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→Theory: Mention that 38df preserves 19edo's 2.3.5.7.13 earlier
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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.
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.
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 }}
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 }}

Revision as of 17:50, 30 May 2026

← 37edo 38edo 39edo →
Prime factorization 2 × 19
Step size 31.5789 ¢ 
Fifth 22\38 (694.737 ¢) (→ 11\19)
Semitones (A1:m2) 2:4 (63.16 ¢ : 126.3 ¢)
Consistency limit 5
Distinct consistency limit 5

38 equal divisions of the octave (abbreviated 38edo or 38ed2), also called 38-tone equal temperament (38tet) or 38 equal temperament (38et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 38 equal parts of about 31.6 ¢ each. Each step represents a frequency ratio of 21/38, or the 38th root of 2.

Theory

Since 38 factors as 2 × 19, 38edo can be thought of as two parallel chains of 19edo. It provides a possible correction to the 11th harmonic of 19edo, which works well with 19edo's flat approximations of the 3rd and 5th harmonics, making it a decent 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 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 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 7th and 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 ⟨38 60 88 106 131 140 155 161] (38df in wart notation) can be used, where the 2.3.5.13-subgroup mapping of 19edo is preserved, while harmonics 11, 17, and 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: 38 22 16 12 10 8 8 6 6 5 5 4 4 4 4 3 3 3 3

[Harmonic series 2-20 in 38df]

Prime harmonics

Approximation of prime harmonics in 38edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -7.2 -7.4 +10.1 -14.5 +12.1 -10.2 -13.3 +3.3 +12.5 -8.2
Relative (%) +0.0 -22.9 -23.3 +32.1 -45.8 +38.3 -32.4 -42.1 +10.5 +39.7 -25.9
Steps
(reduced) 		38
(0) 		60
(22) 		88
(12) 		107
(31) 		131
(17) 		141
(27) 		155
(3) 		161
(9) 		172
(20) 		185
(33) 		188
(36)

Intervals

Step Cents Approximated ratios Ups and downs notation*
(EUs: vvA1 and vvd2)
Ratios of the
2.3.5.11.17.19 subgroup 		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

* 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.

Step offset 0 1 2 3 4 5
Sharp symbol   
 
  
   
  
   
Flat symbol
 
  
   
  
   

Quarter-tone notation

Since a sharp raises by two steps, quarter-tone accidentals can also be used:

Step offset −4 −3 −2 −1 0 +1 +2 +3 +4
Symbol  
 		 
 		 
 		 
 		 
 		 
 		 
 		 
 		 

Sagittal notation

This notation uses the same sagittal sequence as EDOs 17, 24, and 31, is a subset of the notation for 76-EDO, and is a superset of the notation for 19-EDO.

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation33/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation33/32

Evo-SZ flavor

Sagittal notationPeriodic table of EDOs with sagittal notation33/32

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

Approximation to JI

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 38edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 38edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/9, 18/11 0.040 0.1
15/11, 22/15 0.109 0.3
5/3, 6/5 0.148 0.5
13/7, 14/13 1.982 6.3
15/13, 26/15 4.891 15.5
13/11, 22/13 4.999 15.8
13/9, 18/13 5.039 16.0
15/14, 28/15 6.873 21.8
11/7, 14/11 6.982 22.1
9/7, 14/9 7.021 22.2
9/5, 10/9 7.070 22.4
11/10, 20/11 7.109 22.5
3/2, 4/3 7.218 22.9
11/6, 12/11 7.258 23.0
5/4, 8/5 7.366 23.3
7/4, 8/7 10.121 32.1
13/8, 16/13 12.104 38.3
13/10, 20/13 12.109 38.3
13/12, 24/13 12.257 38.8
7/5, 10/7 14.091 44.6
7/6, 12/7 14.239 45.1
9/8, 16/9 14.436 45.7
11/8, 16/11 14.476 45.8
15/8, 16/15 14.585 46.2
15-odd-limit intervals in 38edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/9, 18/11 0.040 0.1
15/11, 22/15 0.109 0.3
5/3, 6/5 0.148 0.5
13/7, 14/13 1.982 6.3
9/5, 10/9 7.070 22.4
11/10, 20/11 7.109 22.5
3/2, 4/3 7.218 22.9
11/6, 12/11 7.258 23.0
5/4, 8/5 7.366 23.3
7/4, 8/7 10.121 32.1
13/8, 16/13 12.104 38.3
9/8, 16/9 14.436 45.7
11/8, 16/11 14.476 45.8
15/8, 16/15 14.585 46.2
7/6, 12/7 17.340 54.9
7/5, 10/7 17.488 55.4
13/12, 24/13 19.322 61.2
13/10, 20/13 19.470 61.7
9/7, 14/9 24.558 77.8
11/7, 14/11 24.597 77.9
15/14, 28/15 24.706 78.2
13/9, 18/13 26.540 84.0
13/11, 22/13 26.580 84.2
15/13, 26/15 26.688 84.5
15-odd-limit intervals by 38df val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
11/9, 18/11 0.040 0.1
15/11, 22/15 0.109 0.3
5/3, 6/5 0.148 0.5
13/7, 14/13 1.982 6.3
15/13, 26/15 4.891 15.5
13/11, 22/13 4.999 15.8
13/9, 18/13 5.039 16.0
15/14, 28/15 6.873 21.8
11/7, 14/11 6.982 22.1
9/7, 14/9 7.021 22.2
9/5, 10/9 7.070 22.4
11/10, 20/11 7.109 22.5
3/2, 4/3 7.218 22.9
11/6, 12/11 7.258 23.0
5/4, 8/5 7.366 23.3
13/10, 20/13 12.109 38.3
13/12, 24/13 12.257 38.8
7/5, 10/7 14.091 44.6
7/6, 12/7 14.239 45.1
9/8, 16/9 14.436 45.7
11/8, 16/11 14.476 45.8
15/8, 16/15 14.585 46.2
13/8, 16/13 19.475 61.7
7/4, 8/7 21.457 67.9

Rank-2 temperaments

Rank-2 temperaments 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 stretching the octave, as in 88ed5, 166zpi or 60edt.

Scales

MOS scales
  • 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 scales
This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.
  • 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
This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

Instruments

Music

Bryan Deister
Claudi Meneghin
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