38edo

← 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 2^1/38, or the 38th root of 2.

Theory

Since 38 = 2 × 19, it can be thought of as two parallel 19edos. While the halving of the step size lowers consistency and leaves it only mediocre in terms of overall 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.

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 11-limit, we can add 121/120 and 176/175.

Using the 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 consistent within the 38df val ⟨38 60 88 106 131 140 155 161].

The harmonic series from 1 to 20 is approximated within 38df by the 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	19-odd-limit ratios, patent val	19-odd-limit ratios, in 38df val	Ups and downs notation* (EUs: vvA1 and vvd2)
0	0.0	1/1	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	20/19, 19/18, 18/17, 17/16	Upaug 1sn, downminor 2nd	^A1, vm2	^D#, vEb
4	126.3	16/15, 15/14, 14/13	16/15, 15/14, 14/13, 13/12	Minor 2nd	m2	Eb
5	157.9	13/12, 12/11, 11/10	12/11, 11/10	Mid 2nd	~2	vE
6	189.5	10/9, 19/17, 9/8	10/9, 19/17, 9/8	Major 2nd	M2	E
7	221.1	17/15, 8/7, 15/13	17/15	Upmajor 2nd	^M2	^E
8	252.6	22/19	8/7, 15/13, 22/19, 7/6	Aug 2nd, Dim 3rd	A2, d3	E#, Fb
9	284.2	7/6, 20/17, 19/16	20/17, 13/11, 19/16	Downminor 3rd	vm3	vF
10	315.8	13/11, 6/5, 17/14	6/5	Minor 3rd	m3	F
11	347.4	16/13, 11/9	17/14, 11/9	Mid 3rd	~3	^F
12	378.9	5/4	16/13, 5/4	Major 3rd	M3	F#
13	410.5	9/7, 24/19, 19/15	24/19, 19/15, 14/11	Upmajor 3rd, Downdim 4th	^M3, vd4	^F#, vGb
14	442.1	14/11, 22/17, 17/13	9/7, 22/17, 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	4/3	Perfect 4th	P4	G
17	536.8	15/11, 11/8, 18/13	19/14, 15/11, 26/19, 11/8	Up 4th	^4	^G
18	568.4	26/19	18/13, 7/5	Aug 4th	A4	G#
19	600.0	7/5, 24/17, 17/12, 10/7	24/17, 17/12	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	16/11, 19/13, 22/15, 28/19	Down 5th	v5	vA
22	694.7	3/2, 28/19	3/2	Perfect 5th	P5	A
23	726.3	20/13	26/17	Up 5th	^5	^A
24	757.9	17/11, 11/7, 26/17	20/13, 17/11, 14/9	Aug 5th, dim 6th	A5, d6	A#
25	789.5	14/9, 19/12, 30/19	11/7, 30/19, 19/12	Upaug 5th, downminor 6th	^A5, vm6	^A#, vBb
26	821.1	8/5	8/5, 13/8	Minor 6th	m6	Bb
27	852.6	18/11, 13/8	18/11, 28/17	Mid 6th	~6	vB
28	884.2	22/13, 5/3, 28/17	5/3	Major 6th	M6	B
29	915.8	32/19, 17/10, 12/7	32/19, 22/13, 17/10	Upmajor 6th	^M6	^B
30	947.4	19/11	12/7, 19/11, 26/15, 7/4	Aug 6th, dim 7th	A6, d7	B#, Cb
31	978.9	30/17, 7/4, 26/15	30/17	Downminor 7th	vm7	vC
32	1010.5	9/5, 16/9, 34/19	16/9, 34/19, 9/5	Minor 7th	m7	C
33	1042.1	11/6, 20/11, 24/13	20/11, 11/6	Mid 7th	~7	^C
34	1073.7	13/7, 15/8, 28/25	24/13, 13/7, 28/15, 15/8	Major 7th	M7	C#
35	1105.3	32/17, 17/9, 36/19, 19/10	32/17, 17/9, 36/19, 19/10	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	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

Revo flavor

Evo-SZ flavor

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.

• 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-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

• Spirit of the Night - Secret of Mana (microtonal cover in 38edo) (2025)
• 38edo improv (2025)
• waltz in 38edo (2026)
  • [short] (demonstrates Lumatone mapping)
  • [full version]

Claudi Meneghin

• Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin (2022)