\53 (701.9{{cent}}), where the large step is identified with the [[Pythagorean comma]], and the small step is identified with the [[41-comma|countercomp comma]].
[[Pythagorean tuning]] can generate this mos scale with a hardness of 1.1822. This mos scale is associated with sharper schismic temperaments and other very accurate fifth-based temperaments, like [[cassandra]], [[gary]], [[counterschismic]], [[cotoneum]], and others. It is known as '''pythomerc''' in [[TAMNAMS Extension]].

== Intervals ==
{{MOS intervals}}

== Scales ==
* [[Pythagorean53]]
* [[Cotoneum53]]

== Scale tree ==
{{MOS tuning spectrum
| 1/1 = [[Mercator]]
| 6/5 = [[Pythagorean tuning]] (701.955{{c}})
| 5/4 = [[Quasipyth]]
| 2/1 = [[Garibaldi]] / [[cassandra]]
| 3/1 = [[Gary]]
| 5/1 = [[Cotoneum]]
}}

MOS scale

2026-06-02T04:11:28Z

Overthink: /* Individual pages for MOS scales */ the cluster temp ones feel more useful

{{interwiki
| en = MOS scale
| de = MOS-Skala
| es =
| ja = MOSスケール
| ro = G2S
}}{{Beginner|Mathematics of MOS}}
A '''moment of symmetry''' ('''MOS''' or '''mos'''<ref group="note">The acronym "MOS" is generally pronounced ''em-oh-ess'', while the {{w|anacronym}} "mos", more common in informal and experimental settings, is generally pronounced ''moss''. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.</ref>) '''scale''' is a [[periodic scale]] where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come in only one size.

MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.

== Examples ==
The most widely used MOS scale is the [[5L 2s|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L 2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).

{| class="wikitable"
|+ style="font-size: 105%;" | Interval classes in the 5L 2s MOS scale
|-
! rowspan="2" | Interval class
! colspan="2" | Small version
! colspan="2" | Large version
|-
! Quality
! Size
! Quality
! Size
|-
! 2nds (1 step)
| minor
| s
| major
| L
|-
! 3rds (2 steps)
| minor
| {{nowrap|1L + 1s}}
| major
| 2L
|-
! 4ths (3 steps)
| perfect
| {{nowrap|2L + 1s}}
| augmented
| 3L
|-
! 5ths (4 steps)
| diminished
| {{nowrap|2L + 2s}}
| perfect
| {{nowrap|3L + 1s}}
|-
! 6ths (5 steps)
| minor
| {{nowrap|3L + 2s}}
| major
| {{nowrap|4L + 1s}}
|-
! 7ths (6 steps)
| minor
| {{nowrap|4L + 2s}}
| major
| {{nowrap|5L + 1s}}
|-
! 8ves (7 steps)
| perfect
| {{nowrap|5L + 2s}}
| colspan="2" | (only one version)
|}

Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

Other MOS scales include [[2L 3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L 4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).

See the [[catalog of MOS]] for other MOS scales.

== Periods and generators ==
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap| C D E F G A C }} does not produces a MOS, because there are more than 2 sizes of each interval class.

The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53, …. However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50, ….

== Step ratio spectrum ==
The [[step ratio]] is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The [[TAMNAMS #Step ratio spectrum|TAMNAMS]] system has names for specific ratios and also ranges of ratios.

When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L 2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.

{| class="wikitable"
|+ style="font-size: 105%;" | 5L 2s step ratios in various edos
|-
! Example edo
! Step ratio
! TAMNAMS name
! Likely temperament interpretations
|-
! 12
| 2:1
| basic
| [[Meantone]] or [[Schismatic]]
|-
! 19
| 3:2
| soft
| [[Meantone]]
|-
! 22
| 4:1
| superhard
| [[Archy]] or [[Superpyth]]
|}

== Naming ==
Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L 2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a 2b" is used (which could refer to either diatonic or {{nowrap| [[2L 5s|anti-diatonic]] {{=}} 2L 5s }}).

By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x27E8;</code> and <code>&#x27E9;</code>) is recommended; using greater-than and less-than signs ("<equave>") can also be done, but this can conflict with HTML and other uses of these symbols.

Several naming systems have been proposed for MOSes, which can be seen at [[MOS naming]].

== History and terminology ==
The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [https://anaphoria.com/mos.pdf ''Moments of Symmetry'']. There is also an introduction by [[Kraig Grady]] here: [https://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].

Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.

With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[pentatonic]]s used in traditional [[Japanese music]] (e.g. {{nowrap| A B C E F A }}), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.

== Equivalent definitions and generalizations ==
A scale is a MOS if and only if it satisfies one of the following equivalent criteria:
# [[Maximum variety]] 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.)
# [[Binary]] and has a [[generator]]: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.
# Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.

Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of [[balanced word|balance]] and [[distributional evenness]] provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see [[Mathematics of MOS]].

== Properties ==
=== Basic properties ===
* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (''L'') step and small (''s'') step are the same size), then by definition it is an {{nowrap| ''x''L (''y'' − ''x'')s }} MOS scale, and the [[basic]] tuning where {{nowrap| ''L'' {{=}} 2''s'' }} is thus {{nowrap|(''x'' + ''y'')}}-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if {{nowrap| ''p'' > 1 }}) you are considering a "multiperiod" MOS scale.
* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[Patent val|patent vals]]) while simultaneously also being used to define the {{nowrap|''px''L (''py'' − ''px'')s}} MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)
* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[Octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament'''*''', we can say that any {{w|natural number|natural}}-coefficient {{w|linear combination}} of vals {{val| ''X'' … }} and {{val| ''Y'' … }} (where {{nowrap| ''X'' < ''Y'' }}) corresponds uniquely to a tuning of the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap| gcd(''a'', ''b'') {{=}} 1 }}, because if {{nowrap| ''k'' {{=}} gcd(''a'', ''b'') > 1 }} then the val {{nowrap| ''a''{{val| ''X'' … }} + ''b''{{val| ''Y'' … }} }} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the {{w|Rational number|rational}} ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).

: The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap| 1{{val| ''X'' … }} + 0{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap| 0{{val| ''X'' … }} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap| 1{{val| ''X'' … ;}} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:

: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying [[step ratio]] {{nowrap| ''r'' {{=}} (''a'' + ''b'')/''b'' ≥ 1 }} for {{w|Natural number|natural}} ''a'' and ''b'', where if {{nowrap| ''b'' {{=}} 0 }} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is ''important to note'' that the correspondence to the {{nowrap| ''X'' & ''Y'' }} rank-2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where {{nowrap| ''X'' & ''Y'' }} describe a contorted temperament on the subgroup given. An example is the {{nowrap| 5 & 19 }} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L 14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.</ref>

* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are {{nowrap| (''a'' + ''b'')L ''a''s }} (generated by generators of soft-of-basic ''a''L ''b''s) and {{nowrap| ''a''L (''a'' + ''b'')s }} (generated by generators of hard-of-basic ''a''L'' b''s).
* Every MOS scale (with a specified [[equave]] ''Ɛ''), excluding {{nowrap|''a''L ''a''s{{angbr|''Ɛ''}} }}, has a ''parent MOS''. If {{nowrap| ''a'' > ''b'' }}, the parent of ''a''L ''b''s is {{nowrap| ''b''L (''a'' − ''b'')s }}; if {{nowrap| ''a'' < ''b'' }}, the parent of ''a''L ''b''s is {{nowrap| ''a''L (''b'' − ''a'')s }}.

=== Advanced discussion ===
See:
* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.
** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.
** [[MOS scale family tree]], a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.
* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.
* [[MOS diagrams]], visualizations of the MOS process.
* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]

== Individual pages for MOS scales ==
=== L ≤ 12, s ≤ 12 ===
{| class="wikitable center-all"
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|s ≤ 12}})
|-
| [[1L 1s]]
| [[1L 2s]]
| [[1L 3s]]
| [[1L 4s]]
| [[1L 5s]]
| [[1L 6s]]
| [[1L 7s]]
| [[1L 8s]]
| [[1L 9s]]
| [[1L 10s]]
| [[1L 11s]]
| [[1L 12s]]
|-
| [[2L 1s]]
| [[2L 2s]]
| [[2L 3s]]
| [[2L 4s]]
| [[2L 5s]]
| [[2L 6s]]
| [[2L 7s]]
| [[2L 8s]]
| [[2L 9s]]
| [[2L 10s]]
| [[2L 11s]]
| [[2L 12s]]
|-
| [[3L 1s]]
| [[3L 2s]]
| [[3L 3s]]
| [[3L 4s]]
| [[3L 5s]]
| [[3L 6s]]
| [[3L 7s]]
| [[3L 8s]]
| [[3L 9s]]
| [[3L 10s]]
| [[3L 11s]]
| [[3L 12s]]
|-
| [[4L 1s]]
| [[4L 2s]]
| [[4L 3s]]
| [[4L 4s]]
| [[4L 5s]]
| [[4L 6s]]
| [[4L 7s]]
| [[4L 8s]]
| [[4L 9s]]
| [[4L 10s]]
| [[4L 11s]]
| [[4L 12s]]
|-
| [[5L 1s]]
| [[5L 2s]]
| [[5L 3s]]
| [[5L 4s]]
| [[5L 5s]]
| [[5L 6s]]
| [[5L 7s]]
| [[5L 8s]]
| [[5L 9s]]
| [[5L 10s]]
| [[5L 11s]]
| [[5L 12s]]
|-
| [[6L 1s]]
| [[6L 2s]]
| [[6L 3s]]
| [[6L 4s]]
| [[6L 5s]]
| [[6L 6s]]
| [[6L 7s]]
| [[6L 8s]]
| [[6L 9s]]
| [[6L 10s]]
| [[6L 11s]]
| [[6L 12s]]
|-
| [[7L 1s]]
| [[7L 2s]]
| [[7L 3s]]
| [[7L 4s]]
| [[7L 5s]]
| [[7L 6s]]
| [[7L 7s]]
| [[7L 8s]]
| [[7L 9s]]
| [[7L 10s]]
| [[7L 11s]]
| [[7L 12s]]
|-
| [[8L 1s]]
| [[8L 2s]]
| [[8L 3s]]
| [[8L 4s]]
| [[8L 5s]]
| [[8L 6s]]
| [[8L 7s]]
| [[8L 8s]]
| [[8L 9s]]
| [[8L 10s]]
| [[8L 11s]]
| [[8L 12s]]
|-
| [[9L 1s]]
| [[9L 2s]]
| [[9L 3s]]
| [[9L 4s]]
| [[9L 5s]]
| [[9L 6s]]
| [[9L 7s]]
| [[9L 8s]]
| [[9L 9s]]
| [[9L 10s]]
| [[9L 11s]]
| [[9L 12s]]
|-
| [[10L 1s]]
| [[10L 2s]]
| [[10L 3s]]
| [[10L 4s]]
| [[10L 5s]]
| [[10L 6s]]
| [[10L 7s]]
| [[10L 8s]]
| [[10L 9s]]
| [[10L 10s]]
| [[10L 11s]]
| [[10L 12s]]
|-
| [[11L 1s]]
| [[11L 2s]]
| [[11L 3s]]
| [[11L 4s]]
| [[11L 5s]]
| [[11L 6s]]
| [[11L 7s]]
| [[11L 8s]]
| [[11L 9s]]
| [[11L 10s]]
| [[11L 11s]]
| [[11L 12s]]
|-
| [[12L 1s]]
| [[12L 2s]]
| [[12L 3s]]
| [[12L 4s]]
| [[12L 5s]]
| [[12L 6s]]
| [[12L 7s]]
| [[12L 8s]]
| [[12L 9s]]
| [[12L 10s]]
| [[12L 11s]]
| [[12L 12s]]
|}

=== L ≤ 12, 13 ≤ s ≤ 24 ===
{| class="wikitable mw-collapsible mw-collapsed center-all"
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|13 ≤ s ≤ 24}})
|-
| [[1L 13s]]
| [[1L 14s]]
| [[1L 15s]]
| [[1L 16s]]
| [[1L 17s]]
| [[1L 18s]]
| [[1L 19s]]
| [[1L 20s]]
| [[1L 21s]]
| [[1L 22s]]
| [[1L 23s]]
| [[1L 24s]]
|-
| [[2L 13s]]
| [[2L 14s]]
| [[2L 15s]]
| [[2L 16s]]
| [[2L 17s]]
| [[2L 18s]]
| [[2L 19s]]
| [[2L 20s]]
| [[2L 21s]]
| [[2L 22s]]
| [[2L 23s]]
| [[2L 24s]]
|-
| [[3L 13s]]
| [[3L 14s]]
| [[3L 15s]]
| [[3L 16s]]
| [[3L 17s]]
| [[3L 18s]]
| [[3L 19s]]
| [[3L 20s]]
| [[3L 21s]]
| [[3L 22s]]
| [[3L 23s]]
| [[3L 24s]]
|-
| [[4L 13s]]
| [[4L 14s]]
| [[4L 15s]]
| [[4L 16s]]
| [[4L 17s]]
| [[4L 18s]]
| [[4L 19s]]
| [[4L 20s]]
| [[4L 21s]]
| [[4L 22s]]
| [[4L 23s]]
| [[4L 24s]]
|-
| [[5L 13s]]
| [[5L 14s]]
| [[5L 15s]]
| [[5L 16s]]
| [[5L 17s]]
| [[5L 18s]]
| [[5L 19s]]
| [[5L 20s]]
| [[5L 21s]]
| [[5L 22s]]
| [[5L 23s]]
| [[5L 24s]]
|-
| [[6L 13s]]
| [[6L 14s]]
| [[6L 15s]]
| [[6L 16s]]
| [[6L 17s]]
| [[6L 18s]]
| [[6L 19s]]
| [[6L 20s]]
| [[6L 21s]]
| [[6L 22s]]
| [[6L 23s]]
| [[6L 24s]]
|-
| [[7L 13s]]
| [[7L 14s]]
| [[7L 15s]]
| [[7L 16s]]
| [[7L 17s]]
| [[7L 18s]]
| [[7L 19s]]
| [[7L 20s]]
| [[7L 21s]]
| [[7L 22s]]
| [[7L 23s]]
| [[7L 24s]]
|-
| [[8L 13s]]
| [[8L 14s]]
| [[8L 15s]]
| [[8L 16s]]
| [[8L 17s]]
| [[8L 18s]]
| [[8L 19s]]
| [[8L 20s]]
| [[8L 21s]]
| [[8L 22s]]
| [[8L 23s]]
| [[8L 24s]]
|-
| [[9L 13s]]
| [[9L 14s]]
| [[9L 15s]]
| [[9L 16s]]
| [[9L 17s]]
| [[9L 18s]]
| [[9L 19s]]
| [[9L 20s]]
| [[9L 21s]]
| [[9L 22s]]
| [[9L 23s]]
| [[9L 24s]]
|-
| [[10L 13s]]
| [[10L 14s]]
| [[10L 15s]]
| [[10L 16s]]
| [[10L 17s]]
| [[10L 18s]]
| [[10L 19s]]
| [[10L 20s]]
| [[10L 21s]]
| [[10L 22s]]
| [[10L 23s]]
| [[10L 24s]]
|-
| [[11L 13s]]
| [[11L 14s]]
| [[11L 15s]]
| [[11L 16s]]
| [[11L 17s]]
| [[11L 18s]]
| [[11L 19s]]
| [[11L 20s]]
| [[11L 21s]]
| [[11L 22s]]
| [[11L 23s]]
| [[11L 24s]]
|-
| [[12L 13s]]
| [[12L 14s]]
| [[12L 15s]]
| [[12L 16s]]
| [[12L 17s]]
| [[12L 18s]]
| [[12L 19s]]
| [[12L 20s]]
| [[12L 21s]]
| [[12L 22s]]
| [[12L 23s]]
| [[12L 24s]]
|}

=== 13 ≤ L ≤ 24, s ≤ 12 ===
{| class="wikitable mw-collapsible mw-collapsed center-all"
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|13 ≤ L ≤ 24|s ≤ 12}})
|-
| [[13L 1s]]
| [[13L 2s]]
| [[13L 3s]]
| [[13L 4s]]
| [[13L 5s]]
| [[13L 6s]]
| [[13L 7s]]
| [[13L 8s]]
| [[13L 9s]]
| [[13L 10s]]
| [[13L 11s]]
| [[13L 12s]]
|-
| [[14L 1s]]
| [[14L 2s]]
| [[14L 3s]]
| [[14L 4s]]
| [[14L 5s]]
| [[14L 6s]]
| [[14L 7s]]
| [[14L 8s]]
| [[14L 9s]]
| [[14L 10s]]
| [[14L 11s]]
| [[14L 12s]]
|-
| [[15L 1s]]
| [[15L 2s]]
| [[15L 3s]]
| [[15L 4s]]
| [[15L 5s]]
| [[15L 6s]]
| [[15L 7s]]
| [[15L 8s]]
| [[15L 9s]]
| [[15L 10s]]
| [[15L 11s]]
| [[15L 12s]]
|-
| [[16L 1s]]
| [[16L 2s]]
| [[16L 3s]]
| [[16L 4s]]
| [[16L 5s]]
| [[16L 6s]]
| [[16L 7s]]
| [[16L 8s]]
| [[16L 9s]]
| [[16L 10s]]
| [[16L 11s]]
| [[16L 12s]]
|-
| [[17L 1s]]
| [[17L 2s]]
| [[17L 3s]]
| [[17L 4s]]
| [[17L 5s]]
| [[17L 6s]]
| [[17L 7s]]
| [[17L 8s]]
| [[17L 9s]]
| [[17L 10s]]
| [[17L 11s]]
| [[17L 12s]]
|-
| [[18L 1s]]
| [[18L 2s]]
| [[18L 3s]]
| [[18L 4s]]
| [[18L 5s]]
| [[18L 6s]]
| [[18L 7s]]
| [[18L 8s]]
| [[18L 9s]]
| [[18L 10s]]
| [[18L 11s]]
| [[18L 12s]]
|-
| [[19L 1s]]
| [[19L 2s]]
| [[19L 3s]]
| [[19L 4s]]
| [[19L 5s]]
| [[19L 6s]]
| [[19L 7s]]
| [[19L 8s]]
| [[19L 9s]]
| [[19L 10s]]
| [[19L 11s]]
| [[19L 12s]]
|-
| [[20L 1s]]
| [[20L 2s]]
| [[20L 3s]]
| [[20L 4s]]
| [[20L 5s]]
| [[20L 6s]]
| [[20L 7s]]
| [[20L 8s]]
| [[20L 9s]]
| [[20L 10s]]
| [[20L 11s]]
| [[20L 12s]]
|-
| [[21L 1s]]
| [[21L 2s]]
| [[21L 3s]]
| [[21L 4s]]
| [[21L 5s]]
| [[21L 6s]]
| [[21L 7s]]
| [[21L 8s]]
| [[21L 9s]]
| [[21L 10s]]
| [[21L 11s]]
| [[21L 12s]]
|-
| [[22L 1s]]
| [[22L 2s]]
| [[22L 3s]]
| [[22L 4s]]
| [[22L 5s]]
| [[22L 6s]]
| [[22L 7s]]
| [[22L 8s]]
| [[22L 9s]]
| [[22L 10s]]
| [[22L 11s]]
| [[22L 12s]]
|-
| [[23L 1s]]
| [[23L 2s]]
| [[23L 3s]]
| [[23L 4s]]
| [[23L 5s]]
| [[23L 6s]]
| [[23L 7s]]
| [[23L 8s]]
| [[23L 9s]]
| [[23L 10s]]
| [[23L 11s]]
| [[23L 12s]]
|-
| [[24L 1s]]
| [[24L 2s]]
| [[24L 3s]]
| [[24L 4s]]
| [[24L 5s]]
| [[24L 6s]]
| [[24L 7s]]
| [[24L 8s]]
| [[24L 9s]]
| [[24L 10s]]
| [[24L 11s]]
| [[24L 12s]]
|}

=== Larger MOS scales ===
[[7L 34s]], [[9L 29s]], [[12L 29s]], [[12L 41s]], [[13L 14s]], [[14L 13s]], [[17L 14s]], [[25L 6s]], [[41L 12s]]

== Variations ==
* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap| L − s }}, the "chroma".
* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.
* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.
* [[Operations on MOSes]]

== Listen ==
This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.

[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}}

== See also ==
* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales
* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]
* [[MOS rhythm]]
* [[:Category:MOS scales|Category:MOS scales]], the category including all MOS-related articles on this wiki
* [[Gallery of MOS patterns]]

== Notes ==
<references group="note" />

== References ==
<references />

[[Category:Math]]
[[Category:MOS scale| ]] 
[[Category:Scale]]
[[Category:Erv Wilson]]

MOS scale

2026-06-02T04:08:53Z

Overthink: /* Larger MOS scales */ add the rest

{{interwiki
| en = MOS scale
| de = MOS-Skala
| es =
| ja = MOSスケール
| ro = G2S
}}{{Beginner|Mathematics of MOS}}
A '''moment of symmetry''' ('''MOS''' or '''mos'''<ref group="note">The acronym "MOS" is generally pronounced ''em-oh-ess'', while the {{w|anacronym}} "mos", more common in informal and experimental settings, is generally pronounced ''moss''. Sometimes "MOSS" or "moss", standing for "moment of symmetry scale", are used instead, although there is no significant difference in meaning.</ref>) '''scale''' is a [[periodic scale]] where every 2nd (that is, every interval formed by ascending a step) is either small or large with no in-between, and the same goes for 3rds, 4ths, etc. Multiples of the period (which is usually the octave or a fraction thereof), however, come in only one size.

MOS scales are often referred to as MOSes, thus MOS can be used as either an adjective or a noun.

== Examples ==
The most widely used MOS scale is the [[5L 2s|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L 2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).

{| class="wikitable"
|+ style="font-size: 105%;" | Interval classes in the 5L 2s MOS scale
|-
! rowspan="2" | Interval class
! colspan="2" | Small version
! colspan="2" | Large version
|-
! Quality
! Size
! Quality
! Size
|-
! 2nds (1 step)
| minor
| s
| major
| L
|-
! 3rds (2 steps)
| minor
| {{nowrap|1L + 1s}}
| major
| 2L
|-
! 4ths (3 steps)
| perfect
| {{nowrap|2L + 1s}}
| augmented
| 3L
|-
! 5ths (4 steps)
| diminished
| {{nowrap|2L + 2s}}
| perfect
| {{nowrap|3L + 1s}}
|-
! 6ths (5 steps)
| minor
| {{nowrap|3L + 2s}}
| major
| {{nowrap|4L + 1s}}
|-
! 7ths (6 steps)
| minor
| {{nowrap|4L + 2s}}
| major
| {{nowrap|5L + 1s}}
|-
! 8ves (7 steps)
| perfect
| {{nowrap|5L + 2s}}
| colspan="2" | (only one version)
|}

Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

Other MOS scales include [[2L 3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L 4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).

See the [[catalog of MOS]] for other MOS scales.

== Periods and generators ==
Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap| C D E F G A C }} does not produces a MOS, because there are more than 2 sizes of each interval class.

The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53, …. However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50, ….

== Step ratio spectrum ==
The [[step ratio]] is the ratio of the larger step size to the smaller step size. MOSes with smaller step ratios sound smooth and soft. MOSes with larger step ratios sound jagged and hard. Different step ratios produce different corresponding potential temperament interpretations. The [[TAMNAMS #Step ratio spectrum|TAMNAMS]] system has names for specific ratios and also ranges of ratios.

When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L 2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.

{| class="wikitable"
|+ style="font-size: 105%;" | 5L 2s step ratios in various edos
|-
! Example edo
! Step ratio
! TAMNAMS name
! Likely temperament interpretations
|-
! 12
| 2:1
| basic
| [[Meantone]] or [[Schismatic]]
|-
! 19
| 3:2
| soft
| [[Meantone]]
|-
! 22
| 4:1
| superhard
| [[Archy]] or [[Superpyth]]
|}

== Naming ==
Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L 2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a 2b" is used (which could refer to either diatonic or {{nowrap| [[2L 5s|anti-diatonic]] {{=}} 2L 5s }}).

By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x27E8;</code> and <code>&#x27E9;</code>) is recommended; using greater-than and less-than signs ("<equave>") can also be done, but this can conflict with HTML and other uses of these symbols.

Several naming systems have been proposed for MOSes, which can be seen at [[MOS naming]].

== History and terminology ==
The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [https://anaphoria.com/mos.pdf ''Moments of Symmetry'']. There is also an introduction by [[Kraig Grady]] here: [https://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].

Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.

With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[pentatonic]]s used in traditional [[Japanese music]] (e.g. {{nowrap| A B C E F A }}), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.

== Equivalent definitions and generalizations ==
A scale is a MOS if and only if it satisfies one of the following equivalent criteria:
# [[Maximum variety]] 2: Ascending by a certain number of steps is equivalent to ascending by one of at most two intervals, and the maximum of two is achieved (i. e. it is not true that ascending by a certain number of steps is always equivalent to ascending by one interval.)
# [[Binary]] and has a [[generator]]: The scale step comes in exactly two sizes, and the scale is formable from stacking some interval called a generator and octave-reducing.
# Mode of a Christoffel word: The scale can be formed by creating a 2D lattice where the period is on the lattice, then taking pitches by travelling vertically and horizontally from the origin, maintaining as close to the line from the origin to the octave as possible without going above it.

Each definition generalizes to scales with three or more step sizes, but these generalizations are not equivalent. The concepts of [[balanced word|balance]] and [[distributional evenness]] provide still different generalizations, although defining MOS through these terms is less helpful. For more information, see [[Mathematics of MOS]].

== Properties ==
=== Basic properties ===
* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (''L'') step and small (''s'') step are the same size), then by definition it is an {{nowrap| ''x''L (''y'' − ''x'')s }} MOS scale, and the [[basic]] tuning where {{nowrap| ''L'' {{=}} 2''s'' }} is thus {{nowrap|(''x'' + ''y'')}}-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if {{nowrap| ''p'' > 1 }}) you are considering a "multiperiod" MOS scale.
* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[Patent val|patent vals]]) while simultaneously also being used to define the {{nowrap|''px''L (''py'' − ''px'')s}} MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)
* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[Octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament'''*''', we can say that any {{w|natural number|natural}}-coefficient {{w|linear combination}} of vals {{val| ''X'' … }} and {{val| ''Y'' … }} (where {{nowrap| ''X'' < ''Y'' }}) corresponds uniquely to a tuning of the {{nowrap| ''X'' & ''Y'' }} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap| gcd(''a'', ''b'') {{=}} 1 }}, because if {{nowrap| ''k'' {{=}} gcd(''a'', ''b'') > 1 }} then the val {{nowrap| ''a''{{val| ''X'' … }} + ''b''{{val| ''Y'' … }} }} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the {{w|Rational number|rational}} ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).

: The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap| 1{{val| ''X'' … }} + 0{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap| 0{{val| ''X'' … }} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap| 1{{val| ''X'' … ;}} + 1{{val| ''Y'' … }} }} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:

: {{nowrap|1/([[step ratio]]) {{=}} ''s''/''L''}} {{nowrap|{{=}} ''b''/(''a'' + ''b'')}} implying [[step ratio]] {{nowrap| ''r'' {{=}} (''a'' + ''b'')/''b'' ≥ 1 }} for {{w|Natural number|natural}} ''a'' and ''b'', where if {{nowrap| ''b'' {{=}} 0 }} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is ''important to note'' that the correspondence to the {{nowrap| ''X'' & ''Y'' }} rank-2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where {{nowrap| ''X'' & ''Y'' }} describe a contorted temperament on the subgroup given. An example is the {{nowrap| 5 & 19 }} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L 14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.</ref>

* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are {{nowrap| (''a'' + ''b'')L ''a''s }} (generated by generators of soft-of-basic ''a''L ''b''s) and {{nowrap| ''a''L (''a'' + ''b'')s }} (generated by generators of hard-of-basic ''a''L'' b''s).
* Every MOS scale (with a specified [[equave]] ''Ɛ''), excluding {{nowrap|''a''L ''a''s{{angbr|''Ɛ''}} }}, has a ''parent MOS''. If {{nowrap| ''a'' > ''b'' }}, the parent of ''a''L ''b''s is {{nowrap| ''b''L (''a'' − ''b'')s }}; if {{nowrap| ''a'' < ''b'' }}, the parent of ''a''L ''b''s is {{nowrap| ''a''L (''b'' − ''a'')s }}.

