100edo
| ← 99edo | 100edo | 101edo → |
(semiconvergent)
100 equal divisions of the octave (abbreviated 100edo or 100ed2), also called 100-tone equal temperament (100tet) or 100 equal temperament (100et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 100 equal parts of exactly 12 ¢ each. Each step represents a frequency ratio of 21/100, or the 100th root of 2.
Theory
It is closely related to 50 EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.
Like 6 EDO, 35 EDO, 47 EDO and 88 EDO, 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to 12 EDO.
Prime harmonics
| Harmonic | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | -5.96 | -2.31 | +3.17 | +0.68 | -0.53 | +3.04 | +2.49 | -4.27 | +2.42 | -5.04 | +0.66 |
| relative (%) | -50 | -19 | +26 | +6 | -4 | +25 | +21 | -36 | +20 | -42 | +5 | |
| Steps (reduced) |
158 (58) |
232 (32) |
281 (81) |
346 (46) |
370 (70) |
409 (9) |
425 (25) |
452 (52) |
486 (86) |
495 (95) |
521 (21) | |
Intervals
| Steps | Cents | Ups and downs notation (dual flat fifth 58\100) |
Ups and downs notation (dual sharp fifth 59\100) |
Approximate ratios |
|---|---|---|---|---|
| 0 | 0 | D | D | 1/1 |
| 1 | 12 | ^D, v3Ebb | ^D, v4Eb | |
| 2 | 24 | ^^D, vvEbb | ^^D, v3Eb | 65/64, 66/65 |
| 3 | 36 | ^3D, vEbb | ^3D, vvEb | |
| 4 | 48 | ^4D, Ebb | ^4D, vEb | 40/39 |
| 5 | 60 | ^5D, v5Eb | ^5D, Eb | |
| 6 | 72 | D#, v4Eb | ^6D, v12E | 25/24, 26/25 |
| 7 | 84 | ^D#, v3Eb | ^7D, v11E | 21/20, 22/21 |
| 8 | 96 | ^^D#, vvEb | ^8D, v10E | 55/52 |
| 9 | 108 | ^3D#, vEb | ^9D, v9E | |
| 10 | 120 | ^4D#, Eb | ^10D, v8E | |
| 11 | 132 | ^5D#, v5E | ^11D, v7E | 14/13 |
| 12 | 144 | Dx, v4E | ^12D, v6E | |
| 13 | 156 | ^Dx, v3E | D#, v5E | 35/32 |
| 14 | 168 | ^^Dx, vvE | ^D#, v4E | 11/10 |
| 15 | 180 | ^3Dx, vE | ^^D#, v3E | |
| 16 | 192 | E | ^3D#, vvE | |
| 17 | 204 | ^E, v3Fb | ^4D#, vE | |
| 18 | 216 | ^^E, vvFb | E | |
| 19 | 228 | ^3E, vFb | ^E, v4F | 8/7 |
| 20 | 240 | ^4E, Fb | ^^E, v3F | 55/48 |
| 21 | 252 | ^5E, v5F | ^3E, vvF | |
| 22 | 264 | E#, v4F | ^4E, vF | 64/55 |
| 23 | 276 | ^E#, v3F | F | |
| 24 | 288 | ^^E#, vvF | ^F, v4Gb | 13/11 |
| 25 | 300 | ^3E#, vF | ^^F, v3Gb | 25/21 |
| 26 | 312 | F | ^3F, vvGb | 6/5 |
| 27 | 324 | ^F, v3Gbb | ^4F, vGb | 77/64 |
| 28 | 336 | ^^F, vvGbb | ^5F, Gb | 40/33 |
| 29 | 348 | ^3F, vGbb | ^6F, v12G | |
| 30 | 360 | ^4F, Gbb | ^7F, v11G | 16/13 |
| 31 | 372 | ^5F, v5Gb | ^8F, v10G | 26/21 |
| 32 | 384 | F#, v4Gb | ^9F, v9G | 5/4 |
| 33 | 396 | ^F#, v3Gb | ^10F, v8G | 44/35, 63/50 |
| 34 | 408 | ^^F#, vvGb | ^11F, v7G | 33/26 |
| 35 | 420 | ^3F#, vGb | ^12F, v6G | 14/11 |
| 36 | 432 | ^4F#, Gb | F#, v5G | 32/25, 50/39 |
| 37 | 444 | ^5F#, v5G | ^F#, v4G | |
| 38 | 456 | Fx, v4G | ^^F#, v3G | 13/10 |
