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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|38}}
{{ED intro}}


== Theory ==
== Theory ==
Since {{nowrap|38 {{=}} 2 × 19}}, it can be thought of as two parallel [[19edo]]s. While the halving of the step size lowers [[consistency]] and leaves it only mediocre in terms of overall [[relative interval error|relative error]], the fact that the 3rd & 5th harmonics are flat by almost exactly the same amount, while the 11th is 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 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. 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 [[11-limit]], we can add 121/120 and 176/175.  
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 [[Warts|38df]] mapping, every [[prime interval]] 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. In other words, all 19-odd-limit intervals are [[consistency|consistent]] within the 38df [[val]] {{val| 38 60 88 106 131 140 155 161 }}.  
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.


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 }}
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]]]
[[File:Harmonic_series_38df.mp3]] [[:File:Harmonic_series_38df.mp3|[Harmonic series 2-20 in 38df]]]
Line 17: Line 19:
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! Step
! rowspan="3" | Step
! Cents
! rowspan="3" | Cents
! 19-odd-limit ratios,<br>in 38df val
! colspan="3" | Approximated ratios
! colspan="3" | [[Ups and downs notation]]*
! 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
| 0.0
| 0.0
|
| [[1/1]]
|
|  
| Perfect 1sn
| Perfect 1sn
| P1
| P1
Line 31: Line 41:
| 1
| 1
| 31.6
| 31.6
|
| [[55/54]], [[45/44]], ''[[33/32]]''
| [[64/63]], ''[[36/35]]''
| [[56/55]]
| Up 1sn
| Up 1sn
| ^1
| ^1
Line 38: Line 50:
| 2
| 2
| 63.2
| 63.2
|
| [[25/24]], [[34/33]]
| [[22/21]]
| [[28/27]], [[26/25]], [[27/26]]
| Aug 1sn, dim 2nd
| Aug 1sn, dim 2nd
| A1, d2
| A1, d2
Line 46: Line 60:
| 94.7
| 94.7
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
| ''[[15/14]]'', [[21/20]]
|
| Upaug 1sn, downminor 2nd
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^A1, vm2
Line 52: Line 68:
| 4
| 4
| 126.3
| 126.3
| [[16/15]], [[15/14]], [[14/13]], [[13/12]]
| [[16/15]]
| [[14/13]]
| [[15/14]], [[14/13]], [[13/12]]
| Minor 2nd
| Minor 2nd
| m2
| m2
Line 60: Line 78:
| 157.9
| 157.9
| [[12/11]], [[11/10]]
| [[12/11]], [[11/10]]
| ''[[13/12]]'', [[35/32]]
|
| Mid 2nd
| Mid 2nd
| ~2
| ~2
Line 67: Line 87:
| 189.5
| 189.5
| [[10/9]], [[19/17]], [[9/8]]
| [[10/9]], [[19/17]], [[9/8]]
|
| [[28/25]]
| Major 2nd
| Major 2nd
| M2
| M2
Line 73: Line 95:
| 7
| 7
| 221.1
| 221.1
| [[17/15]]
| [[25/22]], [[17/15]]
| [[8/7]], ''[[15/13]]''
|
| Upmajor 2nd
| Upmajor 2nd
| ^M2
| ^M2
Line 80: Line 104:
| 8
| 8
| 252.6
| 252.6
| [[8/7]], [[15/13]], [[22/19]], [[7/6]]
| [[22/19]]
|
| ''[[8/7]]'', [[15/13]], [[7/6]]
| Aug 2nd, Dim 3rd
| Aug 2nd, Dim 3rd
| A2, d3
| A2, d3
Line 87: Line 113:
| 9
| 9
| 284.2
| 284.2
| [[20/17]], [[13/11]], [[19/16]]
| [[20/17]], [[19/16]]
| ''[[7/6]]''
| [[13/11]]
| Downminor 3rd
| Downminor 3rd
| vm3
| vm3
Line 95: Line 123:
| 315.8
| 315.8
| [[6/5]]
| [[6/5]]
| ''[[13/11]]'', ''[[17/14]]''
|
| Minor 3rd
| Minor 3rd
| m3
| m3
Line 101: Line 131:
| 11
| 11
| 347.4
| 347.4
| [[17/14]], [[11/9]]
| [[11/9]]
| [[16/13]]
| [[17/14]]
| Mid 3rd
| Mid 3rd
| ~3
| ~3
Line 108: Line 140:
| 12
| 12
| 378.9
| 378.9
| [[16/13]], [[5/4]]
| [[5/4]]
|
| ''[[16/13]]''
| Major 3rd
| Major 3rd
| M3
| M3
Line 115: Line 149:
| 13
| 13
| 410.5
| 410.5
| [[24/19]], [[19/15]], [[14/11]]
| [[24/19]], [[19/15]]
| ''[[9/7]]''
| [[14/11]]
| Upmajor 3rd, Downdim 4th
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^M3, vd4
Line 122: Line 158:
| 14
| 14
| 442.1
| 442.1
| [[9/7]], [[22/17]], [[13/10]]
| [[22/17]], [[32/25]]
| ''[[14/11]]'', ''[[17/13]]''
