41edo: Difference between revisions
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! | | ! | | ||
! | Cents Value | ! | Cents Value | ||
!pions | |||
!7mus | |||
! | Approximate Ratios in the [[11-limit|11-limit]] | ! | Approximate Ratios in the [[11-limit|11-limit]] | ||
! colspan="2" | [[Ups_and_Downs_Notation|Ups and Downs Notation]] | ! colspan="2" | [[Ups_and_Downs_Notation|Ups and Downs Notation]] | ||
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|- | |- | ||
| style="text-align:center;" | 0 | | style="text-align:center;" | 0 | ||
| style="text-align:center;" | 0.00 | | colspan="3" style="text-align:center;" | 0.00 | ||
| | [[1/1|1/1]] | | | [[1/1|1/1]] | ||
| style="text-align:center;" | P1 | | style="text-align:center;" | P1 | ||
| Line 340: | Line 342: | ||
| style="text-align:center;" | 1 | | style="text-align:center;" | 1 | ||
| style="text-align:center;" | 29.27 | | style="text-align:center;" | 29.27 | ||
|31.02 | |||
|37.46 (25.77<sub>16</sub>) | |||
| | [[81/80|81/80]] | | | [[81/80|81/80]] | ||
| style="text-align:center;" | ^1 | | style="text-align:center;" | ^1 | ||
| Line 349: | Line 353: | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
| style="text-align:center;" | 58.54 | | style="text-align:center;" | 58.54 | ||
|62.05 | |||
|74.93 (4A.EB<sub>16</sub>) | |||
| | [[25/24|25/24]], [[28/27|28/27]], [[33/32|33/32]] | | | [[25/24|25/24]], [[28/27|28/27]], [[33/32|33/32]] | ||
| style="text-align:center;" | vm2 | | style="text-align:center;" | vm2 | ||
| Line 357: | Line 363: | ||
|- | |- | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| style="text-align:center;" | 87. | | style="text-align:center;" | 87.805 | ||
|93.07 | |||
|112.39 (70.64<sub>16</sub>) | |||
| | [[21/20|21/20]], [[22/21|22/21]] | | | [[21/20|21/20]], [[22/21|22/21]] | ||
| style="text-align:center;" | m2 | | style="text-align:center;" | m2 | ||
| Line 369: | Line 377: | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| style="text-align:center;" | 117.07 | | style="text-align:center;" | 117.07 | ||
|124.1 | |||
|149.85 (95.DA8<sub>16</sub>) | |||
| | [[16/15|16/15]], [[15/14|15/14]] | | | [[16/15|16/15]], [[15/14|15/14]] | ||
| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| Line 378: | Line 388: | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| style="text-align:center;" | 146.34 | | style="text-align:center;" | 146.34 | ||
|155.12 | |||
|187.32 (BB.51<sub>16</sub>) | |||
| | [[12/11|12/11]] | | | [[12/11|12/11]] | ||
| style="text-align:center;" | ~2 | | style="text-align:center;" | ~2 | ||
| Line 387: | Line 399: | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| style="text-align:center;" | 175.61 | | style="text-align:center;" | 175.61 | ||
|186.15 | |||
|224.78 (E0.C8<sub>16</sub>) | |||
| | [[10/9|10/9]], [[11/10|11/10]] | | | [[10/9|10/9]], [[11/10|11/10]] | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| Line 396: | Line 410: | ||
| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
| style="text-align:center;" | 204.88 | | style="text-align:center;" | 204.88 | ||
|217.17 | |||
|262.24 (106.3E<sub>16</sub>) | |||
| | [[9/8|9/8]] | | | [[9/8|9/8]] | ||
| style="text-align:center;" | M2 | | style="text-align:center;" | M2 | ||
| Line 405: | Line 421: | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| style="text-align:center;" | 234.15 | | style="text-align:center;" | 234.15 | ||
|248.195 | |||
|299.71 (12B.B5<sub>16</sub>) | |||
| | [[8/7|8/7]] | | | [[8/7|8/7]] | ||
| style="text-align:center;" | ^M2 | | style="text-align:center;" | ^M2 | ||
| Line 413: | Line 431: | ||
|- | |- | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| style="text-align:center;" | 263. | | style="text-align:center;" | 263.415 | ||
|279.22 | |||
|337.17 (151.2C<sub>16</sub>) | |||
| | [[7/6|7/6]], [[32/25|32/25]] | | | [[7/6|7/6]], [[32/25|32/25]] | ||
| style="text-align:center;" | vm3 | | style="text-align:center;" | vm3 | ||
| Line 423: | Line 443: | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| style="text-align:center;" | 292.68 | | style="text-align:center;" | 292.68 | ||
