275edo

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275edo

276edo →

Prime factorization	52 × 11
Step size	4.36364¢
Fifth	161\275 (702.545¢)
Semitones (A1:m2)	27:20 (117.8¢ : 87.27¢)
Consistency limit	9
Distinct consistency limit	9

275 equal divisions of the octave (abbreviated 275edo or 275ed2), also called 275-tone equal temperament (275tet) or 275 equal temperament (275et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 275 equal parts of about 4.36 ¢ each. Each step represents a frequency ratio of 2^1/275, or the 275th root of 2.

Theory

If harmonic 5 is used, 275et tends very sharp. It tempers out [24 -21 4⟩ (vulture comma) and [19 10 -15⟩ (trisedodge comma) in the 5-limit; 6144/6125 and 10976/10935 in the 7-limit.

The 275e val ⟨275 436 639 772 952] being the best, tempers out 441/440, 4000/3993, 14700/14641, and 19712/19683. This can be extended to the 13-limit through 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079.

The 275 val ⟨275 436 639 772 951] tempers out 3025/3024, 3773/3750, 8019/8000. This can be extended to the 13-limit through 352/351, 676/675, 1716/1715, 2200/2197, and 3584/3575.

Prime harmonics

Approximation of prime harmonics in 275edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.59	+2.05	-0.10	-1.50	+1.65	-0.23	-0.79	+0.09	+0.24	-1.76
	Relative (%)	+0.0	+13.5	+47.0	-2.3	-34.4	+37.9	-5.2	-18.0	+2.0	+5.5	-40.4
Steps (reduced)		275 (0)	436 (161)	639 (89)	772 (222)	951 (126)	1018 (193)	1124 (24)	1168 (68)	1244 (144)	1336 (236)	1362 (262)

Subsets and supersets

Since 275 factors into 5² × 11, 275edo has subset edos 5, 11, 25 and 55.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[436 -275⟩	[⟨275 436]]	-0.1863	0.1862	4.27
2.3.5	[24 -21 4⟩, [19 10 -15⟩	[⟨275 436 639]]	-0.4184	0.3618	8.29
2.3.5.7	6144/6125, 10976/10935, 9882516/9765625	[⟨275 436 639 772]]	-0.3051	0.3698	8.48
2.3.5.7.11	441/440, 4000/3993, 6144/6125, 10976/10935	[⟨275 436 639 772 952]] (275e)	-0.4096	0.3912	8.97
2.3.5.7.11.13	364/363, 441/440, 676/675, 6144/6125, 10976/10935	[⟨275 436 639 772 952 1018]] (275e)	-0.4158	0.3574	8.19

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	6\275	26.18	1594323/1562500	Sfourth (5-limit)
1	109\275	485.64	320/243	Vulture (5-limit)
1	128\275	558.55	112/81	Condor (275e)
5	17\275	74.18	25/24	Countdown (275e)
11	114\275 (11\275)	497.45 (48.00)	4/3 (36/35)	Hendecatonic

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct