200edo

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200edo

201edo →

Prime factorization	23 × 52
Step size	6 ¢
Fifth	117\200 (702 ¢) (semiconvergent)
Semitones (A1:m2)	19:15 (114 ¢ : 90 ¢)
Consistency limit	9
Distinct consistency limit	9

Theory

200edo contains a perfect fifth of exactly 702 cents and a perfect fourth of exactly 498 cents, which is accurate due to 200 being the denominator of a continued fraction convergent to log₂(3/2). The error is only about 1/22 cents. In light of having its perfect fifth precise and the step divisibly by 9, it is essentially a perfect edo for Carlos Alpha, even up many octaves (the difference between 13 steps of 200edo and 1 step of Carlos Alpha is only 0.03501 cents).

It tempers out the schisma, 32805/32768 and the quartemka, [2 -32 21⟩ in the 5-limit, and the gamelisma, 1029/1024, in the 7-limit, so that it supports the guiron temperament.

One step of 200edo is close to 289/288. Unfortunately, it is not compatible with any 2.3.17 subgroup mapping of 200edo.

Prime harmonics

Approximation of prime harmonics in 200edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.04	-2.31	-2.83	+0.68	-0.53	-2.96	+2.49	+1.73	+2.42	+0.96
	Relative (%)	+0.0	+0.7	-38.6	-47.1	+11.4	-8.8	-49.3	+41.4	+28.8	+40.4	+16.1
Steps (reduced)		200 (0)	317 (117)	464 (64)	561 (161)	692 (92)	740 (140)	817 (17)	850 (50)	905 (105)	972 (172)	991 (191)

Subsets and supersets

200's divisors are: 2, 4, 5, 8, 10, 20, 25, 40, 50, 100. It factorizes as 5² × 2³.

400edo, which doubles it, provides good correction for the harmonics 5 and 7, and makes for a strong 19-limit system.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[317 -200⟩	⟨200 317]	-0.0142	0.0142	0.24
2.3.5	32805/32768, [2 -32 21⟩	⟨200 317 464]	+0.3226	0.4767	7.95
2.3.5.7	1029/1024, 10976/10935, 390625/387072	⟨200 317 464 561]	+0.4937	0.5082	8.47

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	Associated Ratio	Temperaments
1	23\200	138.00	27/25	Quartemka
1	39\200	234.00	8/7	Guiron
1	83\200	498.00	4/3	Helmholtz

Scales

• 34 34 15 34 34 34 15 = Pythagorean tuning

• 32 32 20 32 32 32 20 = Meantone tuning in the same way of 50edo

• 27 27 27 27 27 27 27 11 = Porcupine tuning

• 26 26 26 9 26 26 26 26 9 = Superdiatonic tuning

• 24 24 24 16 24 24 24 24 16 = Superdiatonic tuning in the same way of 25edo

• 22 22 8 22 22 22 8 22 22 22 8 = Sensi

• 16 16 16 8 16 16 16 16 8 16 16 16 16 8 = Ketradektriatoh tuning

Music

Francium

• "On Fire" from Mysteries (2023) Spotify | Bandcamp | YouTube

Claudi Meneghin

• Fugue on Elgar’s Enigma Theme play