|310.24 | |||
|374.63 (176.A2<sub>16</sub>) | |||
| | [[32/27|32/27]] | | | [[32/27|32/27]] | ||
| style="text-align:center;" | m3 | | style="text-align:center;" | m3 | ||
| Line 432: | Line 454: | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| style="text-align:center;" | 321.95 | | style="text-align:center;" | 321.95 | ||
|341.27 | |||
|412.1 (1A0.19<sub>16</sub>) | |||
| | [[6/5|6/5]] | | | [[6/5|6/5]] | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| Line 441: | Line 465: | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| style="text-align:center;" | 351.22 | | style="text-align:center;" | 351.22 | ||
| | [[11/9|11/9]],[[27/22|27/22]] | |372.29 | ||
|449.56 (1C5.9<sub>16</sub>) | |||
| | [[11/9|11/9]], [[27/22|27/22]] | |||
| style="text-align:center;" | ~3 | | style="text-align:center;" | ~3 | ||
| style="text-align:center;" | F^^ | | style="text-align:center;" | F^^ | ||
| Line 450: | Line 476: | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
| style="text-align:center;" | 380.49 | | style="text-align:center;" | 380.49 | ||
|403.32 | |||
|487.02 (1EB.06<sub>16</sub>) | |||
| | [[5/4|5/4]] | | | [[5/4|5/4]] | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| Line 459: | Line 487: | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| style="text-align:center;" | 409.76 | | style="text-align:center;" | 409.76 | ||
|434.34 | |||
|524.49 (20C.7C8<sub>16</sub>) | |||
| | [[14/11|14/11]], [[81/64|81/64]] | | | [[14/11|14/11]], [[81/64|81/64]] | ||
| style="text-align:center;" | M3 | | style="text-align:center;" | M3 | ||
| Line 468: | Line 498: | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
| style="text-align:center;" | 439.02 | | style="text-align:center;" | 439.02 | ||
|465.37 | |||
|561.95 (231.F38<sub>16</sub>) | |||
| | [[9/7|9/7]] | | | [[9/7|9/7]] | ||
| style="text-align:center;" | ^M3 | | style="text-align:center;" | ^M3 | ||
| Line 477: | Line 509: | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| style="text-align:center;" | 468.29 | | style="text-align:center;" | 468.29 | ||
|496.39 | |||
|599.415 (257.6A<sub>16</sub>) | |||
| | [[21/16|21/16]] | | | [[21/16|21/16]] | ||
| style="text-align:center;" | v4 | | style="text-align:center;" | v4 | ||
| Line 486: | Line 520: | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| style="text-align:center;" | 497.56 | | style="text-align:center;" | 497.56 | ||
|527.415 | |||
|636.88 (27C.E1<sub>16</sub>) | |||
| | [[4/3|4/3]] | | | [[4/3|4/3]] | ||
| style="text-align:center;" | P4 | | style="text-align:center;" | P4 | ||
| Line 495: | Line 531: | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
| style="text-align:center;" | 526.83 | | style="text-align:center;" | 526.83 | ||
|558.44 | |||
|674.34 (2A2.578<sub>16</sub>) | |||
| | [[15/11|15/11]], [[27/20|27/20]] | | | [[15/11|15/11]], [[27/20|27/20]] | ||
| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| Line 503: | Line 541: | ||
|- | |- | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
| style="text-align:center;" | 556. | | style="text-align:center;" | 556.1 | ||
|589.46 | |||
|711.805 (2C7.CE<sub>16</sub>) | |||
| | [[11/8|11/8]] | | | [[11/8|11/8]] | ||
| style="text-align:center;" | ^^4 | | style="text-align:center;" | ^^4 | ||
| Line 513: | Line 553: | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
| style="text-align:center;" | 585.37 | | style="text-align:center;" | 585.37 | ||
|620.49 | |||
|749.27 (2ED.45<sub>16</sub>) | |||
| | [[7/5|7/5]] | | | [[7/5|7/5]] | ||
| style="text-align:center;" | vA4, d5 | | style="text-align:center;" | vA4, d5 | ||
| Line 524: | Line 566: | ||
| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
| style="text-align:center;" | 614.63 | | style="text-align:center;" | 614.63 | ||
|651.51 | |||
|786.73 (312.BB<sub>16</sub>) | |||
| | [[10/7|10/7]] | | | [[10/7|10/7]] | ||
| style="text-align:center;" | A4, ^d5 | | style="text-align:center;" | A4, ^d5 | ||
| Line 535: | Line 579: | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| style="text-align:center;" | 643.90 | | style="text-align:center;" | 643.90 | ||
