User:Francium/1031edo

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1031edo

1032edo →

Prime factorization	1031 (prime)
Step size	1.16392 ¢
Fifth	603\1031 (701.843 ¢)
Semitones (A1:m2)	97:78 (112.9 ¢ : 90.79 ¢)
Consistency limit	9
Distinct consistency limit	9

1031 equal divisions of the octave (abbreviated 1031edo or 1031ed2), also called 1031-tone equal temperament (1031tet) or 1031 equal temperament (1031et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1031 equal parts of about 1.16 ¢ each. Each step represents a frequency ratio of 2^1/1031, or the 1031st root of 2.

Theory

1031edo is consistent to the 9-odd-limit, tempering out 420175/419904, 5250987/5242880 and 152946081792/152587890625 in the 7-limit. It is strong in the 2.3.5.13.17 subgroup, tempering out 2601/2600, 1601613/1600000, 33564375/33554432 and 4782969/4780672.

Prime harmonics

Approximation of prime harmonics in 1031edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.112	+0.107	-0.446	+0.379	-0.178	-0.203	+0.450	+0.242	+0.491	+0.260
	Relative (%)	+0.0	-9.6	+9.2	-38.3	+32.6	-15.3	-17.4	+38.7	+20.8	+42.2	+22.4
Steps (reduced)		1031 (0)	1634 (603)	2394 (332)	2894 (832)	3567 (474)	3815 (722)	4214 (90)	4380 (256)	4664 (540)	5009 (885)	5108 (984)

Subsets and supersets

1031edo is the 273rd prime edo. 3093edo, which triples it, gives a good correction to its harmonic 11.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[-1634 1031⟩	[⟨1031 1634]]	0.0354	0.0354	3.04
2.3.5	[38 -2 -15⟩, [-38 65 -28⟩	[⟨1031 1634 2394]]	0.0082	0.0481	4.13
2.3.5.7	420175/419904, 5250987/5242880, 152946081792/152587890625	[⟨1031 1634 2394 2894]]	0.0458	0.0774	6.65

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	166\1031	193.210	262144/234375	Luna

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

Music

Francium

• "sweet gravy" from have nice day (2025) – Spotify | Bandcamp | YouTube – in Sycamore, 1031edo tuning