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

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 & 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. 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.

In 38df, all primes are flat from 3 to 19.

In 38df, the mapping of all 19-odd-limit intervals is identical to their direct best approximation in 38edo, excepting, of course, for 7/4 and 13/8, as well as their octave complements 8/7 and 16/13, which are mapped to their second-best steps within 38df.

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-limit ratios, treated as 38df	Ups and Downs Notation*
0	0.0000		Perfect 1sn	P1	D
1	31.5789		Up 1sn	^1	^D
2	63.1579		Aug 1sn, dim 2nd	A1, d2	D#
3	94.7368	20/19, 19/18, 18/17, 17/16	Upaug 1sn, downminor 2nd	^A1, vm2	^D#, vEb
4	126.3157	16/15, 15/14, 14/13, 13/12	Minor 2nd	m2	Eb
5	157.8947	12/11, 11/10	Mid 2nd	~2	vE
6	189.4737	10/9, 19/17, 9/8	Major 2nd	M2	E
7	221.0526	17/15	Upmajor 2nd	^M2	^E
8	252.6316	8/7, 15/13, 22/19, 7/6	Aug 2nd, Dim 3rd	A2, d3	E#, Fb
9	284.2105	20/17, 13/11, 19/16	Downminor 3rd	vm3	vF
10	315.7895	6/5	Minor 3rd	m3	F
11	347.3684	17/14, 11/9	Mid 3rd	~3	^F
12	378.9474	16/13, 5/4	Major 3rd	M3	F#
13	410.5263	24/19, 19/15, 14/11	Upmajor 3rd, Downdim 4th	^M3, vd4	^F#, vGb
14	442.1053	9/7, 22/17, 13/10	Aug 3rd, dim 4th	A3, d4	Gb
15	473.6843	17/13	Down 4th	v4	vG
16	505.2632	4/3	Perfect 4th	P4	G
17	536.8421	19/14, 15/11, 26/19, 11/8	Up 4th	^4	^G
18	568.4211	18/13, 7/5	Aug 4th	A4	G#
19	600.0000	24/17, 17/12	Upaug 4th, downdim 5th	^A4, vd5	^G#, vAb
20	631.5789	10/7, 13/9	Dim 5th	d5	Ab
21	663.1579	16/11, 19/13, 22/15, 28/19	Down 5th	v5	vA
22	694.7368	3/2	Perfect 5th	P5	A
23	726.3157	26/17	Up 5th	^5	^A
24	757.8947	20/13, 17/11, 14/9	Aug 5th, dim 6th	A5, d6	A#
25	789.4737	11/7, 30/19, 19/12	Upaug 5th, downminor 6th	^A5, vm6	^A#, vBb
26	821.0526	8/5, 13/8	Minor 6th	m6	Bb
27	852.6316	18/11, 28/17	Mid 6th	~6	vB
28	884.2105	5/3	Major 6th	M6	B
29	915.7895	32/19, 22/13, 17/10	Upmajor 6th	^M6	^B
30	947.3684	12/7, 19/11, 26/15, 7/4	Aug 6th, dim 7th	A6, d7	B#, Cb
31	978.9474	30/17	Downminor 7th	vm7	vC
32	1010.5263	16/9, 34/19, 9/5	Minor 7th	m7	C
33	1042.1053	20/11, 11/6	Mid 7th	~7	^C
34	1073.6843	24/13, 13/7, 28/15, 15/8	Major 7th	M7	C#
35	1105.2632	32/17, 17/9, 36/19, 19/10	Upmajor 7th, Downdim 8ve	^M7, vd8	^C#, vDb
36	1136.8421		Aug 7th, dim 8ve	A7, d8	Db
37	1168.4211		Down 8ve	v8	vD
38	1200.0000		Perfect 8ve	P8	D

* Ups and downs may be substituted with semi-sharps and semi-flats, respectively

Instruments

• Lumatone mapping for 38edo
• Skip fretting system 38 2 11

Music

• Canon at the Semitone on The Mother's Malison Theme for Cor Anglais and Violin by Claudi Meneghin