581edo

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581edo

582edo →

Prime factorization	7 × 83
Step size	2.0654 ¢
Fifth	340\581 (702.238 ¢)
Semitones (A1:m2)	56:43 (115.7 ¢ : 88.81 ¢)
Consistency limit	25
Distinct consistency limit	25

Theory

581edo is a very strong 19- and 23-limit system, distinctly consistent to the 25-odd-limit. The equal temperament tempers out 2401/2400 in the 7-limit, 3025/3024, 19712/19683, 151263/151250 in the 11-limit, and 2080/2079, 4096/4095, 4225/4224, 6656/6655 and 10648/10647 in the 13-limit. It supports and gives a good tuning for newt, the 270 & 311 microtemperament, which features a neutral-third generator.

Prime harmonics

Approximation of prime harmonics in 581edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.283	-0.083	-0.151	+0.145	+0.092	+0.380	-0.095	-0.391	-1.006	-0.801
	Relative (%)	+0.0	+13.7	-4.0	-7.3	+7.0	+4.5	+18.4	-4.6	-18.9	-48.7	-38.8
Steps (reduced)		581 (0)	921 (340)	1349 (187)	1631 (469)	2010 (267)	2150 (407)	2375 (51)	2468 (144)	2628 (304)	2822 (498)	2878 (554)

Subsets and supersets

Since 581 factors into 7 × 83, 581edo contains 7edo and 83edo as subsets.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[921 -581⟩	[⟨581 921]]	-0.0891	0.0891	4.32
2.3.5	[-29 -11 20⟩, [33 -34 9⟩	[⟨581 921 1349]]	-0.0475	0.0936	4.53
2.3.5.7	2401/2400, 33554432/33480783, 48828125/48771072	[⟨581 921 1349 1631]]	-0.0222	0.0922	4.46
2.3.5.7.11	2401/2400, 3025/3024, 19712/19683, 234375/234256	[⟨581 921 1349 1631 2010]]	-0.0261	0.0828	4.01
2.3.5.7.11.13	2080/2079, 2401/2400, 3025/3024, 4096/4095, 78125/78078	[⟨581 921 1349 1631 2010 2150]]	-0.0259	0.0756	3.66
2.3.5.7.11.13.17	1225/1224, 2058/2057, 2080/2079, 2401/2400, 4096/4095, 4914/4913	[⟨581 921 1349 1631 2010 2150 2375]]	-0.0355	0.0738	3.58
2.3.5.7.11.13.17.19	1216/1215, 1225/1224, 1540/1539, 1729/1728, 2058/2057, 2080/2079, 4914/4913	[⟨581 921 1349 1631 2010 2150 2375 2468]]	-0.0283	0.0717	3.47
2.3.5.7.11.13.17.19.23	1216/1215, 1225/1224, 1288/1287, 1540/1539, 1729/1728, 2024/2023, 2058/2057, 2080/2079	[⟨581 921 1349 1631 2010 2150 2375 2468 2628]]	-0.0155	0.0800	3.87

• 581et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first after 270 with a lower 19-limit relative error, and the first after 311 with a lower 23-limit relative error. It is only bettered by 742 in terms of either 19-limit absolute error or 19-limit relative error, by 718 in terms of 23-limit absolute error, and not until 1578 do we reach a lower 23-limit relative error.
• 581et is also notable in the 17-limit, where it has a lower absolute error than any previous equal temperaments, past 494 and followed by 742.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	17\581	35.11	1990656/1953125	Gammic (5-limit)
1	64\581	132.19	[-38 5 13⟩	Astro
1	170\581	351.12	49/40	Newt
1	241\581	497.76	4/3	Gary
1	282\581	582.44	7/5	Neptune (7-limit)
1	285\581	588.64	[-14 15 -4⟩	Countritonic (5-limit)

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