|683.54 | |||
|825.195 (347.32<sub>16</sub>) | |||
| | [[16/11|16/11]] | | | [[16/11|16/11]] | ||
| style="text-align:center;" | vv5 | | style="text-align:center;" | vv5 | ||
| Line 544: | Line 590: | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
| style="text-align:center;" | 673.17 | | style="text-align:center;" | 673.17 | ||
|713.56 | |||
|862.68 (35D.B88<sub>16</sub>) | |||
| | [[22/15|22/15]], [[40/27|40/27]] | | | [[22/15|22/15]], [[40/27|40/27]] | ||
| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| Line 553: | Line 601: | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
| style="text-align:center;" | 702.44 | | style="text-align:center;" | 702.44 | ||
|744.585 | |||
|899.12 (383.1F<sub>16</sub>) | |||
| | [[3/2|3/2]] | | | [[3/2|3/2]] | ||
| style="text-align:center;" | P5 | | style="text-align:center;" | P5 | ||
| Line 562: | Line 612: | ||
| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
| style="text-align:center;" | 731.71 | | style="text-align:center;" | 731.71 | ||
|775.61 | |||
|936.585 (3A8.96<sub>16</sub>) | |||
| | [[32/21|32/21]] | | | [[32/21|32/21]] | ||
| style="text-align:center;" | ^5 | | style="text-align:center;" | ^5 | ||
| Line 571: | Line 623: | ||
| style="text-align:center;" | 26 | | style="text-align:center;" | 26 | ||
| style="text-align:center;" | 760.98 | | style="text-align:center;" | 760.98 | ||
|806.63 | |||
|994.05 (3CE.0C8<sub>16</sub>) | |||
| | [[14/9|14/9]], [[25/16|25/16]] | | | [[14/9|14/9]], [[25/16|25/16]] | ||
| style="text-align:center;" | vm6 | | style="text-align:center;" | vm6 | ||
| Line 580: | Line 634: | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
| style="text-align:center;" | 790.24 | | style="text-align:center;" | 790.24 | ||
|837.66 | |||
|1011.51 (3F3.838<sub>16</sub>) | |||
| | [[11/7|11/7]], [[128/81|128/81]] | | | [[11/7|11/7]], [[128/81|128/81]] | ||
| style="text-align:center;" | m6 | | style="text-align:center;" | m6 | ||
| Line 589: | Line 645: | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| style="text-align:center;" | 819.51 | | style="text-align:center;" | 819.51 | ||
|868.68. | |||
|1048.97 (418.FA<sub>16</sub>) | |||
| | [[8/5|8/5]] | | | [[8/5|8/5]] | ||
| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| Line 598: | Line 656: | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| style="text-align:center;" | 848.78 | | style="text-align:center;" | 848.78 | ||
|899.71 | |||
|1086.44 (43D.7<sub>16</sub>) | |||
| | [[18/11|18/11]], [[44/27|44/27]] | | | [[18/11|18/11]], [[44/27|44/27]] | ||
| style="text-align:center;" | ~6 | | style="text-align:center;" | ~6 | ||
| Line 607: | Line 667: | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
| style="text-align:center;" | 878.05 | | style="text-align:center;" | 878.05 | ||
|930.73 | |||
|1123.9 (463.F7<sub>16</sub>) | |||
| | [[5/3|5/3]] | | | [[5/3|5/3]] | ||
| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| Line 616: | Line 678: | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| style="text-align:center;" | 907.32 | | style="text-align:center;" | 907.32 | ||
|961.76 | |||
|1161.37 (489.5E<sub>16</sub>) | |||
| | [[27/16|27/16]] | | | [[27/16|27/16]] | ||
| style="text-align:center;" | M6 | | style="text-align:center;" | M6 | ||
| Line 625: | Line 689: | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
| style="text-align:center;" | 936.59 | | style="text-align:center;" | 936.59 | ||
|992.78 | |||
|1198.83 (4AE.D4<sub>16</sub>) | |||
| | [[12/7|12/7]] | | | [[12/7|12/7]] | ||
| style="text-align:center;" | ^M6 | | style="text-align:center;" | ^M6 | ||
| Line 634: | Line 700: | ||
| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
| style="text-align:center;" | 965.85 | | style="text-align:center;" | 965.85 | ||
|1023.805 | |||
|1236.29 (4D4.4B<sub>16</sub>) | |||
| | [[7/4|7/4]] | | | [[7/4|7/4]] | ||
| style="text-align:center;" | vm7 | | style="text-align:center;" | vm7 | ||
| Line 643: | Line 711: | ||
| style="text-align:center;" | 34 | | style="text-align:center;" | 34 | ||
