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=<span style="background-color: #e5e5e5; color: #008693; font-family: 'Times New Roman',Times,serif; font-size: 113%;">38 tone equal temperament</span>=
{{Infobox ET}}
{{ED intro}}


'''38edo''' divides the octave into 38 equal parts of 31.579 [[cent|cent]]s. Since 38 = 2*19, it can be thought of as two parallel [[19edo|19edo]]s. It [[tempering_out|tempers out]] the same [[5-limit|5-limit]] commas as [[19edo|19]], namely 81/80, 3125/3072 and 15625/15552. In the [[7-limit|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|11-limit]], we can add 121/120 and 176/175.
== 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 and 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.  


===38-EDO Intervals:===
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.


{| class="wikitable"
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 }}.
 
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="2" | '''Step'''
! Step
| colspan="3" | '''Size in'''
! Cents
! 19-odd-limit ratios,<br>in 38df val
! colspan="3" | [[Ups and downs notation]]*
([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and vvd2)
|-
|-
|'''Cents'''
| 0
!pions
| 0.0
!7mus
|
| Perfect 1sn
| P1
| D
|-
|-
| colspan="4"| 0
| 1
| 31.6
|
| Up 1sn
| ^1
| ^D
|-
|-
| | 1
| 2
| | 31.57895
| 63.2
|33.4737
|
|40.42105 (28.6BCA<sub>16</sub>)
| Aug 1sn, dim 2nd
| A1, d2
| D#
|-
|-
| | 2
| 3
| |63.1579
| 94.7
|66.9474
| [[20/19]], [[19/18]], [[18/17]], [[17/16]]
|80.8421 (50.D794<sub>16</sub>)
| Upaug 1sn, downminor 2nd
| ^A1, vm2
| ^D#, vEb
|-
|-
| | 3
| 4
| | 94.7368
| 126.3
|100.42105
| [[16/15]], [[15/14]], [[14/13]], [[13/12]]
|121.2632 (79.435E<sub>16</sub>)
| Minor 2nd
| m2
| Eb
|-
|-
| | 4
| 5
| |126.3157
| 157.9
|133.8947
| [[12/11]], [[11/10]]
|161.6843 (A1.AF29<sub>16</sub>)
| Mid 2nd
| ~2
| vE
|-
|-
| | 5
| 6
| | 157.8947
| 189.5
|167.3684
| [[10/9]], [[19/17]], [[9/8]]
|202.1053 (CA.1AF3<sub>16</sub>)
| Major 2nd
| M2
| E
|-
|-
| | 6
| 7
| |189.4737
| 221.1
|200.8421
| [[17/15]]
|242.5263 (F2.86BD<sub>16</sub>)
| Upmajor 2nd
| ^M2
| ^E
|-
|-
| | 7
| 8
| | 221.0526
| 252.6
|234.3157
| [[8/7]], [[15/13]], [[22/19]], [[7/6]]
|282.9474 (11A.F287<sub>16</sub>)
| Aug 2nd, Dim 3rd
| A2, d3
| E#, Fb
|-
|-
| | 8
| 9
| |252.6316
| 284.2
|267.7895
| [[20/17]], [[13/11]], [[19/16]]
|323.3684 (143.5E51<sub>16</sub>)
| Downminor 3rd
| vm3
| vF
|-
|-
| | 9
| 10
| | 284.2105
| 315.8
|301.2632
| [[6/5]]
|365.7895 (16B.CA1B<sub>16</sub>)
| Minor 3rd
| m3
| F
|-
|-
| | 10
| 11
| |315.7895
| 347.4
|334.7368
| [[17/14]], [[11/9]]
|404.2105 (194.35E5<sub>16</sub>)
| Mid 3rd
| ~3
| ^F
|-
|-
| | 11
| 12
| | 347.3684
| 378.9
|368.2105
| [[16/13]], [[5/4]]
|444.6316 (1BC.A1AF<sub>16</sub>)
| Major 3rd
| M3
| F#
|-
|-
| | 12
| 13
| |378.9474
| 410.5
|401.6843
| [[24/19]], [[19/15]], [[14/11]]
|485.0526 (1E5.0D79<sub>16</sub>)
| Upmajor 3rd, Downdim 4th
| ^M3, vd4
| ^F#, vGb
|-
|-
| | 13
| 14
| | 410.5263
| 442.1
|435.1579
| [[9/7]], [[22/17]], [[13/10]]
|525.4737 (20D.7943<sub>16</sub>)
| Aug 3rd, dim 4th
| A3, d4
| Gb
|-
|-
| | 14
| 15
| |442.1053
| 473.7
|468.6316
| [[17/13]]
|565.8947 (235.E50E<sub>16</sub>)
| Down 4th
| v4
| vG
|-
|-
| | 15
| 16
| | 473.6843
| 505.3
|502.1053
| [[4/3]]
|606.3157 (25E.50D8<sub>16</sub>)
| Perfect 4th
| P4
| G
|-
|-
| | 16
| 17
| |505.2632
| 536.8
|535.57895
| [[19/14]], [[15/11]], [[26/19]], [[11/8]]
|646.7368 (286.BCA2<sub>16</sub>)
