65edo: Difference between revisions
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+ratios for interval table |
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| Line 20: | Line 20: | ||
== Intervals == | == Intervals == | ||
{| class="wikitable center-all right-2 left-3" | |||
{| class="wikitable center-all right-2" | |||
|- | |- | ||
! [[Degree]] | ! [[Degree]] | ||
! [[Cent]]s | ! [[Cent]]s | ||
! colspan="2" |[[Ups and Downs Notation]] | ! Approximate Ratios * | ||
! colspan="2" | [[Ups and Downs Notation]] | |||
|- | |- | ||
| 0 | | 0 | ||
| 0.00 | | 0.00 | ||
| 1/1 | |||
| P1 | | P1 | ||
| D | | D | ||
| Line 34: | Line 35: | ||
| 1 | | 1 | ||
| 18.46 | | 18.46 | ||
| 81/80, 100/99, 121/120 | |||
| ^1 | | ^1 | ||
| ^D | | ^D | ||
| Line 39: | Line 41: | ||
| 2 | | 2 | ||
| 36.92 | | 36.92 | ||
| 45/44, 55/54, 128/125 | |||
| ^^1 | | ^^1 | ||
| ^^D | | ^^D | ||
| Line 44: | Line 47: | ||
| 3 | | 3 | ||
| 55.38 | | 55.38 | ||
| 33/32 | |||
| vvm2 | | vvm2 | ||
| vvEb | | vvEb | ||
| Line 49: | Line 53: | ||
| 4 | | 4 | ||
| 73.85 | | 73.85 | ||
| 25/24 | |||
| vm2 | | vm2 | ||
| vEb | | vEb | ||
| Line 54: | Line 59: | ||
| 5 | | 5 | ||
| 92.31 | | 92.31 | ||
| 135/128, 256/243 | |||
| m2 | | m2 | ||
| Eb | | Eb | ||
| Line 59: | Line 65: | ||
| 6 | | 6 | ||
| 110.77 | | 110.77 | ||
| 16/15 | |||
| A1/^m2 | | A1/^m2 | ||
| D#/^Eb | | D#/^Eb | ||
| Line 64: | Line 71: | ||
| 7 | | 7 | ||
| 129.23 | | 129.23 | ||
| 14/13 | |||
| v~2 | | v~2 | ||
| ^^Eb | | ^^Eb | ||
| Line 69: | Line 77: | ||
| 8 | | 8 | ||
| 147.69 | | 147.69 | ||
| 12/11 | |||
| ~2 | | ~2 | ||
| vvvE | | vvvE | ||
| Line 74: | Line 83: | ||
| 9 | | 9 | ||
| 166.15 | | 166.15 | ||
| 11/10 | |||
| ^~2 | | ^~2 | ||
| vvE | | vvE | ||
| Line 79: | Line 89: | ||
| 10 | | 10 | ||
| 184.62 | | 184.62 | ||
| 10/9 | |||
| vM2 | | vM2 | ||
| vE | | vE | ||
| Line 84: | Line 95: | ||
| 11 | | 11 | ||
| 203.08 | | 203.08 | ||
| 9/8 | |||
| M2 | | M2 | ||
| E | | E | ||
| Line 89: | Line 101: | ||
| 12 | | 12 | ||
| 221.54 | | 221.54 | ||
| 25/22 | |||
| ^M2 | | ^M2 | ||
| ^E | | ^E | ||
| Line 94: | Line 107: | ||
| 13 | | 13 | ||
| 240.00 | | 240.00 | ||
| 55/48 | |||
| ^^M2 | | ^^M2 | ||
| ^^E | | ^^E | ||
| Line 99: | Line 113: | ||
| 14 | | 14 | ||
| 258.46 | | 258.46 | ||
| 64/55 | |||
| vvm3 | | vvm3 | ||
| vvF | | vvF | ||
| Line 104: | Line 119: | ||
| 15 | | 15 | ||
| 276.92 | | 276.92 | ||
| 75/64 | |||
| vm3 | | vm3 | ||
| vF | | vF | ||
| Line 109: | Line 125: | ||
| 16 | | 16 | ||
| 295.38 | | 295.38 | ||
| 32/27 | |||
| m3 | | m3 | ||
| F | | F | ||
| Line 114: | Line 131: | ||
| 17 | | 17 | ||
| 313.85 | | 313.85 | ||
| 6/5 | |||