| style="text-align:center;" | 995.12 | | style="text-align:center;" | 995.12 | ||
|1064.83 | |||
|1273.76 (4F9.C2<sub>16,)</sub> | |||
| | [[16/9|16/9]] | | | [[16/9|16/9]] | ||
| style="text-align:center;" | m7 | | style="text-align:center;" | m7 | ||
| Line 652: | Line 722: | ||
| style="text-align:center;" | 35 | | style="text-align:center;" | 35 | ||
| style="text-align:center;" | 1024.39 | | style="text-align:center;" | 1024.39 | ||
|1085.85 | |||
|1311.22 (51F.38<sub>16</sub>) | |||
| | [[9/5|9/5]], [[20/11|20/11]] | | | [[9/5|9/5]], [[20/11|20/11]] | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| Line 661: | Line 733: | ||
| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
| style="text-align:center;" | 1053.66 | | style="text-align:center;" | 1053.66 | ||
|1116.88 | |||
|1348.32 (544.AF<sub>16</sub>) | |||
| | [[11/6|11/6]] | | | [[11/6|11/6]] | ||
| style="text-align:center;" | ~7 | | style="text-align:center;" | ~7 | ||
| Line 670: | Line 744: | ||
| style="text-align:center;" | 37 | | style="text-align:center;" | 37 | ||
| style="text-align:center;" | 1082.93 | | style="text-align:center;" | 1082.93 | ||
|1147.9 | |||
|1386.15 (57A.258<sub>16</sub>) | |||
| | [[15/8|15/8]] | | | [[15/8|15/8]] | ||
| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| Line 678: | Line 754: | ||
|- | |- | ||
| style="text-align:center;" | 38 | | style="text-align:center;" | 38 | ||
| style="text-align:center;" | 1112. | | style="text-align:center;" | 1112.195 | ||
|1178.93 | |||
|1423.61 (58F.9C<sub>16</sub>) | |||
| | [[40/21|40/21]], [[21/11|21/11]] | | | [[40/21|40/21]], [[21/11|21/11]] | ||
| style="text-align:center;" | M7 | | style="text-align:center;" | M7 | ||
| Line 688: | Line 766: | ||
| style="text-align:center;" | 39 | | style="text-align:center;" | 39 | ||
| style="text-align:center;" | 1141.46 | | style="text-align:center;" | 1141.46 | ||
|1209.95 | |||
|1461.07 (5B5.14<sub>16</sub>) | |||
| | [[48/25|48/25]], [[27/14|27/14]], [[64/33|64/33]] | | | [[48/25|48/25]], [[27/14|27/14]], [[64/33|64/33]] | ||
| style="text-align:center;" | ^M7 | | style="text-align:center;" | ^M7 | ||
| Line 697: | Line 777: | ||
| style="text-align:center;" | 40 | | style="text-align:center;" | 40 | ||
| style="text-align:center;" | 1170.73 | | style="text-align:center;" | 1170.73 | ||
|1240.98 | |||
|1498.54 (5DA.89<sub>16</sub>) | |||
| | [[160/81|160/81]] | | | [[160/81|160/81]] | ||
| style="text-align:center;" | v8 | | style="text-align:center;" | v8 | ||
| Line 706: | Line 788: | ||
| style="text-align:center;" | 41 | | style="text-align:center;" | 41 | ||
| style="text-align:center;" | 1200 | | style="text-align:center;" | 1200 | ||
|1272 | |||
|1536 (600<sub>16</sub>) | |||
| | 2/1 | | | 2/1 | ||
| style="text-align:center;" | P8 | | style="text-align:center;" | P8 | ||
Revision as of 20:19, 2 April 2019
Introduction
The 41-tET, 41-EDO, 41-ET, or 41-Tone Equal Temperament is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 cents, an interval close in size to 64/63, the septimal comma. 41-ET can be seen as a tuning of the Garibaldi temperament [1] , [2] , [3] the Magic temperament [4] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after 29edo) whose perfect fifth is closer to just intonation than that of 12-ET, and is the seventh zeta integral edo after 31; it is not, however, a zeta gap edo. This has to do with the fact that it can deal with the 11-limit fairly well, and the 13-limit perhaps close enough for government work, though its 13/10 is 14 cents sharp. Various 13-limit magic extensions are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.
41edo is consistent in the 15 odd limit. In fact, all of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances. (In comparison, 31edo is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).
41-ET forms the foundation of the H-System, which uses the scale degrees of 41-ET as the basic 13-limit intervals requiring fine tuning +/- 1 average JND from the 41-ET circle in 205edo.
41edo is the 13th prime edo, following 37edo and coming before 43edo.
