131edo

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131edo

132edo →

Prime factorization	131 (prime)
Step size	9.16031 ¢
Fifth	77\131 (705.344 ¢)
Semitones (A1:m2)	15:8 (137.4 ¢ : 73.28 ¢)
Dual sharp fifth	77\131 (705.344 ¢)
Dual flat fifth	76\131 (696.183 ¢)
Dual major 2nd	22\131 (201.527 ¢)
Consistency limit	3
Distinct consistency limit	3

Theory

131edo is inconsistent to the 5-odd-limit and the error of harmonic 3 is quite large. However, it is the next edo after 81edo on the Golden Tone System (Das Goldene Tonsystem) of Thorvald Kornerup, using the 131b val. The patent val has a fifth sharp by 3.389 cents rather than flat like the meantone fifth; rather than tempering out 81/80 it tempers out the immunity comma, 1638400/1594323. In the 7-limit it tempers out 3125/3087 and 245/243, so that it supports bohpier.

131edo is also notable for having a good approximation to acoustic e, at 189\131, which is a semiconvergent. This number of steps, 189, is particularly well-factorizable, and logarithmic divisors of acoustic e form a sequence of rapidly converging approximations to small rationals. Among these are 5/4 (2\9edn = 42\131), 15/13 (1\7edn = 27\131), 19/17 (1\9edn = 21\131), 11/10 (2\21edn = 18\131), 14/13 (2\27edn = 14\131), and 32/31 (2\63edn = 6\131), with accuracy increasing the smaller the fraction.

Odd harmonics

Approximation of odd harmonics in 131edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23	25	27	29	31
Error	Absolute (¢)	+3.39	-1.58	+2.17	-2.38	-1.70	+2.22	+1.81	-4.19	-4.38	-3.61	+3.79	-3.16	+1.01	-3.62	+0.00
	Relative (%)	+37.0	-17.3	+23.7	-26.0	-18.6	+24.2	+19.7	-45.8	-47.9	-39.4	+41.3	-34.5	+11.0	-39.6	+0.0
Steps (reduced)		208 (77)	304 (42)	368 (106)	415 (22)	453 (60)	485 (92)	512 (119)	535 (11)	556 (32)	575 (51)	593 (69)	608 (84)	623 (99)	636 (112)	649 (125)

Subsets and supersets

131edo is the 32nd prime edo, following 127edo and before 137edo.

Scales

Mos scales

33 16 33 33 16	Pentatonic (comparable with 8edo and 99edo)
23 23 8 23 23 23 8	Pythagorean tuning (comparable with 17edo)
21 21 13 21 21 21 13	Meantone tuning (comparable with 50edo)
19 12 19 19 12 19 19 12	Oneirotonic tuning (comparable with 55edo)
18 18 18 18 18 18 18 5	Porcupine tuning (comparable with 29edo and 80edo)
17 17 17 6 17 17 17 17 6	Superdiatonic tuning (comparable with 23edo)
16 16 16 16 16 16 16 16 3	Bohpier tuning (comparable with 41edo)
13 13 9 13 13 13 9 13 13 13 9	Sensi-11 Tuning
11 11 11 11 11 5 11 11 11 11 11 11 5	De Vries 13-tone Tuning
10 10 10 7 10 10 10 10 7 10 10 10 10 7	Ketradektriatoh Tuning
21 17 21 17 17 21 17	mohaha7
4 17 17 17 4 17 17 4 17 17	mohaha10