| ^m3 | | ^m3 | ||
| ^F | | ^F | ||
| Line 119: | Line 137: | ||
| 18 | | 18 | ||
| 332.31 | | 332.31 | ||
| 40/33 | |||
| v~3 | | v~3 | ||
| ^^F | | ^^F | ||
| Line 124: | Line 143: | ||
| 19 | | 19 | ||
| 350.77 | | 350.77 | ||
| 11/9, 27/22 | |||
| ~3 | | ~3 | ||
| ^^^F | | ^^^F | ||
| Line 129: | Line 149: | ||
| 20 | | 20 | ||
| 369.23 | | 369.23 | ||
| 26/21 | |||
| ^~3 | | ^~3 | ||
| vvF# | | vvF# | ||
| Line 134: | Line 155: | ||
| 21 | | 21 | ||
| 387.69 | | 387.69 | ||
| 5/4 | |||
| vM3 | | vM3 | ||
| vF# | | vF# | ||
| Line 139: | Line 161: | ||
| 22 | | 22 | ||
| 406.15 | | 406.15 | ||
| 81/64 | |||
| M3 | | M3 | ||
| F# | | F# | ||
| Line 144: | Line 167: | ||
| 23 | | 23 | ||
| 424.62 | | 424.62 | ||
| 32/25 | |||
| ^M3 | | ^M3 | ||
| ^F# | | ^F# | ||
| Line 149: | Line 173: | ||
| 24 | | 24 | ||
| 443.08 | | 443.08 | ||
| 128/99 | |||
| ^^M3 | | ^^M3 | ||
| ^^F# | | ^^F# | ||
| Line 154: | Line 179: | ||
| 25 | | 25 | ||
| 461.54 | | 461.54 | ||
| 72/55 | |||
| vv4 | | vv4 | ||
| vvG | | vvG | ||
| Line 159: | Line 185: | ||
| 26 | | 26 | ||
| 480.00 | | 480.00 | ||
| 33/25 | |||
| v4 | | v4 | ||
| vG | | vG | ||
| Line 164: | Line 191: | ||
| 27 | | 27 | ||
| 498.46 | | 498.46 | ||
| 4/3 | |||
| P4 | | P4 | ||
| G | | G | ||
| Line 169: | Line 197: | ||
| 28 | | 28 | ||
| 516.92 | | 516.92 | ||
| 27/20 | |||
| ^4 | | ^4 | ||
| ^G | | ^G | ||
| Line 174: | Line 203: | ||
| 29 | | 29 | ||
| 535.38 | | 535.38 | ||
| 15/11 | |||
| v~4 | | v~4 | ||
| ^^G | | ^^G | ||
| Line 179: | Line 209: | ||
| 30 | | 30 | ||
| 553.85 | | 553.85 | ||
| 11/8 | |||
| ~4 | | ~4 | ||
| ^^^G | | ^^^G | ||
| Line 184: | Line 215: | ||
| 31 | | 31 | ||
| 572.31 | | 572.31 | ||
| 25/18 | |||
| ^~4/vd5 | | ^~4/vd5 | ||
| vvG#/vAb | | vvG#/vAb | ||
| Line 189: | Line 221: | ||
| 32 | | 32 | ||
| 590.77 | | 590.77 | ||
| 45/32 | |||
| vA4/d5 | | vA4/d5 | ||
| vG#/Ab | | vG#/Ab | ||
| Line 194: | Line 227: | ||
| 33 | | 33 | ||
| 609.23 | | 609.23 | ||
| 64/45 | |||
| A4/^d5 | | A4/^d5 | ||
| G#/^Ab | | G#/^Ab | ||
| Line 199: | Line 233: | ||
| 34 | | 34 | ||
| 627.69 | | 627.69 | ||
| 36/25 | |||
| ^A4/v~5 | | ^A4/v~5 | ||
| ^G#/^^Ab | | ^G#/^^Ab | ||
| Line 204: | Line 239: | ||
| 35 | | 35 | ||
| 646.15 | | 646.15 | ||
| 16/11 | |||
| ~5 | | ~5 | ||
| vvvA | | vvvA | ||
| Line 209: | Line 245: | ||
| 36 | | 36 | ||
| 664.62 | | 664.62 | ||
| 22/15 | |||
| ^~5 | | ^~5 | ||
| vvA | | vvA | ||
| Line 214: | Line 251: | ||
| 37 | | 37 | ||
| 683.08 | | 683.08 | ||
| 40/27 | |||
| v5 | | v5 | ||
| vA | | vA | ||
| Line 219: | Line 257: | ||
| 38 | | 38 | ||
| 701.54 | | 701.54 | ||
| 3/2 | |||
| P5 | | P5 | ||
| A | | A | ||
