120edo: Difference between revisions
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|120}} | {{Harmonics in equal|120}} | ||
=== Miscellaneous properties === | |||
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]]. | |||
120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes. | |||
== Interval list == | == Interval list == | ||
| Line 619: | Line 624: | ||
|{{UDnote|step=119}} | |{{UDnote|step=119}} | ||
|{{UDnote|fifth=71|step=119}} | |{{UDnote|fifth=71|step=119}} | ||
|- | |||
|120 | |||
|1200 | |||
|{{UDnote|step=120}} | |||
|{{UDnote|fifth=71|step=120}} | |||
|} | |} | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Highly composite]] | [[Category:Highly composite]] | ||
Revision as of 08:27, 16 June 2023
| ← 119edo | 120edo | 121edo → |
Theory
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).
120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out 243/242.
120edo shares the perfect fifth with 12edo, tempering out the Pythagorean comma. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit superpyth temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit surmarvelpyth temperament where it represents 675/448, which is marvel comma sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit decoid temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | +3.69 | +1.17 | -1.32 | -0.53 | -4.96 | +2.49 | +1.73 | +0.42 | +4.96 |
| Relative (%) | +0.0 | -19.6 | +36.9 | +11.7 | -13.2 | -5.3 | -49.6 | +24.9 | +17.3 | +4.2 | +49.6 | |
| Steps (reduced) |
120 (0) |
190 (70) |
279 (39) |
337 (97) |
415 (55) |
444 (84) |
490 (10) |
510 (30) |
543 (63) |
583 (103) |
595 (115) | |
Miscellaneous properties
Being the simplest division of the octave by the Germanic long hundred, it has a unit step which is the fine relative cent of 1edo.
120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.
Interval list
| # | Cents | Ups and downs notation (fifth 7\12) | Ups and downs notation (fifth 71\120) |
|---|---|---|---|
| 0 | 0 | D | D |
| 1 | 10 | ^D, ^E♭♭ | ^D, v4E♭ |
| 2 | 20 | ^^D, ^^E♭♭ | ^^D, v3E♭ |
| 3 | 30 | ^3D, ^3E♭♭ | ^3D, vvE♭ |
