94edo

← 93edo

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.

94edo can be thought of as two sets of 47edo offset by one step of 94edo. It inherits from 47edo's good approximations of primes 5, 7, 13 and 17, while it dramatically improves on prime 3, as well as primes 11, 19 and 23 to a lesser degree. It can also be thought of as the "sum" of 41edo and 53edo (41 + 53 = 94), both of which are known for their approximation of Pythagorean tuning. Therefore 94edo's fifth is the mediant of these two tunings' fifths; it is slightly sharp of just and less accurate than 53edo's fifth, but more accurate than 41edo's.

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 1 step of Carlos Beta and 5 steps of 94edo 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

See also: 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. 'S/s' indicates alteration by the septimal comma, 64/63; 'K/k' indicates alteration by the syntonic comma, 81/80; 'U/u' by the undecimal quartertone, 33/32; 'L/l' by pentacircle comma, 896/891; 'O/o' by 45/44; 'R/r' by the rastma, 243/242; 'T/t' by the tridecimal quartertone, 1053/1024; and finally, 'H/h', by 40/39. Capital letters alter downward, lowercase alter upwards. Important 13-limit intervals approximated that are not associated with the extended diatonic interval names are added in brackets. Multiple alterations by 'K' 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 WOFED interval names

Step	Cents	13-limit	23-limit	Short-form WOFED	Long-form WOFED	Diatonic
1	12.766	896/891, 243/242, (3125/3072, 245/243, 100/99, 99/98)	85/84	L1, R1	large unison, rastma
2	25.532	81/80, 64/63, (50/49)		K1, S1	komma, super unison
3	38.298	45/44, 40/39, (250/243, 49/48)	46/45	O1, H1	on unison, hyper unison
4	51.064	33/32, (128/125, 36/35, 35/34, 34/33)		U1, T1, hm2	uber unison, tall unison, hypo minor second
5	63.830	28/27, 729/704, 27/26, (25/24)		sm2, uA1, tA1, (kkA1)	sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison)	dd3
6	76.596	22/21, (648/625, 26/25)	23/22, 24/23	lm2, oA1	little minor second, off augmented unison
7	89.362	256/243, 135/128, (21/20)	19/18, 20/19	m2, kA1	minor second, komma-down augmented unison	m2
8	102.128	128/121, (35/33)	17/16, 18/17	Rm2, rA1	rastmic minor second, rastmic augmented unison
9	114.894	16/15, (15/14)		Km2, A1	classic minor second, augmented unison	A1
10	127.660	320/297, 189/176, (14/13)		Om2, LA1	oceanic minor second, large augmented unison
11	140.426	88/81, 13/12, 243/224, (27/25)	25/23, 38/35	n2, Tm2, SA1, (KKm2)	lesser neutral second, tall minor second, super augmented unison, (2-komma-up minor second)
12	153.191	12/11, (35/32)	23/21	N2, tM2, HA1	greater netral second, tiny major second, hyper augmented unison	ddd4
13	165.957	11/10		oM2	off major second
14	178.723	10/9	21/19	kM2	komma-down major second	d3
15	191.489	121/108, (49/44, 39/35)	19/17	rM2	rastmic major second
16	204.255	9/8		M2	major second	M2
17	217.021	112/99, (25/22)	17/15, 26/23	LM2	large major second
18	229.787	8/7		SM2	super major second	AA1
19	242.553	15/13	23/20, 38/33	HM2	hyper major second
20	255.319	52/45	22/19	hm3	hypo minor third
21	268.085	7/6, (75/64)		sm3, (kkA2)	sub minor third, (classic augmented second)	dd4
22	280.851	33/28	20/17, 27/23	lm3	little minor third
23	293.617	32/27, (25/21, 13/11)	19/16	m3	minor third	m3
24	306.383	144/121, (81/70)		Rm3	rastmic minor third
25	319.149	6/5		Km3	classic minor third	A2
26	331.915	40/33	17/14, 23/19	Om3	on minor third
27	344.681	11/9, 39/32, (243/200, 60/49)	28/23	n3, Tm3	lesser neutral third, tall minor third	AAA1
28	357.447	27/22, 16/13, (100/81,49/40)		N3, tM3	greater neutral third, tiny major third	ddd5
29	370.213	99/80, (26/21)	21/17	oM3	off major third
30	382.979	5/4		kM3	classic major third	d4
31	395.745	121/96, (34/27)		rM3	rastmic major third
32	408.511	81/64, (33/26)	19/15, 24/19	M3	major third	M3
33	421.277	14/11	23/18	LM3	large major third
34	434.043	9/7, (32/25)		SM3, (KKd4)	super major third, (classic diminished fourth)	AA2