| Line 224: | Line 263: | ||
| 39 | | 39 | ||
| 720.00 | | 720.00 | ||
| 50/33 | |||
| ^5 | | ^5 | ||
| ^A | | ^A | ||
| Line 229: | Line 269: | ||
| 40 | | 40 | ||
| 738.46 | | 738.46 | ||
| 55/36 | |||
| ^^5 | | ^^5 | ||
| ^^A | | ^^A | ||
| Line 234: | Line 275: | ||
| 41 | | 41 | ||
| 756.92 | | 756.92 | ||
| 99/64 | |||
| vvm6 | | vvm6 | ||
| vvBb | | vvBb | ||
| Line 239: | Line 281: | ||
| 42 | | 42 | ||
| 775.38 | | 775.38 | ||
| 25/16 | |||
| vm6 | | vm6 | ||
| vBb | | vBb | ||
| Line 244: | Line 287: | ||
| 43 | | 43 | ||
| 793.85 | | 793.85 | ||
| 128/81 | |||
| m6 | | m6 | ||
| Bb | | Bb | ||
| Line 249: | Line 293: | ||
| 44 | | 44 | ||
| 812.31 | | 812.31 | ||
| 8/5 | |||
| ^m6 | | ^m6 | ||
| ^Bb | | ^Bb | ||
| Line 254: | Line 299: | ||
| 45 | | 45 | ||
| 830.77 | | 830.77 | ||
| 21/13 | |||
| v~6 | | v~6 | ||
| ^^Bb | | ^^Bb | ||
| Line 259: | Line 305: | ||
| 46 | | 46 | ||
| 849.23 | | 849.23 | ||
| 18/11, 44/27 | |||
| ~6 | | ~6 | ||
| vvvB | | vvvB | ||
| Line 264: | Line 311: | ||
| 47 | | 47 | ||
| 867.69 | | 867.69 | ||
| 33/20 | |||
| ^~6 | | ^~6 | ||
| vvB | | vvB | ||
| Line 269: | Line 317: | ||
| 48 | | 48 | ||
| 886.15 | | 886.15 | ||
| 5/3 | |||
| vM6 | | vM6 | ||
| vB | | vB | ||
| Line 274: | Line 323: | ||
| 49 | | 49 | ||
| 904.62 | | 904.62 | ||
| 27/16 | |||
| M6 | | M6 | ||
| B | | B | ||
| Line 279: | Line 329: | ||
| 50 | | 50 | ||
| 923.08 | | 923.08 | ||
| 128/75 | |||
| ^M6 | | ^M6 | ||
| ^B | | ^B | ||
| Line 284: | Line 335: | ||
| 51 | | 51 | ||
| 941.54 | | 941.54 | ||
| 55/32 | |||
| ^^M6 | | ^^M6 | ||
| ^^B | | ^^B | ||
| Line 289: | Line 341: | ||
| 52 | | 52 | ||
| 960.00 | | 960.00 | ||
| 96/55 | |||
| vvm7 | | vvm7 | ||
| vvC | | vvC | ||
| Line 294: | Line 347: | ||
| 53 | | 53 | ||
| 978.46 | | 978.46 | ||
| 44/25 | |||
| vm7 | | vm7 | ||
| vC | | vC | ||
| Line 299: | Line 353: | ||
| 54 | | 54 | ||
| 996.92 | | 996.92 | ||
| 16/9 | |||
| m7 | | m7 | ||
| C | | C | ||
| Line 304: | Line 359: | ||
| 55 | | 55 | ||
| 1015.38 | | 1015.38 | ||
| 9/5 | |||
| ^m7 | | ^m7 | ||
| ^C | | ^C | ||
| Line 309: | Line 365: | ||
| 56 | | 56 | ||
| 1033.85 | | 1033.85 | ||
| 20/11 | |||
| v~7 | | v~7 | ||
| ^^C | | ^^C | ||
| Line 314: | Line 371: | ||
| 57 | | 57 | ||
| 1052.31 | | 1052.31 | ||
| 11/6 | |||
| ~7 | | ~7 | ||
| ^^^C | | ^^^C | ||
| Line 319: | Line 377: | ||
| 58 | | 58 | ||
| 1070.77 | | 1070.77 | ||
| 13/7 | |||
| ^~7 | | ^~7 | ||
| vvC# | | vvC# | ||
| Line 324: | Line 383: | ||
| 59 | | 59 | ||
| 1089.23 | | 1089.23 | ||
| 15/8 | |||
| vM7 | | vM7 | ||
| vC# | | vC# | ||
| Line 329: | Line 389: | ||
| 60 | | 60 | ||
| 1107.69 | | 1107.69 | ||
| 243/128, 256/135 | |||
