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'''38edo''' divides the octave into 38 equal parts of 31.579 [[cent|cents]]. Since 38 = 2*19, it can be thought of as two parallel [[19edo|19edos]]. It [[tempering_out|tempers out]] the same [[5-limit]] commas as [[19edo|19]], 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 [[Meantone_family|injera temperament]], for which 38 is the [[Optimal_patent_val|optimal patent val]]. In the [[11-limit]], we can add 121/120 and 176/175.
{{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 ==
== Intervals ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! Step
! rowspan="3" | Step
! Cents
! rowspan="3" | Cents
! colspan="3" | Approximated ratios
! rowspan="3" colspan="3" | [[Ups and downs notation]]*<br>([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
! rowspan="2" | Ratios of the <br>2.3.5.11.17.19 subgroup
! colspan="2" | Ratios of 7 and 13
|-
! Patent val
! 38df val
|-
|-
| 0
| 0
| 0.0000
| 0.0
| [[1/1]]
|
|
| Perfect 1sn
| P1
| D
|-
|-
| 1
| 1
| 31.5789
| 31.6
| [[55/54]], [[45/44]], ''[[33/32]]''
| [[64/63]], ''[[36/35]]''
| [[56/55]]
| Up 1sn
| ^1
| ^D
|-
|-
| 2
| 2
| 63.1579
| 63.2
| [[25/24]], [[34/33]]
| [[22/21]]
| [[28/27]], [[26/25]], [[27/26]]
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
|-
| 3
| 3
| 94.7368
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]'', [[21/20]]
|
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
|-
| 4
| 4
| 126.3157
| 126.3
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| m2
| Eb
|-
|-
| 5
| 5
| 157.8947
| 157.9
| [[12/11]], [[11/10]]
| ''[[13/12]]'', [[35/32]]
|
| Mid 2nd
| ~2
| vE
|-
|-
| 6
| 6
| 189.4737
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
|
| [[28/25]]
| Major 2nd
| M2
| E
|-
|-
| 7
| 7
| 221.0526
| 221.1
| [[25/22]], [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| ^M2
| ^E
|-
|-
| 8
| 8
| 252.6316
| 252.6
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
|-
| 9
| 9
| 284.2105
| 284.2
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
|-
| 10
| 10
| 315.7895
| 315.8
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| m3
| F
|-
|-
| 11
| 11
| 347.3684
| 347.4
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| ~3
| ^F
|-
|-
| 12
| 12
| 378.9474
| 378.9
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| M3
| F#
|-
|-
| 13
| 13
| 410.5263
| 410.5
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
|-
| 14
| 14
| 442.1053
| 442.1
| [[22/17]], [[32/25]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]], ''[[21/16]]''
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
|-
| 15
| 15
| 473.6843
| 473.7
| [[25/19]]
| [[21/16]], ''[[13/10]]''
| [[17/13]]
| Down 4th
| v4
| vG
|-
|-
| 16
| 16
| 505.2632
| 505.3
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| P4
| G
|-
|-
| 17
| 17
| 536.8421
| 536.8
| [[15/11]], [[11/8]], [[34/25]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| ^4
| ^G
|-
|-
| 18
| 18
| 568.4211
| 568.4
| [[25/18]]
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| Aug 4th
| A4
| G#
|-
|-
| 19
| 19
| 600.0000
| 600.0
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
|-
| 20
| 20
| 631.5789
| 631.6
| [[36/25]]
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| Dim 5th
| d5
| Ab
|-
|-
| 21
| 21
| 663.1579
| 663.2
| [[22/15]], [[16/11]], [[25/17]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| v5
| vA
|-
|-
| 22
| 22
| 694.7368
| 694.7
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| P5
| A
|-
|-
| 23
| 23
| 726.3157
| 726.3
| [[38/25]]
| ''[[20/13]]''
| [[26/17]]
| Up 5th
| ^5
| ^A
|-
|-
| 24
| 24
| 757.8947
| 757.9
| [[17/11]], [[25/16]]
| ''[[26/17]]'', ''[[11/7]]''
| [[14/9]], [[20/13]], ''[[32/21]]''
| Aug 5th, dim 6th
| A5, d6
| A#
|-
|-
| 25
| 25
| 789.4737
| 789.5
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
|-
| 26
| 26
| 821.0526
| 821.1
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| m6
| Bb
|-
|-
| 27
| 27
| 852.6316
| 852.6
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| ~6
| vB
|-
|-
| 28
| 28
| 884.2105
| 884.2
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| M6
| B
|-
|-
| 29
| 29
| 915.7895
| 915.8
| [[32/19]], [[17/10]]
|
| [[22/13]]
| Upmajor 6th
| ^M6
| ^B
|-
|-
| 30
| 30
| 947.3684
| 947.4
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
|-
| 31
| 31
| 978.9474
| 978.9
| [[44/25]], [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| vm7
| vC
|-
|-
| 32
| 32
| 1010.5263
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
|
| [[25/14]]
| Minor 7th
| m7
| C
|-
|-
| 33
| 33
| 1042.1053
| 1042.1
| [[20/11]], [[11/6]]
| ''[[24/13]]'', [[64/35]]
|
| Mid 7th
| ~7
| ^C
|-
|-
| 34
| 34
| 1073.6843
| 1073.7
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| M7
| C#
|-
|-
| 35
| 35
| 1105.2632
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
|-
| 36
| 36
| 1136.8421
| 1136.8
| [[33/17]], [[48/25]]
|
| [[27/14]], [[25/13]], [[52/27]]
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
|-
| 37
| 37
| 1168.4211
| 1168.4
| ''[[64/33]]'', [[88/45]], [[108/55]]
| [[63/32]]
| [[55/28]]
| Down 8ve
| v8
| vD
|-
|-
| 38
| 38
| 1200.0000
| 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
|}
|}


[[Category:Theory]]
== Octave stretch or compression ==
[[Category:Edo]]
38edo's approximation of [[JI]] can be improved by slightly [[octave stretch|stretching the octave]], as in [[ed5|88ed5]], [[zpi|166zpi]] or [[60edt]].
[[Category:38| ]]  <!-- main article -->
 
== 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| ]]  <!-- Main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/38edo"