=== Advanced discussion ===
See:
* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.
** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.
** [[MOS scale family tree]], a tree initially described by Erv Wilson that organizes scales by parent-and-child relationship, which also helps illustrate mos recursion.
* [[Generator ranges of MOS]], organized by number of scale steps and quantity of L/s steps.
* [[MOS diagrams]], visualizations of the MOS process.
* [http://x31eq.com/temper/method.html How to Find Linear Temperaments], by [[Graham Breed]]

== Individual pages for MOS scales ==
=== L ≤ 12, s ≤ 12 ===
{| class="wikitable center-all"
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|s ≤ 12}})
|-
| [[1L 1s]]
| [[1L 2s]]
| [[1L 3s]]
| [[1L 4s]]
| [[1L 5s]]
| [[1L 6s]]
| [[1L 7s]]
| [[1L 8s]]
| [[1L 9s]]
| [[1L 10s]]
| [[1L 11s]]
| [[1L 12s]]
|-
| [[2L 1s]]
| [[2L 2s]]
| [[2L 3s]]
| [[2L 4s]]
| [[2L 5s]]
| [[2L 6s]]
| [[2L 7s]]
| [[2L 8s]]
| [[2L 9s]]
| [[2L 10s]]
| [[2L 11s]]
| [[2L 12s]]
|-
| [[3L 1s]]
| [[3L 2s]]
| [[3L 3s]]
| [[3L 4s]]
| [[3L 5s]]
| [[3L 6s]]
| [[3L 7s]]
| [[3L 8s]]
| [[3L 9s]]
| [[3L 10s]]
| [[3L 11s]]
| [[3L 12s]]
|-
| [[4L 1s]]
| [[4L 2s]]
| [[4L 3s]]
| [[4L 4s]]
| [[4L 5s]]
| [[4L 6s]]
| [[4L 7s]]
| [[4L 8s]]
| [[4L 9s]]
| [[4L 10s]]
| [[4L 11s]]
| [[4L 12s]]
|-
| [[5L 1s]]
| [[5L 2s]]
| [[5L 3s]]
| [[5L 4s]]
| [[5L 5s]]
| [[5L 6s]]
| [[5L 7s]]
| [[5L 8s]]
| [[5L 9s]]
| [[5L 10s]]
| [[5L 11s]]
| [[5L 12s]]
|-
| [[6L 1s]]
| [[6L 2s]]
| [[6L 3s]]
| [[6L 4s]]
| [[6L 5s]]
| [[6L 6s]]
| [[6L 7s]]
| [[6L 8s]]
| [[6L 9s]]
| [[6L 10s]]
| [[6L 11s]]
| [[6L 12s]]
|-
| [[7L 1s]]
| [[7L 2s]]
| [[7L 3s]]
| [[7L 4s]]
| [[7L 5s]]
| [[7L 6s]]
| [[7L 7s]]
| [[7L 8s]]
| [[7L 9s]]
| [[7L 10s]]
| [[7L 11s]]
| [[7L 12s]]
|-
| [[8L 1s]]
| [[8L 2s]]
| [[8L 3s]]
| [[8L 4s]]
| [[8L 5s]]
| [[8L 6s]]
| [[8L 7s]]
| [[8L 8s]]
| [[8L 9s]]
| [[8L 10s]]
| [[8L 11s]]
| [[8L 12s]]
|-
| [[9L 1s]]
| [[9L 2s]]
| [[9L 3s]]
| [[9L 4s]]
| [[9L 5s]]
| [[9L 6s]]
| [[9L 7s]]
| [[9L 8s]]
| [[9L 9s]]
| [[9L 10s]]
| [[9L 11s]]
| [[9L 12s]]
|-
| [[10L 1s]]
| [[10L 2s]]
| [[10L 3s]]
| [[10L 4s]]
| [[10L 5s]]
| [[10L 6s]]
| [[10L 7s]]
| [[10L 8s]]
| [[10L 9s]]
| [[10L 10s]]
| [[10L 11s]]
| [[10L 12s]]
|-
| [[11L 1s]]
| [[11L 2s]]
| [[11L 3s]]
| [[11L 4s]]
| [[11L 5s]]
| [[11L 6s]]
| [[11L 7s]]
| [[11L 8s]]
| [[11L 9s]]
| [[11L 10s]]
| [[11L 11s]]
| [[11L 12s]]
|-
| [[12L 1s]]
| [[12L 2s]]
| [[12L 3s]]
| [[12L 4s]]
| [[12L 5s]]
| [[12L 6s]]
| [[12L 7s]]
| [[12L 8s]]
| [[12L 9s]]
| [[12L 10s]]
| [[12L 11s]]
| [[12L 12s]]
|}

=== 13 ≤ L ≤ 24, s ≤ 12 ===
{| class="wikitable mw-collapsible mw-collapsed center-all"
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|13 ≤ L ≤ 24|s ≤ 12}})
|-
| [[13L 1s]]
| [[13L 2s]]
| [[13L 3s]]
| [[13L 4s]]
| [[13L 5s]]
| [[13L 6s]]
| [[13L 7s]]
| [[13L 8s]]
| [[13L 9s]]
| [[13L 10s]]
| [[13L 11s]]
| [[13L 12s]]
|-
| [[14L 1s]]
| [[14L 2s]]
| [[14L 3s]]
| [[14L 4s]]
| [[14L 5s]]
| [[14L 6s]]
| [[14L 7s]]
| [[14L 8s]]
| [[14L 9s]]
| [[14L 10s]]
| [[14L 11s]]
| [[14L 12s]]
|-
| [[15L 1s]]
| [[15L 2s]]
| [[15L 3s]]
| [[15L 4s]]
| [[15L 5s]]
| [[15L 6s]]
| [[15L 7s]]
| [[15L 8s]]
| [[15L 9s]]
| [[15L 10s]]
| [[15L 11s]]
| [[15L 12s]]
|-
| [[16L 1s]]
| [[16L 2s]]
| [[16L 3s]]
| [[16L 4s]]
| [[16L 5s]]
| [[16L 6s]]
| [[16L 7s]]
| [[16L 8s]]
| [[16L 9s]]
| [[16L 10s]]
| [[16L 11s]]
| [[16L 12s]]
|-
| [[17L 1s]]
| [[17L 2s]]
| [[17L 3s]]
| [[17L 4s]]
| [[17L 5s]]
| [[17L 6s]]
| [[17L 7s]]
| [[17L 8s]]
| [[17L 9s]]
| [[17L 10s]]
| [[17L 11s]]
| [[17L 12s]]
|-
| [[18L 1s]]
| [[18L 2s]]
| [[18L 3s]]
| [[18L 4s]]
| [[18L 5s]]
| [[18L 6s]]
| [[18L 7s]]
| [[18L 8s]]
| [[18L 9s]]
| [[18L 10s]]
| [[18L 11s]]
| [[18L 12s]]
|-
| [[19L 1s]]
| [[19L 2s]]
| [[19L 3s]]
| [[19L 4s]]
| [[19L 5s]]
| [[19L 6s]]
| [[19L 7s]]
| [[19L 8s]]
| [[19L 9s]]
| [[19L 10s]]
| [[19L 11s]]
| [[19L 12s]]
|-
| [[20L 1s]]
| [[20L 2s]]
| [[20L 3s]]
| [[20L 4s]]
| [[20L 5s]]
| [[20L 6s]]
| [[20L 7s]]
| [[20L 8s]]
| [[20L 9s]]
| [[20L 10s]]
| [[20L 11s]]
| [[20L 12s]]
|-
| [[21L 1s]]
| [[21L 2s]]
| [[21L 3s]]
| [[21L 4s]]
| [[21L 5s]]
| [[21L 6s]]
| [[21L 7s]]
| [[21L 8s]]
| [[21L 9s]]
| [[21L 10s]]
| [[21L 11s]]
| [[21L 12s]]
|-
| [[22L 1s]]
| [[22L 2s]]
| [[22L 3s]]
| [[22L 4s]]
| [[22L 5s]]
| [[22L 6s]]
| [[22L 7s]]
| [[22L 8s]]
| [[22L 9s]]
| [[22L 10s]]
| [[22L 11s]]
| [[22L 12s]]
|-
| [[23L 1s]]
| [[23L 2s]]
| [[23L 3s]]
| [[23L 4s]]
| [[23L 5s]]
| [[23L 6s]]
| [[23L 7s]]
| [[23L 8s]]
| [[23L 9s]]
| [[23L 10s]]
| [[23L 11s]]
| [[23L 12s]]
|-
| [[24L 1s]]
| [[24L 2s]]
| [[24L 3s]]
| [[24L 4s]]
| [[24L 5s]]
| [[24L 6s]]
| [[24L 7s]]
| [[24L 8s]]
| [[24L 9s]]
| [[24L 10s]]
| [[24L 11s]]
| [[24L 12s]]
|}

=== L ≤ 12, 13 ≤ s ≤ 24 ===
{| class="wikitable mw-collapsible mw-collapsed center-all"
|+ style="font-size: 105%; white-space: nowrap;" | Pages for MOS scales ({{nowrap|L ≤ 12|13 ≤ s ≤ 24}})
|-
| [[1L 13s]]
| [[1L 14s]]
| [[1L 15s]]
| [[1L 16s]]
| [[1L 17s]]
| [[1L 18s]]
| [[1L 19s]]
| [[1L 20s]]
| [[1L 21s]]
| [[1L 22s]]
| [[1L 23s]]
| [[1L 24s]]
|-
| [[2L 13s]]
| [[2L 14s]]
| [[2L 15s]]
| [[2L 16s]]
| [[2L 17s]]
| [[2L 18s]]
| [[2L 19s]]
| [[2L 20s]]
| [[2L 21s]]
| [[2L 22s]]
| [[2L 23s]]
| [[2L 24s]]
|-
| [[3L 13s]]
| [[3L 14s]]
| [[3L 15s]]
| [[3L 16s]]
| [[3L 17s]]
| [[3L 18s]]
| [[3L 19s]]
| [[3L 20s]]
| [[3L 21s]]
| [[3L 22s]]
| [[3L 23s]]
| [[3L 24s]]
|-
| [[4L 13s]]
| [[4L 14s]]
| [[4L 15s]]
| [[4L 16s]]
| [[4L 17s]]
| [[4L 18s]]
| [[4L 19s]]
| [[4L 20s]]
| [[4L 21s]]
| [[4L 22s]]
| [[4L 23s]]
| [[4L 24s]]
|-
| [[5L 13s]]
| [[5L 14s]]
| [[5L 15s]]
| [[5L 16s]]
| [[5L 17s]]
| [[5L 18s]]
| [[5L 19s]]
| [[5L 20s]]
| [[5L 21s]]
| [[5L 22s]]
| [[5L 23s]]
| [[5L 24s]]
|-
| [[6L 13s]]
| [[6L 14s]]
| [[6L 15s]]
| [[6L 16s]]
| [[6L 17s]]
| [[6L 18s]]
| [[6L 19s]]
| [[6L 20s]]
| [[6L 21s]]
| [[6L 22s]]
| [[6L 23s]]
| [[6L 24s]]
|-
| [[7L 13s]]
| [[7L 14s]]
| [[7L 15s]]
| [[7L 16s]]
| [[7L 17s]]
| [[7L 18s]]
| [[7L 19s]]
| [[7L 20s]]
| [[7L 21s]]
| [[7L 22s]]
| [[7L 23s]]
| [[7L 24s]]
|-
| [[8L 13s]]
| [[8L 14s]]
| [[8L 15s]]
| [[8L 16s]]
| [[8L 17s]]
| [[8L 18s]]
| [[8L 19s]]
| [[8L 20s]]
| [[8L 21s]]
| [[8L 22s]]
| [[8L 23s]]
| [[8L 24s]]
|-
| [[9L 13s]]
| [[9L 14s]]
| [[9L 15s]]
| [[9L 16s]]
| [[9L 17s]]
| [[9L 18s]]
| [[9L 19s]]
| [[9L 20s]]
| [[9L 21s]]
| [[9L 22s]]
| [[9L 23s]]
| [[9L 24s]]
|-
| [[10L 13s]]
| [[10L 14s]]
| [[10L 15s]]
| [[10L 16s]]
| [[10L 17s]]
| [[10L 18s]]
| [[10L 19s]]
| [[10L 20s]]
| [[10L 21s]]
| [[10L 22s]]
| [[10L 23s]]
| [[10L 24s]]
|-
| [[11L 13s]]
| [[11L 14s]]
| [[11L 15s]]
| [[11L 16s]]
| [[11L 17s]]
| [[11L 18s]]
| [[11L 19s]]
| [[11L 20s]]
| [[11L 21s]]
| [[11L 22s]]
| [[11L 23s]]
| [[11L 24s]]
|-
| [[12L 13s]]
| [[12L 14s]]
| [[12L 15s]]
| [[12L 16s]]
| [[12L 17s]]
| [[12L 18s]]
| [[12L 19s]]
| [[12L 20s]]
| [[12L 21s]]
| [[12L 22s]]
| [[12L 23s]]
| [[12L 24s]]
|}

=== Larger MOS scales ===
[[7L 34s]], [[9L 29s]], [[12L 29s]], [[12L 41s]], [[13L 14s]], [[14L 13s]], [[17L 14s]], [[25L 6s]], [[41L 12s]]

== Variations ==
* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap| L − s }}, the "chroma".
* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.
* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.
* [[Operations on MOSes]]

== Listen ==
This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste.

[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}}

== See also ==
* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales
* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]
* [[MOS rhythm]]
* [[:Category:MOS scales|Category:MOS scales]], the category including all MOS-related articles on this wiki
* [[Gallery of MOS patterns]]

== Notes ==
<references group="note" />

== References ==
<references />

[[Category:Math]]
[[Category:MOS scale| ]] 
[[Category:Scale]]
[[Category:Erv Wilson]]

Subgroup temperaments

2026-06-02T03:42:54Z

Overthink: /* Baldi */ revert change for now (see talk)

{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.

For temperaments that omit various prime harmonics, see:
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].

= Composite subgroup temperaments =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.165

==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

: [[gencom]]: [2 9/8; 225/224 325/324 640/637]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.3.25 subgroup ==

=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.

Subgroup: 2.3.25

Edo join: 17 & 12

Comma list: [[2048/2025]]

{{Mapping|legend=2| 1 1 7| 0 1 -4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136

==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''

Edo join: 17 & 12

Comma list: 2048/2025, 576/575, 82/81

{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264

===== Sburb =====
This temperament sets the 413th harmonic (octave-reduced) to the diminished seventh.

Subgroup: 2.3.7.23.25.41.59

Edo join: 17 & 12

Comma list: 64/63, 225/224, 162/161, 82/81, 177/175

{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387

== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

[[Badness]]: 0.00263

=== Joan ===
{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.7.13.17 subgroup ==

=== Novisept ===
Novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives you 7/4.

[[Subgroup]]: 2.9.7.13.17

[[Comma list]]: 729/728, 442/441, 833/832

{{Mapping|legend=2| 1 1 1 -1 3| 0 6 5 13 3 }}

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~9/7 = 433.836

== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

{{Mapping|legend=2|1 1 2|0 3 2}}

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

=== Genius ===

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

{{Mapping|legend=2| 1 2 4 | 0 3 1 }}

: sval mapping generators: ~2, ~21/16

{{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }}

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

=== Tetrameantone ===
{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

=== Tetramagic ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

{{Mapping|legend=2| 1 0 1 | 0 5 1 }}

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

{{Todo|inline=1|cleanup}}

<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)
⟨2399.3973, 193.8643]

TE Step Tunings (cents)
⟨25.21211, 47.81337]

TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]

TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]

Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave

Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩

TE Generator Tunings (cents)
⟨2399.4195, 193.8654]

TE Step Tunings (cents)
⟨25.23883, 47.79592]

TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]

TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave

Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)
⟨2400.1097, 193.9498]

TE Step Tunings (cents)
⟨24.18752, 48.52491]

TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave

Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>

==== 4.6.5.7.11.13 ====

<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩

TE Generator Tunings (cents)
⟨2401.2305, 193.5378]

TE Step Tunings (cents)
⟨42.79107, 35.98524]

TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>

==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)
⟨2400.4701, 193.4599]

TE Step Tunings (cents)
⟨43.39350, 35.55764]

TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>

==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩

TE Generator Tunings (cents)
⟨2399.9219, 193.3952]

TE Step Tunings (cents)
⟨44.14256, 35.03670]

TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>

==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

TE Generator Tunings (cents)
⟨2399.3286, 193.5316]

TE Step Tunings (cents)
⟨37.31613, 39.63311]

TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>

== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

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

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

== 8.9.7 subgroup ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10-3

= Fractional subgroup temperaments =
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

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

: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====
{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

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

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====
{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====
{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===
{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

{{Mapping|legend=2| 1 0 0 | 0 2 1 }}

{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}
: <nowiki/>* wart for 5/3
: † wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

[[Subgroup]]: 2.5/3.13/9

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}

{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])3 = [[13/10]] and ([[15/14]])3 = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.

[[Subgroup]]: 2.5.7/3.11/3.13

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})

{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} using subgroup TE 
: <nowiki/>* wart for 7/3

Badness (Sintel): 0.329

==== Laz ====
{{See also | Chromatic pairs #Laz }}

Laz is related to [[avalokita]] as well as to [[winston]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}

{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}

Kryptonite is related to [[krypton]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12

{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}

Kiribati is related to [[nakika]] as well as to [[octacot]].

[[Subgroup]]: 2.9/5.7/3.11/9

[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})

{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20

{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents

=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.

[[Subgroup]]: 2.7/3.11

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

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

{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents

== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.

[[Subgroup]]: 2.5.9/7

[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})

{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5

{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents

==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[50/49]]

{{Mapping|legend=2| 2 3 1 | 0 1 0 }}

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}

{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830

{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
 based on subgroup TE 

Badness (Sintel): 0.119

==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}

Edson is related to [[pele]] and [[andromeda]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3

{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

{{Optimal ET sequence|legend=1| 12, 17, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

==== Haumea ====
{{See also| Chromatic pairs #Haumea }}

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}

{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: 364/363, 441/440, 1001/1000

{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

[[Subgroup]]: 2.7/5.9/5

[[Comma list]]: [[250047/250000]]

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

{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461

{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents

=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].

[[Subgroup]]: 2.7/5.11/5.13/5

[[Comma list]]: [[847/845]], [[1001/1000]]

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

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents

==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

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

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95t, 124t }}
: t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents

== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

[[Subgroup]]: 2.11/5.13/5

[[Comma list]]: [[2200/2197]]

{{Mapping|legend=2| 1 0 1| 0 3 1 }}

{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012

{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents

==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168, 540/539, 729/728

{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents

=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].

Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].

[[Subgroup]]: 2.3.7.11/5

[[Comma list]]: 1029/1024, 4000/3993

{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742

{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 1029/1024, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918

{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17

[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19

[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
: <nowiki />* wart for 11/7
: † wart for 13/7

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====
{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

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

{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

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

[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5

=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]

[[Subgroup]]: 2.3.49/5

[[Comma list]]: 2401/2400

{{Mapping|legend=2| 1 1 3 | 0 2 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

[[Tp tuning #T2 tuning|RMS error]]: ?

== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

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

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})

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

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

<nowiki/>* wart for 19/7

[[Badness]] (Sintel): 0.082

=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].

[[Subgroup]]: 2.17/7.19/7

[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})

{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}

{{Optimal ET sequence|legend=1| 7, 18, 25 }}

[[Badness]] (Sintel): 0.005

==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

Subgroup: 2.17/7.19/7.23/7

Comma list: [[323/322]], [[392/391]]

Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}

Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}

{{Optimal ET sequence|legend=0| 7, 18, 25 }}

Badness (Sintel): 0.029

==== 2.25/7.17/7.19/7.23/7 subgroup ====

Subgroup: 2.25/7.17/7.19/7.23/7

Comma list: [[323/322]], [[392/391]], [[476/475]]

Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}

Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}

{{Optimal ET sequence|legend=0| 7, 18, 25 }}

Badness (Sintel): 0.053

== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}

=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: [[10976/10935]]

{{Mapping|legend=2| 1 2 3 | 0 3 1 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909

[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

=== Halftone ===
{{Main| Halftone }}

[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: 9604/9375

{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2

[[Comma list]]: 1232/1215, 27783/27500

{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2

[[Comma list]]: 275/273, 1232/1215, 1323/1300

{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2

=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4

[[Comma list]]: 245/243

{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}

: sval mapping generators: ~3/2, ~9/7

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728

[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]

==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 100/99, 245/243

{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771

[[Optimal ET sequence]]: [[8edf]], [[13edf]]

==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 245/243, 441/440

{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939

[[Optimal ET sequence]]: [[8edf]], [[11edf]]

== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''

[[Subgroup]]: 3/2.5/4.11/8

[[Comma list]]: 121/120

{{Mapping|legend=2| 1 1 1 | 0 2 -1 }}]

: [[gencom]]: [3/2 12/11; 121/120]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29

{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}

== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 87808/85293

{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}

: sval mapping generators: ~3/2, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2

=== Doubleton ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 1352/1323

{{Mapping|legend=2| 2 0 3 | 0 1 1 }}

: sval mapping generators: ~26/21, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2

== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11

[[Comma list]]: {{monzo| 11 1 -5 }}

{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}

: [[gencom]]: [5/2 125/88; 341796875/329832448]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2

= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments

[[Category:Subgroup temperaments| ]] 
[[Category:Temperament collections]]
{{Todo| review | cleanup }}

Talk:Subgroup temperaments

2026-06-02T03:42:22Z

Overthink: /* Baldy -> baldi */ reply

{{WSArchiveLink}}

== Page is too cluttered ==

We'd better soon split this page into several, probably by subgroups. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:56, 28 September 2021 (UTC)

Update: Done. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:04, 6 November 2021 (UTC)

== "Breedsmic" ==

I don't suppose meaningful distinction can be made between "breed" and "breedsmic". Iow both should be referring to the same 7-limit rank-3 temperament, not this fractional subgroup thing. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:24, 16 January 2023 (UTC)

: We can find several examples of temperament pairs with different but related names, where they have the same [[comma basis]], differing only by their [[domain basis]]. For example, marveltri on this same page makes only the marvel comma [[225/224]] vanish, which is the same as (7-limit) [[marvel]] temperament does; the only difference between the two temperaments is that marveltri's domain is the nonstandard 2.5.9/7 where marvel's domain is the standard 2.5.7.9. So, it's reasonable to think there could be two different names for the temperaments which both make only the breedsma [[2401/2400]] vanish but have different domain bases: 2.3.5.7 and 2.3.49/5 in this case.

: However, "breed" and "breedsmic" cannot be the names that distinguish these two temperaments. "Breedsmic" has a strong claim to them, because "-ma" → "-mic" is typical for deriving a temperament name from a comma name, in this case "breedsma" → "breedsmic". But "breed" also has a strong claim, because it's typical when we have a temperament family based on tempering out a comma to name the temperament which tempers out only that comma be named the same as the family, in this case "breed family" → "breed". Therefore, I agree with you that both of these names must be synonymous.

: And I believe they both must at least refer to version of the temperament in the standard domain basis. Possibly both "breed" and "breedsmic" could be used for the one in the nonstandard domain basis, too, in the same way we'd name any such temperament. But because this page already has a separate entry for the particular 2.3.49/5 version of this temperament, however, I don't think that's the right thing to do. This 2.3.49/7 temperament should have a different name. Perhaps we could model it after marveltri. I don't exactly understand the meaning of "tri" as a suffix; perhaps it refers to the shift from 3 to 9. So perhaps this would be "breedsept", on account of the shift from 7 → 49? Something like that, maybe. Like I said, though, I don't understand the "tri" so I defer to anyone who does, or anyone who has a better idea. If we can't come up with something, then "2.3.49/5 breed(smic)" works. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:06, 16 January 2023 (UTC)

:: Thank you for supporting my ideas. I don't understand the suffix either. Besides "-tri" I also see "-tet". The only common thing among the "-tet" temps seems to be that they're all kinda supported by 4et. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:34, 16 January 2023 (UTC)

== Clarification needed ==
Why is there "clarification needed" on CTE generators of some 3/2-repeating temperaments? I had calculated the generators using sintel's temperament calculator so they should be correct.
[[User:CompactStar|CompactStar]] ([[User talk:CompactStar|talk]]) 23:47, 29 May 2023 (UTC)

: There's the distinction of subgroup CTE vs inharmonic CTE for fractional subgroup temperaments. If you happen to know which scheme is used in sintel's calculator feel free to clarify. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 06:17, 30 May 2023 (UTC)

:: By comparing Sintel's calculator and x31eq, I have confirmed Sintel's calculator uses subgroup ''TE''. It is very likely Sintel's calculator is consistent in using subgroup tuning for both TE and CTE, but I cannot confirm this because x31eq does not show CTE.

[[User:CompactStar|CompactStar]] ([[User talk:CompactStar|talk]]) 07:28, 30 May 2023 (UTC)

::: Alright. That's a good reason to believe it's subgroup CTE. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:19, 30 May 2023 (UTC)

:::: Just want to confirm that the calculator indeed does subgroup TE for both cases. - [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]])

== Baldy -> baldi ==

This name change would make the temperament's connection to garibaldi more obvious. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 01:40, 2 June 2026 (UTC)

: I personally think it's obvious enough already. "Baldy" is used on many other pages, so the community should decide whether to change before updating it across the wiki. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:42, 2 June 2026 (UTC)

19edo

2026-06-02T00:42:52Z

Overthink: /* Theory */ grammar

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. 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 right-1 right-2"
|-
! [[Degree|#]]
! [[Cent]]s
! [[Interval category|Interval categories]]
! Approximated ratios<ref group="note">As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament.</ref>
|-
| 0
| 0.0
| Unison (prime)
| [[1/1]]
|-
| 1
| 63.2
| Augmented unison
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
|-
| 2
| 126.3
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 189.5
| Major second
| [[9/8]], [[10/9]]
|-
| 4
| 252.6
| Augmented second Diminished third
| [[7/6]], [[8/7]], [[15/13]]
|-
| 5
| 315.8
| Minor third
| [[6/5]]
|-
| 6
| 378.9
| Major third
| [[5/4]], [[16/13]], [[56/45]]
|-
| 7
| 442.1
| Augmented third
| [[9/7]], [[13/10]], [[21/16]], [[32/25]]
|-
| 8
| 505.3
| Perfect fourth
| [[4/3]], [[75/56]]
|-
| 9
| 568.4
| Augmented fourth (Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
|-
| 10
| 631.6
| Diminished fifth (Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
|-
| 11
| 694.7
| Perfect fifth
| [[3/2]], [[112/75]]
|-
| 12
| 757.9
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]], [[32/21]]
|-
| 13
| 821.1
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
|-
| 14
| 884.2
| Major sixth
| [[5/3]]
|-
| 15
| 947.4
| Augmented sixth Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
|-
| 16
| 1010.5
| Minor seventh
| [[9/5]], [[16/9]]
|-
| 17
| 1073.7
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
|-
| 18
| 1136.8
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
|-
| 19
| 1200.0
| Octave
| [[2/1]]
|}
<references group="note"/>

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

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

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

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

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

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

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

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

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

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

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

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

{{Sharpness-sharp1}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

User:Overthink/Table of 388edo intervals

2026-06-02T00:23:37Z

Overthink: blank for now

User:Overthink/Table of 121edo intervals

2026-06-02T00:23:23Z

Overthink: blank for now

23/18

2026-06-01T19:25:35Z

Overthink: /* Approximation */ - extra newline

{{Infobox Interval
| Name = vicesimotertial major third
| Color name = 23o4, twetho 4th
| Sound = jid_23_18_pluck_adu_dr220.mp3
}}

'''23/18''', the '''vicesimoterial major third''', is a [[23-limit]] interval that is the [[mediant]] between [[9/7]] and [[14/11]], giving it a character that is somewhere between the gentle undecimal thirds and the more strident septimal supermajor ones. It is sharp of the [[81/64|Pythagorean major third]] by a vicesimoterial formal comma, [[736/729]].

== Approximation ==
This interval is decently represented by 6 steps of [[17edo]], and near perfectly by 29 steps of [[82edo]]. If used as a generator, it creates [[squares]] temperament.
{{Interval edo approximation}}

== See also ==

* [[36/23]] – its [[octave complement]]
* [[27/23]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Third]]
[[Category:Supermajor third]]

Talk:5120/5103

2026-06-01T15:41:03Z

Overthink: /* Petition to officialize aberschisma, and change hemifamity to aberschismic */ enter

== "Universal" name for 5120/5103 ==

I've noticed the conversation on the XA Discord about picking a name to replace "hemifamity comma". Suggestions I've seen include ''argent comma'' and ''pele comma''. I'm a bit biased towards ''aberschisma'' since I coined the name, but MidnightBlue pointed out that 6¢ is quite wide to be calling it a schisma, which I've also thought about. Maybe ''aberkleisma'' or even ''pentasept comma''?

: I'm not too invested into this comma, but will add my <strike>schisma</strike> 2c after seeing the Discord convo: ''pele comma'' and ''peleisma'' are fine with me, whereas I'm afraid ''argent comma'', while logical, may be confused with the [[argyria]] (which is more of an ''arg''ument for renaming the latter).

: On a side note, 5120/5103 does function like a kleisma for me, particularly because the ratio of the pental kleisma to it is the [[horwell comma]], which is among the staple commas in my 7-limit analysis of edos incl. 53. Because schismic x kleismic product words are among the best ways to make well-tempered 53-note scales, the pental kleisma is a chroma there, and when horwell tempered, it turns into 5120/5103 and is, among other roles, the scale-chroma between the 81/80 and 64/63 steps.

: Meanwhile, the ratio of 5120/5103 to the pental schisma is the [[garischisma]]. So for fans of the latter, which I'm not, it may act like a schisma instead, but that's less likely because the pental schisma flattens the fifth while the garischisma sharpens it, so if anything, the latter and 5120/5103 would be seen as 'negative schismas', which, btw, brings us to the concept of [[counterpyth]].

