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

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.

Prime harmonics

Approximation of prime harmonics in 94edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.17	-3.33	+1.39	-2.38	+2.03	-2.83	-3.90	-2.74	+4.47	+3.90
	Relative (%)	+0.0	+1.4	-26.1	+10.9	-18.7	+15.9	-22.2	-30.5	-21.5	+35.0	+30.6
Steps (reduced)		94 (0)	149 (55)	218 (30)	264 (76)	325 (43)	348 (66)	384 (8)	399 (23)	425 (49)	457 (81)	466 (90)

Intervals

Main article: Table of 94edo intervals

Assuming 23-limit patent val <94 149 218 264 325 348 384 399 425|, here is a table of intervals as approximated by 94edo steps, and their corresponding 13-limit well-ordered extended diatonic interval names. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'k' (representing 81/80), down from augmented and major, or up from diminished and minor intervals are also added in brackets, along with their associated (5-limit) intervals.

94edo well-ordered extended diatonic (WED) interval names

Step	Cents	13-limit	Short-form WED	Diatonic	Long-form	23-limit
1	12.766	896/891, 243/242, (3125/3072, 245/243, 99/98)	L1, R1			85/84
2	25.532	81/80, 64/63, (50/49)	K1, S1
3	38.298	45/44, 40/39, (250/243, 49/48)	O1, H1			46/45
4	51.064	33/32, (128/125, 36/35)	U1, T1
5	63.830	28/27, 729/704, 27/26, (25/24)	sm2, uA1, tA1, (kkA1)	dd3
6	76.596	22/21, (648/625)	lm2, oA1
7	89.362	256/243, 135/128, (21/20)	m2, kA1	m2		19/18
8	102.128	128/121, (35/33)	Rm2, rA1			17/16, 18/17
9	114.894	16/15, (15/14)	Km2, A1	A1
10	127.660	320/297, 189/176, (14/13)	Om2, LA1
11	140.426	88/81, 13/12, 243/224, (27/25)	n2, Tm2, SA1, (kkm2)
12	153.191	12/11, (35/32)	N2, HA1	ddd4
13	165.957	11/10	oM2
14	178.723	10/9	kM2	d3
15	191.489	121/108, (49/44, 39/35)	rM2			19/17
16	204.255	9/8	M2	M2
17	217.021	112/99, (25/22)	LM2			17/15
18	229.787	8/7	SM2	AA1
19	242.553	15/13	HM2			23/20
20	255.319	52/45	hm3			22/19
21	268.085	7/6, (75/64)	sm3, (kkA2)	dd4
22	280.851	33/28	lm3			20/17
23	293.617	32/27, (25/21)	m3	m3		13/11
24	306.383	144/121, (81/70)	Rm3
25	319.149	6/5	Km3	A2
26	331.915	40/33	Om3			17/14, 23/19
27	344.681	11/9, 39/32, (243/200, 60/49)	n3, Tm3	AAA1
28	357.447	27/22, 16/13, (100/81,49/40)	N3, tM3	ddd5
29	370.213	99/80, (26/21)	oM3			21/17
30	382.979	5/4	kM3	d4
31	395.745	121/96, (34/27)	rM3
32	408.511	81/64, (33/26)	M3	M3		19/15
33	421.277	14/11	LM3			23/18
34	434.043	9/7, (32/25)	SM3, (KKd4)	AA2
35	446.809	135/104, (35/27)	HM3	ddd6		22/17
36	459.574	13/10	h4
37	472.340	21/16	s4	dd5
38	485.106	297/224	l4
39	497.872	4/3	P4	P4
40	510.638	162/121, (35/36)	R4
41	523.404	27/20	K4	A3		19/14, 23/17
42	536.170	15/11	O4
43	548.936	11/8	U4, T4	AAA2
44	561.702	243/176, 18/13, (25/18)	uA4, tA4, (kkA4)	dd6
45	574.468	88/63	ld5, oA4
46	587.234	45/32, (7/5)	kA4	d5
47	600.000	363/256, 512/363, (99/70)	rA4, Rd5			17/12, 24/17
48	612.766			A4
49	625.532
50	638.298
51	651.064			ddd7
52	663.830
53	676.596			d6
54	689.362
55	702.128			P5
56	714.894
57	727.660			AA4
58	740.426
59	753.191
60	765.957			dd7
61	778.723
62	791.489			m6
63	804.255
64	817.021			A5
65	829.787
66	842.553			AAA4
67	855.319			ddd8
68	868.085
69	880.851			d7
70	893.617
71	906.383			M6
72	919.149
73	931.915			AA5
74	944.681
75	957.447
76	970.213			dd8
77	982.979
78	995.745			m7
79	1008.511
80	1021.277			A6
81	1034.043
82	1046.809			AAA5
83	1059.574
84	1072.340
85	1085.106			d8
86	1097.872
87	1110.638			M7
88	1123.404
89	1136.170			AA6
90	1148.936
91	1161.702
92	1174.468
93	1187.234
94	1200.000			P1

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 equal temperament that does better in this subgroup is 193.

Rank-2 temperaments

Periods per 8ve	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 nohajira, 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