40ed10
| ← 39ed10 | 40ed10 | 41ed10 → |
The division of the 10th harmonic into 40 equal parts (40ED10) is related to 12EDO, but with 10/1 instead of 2/1 being just. The step size (99.657843 cents) of this equal-step tuning is very close to 1\12 (one step of 12 EDO).
The octave, which is 12\40 = 3\10, is compressed by about 4.1 cents.
Theory
Since 40ED10 has relations to the proximity of 1024 to 1000, just like 12EDO it tempers out the lesser diesis of 128/125. However in this situation the tempering has a different interpretation, namely that "in favor of 1000".
Interval
| degree | cents value | corresponding JI intervals |
comments |
|---|---|---|---|
| 0 | 0.0000 | exact 1/1 | |
| 1 | 99.6578 | 18/17 | |
| 2 | 199.3157 | ||
| 3 | 298.9735 | 19/16 | |
| 4 | 398.6314 | ||
| 5 | 498.2892 | 4/3 | |
| 6 | 597.9471 | 24/17 | |
| 7 | 697.6049 | ||
| 8 | 797.2627 | ||
| 9 | 896.9206 | ||
| 10 | 996.5784 | 16/9 | |
| 11 | 1096.2363 | 32/17 | |
| 12 | 1195.8941 | compressed octave | |
| 13 | 1295.5520 | ||
| 14 | 1395.2098 | 56/25 | |
| 15 | 1494.8676 | ||
| 16 | 1594.5255 | ||
| 17 | 1694.1833 | ||
| 18 | 1793.8412 | ||
| 19 | 1893.4990 | 224/75 | |
| 20 | 1993.1569 | ||
| 21 | 2092.8147 | 375/112 | |
| 22 | 2192.4725 | ||
| 23 | 2292.1304 | ||
| 24 | 2391.7882 | ||
| 25 | 2491.4461 | ||
| 26 | 2591.1039 | 125/28 | |
| 27 | 2690.7618 | ||
| 28 | 2790.4196 | ||
| 29 | 2890.0774 | 85/16 | |
| 30 | 2989.7353 | 45/8 | |
| 31 | 3089.3931 | ||
| 32 | 3189.0510 | ||
| 33 | 3288.7088 | ||
| 34 | 3388.3667 | 85/12 | |
| 35 | 3488.0245 | 15/2 | |
| 36 | 3587.6823 | ||
| 37 | 3687.3402 | ||
| 38 | 3786.9980 | ||
| 39 | 3886.6559 | 85/9 | |
| 40 | 3986.3137 | exact 10/1 |
Miscellany
It is possible to call this division a form of kilobyte tuning, as
[math]2^{10} \approx 10^{3} = 1024 \approx 1000[/math];
which lies in the basis of using a "decimal" prefix to an otherwise binary unit of information.
Regular temperaments
40ED10 can also be thought of as a generator of the 2.3.5.17.19 subgroup temperament which tempers out 4624/4617, 6144/6137, and 6885/6859, which is a cluster temperament with 12 clusters of notes in an octave (quintilischis temperament). This temperament is supported by 12, 253, 265, 277, 289, 301, 313, and 325 EDOs.
Tempering out 400/399 (equating 20/19 and 21/20) leads quintilipyth (12&253), and tempering out 476/475 (equating 19/17 with 28/25) leads quintaschis (12&289).