: Afaik, counterpyth has never been considered under this name without 5120/5103, whereas [[1216/1215]] works well together with other commas that stack slightly sharp fifths, such as the [[wilschisma]] and the [[symbiotic comma]], and the name ''Eratosthenes' comma'' is good, so I disagree with the assignment of the counterpyth family label to any temp with 1216/1215 in sintel's finder. I.e., to me, 5120/5103 is more related to counterpyth than 1216/1215 is. But I can't be sure of my judgment on this without FloraC's opinion. Either way, I don't mind ''counterpyth comma'' for 5120/5103, its 7-limit rank-3 then called counterpyth like its canonical extension to 2.3.5.7.19 already is.

: --[[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 23:55, 1 June 2025 (UTC)

:: I'm all for ''argent comma''. The similarity with ''argyria'' isn't high enough to worry me. I'm against ''pele comma'' cuz that would set pele as canon which I don't think we should ever do. For the same reason I'd hesitate to call it ''counterpyth comma''. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:17, 2 June 2025 (UTC)

::: OK, then I settle on ''argent comma'' too. That matches my view of argent fifths as a distinct region that's roughly [65\111, 17\29] and sharp of the olympic / garischismic / symbiotic / wilschismic fifths region that's roughly [55\94, 65\111]. --[[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 18:38, 2 June 2025 (UTC)

:::: What about 41edo and 46edo? Those are both notable tunings that temper out the comma and have fifths that fall outside of your *argent* range. -- [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 00:35, 12 June 2025 (UTC)

::::: On my part I include 41edo within the argent range; and there's a case to be made that 46edo and 53edo, while notable, aren't truly representative of the intonation of tempering out 5120/5103. "Argent" strictly speaking refers not to a particular tuning range, anyhow, but to a specific tuning that sets the logarithmic ratio of the perfect fifth to the perfect fourth to be sqrt(2):1, for which one can define bands of tolerance around, but which very closely corresponds to the most accurate tunings that temper out this comma. Perhaps "argentisma" -> argentic, argentismic would be clearer, so as not to imply an RTT interpretation for the term "argent temperament" which is already in use.

::::: Compare this to the intonation of counterpyth, which quite distinctly favors tunings of 3/2 far flatter than the optimum of tempering out 5120/5103 by itself: just 19/15 gives us roughly 1/16-comma hemifamity as opposed to just 15/14, 7/5, or 21/20 which provide 1/5, 1/6, and 1/7-comma tunings. For this reason, I oppose seeing counterpyth as a canonical extension to the 7-limit rank-3 {5120/5103} temperament. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 01:24, 12 June 2025 (UTC)

::::: I did think about making 24\41 the boundary instead, as rank-3 microtemps tend to have flatter fifths than that even if 152fg or 111 support them. My flat end of argent is surely not flatter than 24\41 and not sharper than 41\70. Between those are kwai fifths... that I may consider too damaged indeed on second thought, and so belonging to the "slightly exo" range codenamed argent. [[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 20:23, 19 June 2025 (UTC)

:::: I prefer "Saruyo". It's the only name out of all these suggestions that directly indicates 5120/5103. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 09:19, 5 June 2025 (UTC)

: I have a dumb idea. Why not call it the *pell comma* after the Pell sequence of numbers, whose convergent ratio gives the approximate ratio between an octave and a perfect fourth for the optimal tuning of the temperament? [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 20:59, 18 September 2025 (UTC)

:: I think referencing the Pell sequence makes way more sense for a member of the family of commas going 50/49, 289/288, 1682/1681, 9801/9800, etc. The only relation of Pell numbers to 5120/5103 is the edo sequence, which seems rather secondary, as much as I'm a promoter of 239edo. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 22:42, 18 September 2025 (UTC)

: I propose the name "interkleisma", since 5120/5103 is the difference between 64/63 and 81/80 (the main formal commas for primes 5 and 7), and is around a kleisma in size.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:02, 13 February 2026 (UTC)

:: It's a half kleisma in 270edo (and 311edo if you consider other kleismata such as 1029/1024) tho. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:20, 13 February 2026 (UTC)

::: Your essay on the 13-limit JI space considers 5120/5103~352/351~847/845, 325/324~385/384, 364/363~441/440, 540/539~729/728, and 351/350 as kleismas. Even if it is a half-kleisma in 270edo, the comma is close enough to the rough interval region, and also no single edo should decide the name.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 16:58, 13 February 2026 (UTC)

:::: Oh wow, my bad. I'll change them to hemikleismata. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:39, 13 February 2026 (UTC)

Should we do a poll here? The name was basically pre-maturely changed according to a poll on XA Discord. Besides, we need to decide what to do with the temp's name. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 12:51, 22 January 2026 (UTC)

: If enough people want to, then I guess. I like the current name of this comma, and I was thinking of the associated full 7-limit temperament being "argentic" and the 2.3.7/5 subgroup one being "argic". [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 06:12, 27 January 2026 (UTC)

: Why not [[64/63|S8]]/[[81/80|S9]]? Are there many properties of this comma that aren't explained by it being ((8/7)/(9/8)) / ((9/8)/(10/9)) = (64/63) / (81/80) and hence ([[10/7]])/([[9/8]])3? --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 12:27, 16 February 2026 (UTC)

:: Because "ess eight over ess nine" is too many syllables. [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 05:34, 17 February 2026 (UTC)

: I'm fine calling it saruyoma, y'all sort this out. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 10:04, 6 May 2026 (UTC)

=== Petition to officialize ''aberschisma'', and change ''hemifamity'' to ''aberschismic'' ===
At this point, ''aberschisma'' and ''aberschismic'' seem like the most widely liked and used names in xenharmonic communities. Thereby I request ''aberschisma'' be set as the permanent, main name for 5120/5103, and ''hemifamity'' be officially changed to ''aberschismic''.

Please put your "yes" or "no" and signature below. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:56, 1 June 2026 (UTC)

# '''Yes'''. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:56, 1 June 2026 (UTC)
# Yes, sure, why not. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 07:48, 1 June 2026 (UTC)
# '''Yes''', I'm only weakly towards that name, but it feels alright, if the community wants it. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 15:40, 1 June 2026 (UTC)

: I was under the impression that "argent" won out in the community? – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 12:30, 1 June 2026 (UTC)
:: In my opinion it's a bit too confusing with the logarithmic argent tuning, even though it is closely related. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 15:40, 1 June 2026 (UTC)

4:5:6

2026-05-31T23:53:01Z

Overthink: sectioning; brief explanation

{{Infobox Chord|ColorName=yo or y}}
'''4:5:6''' is an otonal [[major triad]], known as the '''just major triad''', '''classical major triad''', or '''Ptolemaic major triad'''. It is the most consonant triad, and it is the most common triad in music. In close voicing root position, it is an [[Delta-rational chord #Isoharmonic chord|isoharmonic chord]]. It occurs as a major triad in the [[meantone]] diatonic scale, as the major third formed by stacking four [[3/2|perfect fifths]] octave reduced, [[81/64]], is equated with [[5/4]]. It serves as the fundamental major root chord in classical music, and approximating is key in [[5-limit]] temperaments. This chord is approximated with all intervals having less than 20{{c}} of error in [[edo]]s {{EDOs|12, 15, 19, 22, 24, 26, 27}}, etc.

This chord is formed by stacking [[5/4]] and [[6/5]] in that order. Swapping the order of these intervals results in [[10:12:15]], the classical minor triad. The minor triad can also be obtained from 4:5:6 by flattening the third by [[25/24]], the diptolemaic chromatic semitone.

== Audio of close voicings ==
[[File:SculpEufaDem4-5-6-onD.mp3|none|thumb|4:5:6, Root position]]
[[File:SculpEufaDem5-6-8-onD.mp3|none|thumb|5:6:8, 1st inversion]][[File:SculpEufaDem3-4-5-onD.mp3|none|thumb|3:4:5, 2nd inversion]]

== Approximation by edos ==
[[7edo]] contains a rough approximation to 4:5:6, which equates the 5/4 and 6/5 steps. The smallest edo to approximate it with acceptable accuracy is [[12edo]], with [[19edo]] being the next edo to improve on it.
{{chord edo approximation}}

== Notable voicings ==
Voicings are arranged from simple to complex using [[Wilson norm]]. AOV and CAOV stand for [[Odd limit #Proposed extensions|all-odd voicing]] and ''condensed'' AOV respectively. Numbers in '''bold''' denote doubled pitches. This list is only a brief overview, see [[Voicings of 4:5:6]] for a more comprehensive list and audio examples.

{| class="wikitable"
! Voices
! [[EFR]]
! [[Kite's thoughts on hi-lo notation|Hi-lo name]]
! Special properties
|-
| rowspan="4" |3 voices
| 1:3:5
| hi3loR
| AOV, [[Delta-rational chord #Isoharmonic chord|isoharmonic]]
|-
| 2:3:5
| hi3
| CAOV
|-
| 3:4:5
| lo5
| 1st inversion, isoharmonic
|-
| 4:5:6
| basic
| isoharmonic
|-
| rowspan="4" |4 voices
| 2:3:'''4''':5
| hi3add8
| isoharmonic
|-
| 2:'''4''':5:6
| addloR
|
|-
| 3:4:5:'''6'''
| addlo5
| isoharmonic
|-
| 4:5:6:'''8'''
| add8
|
|}
== Related chords ==
Melodic inversion: 1/(6:5:4) = [[10:12:15]].

Plausible [[chord homonym|homonyms]]: None.

Lower limit soundalikes: [[64:81:96]] (3-limit)

Notable extensions (5-limit except where noted):
* [[12:15:18:20]] – adds 5/3
* [[8:10:12:15]] – adds 15/8
* [[36:45:54:64]] – adds 16/9
* [[20:25:30:36]] – adds 9/5
* [[4:5:6:7]] – adds 7/4 (7-limit)

[[Category:Major triads|#]]

4:5:6

2026-05-31T23:50:50Z

Overthink: - extra whitespace

{{Infobox Chord|ColorName=yo or y}}
'''4:5:6''' is an otonal [[major triad]], known as the '''just major triad''', '''classical major triad''', or '''Ptolemaic major triad'''. It is the most consonant triad, and it is the most common triad in music. In close voicing root position, it is an [[Delta-rational chord #Isoharmonic chord|isoharmonic chord]]. It occurs as a major triad in the [[meantone]] diatonic scale, as the major third formed by stacking four [[3/2|perfect fifths]] octave reduced, [[81/64]], is equated with [[5/4]]. It serves as the fundamental major root chord in classical music, and approximating is key in [[5-limit]] temperaments. This chord is approximated with all intervals having less than 20{{c}} of error in [[edo]]s {{EDOs|12, 15, 19, 22, 24, 26, 27}}, etc.

This chord is formed by stacking [[5/4]] and [[6/5]] in that order. Swapping the order of these intervals results in [[10:12:15]], the classical minor triad. The minor triad can also be obtained from 4:5:6 by flattening the third by [[25/24]], the diptolemaic chromatic semitone.
{{chord edo approximation}}

== Audio of close voicings ==
[[File:SculpEufaDem4-5-6-onD.mp3|none|thumb|4:5:6, Root position]]
[[File:SculpEufaDem5-6-8-onD.mp3|none|thumb|5:6:8, 1st inversion]][[File:SculpEufaDem3-4-5-onD.mp3|none|thumb|3:4:5, 2nd inversion]]

== Notable voicings ==
Voicings are arranged from simple to complex using [[Wilson norm]]. AOV and CAOV stand for [[Odd limit #Proposed extensions|all-odd voicing]] and ''condensed'' AOV respectively. Numbers in '''bold''' denote doubled pitches. This list is only a brief overview, see [[Voicings of 4:5:6]] for a more comprehensive list and audio examples.

{| class="wikitable"
! Voices
! [[EFR]]
! [[Kite's thoughts on hi-lo notation|Hi-lo name]]
! Special properties
|-
| rowspan="4" |3 voices
| 1:3:5
| hi3loR
| AOV, [[Delta-rational chord #Isoharmonic chord|isoharmonic]]
|-
| 2:3:5
| hi3
| CAOV
|-
| 3:4:5
| lo5
| 1st inversion, isoharmonic
|-
| 4:5:6
| basic
| isoharmonic
|-
| rowspan="4" |4 voices
| 2:3:'''4''':5
| hi3add8
| isoharmonic
|-
| 2:'''4''':5:6
| addloR
|
|-
| 3:4:5:'''6'''
| addlo5
| isoharmonic
|-
| 4:5:6:'''8'''
| add8
|
|}
== Related chords ==
Melodic inversion: 1/(6:5:4) = [[10:12:15]].

Plausible [[chord homonym|homonyms]]: None.

Lower limit soundalikes: [[64:81:96]] (3-limit)

Notable extensions (5-limit except where noted):
* [[12:15:18:20]] – adds 5/3
* [[8:10:12:15]] – adds 15/8
* [[36:45:54:64]] – adds 16/9
* [[20:25:30:36]] – adds 9/5
* [[4:5:6:7]] – adds 7/4 (7-limit)

[[Category:Major triads|#]]

5L 3s

2026-05-31T19:36:33Z

Overthink: /* Ultrahard tunings */ use words for these

{{Interwiki
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
| Neutral = 2L 6s
}}
: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
5L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).

== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.

'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].

== Scale properties ==

=== Intervals ===
{{MOS intervals}}

=== Generator chain ===
{{MOS genchain}}

=== Modes ===
{{MOS mode degrees}}

==== Proposed mode names ====
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}

== Tunings==
=== Simple tunings ===
The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.

{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}

=== Hypohard tunings ===
[[Hypohard]] oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with [[meantone]] diatonic tunings:
* The large step is a "meantone", around the range of [[10/9]] to [[9/8]].
* The major 2-mosstep is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.

With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].

EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
* 13edo has characteristically small 1-mossteps of about 185{{c}}. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all hypohard tunings.
* 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3{{c}}, a perfect 5-mosstep) and falling fifths (666.7{{c}}, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* 31edo can be used to make the major 2-mosstep a near-just 5/4.
* [[44edo]] (generator {{nowrap|17\44 {{=}} 463.64{{c}}}}), [[57edo]] (generator {{nowrap|22\57 {{=}} 463.16{{c}}}}), and [[70edo]] (generator 27\70 {{=}} 462.857{{c}}}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}

=== Hyposoft tunings ===
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,
* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92{{c}} to 114{{c}}.
* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342{{c}}) to 4\13 (369{{c}}).

* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71{{c}}) and Baroque diatonic semitones (114.29{{c}}, close to quarter-comma meantone's 117.11{{c}}).
* [[34edo]]'s 9:10:11:13 is even better.

This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)

{{MOS tunings
| Step Ratios = Hyposoft
| JI Ratios =
1/1;
16/15;
10/9; 11/10;
13/11; 20/17;
11/9;
5/4;
13/10;
18/13; 32/23;
13/9; 23/16;
20/13;
8/5;
18/11;
22/13; 17/10;
9/5;
15/8;
2/1
}}

=== Parasoft and ultrasoft tunings ===
The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo. The generator is close to the optimal range for [[tridec]] temperament, which approximates the 5:7:11:13 chord.

{{MOS tunings
| Step Ratios = 6/5; 3/2; 4/3
| JI Ratios =
1/1;
14/13;
11/10;
9/8;
15/13;
13/11;
14/11;
13/10;
4/3;
15/11;
7/5;
10/7;
22/15;
3/2;
20/13;
11/7;
22/13;
26/15;
16/9;
20/11;
13/7;
2/1
}}

=== Parahard tunings ===
23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).

{{MOS tunings
| JI Ratios =
1/1;
21/17;
17/16;
14/11;
6/5;
21/16;
21/17;
34/21;
32/21;
5/3;
11/7;
32/17;
34/21;
2/1
| Step Ratios = 4/1
}}

=== Ultrahard tunings ===
{{Main|5L 3s/Temperaments#Buzzard}}

[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.

In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] and [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|archipelago]] harmonies, as by dividing the perfect fifth in four it obviously also divides it in two as well.

Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.

{{MOS tunings
| JI Ratios =
1/1;
8/7;
13/10;
21/16;
3/2;
12/7, 22/13;
26/15;
49/25, 160/81;
2/1
| Step Ratios = 7/1; 10/1; 12/1
| Tolerance = 30
}}

== Approaches ==
* [[5L 3s/Temperaments]]

== Samples ==
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]])

[[File:13edo Prelude in J Oneirominor.mp3]]

[[WT13C]] [[:File:13edo Prelude in J Oneirominor.mp3|Prelude II (J Oneirominor)]] ([[:File:13edo Prelude in J Oneirominor Score.pdf|score]]) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

[[File:13edo_1MC.mp3]]

(13edo, first 30 seconds is in J Celephaïsian)

[[File:A Moment of Respite.mp3]]

(13edo, L Ilarnekian)

[[File:Lunar Approach.mp3]]

(by [[Igliashon Jones]], 13edo, J Celephaïsian)

=== 13edo Oneirotonic Modal Studies ===
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Ilarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian

== Scale tree ==
{{MOS tuning spectrum
| 13/8 = Golden oneirotonic (458.3592{{c}})
| 13/5 = Golden A-Team (465.0841{{c}})
}}

[[Category:Oneirotonic| ]] 
[[Category:Pages with internal sound examples]]

5491/5488

2026-05-31T19:32:14Z

Overthink: relation

{{Infobox interval
| Name = Supraminisma
| Comma = yes
}}
'''5491/5488''', the '''supraminisma''', is a [[19-limit]] [[unnoticeable comma]] measuring about 0.95 cents in size. It is the difference between two [[17/14]] supraminor thirds and a [[28/19]] narrow fifth. In terms of other commas, it is the difference between [[323/322]] and [[392/391]].

== Temperaments ==
[[Tempering out]] this comma in the full 19-limit leads to the rank-7 '''supraminismic''' temperament, and tempering out in the 2.7.17.19-subgroup leads to the '''supraminic''' temperament. The most notable temperament tempering out this comma is the 2.17/7.19/7-subgroup '''[[supramin]]''' temperament, which approximates the 14:17:19 triad with low complexity and high accuracy, and is tuned well by [[25edo]].

== Etymology ==
This comma was named by [[User:Overthink|Overthink]] in 2026 together with the supramin temperament, referring to the fact that supramin is generated by a supraminor third.

[[Category:Commas named for their regular temperament properties]]

5L 3s

2026-05-31T19:30:18Z

Overthink: /* Parasoft and ultrasoft tunings */ tridec

{{Interwiki
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
| Neutral = 2L 6s
}}
: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
5L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).

== Name ==
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.

'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].

== Scale properties ==

=== Intervals ===
{{MOS intervals}}

=== Generator chain ===
{{MOS genchain}}

=== Modes ===
{{MOS mode degrees}}

==== Proposed mode names ====
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}

== Tunings==
=== Simple tunings ===
The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.

{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}

=== Hypohard tunings ===
[[Hypohard]] oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with [[meantone]] diatonic tunings:
* The large step is a "meantone", around the range of [[10/9]] to [[9/8]].
* The major 2-mosstep is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.

With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].

EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
* 13edo has characteristically small 1-mossteps of about 185{{c}}. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all hypohard tunings.
* 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3{{c}}, a perfect 5-mosstep) and falling fifths (666.7{{c}}, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* 31edo can be used to make the major 2-mosstep a near-just 5/4.
* [[44edo]] (generator {{nowrap|17\44 {{=}} 463.64{{c}}}}), [[57edo]] (generator {{nowrap|22\57 {{=}} 463.16{{c}}}}), and [[70edo]] (generator 27\70 {{=}} 462.857{{c}}}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.

{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}

=== Hyposoft tunings ===
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,
* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92{{c}} to 114{{c}}.
* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342{{c}}) to 4\13 (369{{c}}).

* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71{{c}}) and Baroque diatonic semitones (114.29{{c}}, close to quarter-comma meantone's 117.11{{c}}).
* [[34edo]]'s 9:10:11:13 is even better.

This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)

{{MOS tunings
| Step Ratios = Hyposoft
| JI Ratios =
1/1;
16/15;
10/9; 11/10;
13/11; 20/17;
11/9;
5/4;
13/10;
18/13; 32/23;
13/9; 23/16;
20/13;
8/5;
18/11;
22/13; 17/10;
9/5;
15/8;
2/1
}}

=== Parasoft and ultrasoft tunings ===
The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo. The generator is close to the optimal range for [[tridec]] temperament, which approximates the 5:7:11:13 chord.

{{MOS tunings
| Step Ratios = 6/5; 3/2; 4/3
| JI Ratios =
1/1;
14/13;
11/10;
9/8;
15/13;
13/11;
14/11;
13/10;
4/3;
15/11;
7/5;
10/7;
22/15;
3/2;
20/13;
11/7;
22/13;
26/15;
16/9;
20/11;
13/7;
2/1
}}

=== Parahard tunings ===
23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).

{{MOS tunings
| JI Ratios =
1/1;
21/17;
17/16;
14/11;
6/5;
21/16;
21/17;
34/21;
32/21;
5/3;
11/7;
32/17;
34/21;
2/1
| Step Ratios = 4/1
}}

=== Ultrahard tunings ===
{{Main|5L 3s/Temperaments#Buzzard}}

[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.

In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] & [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|archipelago]] harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.

Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.

{{MOS tunings
| JI Ratios =
1/1;
8/7;
13/10;
21/16;
3/2;
12/7, 22/13;
26/15;
49/25, 160/81;
2/1
| Step Ratios = 7/1; 10/1; 12/1
| Tolerance = 30
}}

== Approaches ==
* [[5L 3s/Temperaments]]

== Samples ==
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]])

[[File:13edo Prelude in J Oneirominor.mp3]]

[[WT13C]] [[:File:13edo Prelude in J Oneirominor.mp3|Prelude II (J Oneirominor)]] ([[:File:13edo Prelude in J Oneirominor Score.pdf|score]]) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

[[File:13edo_1MC.mp3]]

(13edo, first 30 seconds is in J Celephaïsian)

[[File:A Moment of Respite.mp3]]

(13edo, L Ilarnekian)

[[File:Lunar Approach.mp3]]

(by [[Igliashon Jones]], 13edo, J Celephaïsian)

=== 13edo Oneirotonic Modal Studies ===
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Ilarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian

== Scale tree ==
{{MOS tuning spectrum
| 13/8 = Golden oneirotonic (458.3592{{c}})
| 13/5 = Golden A-Team (465.0841{{c}})
}}

[[Category:Oneirotonic| ]] 
[[Category:Pages with internal sound examples]]

3L 2s

2026-05-31T19:22:28Z

Overthink: /* Scale tree */ fix typo

{{Infobox MOS
| Name = antipentic
| Periods = 1
| nLargeSteps = 3
| nSmallSteps = 2
| Equalized = 3
| Collapsed = 2
| Pattern = LLsLs
}}

: ''For the 3/2-equivalent 3L 2s pattern, see [[3L 2s (fifth-equivalent)]].''

{{MOS intro}}

The only notable low-[[harmonic entropy|harmonic-entropy]] scale for this [[MOSScales|MOS]] pattern is [[Sensipent_family|sensi]], in which two generators make a [[5/3]]. The harmonic entropy of sensi[5] is still fairly high relative to other [[pentatonic]] scales, such as [[meantone]][5]. It's also [[improper]].

== Name ==
The TAMNAMS system applies the name ''antipentic'' as the scale utilizes opposite step sizes to the “classic” pentatonic scale ([[2L 3s]]). The name ''antipentic'' can be used regardless of the equave.

== Intervals ==
{{MOS intervals}}

== Modes ==
{{MOS mode degrees}}

== Scale tree ==
Generator ranges:
* Chroma-positive generator: 720{{c}} (3\5) to 800{{c}} (2\3)
* Chroma-negative generator: 400{{c}} (1\3) to 480{{c}} (2\5)
{{MOS tuning spectrum
| 11/8 = [[Semisept]]
| 13/8 = Golden [[father]]/[[petrtri]]/[[aurora]] (741.6408{{c}})
| | 12/5 = [[Sensi]]
13/5 = Golden [[sentry]] (759.4078{{c}})
| 9/2 = [[Squares]]/[[skwares]]
}}

[[Category:Antipentic]]
[[Category:5-tone scales]]

10edt

2026-05-31T02:48:13Z

Overthink: /* Harmonics */ integer and prime harmonics side-by-side is nonstandard and potentially confusing

{{Infobox ET}}
{{ED intro}}

== Theory ==
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the [[5/4]] inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord.

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.

One step of 10edt can serve as the generator for [[pocus]] temperament, a [[Temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1}}
{{Harmonics in equal|10|3|1|start=12}}

=== Interval table ===
{| class="wikitable"
|-
! Degrees
! [[Cent]]s
! [[Hekt]]s
! Approximate Ratios
|-
| colspan="3" | 0
| [[1/1]]
|-
| 1
| 190.196
| 130
| [[10/9]], [[28/25]]
|-
| 2
| 380.391
| 260
| [[5/4]]
|-
| 3
| 570.587
| 390
| [[7/5]]
|-
| 4
| 760.782
| 520
| [[14/9]]
|-
| 5
| 950.978
| 650
| 45/26, [[26/15]]
|-
| 6
| 1141.173
| 780
| [[27/14]]
|-
| 7
| 1331.369
| 910
| [[15/7]] ([[15/14]] plus an octave)
|-
| 8
| 1521.564
| 1040
| [[12/5]] ([[6/5]] plus an octave)
|-
| 9
| 1711.760
| 1170
| [[27/20|27/10]]
|-
| 10
| 1901.955
| 1300
| [[3/1]]
|}

[[Category:Macrotonal]]

39edo

2026-05-30T22:00:18Z

Overthink: /* Intervals */ - obsolete note

{{Infobox ET}}
{{ED intro}}

== Theory ==
39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]]. We have two choices for a tuning of [[7/1|7]]. The sharp one yields [[superpyth]] temperament, while the flat (patent) one yields [[semaphore]] (and also [[hemifamity]]) temperament.

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39d is [[quasisuper]]. Alternatively, it can be seen as a [[hemifamity]] semaphore (that is, immunity) system in the patent val. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.

Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.

39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).

=== Odd harmonics ===
{{Harmonics in equal|39|columns=11}}
{{Harmonics in equal|39|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 39edo (continued)}}

=== As a tuning of other temperaments ===
39edo, with its 400{{c}} major third, [[tempering out|tempers out]] the [[diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. Alternatively, the patent val tempers out 49/48 to yield semaphore. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].

=== Subsets and supersets ===
Since 39 factors into primes as {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.

== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 left-5 right-9 right-10"
|-
! rowspan="2" | Steps
! rowspan="2" | Cents
! rowspan="2" | Ratios of the [[2.3.5.11 subgroup]]
! colspan="2" | Intervals of 7
! colspan="3" rowspan="2" | [[Ups and downs notation]]
|-
! Patent val
! 39d val
|-
| 0
| 0.0
| colspan=3 | [[1/1]]
| P1
| perfect unison
| D
|-
| 1
| 30.8
| [[55/54]], [[81/80]]
| ''[[28/27]]'', [[64/63]]
| ''[[36/35]]'', [[50/49]], ''[[56/55]]''
| ^1, vm2
| up unison, downminor 2nd
| ^D, vEb
|-
| 2
| 61.5
| [[33/32]]
| ''[[21/20]]'', [[36/35]]
| [[28/27]], ''[[49/48]]''
| m2
| minor 2nd
| Eb
|-
| 3
| 92.3
| ''[[16/15]]'', ''[[25/24]]''
| ''[[50/49]]''
| [[21/20]]
| ^m2
| upminor 2nd
| ^Eb
|-
| 4
| 123.1
|
|
| [[15/14]]
| ^^m2
| dupminor 2nd
| ^^Eb
|-
| 5
| 153.8
| [[11/10]], [[12/11]]
| ''[[15/14]]''
|
| vvM2
| dudmajor 2nd
| vvE
|-
| 6
| 184.6
| [[10/9]]
|
|
| vM2
| downmajor 2nd
| vE
|-
| 7
| 215.4
| [[9/8]]
|
| ''[[8/7]]''
| M2
| major 2nd
| E
|-
| 8
| 246.2
|
| [[8/7]], ''[[7/6]]''
| [[81/70]]
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| ^E, vF
|-
| 9
| 276.9
|
| ''[[81/70]]''
| [[7/6]]
| m3
| minor 3rd
| F
|-
| 10
| 307.7
| [[6/5]]
|
|
| ^m3
| upminor 3rd
| ^F
|-
| 11
| 338.5
| [[11/9]]
|
|
| ^^m3
| dupminor 3rd
| ^^F
|-
| 12
| 369.2
| [[27/22]]
|
|
| vvM3
| dudmajor 3rd
| vvF#
|-
| 13
| 400.0
| [[5/4]]
| ''[[14/11]]''
|
| vM3
| downmajor 3rd
| vF#
|-
| 14
| 430.8
|
| ''[[35/27]]''
| [[9/7]], [[14/11]]
| M3
| major 3rd
| F#
|-
| 15
| 461.5
|
| ''[[9/7]]''
| [[35/27]]
| v4
| down 4th
| vG
|-
| 16
| 492.3
| [[4/3]]
|
|
| P4
| perfect 4th
| G
|-
| 17
| 523.1
| [[27/20]]
|
|
| ^4
| up 4th
| ^G
|-
| 18
| 553.8
| [[11/8]]
| ''[[7/5]]''
|
| ^^4
| dup 4th
| ^^G
|-
| 19
| 584.6
|
|
| [[7/5]]
| vvA4, ^d5
| dudaug 4th, updim 5th
| vvG#, ^Ab
|-
| 20
| 615.4
|
|
| [[10/7]]
| vA4, ^^d5
| downaug 4th, dupdim 5th
| vG#, ^^Ab
|-
| 21
| 646.2
| [[16/11]]
| ''[[10/7]]''
|
| vv5
| dud 5th
| vvA
|-
| 22
| 676.9
| [[40/27]]
|
|
| v5
| down 5th
| vA
|-
| 23
| 707.7
| [[3/2]]
|
|
| P5
| perfect 5th
| A
|-
| 24
| 738.5
|
| ''[[14/9]]''
| [[54/35]]
| ^5
| up 5th
| A^
|-
| 25
| 769.2
|
| ''[[54/35]]''
| [[11/7]], [[14/9]]
| m6
| minor 6th
| Bb
|-
| 26
| 800.0
| [[8/5]]
| ''[[11/7]]''
|
| ^m6
| upminor 6th
| ^Bb
|-
| 27
| 830.8
| [[44/27]]
|
|
| ^^m6
| dupminor 6th
| ^^Bb
|-
| 28
| 861.5
| [[18/11]]
|
|
| vvM6
| dudmajor 6th
| vvB
|-
| 29
| 892.3
| [[5/3]]
|
|
| vM6
| downmajor 6th
| vB
|-
| 30
| 923.1
|
| ''[[140/81]]''
| [[12/7]]
| M6
| major 6th
| B
|-
| 31
| 953.8
|
| [[7/4]], ''[[12/7]]''
| [[140/81]]
| ^M6, vm7
| upmajor 6th, downminor 7th
| ^B, vC
|-
| 32
| 984.6
| [[16/9]]
|
| ''[[7/4]]''
| m7
| minor 7th
| C
|-
| 33
| 1015.4
| [[9/5]]
|
|
| ^m7
| upminor 7th
| ^C
|-
| 34
| 1046.2
| [[11/6]], [[20/11]]
| ''[[28/15]]''
|
| ^^m7
| dupminor 7th
| ^^C
|-
| 35
| 1076.9
|
|
| [[28/15]]
| vvM7
| dudmajor 7th
| vvC#
|-
| 36
| 1107.7
| ''[[15/8]]'', ''[[48/25]]''
| ''[[49/25]]''
| [[40/21]]
| vM7
| downmajor 7th
| vC#
|-
| 37
| 1138.5
| [[64/33]]
| [[35/18]], ''[[40/21]]''
| [[27/14]], ''[[96/49]]''
| M7
| major 7th
| C#
|-
| 38
| 1169.2
| [[108/55]], [[160/81]]
| [[63/32]], ''[[27/14]]''
| ''[[35/18]]'', [[49/25]]
| ^M7, v8
| upmajor 7th, down 8ve
| ^C#, vD
|-
| 39
| 1200.0
| colspan=3 | [[2/1]]
| P8
| perfect 8ve
| D
|}

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].

== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}

=== Kite's ups and downs notation ===
39edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Ups and downs sharpness}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].

==== Evo flavor ====
<imagemap>
File:39-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:39-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:39-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:39-EDO_Revo_Sagittal.svg]]
</imagemap>

=== Armodue notation ===
; Armodue nomenclature 5;2 relation
* '''‡''' = Semisharp (1/5-tone up)
* '''b''' = Flat (3/5-tone down)
* '''#''' = Sharp (3/5-tone up)
* '''v''' = Semiflat (1/5-tone down)

{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
|-
! colspan="2" | #
! Cents
! Armodue notation
! Associated ratios
|-
| 0
|
| 0.0
| 1
| [[1/1]]
|-
| 1
|
| 30.8
| 1‡ (9#)
|
|-
| 2
|
| 61.5
| 2b
|
|-
| 3
|
| 92.3
| 1#
|
|-
| 4
|
| 123.1
| 2v
|
|-
| 5
|
| 153.8
| 2
| 11/10~12/11
|-
| 6
|
| 184.6
| 2‡
|
|-
| 7
| ·
| 215.4
| 3b
| 8/7
|-
| 8
|
| 246.2
| 2#
|
|-
| 9
|
| 276.9
| 3v
|
|-
| 10
|
| 307.7
| 3
| 6/5~7/6
|-
| 11
|
| 338.5
| 3‡
|
|-
| 12
| ·
| 369.2
| 4b
| 5/4
|-
| 13
|
| 400.0
| 3#
|
|-
| 14
|
| 430.8
| 4v (5b)
|
|-
| 15
|
| 461.5
| 4
|
|-
| 16
|
| 492.3
| 4‡ (5v)
|
|-
| 17
| ·
| 523.1
| 5
| 4/3~11/8
|-
| 18
|
| 553.8
| 5‡ (4#)
|
|-
| 19
|
| 584.6
| 6b
| 10/7
|-
| 20
|
| 615.4
| 5#
| 7/5
|-
| 21
|
| 646.2
| 6v
|
|-
| 22
| ·
| 676.9
| 6
| 3/2~16/11
|-
| 23
|
| 707.7
| 6‡
|
|-
| 24
|
| 738.5
| 7b
|
|-
| 25
|
| 769.2
| 6#
|
|-
| 26
|
| 800.0
| 7v
|
|-
| 27
| ·
| 830.8
| 7
| 8/5
|-
| 28
|
| 861.5
| 7‡
|
|-
| 29
|
| 892.3
| 8b
| 5/3~12/7
|-
| 30
|
| 923.1
| 7#
|
|-
| 31
|
| 953.8
| 8v
|
|-
| 32
| ·
| 984.6
| 8
| 7/4
|-
| 33
|
| 1015.4
| 8‡
|
|-
| 34
|
| 1046.2
| 9b
| 11/6~20/11
|-
| 35
|
| 1076.9
| 8#
|
|-
| 36
|
| 1107.7
| 9v (1b)
|
|-
| 37
|
| 1138.5
| 9
|
|-
| 38
|
| 1169.2
| 9‡ (1v)
|
|-
| 39
| ··
| 1200.0
| 1
| 2/1
|}

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|39.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 39df val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 62 -39 }}
| {{Mapping| 39 62 }}
| −1.81
| 1.81
| 5.88
|-
| 2.3.5
| 128/125, 1594323/1562500
| {{Mapping| 39 62 91 }}
| −3.17
| 2.42
| 7.89
|-
| 2.3.5.7
| 64/63, 126/125, 2430/2401
| {{Mapping| 39 62 91 110 }} (39d)
| −3.78
| 2.35
| 7.65
|-
| 2.3.5.7.11
| 64/63, 99/98, 121/120, 126/125
| {{Mapping| 39 62 91 110 135 }} (39d)
| −3.17
| 2.43
| 7.91
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-4 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Temperament
! Mos scales
|-
| 1
| 1\39
| 30.8
|
|
|-
| 1
| 2\39
| 61.5
| [[Unicorn]] (39d)
| [[1L 18s]], [[19L 1s]]
|-
| 1
| 4\39
| 123.1
| [[Negri]] (39c)
| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]
|-
| 1
| 5\39
| 153.8
|
| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]
|-
| 1
| 7\39
| 215.4
| [[Machine]] (39d)
| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]
|-
| 1
| 8\39
| 246.2
| [[Immunity]] (39) / [[immunized]] (39d)
| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]
|-
| 1
| 10\39
| 307.7
| [[Familia]] (39df)
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]
|-
| 1
| 11\39
| 338.5
| [[Amity]] (39) / [[accord]] (39d)
| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]
|-
| 1
| 14\39
| 430.8
| [[Hamity]] (39df)
| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[14L 11s]]
|-
| 1
| 16\39
| 492.3
| [[Quasisuper]] (39d)
| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]
|-
| 1
| 17\39
| 523.1
| [[Mavila]] (39bc)
| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]
|-
| 1
| 19\39
| 584.6
| [[Pluto]] (39d)
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]
|-
| 3
| 1\39
| 30.8
|
|
|-
| 3
| 2\39
| 61.5
|
| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]
|-
| 3
| 6\39
| 184.6
| [[Terrain]] / [[mirkat]] (39df)
| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]
|-
| 3
| 8\39 (5\39)
| 246.2 (153.8)
| [[Triforce]] (39)
| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]
|-
| 3
| 16\39 (3\39)
| 492.3 (92.3)
| [[Augene]] (39d)
| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]
|-
| 3
| 17\39 (4\39)
| 523.1 (123.0)
| [[Deflated]] (39bd)
| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]
|-
| 13
| 16\39 (1\39)
| 492.3 (30.8)
| [[Tridecatonic]]
| [[13L 13s]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
39edo is a [[zeta valley edo]] and is generally poor at approximating primes for its size. Its poor approximations of harmonics 3, 5, 7, and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[ed6|101ed6]] or [[173zpi]].

39edo can be usefully mapped onto the val 39dfgijk. The [[Tenney-Euclidean]] tuning of this regular temperament is 30.67475 cents per step, which is closely approximated by [[62edt]] and 173zpi.

== 39edo and world music ==
Some might consider 39edo a candidate for a "universal tuning" in that it offers reasonable approximations of many different world music [[approaches to musical tuning|traditions]]; it is one of the simplest edos that can make this claim. Because of this, composers wishing to combine multiple world music traditions (for example, [[gamelan]] with [[maqam]] singing) within one unified framework might find 39edo an interesting possibility.

=== Western ===
39edo offers not one, but several different ways to realize the traditional Western diatonic scale. One way is to simply take a [[chain of fifths]] (the diatonic mos: 7 7 2 7 7 7 2). Because 39edo is a [[superpyth]] rather than a [[meantone]] system, this means that the harmonic quality of its diatonic scale will differ somewhat, since "minor" and "major" triads now approximate 6:7:9 and 14:18:21 respectively, rather than 10:12:15 and 4:5:6 as in meantone diatonic systems. Diatonic compositions translated onto this scale thus acquire a wildly different harmonic character, albeit still pleasing.

Another option is to use a [[modmos]], such as 7 6 3 7 6 7 3; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either 7 3 3 3 7 3 3 7 3 (a [[modmos]] of type [[3L 6s]]) or 4 3 6 3 4 3 6 4 3 3. There are other modmos scales that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.

The mos and the modmos scales all have smaller-than-usual [[semitone (interval region)|semitones]], which makes them more effective for melody than their counterparts in 12edo or meantone systems.

Because 39edo and 12edo both have an overall sharp character and share the same major third, they have a relatively similar sound. Thus, 39edo (unlike, say, 22edo or 19edo, which are both "acquired tastes") does not sound all that [[xenharmonic]] to people used to 12edo. Check out [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the 7 6 3 7 6 7 3 modmos), for example.

=== Indian ===
A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L 5s]] MOS (where the generator is a perfect fifth).

=== Arabic, Turkish, Iranian ===
While [[Arabic, Turkish, Persian music|middle-eastern music]] is commonly approximated using [[24edo]], 39edo offers a potentially better alternative. [[17edo]] and 24edo both satisfy the "Level 1" requirements for [[maqam]] tuning systems. 39edo is a Level 2 system because:

* It has two types of "neutral" seconds (154 and 185 cents)
* It has two minor seconds (92 and 123 cents), which when added together give a whole tone (215 cents)

whereas neither 17edo nor 24edo satisfy these properties.

39edo will likely be more suited to some middle-eastern scales than others. Specifically, Turkish music (in which the Rast makam has a "major-like" wide neutral third and a wide "neutral" second approaching 10/9), will likely be especially well suited to 39edo.

=== Blues / Jazz / African-American ===
The [[harmonic seventh]] ("[[barbershop]] seventh") [[tetrad]] is reasonably well approximated in 39edo, and some temperaments (augene in particular) give scales that are liberally supplied with them. John Coltrane might have loved augene (→ [[Wikipedia: Coltrane changes]]).

[[Tritone]] substitution, which is a major part of jazz and blues harmony, is more complicated in 39edo because there are two types of tritones. Therefore, the tritone substitution of one seventh chord will need to be a different type of seventh chord. However, this also opens new possibilities; if the substituted chord is of a more consonant type than the original, then the tritone substitution may function as a ''resolution'' rather than a suspension.

Blue notes, rather than being considered inflections, can be notated as accidentals instead; for example, a "blue major third" can be identified as either of the two neutral thirds. There are two possible [[mapping]]s for [[7/4]] which are about equal in closeness. The sharp mapping is the normal one because it works better with the [[5/4]] and [[3/2]], but using the flat one instead (as an accidental) allows for another type of blue note.

=== Other ===
39edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator. Pelog can also be approximated as 4 5 13 4 13.

It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] mos (which is 9 7 7 9 7). [[Slendro]] can be approximated using that scale or using something like the [[quasi-equal]] 8 8 8 8 7 or 8 8 7 8 8.

One expressive [[pentatonic]] scale is the oneirotonic subset 9 6 9 9 6.

Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

== Scales ==
* [[Quasisuper]][7] [[MOS scale]]: 7 7 2 7 7 7 2
* Quasisuper[7] [[5-limit|pental]] [[modmos]]: 7 6 3 7 6 7 3
* [[3L 6s]] modmos: 7 3 3 3 7 3 3 7 3
* Extended quasisuper: 4 3 6 3 4 3 6 4 3 3
* Quasisuper[22] MOS scale (resembles [[Indian]] [[sruti]]): 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 1 2 2
* Slendro approximations: 9 7 7 9 7 or 8 8 8 8 7 or 8 8 7 8 8
* An expressive [[oneirotonic]] subset: 9 6 9 9 6
* ''The scales listed in: [[User:BudjarnLambeth/Quasipelog theory]]''

== Instruments ==
=== Lumatone mapping ===
See [[Lumatone mapping for 39edo]]

=== Skip fretting ===
'''Skip fretting system 39 2 5''' is a [[skip-fretting]] system for [[39edo]]. All examples on this page are for 7-string [[Guitar|guitar.]]

; Prime harmonics
1/1: string 2 open

2/1: string 5 fret 12 and string 7 fret 7

3/2: string 3 fret 9 and string 5 fret 4

5/4: string 1 fret 9 and string 3 fret 4

7/4: string 5 fret 8 and string 7 fret 3

11/8: string 2 fret 9 and string 4 fret 4

=== Prototypes ===
[[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]]

''An illustrative image of a 39edo keyboard''

[[File:Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|alt=Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|826x203px|Custom_700mm_5-str_Tricesanonaphonic_Guitar.png]]

''39edo fretboard visualization''

== Music ==
=== Modern renderings ===
; {{W|HOYO-MiX}}
* [https://www.youtube.com/shorts/4y11CWLIHNA "Sinner's Finale" from ''Genshin Impact OST''] (2023) – covered by [[Bryan Deister]] (2025)

=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)
* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)
* [https://www.youtube.com/shorts/1T_xrZpUslQ ''39edo improv''] (2025)
* [https://www.youtube.com/watch?v=kYQyRY7xFJs ''Waltz in 39edo''] (2025)
* [https://www.youtube.com/watch?v=Vzife15uUU4 ''Tilt Your Head Down''] (2026)

; [[groundfault]]
* From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube]
** "Resolute Prelude"
** "Residual Soliloquy"

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021)

[[Category:Listen]]
{{Todo|add scales list}}

39edo

2026-05-30T21:59:50Z

Overthink: /* Intervals */ left align those

{{Infobox ET}}
{{ED intro}}

== Theory ==
39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]]. We have two choices for a tuning of [[7/1|7]]. The sharp one yields [[superpyth]] temperament, while the flat (patent) one yields [[semaphore]] (and also [[hemifamity]]) temperament.

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39d is [[quasisuper]]. Alternatively, it can be seen as a [[hemifamity]] semaphore (that is, immunity) system in the patent val. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.

Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.

39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).

=== Odd harmonics ===
{{Harmonics in equal|39|columns=11}}
{{Harmonics in equal|39|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 39edo (continued)}}

=== As a tuning of other temperaments ===
39edo, with its 400{{c}} major third, [[tempering out|tempers out]] the [[diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. Alternatively, the patent val tempers out 49/48 to yield semaphore. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].

=== Subsets and supersets ===
Since 39 factors into primes as {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.

== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 left-5 right-9 right-10"
|-
! rowspan="2" | Steps
! rowspan="2" | Cents
! rowspan="2" | Ratios of the [[2.3.5.11 subgroup]]
! colspan="2" | Intervals of 7
! colspan="3" rowspan="2" | [[Ups and downs notation]]
|-
! Patent val
! 39d val
|-
| 0
| 0.0
| colspan=3 | [[1/1]]
| P1
| perfect unison
| D
|-
| 1
| 30.8
| [[55/54]], [[81/80]]
| ''[[28/27]]'', [[64/63]]
| ''[[36/35]]'', [[50/49]], ''[[56/55]]''
| ^1, vm2
| up unison, downminor 2nd
| ^D, vEb
|-
| 2
| 61.5
| [[33/32]]
| ''[[21/20]]'', [[36/35]]
| [[28/27]], ''[[49/48]]''
| m2
| minor 2nd
| Eb
|-
| 3
| 92.3
| ''[[16/15]]'', ''[[25/24]]''
| ''[[50/49]]''
| [[21/20]]
| ^m2
| upminor 2nd
| ^Eb
|-
| 4
| 123.1
|
|
| [[15/14]]
| ^^m2
| dupminor 2nd
| ^^Eb
|-
| 5
| 153.8
| [[11/10]], [[12/11]]
| ''[[15/14]]''
|
| vvM2
| dudmajor 2nd
| vvE
|-
| 6
| 184.6
| [[10/9]]
|
|
| vM2
| downmajor 2nd
| vE
|-
| 7
| 215.4
| [[9/8]]
|
| ''[[8/7]]''
| M2
| major 2nd
| E
|-
| 8
| 246.2
|
| [[8/7]], ''[[7/6]]''
| [[81/70]]
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| ^E, vF
|-
| 9
| 276.9
|
| ''[[81/70]]''
| [[7/6]]
| m3
| minor 3rd
| F
|-
| 10
| 307.7
| [[6/5]]
|
|
| ^m3
| upminor 3rd
| ^F
|-
| 11
| 338.5
| [[11/9]]
|
|
| ^^m3
| dupminor 3rd
| ^^F
|-
| 12
| 369.2
| [[27/22]]
|
|
| vvM3
| dudmajor 3rd
| vvF#
|-
| 13
| 400.0
| [[5/4]]
| ''[[14/11]]''
|
| vM3
| downmajor 3rd
| vF#
|-
| 14
| 430.8
|
| ''[[35/27]]''
| [[9/7]], [[14/11]]
| M3
| major 3rd
| F#
|-
| 15
| 461.5
|
| ''[[9/7]]''
| [[35/27]]
| v4
| down 4th
| vG
|-
| 16
| 492.3
| [[4/3]]
|
|
| P4
| perfect 4th
| G
|-
| 17
| 523.1
| [[27/20]]
|
|
| ^4
| up 4th
| ^G
|-
| 18
| 553.8
| [[11/8]]
| ''[[7/5]]''
|
| ^^4
| dup 4th
| ^^G
|-
| 19
| 584.6
|
|
| [[7/5]]
| vvA4, ^d5
| dudaug 4th, updim 5th
| vvG#, ^Ab
|-
| 20
| 615.4
|
|
| [[10/7]]
| vA4, ^^d5
| downaug 4th, dupdim 5th
| vG#, ^^Ab
|-
| 21
| 646.2
| [[16/11]]
| ''[[10/7]]''
|
| vv5
| dud 5th
| vvA
|-
| 22
| 676.9
| [[40/27]]
|
|
| v5
| down 5th
| vA
|-
| 23
| 707.7
| [[3/2]]
|
|
| P5
| perfect 5th
| A
|-
| 24
| 738.5
|
| ''[[14/9]]''
| [[54/35]]
| ^5
| up 5th
| A^
|-
| 25
| 769.2
|
| ''[[54/35]]''
| [[11/7]], [[14/9]]
| m6
| minor 6th
| Bb
|-
| 26
| 800.0
| [[8/5]]
| ''[[11/7]]''
|
| ^m6
| upminor 6th
| ^Bb
|-
| 27
| 830.8
| [[44/27]]
|
|
| ^^m6
| dupminor 6th
| ^^Bb
|-
| 28
| 861.5
| [[18/11]]
|
|
| vvM6
| dudmajor 6th
| vvB
|-
| 29
| 892.3
| [[5/3]]
|
|
| vM6
| downmajor 6th
| vB
|-
| 30
| 923.1
|
| ''[[140/81]]''
| [[12/7]]
| M6
| major 6th
| B
|-
| 31
| 953.8
|
| [[7/4]], ''[[12/7]]''
| [[140/81]]
| ^M6, vm7
| upmajor 6th, downminor 7th
| ^B, vC
|-
| 32
| 984.6
| [[16/9]]
|
| ''[[7/4]]''
| m7
| minor 7th
| C
|-
| 33
| 1015.4
| [[9/5]]
|
|
| ^m7
| upminor 7th
| ^C
|-
| 34
| 1046.2
| [[11/6]], [[20/11]]
| ''[[28/15]]''
|
| ^^m7
| dupminor 7th
| ^^C
|-
| 35
| 1076.9
|
|
| [[28/15]]
| vvM7
| dudmajor 7th
| vvC#
|-
| 36
| 1107.7
| ''[[15/8]]'', ''[[48/25]]''
| ''[[49/25]]''
| [[40/21]]
| vM7
| downmajor 7th
| vC#
|-
| 37
| 1138.5
| [[64/33]]
| [[35/18]], ''[[40/21]]''
| [[27/14]], ''[[96/49]]''
| M7
| major 7th
| C#
|-
| 38
| 1169.2
| [[108/55]], [[160/81]]
| [[63/32]], ''[[27/14]]''
| ''[[35/18]]'', [[49/25]]
| ^M7, v8
| upmajor 7th, down 8ve
| ^C#, vD
|-
| 39
| 1200.0
| colspan=3 | [[2/1]]
| P8
| perfect 8ve
| D
|}
<nowiki/>* 11-limit in the 39d val, inconsistent intervals in ''italic''

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].

== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}

=== Kite's ups and downs notation ===
39edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Ups and downs sharpness}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].

==== Evo flavor ====
<imagemap>
File:39-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:39-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:39-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:39-EDO_Revo_Sagittal.svg]]
</imagemap>

=== Armodue notation ===
; Armodue nomenclature 5;2 relation
* '''‡''' = Semisharp (1/5-tone up)
* '''b''' = Flat (3/5-tone down)
* '''#''' = Sharp (3/5-tone up)
* '''v''' = Semiflat (1/5-tone down)

{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
|-
! colspan="2" | #
! Cents
! Armodue notation
! Associated ratios
|-
| 0
|
| 0.0
| 1
| [[1/1]]
|-
| 1
|
| 30.8
| 1‡ (9#)
|
|-
| 2
|
| 61.5
| 2b
|
|-
| 3
|
| 92.3
| 1#
|
|-
| 4
|
| 123.1
| 2v
|
|-
| 5
|
| 153.8
| 2
| 11/10~12/11
|-
| 6
|
| 184.6
| 2‡
|
|-
| 7
| ·
| 215.4
| 3b
| 8/7
|-
| 8
|
| 246.2
| 2#
|
|-
| 9
|
| 276.9
| 3v
|
|-
| 10
|
| 307.7
| 3
| 6/5~7/6
|-
| 11
|
| 338.5
| 3‡
|
|-
| 12
| ·
| 369.2
| 4b
| 5/4
|-
| 13
|
| 400.0
| 3#
|
|-
| 14
|
| 430.8
| 4v (5b)
|
|-
| 15
|
| 461.5
| 4
|
|-
| 16
|
| 492.3
| 4‡ (5v)
|
|-
| 17
| ·
| 523.1
| 5
| 4/3~11/8
|-
| 18
|
| 553.8
| 5‡ (4#)
|
|-
| 19
|
| 584.6
| 6b
| 10/7
|-
| 20
|
| 615.4
| 5#
| 7/5
|-
| 21
|
| 646.2
| 6v
|
|-
| 22
| ·
| 676.9
| 6
| 3/2~16/11
|-
| 23
|
| 707.7
| 6‡
|
|-
| 24
|
| 738.5
| 7b
|
|-
| 25
|
| 769.2
| 6#
|
|-
| 26
|
| 800.0
| 7v
|
|-
| 27
| ·
| 830.8
| 7
| 8/5
|-
| 28
|
| 861.5
| 7‡
|
|-
| 29
|
| 892.3
| 8b
| 5/3~12/7
|-
| 30
|
| 923.1
| 7#
|
|-
| 31
|
| 953.8
| 8v
|
|-
| 32
| ·
| 984.6
| 8
| 7/4
|-
| 33
|
| 1015.4
| 8‡
|
|-
| 34
|
| 1046.2
| 9b
| 11/6~20/11
|-
| 35
|
| 1076.9
| 8#
|
|-
| 36
|
| 1107.7
| 9v (1b)
|
|-
| 37
|
| 1138.5
| 9
|
|-
| 38
|
| 1169.2
| 9‡ (1v)
|
|-
| 39
| ··
| 1200.0
| 1
| 2/1
|}

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|39.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 39df val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 62 -39 }}
| {{Mapping| 39 62 }}
| −1.81
| 1.81
| 5.88
|-
| 2.3.5
| 128/125, 1594323/1562500
| {{Mapping| 39 62 91 }}
| −3.17
| 2.42
| 7.89
|-
| 2.3.5.7
| 64/63, 126/125, 2430/2401
| {{Mapping| 39 62 91 110 }} (39d)
| −3.78
| 2.35
| 7.65
|-
| 2.3.5.7.11
| 64/63, 99/98, 121/120, 126/125
| {{Mapping| 39 62 91 110 135 }} (39d)
| −3.17
| 2.43
| 7.91
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-4 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Temperament
! Mos scales
|-
| 1
| 1\39
| 30.8
|
|
|-
| 1
| 2\39
| 61.5
| [[Unicorn]] (39d)
| [[1L 18s]], [[19L 1s]]
|-
| 1
| 4\39
| 123.1
| [[Negri]] (39c)
| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]
|-
| 1
| 5\39
| 153.8
|
| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]
|-
| 1
| 7\39
| 215.4
| [[Machine]] (39d)
| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]
|-
| 1
| 8\39
| 246.2
| [[Immunity]] (39) / [[immunized]] (39d)
| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]
|-
| 1
| 10\39
| 307.7
| [[Familia]] (39df)
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]
|-
| 1
| 11\39
| 338.5
| [[Amity]] (39) / [[accord]] (39d)
| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]
|-
| 1
| 14\39
| 430.8
| [[Hamity]] (39df)
| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[14L 11s]]
|-
| 1
| 16\39
| 492.3
| [[Quasisuper]] (39d)
| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]
|-
| 1
| 17\39
| 523.1
| [[Mavila]] (39bc)
| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]
|-
| 1
| 19\39
| 584.6
| [[Pluto]] (39d)
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]
|-
| 3
| 1\39
| 30.8
|
|
|-
| 3
| 2\39
| 61.5
|
| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]
|-
| 3
| 6\39
| 184.6
| [[Terrain]] / [[mirkat]] (39df)
| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]
|-
| 3
| 8\39 (5\39)
| 246.2 (153.8)
| [[Triforce]] (39)
| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]
|-
| 3
| 16\39 (3\39)
| 492.3 (92.3)
| [[Augene]] (39d)
| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]
|-
| 3
| 17\39 (4\39)
| 523.1 (123.0)
| [[Deflated]] (39bd)
| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]
|-
| 13
| 16\39 (1\39)
| 492.3 (30.8)
| [[Tridecatonic]]
| [[13L 13s]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
39edo is a [[zeta valley edo]] and is generally poor at approximating primes for its size. Its poor approximations of harmonics 3, 5, 7, and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[ed6|101ed6]] or [[173zpi]].

39edo can be usefully mapped onto the val 39dfgijk. The [[Tenney-Euclidean]] tuning of this regular temperament is 30.67475 cents per step, which is closely approximated by [[62edt]] and 173zpi.

== 39edo and world music ==
Some might consider 39edo a candidate for a "universal tuning" in that it offers reasonable approximations of many different world music [[approaches to musical tuning|traditions]]; it is one of the simplest edos that can make this claim. Because of this, composers wishing to combine multiple world music traditions (for example, [[gamelan]] with [[maqam]] singing) within one unified framework might find 39edo an interesting possibility.

=== Western ===
39edo offers not one, but several different ways to realize the traditional Western diatonic scale. One way is to simply take a [[chain of fifths]] (the diatonic mos: 7 7 2 7 7 7 2). Because 39edo is a [[superpyth]] rather than a [[meantone]] system, this means that the harmonic quality of its diatonic scale will differ somewhat, since "minor" and "major" triads now approximate 6:7:9 and 14:18:21 respectively, rather than 10:12:15 and 4:5:6 as in meantone diatonic systems. Diatonic compositions translated onto this scale thus acquire a wildly different harmonic character, albeit still pleasing.

Another option is to use a [[modmos]], such as 7 6 3 7 6 7 3; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either 7 3 3 3 7 3 3 7 3 (a [[modmos]] of type [[3L 6s]]) or 4 3 6 3 4 3 6 4 3 3. There are other modmos scales that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.

The mos and the modmos scales all have smaller-than-usual [[semitone (interval region)|semitones]], which makes them more effective for melody than their counterparts in 12edo or meantone systems.