Commas
41 EDO tempers out the following commas using its patent val, < 41 65 95 115 142 152 168 174 185 199 203 |.
| Name | Monzo | Ratio | Cents |
|---|---|---|---|
| odiheim | | -1 2 -4 5 -2 > | 0.15 | |
| harmonisma | | 3 -2 0 -1 3 -2 > | 10648/10647 | 0.16 |
| tridecimal schisma, Sagittal schismina | | 12 -2 -1 -1 0 -1/1 > | 4096/4095 | 0.42 |
| Lehmerisma | | -4 -3 2 -1 2 > | 3025/3024 | 0.57 |
| Breedsma | | -5 -1 -2 4 > | 2401/2400 | 0.72 |
| Eratosthenes' comma | | 6 -5 -1 0 0 0 0 1 > | 1216/1215 | 1.42 |
| schisma | | -15 8 1 > | 32805/32768 | 1.95 |
| squbema | | -3 6 0 -1 0 -1 > | 729/728 | 2.38 |
| septendecimal bridge comma | | -1 -1 1 -1 1 1 -1 > | 715/714 | 2.42 |
| Swets' comma, swetisma | | 2 3 1 -2 -1 > | 540/539 | 3.21 |
| undevicesimal comma, Boethius' comma | | -9 3 0 0 0 0 0 1 > | 513/512 | 3.38 |
| moctdel | | -2 0 3 -3 1 > | 1375/1372 | 3.78 |
| Beta 2, septimal schisma, garischisma | | 25 -14 0 -1 > | 3.80 | |
| Werckmeister's undecimal septenarian schisma, werckisma | | -3 2 -1 2 -1 > | 441/440 | 3.93 |
| cuthbert | | 0 0 -1 1 2 -2 > | 847/845 | 4.09 |
| undecimal kleisma, keenanisma | | -7 -1 1 1 1 > | 385/384 | 4.50 |
| gentle comma | | 2 -1 0 1 -2 1 > | 364/363 | 4.76 |
| minthma | | 5 -3 0 0 1 -1 > | 352/351 | 4.93 |
| marveltwin | | -2 -4 2 0 0 1 > | 325/324 | 5.34 |
| Beta 5, Garibaldi comma, hemifamity | | 10 -6 1 -1 > | 5120/5103 | 5.76 |
| hemimage | | 5 -7 -1 3 > | 10976/10935 | 6.48 |
| septendecimal kleisma | | 8 -1 -1 0 0 0 -1 > | 256/255 | 6.78 |
| small BP diesis, mirkwai | | 0 3 4 -5 > | 16875/16807 | 6.99 |
| neutral third comma, rastma | | -1 5 0 0 -2 > | 243/242 | 7.14 |
| kestrel comma | | 2 3 0 -1 1 -2 > | 1188/1183 | 7.30 |
| septimal kleisma, marvel comma | | -5 2 2 -1 > | 225/224 | 7.71 |
| huntma | | 7 0 1 -2 0 -1 > | 640/637 | 8.13 |
| spleen comma | | 1 1 1 1 -1 0 0 -1 > | 210/209 | 8.26 |
| orgonisma | | 16 0 0 -2 -3 > | 65536/65219 | 8.39 |
| gamelan residue, gamelisma | | -10 1 0 3 > | 1029/1024 | 8.43 |
| septendecimal comma | | -7 7 0 0 0 0 -1 > | 2187/2176 | 8.73 |
| mynucuma | | 2 -1 -1 2 0 -1 > | 196/195 | 8.86 |
| quince | | -15 0 -2 7 > | 9.15 | |
| undecimal semicomma, pentacircle (minthma * gentle) | | 7 -4 0 1 -1 > | 896/891 | 9.69 |
| 29th-partial chroma | | -4 -2 1 0 0 0 0 0 0 1 > | 145/144 | 11.98 |
| grossma | | 4 2 0 0 -1 -1 > | 144/143 | 12.06 |
| gassorma | | 0 -1 2 -1 1 -1 > | 275/273 | 12.64 |
| septimal semicomma, octagar | | 5 -4 3 -2 > | 4000/3969 | 13.47 |
| minor BP diesis, sensamagic | | 0 -5 1 2 > | 245/243 | 14.19 |
| secorian | | 12 -7 0 1 0 -1/1 > | 28672/28431 | 14.61 |
| mirwomo comma | | -15 3 2 2 > | 33075/32768 | 16.14 |
| vicesimotertial comma | | 5 -6 0 0 0 0 0 0 1 > | 736/729 | 16.54 |
| small tridecimal comma, animist | | -3 1 1 1 0 -1 > | 105/104 | 16.57 |
| hemimin | | 6 1 0 1 -3 > | 1344/1331 | 16.83 |
| Ptolemy's comma, ptolemisma | | 2 -2 2 0 -1 > | 100/99 | 17.40 |
| '41-tone' comma | | 65 -41 > | 19.84 | |
| tolerma | | 10 -11 2 1 > | 19.95 | |
| major BP diesis, gariboh | | 0 -2 5 -3 > | 3125/3087 | 21.18 |
| cassacot | | -1 0 1 2 -2 > | 245/242 | 21.33 |
| keema | | -5 -3 3 1 > | 875/864 | 21.90 |
| blackjackisma | | -10 7 8 -7 > | 22.41 | |
| roda | | 20 -17 3 > | 25.71 | |