| 4 | 40 | ^4D, ^4E♭♭ | ^4D, vE♭ |
| 5 | 50 | ^5D, v5E♭ | ^5D, E♭ |
| 6 | 60 | v4D♯, v4E♭ | ^6D, ^E♭ |
| 7 | 70 | v3D♯, v3E♭ | ^7D, ^^E♭ |
| 8 | 80 | vvD♯, vvE♭ | ^8D, ^3E♭ |
| 9 | 90 | vD♯, vE♭ | v8D♯, ^4E♭ |
| 10 | 100 | D♯, E♭ | v7D♯, ^5E♭ |
| 11 | 110 | ^D♯, ^E♭ | v6D♯, ^6E♭ |
| 12 | 120 | ^^D♯, ^^E♭ | v5D♯, ^7E♭ |
| 13 | 130 | ^3D♯, ^3E♭ | v4D♯, ^8E♭ |
| 14 | 140 | ^4D♯, ^4E♭ | v3D♯, v8E |
| 15 | 150 | ^5D♯, v5E | vvD♯, v7E |
| 16 | 160 | v4D𝄪, v4E | vD♯, v6E |
| 17 | 170 | v3D𝄪, v3E | D♯, v5E |
| 18 | 180 | vvD𝄪, vvE | ^D♯, v4E |
| 19 | 190 | vD𝄪, vE | ^^D♯, v3E |
| 20 | 200 | E | ^3D♯, vvE |
| 21 | 210 | ^E, ^F♭ | ^4D♯, vE |
| 22 | 220 | ^^E, ^^F♭ | E |
| 23 | 230 | ^3E, ^3F♭ | ^E, v4F |
| 24 | 240 | ^4E, ^4F♭ | ^^E, v3F |
| 25 | 250 | ^5E, v5F | ^3E, vvF |
| 26 | 260 | v4E♯, v4F | ^4E, vF |
| 27 | 270 | v3E♯, v3F | F |
| 28 | 280 | vvE♯, vvF | ^F, v4G♭ |
| 29 | 290 | vE♯, vF | ^^F, v3G♭ |
| 30 | 300 | F | ^3F, vvG♭ |
| 31 | 310 | ^F, ^G♭♭ | ^4F, vG♭ |
| 32 | 320 | ^^F, ^^G♭♭ | ^5F, G♭ |
| 33 | 330 | ^3F, ^3G♭♭ | ^6F, ^G♭ |
| 34 | 340 | ^4F, ^4G♭♭ | ^7F, ^^G♭ |
| 35 | 350 | ^5F, v5G♭ | ^8F, ^3G♭ |
| 36 | 360 | v4F♯, v4G♭ | v8F♯, ^4G♭ |
| 37 | 370 | v3F♯, v3G♭ | v7F♯, ^5G♭ |
| 38 | 380 | vvF♯, vvG♭ | v6F♯, ^6G♭ |
| 39 | 390 | vF♯, vG♭ | v5F♯, ^7G♭ |
| 40 | 400 | F♯, G♭ | v4F♯, ^8G♭ |
| 41 | 410 | ^F♯, ^G♭ | v3F♯, v8G |
| 42 | 420 | ^^F♯, ^^G♭ | vvF♯, v7G |
| 43 | 430 | ^3F♯, ^3G♭ | vF♯, v6G |
| 44 | 440 | ^4F♯, ^4G♭ | F♯, v5G |
| 45 | 450 | ^5F♯, v5G | ^F♯, v4G |
| 46 | 460 | v4F𝄪, v4G | ^^F♯, v3G |
| 47 | 470 | v3F𝄪, v3G | ^3F♯, vvG |
| 48 | 480 | vvF𝄪, vvG | ^4F♯, vG |
| 49 | 490 | vF𝄪, vG | G |
| 50 | 500 | G | ^G, v4A♭ |
| 51 | 510 | ^G, ^A♭♭ | ^^G, v3A♭ |
| 52 | 520 | ^^G, ^^A♭♭ | ^3G, vvA♭ |
| 53 | 530 | ^3G, ^3A♭♭ | ^4G, vA♭ |
| 54 | 540 | ^4G, ^4A♭♭ | ^5G, A♭ |
| 55 | 550 | ^5G, v5A♭ | ^6G, ^A♭ |
| 56 | 560 | v4G♯, v4A♭ | ^7G, ^^A♭ |
| 57 | 570 | v3G♯, v3A♭ | ^8G, ^3A♭ |
| 58 | 580 | vvG♯, vvA♭ | v8G♯, ^4A♭ |
| 59 | 590 | vG♯, vA♭ | v7G♯, ^5A♭ |
| 60 | 600 | G♯, A♭ | v6G♯, ^6A♭ |
| 61 | 610 | ^G♯, ^A♭ | v5G♯, ^7A♭ |
| 62 | 620 | ^^G♯, ^^A♭ | v4G♯, ^8A♭ |