Because 39edo and 12edo both have an overall sharp character and share the same major third, they have a relatively similar sound. Thus, 39edo (unlike, say, 22edo or 19edo, which are both "acquired tastes") does not sound all that [[xenharmonic]] to people used to 12edo. Check out [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the 7 6 3 7 6 7 3 modmos), for example.

=== Indian ===
A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L 5s]] MOS (where the generator is a perfect fifth).

=== Arabic, Turkish, Iranian ===
While [[Arabic, Turkish, Persian music|middle-eastern music]] is commonly approximated using [[24edo]], 39edo offers a potentially better alternative. [[17edo]] and 24edo both satisfy the "Level 1" requirements for [[maqam]] tuning systems. 39edo is a Level 2 system because:

* It has two types of "neutral" seconds (154 and 185 cents)
* It has two minor seconds (92 and 123 cents), which when added together give a whole tone (215 cents)

whereas neither 17edo nor 24edo satisfy these properties.

39edo will likely be more suited to some middle-eastern scales than others. Specifically, Turkish music (in which the Rast makam has a "major-like" wide neutral third and a wide "neutral" second approaching 10/9), will likely be especially well suited to 39edo.

=== Blues / Jazz / African-American ===
The [[harmonic seventh]] ("[[barbershop]] seventh") [[tetrad]] is reasonably well approximated in 39edo, and some temperaments (augene in particular) give scales that are liberally supplied with them. John Coltrane might have loved augene (→ [[Wikipedia: Coltrane changes]]).

[[Tritone]] substitution, which is a major part of jazz and blues harmony, is more complicated in 39edo because there are two types of tritones. Therefore, the tritone substitution of one seventh chord will need to be a different type of seventh chord. However, this also opens new possibilities; if the substituted chord is of a more consonant type than the original, then the tritone substitution may function as a ''resolution'' rather than a suspension.

Blue notes, rather than being considered inflections, can be notated as accidentals instead; for example, a "blue major third" can be identified as either of the two neutral thirds. There are two possible [[mapping]]s for [[7/4]] which are about equal in closeness. The sharp mapping is the normal one because it works better with the [[5/4]] and [[3/2]], but using the flat one instead (as an accidental) allows for another type of blue note.

=== Other ===
39edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator. Pelog can also be approximated as 4 5 13 4 13.

It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] mos (which is 9 7 7 9 7). [[Slendro]] can be approximated using that scale or using something like the [[quasi-equal]] 8 8 8 8 7 or 8 8 7 8 8.

One expressive [[pentatonic]] scale is the oneirotonic subset 9 6 9 9 6.

Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

== Scales ==
* [[Quasisuper]][7] [[MOS scale]]: 7 7 2 7 7 7 2
* Quasisuper[7] [[5-limit|pental]] [[modmos]]: 7 6 3 7 6 7 3
* [[3L 6s]] modmos: 7 3 3 3 7 3 3 7 3
* Extended quasisuper: 4 3 6 3 4 3 6 4 3 3
* Quasisuper[22] MOS scale (resembles [[Indian]] [[sruti]]): 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 1 2 2
* Slendro approximations: 9 7 7 9 7 or 8 8 8 8 7 or 8 8 7 8 8
* An expressive [[oneirotonic]] subset: 9 6 9 9 6
* ''The scales listed in: [[User:BudjarnLambeth/Quasipelog theory]]''

== Instruments ==
=== Lumatone mapping ===
See [[Lumatone mapping for 39edo]]

=== Skip fretting ===
'''Skip fretting system 39 2 5''' is a [[skip-fretting]] system for [[39edo]]. All examples on this page are for 7-string [[Guitar|guitar.]]

; Prime harmonics
1/1: string 2 open

2/1: string 5 fret 12 and string 7 fret 7

3/2: string 3 fret 9 and string 5 fret 4

5/4: string 1 fret 9 and string 3 fret 4

7/4: string 5 fret 8 and string 7 fret 3

11/8: string 2 fret 9 and string 4 fret 4

=== Prototypes ===
[[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]]

''An illustrative image of a 39edo keyboard''

[[File:Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|alt=Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|826x203px|Custom_700mm_5-str_Tricesanonaphonic_Guitar.png]]

''39edo fretboard visualization''

== Music ==
=== Modern renderings ===
; {{W|HOYO-MiX}}
* [https://www.youtube.com/shorts/4y11CWLIHNA "Sinner's Finale" from ''Genshin Impact OST''] (2023) – covered by [[Bryan Deister]] (2025)

=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)
* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)
* [https://www.youtube.com/shorts/1T_xrZpUslQ ''39edo improv''] (2025)
* [https://www.youtube.com/watch?v=kYQyRY7xFJs ''Waltz in 39edo''] (2025)
* [https://www.youtube.com/watch?v=Vzife15uUU4 ''Tilt Your Head Down''] (2026)

; [[groundfault]]
* From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube]
** "Resolute Prelude"
** "Residual Soliloquy"

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021)

[[Category:Listen]]
{{Todo|add scales list}}

39edo

2026-05-30T21:59:01Z

Overthink: /* Intervals */ cleanup and links (minor note: don't italicize ","); Remove "nearest just interval" (generally not preferred).

{{Infobox ET}}
{{ED intro}}

== Theory ==
39edo's [[3/2|perfect fifth]] is 5.8{{c}} sharp. Together with its best [[5/4|classical major third]] which is the familiar 400{{c}} of [[12edo]]. We have two choices for a tuning of [[7/1|7]]. The sharp one yields [[superpyth]] temperament, while the flat (patent) one yields [[semaphore]] (and also [[hemifamity]]) temperament.

As a [[superpyth]] system, 39edo is intermediate between [[17edo]] and [[22edo]] {{nowrap|(39 {{=}} 17 + 22)}}; its fifth thus falls in the "shrub region" where the diatonic thirds are between standard neogothic thirds and septimal thirds. The specific 7-limit variant supported by 39d is [[quasisuper]]. Alternatively, it can be seen as a [[hemifamity]] semaphore (that is, immunity) system in the patent val. While 17edo is superb for melody (as documented by [[George Secor]]), it does not approximate the 5th harmonic at all and only poorly approximates the 7th. 22edo is much better for 5-limit and 7-limit harmony but is less effective for melody assuming that the MOS diatonic is used, because the [[diatonic semitone]] is [[quartertone]]-sized. 39edo offers a compromise, since it still supports good 5- and 7-limit harmonies (though less close than 22edo), while at the same time having a diatonic semitone of 61.5 cents, as the ideal diatonic semitone for melody is somewhere in between 60 and 80 cents, i.e. a third tone, by Secor's estimates.

Alternatively, if we take 22\39 as a fifth, 39edo can be used as a tuning of [[mavila]], and from that point of view it seems to have attracted the attention of the [[Armodue]] school, an Italian group that use the scheme of [[7L 2s|superdiatonic]] LLLsLLLLs like a base scale for notation and theory, suited in [[16edo]], and allied systems: [[25edo]] [1/3-tone 3;2]; [[41edo]] [1/5-tone 5;3]; and [[57edo]] [1/7-tone 7;4]. The [[hornbostel]] temperament is included too with: [[23edo]] [1/3-tone 3;1]; 39edo [1/5-tone 5;2] & [[62edo]] [1/8-tone 8;3]. The mavila fifth in 39edo like all mavila fifths is very, very flat, in this case, 25{{c}} flat.

39edo offers not one, but many, possible ways of extending tonality beyond the diatonic scale, even if it does not do as good of a job at approximating [[JI]] as some other systems do. Because it can also approximate [[mavila]] as well as "anti-mavila" ([[oneirotonic]]), the latter of which it inherits from [[13edo]], this makes 39edo an extremely versatile temperament usable in a wide range of situations (both harmonic and inharmonic).

=== Odd harmonics ===
{{Harmonics in equal|39|columns=11}}
{{Harmonics in equal|39|columns=11|start=12|collapsed=1|title=Approximation of odd harmonics in 39edo (continued)}}

=== As a tuning of other temperaments ===
39edo, with its 400{{c}} major third, [[tempering out|tempers out]] the [[diesis]] (128/125), and using the 39d val, the septimal comma, [[64/63]], as well as [[126/125]] are added to the comma list. Alternatively, the patent val tempers out 49/48 to yield semaphore. In the 11-limit we find that the equal temperament tempers out [[99/98]] and [[121/120]]. Tempering out both 128/125 and 64/63 makes 39et, in some few ways, allied to [[12edo|12et]] in [[support]]ing [[augene]], and is in fact, an excellent choice for a 7-limit augene tuning. It also tempers out the [[amity comma]] (1600000/1594323), and supports the variant of amity known as [[accord]].

=== Subsets and supersets ===
Since 39 factors into primes as {{nowrap| 3 × 13 }}, 39edo contains [[3edo]] and [[13edo]] as subsets. Multiplying 39edo by 2 yields [[78edo]], which corrects several harmonics.

== Intervals ==
{| class="wikitable center-all right-2 left-3 right-9 right-10"
|-
! rowspan="2" | Steps
! rowspan="2" | Cents
! rowspan="2" | Ratios of the [[2.3.5.11 subgroup]]
! colspan="2" | Intervals of 7
! colspan="3" rowspan="2" | [[Ups and downs notation]]
|-
! Patent val
! 39d val
|-
| 0
| 0.0
| colspan=3 | [[1/1]]
| P1
| perfect unison
| D
|-
| 1
| 30.8
| [[55/54]], [[81/80]]
| ''[[28/27]]'', [[64/63]]
| ''[[36/35]]'', [[50/49]], ''[[56/55]]''
| ^1, vm2
| up unison, downminor 2nd
| ^D, vEb
|-
| 2
| 61.5
| [[33/32]]
| ''[[21/20]]'', [[36/35]]
| [[28/27]], ''[[49/48]]''
| m2
| minor 2nd
| Eb
|-
| 3
| 92.3
| ''[[16/15]]'', ''[[25/24]]''
| ''[[50/49]]''
| [[21/20]]
| ^m2
| upminor 2nd
| ^Eb
|-
| 4
| 123.1
|
|
| [[15/14]]
| ^^m2
| dupminor 2nd
| ^^Eb
|-
| 5
| 153.8
| [[11/10]], [[12/11]]
| ''[[15/14]]''
|
| vvM2
| dudmajor 2nd
| vvE
|-
| 6
| 184.6
| [[10/9]]
|
|
| vM2
| downmajor 2nd
| vE
|-
| 7
| 215.4
| [[9/8]]
|
| ''[[8/7]]''
| M2
| major 2nd
| E
|-
| 8
| 246.2
|
| [[8/7]], ''[[7/6]]''
| [[81/70]]
| ^M2, vm3
| upmajor 2nd, downminor 3rd
| ^E, vF
|-
| 9
| 276.9
|
| ''[[81/70]]''
| [[7/6]]
| m3
| minor 3rd
| F
|-
| 10
| 307.7
| [[6/5]]
|
|
| ^m3
| upminor 3rd
| ^F
|-
| 11
| 338.5
| [[11/9]]
|
|
| ^^m3
| dupminor 3rd
| ^^F
|-
| 12
| 369.2
| [[27/22]]
|
|
| vvM3
| dudmajor 3rd
| vvF#
|-
| 13
| 400.0
| [[5/4]]
| ''[[14/11]]''
|
| vM3
| downmajor 3rd
| vF#
|-
| 14
| 430.8
|
| ''[[35/27]]''
| [[9/7]], [[14/11]]
| M3
| major 3rd
| F#
|-
| 15
| 461.5
|
| ''[[9/7]]''
| [[35/27]]
| v4
| down 4th
| vG
|-
| 16
| 492.3
| [[4/3]]
|
|
| P4
| perfect 4th
| G
|-
| 17
| 523.1
| [[27/20]]
|
|
| ^4
| up 4th
| ^G
|-
| 18
| 553.8
| [[11/8]]
| ''[[7/5]]''
|
| ^^4
| dup 4th
| ^^G
|-
| 19
| 584.6
|
|
| [[7/5]]
| vvA4, ^d5
| dudaug 4th, updim 5th
| vvG#, ^Ab
|-
| 20
| 615.4
|
|
| [[10/7]]
| vA4, ^^d5
| downaug 4th, dupdim 5th
| vG#, ^^Ab
|-
| 21
| 646.2
| [[16/11]]
| ''[[10/7]]''
|
| vv5
| dud 5th
| vvA
|-
| 22
| 676.9
| [[40/27]]
|
|
| v5
| down 5th
| vA
|-
| 23
| 707.7
| [[3/2]]
|
|
| P5
| perfect 5th
| A
|-
| 24
| 738.5
|
| ''[[14/9]]''
| [[54/35]]
| ^5
| up 5th
| A^
|-
| 25
| 769.2
|
| ''[[54/35]]''
| [[11/7]], [[14/9]]
| m6
| minor 6th
| Bb
|-
| 26
| 800.0
| [[8/5]]
| ''[[11/7]]''
|
| ^m6
| upminor 6th
| ^Bb
|-
| 27
| 830.8
| [[44/27]]
|
|
| ^^m6
| dupminor 6th
| ^^Bb
|-
| 28
| 861.5
| [[18/11]]
|
|
| vvM6
| dudmajor 6th
| vvB
|-
| 29
| 892.3
| [[5/3]]
|
|
| vM6
| downmajor 6th
| vB
|-
| 30
| 923.1
|
| ''[[140/81]]''
| [[12/7]]
| M6
| major 6th
| B
|-
| 31
| 953.8
|
| [[7/4]], ''[[12/7]]''
| [[140/81]]
| ^M6, vm7
| upmajor 6th, downminor 7th
| ^B, vC
|-
| 32
| 984.6
| [[16/9]]
|
| ''[[7/4]]''
| m7
| minor 7th
| C
|-
| 33
| 1015.4
| [[9/5]]
|
|
| ^m7
| upminor 7th
| ^C
|-
| 34
| 1046.2
| [[11/6]], [[20/11]]
| ''[[28/15]]''
|
| ^^m7
| dupminor 7th
| ^^C
|-
| 35
| 1076.9
|
|
| [[28/15]]
| vvM7
| dudmajor 7th
| vvC#
|-
| 36
| 1107.7
| ''[[15/8]]'', ''[[48/25]]''
| ''[[49/25]]''
| [[40/21]]
| vM7
| downmajor 7th
| vC#
|-
| 37
| 1138.5
| [[64/33]]
| [[35/18]], ''[[40/21]]''
| [[27/14]], ''[[96/49]]''
| M7
| major 7th
| C#
|-
| 38
| 1169.2
| [[108/55]], [[160/81]]
| [[63/32]], ''[[27/14]]''
| ''[[35/18]]'', [[49/25]]
| ^M7, v8
| upmajor 7th, down 8ve
| ^C#, vD
|-
| 39
| 1200.0
| colspan=3 | [[2/1]]
| P8
| perfect 8ve
| D
|}
<nowiki/>* 11-limit in the 39d val, inconsistent intervals in ''italic''

Chords can be named using ups and downs as C upminor, D downmajor seven, etc. See [[Ups and downs notation #Chords and chord progressions]].

== Notation ==
=== Stein–Zimmermann–Gould notation ===
[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:
{{Sharpness-sharp5-szg}}

=== Kite's ups and downs notation ===
39edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, dudsharp, downsharp, sharp, upsharp etc. and down, dud, dupflat etc. Note that dudsharp is equivalent to trup (triple-up) and dupflat is equivalent to trud (triple-down).
{{Ups and downs sharpness}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as [[46edo #Sagittal notation|46edo]].

==== Evo flavor ====
<imagemap>
File:39-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:39-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:39-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 240 106 [[33/32]]
default [[File:39-EDO_Revo_Sagittal.svg]]
</imagemap>

=== Armodue notation ===
; Armodue nomenclature 5;2 relation
* '''‡''' = Semisharp (1/5-tone up)
* '''b''' = Flat (3/5-tone down)
* '''#''' = Sharp (3/5-tone up)
* '''v''' = Semiflat (1/5-tone down)

{| class="wikitable center-all right-3 left-5 mw-collapsible mw-collapsed"
|-
! colspan="2" | #
! Cents
! Armodue notation
! Associated ratios
|-
| 0
|
| 0.0
| 1
| [[1/1]]
|-
| 1
|
| 30.8
| 1‡ (9#)
|
|-
| 2
|
| 61.5
| 2b
|
|-
| 3
|
| 92.3
| 1#
|
|-
| 4
|
| 123.1
| 2v
|
|-
| 5
|
| 153.8
| 2
| 11/10~12/11
|-
| 6
|
| 184.6
| 2‡
|
|-
| 7
| ·
| 215.4
| 3b
| 8/7
|-
| 8
|
| 246.2
| 2#
|
|-
| 9
|
| 276.9
| 3v
|
|-
| 10
|
| 307.7
| 3
| 6/5~7/6
|-
| 11
|
| 338.5
| 3‡
|
|-
| 12
| ·
| 369.2
| 4b
| 5/4
|-
| 13
|
| 400.0
| 3#
|
|-
| 14
|
| 430.8
| 4v (5b)
|
|-
| 15
|
| 461.5
| 4
|
|-
| 16
|
| 492.3
| 4‡ (5v)
|
|-
| 17
| ·
| 523.1
| 5
| 4/3~11/8
|-
| 18
|
| 553.8
| 5‡ (4#)
|
|-
| 19
|
| 584.6
| 6b
| 10/7
|-
| 20
|
| 615.4
| 5#
| 7/5
|-
| 21
|
| 646.2
| 6v
|
|-
| 22
| ·
| 676.9
| 6
| 3/2~16/11
|-
| 23
|
| 707.7
| 6‡
|
|-
| 24
|
| 738.5
| 7b
|
|-
| 25
|
| 769.2
| 6#
|
|-
| 26
|
| 800.0
| 7v
|
|-
| 27
| ·
| 830.8
| 7
| 8/5
|-
| 28
|
| 861.5
| 7‡
|
|-
| 29
|
| 892.3
| 8b
| 5/3~12/7
|-
| 30
|
| 923.1
| 7#
|
|-
| 31
|
| 953.8
| 8v
|
|-
| 32
| ·
| 984.6
| 8
| 7/4
|-
| 33
|
| 1015.4
| 8‡
|
|-
| 34
|
| 1046.2
| 9b
| 11/6~20/11
|-
| 35
|
| 1076.9
| 8#
|
|-
| 36
|
| 1107.7
| 9v (1b)
|
|-
| 37
|
| 1138.5
| 9
|
|-
| 38
|
| 1169.2
| 9‡ (1v)
|
|-
| 39
| ··
| 1200.0
| 1
| 2/1
|}

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|39.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 39df val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 62 -39 }}
| {{Mapping| 39 62 }}
| −1.81
| 1.81
| 5.88
|-
| 2.3.5
| 128/125, 1594323/1562500
| {{Mapping| 39 62 91 }}
| −3.17
| 2.42
| 7.89
|-
| 2.3.5.7
| 64/63, 126/125, 2430/2401
| {{Mapping| 39 62 91 110 }} (39d)
| −3.78
| 2.35
| 7.65
|-
| 2.3.5.7.11
| 64/63, 99/98, 121/120, 126/125
| {{Mapping| 39 62 91 110 135 }} (39d)
| −3.17
| 2.43
| 7.91
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-4 left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Temperament
! Mos scales
|-
| 1
| 1\39
| 30.8
|
|
|-
| 1
| 2\39
| 61.5
| [[Unicorn]] (39d)
| [[1L 18s]], [[19L 1s]]
|-
| 1
| 4\39
| 123.1
| [[Negri]] (39c)
| [[1L 8s]], [[9L 1s]], [[10L 9s]], [[10L 19s]]
|-
| 1
| 5\39
| 153.8
|
| [[1L 6s]], [[7L 1s]], [[8L 7s]], [[8L 15s]], [[8L 23s]]
|-
| 1
| 7\39
| 215.4
| [[Machine]] (39d)
| [[1L 4s]], [[5L 1s]], [[6L 5s]], [[11L 6s]], [[11L 17s]]
|-
| 1
| 8\39
| 246.2
| [[Immunity]] (39) / [[immunized]] (39d)
| [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]], [[5L 19s]], [[5L 24s]], [[5L 29s]]
|-
| 1
| 10\39
| 307.7
| [[Familia]] (39df)
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[4L 23s]], [[4L 27s]], [[4L 31s]]
|-
| 1
| 11\39
| 338.5
| [[Amity]] (39) / [[accord]] (39d)
| [[3L 1s]], [[4L 3s]], [[7L 4s]], [[7L 11s]], [[7L 18s]], [[7L 25s]]
|-
| 1
| 14\39
| 430.8
| [[Hamity]] (39df)
| [[3L 2s]], [[3L 5s]], [[3L 8s]], [[11L 3s]], [[14L 11s]]
|-
| 1
| 16\39
| 492.3
| [[Quasisuper]] (39d)
| [[2L 3s]], [[5L 2s]], [[5L 7s]], [[5L 12s]], [[17L 5s]]
|-
| 1
| 17\39
| 523.1
| [[Mavila]] (39bc)
| [[2L 3s]], [[2L 5s]], [[7L 2s]], [[7L 9s]], [[16L 7s]]
|-
| 1
| 19\39
| 584.6
| [[Pluto]] (39d)
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]] etc. … [[2L 35s]]
|-
| 3
| 1\39
| 30.8
|
|
|-
| 3
| 2\39
| 61.5
|
| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[3L 12s]], [[3L 15s]], [[18L 3s]]
|-
| 3
| 6\39
| 184.6
| [[Terrain]] / [[mirkat]] (39df)
| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15]], [[6L 21s]], [[6L 27s]]
|-
| 3
| 8\39 (5\39)
| 246.2 (153.8)
| [[Triforce]] (39)
| [[3L 3s]], [[6L 3s]], [[9L 6s]], [[15L 9s]]
|-
| 3
| 16\39 (3\39)
| 492.3 (92.3)
| [[Augene]] (39d)
| [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], [[12L 15s]]
|-
| 3
| 17\39 (4\39)
| 523.1 (123.0)
| [[Deflated]] (39bd)
| [[3L 3s]], [[3L 6s]], [[9L 3s]], [[9L 12s]], [[9L 21s]]
|-
| 13
| 16\39 (1\39)
| 492.3 (30.8)
| [[Tridecatonic]]
| [[13L 13s]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
39edo is a [[zeta valley edo]] and is generally poor at approximating primes for its size. Its poor approximations of harmonics 3, 5, 7, and 13 can all be improved by slightly [[octave shrinking|compressing the octave]], to get a tuning like [[ed6|101ed6]] or [[173zpi]].

39edo can be usefully mapped onto the val 39dfgijk. The [[Tenney-Euclidean]] tuning of this regular temperament is 30.67475 cents per step, which is closely approximated by [[62edt]] and 173zpi.

== 39edo and world music ==
Some might consider 39edo a candidate for a "universal tuning" in that it offers reasonable approximations of many different world music [[approaches to musical tuning|traditions]]; it is one of the simplest edos that can make this claim. Because of this, composers wishing to combine multiple world music traditions (for example, [[gamelan]] with [[maqam]] singing) within one unified framework might find 39edo an interesting possibility.

=== Western ===
39edo offers not one, but several different ways to realize the traditional Western diatonic scale. One way is to simply take a [[chain of fifths]] (the diatonic mos: 7 7 2 7 7 7 2). Because 39edo is a [[superpyth]] rather than a [[meantone]] system, this means that the harmonic quality of its diatonic scale will differ somewhat, since "minor" and "major" triads now approximate 6:7:9 and 14:18:21 respectively, rather than 10:12:15 and 4:5:6 as in meantone diatonic systems. Diatonic compositions translated onto this scale thus acquire a wildly different harmonic character, albeit still pleasing.

Another option is to use a [[modmos]], such as 7 6 3 7 6 7 3; this scale enables us to continue using [[5-limit|pental]] rather than [[7-limit|septimal]] thirds, but it has a false ([[wolf interval|wolf]]) fifth. When translating diatonic compositions into this scale, it is possible to avoid the wolf fifth by introducing accidental notes when necessary. It is also possible to avoid the wolf fifth by extending the scale to either 7 3 3 3 7 3 3 7 3 (a [[modmos]] of type [[3L 6s]]) or 4 3 6 3 4 3 6 4 3 3. There are other modmos scales that combine both pental and septimal harmonies. As such, a single Western classical or pop composition can be translated into 39edo in ''many'' different ways, acquiring a distinctly different but still harmonious character each time.

The mos and the modmos scales all have smaller-than-usual [[semitone (interval region)|semitones]], which makes them more effective for melody than their counterparts in 12edo or meantone systems.

Because 39edo and 12edo both have an overall sharp character and share the same major third, they have a relatively similar sound. Thus, 39edo (unlike, say, 22edo or 19edo, which are both "acquired tastes") does not sound all that [[xenharmonic]] to people used to 12edo. Check out [https://www.prismnet.com/~hmiller/midi/canon39.mid Pachelbel's Canon in 39edo] (using the 7 6 3 7 6 7 3 modmos), for example.

=== Indian ===
A similar situation arises with [[Indian music]] since the sruti system, like the Western system, also has multiple possible mappings in 39edo. Many of these are modified versions of the [[17L 5s]] MOS (where the generator is a perfect fifth).

=== Arabic, Turkish, Iranian ===
While [[Arabic, Turkish, Persian music|middle-eastern music]] is commonly approximated using [[24edo]], 39edo offers a potentially better alternative. [[17edo]] and 24edo both satisfy the "Level 1" requirements for [[maqam]] tuning systems. 39edo is a Level 2 system because:

* It has two types of "neutral" seconds (154 and 185 cents)
* It has two minor seconds (92 and 123 cents), which when added together give a whole tone (215 cents)

whereas neither 17edo nor 24edo satisfy these properties.

39edo will likely be more suited to some middle-eastern scales than others. Specifically, Turkish music (in which the Rast makam has a "major-like" wide neutral third and a wide "neutral" second approaching 10/9), will likely be especially well suited to 39edo.

=== Blues / Jazz / African-American ===
The [[harmonic seventh]] ("[[barbershop]] seventh") [[tetrad]] is reasonably well approximated in 39edo, and some temperaments (augene in particular) give scales that are liberally supplied with them. John Coltrane might have loved augene (→ [[Wikipedia: Coltrane changes]]).

[[Tritone]] substitution, which is a major part of jazz and blues harmony, is more complicated in 39edo because there are two types of tritones. Therefore, the tritone substitution of one seventh chord will need to be a different type of seventh chord. However, this also opens new possibilities; if the substituted chord is of a more consonant type than the original, then the tritone substitution may function as a ''resolution'' rather than a suspension.

Blue notes, rather than being considered inflections, can be notated as accidentals instead; for example, a "blue major third" can be identified as either of the two neutral thirds. There are two possible [[mapping]]s for [[7/4]] which are about equal in closeness. The sharp mapping is the normal one because it works better with the [[5/4]] and [[3/2]], but using the flat one instead (as an accidental) allows for another type of blue note.

=== Other ===
39edo offers approximations of [[pelog]] and [[mavila]] using the flat fifth as a generator. Pelog can also be approximated as 4 5 13 4 13.

It also offers ''many'' possible [[pentatonic]] scales, including the [[2L 3s]] mos (which is 9 7 7 9 7). [[Slendro]] can be approximated using that scale or using something like the [[quasi-equal]] 8 8 8 8 7 or 8 8 7 8 8.

One expressive [[pentatonic]] scale is the oneirotonic subset 9 6 9 9 6.

Many Asian{{clarify|which ones specifically}} and [[African music|African]] {{clarify|which ones specifically}} musical styles can thus be accommodated.

== Scales ==
* [[Quasisuper]][7] [[MOS scale]]: 7 7 2 7 7 7 2
* Quasisuper[7] [[5-limit|pental]] [[modmos]]: 7 6 3 7 6 7 3
* [[3L 6s]] modmos: 7 3 3 3 7 3 3 7 3
* Extended quasisuper: 4 3 6 3 4 3 6 4 3 3
* Quasisuper[22] MOS scale (resembles [[Indian]] [[sruti]]): 2 2 1 2 2 2 1 2 2 2 1 2 2 2 2 1 2 2 2 1 2 2
* Slendro approximations: 9 7 7 9 7 or 8 8 8 8 7 or 8 8 7 8 8
* An expressive [[oneirotonic]] subset: 9 6 9 9 6
* ''The scales listed in: [[User:BudjarnLambeth/Quasipelog theory]]''

== Instruments ==
=== Lumatone mapping ===
See [[Lumatone mapping for 39edo]]

=== Skip fretting ===
'''Skip fretting system 39 2 5''' is a [[skip-fretting]] system for [[39edo]]. All examples on this page are for 7-string [[Guitar|guitar.]]

