94edo
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
| 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
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' indicates alteration by the septimal comma, 64/63; 'K' indicates alteration by the syntonic comma, 81/80; 'U' by the undecimal quatertone, 33/32; 'L' by pentacircle comma, 896/891; 'O' by 45/44; 'R' by the rastma, 243/242; 'T' by the tridecimal quartertone, 1053/1024; and finally, '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.
| Step | Cents | 13-limit | Short-form WED | Diatonic | Long-form WED | 23-limit |
|---|---|---|---|---|---|---|
| 1 | 12.766 | 896/891, 243/242, (3125/3072, 245/243, 100/99, 99/98) | L1, R1 | large unison, rastma | 85/84 | |
| 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) | O1, H1 | on unison, hyper unison | 46/45 | |
| 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) | dd3 | sub minor second, unter augmented unison, tiny augmented unison, (classic augmented unison) | |
| 6 | 76.596 | 22/21, (648/625, 26/25) | lm2, oA1 | little minor second, off augmented unison | 23/22, 24/23 | |
| 7 | 89.362 | 256/243, 135/128, (21/20) | m2, kA1 | m2 | minor second, komma-down augmented unison | 19/18, 20/19 |
| 8 | 102.128 | 128/121, (35/33) | Rm2, rA1 | rastmic minor second, rastmic augmented unison | 17/16, 18/17 | |
| 9 | 114.894 | 16/15, (15/14) | Km2, A1 | A1 | classic minor second, augmented unison | |
| 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) | n2, Tm2, SA1, (KKm2) | lesser neutral second, tall minor second, super augmented unison, (2-komma-up minor second) | 25/23, 38/35 | |
| 12 | 153.191 | 12/11, (35/32) | N2, tM2, HA1 | ddd4 | greater netral second, tiny major second, hyper augmented unison | 23/21 |
| 13 | 165.957 | 11/10 | oM2 | off major second | ||
| 14 | 178.723 | 10/9 | kM2 | d3 | komma-down major second | 21/19 |
| 15 | 191.489 | 121/108, (49/44, 39/35) | rM2 | rastmic major second | 19/17 | |
| 16 | 204.255 | 9/8 | M2 | M2 | major second | |
| 17 | 217.021 | 112/99, (25/22) | LM2 | large major second | 17/15, 26/23 | |
| 18 | 229.787 | 8/7 | SM2 | AA1 | super major second | |
| 19 | 242.553 | 15/13 | HM2 | hyper major second | 23/20, 38/33 | |
| 20 | 255.319 | 52/45 | hm3 | hypo minor third | 22/19 | |
| 21 | 268.085 | 7/6, (75/64) | sm3, (kkA2) | dd4 | sub minor third, (classic augmented second) | |
| 22 | 280.851 | 33/28 | lm3 | little minor third | 20/17, 27/23 | |
| 23 | 293.617 | 32/27, (25/21, 13/11) | m3 | m3 | minor third | 19/16 |
| 24 | 306.383 | 144/121, (81/70) | Rm3 | rastmic minor third | ||
| 25 | 319.149 | 6/5 | Km3 | A2 | classic minor third | |
| 26 | 331.915 | 40/33 | Om3 | on minor third | 17/14, 23/19 | |
| 27 | 344.681 | 11/9, 39/32, (243/200, 60/49) | n3, Tm3 | AAA1 | lesser neutral third, tall minor third | 28/23 |
| 28 | 357.447 | 27/22, 16/13, (100/81,49/40) | N3, tM3 | ddd5 | greater neutral third, tiny major third | |
| 29 | 370.213 | 99/80, (26/21) | oM3 | off major third | 21/17 | |
| 30 | 382.979 | 5/4 | kM3 | d4 | classic major third | |
| 31 | 395.745 | 121/96, (34/27) | rM3 | rastmic major third | ||
| 32 | 408.511 | 81/64, (33/26) | M3 | M3 | major third | 19/15, 24/19 |
| 33 | 421.277 | 14/11 | LM3 | large major third | 23/18 | |
| 34 | 434.043 | 9/7, (32/25) | SM3, (KKd4) | AA2 | super major third, (classic diminished fourth) | |
| 35 | 446.809 | 135/104, (35/27) | HM3 | ddd6 | hyper major third | 22/17 |
| 36 | 459.574 | 13/10 | h4 | hypo fourth | 17/13, 30/23 | |
| 37 | 472.340 | 21/16 | s4 | dd5 | sub fourth | 25/19, 46/35 |
| 38 | 485.106 | 297/224 | l4 | little fourth | ||
| 39 | 497.872 | 4/3 | P4 | P4 | perfect fourth | |
| 40 | 510.638 | 162/121, (35/26) | R4 | rastmic fourth | ||
| 41 | 523.404 | 27/20 | K4 | A3 | komma-up fourth | 19/14, 23/17 |
| 42 | 536.170 | 15/11 | O4 | on fourth | 34/25 | |
| 43 | 548.936 | 11/8 | U4, T4 | AAA2 | uber/undecimal fourth, tall fourth | 26/19 |
| 44 | 561.702 | 18/13, (25/18) | tA4, uA4, (kkA4) | dd6 | tiny augmented fourth, unter augmented fourth, (classic augmented fourth) | |