| 39 | 468 | ^Fx, v3G | ^3F#, vvG | 21/16, 55/42 |
| 40 | 480 | ^^Fx, vvG | ^4F#, vG | 33/25 |
| 41 | 492 | ^3Fx, vG | G | |
| 42 | 504 | G | ^G, v4Ab | |
| 43 | 516 | ^G, v3Abb | ^^G, v3Ab | 35/26 |
| 44 | 528 | ^^G, vvAbb | ^3G, vvAb | 65/48 |
| 45 | 540 | ^3G, vAbb | ^4G, vAb | |
| 46 | 552 | ^4G, Abb | ^5G, Ab | 11/8 |
| 47 | 564 | ^5G, v5Ab | ^6G, v12A | |
| 48 | 576 | G#, v4Ab | ^7G, v11A | |
| 49 | 588 | ^G#, v3Ab | ^8G, v10A | |
| 50 | 600 | ^^G#, vvAb | ^9G, v9A | |
| 51 | 612 | ^3G#, vAb | ^10G, v8A | |
| 52 | 624 | ^4G#, Ab | ^11G, v7A | |
| 53 | 636 | ^5G#, v5A | ^12G, v6A | |
| 54 | 648 | Gx, v4A | G#, v5A | 16/11 |
| 55 | 660 | ^Gx, v3A | ^G#, v4A | |
| 56 | 672 | ^^Gx, vvA | ^^G#, v3A | 65/44 |
| 57 | 684 | ^3Gx, vA | ^3G#, vvA | 52/35, 77/52 |
| 58 | 696 | A | ^4G#, vA | |
| 59 | 708 | ^A, v3Bbb | A | |
| 60 | 720 | ^^A, vvBbb | ^A, v4Bb | 50/33 |
| 61 | 732 | ^3A, vBbb | ^^A, v3Bb | 32/21 |
| 62 | 744 | ^4A, Bbb | ^3A, vvBb | 20/13 |
| 63 | 756 | ^5A, v5Bb | ^4A, vBb | 65/42 |
| 64 | 768 | A#, v4Bb | ^5A, Bb | 25/16, 39/25 |
| 65 | 780 | ^A#, v3Bb | ^6A, v12B | 11/7 |
| 66 | 792 | ^^A#, vvBb | ^7A, v11B | 52/33 |
| 67 | 804 | ^3A#, vBb | ^8A, v10B | 35/22 |
| 68 | 816 | ^4A#, Bb | ^9A, v9B | 8/5 |
| 69 | 828 | ^5A#, v5B | ^10A, v8B | 21/13 |
| 70 | 840 | Ax, v4B | ^11A, v7B | 13/8 |
| 71 | 852 | ^Ax, v3B | ^12A, v6B | |
| 72 | 864 | ^^Ax, vvB | A#, v5B | 33/20 |
| 73 | 876 | ^3Ax, vB | ^A#, v4B | |
| 74 | 888 | B | ^^A#, v3B | 5/3 |
| 75 | 900 | ^B, v3Cb | ^3A#, vvB | 42/25 |
| 76 | 912 | ^^B, vvCb | ^4A#, vB | 22/13 |
| 77 | 924 | ^3B, vCb | B | |
| 78 | 936 | ^4B, Cb | ^B, v4C | 55/32 |
| 79 | 948 | ^5B, v5C | ^^B, v3C | |
| 80 | 960 | B#, v4C | ^3B, vvC | |
| 81 | 972 | ^B#, v3C | ^4B, vC | 7/4 |
| 82 | 984 | ^^B#, vvC | C | |
| 83 | 996 | ^3B#, vC | ^C, v4Db | |
| 84 | 1008 | C | ^^C, v3Db | |
| 85 | 1020 | ^C, v3Dbb | ^3C, vvDb | |
| 86 | 1032 | ^^C, vvDbb | ^4C, vDb | 20/11 |
| 87 | 1044 | ^3C, vDbb | ^5C, Db | 64/35 |
| 88 | 1056 | ^4C, Dbb | ^6C, v12D | |
| 89 | 1068 | ^5C, v5Db | ^7C, v11D | 13/7 |
| 90 | 1080 | C#, v4Db | ^8C, v10D | |
| 91 | 1092 | ^C#, v3Db | ^9C, v9D | |
| 92 | 1104 | ^^C#, vvDb | ^10C, v8D | |
| 93 | 1116 | ^3C#, vDb | ^11C, v7D | 21/11, 40/21 |
| 94 | 1128 | ^4C#, Db | ^12C, v6D | 25/13, 48/25 |
| 95 | 1140 | ^5C#, v5D | C#, v5D | |
| 96 | 1152 | Cx, v4D | ^C#, v4D | 39/20 |
| 97 | 1164 | ^Cx, v3D | ^^C#, v3D | |
| 98 | 1176 | ^^Cx, vvD | ^3C#, vvD | 65/33 |
| 99 | 1188 | ^3Cx, vD | ^4C#, vD | |
| 100 | 1200 | D | D | 2/1 |
Scales
100bddd and the 22-note scales
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to 22 EDO for pajara temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range favored by George Secor for neomedieval compositions.
