72edo: Difference between revisions
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| Line 291: | Line 291: | ||
| | degrees | | | degrees | ||
| | cents value | | | cents value | ||
|pions | |||
|7mus | |||
| | approximate ratios (11-limit) | | | approximate ratios (11-limit) | ||
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]] | | colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]] | ||
|- | |- | ||
| | | colspan="4"| 0 | ||
| | 1/1 | | | 1/1 | ||
| style="text-align:center;" | P1 | | style="text-align:center;" | P1 | ||
| Line 303: | Line 304: | ||
| | 1 | | | 1 | ||
| | 16.667 | | | 16.667 | ||
|17.667 | |||
|21.333 (15.555<sub>16</sub>) | |||
| | 81/80 | | | 81/80 | ||
| style="text-align:center;" | ^1 | | style="text-align:center;" | ^1 | ||
| Line 310: | Line 313: | ||
| | 2 | | | 2 | ||
| | 33.333 | | | 33.333 | ||
|35.333 | |||
|42.667 (2A.AAB<sub>16</sub>) | |||
| | 45/44 | | | 45/44 | ||
| style="text-align:center;" | ^^ | | style="text-align:center;" | ^^ | ||
| Line 317: | Line 322: | ||
| | 3 | | | 3 | ||
| | 50 | | | 50 | ||
|53 | |||
|64 (40<sub>16</sub>) | |||
| | 33/32 | | | 33/32 | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>1, v<span style="font-size: 90%; vertical-align: super;">3</span>m2 | ||
| Line 326: | Line 333: | ||
| | 4 | | | 4 | ||
| | 66.667 | | | 66.667 | ||
|70.667 | |||
|85.333 (55.555<sub>16</sub>) | |||
| | 25/24 | | | 25/24 | ||
| style="text-align:center;" | vvm2 | | style="text-align:center;" | vvm2 | ||
| Line 333: | Line 342: | ||
| | 5 | | | 5 | ||
| | 83.333 | | | 83.333 | ||
|88.333 | |||
|106.667 (6A.AAB<sub>16</sub>) | |||
| | 21/20 | | | 21/20 | ||
| style="text-align:center;" | vm2 | | style="text-align:center;" | vm2 | ||
| Line 340: | Line 351: | ||
| | 6 | | | 6 | ||
| | 100 | | | 100 | ||
|106 | |||
|128 (80<sub>16</sub>) | |||
| | 35/33 | | | 35/33 | ||
| style="text-align:center;" | m2 | | style="text-align:center;" | m2 | ||
| Line 347: | Line 360: | ||
| | 7 | | | 7 | ||
| | 116.667 | | | 116.667 | ||
|123.667 | |||
|149.333 (95.555<sub>16</sub>) | |||
| | 15/14 | | | 15/14 | ||
| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| Line 354: | Line 369: | ||
| | 8 | | | 8 | ||
| | 133.333 | | | 133.333 | ||
|141.333 | |||
|170.667 (AA.AAB<sub>16</sub>) | |||
| | 27/25 | | | 27/25 | ||
| style="text-align:center;" | v~2 | | style="text-align:center;" | v~2 | ||
| Line 361: | Line 378: | ||
| | 9 | | | 9 | ||
| | 150 | | | 150 | ||
|159 | |||
|192 (C0<sub>16</sub>) | |||
| | 12/11 | | | 12/11 | ||
| style="text-align:center;" | ~2 | | style="text-align:center;" | ~2 | ||
| Line 368: | Line 387: | ||
| | 10 | | | 10 | ||
| | 166.667 | | | 166.667 | ||
|176.667 | |||
|213.333 (D5.555<sub>16</sub>) | |||
| | 11/10 | | | 11/10 | ||
| style="text-align:center;" | ^~2 | | style="text-align:center;" | ^~2 | ||
| Line 375: | Line 396: | ||
| | 11 | | | 11 | ||
| | 183.333 | | | 183.333 | ||
|194.333 | |||
|234.667 (EA.AAB<sub>16</sub>) | |||
| | 10/9 | | | 10/9 | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| Line 382: | Line 405: | ||
| | 12 | | | 12 | ||
| | 200 | | | 200 | ||
|212 | |||
|256 (100<sub>16</sub>) | |||
| | 9/8 | | | 9/8 | ||
| style="text-align:center;" | M2 | | style="text-align:center;" | M2 | ||
| Line 389: | Line 414: | ||
| | 13 | | | 13 | ||
| | 216.667 | | | 216.667 | ||
|229.667 | |||
|277.333 (115.555<sub>16</sub>) | |||
| | 25/22 | | | 25/22 | ||
| style="text-align:center;" | ^M2 | | style="text-align:center;" | ^M2 | ||
| Line 396: | Line 423: | ||
| | 14 | | | 14 | ||
| | 233.333 | | | 233.333 | ||
|247.333 | |||
|298.667 (12A.AAB<sub>16</sub>) | |||
| | 8/7 | | | 8/7 | ||
| style="text-align:center;" | ^^M2 | | style="text-align:center;" | ^^M2 | ||
| Line 403: | Line 432: | ||
| | 15 | | | 15 | ||
| | 250 | | | 250 | ||
|265 | |||
|320 (140<sub>16</sub>) | |||
| | 81/70 | | | 81/70 | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M2, v<span style="font-size: 90%; vertical-align: super;">3</span>m3 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M2, v<span style="font-size: 90%; vertical-align: super;">3</span>m3 | ||
