525edo

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525edo

526edo →

Prime factorization	3 × 52 × 7
Step size	2.28571 ¢
Fifth	307\525 (701.714 ¢)
Semitones (A1:m2)	49:40 (112 ¢ : 91.43 ¢)
Consistency limit	25
Distinct consistency limit	25

Theory

525edo is distinctly consistent through the 25-odd-limit. It tempers out the schisma, 32805/32768, and [8 77 -56⟩ in the 5-limit; 250047/250000, 703125/702464 and [21 3 1 -10⟩ in the 7-limit; 3025/3024, 24057/24010, 102487/102400 and 180224/180075 in the 11-limit; 729/728, 1716/1715, 2200/2197, 4096/4095 and 14641/14625 in the 13-limit.

525's divisors are 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175.

Fractional-octave temperaments.

It supports the 140 & 525 temperament, with period 35 which sets 7/5 and 10/7 to two "legs" of 35edo (17\35 and 18\35) opposing the tonic and tempers out [34 0 70 -70⟩, setting a circle of thirty-five 50/49's equal with the octave.

525edo supports 21st-octave period called akjayland, and the 23-limit extension of akjayland called vasca, defined as 357 & 525. It is more suitable to view this temperament as vasca in 525edo as opposed to simply akjayland, since 525edo is consistent in the 23-limit, while other EDOs which support akjayland are not. In addition to splitting octave into 21 parts, 525edo also supports the relationship that sets 11\21 to 23/16.

Prime harmonics

Approximation of prime harmonics in 525edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.24	-0.03	+0.32	-0.46	+0.62	+0.19	-0.37	+0.30	-1.01	+0.11
	Relative (%)	+0.0	-10.5	-1.2	+13.9	-20.2	+26.9	+8.2	-16.2	+13.0	-44.0	+4.7
Steps (reduced)		525 (0)	832 (307)	1219 (169)	1474 (424)	1816 (241)	1943 (368)	2146 (46)	2230 (130)	2375 (275)	2550 (450)	2601 (501)

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[512 -323⟩	[⟨525 832]]	+0.0759	0.0759	3.32
2.3.5	32805/32768, [8 77 -56⟩	[⟨525 832 1219]]	+0.0546	0.0689	3.02
2.3.5.7	32805/32768, 250047/250000, [21 3 1 -10⟩	[⟨525 832 1219 1474]]	+0.0128	0.0940	4.11
2.3.5.7.11	3025/3024, 24057/24010, 32805/32768, 102487/102400	[⟨525 832 1219 1474 1816]]	+0.0368	0.0969	4.24
2.3.5.7.11.13	729/728, 1716/1715, 2200/2197, 3025/3024, 14641/14625	[⟨525 832 1219 1474 1816 1943]]	+0.0030	0.1164	5.09
2.3.5.7.11.13.17	729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197	[⟨525 832 1219 1474 1816 1943 2146]]	-0.0040	0.1091	4.77
2.3.5.7.11.13.17.19	729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197, 2376/2375	[⟨525 832 1219 1474 1816 1943 2146 2230]]	+0.0074	0.1064	4.66

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per Octave	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	218\525	498.29	4/3	Helmholtz
3	218\525 (43\525)	498.29 (98.29)	4/3 (18/17)	Term
3	109\525 (66\525)	249.14 (150.86)	15/13 (12/11)	Hemiterm (525f)
7	218\525 (7\525)	498.29 (16.00)	4/3 (99/98)	Septant
21	256\525 (6\525)	585.14 (13.71)	91875/65536 (126/125)	Akjayland / vasca