| 45 | 574.468 | 88/63 | ld5, oA4 | little diminished fifth, off augmented fourth | 32/23, 46/33 | |
| 46 | 587.234 | 45/32, (7/5) | kA4 | d5 | komma-down augmented fourth | 38/27 |
| 47 | 600.000 | 363/256, 512/363, (99/70) | rA4, Rd5 | rastmic augmented fourth, rastmic diminished fifth | 17/12, 24/17 | |
| 48 | 612.766 | 64/45, (10/7) | Kd5 | A4 | komma-up diminished fifth | 27/19 |
| 49 | 625.532 | 63/44 | LA4, Od5 | large augmented fourth, off diminished fifth | 23/16, 33/23 | |
| 50 | 638.298 | 13/9, (36/25) | Td5, Ud5, (KKd5) | AA3 | tall diminished fifth, uber diminished fifth, (classic diminished fifth) | |
| 51 | 651.064 | 16/11 | u5, t5 | ddd7 | unter/undecimal fifth, tiny fifth | 19/13 |
| 52 | 663.830 | 22/15 | o5 | off fifth | 25/17 | |
| 53 | 676.596 | 40/27 | k5 | d6 | komma-down fifth | 28/19, 34/23 |
| 54 | 689.362 | 121/81, (52/35) | r5 | rastmic fifth | ||
| 55 | 702.128 | 3/2 | P5 | P5 | perfect fifth | |
| 56 | 714.894 | 448/297 | L5 | large fifth | ||
| 57 | 727.660 | 32/21 | S5 | AA4 | super fifth | 38/25, 35/23 |
| 58 | 740.426 | 20/13 | H5 | hyper fifth | 26/17, 23/15 | |
| 59 | 753.191 | 208/135 | hm6 | AAA3 | hypo minor sixth | 17/11 |
| 60 | 765.957 | 14/9, (128/75) | sm6, (kkA5) | dd7 | sub minor sixth, (classic augmented fifth) | |
| 61 | 778.723 | 11/7 | lm6 | little minor sixth | 36/23 | |
| 62 | 791.489 | 128/81 | m6 | m6 | minor sixth | 19/12, 30/19 |
| 63 | 804.255 | 192/121 | Rm6 | rastmic minor sixth | 27/17 | |
| 64 | 817.021 | 8/5 | Km6 | A5 | classic minor sixth | |
| 65 | 829.787 | 160/99, (21/13) | Om6 | on minor sixth | 34/21 | |
| 66 | 842.553 | 44/27, 13/8, (81/50, 80/49) | n6, Tm6 | AAA4 | less neutral sixth, tall minor sixth | |
| 67 | 855.319 | 18/11, 64/39, (400/243, 49/30) | N6, tM6 | ddd8 | greater neutral sixth, tiny minor sixth | 23/14 |
| 68 | 868.085 | 33/20 | oM6 | off major sixth | 28/17, 38/23 | |
| 69 | 880.851 | 5/3 | kM6 | d7 | classic major sixth | |
| 70 | 893.617 | 121/72 | rM6 | rastmic major sixth | ||
| 71 | 906.383 | 27/16, (42/35, 22/13) | M6 | M6 | major sixth | 32/19 |
| 72 | 919.149 | 56/33 | LM6 | large major sixth | 17/10, 46/27 | |
| 73 | 931.915 | 12/7, 128/75 | SM6, (KKd7) | AA5 | super major sixth (classic diminished seventh) | |
| 74 | 944.681 | 45/26 | HM6 | hyper major sixth | 19/11 | |
| 75 | 957.447 | 26/15 | hm7 | hypo minor seventh | 40/23, 33/19 | |
| 76 | 970.213 | 7/4 | sm7 | dd8 | sub minor seventh | |
| 77 | 982.979 | 99/56, (44/25) | lm7 | little minor seventh | 30/17, 23/13 | |
| 78 | 995.745 | 16/9 | m7 | m7 | minor seventh | |
| 79 | 1008.511 | 216/121 | Rm7 | rastmic minor seventh | 34/19 | |
| 80 | 1021.277 | 9/5 | Km7 | A6 | classic minor seventh | 38/21 |
| 81 | 1034.043 | 20/11 | Om7 | on minor seventh | ||
| 82 | 1046.809 | 11/6, (64/35) | n7, Tm7, hd8 | AAA5 | less neutral seventh, tall minor seventh, hypo diminished octave | 42/23 |
| 83 | 1059.574 | 81/44, 24/13, (50/27) | N7, tM7, sd8, (kkM7) | greater neutral seventh, tiny major seventh, sub diminished octave, (2-comma down major seventh) | 46/25, 35/19 | |
| 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 | d8 | classic major seventh, diminished octave | |
| 86 | 1097.872 | 121/64 | rM7, Rd8 | rastmic major seventh, rastmic diminished octave | 32/17, 17/9 | |
| 87 | 1110.638 | 243/128, 256/135, (40/21) | M7, Kd8 | M7 | major seventh, komma-up diminished octave | 36/19, 19/10 |
| 88 | 1123.404 | 21/11, (25/13) | LM7, Od8 | large major seventh, on diminished octave | 44/23, 23/12 | |
| 89 | 1136.170 | 27/14, 52/27, (48/25) | SM7, Td8, Ud8, (KKd8) | AA6 | super major seventh, tall diminished octave, unter diminished octave, (classic diminished octave) | |
| 90 | 1148.936 | 64/33, (35/18, 68/35, 33/17) | u8, t8, HM7 | unter octave, tiny octave, hyper major seventh | 33/17 | |
| 91 | 1161.702 | 88/45, 39/20 | o8, h8 | off octave, hypo octave | 45/23 | |
| 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 | P8 | perfect octave |
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 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 … ]]