- Quintilischis (12&289)
2.3.5.17.19 subgroup
Comma list: 4624/4617, 6144/6137, 6885/6859
Gencom: [2 18/17; 4624/4617 6144/6137 6885/6859]
Gencom mapping: [⟨1 2 -1 5 4], ⟨0 -5 40 -11 3]]
POTE generator: ~18/17 = 99.652
Optimal ET sequence: 12, 253, 265, 277, 289
RMS error: 0.1636 cents
- Quintilipyth (12 & 253)
7-limit
Comma list: 32805/32768, 9765625/9680832
Mapping: [⟨1 2 -1 -4], ⟨0 -5 40 82]]
POTE generator: ~625/588 = 99.625
Optimal ET sequence: 12, 253, 265
Badness: 0.253966
11-limit
Comma list: 1375/1372, 4375/4356, 32805/32768
Mapping: [⟨1 2 -1 -4 -7], ⟨0 -5 40 82 126]]
POTE generator: ~35/33 = 99.616
Optimal ET sequence: 12, 253, 265, 518c, 783cc
Badness: 0.113044
13-limit
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
Mapping: [⟨1 2 -1 -4 -7 -9], ⟨0 -5 40 82 126 153]]
POTE generator: ~35/33 = 99.612
Optimal ET sequence: 12f, 253, 518c, 771cc
Badness: 0.069127
17-limit
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Mapping: [⟨1 2 -1 -4 -7 -9 5], ⟨0 -5 40 82 126 153 -11]]
POTE generator: ~18/17 = 99.612
Optimal ET sequence: 12f, 253, 518c, 771cc
Badness: 0.045992
19-limit
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Mapping: [⟨1 2 -1 -4 -7 -9 5 4], ⟨0 -5 40 82 126 153 -11 3]]
POTE generator: ~18/17 = 99.615
Optimal ET sequence: 12f, 253, 265, 518ch
Badness: 0.038155
- Quintaschis (12 & 289)
7-limit
Comma list: 32805/32768, 49009212/48828125
Mapping: [⟨1 2 -1 -5], ⟨0 -5 40 94]]
POTE generator: ~200/189 = 99.664
Optimal ET sequence: 12, 277d, 289
Badness: 0.132890
11-limit
Comma list: 441/440, 32805/32768, 1953125/1951488
Mapping: [⟨1 2 -1 -5 -8], ⟨0 -5 40 94 138]]
POTE generator: ~35/33 = 99.653
Optimal ET sequence: 12, 277d, 289
Badness: 0.111477
13-limit
Comma list: 364/363, 441/440, 32805/32768, 109512/109375
Mapping: [⟨1 2 -1 -5 -8 -11], ⟨0 -5 40 94 138 177]]
POTE generator: ~35/33 = 99.658
Optimal ET sequence: 12f, 277df, 289
Badness: 0.074218
17-limit
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
Mapping: [⟨1 2 -1 -5 -8 -11 5], ⟨0 -5 40 94 138 177 -11]]
POTE generator: ~18/17 = 99.656
Optimal ET sequence: 12f, 277df, 289
Badness: 0.050571
19-limit
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
Mapping: [⟨1 2 -1 -5 -8 -11 5 4], ⟨0 -5 40 94 138 177 -11 3]]
POTE generator: ~18/17 = 99.659
Optimal ET sequence: 12f, 277df, 289
Badness: 0.042120
- Quintahelenic (12 & 301)
11-limit
Comma list: 5632/5625, 8019/8000, 151263/151250
Mapping: [⟨1 2 -1 -5 -9], ⟨0 -5 40 94 150]]
POTE generator: ~200/189 = 99.671
Optimal ET sequence: 12, 289e, 301, 915, 1216ce
Badness: 0.082225
13-limit
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
Mapping: [⟨1 2 -1 -5 -9 -11], ⟨0 -5 40 94 150 177]]
POTE generator: ~200/189 = 99.661
Optimal ET sequence: 12f, 289e, 301
Badness: 0.055570
17-limit
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
Mapping: [⟨1 2 -1 -5 -9 -11 5], ⟨0 -5 40 94 150 177 -11]]
POTE generator: ~18/17 = 99.665
Optimal ET sequence: 12f, 289e, 301
Badness: 0.040412
19-limit
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
Mapping: [⟨1 2 -1 -5 -9 -11 5 4], ⟨0 -5 40 94 150 177 -11 3]]
POTE generator: ~18/17 = 99.668
Optimal ET sequence: 12f, 289e, 301
Badness: 0.036840
- Quintahelenoid (12 & 301)
13-limit
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
Mapping: [⟨1 2 -1 -5 -9 14], ⟨0 -5 40 94 150 -124]]
POTE generator: ~200/189 = 99.672
Optimal ET sequence: 12, 301, 614, 915
Badness: 0.066108
17-limit
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Mapping: [⟨1 2 -1 -5 -9 14 5], ⟨0 -5 40 94 150 -124 -11]]
POTE generator: ~18/17 = 99.671
Optimal ET sequence: 12, 301, 915gg, 1216cegg
Badness: 0.047908
19-limit
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
Mapping: [⟨1 2 -1 -5 -9 14 5 4], ⟨0 -5 40 94 150 -124 -11 3]]
POTE generator: ~18/17 = 99.672
Optimal ET sequence: 12, 301, 614gh, 915gghh
Badness: 0.039542