User:Francium/4597edo

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4597edo

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Prime factorization	4597 (prime)
Step size	0.26104 ¢
Fifth	2689\4597 (701.936 ¢)
Semitones (A1:m2)	435:346 (113.6 ¢ : 90.32 ¢)
Consistency limit	5
Distinct consistency limit	5

4597 equal divisions of the octave (abbreviated 4597edo or 4597ed2), also called 4597-tone equal temperament (4597tet) or 4597 equal temperament (4597et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 4597 equal parts of about 0.261 ¢ each. Each step represents a frequency ratio of 2^1/4597, or the 4597th root of 2.

Theory

4597edo is consistent to the 5-limit, its harmonic 7 is about halfway its steps. It is strong in the 2.3.5.11.13.17.23.29 subgroup, tempering out 20736/20735, 359424/359375, 3408075/3407872, 13311/13310, 486243/486200, 2460375/2460172, 1069056/1068925. Using the 2.5.11.29.37 subgroup, it tempers out 9251/9250.

Prime harmonics

Approximation of prime harmonics in 4597edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.019	+0.025	-0.107	-0.002	+0.021	-0.017	+0.072	+0.048	-0.036	-0.115
	Relative (%)	+0.0	-7.3	+9.7	-41.1	-0.7	+7.9	-6.7	+27.7	+18.6	-13.9	-44.0
Steps (reduced)		4597 (0)	7286 (2689)	10674 (1480)	12905 (3711)	15903 (2112)	17011 (3220)	18790 (402)	19528 (1140)	20795 (2407)	22332 (3944)	22774 (4386)

Subsets and supersets

4597edo is the 622nd prime edo. 9194edo, which doubles it, gives a good correction to its harmonic 7.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[-7286 4597⟩	[⟨4597 7286]]	0.0060	0.0060	2.30
2.3.5	[-16 35 -17⟩, [342 -81 -92⟩	[⟨4597 7286 10674]]	0.0004	0.0093	3.56
2.3.5.7	200120949/200000000, [-7 30 -9 -7⟩, [47 -18 -14 5⟩	[⟨4597 7286 10674 12905]]	0.0098	0.0183	7.01
2.3.5.7.11	117649/117612, 50014503/50000000, 14348907/14348180, 6576668672/6576582375	[⟨4597 7286 10674 12905 15903]]	0.0080	0.0167	6.40
2.3.5.7.11.13	123201/123200, 1990656/1990625, 117649/117612, 1574640/1574573, 3195731/3194880	[⟨4597 7286 10674 12905 15903 17011]]	0.0057	0.0161	6.17
2.3.5.7.11.13.17	12376/12375, 123201/123200, 1990656/1990625, 790272/790075, 637637/637500, 163863/163840	[⟨4597 7286 10674 12905 15903 17011 18790]]	0.0055	0.0149	5.71
2.3.5.7.11.13.17.19	12376/12375, 10241/10240, 5929/5928, 123201/123200, 1549184/1549125, 203148/203125, 1285956/1285625	[⟨4597 7286 10674 12905 15903 17011 18790 19528]]	0.0027	0.0158	4.44

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	699\4597	182.4668	10/9	Minortone