| 63 | 630 | ^3G♯, ^3A♭ | v3G♯, v8A |
| 64 | 640 | ^4G♯, ^4A♭ | vvG♯, v7A |
| 65 | 650 | ^5G♯, v5A | vG♯, v6A |
| 66 | 660 | v4G𝄪, v4A | G♯, v5A |
| 67 | 670 | v3G𝄪, v3A | ^G♯, v4A |
| 68 | 680 | vvG𝄪, vvA | ^^G♯, v3A |
| 69 | 690 | vG𝄪, vA | ^3G♯, vvA |
| 70 | 700 | A | ^4G♯, vA |
| 71 | 710 | ^A, ^B♭♭ | A |
| 72 | 720 | ^^A, ^^B♭♭ | ^A, v4B♭ |
| 73 | 730 | ^3A, ^3B♭♭ | ^^A, v3B♭ |
| 74 | 740 | ^4A, ^4B♭♭ | ^3A, vvB♭ |
| 75 | 750 | ^5A, v5B♭ | ^4A, vB♭ |
| 76 | 760 | v4A♯, v4B♭ | ^5A, B♭ |
| 77 | 770 | v3A♯, v3B♭ | ^6A, ^B♭ |
| 78 | 780 | vvA♯, vvB♭ | ^7A, ^^B♭ |
| 79 | 790 | vA♯, vB♭ | ^8A, ^3B♭ |
| 80 | 800 | A♯, B♭ | v8A♯, ^4B♭ |
| 81 | 810 | ^A♯, ^B♭ | v7A♯, ^5B♭ |
| 82 | 820 | ^^A♯, ^^B♭ | v6A♯, ^6B♭ |
| 83 | 830 | ^3A♯, ^3B♭ | v5A♯, ^7B♭ |
| 84 | 840 | ^4A♯, ^4B♭ | v4A♯, ^8B♭ |
| 85 | 850 | ^5A♯, v5B | v3A♯, v8B |
| 86 | 860 | v4A𝄪, v4B | vvA♯, v7B |
| 87 | 870 | v3A𝄪, v3B | vA♯, v6B |
| 88 | 880 | vvA𝄪, vvB | A♯, v5B |
| 89 | 890 | vA𝄪, vB | ^A♯, v4B |
| 90 | 900 | B | ^^A♯, v3B |
| 91 | 910 | ^B, ^C♭ | ^3A♯, vvB |
| 92 | 920 | ^^B, ^^C♭ | ^4A♯, vB |
| 93 | 930 | ^3B, ^3C♭ | B |
| 94 | 940 | ^4B, ^4C♭ | ^B, v4C |
| 95 | 950 | ^5B, v5C | ^^B, v3C |
| 96 | 960 | v4B♯, v4C | ^3B, vvC |
| 97 | 970 | v3B♯, v3C | ^4B, vC |
| 98 | 980 | vvB♯, vvC | C |
| 99 | 990 | vB♯, vC | ^C, v4D♭ |
| 100 | 1000 | C | ^^C, v3D♭ |
| 101 | 1010 | ^C, ^D♭♭ | ^3C, vvD♭ |
| 102 | 1020 | ^^C, ^^D♭♭ | ^4C, vD♭ |
| 103 | 1030 | ^3C, ^3D♭♭ | ^5C, D♭ |
| 104 | 1040 | ^4C, ^4D♭♭ | ^6C, ^D♭ |
| 105 | 1050 | ^5C, v5D♭ | ^7C, ^^D♭ |
| 106 | 1060 | v4C♯, v4D♭ | ^8C, ^3D♭ |
| 107 | 1070 | v3C♯, v3D♭ | v8C♯, ^4D♭ |
| 108 | 1080 | vvC♯, vvD♭ | v7C♯, ^5D♭ |
| 109 | 1090 | vC♯, vD♭ | v6C♯, ^6D♭ |
| 110 | 1100 | C♯, D♭ | v5C♯, ^7D♭ |
| 111 | 1110 | ^C♯, ^D♭ | v4C♯, ^8D♭ |
| 112 | 1120 | ^^C♯, ^^D♭ | v3C♯, v8D |
| 113 | 1130 | ^3C♯, ^3D♭ | vvC♯, v7D |
| 114 | 1140 | ^4C♯, ^4D♭ | vC♯, v6D |
| 115 | 1150 | ^5C♯, v5D | C♯, v5D |
| 116 | 1160 | v4C𝄪, v4D | ^C♯, v4D |
| 117 | 1170 | v3C𝄪, v3D | ^^C♯, v3D |
| 118 | 1180 | vvC𝄪, vvD | ^3C♯, vvD |
| 119 | 1190 | vC𝄪, vD | ^4C♯, vD |
| 120 | 1200 | D | D |