User:Francium/3631edo

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3631edo

3632edo →

Prime factorization	3631 (prime)
Step size	0.330487 ¢
Fifth	2124\3631 (701.955 ¢) (semiconvergent)
Semitones (A1:m2)	344:273 (113.7 ¢ : 90.22 ¢)
Consistency limit	9
Distinct consistency limit	9

3631 equal divisions of the octave (abbreviated 3631edo or 3631ed2), also called 3631-tone equal temperament (3631tet) or 3631 equal temperament (3631et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 3631 equal parts of about 0.33 ¢ each. Each step represents a frequency ratio of 2^1/3631, or the 3631st root of 2.

Theory

3631edo is consistent to the 9-odd-limit, tempering out 43046721/43025920, 78125000/78121827 and [79 -6 -13 -14⟩ in the 7-limit. Its harmonic 3 has an extremely low error of 0.1 percent. It is strong in the 2.3.5.11.23 subgroup, tempering out 64009/64000, 19133125/19131876, 184549376/184528125, 510526328421483/510320967680000. As an equal temperament, 3631edo supports quectismic in the 5-limit.

Prime harmonics

Approximation of prime harmonics in 3631edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.000	+0.026	+0.163	-0.065	-0.098	+0.140	-0.074	-0.018	-0.109	+0.104
	Relative (%)	+0.0	+0.1	+7.9	+49.4	-19.6	-29.7	+42.2	-22.5	-5.3	-32.9	+31.3
Steps (reduced)		3631 (0)	5755 (2124)	8431 (1169)	10194 (2932)	12561 (1668)	13436 (2543)	14842 (318)	15424 (900)	16425 (1901)	17639 (3115)	17989 (3465)

Subsets and supersets

3631edo is the 508th prime edo.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[5755 -3631⟩	[⟨3631 5755]]	−0.0001	0.0001	0.03
2.3.5	[91 -12- 31⟩, [-33 97 -52⟩	[⟨3631 5755 8431]]	−0.0038	0.0053	1.60

Music

Francium

• "Wings and Toasties" from Unsuspecting Tyrant Double-Decker Beef Fort (2026) – Spotify | Bandcamp | YouTube – in Quectismic, 3631edo tuning