41edt
| ← 40edt | 41edt | 42edt → |
41 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 41edt or 41ed3), is a nonoctave tuning system that divides the interval of 3/1 into 41 equal parts of about 46.4 ¢ each. Each step represents a frequency ratio of 31/41, or the 41st root of 3.
41edt is related to 26edo, but with the 3/1 rather than the 2/1 being just. The octave is about 6.12 cents stretched and the step size is about 46.3891 cents. Unlike 26edo, it is only consistent up to the 10-integer-limit, with discrepancy for the 11th harmonic.
41edt is related to the regular temperament which tempers out 823543/820125 and 2199023255552/2197176384375 in the 7-limit, which is supported by 181, 207, 388, 569, and 595 EDOs.
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 46.4 | 31.7 | |
| 2 | 92.8 | 63.4 | 18/17, 19/18, 20/19 |
| 3 | 139.2 | 95.1 | 13/12, 25/23, 27/25 |
| 4 | 185.6 | 126.8 | 10/9, 19/17, 29/26 |
| 5 | 231.9 | 158.5 | 8/7 |
| 6 | 278.3 | 190.2 | 20/17, 27/23 |
| 7 | 324.7 | 222 | 23/19, 29/24 |
| 8 | 371.1 | 253.7 | 21/17, 26/21 |
| 9 | 417.5 | 285.4 | 23/18 |
| 10 | 463.9 | 317.1 | 17/13, 21/16, 30/23 |
| 11 | 510.3 | 348.8 | |
| 12 | 556.7 | 380.5 | 18/13, 29/21 |
| 13 | 603.1 | 412.2 | 17/12, 24/17, 27/19 |
| 14 | 649.4 | 443.9 | 29/20 |
| 15 | 695.8 | 475.6 | 3/2 |
| 16 | 742.2 | 507.3 | 20/13, 23/15, 26/17 |
| 17 | 788.6 | 539 | 19/12, 30/19 |
| 18 | 835 | 570.7 | 13/8, 21/13 |
| 19 | 881.4 | 602.4 | 5/3 |
| 20 | 927.8 | 634.1 | 12/7, 29/17 |
| 21 | 974.2 | 665.9 | 7/4 |
| 22 | 1020.6 | 697.6 | 9/5 |
| 23 | 1067 | 729.3 | 13/7, 24/13 |
| 24 | 1113.3 | 761 | 19/10 |
| 25 | 1159.7 | 792.7 | |
| 26 | 1206.1 | 824.4 | 2/1 |
| 27 | 1252.5 | 856.1 | |
| 28 | 1298.9 | 887.8 | 17/8, 19/9 |
| 29 | 1345.3 | 919.5 | 13/6 |
| 30 | 1391.7 | 951.2 | 29/13 |
| 31 | 1438.1 | 982.9 | 16/7, 23/10 |
| 32 | 1484.5 | 1014.6 | |
| 33 | 1530.8 | 1046.3 | 17/7, 29/12 |
| 34 | 1577.2 | 1078 | |
| 35 | 1623.6 | 1109.8 | 23/9 |
| 36 | 1670 | 1141.5 | 21/8 |
| 37 | 1716.4 | 1173.2 | 27/10 |
| 38 | 1762.8 | 1204.9 | 25/9 |
| 39 | 1809.2 | 1236.6 | 17/6 |
| 40 | 1855.6 | 1268.3 | |
| 41 | 1902 | 1300 | 3/1 |
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.1 | +0.0 | -3.0 | +17.6 | -22.7 | +12.8 | +12.3 | +5.3 | -0.7 | +15.5 | -7.2 |
| Relative (%) | +13.2 | +0.0 | -6.4 | +37.9 | -48.9 | +27.7 | +26.5 | +11.4 | -1.6 | +33.3 | -15.6 | |
| Steps (reduced) |
26 (26) |
41 (0) |
60 (19) |
73 (32) |
89 (7) |
96 (14) |
106 (24) |
110 (28) |
117 (35) |
126 (3) |
128 (5) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +11.2 | +19.0 | -17.0 | +14.5 | -7.9 | -8.0 | -19.3 | +3.8 | -3.8 | -5.5 | -3.1 |
| Relative (%) | +24.1 | +41.0 | -36.7 | +31.3 | -17.1 | -17.3 | -41.7 | +8.2 | -8.2 | -11.9 | -6.7 | |
| Steps (reduced) |
135 (12) |
139 (16) |
140 (17) |
144 (21) |
148 (25) |
152 (29) |
153 (30) |
157 (34) |
159 (36) |
160 (37) |
163 (40) | |
Related regular temperaments
181 & 207 temperament
5-limit
Comma: [287 -121 -41⟩
POTE generator: ~[140 -59 -20⟩ = 46.3927
Mapping: [⟨1 0 7], ⟨0 41 -121]]
EDOs: 181, 207, 388, 569, 595, 957, 1345
Badness: 17.5651
7-limit
Commas: 823543/820125, 2199023255552/2197176384375
POTE generator: ~131072/127575 = 46.3932
Mapping: [⟨1 0 7 3], ⟨0 41 -121 -5]]
Badness: 0.6461
11-limit
Commas: 42592/42525, 43923/43904, 184877/184320
POTE generator: ~352/343 = 46.3934
Mapping: [⟨1 0 7 3 4], ⟨0 41 -121 -5 -14]]
Badness: 0.1362
13-limit
Commas: 847/845, 4096/4095, 4459/4455, 17303/17280
POTE generator: ~352/343 = 46.3921
Mapping: [⟨1 0 7 3 4 2], ⟨0 41 -121 -5 -14 44]]
Badness: 0.0707
17-limit
Commas: 833/832, 847/845, 1089/1088, 2058/2057, 2431/2430
POTE generator: ~187/182 = 46.3918
Mapping: [⟨1 0 7 3 4 2 2], ⟨0 41 -121 -5 -14 44 54]]
Badness: 0.0411
26 & 388 temperament
5-limit
Comma: [-41 146 -82⟩
POTE generator: ~[-16 57 -32⟩ = 46.3883
Mapping: [⟨2 0 -1], ⟨0 41 73]]
EDOs: 26, 388, 414, 802, 1190, 1578, 1966, 2354
Badness: 3.9285
7-limit
Commas: 4375/4374, [-62 -1 2 21⟩
POTE generator: ~17294403/16777216 = 46.3835
Mapping: [⟨2 0 -1 6], ⟨0 41 73 -5]]
Badness: 0.4543
11-limit
Commas: 3025/3024, 4375/4374, 5931980229/5905580032
POTE generator: ~352/343 = 46.3827
Mapping: [⟨2 0 -1 6 8], ⟨0 41 73 -5 -14]]
Badness: 0.1020
13-limit
Commas: 2200/2197, 3025/3024, 4375/4374, 50421/50336
POTE generator: ~352/343 = 46.3825
Mapping: [⟨2 0 -1 6 8 4], ⟨0 41 73 -5 -14 44]]
Badness: 0.0595
17-limit
Commas: 833/832, 1089/1088, 1225/1224, 1701/1700, 2200/2197
POTE generator: ~187/182 = 46.3824
Mapping: [⟨2 0 -1 6 8 4 4], ⟨0 41 73 -5 -14 44 54]]
Badness: 0.0326