388edo: Difference between revisions

← 387edo 388edo 389edo →
Prime factorization	22 × 97
Step size	3.09278 ¢
Fifth	227\388 (702.062 ¢)
Semitones (A1:m2)	37:29 (114.4 ¢ : 89.69 ¢)
Consistency limit	37
Distinct consistency limit	27

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Revision as of 10:41, 20 October 2025

388 equal divisions of the octave (abbreviated 388edo or 388ed2), also called 388-tone equal temperament (388tet) or 388 equal temperament (388et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 388 equal parts of about 3.09 ¢ each. Each step represents a frequency ratio of 2^1/388, or the 388th root of 2.

Theory

388edo is the first edo that is distinctly consistent through to the 27-odd-limit; it is also consistent through the 37-odd-limit.

The equal temperament tempers out the vishnuzma, [23 6 -14⟩, the alphatricot comma, [39 -29 3⟩, the minortone comma, [-16 35 -17⟩, and the raider comma, [71 -99 31⟩, in the 5-limit, giving a strong tuning. It tempers out 4375/4374 and 235298/234375 in the 7-limit, and 3025/3024, 5632/5625 and 9801/9800 in the 11-limit and 847/845, 1001/1000 and 4096/4095 in the 13-limit.

It provides the optimal patent val for the rank-5 cuthbert temperament, which tempers out 847/845, the cuthbert comma, and for a number of other temperaments tempering it out, e.g. neusec, the 190 & 198 temperament. By tempering out cuthbert it supports cuthbert chords, in addition to sinbadmic chords.

Prime harmonics

Approximation of prime harmonics in 388edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31	37
Error	Absolute (¢)	+0.00	+0.11	+0.28	-0.78	-0.80	+0.71	+0.20	-0.61	-0.44	+0.32	-0.71	-0.83
Error	Relative (%)	+0.0	+3.5	+9.2	-25.4	-25.9	+22.9	+6.4	-19.6	-14.2	+10.3	-22.8	-26.8
Steps (reduced)		388 (0)	615 (227)	901 (125)	1089 (313)	1342 (178)	1436 (272)	1586 (34)	1648 (96)	1755 (203)	1885 (333)	1922 (370)	2021 (81)

Approximation of prime harmonics in 388edo
Harmonic		41	43	47	53	59	61	67	71	73	79	83	89
Error	Absolute (¢)	+0.83	-1.21	-0.56	-1.34	-1.44	-0.39	+1.11	-0.32	+1.08	+0.41	+1.50	+1.28
Error	Relative (%)	+27.0	-39.1	-18.0	-43.3	-46.6	-12.6	+35.7	-10.2	+34.8	+13.3	+48.5	+41.5
Steps (reduced)		2079 (139)	2105 (165)	2155 (215)	2222 (282)	2282 (342)	2301 (361)	2354 (26)	2386 (58)	2402 (74)	2446 (118)	2474 (146)	2513 (185)

Approximation of prime harmonics in 388edo
Harmonic		97	101	103	107	109	113	127	131	137	139	149	151
Error	Absolute (¢)	+0.72	-1.19	-1.12	+0.96	-0.17	-0.71	+1.21	+0.10	-0.11	-0.46	-0.12	+1.54
Error	Relative (%)	+23.4	-38.6	-36.2	+31.1	-5.6	-22.9	+39.0	+3.2	-3.6	-14.9	-3.7	+49.9
Steps (reduced)		2561 (233)	2583 (255)	2594 (266)	2616 (288)	2626 (298)	2646 (318)	2712 (384)	2729 (13)	2754 (38)	2762 (46)	2801 (85)	2809 (93)

Subsets and supersets

Since 388 factors into primes as 2² × 97, 388edo has subset edos 2, 4, 97, and 194.

Approximation to JI

This edo has a high consistency limit, although due to 311edo having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.

388edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic 7. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.

311edo also deals better with composite harmonics than 388. 311edo is consistent to the 41-limit 77-odd-limit, while 388edo has inconsistencies involving composite harmonics as low as 39, and harmonic 49 itself is inconsistent. The 7th and 11th harmonics both being flat by just over 25% of a step is less than ideal. However, it approximates some higher primes better than 311 does. The only inconsistencies in the 41-odd-limit in 388edo are 39/28, 39/22 ,39/37, 41/28, 41/22, 41/37 and their octave complements. This is due to the fact that harmonics 39 and 47 are quite sharp, both just over 1/4 of a step. 311edo misses most primes after 41, though it hits 73, 89, (101,) 109, 113, 139, 149, and 151. 388, on the other hand, hits primes 47, 61, 71, 79, 97, 109, 113, 131, 137, 139, and 149. Still, 311 does much better at composite harmonics due to having much lower error in the 13-limit, which is also important to note by itself, though if one wants to approximate the 13-limit specifically they may prefer 270edo. Note that 311 has generally higher absolute errors than 388 due to its smaller size, but smaller size also means the system is easier to handle.

Another system notable in high limits around this size is 422edo.

The following table shows how 37-odd-limit intervals are represented in 388edo. Prime harmonics are in bold.

As 388edo is consistent in the 37-odd-limit, the mappings by direct approximation and through the patent val are identical.

37-odd-limit intervals in 388edo
Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
29/27, 54/29	0.001	0.0
33/31, 62/33	0.010	0.3
17/9, 18/17	0.014	0.5
11/7, 14/11	0.018	0.6
37/22, 44/37	0.026	0.8
31/21, 42/31	0.028	0.9
29/20, 40/29	0.035	1.1
27/20, 40/27	0.036	1.2
37/28, 56/37	0.044	1.4
35/23, 46/35	0.061	2.0
9/5, 10/9	0.071	2.3
29/15, 30/29	0.071	2.3
21/19, 38/21	0.072	2.3
31/28, 56/31	0.079	2.6
17/10, 20/17	0.085	2.7
33/28, 56/33	0.089	2.9
33/19, 38/33	0.090	2.9
17/12, 24/17	0.092	3.0
31/22, 44/31	0.097	3.1
31/19, 38/31	0.100	3.2
35/19, 38/35	0.105	3.4
29/18, 36/29	0.106	3.4
3/2, 4/3	0.107	3.5
29/17, 34/29	0.120	3.9
27/17, 34/27	0.121	3.9
37/31, 62/37	0.123	4.0
21/11, 22/21	0.125	4.0
37/33, 66/37	0.133	4.3
25/13, 26/25	0.141	4.6
37/21, 42/37	0.151	4.9
23/19, 38/23	0.167	5.4
5/3, 6/5	0.177	5.7
19/14, 28/19	0.179	5.8
17/15, 30/17	0.192	6.2
35/33, 66/35	0.195	6.3
19/11, 22/19	0.197	6.4
17/16, 32/17	0.199	6.4
35/31, 62/35	0.205	6.6
29/24, 48/29	0.213	6.9
9/8, 16/9	0.214	6.9
37/19, 38/37	0.223	7.2
23/21, 42/23	0.239	7.7
27/25, 50/27	0.248	8.0
29/25, 50/29	0.249	8.0
33/23, 46/33	0.256	8.3
31/23, 46/31	0.266	8.6
5/4, 8/5	0.284	9.2
35/22, 44/35	0.302	9.8
15/13, 26/15	0.318	10.3
29/16, 32/29	0.320	10.3
27/16, 32/27	0.321	10.4
37/35, 70/37	0.328	10.6
23/14, 28/23	0.345	11.2
25/18, 36/25	0.355	11.5
23/22, 44/23	0.363	11.7
25/17, 34/25	0.369	11.9
27/26, 52/27	0.389	12.6
37/23, 46/37	0.389	12.6
29/26, 52/29	0.390	12.6
15/8, 16/15	0.391	12.6
13/10, 20/13	0.425	13.7
23/16, 32/23	0.439	14.2
25/24, 48/25	0.462	14.9
13/9, 18/13	0.496	16.0
35/32, 64/35	0.500	16.2
17/13, 26/17	0.510	16.5
23/12, 24/23	0.546	17.7
25/16, 32/25	0.568	18.4
13/12, 24/13	0.603	19.5
19/16, 32/19	0.606	19.6
35/24, 48/35	0.607	19.6
23/17, 34/23	0.639	20.6
23/18, 36/23	0.653	21.1
21/16, 32/21	0.678	21.9
33/32, 64/33	0.696	22.5
35/34, 68/35	0.700	22.6
31/16, 32/31	0.706	22.8
13/8, 16/13	0.709	22.9
19/12, 24/19	0.713	23.0
35/18, 36/35	0.714	23.1
23/20, 40/23	0.724	23.4
29/23, 46/29	0.759	24.5
27/23, 46/27	0.760	24.6
7/4, 8/7	0.785	25.4
11/8, 16/11	0.802	25.9
19/17, 34/19	0.805	26.0
31/24, 48/31	0.813	26.3
19/18, 36/19	0.820	26.5
35/29, 58/35	0.820	26.5
35/27, 54/35	0.821	26.5
37/32, 64/37	0.829	26.8
23/15, 30/23	0.830	26.8
21/17, 34/21	0.877	28.4
19/10, 20/19	0.890	28.8
7/6, 12/7	0.892	28.8
33/17, 34/33	0.895	28.9
31/17, 34/31	0.905	29.3
11/6, 12/11	0.909	29.4
31/18, 36/31	0.919	29.7
29/19, 38/29	0.926	29.9
27/19, 38/27	0.926	30.0
37/24, 48/37	0.935	30.2
21/20, 40/21	0.962	31.1
33/20, 40/33	0.980	31.7
17/14, 28/17	0.984	31.8
31/20, 40/31	0.990	32.0
19/15, 30/19	0.997	32.2
29/21, 42/29	0.998	32.3
9/7, 14/9	0.998	32.3
17/11, 22/17	1.002	32.4
25/23, 46/25	1.008	32.6
33/29, 58/33	1.015	32.8
11/9, 18/11	1.016	32.9
31/29, 58/31	1.025	33.2
31/27, 54/31	1.026	33.2
37/34, 68/37	1.028	33.2
37/36, 72/37	1.042	33.7
7/5, 10/7	1.069	34.6
11/10, 20/11	1.087	35.1
31/30, 60/31	1.097	35.5
29/28, 56/29	1.104	35.7
27/14, 28/27	1.105	35.7
37/20, 40/37	1.113	36.0
29/22, 44/29	1.122	36.3
27/22, 44/27	1.123	36.3
37/29, 58/37	1.148	37.1
23/13, 26/23	1.149	37.1
37/27, 54/37	1.149	37.2
25/19, 38/25	1.174	38.0
15/14, 28/15	1.176	38.0
15/11, 22/15	1.194	38.6
35/26, 52/35	1.210	39.1
37/30, 60/37	1.220	39.4
25/21, 42/25	1.246	40.3
33/25, 50/33	1.264	40.9
31/25, 50/31	1.274	41.2
19/13, 26/19	1.315	42.5
25/14, 28/25	1.353	43.8
25/22, 44/25	1.371	44.3
21/13, 26/21	1.387	44.9
37/25, 50/37	1.397	45.2
33/26, 52/33	1.405	45.4
31/26, 52/31	1.415	45.8
13/7, 14/13	1.494	48.3
13/11, 22/13	1.512	48.9
37/26, 52/37	1.538	49.7