| Line 412: | Line 443: | ||
| | 16 | | | 16 | ||
| | 266.667 | | | 266.667 | ||
|282.333 | |||
|341.333 (155.555<sub>16</sub>) | |||
| | 7/6 | | | 7/6 | ||
| style="text-align:center;" | vvm3 | | style="text-align:center;" | vvm3 | ||
| Line 419: | Line 452: | ||
| | 17 | | | 17 | ||
| | 283.333 | | | 283.333 | ||
|300.333 | |||
|362.667 (16A.AAB<sub>16</sub>) | |||
| | 33/28 | | | 33/28 | ||
| style="text-align:center;" | vm3 | | style="text-align:center;" | vm3 | ||
| Line 426: | Line 461: | ||
| | 18 | | | 18 | ||
| | 300 | | | 300 | ||
|318 | |||
|384 (180<sub>16</sub>) | |||
| | 25/21 | | | 25/21 | ||
| style="text-align:center;" | m3 | | style="text-align:center;" | m3 | ||
| Line 433: | Line 470: | ||
| | 19 | | | 19 | ||
| | 316.667 | | | 316.667 | ||
|335.667 | |||
|405.333 (195.555<sub>16</sub>) | |||
| | 6/5 | | | 6/5 | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| Line 440: | Line 479: | ||
| | 20 | | | 20 | ||
| | 333.333 | | | 333.333 | ||
|353.333 | |||
|426.667 (1AA.AAB<sub>16</sub>) | |||
| | 40/33 | | | 40/33 | ||
| style="text-align:center;" | v~3 | | style="text-align:center;" | v~3 | ||
| Line 447: | Line 488: | ||
| | 21 | | | 21 | ||
| | 350 | | | 350 | ||
|371 | |||
|448 (1C0<sub>16</sub>) | |||
| | 11/9 | | | 11/9 | ||
| style="text-align:center;" | ~3 | | style="text-align:center;" | ~3 | ||
| Line 454: | Line 497: | ||
| | 22 | | | 22 | ||
| | 366.667 | | | 366.667 | ||
|388.667 | |||
|469.333 (1D5.555<sub>16</sub>) | |||
| | 99/80 | | | 99/80 | ||
| style="text-align:center;" | ^~3 | | style="text-align:center;" | ^~3 | ||
| Line 461: | Line 506: | ||
| | 23 | | | 23 | ||
| | 383.333 | | | 383.333 | ||
|406.333 | |||
|490.667 (1EA.AAB<sub>16</sub>) | |||
| | 5/4 | | | 5/4 | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| Line 468: | Line 515: | ||
| | 24 | | | 24 | ||
| | 400 | | | 400 | ||
|424 | |||
|512 (200<sub>16</sub>) | |||
| | 44/35 | | | 44/35 | ||
| style="text-align:center;" | M3 | | style="text-align:center;" | M3 | ||
| Line 475: | Line 524: | ||
| | 25 | | | 25 | ||
| | 416.667 | | | 416.667 | ||
|441.667 | |||
|533.333 (215.555<sub>16</sub>) | |||
| | 14/11 | | | 14/11 | ||
| style="text-align:center;" | ^M3 | | style="text-align:center;" | ^M3 | ||
| Line 482: | Line 533: | ||
| | 26 | | | 26 | ||
| | 433.333 | | | 433.333 | ||
|459.333 | |||
|554.667 (22A.AAB<sub>16</sub>) | |||
| | 9/7 | | | 9/7 | ||
| style="text-align:center;" | ^^M3 | | style="text-align:center;" | ^^M3 | ||
| Line 489: | Line 542: | ||
| | 27 | | | 27 | ||
| | 450 | | | 450 | ||
|477 | |||
|576 (240<sub>16</sub>) | |||
| | 35/27 | | | 35/27 | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M3, v<span style="font-size: 90%; vertical-align: super;">3</span>4 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M3, v<span style="font-size: 90%; vertical-align: super;">3</span>4 | ||
| Line 498: | Line 553: | ||
| | 28 | | | 28 | ||
| | 466.667 | | | 466.667 | ||
|494.667 | |||
|597.333 (255.555<sub>16</sub>) | |||
| | 21/16 | | | 21/16 | ||
| style="text-align:center;" | vv4 | | style="text-align:center;" | vv4 | ||
| Line 505: | Line 562: | ||
| | 29 | | | 29 | ||
| | 483.333 | | | 483.333 | ||
|512.333 | |||
|618.667 (26A.AAB<sub>16</sub>) | |||
| | 33/25 | | | 33/25 | ||
| style="text-align:center;" | v4 | | style="text-align:center;" | v4 | ||
| Line 512: | Line 571: | ||
| | 30 | | | 30 | ||
| | 500 | | | 500 | ||
|530 | |||
|640 (280<sub>16</sub>) | |||
| | 4/3 | | | 4/3 | ||
| style="text-align:center;" | P4 | | style="text-align:center;" | P4 | ||
| Line 519: | Line 580: | ||
| | 31 | | | 31 | ||
| | 516.667 | | | 516.667 | ||
|547.667 | |||
|661.333 (295.555<sub>16</sub>) | |||
| | 27/20 | | | 27/20 | ||
| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| Line 526: | Line 589: | ||
| | 32 | | | 32 | ||
| | 533.333 | | | 533.333 | ||
|565.333 | |||
|682.667 (2AA.AAB<sub>16</sub>) | |||
