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 divisible 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).

The equal temperament 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 obvious 2.3.17 subgroup mappings 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 factorizes as 5² × 2³. 200edo's subset edos are: 2, 4, 5, 8, 10, 20, 25, 40, 50, 100.

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

Regular temperament properties

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 1 | 23\200 | 138.00 | 27/25 | Quartemka |- | 1 | 39\200 | 234.00 | 8/7 | Guiron |- | 1 | 83\200 | 498.00 | 4/3 | Helmholtz Template:Rank-2 end Template:Orf

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 – YouTube | soonlabel archive^{[dead link]} | play^{[dead link]}