; Prime harmonics
1/1: string 2 open

2/1: string 5 fret 12 and string 7 fret 7

3/2: string 3 fret 9 and string 5 fret 4

5/4: string 1 fret 9 and string 3 fret 4

7/4: string 5 fret 8 and string 7 fret 3

11/8: string 2 fret 9 and string 4 fret 4

=== Prototypes ===
[[File:TECLADO_39-EDD.PNG|alt=TECLADO 39-EDD.PNG|800x467px|TECLADO 39-EDD.PNG]]

''An illustrative image of a 39edo keyboard''

[[File:Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|alt=Custom_700mm_5-str_Tricesanonaphonic_Guitar.png|826x203px|Custom_700mm_5-str_Tricesanonaphonic_Guitar.png]]

''39edo fretboard visualization''

== Music ==
=== Modern renderings ===
; {{W|HOYO-MiX}}
* [https://www.youtube.com/shorts/4y11CWLIHNA "Sinner's Finale" from ''Genshin Impact OST''] (2023) – covered by [[Bryan Deister]] (2025)

=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/oeFI957W-xg ''39edo''] (2023)
* [https://www.youtube.com/watch?v=XLRaG_pBN7k ''39edo jam''] (2025)
* [https://www.youtube.com/shorts/1T_xrZpUslQ ''39edo improv''] (2025)
* [https://www.youtube.com/watch?v=kYQyRY7xFJs ''Waltz in 39edo''] (2025)
* [https://www.youtube.com/watch?v=Vzife15uUU4 ''Tilt Your Head Down''] (2026)

; [[groundfault]]
* From ''Souvenirs of the Affliction'' (2025) – [https://groundfco.bandcamp.com/album/souvenirs-of-the-affliction Bandcamp] | [https://www.youtube.com/watch?v=rrjuGmmodn0 YouTube]
** "Resolute Prelude"
** "Residual Soliloquy"

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=Q9wQV1J5eLE ''Romance On Other Planets''] (2021)

[[Category:Listen]]
{{Todo|add scales list}}

Garibaldi

2026-05-30T21:43:06Z

Overthink: /* As a detemperament of 12et */ I don't know about linking "MOS" as MOSDiagrams. Also generally lowercase is preferred.

{{Infobox regtemp
| Title = Garibaldi
| Subgroups = 2.3.5.7, 2.3.5.7.19
| Comma basis = [[225/224]], [[3125/3087]] (7-limit); [[190/189]], [[225/224]], [[361/360]] (2.3.5.7.19)
| Mapping = 1; 1 -8 -14 -3
| Edo join 1 = 41 | Edo join 2 = 53
| Generators = 3/2
| Generators tuning = 702.10
| Optimization method = CWE
| Pergen = (P8, P5)
| MOS scales = [[5L 2s]], [[5L 7s]], [[12L 5s]], [[12L 17s]]
| Odd limit 1 = 9 | Mistuning 1 = 4.33 | Complexity 1 = 17
| Odd limit 2 = 2.3.5.7.19 21 | Mistuning 2 = 4.65 | Complexity 2 = 17
}}
'''Garibaldi''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family #Garibaldi|schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|helmholtz]] temperament beyond the 5-limit but with the same simple [[chain of fifths|chain-of-fifths]] structure (so that [[chain-of-fifths notation|standard notation]] may be used). The garibaldi temperament tempers together the Pythagorean, syntonic, and archytas commas into a jack-of-all-trades "generic comma", which can be used to reach intervals of 3, 5, and 7. As in helmholtz temperament, [[5/4]] is mapped to the diminished fourth (e.g. C–F♭; a comma-flat major third), and the new mapping specific to garibaldi is that [[7/4]] is mapped to the double-diminished octave (e.g. C–C𝄫; a comma-flat minor seventh). This makes garibaldi a [[marvel temperaments|marvel]] and [[hemifamity temperaments|hemifamity]] temperament. Tuning the fifth a fraction of a cent sharp gives the best tunings.

Immediate 11-limit extensions include '''cassandra''' ({{nowrap| 41 & 53 }}), mapping 11/8 to +23 fifths, '''andromeda''' ({{nowrap| 29 & 41 }}), mapping 11/8 to −18 fifths, and '''helenus''' ({{nowrap| 53 & 65d }}), mapping 11/8 to −30 fifths. Garibaldi is most naturally a 2.3.5.7.19-[[subgroup]] temperament due to its immediate availability of [[19/16]] at the minor third (C–E♭). This is sometimes known as ''garibaldi nestoria.''

Garibaldi was named in honor of [[Eduardo Sábat-Garibaldi]], who developed the [[dinarra]], a 53-tone [[microtonal guitar]] in the 1/9-schisma tuning.

See [[Schismatic family #Garibaldi]] for technical data.

== Interval chain ==
In the following table, odd harmonics 1–21 and their inverses are in '''bold'''.

{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="4" | Approximate ratios
|-
! rowspan="2" | 2.3.5.7.19 subgroup
! colspan="3" | 13-limit extensions
|-
! Cassandra
! Andromeda
! Helenus
|-
| 0
| 0.00
| '''1/1'''
|
|
|
|-
| 1
| 702.10
| '''3/2'''
|
|
|
|-
| 2
| 204.20
| '''9/8'''
|
|
|
|-
| 3
| 906.30
| 27/16, '''32/19''', 42/25
| 22/13
| 22/13
| 22/13
|-
| 4
| 408.40
| 19/15, 24/19
|
| 14/11
|
|-
| 5
| 1110.50
| 19/10, 36/19, 40/21
|
| 21/11
|
|-
| 6
| 612.60
| 10/7
|
|
|
|-
| 7
| 114.70
| 15/14, '''16/15'''
|
| 14/13
|
|-
| 8
| 816.80
| '''8/5'''
|
| 21/13
|
|-
| 9
| 318.90
| 6/5
|
| 40/33
|
|-
| 10
| 1021.00
| 9/5, 38/21
|
| 20/11
|
|-
| 11
| 523.09
| 19/14, 27/20
|
| 15/11
|
|-
| 12
| 25.19
| 50/49, 57/56, 64/63, 81/80
|
| 40/39, 45/44
|
|-
| 13
| 727.29
| '''32/21'''
|
| 20/13
|
|-
| 14
| 229.39
| '''8/7'''
|
| 15/13
|
|-
| 15
| 931.49
| 12/7
|
| 19/11
|
|-
| 16
| 433.59
| 9/7
|
|
| 14/11
|-
| 17
| 1135.69
| 27/14, 48/25
| 52/27
| 64/33
| 21/11
|-
| 18
| 637.79
| 36/25, 81/56
| 13/9
| '''16/11''', 19/13
|
|-
| 19
| 139.89
| 27/25
| 13/12
| 12/11
| 14/13
|-
| 20
| 841.99
| 57/35, 80/49
| '''13/8''', 44/27
| 18/11, 64/39
| 21/13
|-
| 21
| 344.09
| 60/49
| 11/9, 39/32
| '''16/13''', 27/22
| 40/33
|-
| 22
| 1046.19
| 64/35
| 11/6
| 24/13
| 20/11
|-
| 23
| 548.29
| 48/35
| '''11/8''', 26/19
| 18/13
| 15/11
|-
| 24
| 50.39
| 36/35
| 33/32
| 27/26
| 40/39, 45/44
|-
| 25
| 752.49
| 54/35
|
|
| 20/13
|-
| 26
| 254.59
| 57/49, 81/70, 144/125
| 22/19
|
| 15/13
|-
| 27
| 956.69
| 171/98, 216/125, 256/147
| 26/15
|
| 19/11
|-
| 28
| 458.79
| 64/49
| 13/10
|
|
|-
| 29
| 1160.89
| 96/49
| 39/20, 88/45
|
| 64/33
|-
| 30
| 662.99
| 72/49
| 22/15
|
| '''16/11''', 19/13
|-
| 31
| 165.08
| 54/49
| 11/10
|
| 12/11
|-
| 32
| 867.18
| 81/49
| 33/20
|
| 18/11, 64/39
|-
| 33
| 369.28
| 216/175
| 26/21
|
| '''16/13''', 27/22
|-
| 34
| 1071.38
| 324/175
| 13/7
|
| 24/13
|-
| 35
| 573.48
| 243/175
|
|
| 18/13
|-
| 36
| 75.58
| 256/245
| 22/21
|
| 27/26
|-
| 37
| 777.68
| 384/245
| 11/7
|
|
|-
| 38
| 279.78
| 288/245
|
|
|
|-
| 39
| 981.88
| 432/245
|
|
|
|-
| 40
| 483.98
| 324/245
|
|
|
|-
| 41
| 1186.08
| 486/245
|
|
|
|}
<nowiki/>* In 2.3.5.7.19-subgroup CWE tuning

=== As a detemperament of 12et ===
[[File:Garibaldi 12et Detempering.png|thumb|Garibaldi as a 41-tone 12et detempering]]
[[File:Garibaldi-cassandra 12et Detempering.png|thumb|Garibaldi/cassandra as a 53-tone 12et detempering]]

Garibaldi is very naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]] (12et), where the chromatic scale becomes a near-equal [[5L 7s]]. The diagram on the right shows a 53-tone detempered scale, with a generator range of -26 to +26. 53 is the largest number of tones for a mos where the 12 categories never overlap.

Each pitch category of 12et is further divided into four or five qualities, separated by a [[pythagorean comma]], which represents the syntonic~septimal comma. Combining this division with the minor and major diatonic qualities of 12et, garibaldi can give up to ''eight'' qualities for each diatonic category. Taking thirds as an example:

In 12tet:

* 7/6~19/16~6/5 (minor)
* 5/4~19/15~9/7 (major)

In garibaldi (cassandra)

* ~[[7/6]] (subminor)
* '''~[[19/16]] (minor)'''
* ~[[6/5]] (superminor)
* ~[[11/9]] (artoneutral)
* ~[[27/22]] (tendoneutral)
* ~[[5/4]] (submajor)
* '''~[[19/15]] (major)'''
* ~[[9/7]] (supermajor)

Notice also the little interval between artoneutral and tendoneutral, ~[[243/242]]. This interval spans 41 generator steps. 41edo tempers it out so that it merges artoneutral and tendoneutral into a [[Sqrt(3/2)|hemififth]] whereas 53edo exaggerates it to the size of the generic comma. 94edo tunes it to one half the size of the general comma, which can be seen as a good compromise.

On another note, excluding 41edo, the two neutral intervals also have natural 13-limit interpretations in cassandra: 11/9~[[39/32]] and 27/22~[[16/13]], tempering out [[352/351]].

== Notation ==
Like in [[schismic]], it is recommended to adopt an additional module of accidentals such as arrows to represent the comma step. Garibaldi further benefits from this as the arrow also stands in for the septimal comma, so that the same inflection can be used to reach classical and septimal intervals alike.

The following tables show how to notate 11- and 13-limit intervals in each extension of garibaldi.

{| class="wikitable center-1 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Cassandra nomenclature for selected intervals
|-
! Ratio
! Nominal
! Example
|-
| 3/2
| Perfect fifth
| C–G
|-
| 5/4
| Downmajor third
| C–vE
|-
| 7/4
| Downminor seventh
| C–vBb
|-
| 11/8
| Double-up fourth
| C–^^F
|-
| 13/8
| Double-up minor sixth
| C–^^Ab
|-
| 19/16
| Minor third
| C–Eb
|}

{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Andromeda nomenclature for selected intervals
|-
! Ratio
! Nominal
! Example
|-
| 11/8
| Down diminished fifth Double-down augmented fourth
| C–vGb C–vvF#
|-
| 13/8
| Double downmajor sixth
| C–vvA
|}

{| class="wikitable center-1 center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Helenus nomenclature for selected intervals
|-
! Ratio
! Nominal
! Example
|-
| 11/8
| Double-down diminished fifth Triple-down augmented fourth
| C–vvGb C–v3F#
|-
| 13/8
| Triple-down major sixth
| C–v3A
|}

== Chords and harmony ==
Traditional tertian harmony is effective. The default triads on the Pythagorean spine are undevicesimal in quality:
* 1–19/15–3/2 (C–E–G)
* 1–19/16–3/2 (C–Eb–G)

Note that the major third also represents [[24/19]], and the minor third, [[13/11]]. These chords are typically associated with a sort of coldness and metalness, like those in [[12edo]] if not more so.

If a warm, sweet, laid-back sound is desired, the thirds can be inflected inwards by a comma to yield
* 1–5/4–3/2 (C–vE–G)
* 1–6/5–3/2 (C–^Eb–G)

Contrarily, for a more sour and active sound, they can be inflected outwards by a comma to yield
* 1–9/7–3/2 (C–^E-G)
* 1–7/6–3/2 (C–vEb-G)

== Scales ==
* [[Garibaldi5]] – proper [[2L 3s]]
* [[Garibaldi7]] – improper [[5L 2s]]
* [[Garibaldi12]] – proper [[5L 7s]]
* [[Garibaldi17]] – improper [[12L 5s]]
* [[Garibaldi24opt]] – optimized 24-note scale for 13-limit

== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.0589{{c}}
| CWE: ~3/2 = 702.0774{{c}}
| POTE: ~3/2 = 702.0852{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings (cassandra)
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 702.1192{{c}}
| CWE: ~3/2 = 702.1135{{c}}
| POTE: ~3/2 = 702.1125{{c}}
|}

=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (garibaldi)
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~3/2 = 702.2086{{c}}
| 7/6
| ~3/2 = 702.140{{c}}
| {{Monzo| 0 -25 11 35 }}
|-
| 9-odd-limit
| ~3/2 = 702.1928{{c}}
| 9/7
| ~3/2 = 702.114{{c}}
| {{Monzo| 0 -27 7 17 }}
|}

{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (cassandra)
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 702.1928{{c}}
| 9/7
| ~3/2 = 702.183{{c}}
| {{Monzo| 0 17 -52 -88 134 }}
|-
| 13-odd-limit
| ~3/2 = 702.1089{{c}}
| 13/7
| ~3/2 = 702.128{{c}}
| {{Monzo| 0 -38 -80 -122 137 116 }}
|-
| 15-odd-limit
| ~3/2 = 702.1089{{c}}
| 13/7
| ~3/2 = 702.112{{c}}
| {{Monzo| 0 -95 -137 -129 167 143 }}
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (andromeda)
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 702.6296{{c}}
| 11/9
|-
| 13-odd-limit
| ~3/2 = 702.7558{{c}}
| 13/9
|-
| 15-odd-limit
| ~3/2 = 702.7558{{c}}
| 13/9
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings (helenus)
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 11-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|-
| 13-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|-
| 15-odd-limit
| ~3/2 = 701.6435{{c}}
| 11/9
|}

=== Tuning spectra ===
==== Garibaldi ====
{| class="wikitable center-all left-4"
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9-odd-limit, 2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|
| 701.5385
| 65d val
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|
| 5/4
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|
| 9/5
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
| [[53edo|31\53]]
|
| 701.8868
|
|-
|
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
| [[94edo|55\94]]
|
| 702.1277
|
|-
|
| 9/7
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|
| 7/6
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| [[41edo|24\41]]
|
| 702.4390
|
|-
|
| 19/14
| 702.6079
|
|-
|
| 21/19
| 702.6732
|
|-
|
| 15/14
| 702.7775
|
|-
|
| 7/5
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''Upper bound of 9-odd-limit, 2.3.5.7.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 13/11
| 703.5968
|
|}

==== Cassandra ====
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|
| 701.5385
| 65def val
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|
| 5/4
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|
| 9/5
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
|
| 19/13
| 701.8702
|
|-
| '''[[53edo|31\53]]'''
|
| '''701.8868'''
| '''Lower bound of 11-, 13-, 15-odd-limit, 2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 15/13
| 701.9355
|
|-
|
| 13/10
| 701.9362
|
|-
|
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 13/8
| 702.0264
|
|-
|
| 13/12
| 702.0301
|
|-
|
| 36/35
| 702.0321
|
|-
|
| 13/9
| 702.0343
|
|-
|
| 19/11
| 702.0694
|
|-
|
| 11/10
| 702.0969
|
|-
|
| 15/11
| 702.1016
|
|-
|
| 13/7
| 702.1089
| 13- and 15-odd-limit minimax
|-
|
| 21/13
| 702.1135
|
|-
| [[94edo|55\94]]
|
| 702.1277
|
|-
|
| 9/7
| 702.1928
| 9- and 11-odd-limit minimax, 1/16 septimal schisma
|-
|
| 7/6
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 11/7
| 702.2295
|
|-
|
| 11/8
| 702.2312
|
|-
|
| 21/11
| 702.2371
|
|-
|
| 19/10
| 702.2399
|
|-
|
| 11/6
| 702.2438
|
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|
| 11/9
| 702.2575
|
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| '''[[41edo|24\41]]'''
|
| '''702.4390'''
| '''Upper bound of 11-, 13-, 15-odd-limit, 2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/14
| 702.6079
|
|-
|
| 21/19
| 702.6732
|
|-
|
| 15/14
| 702.7775
|
|-
|
| 7/5
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''29ef val, upper bound of 9-odd-limit diamond monotone'''
|-
|
| 13/11
| 703.5968
|
|}

==== Andromeda ====
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo generator
! Unchanged interval (eigenmonzo)*
! Generator (¢)
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
| [[65edo|38\65]]
|
| 701.5385
| 65deeff val
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|
| 5/4
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|
| 9/5
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
| [[53edo|31\53]]
|
| 701.8868
| 53ef val
|-
|
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
|
| 9/7
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|
| 7/6
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| '''[[41edo|24\41]]'''
|
| '''702.4390'''
| '''Lower bound of 13-, 15-odd-limit, 2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/14
| 702.6079
|
|-
|
| 11/9
| 702.6296
| 11-odd-limit minimax
|-
|
| 11/6
| 702.6651
|
|-
|
| 21/19
| 702.6732
|
|-
|
| 11/8
| 702.7046
|
|-
|
| 13/9
| 702.7558
| 13- and 15-odd-limit minimax
|-
|
| 15/14
| 702.7775
|
|-
|
| 13/12
| 702.7922
|
|-
|
| 13/8
| 702.8320
|
|-
|
| 7/5
| 702.9146
|
|-
|
| 19/11
| 703.0797
|
|-
|
| 21/20
| 703.1066
|
|-
|
| 19/13
| 703.1659
|
|-
|
| 15/11
| 703.3592
|
|-
|
| 15/13
| 703.4101
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''Upper bound of 9-, 11-, 13-, 15-odd-limit, 2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 11/10
| 703.4996
|
|-
|
| 13/10
| 703.5220
|
|-
|
| 13/11
| 703.5968
|
|-
|
| 21/13
| 701.7817
|
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/11
| 703.8926
|
|-
|
| 13/7
| 704.0426
|
|-
|
| 11/7
| 704.3770
|
|}

==== Helenus ====
{| class="wikitable mw-collapsible mw-collapsed center-all left-4"
! Edo generator
! Unchanged interval (eigenmonzo)*
! Generator (¢)
! Comments
|-
| '''[[12edo|7\12]]'''
|
| '''700.0000'''
| '''Lower bound of 9- and 11-odd-limit diamond monotone'''
|-
|
| 19/16
| 700.8290
| 1/3 undevicesimal schisma
|-
|
| 11/7
| 701.0942
|
|-
|
| 19/12
| 701.1105
| 1/4 undevicesimal schisma
|-
|
| 21/11
| 701.1149
|
|-
|
| 13/7
| 701.4894
|
|-
|
| 21/13
| 701.5127
|
|-
| '''[[65edo|38\65]]'''
|
| '''701.5385'''
| '''65d val, lower bound of 13-, 15-odd-limit, 2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 11/10
| 701.5907
|
|-
|
| 15/11
| 701.6066
|
|-
|
| 11/8
| 701.6227
|
|-
|
| 11/6
| 701.6335
|
|-
|
| 11/9
| 701.6435
| 11-, 13-, and 15-odd-limit minimax
|-
|
| 15/8
| 701.6759
| 1/7 schisma
|-
|
| 19/11
| 701.7109
|
|-
|
| 5/4
| 701.7108
| 1/8 schisma
|-
|
| 25/24
| 701.7252
| 2/17 schisma
|-
|
| 5/3
| 701.7379
| 5-odd-limit minimax, 1/9 schisma
|-
|
| 9/5
| 701.7596
| 1/10 schisma
|-
|
| 81/80
| 701.7922
| 1/12 schisma
|-
|
| 13/8
| 701.8022
|
|-
|
| 13/12
| 701.8067
|
|-
|
| 13/9
| 701.8109
|
|-
|
| 13/10
| 701.8314
|
|-
|
| 15/13
| 701.8362
|
|-
| '''[[53edo|31\53]]'''
|
| '''701.8868'''
| '''Upper bound of 11-, 13-, 15-odd-limit, 2.3.5.7.11.13.19 subgroup 19- and 21-odd-limit diamond monotone'''
|-
|
| 19/13
| 701.8995
|
|-
|
| 3/2
| 701.9550
| Pythagorean tuning
|-
|
| 36/35
| 702.0321
|
|-
|
| 9/7
| 702.1928
| 9-odd-limit minimax, 1/16 septimal schisma
|-
|
| 7/6
| 702.2086
| 7-odd-limit minimax, 1/15 septimal schisma
|-
|
| 49/48
| 702.2174
| 2/29 septimal schisma
|-
|
| 7/4
| 702.2267
| 1/14 septimal schisma
|-
|
| 19/10
| 702.2399
|
|-
|
| 21/16
| 702.2476
| 1/13 septimal schisma
|-
|
| 64/63
| 702.2720
| 1/12 septimal schisma
|-
|
| 19/15
| 702.3111
|
|-
| [[41edo|24\41]]
|
| 702.4390
| 41ef val
|-
|
| 19/14
| 702.6079
|
|-
|
| 21/19
| 702.6732
|
|-
|
| 15/14
| 702.7775
|
|-
|
| 7/5
| 702.9146
|
|-
|
| 21/20
| 703.1066
|
|-
| '''[[29edo|17\29]]'''
|
| '''703.4483'''
| '''29eeff val, upper bound of 9-odd-limit diamond monotone'''
|-
|
| 13/11
| 703.5968
|
|}
<nowiki/>* Besides the octave

[[Category:Garibaldi| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Schismatic family]]
[[Category:Marvel temperaments]]
[[Category:Gariboh clan]]
[[Category:Hemifamity temperaments]]

4L 3s

2026-05-30T20:06:33Z

Overthink: /* Scale tree */ add this

{{Interwiki
|en=4L 3s
|es=
|de=
|ja=4L 3s
}}
{{Infobox MOS}}

{{MOS intro}}
4L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], where one large step of diatonic ([[5L 2s]]) is replaced with a small step.

== Name ==
{{TAMNAMS name}}

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
{{MOS intervals}}

=== Generator chain ===
{{MOS genchain}}

=== Modes ===
{{MOS mode degrees}}

==== Proposed names ====
Alexandru Ianu ([[User:Ayceman|Ayceman]])<ref>Description of ''Sylvian Moon Dance'' mentioning the naming proposal https://musescore.com/user/36772625/scores/6700443 – The theme relates to the mystical nature of the Tribunal and TES lore, which fits smitonic.</ref> has proposed the following mode names relating to the Almsivi in Morrowind (TES):
{{MOS modes
| Mode Names=Nerevarine $
Vivecan $
Lorkhanic $
Sothic $
Kagrenacan $
Almalexian $
Dagothic $
}}

== Theory ==
=== Low harmonic entropy scales ===
There are two notable harmonic entropy minima:
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1.
* [[myna|Myna temperament]], in which the generator is also 6/5 but it takes 10 of them to make a 6/1, meaning that a larger MOS than 4L 3s is required to reach 3/2 or 4/3.

=== Temperament interpretations ===
{{main|4L 3s/Temperaments}}
4L 3s has the following temperament interpretations:
* [[Sixix]], with generators around 338.6{{c}}.
* [[Orgone]], with generators around 323.4{{c}}.
* [[Kleismic]], with generators around 317{{c}}.

Other temperaments, such as [[amity]] and [[myna]], require more than 7 pitches to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.

== Tuning ranges ==
{{Todo|Populate|comment=Populate with JI ratios from prior edits of this page.|inline=1}}

=== Simple tunings ===
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 11edo, 15edo, and 18edo, respectively.
{{MOS tunings}}

=== Parasoft tunings ===
Parasoft smitonic tunings can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:

* The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
* The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.

These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702{{c}}), and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.

Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:

* 18edo can be used to make the large and small steps more distinct, or can be considered a distorted 19edo diatonic.
** 18edo has a major 1-mosstep that is close to 9/8 (203{{c}}).
** 18edo's major and minor 4-mossteps are both equally off from 12edo's diatonic perfect 5th (700{{c}}) by 33.3{{c}}.
** 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* The augmented 2-mosstep of 25edo is very close to 5/4 (386{{c}}).
{{MOS tunings|Step Ratios=3/2; 7/5; 4/3}}

=== Hyposoft tunings ===
Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327{{c}} and 333{{c}}. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic".

Edos include [[11edo]] (not shown), [[18edo]], and [[29edo]].

{{MOS tunings|Step Ratios=3/2; 5/3; 7/4}}

=== Hypohard tunings===
Hypohard smitonic tunings (2:1 to 3:1) have generators between 320{{c}} and 327{{c}}. The major 1-mosstep, or large step, tends to approximate [[8/7]] (231{{c}}) and the major 3-mosstep tends to approximate [[11/8]] (551{{c}}). [[26edo]] approximates these two intervals very well. These JI approximations are associated with [[orgone]] temperament.

Other hypohard edos include [[11edo]] (not shown), [[15edo]] and [[37edo]].

{{MOS tunings|Step Ratios=3/1; 5/2; 7/3}}

=== Parahard tunings ===
Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9{{c}} and 320{{c}}, putting it close to a pure 6/5 (316{{c}}). Stacking six generators and octave-reducing approximates 3/2 (702{{c}}), a diatonic perfect 5th, represented by the diminished 5-mosstep.

This range contains very accurate edos such as [[53edo]] and [[72edo]], and has very accurate approximations to many [[low-overtone JI]] intervals, namely basic [[5-limit]] ratios and some ratios involving 13. However, 4L 3s only has one interval of 3/2, so it's suggested to use a larger MOS, such as [[4L 7s]], to achieve 5-limit harmony.

These JI approximations are associated with [[kleismic]] temperament, through the 2.3.5.13 extension known as [[Kleismic family#Cata|cata]].

Parahard edos smaller than 53edo include [[15edo]] (not shown), [[19edo]], and [[34edo]].

{{MOS tunings|Step Ratios=4/1; 11/3; 7/2}}

== Scales ==
* [[Orgone7]]
* [[Cata7]]
* [[Myna7]]

== Scale tree==
{{MOS tuning spectrum
| 6/5 = [[Amity]]/[[hitchcock]] ↑
| 5/4 = [[Sixix]]
| 4/3 = [[Supramin]]
| 13/8 = Golden 4L 3s (868.3282{{c}})
| 12/5 = [[Hyperkleismic]]
| 5/2 = [[Orgone]]
| 13/5 = Golden superkleismic
| 8/3 = [[Superkleismic]]
| 11/3 = [[Hanson]]/[[keemun]]
| 6/1 = [[Oolong]]/[[myna]] ↓
}}

== Music ==
* [[City of the Asleep]], [https://cityoftheasleep.bandcamp.com/album/an-amputated-elliptic-knob-of-the-cryptocurve-regenerates "An Amputated Elliptic Knob of the Cryptocurve Regenerates"] (Various orgone edos)
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

== References ==
<references />

[[Category:Smitonic|*]] 
[[Category:7-tone scales]]

392/391

2026-05-30T20:02:30Z

Overthink: /* Etymology */ link

{{Infobox Interval
| Name = minor semivicema
| Color name = 23u17uzz1, twethusuzozo 1sn, Twethusuzozo comma
| Comma = yes
}}
'''392/391''', the '''minor semivicema''' is a [[23-limit]] [[Superparticular ratio|superparticular comma]] measuring about 4.42 [[cent]]s. It forms the difference between [[17/14]] (septendecimal semi-minor third) and [[28/23]] (vicesimotertial neutral third).

== Etymology ==
The name ''minor semivicema'' is contrast to the ''major semivicema'', [[323/322]], measuring about 5.37¢, which is the difference between 17/14 and [[23/19]]. 392/391 is smaller than 323/322 by a 19-limit [[unnoticeable comma]], [[5491/5488]]. The word ''semivicema'' was introduced by [[User:Xenllium|Xenllium]] in 2023. It is a contraction of ''semi-minor vicesimotertial comma'' into a single word.

== See also ==
* [[Small comma]]
* [[List of superparticular intervals]]

[[Category:Semivicemic]]
[[Category:Commas named for the intervals they stack]]

323/322

2026-05-30T20:02:15Z

Overthink: /* Etymology */ link

{{Infobox Interval
| Name = major semivicema
| Color name = 23u19o17or1, twethunosoru 1sn, Twethunosoru comma
| Comma = yes
}}
'''323/322''', the '''major semivicema''' is a [[23-limit]] [[Superparticular ratio|superparticular comma]] measuring about 5.37 [[cent]]s. It forms the difference between [[17/14]] (septendecimal semi-minor third) and [[23/19]] (vicesimotertial semi-minor third).

== Etymology ==
The name ''major semivicema'' is contrast to the ''minor semivicema'', [[392/391]], measuring about 4.42¢, which is the difference between 17/14 and [[28/23]]. 323/322 is larger than 392/391 by a 19-limit [[unnoticeable comma]], [[5491/5488]]. The word ''semivicema'' was introduced by [[User:Xenllium|Xenllium]] in 2023. It is a contraction of ''semi-minor vicesimotertial comma'' into a single word.

