40ed10

← 39ed10

40ed10

41ed10 →

Prime factorization	23 × 5
Step size	99.6578 ¢
Octave	12\40ed10 (1195.89 ¢) (→ 3\10ed10)
Twelfth	19\40ed10 (1893.5 ¢)
Consistency limit	10
Distinct consistency limit	6

40 equal divisions of the 10th harmonic (abbreviated 40ed10) is a nonoctave tuning system that divides the interval of 10/1 into 40 equal parts of about 99.7 ¢ each. Each step represents a frequency ratio of 10^1/40, or the 40th root of 10.

Theory

40ed10 is related to 12edo, but with 10/1 instead of 2/1 being just. The octave, which comes from 10ed10, is compressed from pure by about 4.1 cents.

Harmonics

Approximation of harmonics in 40ed10

Harmonic		2	3	4	5	6	7	8	9	10	11	12
Error	Absolute (¢)	-4.1	-8.5	-8.2	+4.1	-12.6	+19.5	-12.3	-16.9	+0.0	+34.3	-16.7
	Relative (%)	-4.1	-8.5	-8.2	+4.1	-12.6	+19.6	-12.4	-17.0	+0.0	+34.4	-16.7
Steps (reduced)		12 (12)	19 (19)	24 (24)	28 (28)	31 (31)	34 (34)	36 (36)	38 (38)	40 (0)	42 (2)	43 (3)

Approximation of harmonics in 40ed10 (continued)

Harmonic		13	14	15	16	17	18	19	20	21	22	23	24
Error	Absolute (¢)	+44.1	+15.4	-4.4	-16.4	-21.7	-21.0	-15.0	-4.1	+11.1	+30.2	-46.8	-20.8
	Relative (%)	+44.2	+15.5	-4.4	-16.5	-21.8	-21.1	-15.0	-4.1	+11.1	+30.3	-46.9	-20.8
Steps (reduced)		45 (5)	46 (6)	47 (7)	48 (8)	49 (9)	50 (10)	51 (11)	52 (12)	53 (13)	54 (14)	54 (14)	55 (15)

Intervals

#	Cents value	Approximate ratio
0	0.0000	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	2/1
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	10/1

Miscellany

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

[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.

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 325edo.

Tempering out 400/399 (equating 20/19 and 21/20) leads to quintilipyth (12 & 253), and tempering out 476/475 (equating 19/17 with 28/25) leads to quintaschis (12 & 289).

See also

• 12edo – relative edo
• 19ed3 – relative ed3
• 28ed5 – relative ed5
• 31ed6 – relative ed6
• 34ed7 – relative ed7
• 42ed11 – relative ed11
• AS18/17 – relative ambitonal sequence