| | 15/11 | | | 15/11 | ||
| style="text-align:center;" | ^^4 | | style="text-align:center;" | ^^4 | ||
| Line 533: | Line 598: | ||
| | 33 | | | 33 | ||
| | 550 | | | 550 | ||
|583 | |||
|704 (2C0<sub>16</sub>) | |||
| | 11/8 | | | 11/8 | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>4 | ||
| Line 540: | Line 607: | ||
| | 34 | | | 34 | ||
| | 566.667 | | | 566.667 | ||
|600.667 | |||
|725.333 (2D5.555<sub>16</sub>) | |||
| | 25/18 | | | 25/18 | ||
| style="text-align:center;" | vvA4 | | style="text-align:center;" | vvA4 | ||
| Line 547: | Line 616: | ||
| | 35 | | | 35 | ||
| | 583.333 | | | 583.333 | ||
|618.333 | |||
|746.667 (2EA.AAB<sub>16</sub>) | |||
| | 7/5 | | | 7/5 | ||
| style="text-align:center;" | vA4, vd5 | | style="text-align:center;" | vA4, vd5 | ||
| Line 554: | Line 625: | ||
| | 36 | | | 36 | ||
| | 600 | | | 600 | ||
|636 | |||
|768 (300<sub>16</sub>) | |||
| | 99/70 | | | 99/70 | ||
| style="text-align:center;" | A4, d5 | | style="text-align:center;" | A4, d5 | ||
| Line 561: | Line 634: | ||
| | 37 | | | 37 | ||
| | 616.667 | | | 616.667 | ||
|653.667 | |||
|789.333 (315.555<sub>16</sub>) | |||
| | 10/7 | | | 10/7 | ||
| style="text-align:center;" | ^A4, ^d5 | | style="text-align:center;" | ^A4, ^d5 | ||
| Line 568: | Line 643: | ||
| | 38 | | | 38 | ||
| | 633.333 | | | 633.333 | ||
|671.333 | |||
|810.667 (32A.AAB<sub>16</sub>) | |||
| | 36/25 | | | 36/25 | ||
| style="text-align:center;" | ^^d5 | | style="text-align:center;" | ^^d5 | ||
| Line 575: | Line 652: | ||
| | 39 | | | 39 | ||
| | 650 | | | 650 | ||
|689 | |||
|832 (340<sub>16</sub>) | |||
| | 16/11 | | | 16/11 | ||
| style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>5 | | style="text-align:center;" | v<span style="font-size: 90%; vertical-align: super;">3</span>5 | ||
| Line 582: | Line 661: | ||
| | 40 | | | 40 | ||
| | 666.667 | | | 666.667 | ||
|706.667 | |||
|853.333 (355.555<sub>16</sub>) | |||
| | 22/15 | | | 22/15 | ||
| style="text-align:center;" | vv5 | | style="text-align:center;" | vv5 | ||
| Line 589: | Line 670: | ||
| | 41 | | | 41 | ||
| | 683.333 | | | 683.333 | ||
|724.333 | |||
|874.667 (36A.AAB<sub>16</sub>) | |||
| | 40/27 | | | 40/27 | ||
| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| Line 596: | Line 679: | ||
| | 42 | | | 42 | ||
| | 700 | | | 700 | ||
|742 | |||
|896 (380<sub>16</sub>) | |||
| | 3/2 | | | 3/2 | ||
| style="text-align:center;" | P5 | | style="text-align:center;" | P5 | ||
| Line 603: | Line 688: | ||
| | 43 | | | 43 | ||
| | 716.667 | | | 716.667 | ||
|759.667 | |||
|917.333 (395.555<sub>16</sub>) | |||
| | 50/33 | | | 50/33 | ||
| style="text-align:center;" | ^5 | | style="text-align:center;" | ^5 | ||
| Line 610: | Line 697: | ||
| | 44 | | | 44 | ||
| | 733.333 | | | 733.333 | ||
|777.333 | |||
|948.667 (3AA.AAB<sub>16</sub>) | |||
| | 32/21 | | | 32/21 | ||
| style="text-align:center;" | ^^5 | | style="text-align:center;" | ^^5 | ||
| Line 617: | Line 706: | ||
| | 45 | | | 45 | ||
| | 750 | | | 750 | ||
|795 | |||
|960 (3C0<sub>16</sub>) | |||
| | 54/35 | | | 54/35 | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>5, v<span style="font-size: 90%; vertical-align: super;">3</span>m6 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>5, v<span style="font-size: 90%; vertical-align: super;">3</span>m6 | ||
| Line 626: | Line 717: | ||
| | 46 | | | 46 | ||
| | 766.667 | | | 766.667 | ||
|812.667 | |||
|981.333 (3D5.555<sub>16</sub>) | |||
| | 14/9 | | | 14/9 | ||
| style="text-align:center;" | vvm6 | | style="text-align:center;" | vvm6 | ||
| Line 633: | Line 726: | ||
| | 47 | | | 47 | ||
| | 783.333 | | | 783.333 | ||
|830.333 | |||
|1002.667 (2EA.AAB<sub>16</sub>) | |||
| | 11/7 | | | 11/7 | ||
| style="text-align:center;" | vm6 | | style="text-align:center;" | vm6 | ||
| Line 640: | Line 735: | ||
| | 48 | | | 48 | ||
| | 800 | | | 800 | ||
|848 | |||
|1024 (400<sub>16</sub>) | |||
| | 35/22 | | | 35/22 | ||
| style="text-align:center;" | m6 | | style="text-align:center;" | m6 | ||