== See also ==
* [[Small comma]]
* [[List of superparticular intervals]]

[[Category:Semivicemic]]
[[Category:Commas named for the intervals they stack]]

1308edo

2026-05-30T20:01:41Z

Overthink: links

{{Infobox ET}}
{{ED intro}}

1308edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], and is the 15th [[zeta gap edo]]. With [[23/17]] and [[27/23]] barely missing the line, it has reasonable approximations up to the 37-limit.

It [[tempering out|tempers out]] {{monzo| 37 25 -33 }} (whoosh comma) and {{monzo| -46 51 -15 }} (171 & 1137 comma) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and {{monzo| 47 4 0 -19 }} in the [[7-limit]]; [[9801/9800]], [[151263/151250]], [[234375/234256]], and 67110351/67108864 in the [[11-limit]]; [[4225/4224]], [[6656/6655]], 50193/50176, 91125/91091, and 655473/655360 in the 13-limit; [[2601/2600]], [[5832/5831]], [[11016/11011]], 11271/11264, [[12376/12375]], and 108086/108045 in the 17-limit; [[5491/5488]], [[5776/5775]], [[5985/5984]], [[6175/6174]], 10241/10240, and 10830/10829 in the 19-limit.

=== Prime harmonics ===
{{Harmonics in equal|1308|columns=12}}

=== Subsets and supersets ===
Since 1308 factors into primes as {{nowrap| 22 × 3 × 109 }}, 1308edo has subset edos {{EDOs| 2, 3, 4, 6, 12, 109, 218, 327, 436, and 654 }}.

5491/5488

2026-05-30T20:00:28Z

Overthink: create comma page

{{Infobox interval
| Name = Supraminisma
| Comma = yes
}}
'''5491/5488''', the '''supraminisma''', is a [[19-limit]] [[unnoticeable comma]] measuring about 0.95 cents in size. It is the difference between two [[17/14]] supraminor thirds and a [[28/19]] narrow fifth.

== Temperaments ==
[[Tempering out]] this comma in the full 19-limit leads to the rank-7 '''supraminismic''' temperament, and tempering out in the 2.7.17.19-subgroup leads to the '''supraminic''' temperament. The most notable temperament tempering out this comma is the 2.17/7.19/7-subgroup '''[[supramin]]''' temperament, which approximates the 14:17:19 triad with low complexity and high accuracy, and is tuned well by [[25edo]].

== Etymology ==
This comma was named by [[User:Overthink|Overthink]] in 2026 together with the supramin temperament, referring to the fact that supramin is generated by a supraminor third.

[[Category:Commas named for their regular temperament properties]]

Subgroup temperaments

2026-05-30T19:51:28Z

Overthink: /* 2.….19/7.… subgroups */ add a temperament

{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.

For temperaments that omit various prime harmonics, see:
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].

= Composite subgroup temperaments =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.165

==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

: [[gencom]]: [2 9/8; 225/224 325/324 640/637]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.3.25 subgroup ==

=== Shrub ===
This is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a [[shrubmajor]] third (~425c) in quarter-comma tuning.

Subgroup: 2.3.25

Edo join: 17 & 12

Comma list: [[2048/2025]]

{{Mapping|legend=2| 1 1 7| 0 1 -4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.136

==== 2.3.23.25.41 subgroup ====
''See also: [[Reversed meantone]]''

Edo join: 17 & 12

Comma list: 2048/2025, 576/575, 82/81

{{Mapping|legend=2| 1 1 1 7 3| 0 1 6 -4 4}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 705.264

===== Sburb =====
This temperament sets the 413th harmonic (octave-reduced) to the diminished seventh.

Subgroup: 2.3.7.23.25.41.59

Edo join: 17 & 12

Comma list: 64/63, 225/224, 162/161, 82/81, 177/175

{{Mapping|legend=2| 1 1 4 1 7 3 10| 0 1 -2 6 -4 4 -7}}

Optimal tuning (CTE): ~2 = 1\1, ~3/2 = 706.387

== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

[[Badness]]: 0.00263

=== Joan ===
{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

{{Mapping|legend=2|1 1 2|0 3 2}}

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

=== Genius ===

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

{{Mapping|legend=2| 1 2 4 | 0 3 1 }}

: sval mapping generators: ~2, ~21/16

{{Mapping|legend=3| 1 1 0 3 | 0 3/2 0 -1/2 }}

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

=== Tetrameantone ===
{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

=== Tetramagic ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

{{Mapping|legend=2| 1 0 1 | 0 5 1 }}

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

{{Todo|inline=1|cleanup}}

<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)
⟨2399.3973, 193.8643]

TE Step Tunings (cents)
⟨25.21211, 47.81337]

TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]

TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]

Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave

Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩

TE Generator Tunings (cents)
⟨2399.4195, 193.8654]

TE Step Tunings (cents)
⟨25.23883, 47.79592]

TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]

TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave

Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)
⟨2400.1097, 193.9498]

TE Step Tunings (cents)
⟨24.18752, 48.52491]

TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave

Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>

==== 4.6.5.7.11.13 ====

<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩

TE Generator Tunings (cents)
⟨2401.2305, 193.5378]

TE Step Tunings (cents)
⟨42.79107, 35.98524]

TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>

==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)
⟨2400.4701, 193.4599]

TE Step Tunings (cents)
⟨43.39350, 35.55764]

TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>

==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩

TE Generator Tunings (cents)
⟨2399.9219, 193.3952]

TE Step Tunings (cents)
⟨44.14256, 35.03670]

TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>

==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

TE Generator Tunings (cents)
⟨2399.3286, 193.5316]

TE Step Tunings (cents)
⟨37.31613, 39.63311]

TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>

== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

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

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

== 8.9.7 subgroup ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

{{Mapping|legend=2| 1 0 2 | 0 1 -1 }}

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10-3

= Fractional subgroup temperaments =
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

{{Mapping|legend=2| 1 0 5 2 | 0 1 -3 2 }}
: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

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

: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====
{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

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

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====
{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====
{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===
{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

{{Mapping|legend=2| 1 0 0 | 0 2 1 }}

{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}
: <nowiki/>* wart for 5/3
: † wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

[[Subgroup]]: 2.5/3.13/9

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

{{Mapping|legend=2| 1 0 2 | 0 1 -2 }}

{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])3 = [[13/10]] and ([[15/14]])3 = [[16/13]], therefore tempering out {S11/S12/S14/S15}. However, note that it is not supported by the 31 & 53 orwell extension dubbed "tridecimal orwell", but instead the less accurate [[winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.

[[Subgroup]]: 2.5.7/3.11/3.13

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})

{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} using subgroup TE 
: <nowiki/>* wart for 7/3

Badness (Sintel): 0.329

==== Laz ====
{{See also | Chromatic pairs #Laz }}

Laz is related to [[avalokita]] as well as to [[winston]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}

{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}

Kryptonite is related to [[krypton]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12

{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}

Kiribati is related to [[nakika]] as well as to [[octacot]].

[[Subgroup]]: 2.9/5.7/3.11/9

[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})

{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20

{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents

=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.

[[Subgroup]]: 2.7/3.11

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

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

{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents

== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.

[[Subgroup]]: 2.5.9/7

[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})

{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5

{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents

==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[50/49]]

{{Mapping|legend=2| 2 3 1 | 0 1 0 }}

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}

{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830

{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
 based on subgroup TE 

Badness (Sintel): 0.119

==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}

Edson is related to [[pele]] and [[andromeda]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3

{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

{{Optimal ET sequence|legend=1| 12, 17, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

==== Haumea ====
{{See also| Chromatic pairs #Haumea }}

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}

{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: 364/363, 441/440, 1001/1000

{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

[[Subgroup]]: 2.7/5.9/5

[[Comma list]]: [[250047/250000]]

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

{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461

{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents

=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].

[[Subgroup]]: 2.7/5.11/5.13/5

[[Comma list]]: [[847/845]], [[1001/1000]]

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

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents

==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

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

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95t, 124t }}
: t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents

== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

[[Subgroup]]: 2.11/5.13/5

[[Comma list]]: [[2200/2197]]

{{Mapping|legend=2| 1 0 1| 0 3 1 }}

{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012

{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents

==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168, 540/539, 729/728

{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents

=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].

Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].

[[Subgroup]]: 2.3.7.11/5

[[Comma list]]: 1029/1024, 4000/3993

{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742

{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 1029/1024, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918

{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17

[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19

[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
: <nowiki />* wart for 11/7
: † wart for 13/7

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====
{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

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

{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

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

[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5

=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]

[[Subgroup]]: 2.3.49/5

[[Comma list]]: 2401/2400

{{Mapping|legend=2| 1 1 3 | 0 2 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

[[Tp tuning #T2 tuning|RMS error]]: ?

== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

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

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{Monzo| -3 1 1 }})

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

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

<nowiki/>* wart for 19/7

[[Badness]] (Sintel): 0.082

=== Supramin ===
This is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by [[User:Overthink|Overthink]] in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]].

[[Subgroup]]: 2.17/7.19/7

[[Comma list]]: [[5491/5488]] ({{Monzo| -4 2 1 }})

{{Mapping|legend=2| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.022{{c}}, ~17/14 = 335.793{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000{{c}}, ~17/14 = 335.785{{c}}

{{Optimal ET sequence|legend=1| 7, 18, 25 }}

[[Badness]] (Sintel): 0.005

==== Supramine ====
This extension approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

Subgroup: 2.17/7.19/7.23/7

Comma list: [[323/322]], [[392/391]]

Subgroup-val mapping: {{Mapping| 1 0 4 3 | 0 1 -2 -1 }}

Optimal tunings:
* Subgroup WE: ~2 = 1199.871{{c}}, ~17/14 = 336.243{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 336.296{{c}}

{{Optimal ET sequence|legend=0| 7, 18, 25 }}

Badness (Sintel): 0.029

==== 2.25/7.17/7.19/7.23/7 subgroup ====

Subgroup: 2.25/7.17/7.19/7.23/7

Comma list: [[323/322]], [[392/391]], [[476/475]]

Subgroup-val mapping: {{Mapping| 1 -2 0 4 3 | 0 3 1 -2 -1 }}

Optimal tunings:
* Subgroup WE: ~2 = 1199.757{{c}}, ~17/14 = 335.428{{c}}
* Subgroup CWE: ~2 = 1200.000{{c}}, ~17/14 = 335.479{{c}}

{{Optimal ET sequence|legend=0| 7, 18, 25 }}

Badness (Sintel): 0.053

== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}

=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: [[10976/10935]]

{{Mapping|legend=2| 1 2 3 | 0 3 1 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909

[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

=== Halftone ===
{{Main| Halftone }}

[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: 9604/9375

{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2

[[Comma list]]: 1232/1215, 27783/27500

{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2

[[Comma list]]: 275/273, 1232/1215, 1323/1300

{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2

=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4

[[Comma list]]: 245/243

{{Mapping|legend=2| 1 1 2 | 0 2 -1 }}

: sval mapping generators: ~3/2, ~9/7

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728

[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]

==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 100/99, 245/243

{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771

[[Optimal ET sequence]]: [[8edf]], [[13edf]]

==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 245/243, 441/440

{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939

[[Optimal ET sequence]]: [[8edf]], [[11edf]]

== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''

[[Subgroup]]: 3/2.5/4.11/8

[[Comma list]]: 121/120

{{Mapping|legend=2| 1 1 1 | 0 2 -1 }}]

: [[gencom]]: [3/2 12/11; 121/120]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29

{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}

== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 87808/85293

{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}

: sval mapping generators: ~3/2, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2

=== Doubleton ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 1352/1323

{{Mapping|legend=2| 2 0 3 | 0 1 1 }}

: sval mapping generators: ~26/21, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2

== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11

[[Comma list]]: {{monzo| 11 1 -5 }}

{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}

: [[gencom]]: [5/2 125/88; 341796875/329832448]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2

= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments

[[Category:Subgroup temperaments| ]] 
[[Category:Temperament collections]]
{{Todo| review | cleanup }}

38edo

2026-05-30T17:55:25Z

Overthink: /* Theory */Grammar changes

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]], [[25/22]], and their [[octave complement]]s, while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are corrected. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]]s of all [[19-odd-limit]] intervals in 38df align with their closest approximations in 38edo, except for 7/4, 13/8, and their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo.

The harmonic series from 1 to 20 is approximated within 38df by the step sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
|
|
|
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
|
|
|
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]''
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]''
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
|
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]]
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
|
| ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
|
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
|
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[16/11]], [[22/15]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
|
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]]
| ''[[26/17]]'', ''[[11/7]]''
|
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
|
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
|
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]''
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
|
|
|
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
|
|
|
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

38edo

2026-05-30T17:50:53Z

Overthink: /* Theory */Mention that 38df preserves 19edo's 2.3.5.7.13 earlier

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] and [[25/22]], (and their inversions), while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where the [[2.3.5.7.13 subgroup|2.3.5.13-subgroup]] mapping of 19edo is preserved, while harmonics [[11/1|11]], [[17/1|17]], and [[19/1|19]] are corrected. In 38df, every [[odd harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, except for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping thus creates a natural full [[19-limit]] extension to the 2.3.5.7.13-subgroup mapping of 19edo.

The harmonic series from 1 to 20 is approximated within 38df by the step sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
|
|
|
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
|
|
|
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]''
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]''
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
|
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]]
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
|
| ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
|
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
|
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[16/11]], [[22/15]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
|
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]]
| ''[[26/17]]'', ''[[11/7]]''
|
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
|
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
|
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]''
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
|
|
|
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
|
|
|
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

38edo

2026-05-30T17:46:01Z

Overthink: /* Theory */Improve writing in 38df paragraph

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] and [[25/22]], (and their inversions), while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[prime harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, except for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their second-closest steps within 38df. The 38df mapping this creates a natural full [[19-limit]] extension to the [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping of 19edo.

The harmonic series from 1 to 20 is approximated within 38df by the step sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
|
|
|
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
|
|
|
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]''
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]''
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
|
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]]
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
|
| ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
|
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
|
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[16/11]], [[22/15]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
|
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]]
| ''[[26/17]]'', ''[[11/7]]''
|
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
|
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
|
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]''
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
|
|
|
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
|
|
|
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

38edo

2026-05-30T17:41:31Z

Overthink: /* Theory */Rewrite

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 38 factors as {{nowrap|2 × 19}}, 38edo can be thought of as two parallel chains of [[19edo]]. It provides a possible correction to the [[11/1|11th harmonic]] of 19edo, which works well with 19edo's flat approximations of the [[3/1|3rd]] and [[5/1|5th]] harmonics, making it a decent [[2.3.5.11 subgroup|2.3.5.11-subgroup]] system. Compared to 19edo, the halving of the step size lowers [[consistency]], and leaves it only mediocre in terms of overall [[relative interval error|relative error]]. However, the fact that the 3rd and 5th harmonics are flat by almost exactly the same amount, while the 11th is close to double that, means there are quite a few near-perfect composite ratios, such as the the [[6/5]] it shares with 19edo, plus [[11/9]], [[15/11]] and [[25/22]], (and their inversions), while a single step nears [[55/54]]. The approximation to [[11/9]] in particular should be noted for forming a 10-strong [[consistent circle]]. This gives several interesting possibilities for unusual near-just chords such as 15:18:22:25:30.

Using the [[patent val]], it [[tempering out|tempers out]] the same [[5-limit]] commas as 19edo, namely [[81/80]], [[3125/3072]] and [[15625/15552]]. In the [[7-limit]], we can add [[50/49]], and tempering out 81/80 and 50/49 gives [[injera]] temperament, for which 38 is the [[optimal patent val]] in the 7-limit. In the [[11-limit]], we can add [[121/120]] and [[176/175]], and in the [[13-limit]] we can add [[66/65]] and [[144/143]]. 38edo patently supports [[mohajira]] up to the 13-limit. While the [[7/1|7th]] and [[13/1|13th]] harmonics themselves are improved compared to 19edo, many other intervals involving these harmonics become less accurate, so whether 38edo actually corrects them is debatable.

Instead, the [[val]] {{val| 38 60 88 '''106''' 131 '''140''' 155 161 }} (38df in [[wart notation]]) can be used, where every [[prime harmonic]] from 3 to 19 is characterized by a flat intonation. Furthermore, the [[mapping]] of all [[19-odd-limit]] intervals in 38df aligns with their closest approximations in 38edo, excepting for 7/4 and 13/8, along with their octave complements 8/7 and 16/13, which are by definition mapped to their secondary optimal steps within 38df. Thus 38df creates a natural full [[19-limit]] extension to the [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] mapping of 19edo.

The harmonic series from 1 to 20 is approximated within 38df by the sequence: {{nowrap| 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3 }}

[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]

=== Prime harmonics ===
{{Harmonics in equal|38}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | Step
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]* ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the 2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
| 0
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
| 1
| 31.6
|
|
|
| Up 1sn
| ^1
| ^D
|-
| 2
| 63.2
|
|
|
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
| 3
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]''
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
| 4
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
| 5
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]''
|
| Mid 2nd
| ~2
| vE
|-
| 6
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
|
| Major 2nd
| M2
| E
|-
| 7
| 221.1
| [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
| 8
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
| 9
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 10
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
| 11
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
| 12
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
| 13
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
| 14
| 442.1
| [[22/17]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]]
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
| 15
| 473.7
|
| ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
| 16
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
| 17
| 536.8
| [[15/11]], [[11/8]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
| 18
| 568.4
|
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
| 19
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 20
| 631.6
|
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
| 21
| 663.2
| [[16/11]], [[22/15]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
| 22
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
| 23
| 726.3
|
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
| 24
| 757.9
| [[17/11]]
| ''[[26/17]]'', ''[[11/7]]''
|
| Aug 5th, dim 6th
| A5, d6
| A#
|-
| 25
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
| 26
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
| 27
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
| 28
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
| 29
| 915.8
| [[32/19]], [[17/10]]
|
|
| Upmajor 6th
| ^M6
| ^B
|-
| 30
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
| 31
| 978.9
| [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
| 32
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
|
| Minor 7th
| m7
| C
|-
| 33
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]''
|
| Mid 7th
| ~7
| ^C
|-
| 34
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
| 35
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
| 36
| 1136.8
|
|
|
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
| 37
| 1168.4
|
|
|
| Down 8ve
| v8
| vD
|-
| 38
| 1200.0
| [[2/1]]
|
|
| Perfect 8ve
| P8
| D
|}
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{Ups and downs sharpness}}

=== Quarter-tone notation ===
Since a sharp raises by two steps, [[24edo#Notation|quarter-tone accidentals]] can also be used:
{{sharpness-sharp2}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[17edo#Sagittal notation|17]], [[24edo#Sagittal notation|24]], and [[31edo#Sagittal notation|31]], is a subset of the notation for [[76edo#Sagittal notation|76-EDO]], and is a superset of the notation for [[19edo#Sagittal notation|19-EDO]].

==== Evo flavor ====
<imagemap>
File:38-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:38-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:38-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[33/32]]
default [[File:38-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}
{{Q-odd-limit intervals|37.9|apx=val|header=none|tag=none|title=15-odd-limit intervals by 38df val mapping}}

== Rank-2 temperaments ==
{| class="wikitable"
|+ [[Rank-2 temperament]]s in 38edo
|-
! Temperament !! Generator !! Periods per octave
|-
| [[Opossum]] || 5\38 || 1
|-
| [[Hemisensi]] || 7\38 || 1
|-
| [[Delorean]] / [[subkla]] || 9\38 || 1
|-
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
|-
| [[Hocus]] || 13\38 || 1
|-
| [[Buzzard]] || 15\38 || 1
|-
| [[Maquila]] / [[wilsec]] || 17\38 || 1
|-
| [[Bimeantone]] / [[injera]] || 3\38 || 2
|-
| [[Bison]] / [[hemikleismic]] || 5\38 || 2
|-
| [[Astrology]] / [[divination]] / [[horoscope]] || 7\38 || 2
|-
| [[Decimal]] || 8\38 || 2
|}

== Octave stretch or compression ==
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].

== Scales ==
; [[MOS scale]]s
* Astrology[22]: 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 2 1 2 2 2 1 2
* Buzzard[8]: 7 1 7 7 1 7 1 7
* Buzzard[13] 1 6 1 6 1 1 6 1 1 6 1 6 1
* Buzzard[18]: 1 5 1 1 1 5 1 1 1 5 1 1 5 1 1 1 5 1
* Buzzard[23]: 1 1 4 1 1 1 4 1 1 1 1 4 1 1 1 1 4 1 1 1 4 1 1
* Decimal[10]: 3 5 3 5 3 3 5 3 5 3
* Decimal[14]: 3 2 3 3 3 2 3 3 2 3 3 3 2 3
* Decimal[24]: 2 1 2 1 2 2 1 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 1 2
* Hocus[23]: 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 6 1 1 1
* Injera[6]: 3 13 3 3 13 3
* Injera[8]: 3 3 10 3 3 3 10 3
* Injera[10]: 3 3 7 3 3 3 3 7 3 3
* Injera[12]: 3 3 3 4 3 3 3 3 3 4 3 3
* Injera[14]: 3 3 3 1 3 3 3 3 3 3 1 3 3 3
* Injera[26]: 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1
* Maquila[20]: 1 3 1 3 1 3 1 3 1 1 3 1 3 1 3 1 3 1 3 1
* Mohajira[7] (''a.k.a. quasi-[[equiheptatonic]]''): 5 6 5 6 5 6 5
* Mohajira[10]: 5 1 5 5 1 5 5 5 1 5
* Mohajira[17]: 1 4 1 4 1 1 4 1 4 1 4 1 1 4 1 4 1
* Mohajira[24]: 1 3 1 1 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 1 3 1
* Subkla[13]: 2 5 2 2 5 2 2 2 5 2 2 5 2
* Subkla[17]: 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2
* Subkla[21]: 2 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 2 2 1 2 2

; MOS subsets
* ''of injera[12]''
** Quasi-major: 6 7 3 6 6 7 3
** Quasi-minor: 6 3 7 6 3 7 6

; [[MODMOS|MODMOS scales]]
{{Idiosyncratic terms}}
* ''of bison[22]''
** Tame bison: 3 1 1 1 1 3 3 1 1 1 3 3 1 1 1 3 3 1 1 1 1 3

* ''of hemisensi[11]''
** Hemisettled11: 3 3 3 4 3 6 3 3 3 4 3

* ''of hemisensi[16]''
** Hemisettled16: 5 1 3 3 1 3 1 1 3 1 5 1 3 3 1 3

* ''of opossum[23]''
** Tame possum: 3 3 2 2 2 3 2 2 2 3 2 2 2 3 3

; Others
{{Idiosyncratic terms}}
* [[Antipental blues]]: 9 7 2 4 9 7
* Ninteenplus: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2
* Quasi-[[equipentatonic]]: 8 8 6 8 8
* [[Well temperament|Well-tempered]] 19-in-38: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 2 2 2

== Instruments ==
* [[Lumatone mapping for 38edo]]
* [[Skip fretting system 38 2 11]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/rewy-32BfRs ''Spirit of the Night - Secret of Mana (microtonal cover in 38edo)''] (2025)
* [https://www.youtube.com/shorts/QcFEW45uxHY ''38edo improv''] (2025)
* ''waltz in 38edo'' (2026)
** [https://www.youtube.com/shorts/Gdx4hk7FKU0 <nowiki>[short]</nowiki>] (demonstrates Lumatone mapping)
** [https://www.youtube.com/watch?v=amukQrZuseY <nowiki>[full version]</nowiki>]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] (2022)

[[Category:38edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Listen]]

13/8

2026-05-30T17:13:51Z

Overthink: /* Approximation */- redundant parameter

{{Infobox Interval
| Name = (lesser) tridecimal neutral sixth
| Color name = 3o6, tho 6th
| Sound = jid_13_8_pluck_adu_dr220.mp3
}}

'''13/8''' is the '''(lesser) tridecimal neutral sixth''', which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In [[13-limit]] [[just intonation]], 13/8, as the octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of [[3/2]] and [[7/4]], separated from each by the [[superparticular]] ratios [[13/12]] and [[14/13]], respectively. The word "lesser" is added when necessary to differentiate it from [[64/39]], another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of [[5/4]] and [[13/10]].

13/8 differs from the Pythagorean minor sixth [[128/81]] by [[1053/1024]], about 48¢, from the classic minor sixth [[8/5]] by [[65/64]], about 27¢, from the undecimal neutral sixth [[18/11]] by [[144/143]], about 12¢, and from the rastmic neutral sixth [[44/27]] by [[352/351]], about 4.9¢.

== Approximation ==
13/8 is a fraction of a cent away from the neutral sixth found in the [[10edo|10''n''-edo]] family (7\10).

This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the [[golden ratio]]. In this case, 13/8 is ~7.4 [[cent|¢]] sharp of the golden ratio.
{{Interval edo approximation}}

== See also ==
* [[16/13]] – its [[octave complement]]
* [[64/39]] – the greater tridecimal neutral sixth
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Neutral sixth]]
[[Category:Golden ratio approximations]]

13/8

2026-05-30T17:12:55Z

Overthink: Grammar

{{Infobox Interval
| Name = (lesser) tridecimal neutral sixth
| Color name = 3o6, tho 6th
| Sound = jid_13_8_pluck_adu_dr220.mp3
}}

'''13/8''' is the '''(lesser) tridecimal neutral sixth''', which measures about 840.5¢, falling between the categories of minor sixth and major sixth. In [[13-limit]] [[just intonation]], 13/8, as the octave-reduced 13th harmonic, is treated as a basic component of harmony. In the harmonic series and in chords based on it, 13/8 sits between the more familiar consonances of [[3/2]] and [[7/4]], separated from each by the [[superparticular]] ratios [[13/12]] and [[14/13]], respectively. The word "lesser" is added when necessary to differentiate it from [[64/39]], another tridecimal neutral sixth. It may also be treated as a type of augmented fifth, as the sum of [[5/4]] and [[13/10]].

13/8 differs from the Pythagorean minor sixth [[128/81]] by [[1053/1024]], about 48¢, from the classic minor sixth [[8/5]] by [[65/64]], about 27¢, from the undecimal neutral sixth [[18/11]] by [[144/143]], about 12¢, and from the rastmic neutral sixth [[44/27]] by [[352/351]], about 4.9¢.

== Approximation ==
13/8 is a fraction of a cent away from the neutral sixth found in the [[10edo|10''n''-edo]] family (7\10).

This interval is a ratio of two consecutive Fibonacci numbers, therefore it approximates the [[golden ratio]]. In this case, 13/8 is ~7.4 [[cent|¢]] sharp of the golden ratio.
{{Interval edo approximation|{{PAGENAME}}}}
== See also ==
* [[16/13]] – its [[octave complement]]
* [[64/39]] – the greater tridecimal neutral sixth
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Neutral sixth]]
[[Category:Golden ratio approximations]]

16/13

2026-05-30T17:11:54Z

Overthink: 13/8 is the octave-reduced form of 13/1

{{Infobox Interval
| Name = (greater) tridecimal neutral third, octave-reduced 13th subharmonic
| Color name = 3u3, thu 3rd
| Sound = jid_16_13_pluck_adu_dr220.mp3
}}

In [[13-limit]] [[just intonation]], '''16/13''', the '''(greater) tridecimal neutral third''', is a 13-limit-based interval measuring about 359.5¢. It is the inversion of [[13/8]], the [[octave reduction|octave-reduced]] 13th harmonic.

16/13 differs from the Pythagorean major third [[81/64]] by [[1053/1024]], about 48¢, from the classic major third [[5/4]] by [[65/64]], about 27¢, from the undecimal neutral third [[11/9]] by [[144/143]], about 12¢, and from the rastmic neutral third [[27/22]] by [[352/351]], about 4.9¢. A [[List_of_root-3rd-P5_triads_in_JI|root-3rd-P5 triad]] featuring 16/13 is 26:32:39, which introduces another tridecimal neutral third, [[39/32]], which measures about 342.5¢. The interval between these two intervals is [[512/507]], about 17¢. While 16/13 is utonal, 39/32 is otonal, as it is the 39th harmonic of the [[harmonic series]].

16/13 is a fraction of a cent away from the neutral third found in the 10''n'' family of edos.

16/13 is near the border-region between neutral thirds and submajor thirds, so it has a bright edge to it compared to narrower neutral thirds, while still sounding slightly darker than a major third like [[5/4]].

== Approximation ==
{{Interval edo approximation|16/13}}
== See also ==
* [[13/8]] – its [[octave complement]]
* [[39/32]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]

[[Category:Third]]
[[Category:Neutral third]]

Talk:39edo

2026-05-30T17:08:53Z

Overthink: /* 39 isn't a dual-7 edo */Reply

12L 12s

2026-05-29T04:29:23Z

Overthink: /* Intervals */ grammar; change some writing

{{Infobox MOS}}
{{MOS intro}}

It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. All of these temperaments map [[3/2]] to 7 steps of 12edo, thus tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo, and adds an independent [[generator]] for [[5/4]] to improve the accuracy of [[5-limit]] harmony. Catler additionally maps 5/4 to 4\12, thus preserving the 5-limit of 12edo, and adds [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12, thus keeping the full [[7-limit]] of 12edo, and uses [[11/8]] as an independent generator.

Using the [[TAMNAMS extension]], it can be named '''dodecawood''', since it has 12 periods per octave, each with one large step and one small step.

