613edo

← 612edo

613edo

614edo →

Prime factorization	613 (prime)
Step size	1.95759¢
Fifth	359\613 (702.773¢)
Semitones (A1:m2)	61:44 (119.4¢ : 86.13¢)
Dual sharp fifth	359\613 (702.773¢)
Dual flat fifth	358\613 (700.816¢)
Dual major 2nd	104\613 (203.589¢)
Consistency limit	3
Distinct consistency limit	3

613 equal divisions of the octave (abbreviated 613edo or 613ed2), also called 613-tone equal temperament (613tet) or 613 equal temperament (613et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 613 equal parts of about 1.96 ¢ each. Each step represents a frequency ratio of 2^1/613, or the 613th root of 2.

Theory

613edo is only consistent to the 3-odd-limit and the harmonic 3 is about halfway its steps. It can be used in the 2.9.5.7.13.19.23.29.31.37 subgroup, tempering out 1521/1520, 875/874, 1863/1862, 38475/38416, 2205/2204, 1520/1519, 186875/186624, 1665/1664 and 194560/194481.

Odd harmonics

Approximation of odd harmonics in 613edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.818	-0.669	+0.179	-0.321	+0.721	-0.723	+0.149	+0.754	+0.040	-0.960	+0.111
	Relative (%)	+41.8	-34.2	+9.1	-16.4	+36.8	-37.0	+7.6	+38.5	+2.0	-49.1	+5.7
Steps (reduced)		972 (359)	1423 (197)	1721 (495)	1943 (104)	2121 (282)	2268 (429)	2395 (556)	2506 (54)	2604 (152)	2692 (240)	2773 (321)

Subsets and supersets

613edo is the 112th prime EDO. 1226edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.9	[-1943 613⟩	[⟨613 1943]]	0.0506	0.0506	2.58
2.9.5	32805/32768, [-97 -69 136⟩	[⟨613 1943 1423]]	0.1299	0.1194	6.10
2.9.5.7	32805/32768, 40500000/40353607, [-23 -7 11 7⟩	[⟨613 1943 1423 1721]]	0.0814	0.1331	6.80

Music

Francium

• "Donut" from You Are A... (2024) – Spotify | Bandcamp | YouTube – porcupinefish[15] in 613edo tuning