| Line 647: | Line 744: | ||
| | 49 | | | 49 | ||
| | 816.667 | | | 816.667 | ||
|865.667 | |||
|1045.333 (415.555<sub>16</sub>) | |||
| | 8/5 | | | 8/5 | ||
| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| Line 654: | Line 753: | ||
| | 50 | | | 50 | ||
| | 833.333 | | | 833.333 | ||
|883.333 | |||
|1066.667 (42A.AAB<sub>16</sub>) | |||
| | 81/50 | | | 81/50 | ||
| style="text-align:center;" | v~6 | | style="text-align:center;" | v~6 | ||
| Line 661: | Line 762: | ||
| | 51 | | | 51 | ||
| | 850 | | | 850 | ||
|901 | |||
|1088 (440<sub>16</sub>) | |||
| | 18/11 | | | 18/11 | ||
| style="text-align:center;" | ~6 | | style="text-align:center;" | ~6 | ||
| Line 668: | Line 771: | ||
| | 52 | | | 52 | ||
| | 866.667 | | | 866.667 | ||
|918.667 | |||
|1109.333 (455.555<sub>16</sub>) | |||
| | 33/20 | | | 33/20 | ||
| style="text-align:center;" | ^~6 | | style="text-align:center;" | ^~6 | ||
| Line 675: | Line 780: | ||
| | 53 | | | 53 | ||
| | 883.333 | | | 883.333 | ||
|936.333 | |||
|1130.667 (46A.AAB<sub>16</sub>) | |||
| | 5/3 | | | 5/3 | ||
| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| Line 682: | Line 789: | ||
| | 54 | | | 54 | ||
| | 900 | | | 900 | ||
|954 | |||
|1152 (480<sub>16</sub>) | |||
| | 27/16 | | | 27/16 | ||
| style="text-align:center;" | M6 | | style="text-align:center;" | M6 | ||
| Line 689: | Line 798: | ||
| | 55 | | | 55 | ||
| | 916.667 | | | 916.667 | ||
|971.667 | |||
|1173.333 (495.555<sub>16</sub>) | |||
| | 56/33 | | | 56/33 | ||
| style="text-align:center;" | ^M6 | | style="text-align:center;" | ^M6 | ||
| Line 696: | Line 807: | ||
| | 56 | | | 56 | ||
| | 933.333 | | | 933.333 | ||
|989.333 | |||
|1194.667 (4AA.AAB<sub>16</sub>) | |||
| | 12/7 | | | 12/7 | ||
| style="text-align:center;" | ^^M6 | | style="text-align:center;" | ^^M6 | ||
| Line 703: | Line 816: | ||
| | 57 | | | 57 | ||
| | 950 | | | 950 | ||
|1007 | |||
|1216 (4C0<sub>16</sub>) | |||
| | 121/70 | | | 121/70 | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M6, v<span style="font-size: 90%; vertical-align: super;">3</span>m7 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M6, v<span style="font-size: 90%; vertical-align: super;">3</span>m7 | ||
| Line 712: | Line 827: | ||
| | 58 | | | 58 | ||
| | 966.667 | | | 966.667 | ||
|1024.667 | |||
|1237.333 (4D5.555<sub>16</sub>) | |||
| | 7/4 | | | 7/4 | ||
| style="text-align:center;" | vvm7 | | style="text-align:center;" | vvm7 | ||
| Line 719: | Line 836: | ||
| | 59 | | | 59 | ||
| | 983.333 | | | 983.333 | ||
|1042.333 | |||
|1258.667 (4EA.AAB<sub>16</sub>) | |||
| | 44/25 | | | 44/25 | ||
| style="text-align:center;" | vm7 | | style="text-align:center;" | vm7 | ||
| Line 726: | Line 845: | ||
| | 60 | | | 60 | ||
| | 1000 | | | 1000 | ||
|1060 | |||
|1280 (500<sub>16</sub>) | |||
| | 16/9 | | | 16/9 | ||
| style="text-align:center;" | m7 | | style="text-align:center;" | m7 | ||
| Line 733: | Line 854: | ||
| | 61 | | | 61 | ||
| | 1016.667 | | | 1016.667 | ||
|1077.667 | |||
|1301.333 (515.555<sub>16</sub>) | |||
| | 9/5 | | | 9/5 | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| Line 740: | Line 863: | ||
| | 62 | | | 62 | ||
| | 1033.333 | | | 1033.333 | ||
|1095.333 | |||
|1322.667 (52A.AAB<sub>16</sub>) | |||
| | 20/11 | | | 20/11 | ||
| style="text-align:center;" | v~7 | | style="text-align:center;" | v~7 | ||
| Line 747: | Line 872: | ||
| | 63 | | | 63 | ||
| | 1050 | | | 1050 | ||
|1113 | |||
|1344 (540<sub>16</sub>) | |||
| | 11/6 | | | 11/6 | ||
| style="text-align:center;" | ~7 | | style="text-align:center;" | ~7 | ||
| Line 754: | Line 881: | ||
| | 64 | | | 64 | ||
| | 1066.667 | | | 1066.667 | ||
|1130.667 | |||
|1365.333 (555.555<sub>16</sub>) | |||
| | 50/27 | | | 50/27 | ||
| style="text-align:center;" | ^~7 | | style="text-align:center;" | ^~7 | ||
| Line 761: | Line 890: | ||
| | 65 | | | 65 | ||
| | 1083.333 | | | 1083.333 | ||
|1148.333 | |||
|1386.667 (56A.AAB<sub>16</sub>) | |||
| | 15/8 | | | 15/8 | ||
| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| Line 768: | Line 899: | ||