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[615 -388⟩	[⟨388 615]]	+0.0337	0.0337	1.09
2.3.5	[23 6 -14⟩, [39 -29 3⟩	[⟨388 615 901]]	−0.0633	0.0501	1.62
2.3.5.7	4375/4374, 235298/234375, 2100875/2097152	[⟨388 615 901 1089]]	+0.0224	0.1546	5.00
2.3.5.7.11	3025/3024, 4375/4374, 5632/5625, 235298/234375	[⟨388 615 901 1089 1342]]	+0.0643	0.1617	5.23
2.3.5.7.11.13	847/845, 1001/1000, 3025/3024, 4096/4095, 4375/4374	[⟨388 615 901 1089 1342 1436]]	+0.0216	0.1758	5.68
2.3.5.7.11.13.17	833/832, 847/845, 1001/1000, 1089/1088, 1225/1224, 1701/1700	[⟨388 615 901 1089 1342 1436 1586]]	+0.0116	0.1646	5.32
2.3.5.7.11.13.17.19	833/832, 847/845, 1001/1000, 1089/1088, 1216/1215, 1225/1224, 1331/1330	[⟨388 615 901 1089 1342 1436 1586 1648]]	+0.0280	0.1600	5.17

388et has a lower absolute error in the 5-limit than any previous equal temperaments, past 323 and followed by 441.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	59\388	182.47	10/9	Mitonic
1	111\388	343.30	8000/6561	Raider
1	145\388	448.45	35/27	Semidimfourth
1	183\388	565.97	75/52	Alphatrillium / pseudotrillium
2	23\388	71.13	25/24	Vishnu / ananta
2	49\388	151.54	12/11	Neusec
4	123\388 (26\388)	380.41 (80.41)	81/65 (22/21)	Quasithird
97	161\388 (1\388)	497.938 (3.09)	4/3 (?)	Berkelium

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

@@ Line 14: / Line 14: @@
 {{Harmonics in equal|388|start=25|columns=12|collapsed=1}}
-=== Approximation to JI ===
+=== Subsets and supersets ===
-This EDO has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.
+Since 388 factors into primes as {{nowrap| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.
+== Approximation to JI ==
+This edo has a high consistency limit, although due to [[311edo]] having a higher consistency limit, among other things, it is mostly unexplored. However, it still makes for an interesting comparison.
 edo has a consistency limit of 37 compared to 311edo's consistency limit of 41, so most people wishing to approximate higher-limit JI choose the latter. 311edo also approximates harmonics up to 42 with under 25% error, while 388edo fails as early as harmonic [[7/1|7]]. One thing to note is that 388edo's distinct consistency limit is 27 compared to 311edo's 23, so those who want distinct consistency in the 27-odd-limit may choose 388.
@@ Line 24: / Line 27: @@
 {{Q-odd-limit intervals|388|limit=37}}
-=== Subsets and supersets ===
-Since 388 factors into {{nowrap| 2<sup>2</sup> × 97 }}, 388edo has subset edos {{EDOs| 2, 4, 97, and 194 }}.
 == Regular temperament properties ==

388edo: Difference between revisions

Revision as of 10:41, 20 October 2025

Contents

Theory

Prime harmonics

Subsets and supersets

Approximation to JI

Regular temperament properties

Rank-2 temperaments

Navigation menu

388edo: Difference between revisions

Revision as of 10:41, 20 October 2025

Theory

Prime harmonics

Subsets and supersets

Approximation to JI

Regular temperament properties

Rank-2 temperaments

Navigation menu

Search