| minimal diesis, tetracot comma | | 5 -9 4 > | 20000/19683 | 27.66 |
| small diesis, magic comma | | -10 -1 5 > | 3125/3072 | 29.61 |
| thuja comma | | 15 0 1 0 -5 > | 29.72 | |
| Ampersand's comma | | -25 7 6 > | 31.57 | |
| great BP diesis | | 0 -7 6 -1 > | 15625/15309 | 35.37 |
| shibboleth | | -5 -10 9 > | 57.27 |
Temperaments
List of edo-distinct 41et rank two temperaments
Intervals
| Cents Value | pions | 7mus | Approximate Ratios in the 11-limit | Ups and Downs Notation | Andrew's Solfege Syllables | Generator | Some MOS and MODMOS Scales Available | ||
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.00 | 1/1 | P1 | D | do | ||||
| 1 | 29.27 | 31.02 | 37.46 (25.7716) | 81/80 | ^1 | D^ | di | ||
| 2 | 58.54 | 62.05 | 74.93 (4A.EB16) | 25/24, 28/27, 33/32 | vm2 | Ebv | ro | Hemimiracle | |
| 3 | 87.805 | 93.07 | 112.39 (70.6416) | 21/20, 22/21 | m2 | Eb | rih | 88cET (approx), | |
| 4 | 117.07 | 124.1 | 149.85 (95.DA816) | 16/15, 15/14 | ^m2 | Eb^ | ra | Miracle | |
| 5 | 146.34 | 155.12 | 187.32 (BB.5116) | 12/11 | ~2 | Evv | ru | Bohlen-Pierce/bohpier | |
| 6 | 175.61 | 186.15 | 224.78 (E0.C816) | 10/9, 11/10 | vM2 | Ev | reh | Tetracot/bunya/monkey | 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5 |
| 7 | 204.88 | 217.17 | 262.24 (106.3E16) | 9/8 | M2 | E | re | Baldy | 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1 |
| 8 | 234.15 | 248.195 | 299.71 (12B.B516) | 8/7 | ^M2 | E^ | ri | Rodan/guiron | 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1 |
| 9 | 263.415 | 279.22 | 337.17 (151.2C16) | 7/6, 32/25 | vm3 | Fv | ma | Septimin | 9-tone MOS: 5 4 5 5 4 5 4 5 4 |
| 10 | 292.68 | 310.24 | 374.63 (176.A216) | 32/27 | m3 | F | meh | Quasitemp | |
| 11 | 321.95 | 341.27 | 412.1 (1A0.1916) | 6/5 | ^m3 | F^ | me | Superkleismic | 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3 |
| 12 | 351.22 | 372.29 | 449.56 (1C5.916) | 11/9, 27/22 | ~3 | F^^ | mu | Hemififths/karadeniz | 10-tone MOS: 5 2 5 5 2 5 5 5 2 5 |
| 13 | 380.49 | 403.32 | 487.02 (1EB.0616) | 5/4 | vM3 | F#v | mi | Magic/witchcraft | 10-tone MOS: 2 9 2 2 9 2 2 9 2 2 |
| 14 | 409.76 | 434.34 | 524.49 (20C.7C816) | 14/11, 81/64 | M3 | F# | maa | Hocus | |
| 15 | 439.02 | 465.37 | 561.95 (231.F3816) | 9/7 | ^M3 | F#^ | mo | 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4 | |
| 16 | 468.29 | 496.39 | 599.415 (257.6A16) | 21/16 | v4 | Gv | fe | Barbad | |
| 17 | 497.56 | 527.415 | 636.88 (27C.E116) | 4/3 | P4 | G | fa | Schismatic (helmholtz, garibaldi, cassandra) | |
| 18 | 526.83 | 558.44 | 674.34 (2A2.57816) | 15/11, 27/20 | ^4 | G^ | fih | Trismegistus | 9-tone MOS: 5 5 3 5 5 5 5 3 5 |
| 19 | 556.1 | 589.46 | 711.805 (2C7.CE16) | 11/8 | ^^4 | G^^ | fu | ||
| 20 | 585.37 | 620.49 | 749.27 (2ED.4516) | 7/5 | vA4, d5 | G#v,
Ab |
fi | Pluto | |
| 21 | 614.63 | 651.51 | 786.73 (312.BB16) | 10/7 | A4, ^d5 | G#,
Ab^ |
se | ||
| 22 | 643.90 | 683.54 | 825.195 (347.3216) | 16/11 | vv5 | Avv | su | ||
| 23 | 673.17 | 713.56 | 862.68 (35D.B8816) | 22/15, 40/27 | v5 | Av | sih | ||
| 24 | 702.44 | 744.585 | 899.12 (383.1F16) | 3/2 | P5 | A | sol | ||
| 25 | 731.71 | 775.61 | 936.585 (3A8.9616) | 32/21 | ^5 | A^ | si | ||
| 26 | 760.98 | 806.63 | 994.05 (3CE.0C816) | 14/9, 25/16 | vm6 | Bbv | lo | ||