| | 66 | | | 66 | ||
| | 1100 | | | 1100 | ||
|1166 | |||
|1408 (580<sub>16</sub>) | |||
| | 66/35 | | | 66/35 | ||
| style="text-align:center;" | M7 | | style="text-align:center;" | M7 | ||
| Line 775: | Line 908: | ||
| | 67 | | | 67 | ||
| | 1116.667 | | | 1116.667 | ||
|1183.667 | |||
|1429.333 (595.555<sub>16</sub>) | |||
| | 21/11 | | | 21/11 | ||
| style="text-align:center;" | ^M7 | | style="text-align:center;" | ^M7 | ||
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|1201.333 | |||
|1450.667 (5AA.AAB<sub>16</sub>) | |||
| | 27/14 | | | 27/14 | ||
| style="text-align:center;" | ^^M7 | | style="text-align:center;" | ^^M7 | ||
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| | 1150 | | | 1150 | ||
|1219 | |||
|1472 (5C0<sub>16</sub>) | |||
| | 35/18 | | | 35/18 | ||
| style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M7, v<span style="font-size: 90%; vertical-align: super;">3</span>8 | | style="text-align:center;" | ^<span style="font-size: 90%; vertical-align: super;">3</span>M7, v<span style="font-size: 90%; vertical-align: super;">3</span>8 | ||
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|1236.667 | |||
|1493.333 (5D5.555<sub>16</sub>) | |||
| | 49/25 | | | 49/25 | ||
| style="text-align:center;" | vv8 | | style="text-align:center;" | vv8 | ||
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|1254.333 | |||
|1514.667 (5EA.AAB<sub>16</sub>) | |||
| | 99/50 | | | 99/50 | ||
| style="text-align:center;" | v8 | | style="text-align:center;" | v8 | ||
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| | 72 | | | 72 | ||
| | 1200 | | | 1200 | ||
|1272 | |||
|1536 (600<sub>16</sub>) | |||
| | 2/1 | | | 2/1 | ||
| style="text-align:center;" | P8 | | style="text-align:center;" | P8 | ||
Revision as of 19:55, 19 May 2019
72-tone equal temperament, or 72-edo, divides the octave into 72 steps or moria. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of 24-tone equal temperament, a common and standard tuning of Arabic music, and has itself been used to tune Turkish music.
Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with 96-edo), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
72-tone equal temperament approximates 11-limit just intonation exceptionally well, is consistent in the 17-limit, and is the ninth Zeta integral tuning. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third (5/4) measures 23 steps, not 24, and other major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh (7/4) is 58 steps, while the undecimal semiaugmented fourth (11/8) is 33.
72 is an excellent tuning for miracle temperament, especially the 11-limit version, and the related rank three temperament prodigy, and is a good tuning for other temperaments and scales, including wizard, harry, catakleismic, compton, unidec and tritikleismic.
Commas
Commas tempered out by 72edo include...
| 3-limit |
|---|
| Pythagorean comma = 531441/524288 = |-19 12> |
| 5-limit |
|---|
| kleisma = 15625/15552 = |-6 -5 6>
ampersand = 34171875/33554432 = |-25 7 6> graviton = 129140163/128000000 = |-13 17 -6> ennealimma = 7629394531250/7625597484987 = |1 -27 18> |
| 7-limit | 11-limit | 13-limit |
|---|---|---|
| ...............................
225/224 1029/1024 2401/2400 4375/4374 16875/16807 19683/19600 420175/419904 250047/250000 |
.......................
243/242 385/384 441/440 540/539 1375/1372 3025/3024 4000/3993 6250/6237 9801/9800 |
.......................
169/168 325/324 351/350 364/363 625/624 676/675 729/728 1001/1000 1575/1573 1716/1715 2080/2079 6656/6655 |
Temperaments
It provides the optimal patent val for miracle and wizard in the 7-limit, miracle, catakleismic, bikleismic, compton, ennealimnic, ennealiminal, enneaportent, marvolo and catalytic in the 11-limit, and catakleismic, bikleismic, compton, comptone, enneaportent, ennealim, catalytic, marvolo, manna, hendec, lizard, neominor, hours, and semimiracle in the 13-limit.
See also List of edo-distinct 72et rank two temperaments.
