94edo

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94edo

95edo →

Prime factorization	2 × 47
Step size	12.766 ¢
Fifth	55\94 (702.128 ¢) (semiconvergent)
Semitones (A1:m2)	9:7 (114.9 ¢ : 89.36 ¢)
Consistency limit	23
Distinct consistency limit	13

The 94 equal divisions of the octave (94edo), or the 94(-tone) equal temperament (94tet, 94et) when viewed from a regular temperament perspective, results from dividing the octave into 94 equally-sized steps, where each step is about 12.8 cents.

Theory

94edo is a remarkable all-around utility tuning system, good from low prime limit to very high prime limit situations. It is the first edo to be consistent through the 23-odd-limit, and no other edo is so consistent until 282 and 311 make their appearance.

The list of 23-limit commas it tempers out is huge, but it is worth noting that it tempers out 32805/32768 and is thus a schismatic system, that it tempers out 225/224 and 385/384 and so is a marvel system, and that it also tempers out 3125/3087, 4000/3969, 5120/5103 and 540/539. It provides the optimal patent val for the rank-5 temperament tempering out 275/273, and for a number of other temperaments, such as isis.

94edo is an excellent EDO for Carlos Beta scale, since the difference between 5 steps of 94edo and 1 step of Carlos Beta is only -0.00314534 cents.

Intervals

Main article: Table of 94edo intervals

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[149 -94⟩	[⟨94 149]]	-0.054	0.054	0.43
2.3.5	32805/32768, 9765625/9565938	[⟨94 149 218]]	+0.442	0.704	5.52
2.3.5.7	225/224, 3125/3087, 118098/117649	[⟨94 149 218 264]]	+0.208	0.732	5.74
2.3.5.7.11	225/224, 385/384, 1331/1323, 2200/2187	[⟨94 149 218 264 325]]	+0.304	0.683	5.35
2.3.5.7.11.13	225/224, 275/273, 325/324, 385/384, 1331/1323	[⟨94 149 218 264 325 348]]	+0.162	0.699	5.48
2.3.5.7.11.13.17	170/169, 225/224, 275/273, 289/288, 325/324, 385/384	[⟨94 149 218 264 325 348 384]]	+0.238	0.674	5.28
2.3.5.7.11.13.17.19	170/169, 190/189, 225/224, 275/273, 289/288, 325/324, 385/384	[⟨94 149 218 264 325 348 384 399]]	+0.323	0.669	5.24
2.3.5.7.11.13.17.19.23	170/169, 190/189, 209/208, 225/224, 275/273, 289/288, 300/299, 323/322	[⟨94 149 218 264 325 348 384 399 425]]	+0.354	0.637	4.99

94et is lower in relative error than any previous equal temperaments in the 23-limit, and the next ET that does better in this subgroup is 193.

Rank-2 temperaments

Periods per octave	Generator	Cents	Associated ratio	Temperament
1	3\94	38.30	49/48	Slender
1	5\94	63.83	25/24	Sycamore / betic
1	11\94	140.43	243/224	Tsaharuk / quanic
1	13\94	165.96	11/10	Tertiaschis
1	19\94	242.55	147/128	Septiquarter
1	39\94	497.87	4/3	Helmholtz / garibaldi / cassandra
2	2\94	25.53	64/63	Ketchup
2	11\94	140.43	27/25	Fifive
2	30\94	382.98	5/4	Wizard / gizzard
2	34\94	434.04	9/7	Pogo / supers
2	43\94	548.94	11/8	Kleischismic

Below are some 23-limit temperaments supported by 94et. It might be noted that 94, a very good tuning for garibaldi temperament, shows us how to extend it to the 23-limit.

• 46&94 ⟨⟨ 8 30 -18 -4 -28 8 -24 2 … ]]
• 68&94 ⟨⟨ 20 28 2 -10 24 20 34 52 … ]]
• 53&94 ⟨⟨ 1 -8 -14 23 20 -46 -3 -35 … ]] (one garibaldi)
• 41&94 ⟨⟨ 1 -8 -14 23 20 48 -3 -35 … ]] (another garibaldi, only differing in the mappings of 17 and 23)
• 135&94 ⟨⟨ 1 -8 -14 23 20 48 -3 59 … ]] (another garibaldi)
• 130&94 ⟨⟨ 6 -48 10 -50 26 6 -18 -22 … ]] (a pogo extension)
• 58&94 ⟨⟨ 6 46 10 44 26 6 -18 -22 … ]] (a supers extension)
• 50&94 ⟨⟨ 24 -4 40 -12 10 24 22 6 … ]]
• 72&94 ⟨⟨ 12 -2 20 -6 52 12 -36 -44 … ]] (a gizzard extension)
• 80&94 ⟨⟨ 18 44 30 38 -16 18 40 28 … ]]
• 94 solo ⟨⟨ 12 -2 20 -6 -42 12 -36 -44 … ]] (a rank one temperament!)

Temperaments to which 94et can be detempered:

• Satin (94&311) ⟨⟨ 3 70 -42 69 -34 50 85 83 … ]]
• 94&422 ⟨⟨ 8 124 -18 90 -28 102 164 96 … ]]

Scales

• Garibaldi12
• Garibaldi17

Since 94edo has a step of 12.766 cents, it also allows one to use its MOS scales as circulating temperaments and is the first edo to allows one to use a Mohajira, Pajara or Miracle MOS scale a as circulating temperament^{[clarification needed]}.

Circulating temperaments in 94edo

Tones	Pattern	L:s
5	4L 1s	19:18
6	4L 2s	16:15
7	3L 4s	14:13
8	6L 2s	12:11
9	4L 5s	11:10
10	4L 6s	10:9
11	6L 5s	9:8
12	10L 2s	8:7
13	3L 10s
14	10L 4s	7:6
15	4L 11s
16	14L 2s	6:5
17	9L 8s
18	4L 14s
19	18L 1s	5:4
20	14L 6s
21	10L 11s
22	6L 16s
23	2L 21s
24	22L 2s	4:3
25	19L 6s
26	16L 10s
27	13L 14s
28	10L 18s
29	7L 22s
30	4L 22s
31	1L 30s
32	30L 2s	3:2
33	28L 5s
34	26L 8s
35	24L 11s
36	22L 14s
37	20L 17s
38	18L 20s
39	16L 23s
40	14L 26s
41	13L 28s
42	10L 32s
43	8L 35s
44	6L 38s
45	4L 41s
46	2L 44s
47	47edo	equal
48	46L 2s	2:1
49	45L 4s
50	44L 6s
51	43L 8s
52	42L 10s
53	41L 12s
54	40L 14s
55	39L 16s
56	38L 18s
57	37L 20s
58	36L 22s
59	35L 24s
60	34L 26s
61	33L 28s
62	32L 30s
63	31L 32s
64	30L 34s
65	29L 36s
66	28L 38s
67	27L 40s
68	26L 42s
69	25L 44s
70	24L 46s
71	23L 48s
72	22L 50s
73	21L 52s
74	20L 54s
75	19L 56s