| [[9/7]], [[13/10]], ''[[21/16]]''
| Aug 3rd, dim 4th
| Aug 3rd, dim 4th
| A3, d4
| A3, d4
Line 129: Line 167:
| 15
| 15
| 473.7
| 473.7
| [[25/19]]
| [[21/16]], ''[[13/10]]''
| [[17/13]]
| [[17/13]]
| Down 4th
| Down 4th
Line 137: Line 177:
| 505.3
| 505.3
| [[4/3]]
| [[4/3]]
| ''[[19/14]]''
|
| Perfect 4th
| Perfect 4th
| P4
| P4
Line 143: Line 185:
| 17
| 17
| 536.8
| 536.8
| [[19/14]], [[15/11]], [[26/19]], [[11/8]]
| [[15/11]], [[11/8]], [[34/25]]
| ''[[18/13]]''
| [[19/14]], [[26/19]]
| Up 4th
| Up 4th
| ^4
| ^4
Line 150: Line 194:
| 18
| 18
| 568.4
| 568.4
| [[25/18]]
| ''[[26/19]]''
| [[18/13]], [[7/5]]
| [[18/13]], [[7/5]]
| Aug 4th
| Aug 4th
Line 158: Line 204:
| 600.0
| 600.0
| [[24/17]], [[17/12]]
| [[24/17]], [[17/12]]
| [[7/5]], [[10/7]]
|
| Upaug 4th, downdim 5th
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^A4, vd5
Line 164: Line 212:
| 20
| 20
| 631.6
| 631.6
| [[36/25]]
| ''[[19/13]]''
| [[10/7]], [[13/9]]
| [[10/7]], [[13/9]]
| Dim 5th
| Dim 5th
Line 171: Line 221:
| 21
| 21
| 663.2
| 663.2
| [[16/11]], [[19/13]], [[22/15]], [[28/19]]
| [[22/15]], [[16/11]], [[25/17]]
| ''[[13/9]]''
| [[19/13]], [[28/19]]
| Down 5th
| Down 5th
| v5
| v5
Line 179: Line 231:
| 694.7
| 694.7
| [[3/2]]
| [[3/2]]
| ''[[28/19]]''
|
| Perfect 5th
| Perfect 5th
| P5
| P5
Line 185: Line 239:
| 23
| 23
| 726.3
| 726.3
| [[38/25]]
| ''[[20/13]]''
| [[26/17]]
| [[26/17]]
| Up 5th
| Up 5th
Line 192: Line 248:
| 24
| 24
| 757.9
| 757.9
| [[20/13]], [[17/11]], [[14/9]]
| [[17/11]], [[25/16]]
| ''[[26/17]]'', ''[[11/7]]''
| [[14/9]], [[20/13]], ''[[32/21]]''
| Aug 5th, dim 6th
| Aug 5th, dim 6th
| A5, d6
| A5, d6
Line 199: Line 257:
| 25
| 25
| 789.5
| 789.5
| [[11/7]], [[30/19]], [[19/12]]
| [[30/19]], [[19/12]]
| ''[[14/9]]''
| [[11/7]]
| Upaug 5th, downminor 6th
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A5, vm6
Line 206: Line 266:
| 26
| 26
| 821.1
| 821.1
| [[8/5]], [[13/8]]
| [[8/5]]
|
| ''[[13/8]]''
| Minor 6th
| Minor 6th
| m6
| m6
Line 213: Line 275:
| 27
| 27
| 852.6
| 852.6
| [[18/11]], [[28/17]]
| [[18/11]]
| [[13/8]]
| [[28/17]]
| Mid 6th
| Mid 6th
| ~6
| ~6
Line 221: Line 285:
| 884.2
| 884.2
| [[5/3]]
| [[5/3]]
| ''[[28/17]]'', ''[[22/13]]''
|
| Major 6th
| Major 6th
| M6
| M6
Line 227: Line 293:
| 29
| 29
| 915.8
| 915.8
| [[32/19]], [[22/13]], [[17/10]]
| [[32/19]], [[17/10]]
|
| [[22/13]]
| Upmajor 6th
| Upmajor 6th
| ^M6
| ^M6
Line 234: Line 302:
| 30
| 30
| 947.4
| 947.4
| [[12/7]], [[19/11]], [[26/15]], [[7/4]]
| [[19/11]]
|
| [[12/7]], [[26/15]], ''[[7/4]]''
| Aug 6th, dim 7th
| Aug 6th, dim 7th
| A6, d7
| A6, d7
Line 241: Line 311:
| 31
| 31
| 978.9
| 978.9
| [[30/17]]
| [[44/25]], [[30/17]]
| ''[[26/15]]'', [[7/4]]
|
| Downminor 7th
| Downminor 7th
| vm7
| vm7
Line 249: Line 321:
| 1010.5
| 1010.5
| [[16/9]], [[34/19]], [[9/5]]
| [[16/9]], [[34/19]], [[9/5]]
|
| [[25/14]]
| Minor 7th
| Minor 7th
| m7
| m7
Line 256: Line 330:
| 1042.1
| 1042.1
| [[20/11]], [[11/6]]
| [[20/11]], [[11/6]]
| ''[[24/13]]'', [[64/35]]
|
| Mid 7th
| Mid 7th
| ~7
| ~7
Line 262: Line 338:
| 34
| 34
| 1073.7
| 1073.7
| [[24/13]], [[13/7]], [[28/15]], [[15/8]]
| [[15/8]]
| [[13/7]]
| [[24/13]], [[13/7]], [[28/15]]
| Major 7th
| Major 7th
| M7
| M7
Line 270: Line 348:
| 1105.3
| 1105.3
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
| ''[[28/15]]''
|
| Upmajor 7th, Downdim 8ve
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^M7, vd8
Line 276: Line 356:
| 36
| 36
| 1136.8
| 1136.8
|
| [[33/17]], [[48/25]]
|
| [[27/14]], [[25/13]], [[52/27]]
| Aug 7th, dim 8ve
| Aug 7th, dim 8ve
| A7, d8
| A7, d8
Line 283: Line 365:
| 37
| 37
| 1168.4
| 1168.4
|
| ''[[64/33]]'', [[88/45]], [[108/55]]
| [[63/32]]
| [[55/28]]
| Down 8ve
| Down 8ve
| v8
| v8
Line 290: Line 374:
| 38
| 38
| 1200.0
| 1200.0
|
| [[2/1]]
|
|  
| Perfect 8ve
| Perfect 8ve
| P8
| P8
Line 298: Line 384:


