40ed10

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]\displaystyle{ 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]\displaystyle{ 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