94edo
| ← 93edo | 94edo | 95edo → |
(semiconvergent)
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. 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.
| 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
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 |