== Notation ==
== Notation ==
 
=== Ups and downs notation ===
=== Ups and Downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp. The gamut runs D, ^D/vD#, D#, ^D#/vEb, Eb, ^Eb/vE, E, ^E, E#/Fb, vF, F etc.  
{{Ups and downs sharpness}}
{{sharpness-sharp2a}}


=== Quarter-tone notation ===
=== Quarter-tone notation ===
Line 343: Line 428:


== Approximation to JI ==
== Approximation to JI ==
=== Zeta peak index ===
=== Interval mappings ===
{| class="wikitable center-all"
{{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
|-
|-
! colspan="3" | Tuning
! Temperament !! Generator !! Periods per octave
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
| [[Opossum]] || 5\38 || 1
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
|-
| [[166zpi]]
| [[Hemisensi]] || 7\38 || 1
| 37.8901105027757
|-
| 31.6705331305933
| [[Delorean]] / [[subkla]] || 9\38 || 1
| 5.808723
|-
| 0.986660
| [[Migration]] / [[mohajira]] / [[nethertone]] / [[ptolemy]] / [[subklei]] || 11\38 || 1
| 15.046792
|-
| 38edo
| [[Hocus]] || 13\38 || 1
| 1203.48025896255
|-
| 6
| [[Buzzard]] || 15\38 || 1
| 6
|-
| [[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 ==
== Instruments ==
Line 379: Line 521:


== Music ==
== Music ==
* [https://www.youtube.com/watch?v=Cw1Cz1ojoSw Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin] by [[Claudi Meneghin]]
; [[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:38edo| ]]  <!-- Main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Listen]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/38edo"