227edo

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227edo

228edo →

Prime factorization	227 (prime)
Step size	5.28634 ¢
Fifth	133\227 (703.084 ¢)
Semitones (A1:m2)	23:16 (121.6 ¢ : 84.58 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

The equal temperament tempers out 15625/15552 (kleisma) and [61 -37 -1⟩ in the 5-limit; 5120/5103, 65625/65536, and 117649/116640 in the 7-limit, so that it supports countercata. In the 11-limit, it tempers out 385/384, 2200/2187, 3388/3375, and 12005/11979, so that it provides the optimal patent val for 11-limit countercata. In the 13-limit, it tempers out 325/324, 352/351, 625/624, 676/675, and 847/845, and again supplies a good tuning for 13-limit countercata, although 140edo tunes it better in this case.

227edo is accurate for the 13th harmonic, as the denominator of a convergent to log₂13, after 10 and before 5231.

Prime harmonics

Approximation of prime harmonics in 227edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+1.13	-0.41	-1.43	-1.54	+0.00	+0.77	-1.48	+0.80	+1.26	+2.10
	Relative (%)	+0.0	+21.4	-7.8	-27.0	-29.1	+0.0	+14.6	-28.0	+15.1	+23.8	+39.7
Steps (reduced)		227 (0)	360 (133)	527 (73)	637 (183)	785 (104)	840 (159)	928 (20)	964 (56)	1027 (119)	1103 (195)	1125 (217)

Subsets and supersets

227edo is the 49th prime edo.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[360 -227⟩	⟨227 360]	-0.3561	0.3560	6.73
2.3.5	15625/15552, [61 -37 -1⟩	⟨227 360 527]	-0.1785	0.3842	7.27
2.3.5.7	5120/5103, 15625/15552, 65625/65536	⟨227 360 527 637]	-0.0071	0.4461	8.44
2.3.5.7.11	385/384, 2200/2187, 3388/3375, 5120/5103	⟨227 360 527 637 785]	+0.0832	0.4380	8.29
2.3.5.7.11.13	325/324, 352/351, 385/384, 625/624, 847/845	⟨227 360 527 637 785 840]	+0.0693	0.4010	7.59
2.3.5.7.11.13.17	352/351, 595/594, 715/714, 847/845, 1001/1000, 3185/3179	⟨227 360 527 637 785 840 928]	+0.0324	0.3821	7.23

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	25\227	132.16	[-38 5 13⟩	Astro / kastro
1	60\227	317.18	6/5	Hanson / countercata
1	94\227	496.92	4/3	Undecental

Music

Francium

• "little hope" from hope in dark times (2024) – Spotify | Bandcamp | YouTube