41edt
| ← 40edt | 41edt | 42edt → |
Division of the third harmonic into 41 equal parts (41edt) is related to 26 edo, but with the 3/1 rather than the 2/1 being just. The octave is about 6.1178 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 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | 97 | 101 | 103 | 107 | 109 | 113 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.1 | +0.0 | -3.0 | +17.6 | -22.7 | +12.8 | +12.3 | +5.3 | -0.7 | +15.5 | -7.2 | +11.2 | +19.0 | -17.0 | +14.5 | -7.9 | -8.0 | -19.3 | +3.8 | -3.8 | -5.5 | -3.1 | +4.2 | +22.5 | +12.6 | -10.9 | +1.5 | -18.0 | -3.7 | -19.7 |
| Relative (%) | +13.2 | +0.0 | -6.4 | +37.9 | -48.9 | +27.7 | +26.5 | +11.4 | -1.6 | +33.3 | -15.6 | +24.1 | +41.0 | -36.7 | +31.3 | -17.1 | -17.3 | -41.7 | +8.2 | -8.2 | -11.9 | -6.7 | +9.0 | +48.5 | +27.3 | -23.5 | +3.3 | -38.9 | -8.0 | -42.5 | |
| Steps (reduced) |
26 (26) |
41 (0) |
60 (19) |
73 (32) |
89 (7) |
96 (14) |
106 (24) |
110 (28) |
117 (35) |
126 (3) |
128 (5) |
135 (12) |
139 (16) |
140 (17) |
144 (21) |
148 (25) |
152 (29) |
153 (30) |
157 (34) |
159 (36) |
160 (37) |
163 (40) |
165 (1) |
168 (4) |
171 (7) |
172 (8) |
173 (9) |
174 (10) |
175 (11) |
176 (12) | |
| Harmonic | 127 | 131 | 137 | 139 | 149 | 151 | 157 | 163 | 167 | 173 | 179 | 181 | 191 | 193 | 197 | 199 | 211 | 223 | 227 | 229 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.0 | +2.7 | +18.0 | -7.1 | +11.8 | -11.3 | +14.0 | -4.5 | -0.1 | -14.8 | +18.9 | -0.3 | -0.6 | -18.7 | -7.8 | +21.1 | +12.5 | +9.5 | -21.3 | +10.0 |
| Relative (%) | +21.6 | +5.9 | +38.7 | -15.4 | +25.4 | -24.4 | +30.2 | -9.8 | -0.3 | -32.0 | +40.8 | -0.7 | -1.4 | -40.3 | -16.8 | +45.5 | +27.0 | +20.5 | -45.8 | +21.5 | |
| Steps (reduced) |
181 (17) |
182 (18) |
184 (20) |
184 (20) |
187 (23) |
187 (23) |
189 (25) |
190 (26) |
191 (27) |
192 (28) |
194 (30) |
194 (30) |
196 (32) |
196 (32) |
197 (33) |
198 (34) |
200 (36) |
202 (38) |
202 (38) |
203 (39) | |
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