Harmonic Scale
Mode 8 of the harmonic series -- overtones 8 through 16, octave repeating -- is well-represented in 72edo. Note that all the different step sizes are distinguished, except for 13:12 and 14:13 (conflated to 8\72edo, 133.3 cents) and 15:14 and 16:15 (conflated to 7\72edo, 116.7 cents, the generator for miracle temperament).
| Overtones in "Mode 8": | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | ||||||||
| ...as JI Ratio from 1/1: | 1/1 | 9/8 | 5/4 | 11/8 | 3/2 | 13/8 | 7/4 | 15/8 | 2/1 | ||||||||
| ...in cents: | 0 | 203.9 | 386.3 | 551.3 | 702.0 | 840.5 | 968.8 | 1088.3 | 1200.0 | ||||||||
| Nearest degree of 72edo: | 0 | 12 | 23 | 33 | 42 | 50 | 58 | 65 | 72 | ||||||||
| ...in cents: | 0 | 200.0 | 383.3 | 550.0 | 700.0 | 833.3 | 966.7 | 1083.3 | 1200.0 | ||||||||
| Steps as Freq. Ratio: | 9:8 | 10:9 | 11:10 | 12:11 | 13:12 | 14:13 | 15:14 | 16:15 | |||||||||
| ...in cents: | 203.9 | 182.4 | 165.0 | 150.6 | 138.6 | 128.3 | 119.4 | 111.7 | |||||||||
| Nearest degree of 72edo: | 12 | 11 | 10 | 9 | 8 | 8 | 7 | 7 | |||||||||
| ...in cents: | 200.0 | 183.3 | 166.7 | 150.0 | 133.3 | 133.3 | 116.7 | 116.7 |
Intervals
| degrees | cents value | pions | 7mus | approximate ratios (11-limit) | ups and downs notation | ||
| 0 | 1/1 | P1 | perfect unison | D | |||
| 1 | 16.667 | 17.667 | 21.333 (15.55516) | 81/80 | ^1 | up unison | D^ |
| 2 | 33.333 | 35.333 | 42.667 (2A.AAB16) | 45/44 | ^^ | double-up unison | D^^ |
| 3 | 50 | 53 | 64 (4016) | 33/32 | ^31, v3m2 | triple-up unison,
triple-down minor 2nd |
D^3, Ebv3 |
| 4 | 66.667 | 70.667 | 85.333 (55.55516) | 25/24 | vvm2 | double-downminor 2nd | Ebvv |
| 5 | 83.333 | 88.333 | 106.667 (6A.AAB16) | 21/20 | vm2 | downminor 2nd | Ebv |
| 6 | 100 | 106 | 128 (8016) | 35/33 | m2 | minor 2nd | Eb |
| 7 | 116.667 | 123.667 | 149.333 (95.55516) | 15/14 | ^m2 | upminor 2nd | Eb^ |
| 8 | 133.333 | 141.333 | 170.667 (AA.AAB16) | 27/25 | v~2 | downmid 2nd | Eb^^ |
| 9 | 150 | 159 | 192 (C016) | 12/11 | ~2 | mid 2nd | Ev3 |
| 10 | 166.667 | 176.667 | 213.333 (D5.55516) | 11/10 | ^~2 | upmid 2nd | Evv |
| 11 | 183.333 | 194.333 | 234.667 (EA.AAB16) | 10/9 | vM2 | downmajor 2nd | Ev |
| 12 | 200 | 212 | 256 (10016) | 9/8 | M2 | major 2nd | E |
| 13 | 216.667 | 229.667 | 277.333 (115.55516) | 25/22 | ^M2 | upmajor 2nd | E^ |
| 14 | 233.333 | 247.333 | 298.667 (12A.AAB16) | 8/7 | ^^M2 | double-upmajor 2nd | E^^ |
| 15 | 250 | 265 | 320 (14016) | 81/70 | ^3M2, v3m3 | triple-up major 2nd,
triple-down minor 3rd |
E^3, Fv3 |
| 16 | 266.667 | 282.333 | 341.333 (155.55516) | 7/6 | vvm3 | double-downminor 3rd | Fvv |
| 17 | 283.333 | 300.333 | 362.667 (16A.AAB16) | 33/28 | vm3 | downminor 3rd | Fv |
| 18 | 300 | 318 | 384 (18016) | 25/21 | m3 | minor 3rd | F |
| 19 | 316.667 | 335.667 | 405.333 (195.55516) | 6/5 | ^m3 | upminor 3rd | F^ |
| 20 | 333.333 | 353.333 | 426.667 (1AA.AAB16) | 40/33 | v~3 | downmid 3rd | F^^ |
| 21 | 350 | 371 | 448 (1C016) | 11/9 | ~3 | mid 3rd | F^3 |
| 22 | 366.667 | 388.667 | 469.333 (1D5.55516) | 99/80 | ^~3 | upmid 3rd | F#vv |
| 23 | 383.333 | 406.333 | 490.667 (1EA.AAB16) | 5/4 | vM3 | downmajor 3rd | F#v |
| 24 | 400 | 424 | 512 (20016) | 44/35 | M3 | major 3rd | F# |
| 25 | 416.667 | 441.667 | 533.333 (215.55516) | 14/11 | ^M3 | upmajor 3rd | F#^ |
| 26 | 433.333 | 459.333 | 554.667 (22A.AAB16) | 9/7 | ^^M3 | double-upmajor 3rd | F#^^ |
| 27 | 450 | 477 | 576 (24016) | 35/27 | ^3M3, v34 | triple-up major 3rd,
triple-down 4th |
F#^3, Gv3 |
| 28 | 466.667 | 494.667 | 597.333 (255.55516) | 21/16 | vv4 | double-down 4th | Gvv |
| 29 | 483.333 | 512.333 | 618.667 (26A.AAB16) | 33/25 | v4 | down 4th | Gv |
| 30 | 500 | 530 | 640 (28016) | 4/3 | P4 | perfect 4th | G |
| 31 | 516.667 | 547.667 | 661.333 (295.55516) | 27/20 | ^4 | up 4th | G^ |
| 32 | 533.333 | 565.333 | 682.667 (2AA.AAB16) | 15/11 | ^^4 | double-up 4th | G^^ |
| 33 | 550 | 583 | 704 (2C016) | 11/8 | ^34 | triple-up 4th | G^3 |
| 34 | 566.667 | 600.667 | 725.333 (2D5.55516) | 25/18 | vvA4 | double-down aug 4th | G#vv |
| 35 | 583.333 | 618.333 | 746.667 (2EA.AAB16) | 7/5 | vA4, vd5 | downaug 4th, updim 5th | G#v, Abv |
| 36 | 600 | 636 | 768 (30016) | 99/70 | A4, d5 | aug 4th, dim 5th | G#, Ab |
| 37 | 616.667 | 653.667 | 789.333 (315.55516) | 10/7 | ^A4, ^d5 | upaug 4th, downdim 5th | G#^, Ab^ |
| 38 | 633.333 | 671.333 | 810.667 (32A.AAB16) | 36/25 | ^^d5 | double-updim 5th | Ab^^ |
| 39 | 650 | 689 | 832 (34016) | 16/11 | v35 | triple-down 5th | Av3 |
| 40 | 666.667 | 706.667 | 853.333 (355.55516) | 22/15 | vv5 | double-down 5th | Avv |
| 41 | 683.333 | 724.333 | 874.667 (36A.AAB16) | 40/27 | v5 | down 5th | Av |
| 42 | 700 | 742 | 896 (38016) | 3/2 | P5 | perfect 5th | A |
| 43 | 716.667 | 759.667 | 917.333 (395.55516) | 50/33 | ^5 | up 5th | A^ |
| 44 | 733.333 | 777.333 | 948.667 (3AA.AAB16) | 32/21 | ^^5 | double-up 5th | A^^ |
| 45 | 750 | 795 | 960 (3C016) | 54/35 | ^35, v3m6 | triple-up 5th,
triple-down minor 6th |
A^3, Bbv3 |
| 46 | 766.667 | 812.667 | 981.333 (3D5.55516) | 14/9 | vvm6 | double-downminor 6th | Bbvv |
| 47 | 783.333 | 830.333 | 1002.667 (2EA.AAB16) | 11/7 | vm6 | downminor 6th | Bbv |
| 48 | 800 | 848 | 1024 (40016) | 35/22 | m6 | minor 6th | Bb |
| 49 | 816.667 | 865.667 | 1045.333 (415.55516) | 8/5 | ^m6 | upminor 6th | Bb^ |
| 50 | 833.333 | 883.333 | 1066.667 (42A.AAB16) | 81/50 | v~6 | downmid 6th | Bb^^ |
| 51 | 850 | 901 | 1088 (44016) | 18/11 | ~6 | mid 6th | Bv3 |
| 52 | 866.667 | 918.667 | 1109.333 (455.55516) | 33/20 | ^~6 | upmid 6th | Bvv |
| 53 | 883.333 | 936.333 | 1130.667 (46A.AAB16) | 5/3 | vM6 | downmajor 6th | Bv |
| 54 | 900 | 954 | 1152 (48016) | 27/16 | M6 | major 6th | B |
| 55 | 916.667 | 971.667 | 1173.333 (495.55516) | 56/33 | ^M6 | upmajor 6th | B^ |
| 56 | 933.333 | 989.333 | 1194.667 (4AA.AAB16) | 12/7 | ^^M6 | double-upmajor 6th | B^^ |
| 57 | 950 | 1007 | 1216 (4C016) | 121/70 | ^3M6, v3m7 | triple-up major 6th,
triple-down minor 7th |
B^3, Cv3 |
| 58 | 966.667 | 1024.667 | 1237.333 (4D5.55516) | 7/4 | vvm7 | double-downminor 7th | Cvv |
| 59 | 983.333 | 1042.333 | 1258.667 (4EA.AAB16) | 44/25 | vm7 | downminor 7th | Cv |
| 60 | 1000 | 1060 | 1280 (50016) | 16/9 | m7 | minor 7th | C |
| 61 | 1016.667 | 1077.667 | 1301.333 (515.55516) | 9/5 | ^m7 | upminor 7th | C^ |
| 62 | 1033.333 | 1095.333 | 1322.667 (52A.AAB16) | 20/11 | v~7 | downmid 7th | C^^ |
| 63 | 1050 | 1113 | 1344 (54016) | 11/6 | ~7 | mid 7th | C^3 |
| 64 | 1066.667 | 1130.667 | 1365.333 (555.55516) | 50/27 | ^~7 | upmid 7th | C#vv |
| 65 | 1083.333 | 1148.333 | 1386.667 (56A.AAB16) | 15/8 | vM7 | downmajor 7th | C#v |
| 66 | 1100 | 1166 | 1408 (58016) | 66/35 | M7 | major 7th | C# |
| 67 | 1116.667 | 1183.667 | 1429.333 (595.55516) | 21/11 | ^M7 | upmajor 7th | C#^ |
| 68 | 1133.333 | 1201.333 | 1450.667 (5AA.AAB16) | 27/14 | ^^M7 | double-upmajor 7th | C#^^ |
| 69 | 1150 | 1219 | 1472 (5C016) | 35/18 | ^3M7, v38 | triple-up major 7th,
triple-down octave |
C#^3, Dv3 |
| 70 | 1166.667 | 1236.667 | 1493.333 (5D5.55516) | 49/25 | vv8 | double-down octave | Dvv |
| 71 | 1183.333 | 1254.333 | 1514.667 (5EA.AAB16) | 99/50 | v8 | down octave | Dv |
| 72 | 1200 | 1272 | 1536 (60016) | 2/1 | P8 | perfect octave | D |
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color | monzo format | examples |
|---|---|---|---|
| double-down minor | zo | {a, b, 0, 1} | 7/6, 7/4 |
| minor | fourthward wa | {a, b}, b < -1 | 32/27, 16/9 |
| upminor | gu | {a, b, -1} | 6/5, 9/5 |
| mid | ilo | {a, b, 0, 0, 1} | 11/9, 11/6 |
| " | lu | {a, b, 0, 0, -1} | 12/11, 18/11 |
| downmajor | yo | {a, b, 1} | 5/4, 5/3 |
| major | fifthward wa | {a, b}, b > 1 | 9/8, 27/16 |
| double-up major | ru | {a, b, 0, -1} | 9/7, 12/7 |
All 72edo chords can be named using ups and downs. Here are the zo, gu, ilo, yo and ru triads:
| color of the 3rd | JI chord | notes as edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| zo | 6:7:9 | 0-16-42 | C Ebvv G | C.vvm | C double-down minor |
| gu | 10:12:15 | 0-19-42 | C Eb^ G | C.^m | C upminor |
| ilo | 18:22:27 | 0-21-42 | C Ev3 G | C~ | C mid |
| yo | 4:5:6 | 0-23-42 | C Ev G | C.v | C downmajor or C dot down |
| ru | 14:18:27 | 0-26-42 | C E^^ G | C.^^ | C double-upmajor or C dot double-up |
For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.
Linear temperaments
| Periods per octave | Generator | Names |
|---|---|---|
| 1 | 1\72 | quincy |
| 1 | 5\72 | marvolo |
| 1 | 7\72 | miracle/benediction/manna |
| 1 | 11\72 | |
| 1 | 13\72 | |
| 1 | 17\72 | neominor |
| 1 | 19\72 | catakleismic |
| 1 | 23\72 | |
| 1 | 25\72 | sqrtphi |
| 1 | 29\72 | |
| 1 | 31\72 | marvo/zarvo |
| 1 | 35\72 | cotritone |
| 2 | 1\72 | |
| 2 | 5\72 | harry |
| 2 | 7\72 | |
| 2 | 11\72 | unidec/hendec |
| 2 | 13\72 | wizard/lizard/gizzard |
| 2 | 17\72 | |
| 3 | 1\72 | |
| 3 | 5\72 | tritikleismic |
| 3 | 7\72 | |
| 3 | 11\72 | mirkat |
| 4 | 1\72 | quadritikleismic |
| 4 | 5\72 | |
| 4 | 7\72 | |
| 6 | 1\72 | |
| 6 | 5\72 | |
| 8 | 1\72 | octoid |
| 8 | 2\72 | octowerck |
| 8 | 4\72 | |
| 9 | 1\72 | |
| 9 | 3\72 | ennealimmal/ennealimmic |
| 12 | 1\72 | compton |
| 18 | 1\72 | hemiennealimmal |
| 24 | 1\72 | hours |
| 36 | 1\72 |
Z function
72edo is the ninth zeta integral edo, as well as being a peak and gap edo, and the maximum value of the Z function in the region near 72 occurs at 71.9506, giving an octave of 1200.824 cents, the stretched octaves of the zeta tuning. Below is a plot of Z in the region around 72.
Music
Kotekant play by Gene Ward Smith
Twinkle canon – 72 edo by Claudi Meneghin
Lazy Sunday by Jake Freivald in the lazysunday scale.
June Gloom #9 by Prent Rodgers
Scales
smithgw72a, smithgw72b, smithgw72c, smithgw72d, smithgw72e, smithgw72f, smithgw72g, smithgw72h, smithgw72i, smithgw72j
blackjack, miracle_8, miracle_10, miracle_12, miracle_12a, miracle_24hi, miracle_24lo
keenanmarvel, xenakis_chrome, xenakis_diat, xenakis_schrome
Euler(24255) genus in 72 equal
External links
- Wikipedia article on 72edo
- OrthodoxWiki Article on Byzantine chant, which uses 72edo
- Wikipedia article on Joe Maneri (1927-2009)
- Ekmelic Music Society/Gesellschaft für Ekmelische Musik, a group of composers and researchers dedicated to 72edo music
- Rick Tagawa's 72edo site, including theory and composers' list
- Danny Wier, composer and musician who specializes in 72-edo