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.
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
| 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 | 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) | 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) | lm2, oA1 | little minor second, off augmented unison | ||
| 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) | less neutral second, tall minor second, super augmented unison, (2-komma-up minor second) | ||
| 12 | 153.191 | 12/11, (35/32) | N2, tM2, HA1 | ddd4 | greater netral second, tiny major second, hyper augmented unison | |
| 13 | 165.957 | 11/10 | oM2 | off major second | ||
| 14 | 178.723 | 10/9 | kM2 | d3 | komma-down major second | |
| 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 | |
| 18 | 229.787 | 8/7 | SM2 | AA1 | super major second | |
| 19 | 242.553 | 15/13 | HM2 | hyper major second | 23/20 | |
| 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 | |
| 23 | 293.617 | 32/27, (25/21, 13/11) | m3 | m3 | minor third | |
| 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 | 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 | ||
| 37 | 472.340 | 21/16 | s4 | dd5 | sub fourth | |
| 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 | ||
| 43 | 548.936 | 11/8 | U4, T4 | AAA2 | uber/undecimal fourth, tall fourth | |
| 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 | ||
| 46 | 587.234 | 45/32, (7/5) | kA4 | d5 | komma-down augmented fourth | |
| 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 | |
| 49 | 625.532 | 63/44 | LA4, Od5 | large augmented fourth, off diminished fifth | ||
| 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 | |
| 52 | 663.830 | 22/15 | o5 | off fifth | ||
| 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 | |
| 58 | 740.426 | 20/13 | H5 | hyper fifth | ||
| 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 | |
| 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 | |
| 72 | 919.149 | 56/33 | LM6 | large major sixth | 17/10 | |
| 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 | |
| 76 | 970.213 | 7/4 | sm7 | dd8 | sub minor seventh | |
| 77 | 982.979 | 99/56, (44/25) | lm7 | little minor seventh | 30/17 | |
| 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 | |
| 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 | |
| 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) | ||
| 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 | LM7, Od8 | large major seventh, on diminished octave | ||
| 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) | 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 |
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
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].
| 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 |