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide closer matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.
| Steps of 22-note MODMOS | Interval name (decatonic) | Interval name (superpyth diatonic) | Pure interval size [multiplicity] Difference from 22 EDO |
Dog interval size [multiplicity] Difference from 22 EDO |
| 1 | Diminished 2nd10 | Minor second | 60¢ [12] 5.4545¢ |
48¢ [10] -6.5455¢ |
| 2 | Minor 2nd10 | Augmented seventh | 108¢ [20] -1.091¢ |
120¢ [2] 10.909¢ |
| 3 | Major 2nd10 | Augmented unison | 168¢ [14] 4.364¢ |
156¢ [8] -7.636¢ |
| 4 | Minor 3rd10 | Major second | 216¢ [18] -2.182¢ |
228¢ [4] 9.818¢ |
| 5 | Major 3rd10 | Minor third | 276¢ [16] 3.273¢ |
264¢ [6] -8.727¢ |
| 6 | Minor 4th10 | Diminished fourth | 324¢ [16] -3.273¢ |
336¢ [6] 8.727¢ |
| 7 | Major 4th10 | Augmented second | 384¢ [18] 2.182¢ |
372¢ [4] -9.818¢ |
| 8 | Augmented 4th10 Diminished 5th10 |
Major third | 432¢ [14] -4.364¢ |
444¢ [8] 7.636¢ |
| 9 | Perfect 5th10 | Perfect fourth | 492¢ [20] 1.091¢ |
480¢ [2] -10.909¢ |
| 10 | Augmented 5th10 Diminished 6th10 |
Diminished fifth | 540¢ [12] -5.4545¢ |
552¢ [10] 6.5455¢ |
| 11 | Perfect 6th10 | Augmented third Diminished sixth |
600¢ [20] | 588¢ [1] -12¢ 612¢ [1] 12¢ |
| 12 | Augmented 6th10 Diminished 7th10 |
Augmented fourth | 660¢ [12] 6.5455¢ |
648¢ [10] -5.4545¢ |
| 13 | Perfect 7th10 | Perfect fifth | 708¢ [20] -1.091¢ |
720¢ [2] 10.909¢ |
| 14 | Augmented 7th10
Diminished 8th10 |
Minor sixth | 768¢ [14] 4.364¢ |
756¢ [8] -7.636¢ |
| 15 | Minor 8th10 | Diminished seventh | 816¢ [18] -2.182¢ |
828¢ [4] 9.818¢ |
| 16 | Major 8th10 | Augmented fifth | 876¢ [16] 3.273¢ |
864¢ [6] -8.727¢ |
| 17 | Minor 9th10 | Major sixth | 924¢ [16] -3.273¢ |
936¢ [6] 8.727¢ |
| 18 | Major 9th10 | Minor seventh | 984¢ [18] 2.182¢ |
972¢ [4] -9.818¢ |
| 19 | Minor 10th10 | Diminished octave | 1032¢ [14] -4.364¢ |
1044¢ [8] 7.636¢ |
| 20 | Major 10th10 | Diminished second | 1092¢ [20] 1.091¢ |
1080¢ [2] -10.909¢ |
| 21 | Augmented 10th10 Diminished 11th10 |
Major seventh | 1140¢ [12] -5.4545¢ |
1152¢ [10] 6.5455¢ |
| 22 | 11th10 | Octave | 1200¢ [22] | N/A |
Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th10 is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are tuned in tritones. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow almost all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the Indian sruti system.
Other, "gentle" alternatives to 22 EDO for pajara include 78ddd and 56d. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&34d temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include 58d and 46d. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&34d temperaments up to 58d is also usable.