| 27 | 790.24 | 837.66 | 1011.51 (3F3.83816) | 11/7, 128/81 | m6 | Bb | leh | ||
| 28 | 819.51 | 868.68. | 1048.97 (418.FA16) | 8/5 | ^m6 | Bb^ | le | ||
| 29 | 848.78 | 899.71 | 1086.44 (43D.716) | 18/11, 44/27 | ~6 | Bvv | lu | ||
| 30 | 878.05 | 930.73 | 1123.9 (463.F716) | 5/3 | vM6 | Bv | la | ||
| 31 | 907.32 | 961.76 | 1161.37 (489.5E16) | 27/16 | M6 | B | laa | ||
| 32 | 936.59 | 992.78 | 1198.83 (4AE.D416) | 12/7 | ^M6 | B^ | li | ||
| 33 | 965.85 | 1023.805 | 1236.29 (4D4.4B16) | 7/4 | vm7 | Cv | ta | ||
| 34 | 995.12 | 1064.83 | 1273.76 (4F9.C216,) | 16/9 | m7 | C | teh | ||
| 35 | 1024.39 | 1085.85 | 1311.22 (51F.3816) | 9/5, 20/11 | ^m7 | C^ | te | ||
| 36 | 1053.66 | 1116.88 | 1348.32 (544.AF16) | 11/6 | ~7 | C^^ | tu | ||
| 37 | 1082.93 | 1147.9 | 1386.15 (57A.25816) | 15/8 | vM7 | C#v | ti | ||
| 38 | 1112.195 | 1178.93 | 1423.61 (58F.9C16) | 40/21, 21/11 | M7 | C# | taa | ||
| 39 | 1141.46 | 1209.95 | 1461.07 (5B5.1416) | 48/25, 27/14, 64/33 | ^M7 | C#^ | to | ||
| 40 | 1170.73 | 1240.98 | 1498.54 (5DA.8916) | 160/81 | v8 | Dv | da | ||
| 41 | 1200 | 1272 | 1536 (60016) | 2/1 | P8 | D | do | ||
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color | monzo format | examples |
|---|---|---|---|
| downminor | 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 |
| upmajor | ru | {a, b, 0, -1} | 9/7, 12/7 |
All 41edo 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 | 0-9-24 | C Ebv G | C.vm | C downminor |
| gu | 10:12:15 | 0-11-24 | C Eb^ G | C.^m | C upminor |
| ilo | 18:22:27 | 0-12-24 | C Evv G | C~ | C mid |
| yo | 4:5:6 | 0-13-24 | C Ev G | C.v | C downmajor or C dot down |
| ru | 14:18:27 | 0-15-24 | C E^ G | C.^ | C upmajor or C dot up |
0-10-20 = D F Ab = Ddim = "D dim"
0-10-21 = D F Ab^ = Ddim(^5) = "D dim up-five"
0-10-22 = D F Avv = Dm(vv5) = "D minor double-down five", or possibly Ddim(^^5)
0-10-23 = D F Av = Dm(v5) = "D minor down-five"
0-10-24 = D F A = Dm = "D minor"
0-14-24 = D F# A = D = "D" or "D major"
0-14-25 = D F# A^ = D(^5) = "D up-five"
0-14-26 = D F# A^^ = D(^^5) = "D double-up-five", or possibly Daug(vv5)
0-14-27 = D F# A#v = Daug(v5) = "D aug down-five"
0-14-28 = D F# A# is Daug = "D aug"
etc.
For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.
Selected just intervals by error
The following table shows how some prominent just intervals are represented in 41edo (ordered by absolute error).
| Interval, complement | Error (abs., in cents) |
| 4/3, 3/2 | 0.484 |
| 9/8, 16/9 | 0.968 |
| 15/14, 28/15 | 2.370 |
| 7/5, 10/7 | 2.854 |
| 8/7, 7/4 | 2.972 |
| 7/6, 12/7 | 3.456 |
| 13/11, 22/13 | 3.473 |
| 11/9, 18/11 | 3.812 |
| 9/7, 14/9 | 3.940 |
| 12/11, 11/6 | 4.296 |
| 11/8, 16/11 | 4.780 |
| 16/15, 15/8 | 5.342 |
| 5/4, 8/5 | 5.826 |
| 6/5, 5/3 | 6.310 |
| 10/9, 9/5 | 6.794 |
| 18/13, 13/9 | 7.285 |
| 14/11, 11/7 | 7.752 |
| 13/12, 24/13 | 7.769 |
| 16/13, 13/8 | 8.253 |
| 15/11, 22/15 | 10.122 |
| 11/10, 20/11 | 10.606 |
| 14/13, 13/7 | 11.225 |
| 15/13, 26/15 | 13.595 |
| 13/10, 20/13 | 14.079 |
Instruments
41-EDO Electric guitar, by Gregory Sanchez.
41-EDO Classical guitar, by Ron Sword.
A possible system to tune keyboards in 41EDO is discussed in http://launch.groups.yahoo.com/group/tuning/message/74155.
Scales and modes
A list of 41edo modes (MOS and others).