| M7 | | M7 | ||
| C# | | C# | ||
| Line 334: | Line 395: | ||
| 61 | | 61 | ||
| 1126.15 | | 1126.15 | ||
| 48/25 | |||
| ^M7 | | ^M7 | ||
| ^C# | | ^C# | ||
| Line 339: | Line 401: | ||
| 62 | | 62 | ||
| 1144.62 | | 1144.62 | ||
| 64/33 | |||
| ^^M7 | | ^^M7 | ||
| ^^C# | | ^^C# | ||
| Line 344: | Line 407: | ||
| 63 | | 63 | ||
| 1163.08 | | 1163.08 | ||
| 88/45, 108/55, 125/64 | |||
| vv8 | | vv8 | ||
| vvD | | vvD | ||
| Line 349: | Line 413: | ||
| 64 | | 64 | ||
| 1181.54 | | 1181.54 | ||
| 160/81, 99/50, 240/121 | |||
| v8 | | v8 | ||
| vD | | vD | ||
| Line 354: | Line 419: | ||
| 65 | | 65 | ||
| 1200.00 | | 1200.00 | ||
| 2/1 | |||
| P8 | | P8 | ||
| D | | D | ||
|} | |} | ||
<nowiki>*</nowiki> based on treating 65edo as a 2.3.5.11.13/7 subgroup temperament. | |||
== Scales == | == Scales == | ||
Revision as of 11:28, 28 July 2021
| ← 64edo | 65edo | 66edo → |
The 65 equal divisions of the octave (65edo), or 65(-tone) equal temperament (65tet, 65et) when viewed from a regular temperament perspective, divides the octave into 65 equal parts of about 18.5 cents each.
Theory
65et can be characterized as the temperament which tempers out the schisma, 32805/32768, the sensipent comma, 78732/78125, and the würschmidt comma. In the 7-limit, there are two different maps; the first is ⟨65 103 151 182], tempering out 126/125, 245/243 and 686/675, so that it supports sensi temperament, and the second is ⟨65 103 151 183] (65d), tempering out 225/224, 3125/3087, 4000/3969 and 5120/5103, so that it supports garibaldi temperament. In both cases, the tuning privileges the 5-limit over the 7-limit, as the 5-limit of 65 is quite accurate. The same can be said for the two different versions of 7-limit würschmidt temperament (wurschmidt and worschmidt) these two mappings provide.
65edo approximates the intervals 3/2, 5/4, 11/8, 19/16, 23/16 and 31/16 well, so that it does a good job representing the 2.3.5.11.19.23.31 just intonation subgroup. To this one may want to add 17/16 and 29/16, giving the 31-limit no-7's no-13's subgroup 2.3.5.11.17.19.23.29.31. Also of interest is the 19-limit 2*65 subgroup 2.3.5.49.11.91.119.19, on which 65 has the same tuning and commas as the zeta edo 130edo.
65edo contains 13edo as a subset. The offset between a just perfect fifth at 702 cents and the 13edo superfifth at 738.5 cents, is approximately 2 degrees of 65edo. Therefore, an instrument fretted to 13edo, with open strings tuned to 3-limit intervals such as 4/3, 3/2, 9/8, 16/9 etc, will approximate a subset of 65edo. For an example of this, see Rubble: a Xenuke Unfolded.
Prime harmonics
Script error: No such module "primes_in_edo".
