40ed10

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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).

It is possible to call this division a form of decibel tuning or kilobyte tuning, as

[math]10^{\frac{1}{10}} \approx 2^{\frac{1}{3}} = 1.2589254 \approx 1.2599210[/math];

which lies in the basis of the definition of decibel. In addition, as a consequence of the previous formula,

[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. 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

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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 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
Vals: 12, 301, 614gh, 915gghh
Badness: 0.039542

See also