35	446.809	135/104, (35/27)	22/17	HM3	hyper major third	ddd6
36	459.574	13/10	17/13, 30/23	h4	hypo fourth
37	472.340	21/16	25/19, 46/35	s4	sub fourth	dd5
38	485.106	297/224		l4	little fourth
39	497.872	4/3		P4	perfect fourth	P4
40	510.638	162/121, (35/26)		R4	rastmic fourth
41	523.404	27/20	19/14, 23/17	K4	komma-up fourth	A3
42	536.170	15/11	34/25	O4	on fourth
43	548.936	11/8	26/19	U4, T4	uber/undecimal fourth, tall fourth	AAA2
44	561.702	18/13, (25/18)		tA4, uA4, (kkA4)	tiny augmented fourth, unter augmented fourth, (classic augmented fourth)	dd6
45	574.468	88/63	32/23, 46/33	ld5, oA4	little diminished fifth, off augmented fourth
46	587.234	45/32, (7/5)	38/27	kA4	komma-down augmented fourth	d5
47	600.000	363/256, 512/363, (99/70)	17/12, 24/17	rA4, Rd5	rastmic augmented fourth, rastmic diminished fifth
48	612.766	64/45, (10/7)	27/19	Kd5	komma-up diminished fifth	A4
49	625.532	63/44	23/16, 33/23	LA4, Od5	large augmented fourth, off diminished fifth
50	638.298	13/9, (36/25)		Td5, Ud5, (KKd5)	tall diminished fifth, uber diminished fifth, (classic diminished fifth)	AA3
51	651.064	16/11	19/13	u5, t5	unter/undecimal fifth, tiny fifth	ddd7
52	663.830	22/15	25/17	o5	off fifth
53	676.596	40/27	28/19, 34/23	k5	komma-down fifth	d6
54	689.362	121/81, (52/35)		r5	rastmic fifth
55	702.128	3/2		P5	perfect fifth	P5
56	714.894	448/297		L5	large fifth
57	727.660	32/21	38/25, 35/23	S5	super fifth	AA4
58	740.426	20/13	26/17, 23/15	H5	hyper fifth
59	753.191	208/135	17/11	hm6	hypo minor sixth	AAA3
60	765.957	14/9, (128/75)		sm6, (kkA5)	sub minor sixth, (classic augmented fifth)	dd7
61	778.723	11/7	36/23	lm6	little minor sixth
62	791.489	128/81	19/12, 30/19	m6	minor sixth	m6
63	804.255	192/121	27/17	Rm6	rastmic minor sixth
64	817.021	8/5		Km6	classic minor sixth	A5
65	829.787	160/99, (21/13)	34/21	Om6	on minor sixth
66	842.553	44/27, 13/8, (81/50, 80/49)		n6, Tm6	less neutral sixth, tall minor sixth	AAA4
67	855.319	18/11, 64/39, (400/243, 49/30)	23/14	N6, tM6	greater neutral sixth, tiny minor sixth	ddd8
68	868.085	33/20	28/17, 38/23	oM6	off major sixth
69	880.851	5/3		kM6	classic major sixth	d7
70	893.617	121/72		rM6	rastmic major sixth
71	906.383	27/16, (42/35, 22/13)	32/19	M6	major sixth	M6
72	919.149	56/33	17/10, 46/27	LM6	large major sixth
73	931.915	12/7, 128/75		SM6, (KKd7)	super major sixth (classic diminished seventh)	AA5
74	944.681	45/26	19/11	HM6	hyper major sixth
75	957.447	26/15	40/23, 33/19	hm7	hypo minor seventh
76	970.213	7/4		sm7	sub minor seventh	dd8
77	982.979	99/56, (44/25)	30/17, 23/13	lm7	little minor seventh
78	995.745	16/9		m7	minor seventh	m7
79	1008.511	216/121	34/19	Rm7	rastmic minor seventh
80	1021.277	9/5	38/21	Km7	classic minor seventh	A6
81	1034.043	20/11		Om7	on minor seventh
82	1046.809	11/6, (64/35)	42/23	n7, Tm7, hd8	less neutral seventh, tall minor seventh, hypo diminished octave	AAA5
83	1059.574	81/44, 24/13, (50/27)	46/25, 35/19	N7, tM7, sd8, (kkM7)	greater neutral seventh, tiny major seventh, sub diminished octave, (2-comma down major seventh)
84	1072.340	297/160, 144/91, (13/7)		oM7, ld8	off major seventh, little diminished octave
85	1085.106	15/8, (28/15)		kM7, d8	classic major seventh, diminished octave	d8
86	1097.872	121/64	32/17, 17/9	rM7, Rd8	rastmic major seventh, rastmic diminished octave
87	1110.638	243/128, 256/135, (40/21)	36/19, 19/10	M7, Kd8	major seventh, komma-up diminished octave	M7
88	1123.404	21/11, (25/13)	44/23, 23/12	LM7, Od8	large major seventh, on diminished octave
89	1136.170	27/14, 52/27, (48/25)		SM7, Td8, Ud8, (KKd8)	super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave)	AA6
90	1148.936	64/33, (35/18, 68/35, 33/17)	33/17	u8, t8, HM7	unter octave, tiny octave, hyper major seventh
91	1161.702	88/45, 39/20	45/23	o8, h8	off octave, hypo octave
92	1174.468	160/81, 63/32, (49/25)		k8, s8	komma-down octave, sub octave
93	1187.234	891/448, 484/243, (486/245, 99/50, 196/99)		l8, r8	little octave, octave - rastma
94	1200.000	2/1		P8	perfect octave	P8