| Up 4th
| ^4
| ^G
|-
|-
| | 17
| 18
| | 536.8421
| 568.4
|569.0526
| [[18/13]], [[7/5]]
|687.1579 (2AF.286C<sub>16</sub>)
| Aug 4th
| A4
| G#
|-
|-
| | 18
| 19
| |568.42105
| 600.0
|602.5263
| [[24/17]], [[17/12]]
|727.57895 (2D7.9436<sub>16</sub>)
| Upaug 4th, downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
|-
| | 19
| 20
| | 600
| 631.6
|636
| [[10/7]], [[13/9]]
|768 (300<sub>16</sub>)
| Dim 5th
| d5
| Ab
|-
|-
| | 20
| 21
| |631.57895
| 663.2
|669.4737
| [[16/11]], [[19/13]], [[22/15]], [[28/19]]
|808.42105 (328.6BCA<sub>16</sub>)
| Down 5th
| v5
| vA
|-
|-
| | 21
| 22
| | 663.1579
| 694.7
|702.9474
| [[3/2]]
|848.8421 (350.D794<sub>16</sub>)
| Perfect 5th
| P5
| A
|-
|-
| | 22
| 23
| |694.7368
| 726.3
|736.42105
| [[26/17]]
|889.2632 (379.435E<sub>16</sub>)
| Up 5th
| ^5
| ^A
|-
|-
| | 23
| 24
| | 726.3157
| 757.9
|769.8947
| [[20/13]], [[17/11]], [[14/9]]
|929.6843 (3A1.AF29<sub>16</sub>)
| Aug 5th, dim 6th
| A5, d6
| A#
|-
|-
| | 24
| 25
| |757.8947
| 789.5
|803.3684
| [[11/7]], [[30/19]], [[19/12]]
|970.1053 (3CA.1AF3<sub>16</sub>)
| Upaug 5th, downminor 6th
| ^A5, vm6
| ^A#, vBb
|-
|-
| | 25
| 26
| | 789.4737
| 821.1
|836.8421
| [[8/5]], [[13/8]]
|1010.5263 (3F2.86BD<sub>16</sub>)
| Minor 6th
| m6
| Bb
|-
|-
| | 26
| 27
| |821.0526
| 852.6
|870.3157
| [[18/11]], [[28/17]]
|1050.9474 (41A.F287<sub>16</sub>)
| Mid 6th
| ~6
| vB
|-
|-
| | 27
| 28
| | 852.6316
| 884.2
|903.7895
| [[5/3]]
|1091.3684 (443.5E51<sub>16</sub>)
| Major 6th
| M6
| B
|-
|-
| | 28
| 29
| |884.2105
| 915.8
|937.2632
| [[32/19]], [[22/13]], [[17/10]]
|1131.7895 (46B.CA1B<sub>16</sub>)
| Upmajor 6th
| ^M6
| ^B
|-
|-
| | 29
| 30
| | 915.7895
| 947.4
|970.7368
| [[12/7]], [[19/11]], [[26/15]], [[7/4]]
|1172.2105 (494.35E5<sub>16</sub>)
| Aug 6th, dim 7th
| A6, d7
| B#, Cb
|-
|-
| | 30
| 31
| |947.3684
| 978.9
|1004.2105
| [[30/17]]
|1212.6316 (4BC.A1AF<sub>16</sub>)
| Downminor 7th
| vm7
| vC
|-
|-
| | 31
| 32
| | 978.9474
| 1010.5
|1037.6843
| [[16/9]], [[34/19]], [[9/5]]
|1253.0526 (4E5.0D79<sub>16</sub>)
| Minor 7th
| m7
| C
|-
|-
| | 32
| 33
| |1010.5263
| 1042.1
|1071.1579
| [[20/11]], [[11/6]]
|1293.4737 (50D.7943<sub>16</sub>)
| Mid 7th
| ~7
| ^C
|-
|-
| | 33
| 34
| | 1042.1053
| 1073.7
|1104.6316
| [[24/13]], [[13/7]], [[28/15]], [[15/8]]
|1333.8947 (535.E50E<sub>16</sub>)
| Major 7th
| M7
| C#
|-
|-
| | 34
| 35
| |1073.6843
| 1105.3
|1138.1053
| [[32/17]], [[17/9]], [[36/19]], [[19/10]]
|1374.3157 (55E.50D8<sub>16</sub>)
| Upmajor 7th, Downdim 8ve
| ^M7, vd8
| ^C#, vDb
|-
|-
| | 35
| 36
| | 1105.2632
| 1136.8
|1171.57895
|
|1414.7368 (586.BCA2<sub>16</sub>)
| Aug 7th, dim 8ve
| A7, d8
| Db
|-
|-
| | 36
| 37
| |1136.8421
| 1168.4
|1205.0526
|
|1455.1579 (5AF.286C<sub>16</sub>)
| Down 8ve
| v8
| vD
|-
|-
| | 37
| 38
| | 1168.42105
| 1200.0
|1235.5263
|
|1495.57895 (5D7.9436<sub>16</sub>)
| Perfect 8ve
|-
| P8
|38
| D
|1200
|1272
|1536 (600<sub>16</sub>)
|}
|}
[[Category:edo]]
<nowiki/>* Ups and downs may be substituted with semi-sharps and semi-flats, respectively
[[Category:theory]]
 
== Notation ==
=== Ups and downs notation ===
Spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.
{{sharpness-sharp2a}}
 
=== 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}}
 
== 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)
 
; [[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]]
[[Category:Todo:add rank 2 temperaments table]]
Retrieved from "https://en.xen.wiki/w/38edo"