Intervals
| Degree | Cents | Approximate Ratios * | Ups and Downs Notation | |
|---|---|---|---|---|
| 0 | 0.00 | 1/1 | P1 | D |
| 1 | 18.46 | 81/80, 100/99, 121/120 | ^1 | ^D |
| 2 | 36.92 | 45/44, 55/54, 128/125 | ^^1 | ^^D |
| 3 | 55.38 | 33/32 | vvm2 | vvEb |
| 4 | 73.85 | 25/24 | vm2 | vEb |
| 5 | 92.31 | 135/128, 256/243 | m2 | Eb |
| 6 | 110.77 | 16/15 | A1/^m2 | D#/^Eb |
| 7 | 129.23 | 14/13 | v~2 | ^^Eb |
| 8 | 147.69 | 12/11 | ~2 | vvvE |
| 9 | 166.15 | 11/10 | ^~2 | vvE |
| 10 | 184.62 | 10/9 | vM2 | vE |
| 11 | 203.08 | 9/8 | M2 | E |
| 12 | 221.54 | 25/22 | ^M2 | ^E |
| 13 | 240.00 | 55/48 | ^^M2 | ^^E |
| 14 | 258.46 | 64/55 | vvm3 | vvF |
| 15 | 276.92 | 75/64 | vm3 | vF |
| 16 | 295.38 | 32/27 | m3 | F |
| 17 | 313.85 | 6/5 | ^m3 | ^F |
| 18 | 332.31 | 40/33 | v~3 | ^^F |
| 19 | 350.77 | 11/9, 27/22 | ~3 | ^^^F |
| 20 | 369.23 | 26/21 | ^~3 | vvF# |
| 21 | 387.69 | 5/4 | vM3 | vF# |
| 22 | 406.15 | 81/64 | M3 | F# |
| 23 | 424.62 | 32/25 | ^M3 | ^F# |
| 24 | 443.08 | 128/99 | ^^M3 | ^^F# |
| 25 | 461.54 | 72/55 | vv4 | vvG |
| 26 | 480.00 | 33/25 | v4 | vG |
| 27 | 498.46 | 4/3 | P4 | G |
| 28 | 516.92 | 27/20 | ^4 | ^G |
| 29 | 535.38 | 15/11 | v~4 | ^^G |
| 30 | 553.85 | 11/8 | ~4 | ^^^G |
| 31 | 572.31 | 25/18 | ^~4/vd5 | vvG#/vAb |
| 32 | 590.77 | 45/32 | vA4/d5 | vG#/Ab |
| 33 | 609.23 | 64/45 | A4/^d5 | G#/^Ab |
| 34 | 627.69 | 36/25 | ^A4/v~5 | ^G#/^^Ab |
| 35 | 646.15 | 16/11 | ~5 | vvvA |
| 36 | 664.62 | 22/15 | ^~5 | vvA |
| 37 | 683.08 | 40/27 | v5 | vA |
| 38 | 701.54 | 3/2 | P5 | A |
| 39 | 720.00 | 50/33 | ^5 | ^A |
| 40 | 738.46 | 55/36 | ^^5 | ^^A |
| 41 | 756.92 | 99/64 | vvm6 | vvBb |
| 42 | 775.38 | 25/16 | vm6 | vBb |
| 43 | 793.85 | 128/81 | m6 | Bb |
| 44 | 812.31 | 8/5 | ^m6 | ^Bb |
| 45 | 830.77 | 21/13 | v~6 | ^^Bb |
| 46 | 849.23 | 18/11, 44/27 | ~6 | vvvB |
| 47 | 867.69 | 33/20 | ^~6 | vvB |
| 48 | 886.15 | 5/3 | vM6 | vB |
| 49 | 904.62 | 27/16 | M6 | B |
| 50 | 923.08 | 128/75 | ^M6 | ^B |
| 51 | 941.54 | 55/32 | ^^M6 | ^^B |
| 52 | 960.00 | 96/55 | vvm7 | vvC |
| 53 | 978.46 | 44/25 | vm7 | vC |
| 54 | 996.92 | 16/9 | m7 | C |
| 55 | 1015.38 | 9/5 | ^m7 | ^C |
| 56 | 1033.85 | 20/11 | v~7 | ^^C |
| 57 | 1052.31 | 11/6 | ~7 | ^^^C |
| 58 | 1070.77 | 13/7 | ^~7 | vvC# |
| 59 | 1089.23 | 15/8 | vM7 | vC# |
| 60 | 1107.69 | 243/128, 256/135 | M7 | C# |
| 61 | 1126.15 | 48/25 | ^M7 | ^C# |
| 62 | 1144.62 | 64/33 | ^^M7 | ^^C# |
| 63 | 1163.08 | 88/45, 108/55, 125/64 | vv8 | vvD |
| 64 | 1181.54 | 160/81, 99/50, 240/121 | v8 | vD |
| 65 | 1200.00 | 2/1 | P8 | D |
* based on treating 65edo as a 2.3.5.11.13/7 subgroup temperament.