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
While 12L 12s can be treated as a 24-form system due to the scale containing 24 notes, it can also make sense as two rings of 12edo that differ by a small step. For example, in compton, the small step represents the [[81/80]] comma, and inflecting 12edo intervals by this step produces more accurate approximations of 5-limit intervals. In catler, the small step represents [[64/63]] and [[36/35]], and inflecting a 12edo minor seventh down by this step gives a more accurate ~7/4, with other ratios involving harmonic 7 also improved, while intervals within the 5-limit are represented as in 12edo.
{{MOS intervals}}

=== Modes ===
Since 12L 12s has only one large step and one small step per period, there are only two modes, which can be called the major and minor modes. In compton, catler, and duodecim, the major mode favors otonalities above the root, while the minor mode favors utonalities above the root. This is because [[5/4]], [[7/4]], and [[11/8]] are all closer to the 12edo step above it than the step below it.
{{MOS mode degrees}}

== Scale tree ==
Softer tunings of 12L 12s are closer to an unequal derivative of [[24edo]], while harder tunings are closer to two rings of 12edo a comma step apart.
{{MOS tuning spectrum
| 3/2 = [[Duodecim]]
| 5/2 = [[Catler]]
| 6/1 = [[Compton]]
}}

12L 12s

2026-05-29T04:25:49Z

Overthink: /* Scale tree */ more optimal tunings (7-limit catler and 5-limit compton, since higher limits are less accessible within 12L 12s)

{{Infobox MOS}}
{{MOS intro}}

It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. All of these temperaments map [[3/2]] to 7 steps of 12edo, thus tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo, and adds an independent [[generator]] for [[5/4]] to improve the accuracy of [[5-limit]] harmony. Catler additionally maps 5/4 to 4\12, thus preserving the 5-limit of 12edo, and adds [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12, thus keeping the full [[7-limit]] of 12edo, and uses [[11/8]] as an independent generator.

Using the [[TAMNAMS extension]], it can be named '''dodecawood''', since it has 12 periods per octave, each with one large step and one small step.

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
While 12L 12s can be treated as a 24-form system due to the scale containing 24 notes, it can also sense as two rings of 12edo that differ by a small step. For example, in compton, the small step represents the [[81/80]] comma, which can be used to achieve accurate 5-limit harmony. In catler, the small step represents [[64/63]] and [[36/35]], and inflecting a 12edo minor seventh down by this step gives a more accurate ~[[7/4]], while 5-limit intervals are represented as in 12edo.
{{MOS intervals}}

=== Modes ===
Since 12L 12s has only one large step and one small step per period, there are only two modes, which can be called the major and minor modes. In compton, catler, and duodecim, the major mode favors otonalities above the root, while the minor mode favors utonalities above the root. This is because [[5/4]], [[7/4]], and [[11/8]] are all closer to the 12edo step above it than the step below it.
{{MOS mode degrees}}

== Scale tree ==
Softer tunings of 12L 12s are closer to an unequal derivative of [[24edo]], while harder tunings are closer to two rings of 12edo a comma step apart.
{{MOS tuning spectrum
| 3/2 = [[Duodecim]]
| 5/2 = [[Catler]]
| 6/1 = [[Compton]]
}}

12L 12s

2026-05-29T04:23:42Z

Overthink: /* Scale tree */ description

{{Infobox MOS}}
{{MOS intro}}

It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. All of these temperaments map [[3/2]] to 7 steps of 12edo, thus tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo, and adds an independent [[generator]] for [[5/4]] to improve the accuracy of [[5-limit]] harmony. Catler additionally maps 5/4 to 4\12, thus preserving the 5-limit of 12edo, and adds [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12, thus keeping the full [[7-limit]] of 12edo, and uses [[11/8]] as an independent generator.

Using the [[TAMNAMS extension]], it can be named '''dodecawood''', since it has 12 periods per octave, each with one large step and one small step.

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
While 12L 12s can be treated as a 24-form system due to the scale containing 24 notes, it can also sense as two rings of 12edo that differ by a small step. For example, in compton, the small step represents the [[81/80]] comma, which can be used to achieve accurate 5-limit harmony. In catler, the small step represents [[64/63]] and [[36/35]], and inflecting a 12edo minor seventh down by this step gives a more accurate ~[[7/4]], while 5-limit intervals are represented as in 12edo.
{{MOS intervals}}

=== Modes ===
Since 12L 12s has only one large step and one small step per period, there are only two modes, which can be called the major and minor modes. In compton, catler, and duodecim, the major mode favors otonalities above the root, while the minor mode favors utonalities above the root. This is because [[5/4]], [[7/4]], and [[11/8]] are all closer to the 12edo step above it than the step below it.
{{MOS mode degrees}}

== Scale tree ==
Softer tunings of 12L 12s are closer to an unequal derivative of [[24edo]], while harder tunings are closer to two rings of 12edo a comma step apart.
{{MOS tuning spectrum
| 3/2 = [[Duodecim]]
| 3/1 = [[Catler]]
| 5/1 = [[Compton]]
}}

12L 12s

2026-05-29T04:21:40Z

Overthink: /* Scale properties */ Remove section which doesn't apply well to 12L 12s (also the table is too wide). Description for intervals

{{Infobox MOS}}
{{MOS intro}}

It is the 24-note mos scale of the [[compton]], [[catler]], and [[duodecim]] temperaments of the [[compton family]], which divide the octave into 12 parts. As such, this mos scale can be considered to be 2 rings of [[12edo]], and can be replicated with two 12edo instruments detuned from each other by a fixed amount. All of these temperaments map [[3/2]] to 7 steps of 12edo, thus tempering out the [[Pythagorean comma]]. Compton uses the [[3-limit]] of 12edo, and adds an independent [[generator]] for [[5/4]] to improve the accuracy of [[5-limit]] harmony. Catler additionally maps 5/4 to 4\12, thus preserving the 5-limit of 12edo, and adds [[7/4]] as an independent generator. Duodecim is a low-accuracy temperament that further maps 7/4 to 10\12, thus keeping the full [[7-limit]] of 12edo, and uses [[11/8]] as an independent generator.

Using the [[TAMNAMS extension]], it can be named '''dodecawood''', since it has 12 periods per octave, each with one large step and one small step.

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
While 12L 12s can be treated as a 24-form system due to the scale containing 24 notes, it can also sense as two rings of 12edo that differ by a small step. For example, in compton, the small step represents the [[81/80]] comma, which can be used to achieve accurate 5-limit harmony. In catler, the small step represents [[64/63]] and [[36/35]], and inflecting a 12edo minor seventh down by this step gives a more accurate ~[[7/4]], while 5-limit intervals are represented as in 12edo.
{{MOS intervals}}

=== Modes ===
Since 12L 12s has only one large step and one small step per period, there are only two modes, which can be called the major and minor modes. In compton, catler, and duodecim, the major mode favors otonalities above the root, while the minor mode favors utonalities above the root. This is because [[5/4]], [[7/4]], and [[11/8]] are all closer to the 12edo step above it than the step below it.
{{MOS mode degrees}}

== Scale tree ==
{{MOS tuning spectrum
| 3/2 = [[Duodecim]]
| 3/1 = [[Catler]]
| 5/1 = [[Compton]]
}}

12L 12s

2026-05-29T04:11:36Z

Overthink: /* Modes */ add description

50edo

2026-05-29T04:07:16Z

Overthink: /* Instruments */ - extra newline; punctuation

{{Infobox ET}}
{{ED intro}}

== Theory ==
As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice; it is a somewhat sharp approximation of [[2/7-comma meantone]] (and is almost exactly 5/18-comma meantone). In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo where the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It is also almost consistent to the no-21s [[25-odd-limit]], only barely missing consistent mapping of [[11/9]] and [[18/11]].

50edo is also quite strong in the realm of tertian harmony for a meantone system, as the errors on [[7/6]], [[6/5]], [[5/4]], and [[9/7]] are all balanced to be roughly half as flat as the fifth, meaning that this set of thirds taken as a whole is minimally out-of-tune given the damage induced by meantone. Though it fails to approximate [[11/9]] well by virtue of not having a perfect hemififth, it inherits the excellent [[16/13]] from [[10edo]] and additionally has a 1.2{{c}} flat [[13/11]], providing even more qualities of roughly just thirds alongside their more complex [[fifth complement]]s.

=== Odd harmonics ===
{{Harmonics in equal|50|intervals=odd|columns=11}}
{{Harmonics in equal|50|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 50edo (continued)}}

=== As a tuning of other temperaments ===
50et tempers out [[126/125]], [[225/224]] and [[3136/3125]] in the [[7-limit]], indicating it [[support]]s septimal meantone; [[245/242]], [[385/384]] and [[540/539]] in the [[11-limit]] and [[105/104]], [[144/143]] and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension [[meanpop]], it can be used to advantage for the [[coblack]] temperament (15 & 50), and provides the optimal patent val for 11- and 13-limit [[Meantone family #Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], {{monzo| 23 6 -14 }}, so that in 50edo seven chromatic semitones stack to a perfect fourth. By comparison, this gives a perfect fifth in 12edo, a doubly diminished fifth in 31edo, and a diminished fourth in 19edo.

=== Relations ===
The 50edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "[[Golden meantone|Golden Tone System]]" ([[Das Goldene Tonsystem]]) of [[Thorvald Kornerup]] (and similarly as the next step from 31edo in [[Joseph Yasser]]'s "[https://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y A Theory of Evolving Tonality]").

== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! #
! Cents
! Ratios<ref group="note">{{sg|50edo|limit=13-limit}}</ref>
! colspan="3" | [[Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and vvd2)
|-
| 0
| 0
| [[1/1]]
| Perfect 1sn
| P1
| D
|-
| 1
| 24
| ''[[45/44]]'', [[49/48]], [[56/55]], [[65/64]], [[66/65]], [[78/77]], [[91/90]], [[99/98]], [[100/99]], [[121/120]], ''[[169/168]]''
| Up 1sn
| ^1
| ^D
|-
| 2
| 48
| ''[[27/26]]'', [[33/32]], [[36/35]], ''[[50/49]]'', ''[[55/54]]'', ''[[64/63]]''
| Dim 2nd, Downaug 1sn
| d2, vA1
| Ebb, vD#
|-
| 3
| 72
| ''[[21/20]]'', [[25/24]], [[26/25]], [[28/27]]
| Aug 1sn, Updim 2nd
| A1, ^d2
| D#, ^Ebb
|-
| 4
| 96
| ''[[22/21]]''
| Downminor 2nd
| vm2
| vEb
|-
| 5
| 120
| [[16/15]], [[15/14]], [[14/13]]
| Minor 2nd
| m2
| Eb
|-
| 6
| 144
| [[13/12]], [[12/11]]
| Upminor 2nd
| ^m2
| ^Eb
|-
| 7
| 168
| [[11/10]]
| Downmajor 2nd
| vM2
| vE
|-
| 8
| 192
| [[9/8]], [[10/9]]
| Major 2nd
| M2
| E
|-
| 9
| 216
| [[25/22]]
| Upmajor 2nd
| ^M2
| ^E
|-
| 10
| 240
| [[8/7]], [[15/13]]
| Downaug 2nd, Dim 3rd
| vA2, d3
| vE#, Fb
|-
| 11
| 264
| [[7/6]]
| Updim 3rd, Aug 2nd
| ^d3, A2
| ^Fb, E#
|-
| 12
| 288
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 13
| 312
| [[6/5]]
| Minor 3rd
| m3
| F
|-
| 14
| 336
| ''[[27/22]]'', [[39/32]], [[40/33]], ''[[49/40]]''
| Upminor 3rd
| ^m3
| ^F
|-
| 15
| 360
| [[16/13]], ''[[11/9]]''
| Downmajor 3rd
| vM3
| vF#
|-
| 16
| 384
| [[5/4]]
| Major 3rd
| M3
| F#
|-
| 17
| 408
| [[14/11]]
| Upmajor 3rd
| ^M3
| ^F#
|-
| 18
| 432
| [[9/7]]
| Downaug 3rd, Dim 4th
| vA3, d4
| vFx, Gb
|-
| 19
| 456
| [[13/10]]
| Updim 4th, Aug 3rd
| A3, ^d4
| ^Gb, Fx
|-
| 20
| 480
| [[33/25]], ''[[55/42]]'', ''[[64/49]]''
| Down 4th
| v4
| vG
|-
| 21
| 504
| [[4/3]]
| Perfect 4th
| P4
| G
|-
| 22
| 528
| [[15/11]]
| Up 4th
| ^4
| ^G
|-
| 23
| 552
| [[11/8]], [[18/13]]
| Downaug 4th
| vA4
| vG#
|-
| 24
| 576
| [[7/5]]
| Aug 4th
| A4
| G#
|-
| 25
| 600
| ''[[63/44]]'', ''[[88/63]]'', [[78/55]], [[55/39]]
| Upaug 4th, Downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 26
| 624
| [[10/7]]
| Dim 5th
| d5
| Ab
|-
| 27
| 648
| [[16/11]], [[13/9]]
| Updim 5th
| ^d5
| ^Ab
|-
| 28
| 672
| [[22/15]]
| Down 5th
| v5
| vA
|-
| 29
| 696
| [[3/2]]
| Perfect 5th
| P5
| A
|-
| 30
| 720
| [[50/33]], ''[[84/55]]'', ''[[49/32]]''
| Up 5th
| ^5
| ^A
|-
| 31
| 744
| [[20/13]]
| Downaug 5th, Dim 6th
| vA5, d6
| vA#, Bbb
|-
| 32
| 768
| [[14/9]]
| Updim 6th, Aug 5th
| ^d6, A5
| ^Bbb, A#
|-
| 33
| 792
| [[11/7]]
| Downminor 6th
| vm6
| vBb
|-
| 34
| 816
| [[8/5]]
| Minor 6th
| m6
| Bb
|-
| 35
| 840
| [[13/8]], ''[[18/11]]''
| Upminor 6th
| ^m6
| ^Bb
|-
| 36
| 864
| ''[[44/27]]'', [[64/39]], [[33/20]], ''[[80/49]]''
| Downmajor 6th
| vM6
| vB
|-
| 37
| 888
| [[5/3]]
| Major 6th
| M6
| B
|-
| 38
| 912
| [[22/13]]
| Upmajor 6th
| ^M6
| ^B
|-
| 39
| 936
| [[12/7]]
| Downaug 6th, Dim 7th
| vA6, d7
| vB#, Cb
|-
| 40
| 960
| [[7/4]]
| Updim 7th, Aug 6th
| ^d7, A6
| ^Cb, B#
|-
| 41
| 984
| [[44/25]]
| Downminor 7th
| vm7
| vC
|-
| 42
| 1008
| [[16/9]], [[9/5]]
| Minor 7th
| m7
| C
|-
| 43
| 1032
| [[20/11]]
| Upminor 7th
| ^m7
| ^C
|-
| 44
| 1056
| [[24/13]], [[11/6]]
| Downmajor 7th
| vM7
| vC#
|-
| 45
| 1080
| [[15/8]], [[28/15]], [[13/7]]
| Major 7th
| M7
| C#
|-
| 46
| 1104
| ''[[21/11]]''
| Upmajor 7th
| ^M7
| ^C#
|-
| 47
| 1128
| ''[[40/21]]'', [[48/25]], [[25/13]], [[27/14]]
| Downaug 7th, Dim 8ve
| vA7, d8
| vCx, Db
|-
| 48
| 1152
| ''[[52/27]]'', [[64/33]], [[35/18]], ''[[49/25]]'', ''[[108/55]]'', ''[[63/32]]''
| Updim 8ve, Aug 7th
| ^d8, A7
| ^Db, Cx
|-
| 49
| 1176
| ''[[88/45]]'', [[96/49]], [[55/28]], [[128/65]], [[65/33]], [[77/39]], [[180/91]], [[196/99]], [[99/50]], [[240/121]], ''[[336/169]]''
| Down 8ve
| v8
| vD
|-
| 50
| 1200
| [[2/1]]
| Perfect 8ve
| P8
| D
|}
<references group="note" />

== Notation ==
=== Stein–Zimmermann–Gould notation ===
50edo can be notated with [[Stein–Zimmermann–Gould notation]]:
{{Sharpness-sharp3-szg}}

Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.

=== Kite's ups and downs notation ===
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{Ups and downs sharpness}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[57edo #Sagittal notation|57]], [[64edo #Sagittal notation|64]], and [[71edo #Second-best fifth notation|71b]].

==== Evo flavor ====
<imagemap>
File:50-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:50-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:50-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:50-EDO_Revo_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

== Approximation to JI ==
[[File:50ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 29-limit intervals approximated in 50edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|50|15}}

== Regular temperament properties ==
=== Temperament measures ===
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -79 50 }}
| {{mapping| 50 79 }}
| +1.88
| 1.88
| 7.83
|-
| 2.3.5
| 81/80, {{monzo| -27 -2 13 }}
| {{mapping| 50 79 116 }}
| +1.58
| 1.59
| 6.62
|-
| 2.3.5.7
| 81/80, 126/125, 84035/82944
| {{mapping| 50 79 116 140 }}
| +1.98
| 1.54
| 6.39
|-
| 2.3.5.7.11
| 81/80, 126/125, 245/242, 385/384
| {{mapping| 50 79 116 140 173 }}
| +1.54
| 1.63
| 6.76
|-
| 2.3.5.7.11.13
| 81/80, 105/104, 126/125, 144/143, 245/242
| {{mapping| 50 79 116 140 173 185 }}
| +1.31
| 1.57
| 6.54
|}

=== Commas ===
50et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 50 79 116 140 173 185 204 212 226 }}, comma values in cents rounded to 2 decimal places. This list is not all-inclusive, and is based on the interval table from Scala version 2.2.

{| class="commatable wikitable center-all left-3 right-4 left-5"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! Name
|-
| 3
| <abbr title="717897987691852588770249/604462909807314587353088">(20 digits)</abbr>
| {{monzo| -79 50 }}
| 297.75
| 50-comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Syntonic comma
|-
| 5
| <abbr title="1220703125/1207959552">(20 digits)</abbr>
| {{monzo| -27 -2 13 }}
| 18.17
| [[Ditonma]]
|-
| 5
| [[6115295232/6103515625|(20 digits)]]
| {{monzo| 23 6 -14 }}
| 3.34
| [[Vishnuzma]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Harrison's comma
|-
| 7
| [[16807/16384]]
| {{monzo| -14 0 0 5}}
| 44.13
| Cloudy comma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Schismean comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Hemimean comma
|-
| 7
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
| {{monzo| 11 -10 -10 10 }}
| 5.57
| [[Linus comma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| [[Meter]]
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{monzo| -6 -8 2 5 }}
| 1.12
| [[Wizma]]
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Frostma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Swetisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| Wizardharry comma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Kalisma
|-
| 13
| [[105/104]]
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| Animist comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Grossma
|-
| 13
| [[196/195]]
| {{monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Mynucuma
|-
| 13
| [[1188/1183]]
| {{monzo| 2 3 0 -1 1 -2 }}
| 7.30
| Kestrel comma
|-
| 13
| [[31213/31104]]
| {{monzo| -7 -5 0 4 0 1 }}
| 6.06
| Praveensma
|-
| 13
| [[364/363]]
| {{monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Minor minthma
|-
| 13
| [[2200/2197]]
| {{monzo| 3 0 2 0 1 -3 }}
| 2.36
| Petrma
|-
| 17
| [[170/169]]
| {{monzo| 1 0 1 0 0 -2 1 }}
| 10.21
| Major naiadma
|-
| 17
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| Minor naiadma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Semitonisma
|-
| 17
| [[375/374]]
| {{monzo| -1 1 3 0 -1 0 -1 }}
| 4.62
| Ursulisma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1 }}
| 11.35
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1}}
| 10.15
| Malcolmisma
|-
| 19
| [[210/209]]
| {{monzo| 1 1 1 1 -1 0 0 1}}
| 8.26
| Spleen comma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Go comma
|-
| 19
| [[495/494]]
| {{monzo| -1 2 1 0 1 -1 0 -1 }}
| 3.50
| Eulalisma
|-
| 23
| [[507/506]]
| 2.3.11.13.23 {{monzo| -1 1 -1 2 -1 }}
| 3.42
| Laodicisma
|-
| 23
| [[529/528]]
| 2.3.11.23 {{monzo| -4 -1 -1 2 }}
| 3.28
| Preziosisma
|-
| 23
| [[576/575]]
| 2.3.5.23 {{monzo| 6 2 -2 -1 }}
| 3.01
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Triaphonisma
|}
<references group="note" />

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 1\50
| 24.0
| 686/675
| [[Sengagen]]
|-
| 1
| 9\50
| 216.0
| 17/15
| [[Tremka]]
|-
| 1
| 11\50
| 264.0
| 7/6
| [[Septimin]]
|-
| 1
| 13\50
| 312.0
| 6/5
| [[Oolong]]
|-
| 1
| 17\50
| 408.0
| 325/256
| [[Coditone]]
|-
| 1
| 19\50
| 456.0
| 125/96
| [[Qak]]
|-
| 1
| 21\50
| 504.0
| 4/3
| [[Meantone]] / [[meanpop]]
|-
| 1
| 23\50
| 552.0
| 11/8
| [[Emka]]
|-
| 2
| 2\50
| 48.0
| 36/35
| [[Pombe]]
|-
| 2
| 3\50
| 72.0
| 25/24
| [[Vishnu]] / [[vishnean]]
|-
| 2
| 6\50
| 144.0
| 12/11
| [[Bisemidim]]
|-
| 2
| 9\50
| 216.0
| 17/15
| [[Wizard]] / [[lizard]] / [[gizzard]]
|-
| 2
| 12\50
| 288.0
| 13/11
| [[Vines]]
|-
| 2
| 21\50 (4\50)
| 504.0 (96.0)
| 4/3 (35/33)
| [[Bimeantone]]
|-
| 5
| 21\50 (1\50)
| 504.0 (24.0)
| 4/3 (49/48)
| [[Cloudtone]]
|-
| 5
| 23\50 (3\50)
| 552.0 (72.0)
| 11/8 (21/20)
| [[Coblack]]
|-
| 10
| 7\50 (3\50)
| 168.0 (72.0)
| 54/49 (25/24)
| [[Decavish]]
|-
| 10
| 21\50 (1\50)
| 504.0 (24.0)
| 4/3 (78/77)
| [[Decic]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
50edo's [[prime]]s 3, 5, 7, 17, 19, and 23 are all tuned flat and its 11 and 13 have close to no error, so 50edo can benefit from slight [[octave stretching]]. Some slightly stretched-octave tunings of 50edo include (least to most stretch): [[equal tuning|166ed10]], [[ed5|116ed5]], [[zpi|238zpi]] and [[ed12|179ed12]].

== Instruments ==
; Lumatone

See [[Lumatone mapping for 50edo]].

; Piano

A [[:Category:Piano|piano]] playing with a 50edo ensemble may wish to use the tuning [[116ed5]]. This tuning is almost exactly the same as 50edo, but with octaves [[octave stretch|stretched]] by 1 cent. Because pianos usually use stretched octaves, this tuning will sit better with the [[timbre]] of the piano, while still being close enough that it sounds perfectly in-tune with the other instruments tuned to 50edo.

== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=RnYqc0NKMLM "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=e6fMO-sue4Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024, organ sound rendering)
* [https://www.youtube.com/watch?v=qjb9DDM32Ic "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742-1749) — rendered by Claudi Meneghin (2025, harpsichord sound rendering)

; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=yrM50pvmD5c ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)

; {{W|John Bull (composer)|John Bull}}
* [https://www.youtube.com/watch?v=6RewllRJ5rU ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2026)

; {{w|Frédéric Chopin}}
* [https://www.youtube.com/shorts/7Bisk0I2H4o ''Prelude Op. 28, No. 7 in A major''] (1839), arranged for fortepiano, tuned into 50-edo – rendered by [[Claudi Meneghin]] (2025)

; {{W|Louis Couperin}}
* [https://www.youtube.com/shorts/NSzakO66Roc ''«La Piémontoise»''] (1658?) – rendered by Claudi Meneghin (2026)

; {{W|Gabriel Fauré}}
* [https://www.youtube.com/watch?v=7djfrUlw2ck ''Pavane'', op. 50] (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2020)

; {{W|Akira Kamiya}}
* [https://www.youtube.com/watch?v=5UnPAhRqmb4 ''funfunfun ta yo''] (2007) – rendered by MortisTheneRd (2024)

; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=YK_kFs4PL2g&list=WL&index=347 ''Gigue KV 574 («Leipziger Gigue»)''] (1789) – rendered by Claudi Meneghin (2026)

=== 21st century===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/zCsc5n6dr_I ''microtonal improv in 50edo''] (2024)
* [https://www.youtube.com/shorts/ynz5XvJOHiE ''Piano that may not be played that well - Deltarune (microtonal cover in 50edo)''] (2025)
* [https://www.youtube.com/shorts/J34qt45jZW4 ''Snow White - Laufey (microtonal cover in 50edo)''] (2025)
* [https://www.youtube.com/shorts/dAyMY-14yZo ''50edo improv''] (2025-10-13)
* [https://www.youtube.com/shorts/L6jF5_HEGkM ''Heat Abnormal - Iyowa (microtonal cover in 50edo)''] (2025)
* [https://www.youtube.com/shorts/WcExL9W2Gyc ''The Prettiest Little Song Of All - Belasco (microtonal cover in 50edo)''] (2025)
* [https://www.youtube.com/shorts/eXGcC52YMh8 ''Mother - Umineko (microtonal cover in 50edo)''] (2026)
* [https://www.youtube.com/shorts/DIiLORDPfUI ''50edo improv''] (2026-05-25)

; [[Francium]]
* [https://www.youtube.com/watch?v=pH6E35hwUnM ''On My Way To Somewhere''] (2023)

; [[Claudi Meneghin]]
* [https://www.youtube.com/shorts/IhKVro5YEcA ''Canon on «Twinkle Twinkle Little Star» in 50-edo, for Organ''] (≤2014, restored and re-hosted 2025)
* [https://www.youtube.com/watch?v=TRXy0FJOKIA ''Fugue on the Dragnet theme''] (2014)
* [https://www.youtube.com/watch?v=wcTVED9zFrU ''Blue Fugue for Organ''] (2018)
* [https://www.youtube.com/watch?v=Zh2jWoIXAf8 ''La Petite Poule Grise - Fugue''] (2019)
* [https://www.youtube.com/watch?v=28x3vqw9kDI ''Happy Birthday Canon'', 6-in-1 Canon in 50edo] (2019)
* [https://www.youtube.com/watch?v=szUpO3FAOes ''Fantasia Catalana''] (2020)
* [https://www.youtube.com/watch?v=38UMa3oWSIE ''Preludi Nocturn i Fuga sobre la Lluna la Pruna''] (2020)
* [https://www.youtube.com/watch?v=C4EkNEu4EeU ''Canon at the Semitone on The Mother's Malison Theme'', for Organ] (2022)
* [https://www.youtube.com/watch?v=FyDKSjS9Qtg ''Fugue on an Original Theme'', for Baroque Ensemble] (2023) ([https://www.youtube.com/watch?v=TXwlLV2TCsw for Organ])
* [https://www.youtube.com/watch?v=2nD_7Ot8-0A ''Catalan Fugue (La Santa Espina)''] (2023)
* [https://www.youtube.com/watch?v=TBxDmpM9Xa8 ''Canon in C='' for Baroque Wind Ensemble] (2023)
* [https://www.youtube.com/watch?v=sIr394fGEEg ''Fantasia Catalana'', for Baroque Ensemble] (2023)
* ''Chord Progression: The Octave Divided into Five Parts in 50 edo'' (intended for demonstrating chord progressions, as the title indicates, but actually works as a short composition)
** [https://www.youtube.com/shorts/7_kROtWc4Sw <nowiki>organ rendition</nowiki>] (2023)
** [https://www.youtube.com/shorts/qsuM1sA2-A0 <nowiki>harpsichord rendition</nowiki>] (2024)
* [https://www.youtube.com/shorts/2x5atFuN6WA ''Baroque Blues - 4-Part Fugue for Baroque Consort''] (2026)
* ''Fugue on the French Lullaby «La Petite Poule Grise»'' (2026)
** [https://www.youtube.com/watch?v=RrsZ-bzf1mE Baroque ensemble rendition]
** [https://www.youtube.com/watch?v=wgsWzn_vfIM organ rendition]

; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album ''the late little xmas album''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone ''Harpsichord meantone improvisation 1 in 50EDO''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo ''Long improvisation 2 in 50EDO''] (2014)
* [https://soundcloud.com/camtaylor-1/chord-sequence-for-difference ''Chord sequence for Difference tones in 50EDO''] (2014)
* [https://soundcloud.com/camtaylor-1/enharmonic-modulations-in ''Enharmonic Modulations in 50EDO''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq ''Harmonic Clusters on 50EDO Harpsichord''] (2014)
* [https://soundcloud.com/camtaylor-1/fragment-in-fifty ''Fragment in Fifty''] (2014)

== Additional reading ==
* [http://www.archive.org/details/harmonicsorphilo00smit Robert Smith's book online]
* [http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html More information about Robert Smith's temperament]{{Dead link}}
* [https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]{{Dead link}}
* [http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]

[[Category:50edo]]
[[Category:Equal divisions of the octave|##]] 
[[Category:Golden meantone]]
[[Category:Historical]]
[[Category:Listen]]
[[Category:Meantone]]
[[Category:Meanpop]]