There are perhaps nine functional minor thirds varying between 242.553 cents and 344.681 cents, and one can even go beyond those boundaries under the right conditions, so musicians playing in 94edo have a lot more flexibility in terms of the particular interval shadings they might use depending on context.

The perfect fifth has three, or perhaps even five, functional options, each differing by one step. Although in most timbres only the central perfect fifth at 702.128 cents sounds consonant and stable, the lower and higher variants provide a change in interval quality, and can be helpful in creating subsets which mimic other edos, and close the circle of fifths in different numbers of pitches. For example, a close approximation to 41edo can be made using a chain of forty 702.128 cent fifths and one wide fifth at 714.894 cents, with an improvement on the tuning of most simple consonances in close keys, but a 1-step variation in interval quality as one modulates to more distant keys.

Every odd-numbered interval can generate the entire tuning of 94edo except for the 600-cent tritone (47\94), which divides the octave exactly in half.

The regular major second divisible into 16 equal parts can be helpful for realising some of the subtle tunings of Ancient Greek tetrachordal theory, Indian raga and Turkish maqam, though it has not been used historically as a division in those musical cultures.

While having the whole gamut of 94 intervals available on a keyboard or other instrument would be quite a feat, one can get a lot out of a 41-tone chain of fifths (with the odd fifth one degree wide) or a 53-tone chain of fifths (with the odd fifth one degree narrow), where the subset behaves much like a well-temperament, arguably usable in all keys but with some interval size variation between closer and more distant keys.

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 equal temperament that does better in this subgroup is 190g.

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	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	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

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

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

• Garibaldi5
• Garibaldi7
• Garibaldi12
• Garibaldi17

Music

Cam Taylor

• 4 Improvisations Saturday 9th September 2017
• Feeling Sad But Warming Up (in 2 parts)
• Playing with the 13-limit