Harmonic Scale
41edo is the first edo to do some justice to Mode 8 of the harmonic series, which Dante Rosati calls the "Diatonic Harmonic Series Scale," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).
| 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 41edo: | 0 | 7 | 13 | 19 | 24 | 29 | 33 | 37 | 41 |
| ...in cents: | 0 | 204.9 | 380.5 | 556.1 | 702.4 | 848.8 | 965.9 | 1082.9 | 1200.0 |
While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)
7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.
Nonoctave Temperaments
Taking every third degree of 41edo produces a scale extremely close to 88cET or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered Bohlen-Pierce Scale (or the 13th root of 3). See chart:
| 3 degrees of 41edo (near 88cET) | overlap | 5 degrees of 41edo (near BP) | ||||
| deg of 41edo | deg of 88cET | cents | cents | cents | deg of BP | deg of 41edo |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | ||
| 3 | 1 | 87.8 | ||||
| 146.3 | 1 | 5 | ||||
| 6 | 2 | 175.6 | ||||
| 9 | 3 | 263.4 | ||||
| 292.7 | 2 | 10 | ||||
| 12 | 4 | 351.2 | ||||
| 15 | 5 | 439.0 | 3 | 15 | ||
| 18 | 6 | 526.8 | ||||
| 585.4 | 4 | 20 | ||||
| 21 | 7 | 614.6 | ||||
| 24 | 8 | 702.4 | ||||
| 731.7 | 5 | 25 | ||||
| 27 | 9 | 790.2 | ||||
| 30 | 10 | 878.0 | 6 | 30 | ||
| 33 | 11 | 965.9 | ||||
| 1024.4 | 7 | 35 | ||||
| 36 | 12 | 1053.7 | ||||
| 39 | 13 | 1141.5 | ||||
| 1170.7 | 8 | 40 | ||||
| [ second octave ] | ||||||
| 1 | 14 | 29.2 | ||||
| 4 | 15 | 117.1 | 9 | 4 | ||
| 7 | 16 | 204.9 | ||||
| 263.4 | 10 | 9 | ||||
| 10 | 17 | 292.7 | ||||
| 13 | 18 | 380.5 | ||||
| 409.8 | 11 | 14 | ||||
| 16 | 19 | 468.3 | ||||
| 19 | 20 | 556.1 | 12 | 19 | ||
| 22 | 21 | 643.9 | ||||
| 702.4 | 13 | 24 | ||||
| 25 | 22 | 731.7 | ||||
| 28 | 23 | 819.5 | ||||
| 848.8 | 14 | 29 | ||||
| 31 | 24 | 907.3 | ||||
| 34 | 25 | 995.1 | 15 | 34 | ||
| 37 | 26 | 1082.9 | ||||
| 1141.5 | 16 | 39 | ||||
| 40 | 27 | 1170.7 | ||||
| [ third octave ] | ||||||
| 2 | 28 | 58.5 | ||||
| 87.8 | 17 | 3 | ||||
| 5 | 29 | 146.3 | ||||
| 8 | 30 | 234.1 | 18 | 8 | ||
| 11 | 31 | 322.0 | ||||
| 380.5 | 19 | 13 | ||||
| 14 | 32 | 409.8 | ||||
| 17 | 33 | 497.6 | ||||
| 526.8 | 20 | 18 | ||||
| 20 | 34 | 585.3 | ||||
| 23 | 35 | 673.2 | 21 | 23 | ||
| 26 | 36 | 761.0 | ||||
| 819.5 | 22 | 28 | ||||
| 29 | 37 | 848.8 | ||||
| 32 | 38 | 936.6 | ||||
| 965.9 | 23 | 33 | ||||
| 35 | 39 | 1024.4 | ||||
| 38 | 40 | 1112.2 | 24 | 38 | ||
Notation
A red-note/blue-note system, similar to the one proposed for 36edo, is one option for notating 41edo. (This is separate from and not compatible with Kite's color notation.) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:
A, red A, blue Bb, Bb, A#, red A#, blue B, B, red B, blue C, C, red C, blue Db, Db, C#, red C#, blue D, D, red D, blue Eb, Eb, D#, red D#, blue E, E, red E, blue F, F, red F, blue Gb, Gb, F#, red F#, blue G, G, red G, blue Ab, Ab, G#, red G#, blue A, A.
Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.
The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.
If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as ups and downs notation. The only difference is the use of minor tritone and major tritone.
Music
EveningHorizon play by Cameron Bobro
Links
- Wikipedia article on 41edo
- Magic22 as srutis describes a possible use of 41edo for indian music.
- see also Magic family
- Sword, Ron. "Tetracontamonophonic Scales for Guitar"
- Taylor, Cam. Intervals, Scales and Chords in 41EDO, a work in progress using just intonation concepts and simplified Sagittal notation.
- ^ "Schismic Temperaments" at x31eq.com the website of Graham Breed
- ^ "Lattices with Decimal Notation" at x31eq.com
- ^ Schismatic temperament
- ^ Magic temperament