400edo: Difference between revisions

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400edo is consistent in the [[21-odd-limit]]. It tempers out the unidecma, {{monzo| -7 22 -12 }}, and the qintosec comma, {{monzo| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, supporting the [[decoid]] temperament and the [[quinmite]] temperament.
400edo is consistent in the [[21-odd-limit]]. It tempers out the unidecma, {{monzo| -7 22 -12 }}, and the qintosec comma, {{monzo| 47 -15 -10 }}, in the 5-limit; [[2401/2400]], 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, [[9801/9800]], 117649/117612, and [[131072/130977]] in the 11-limit; [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], [[4096/4095]], [[4225/4224]] and 39366/39325 in the 13-limit, supporting the [[decoid]] temperament and the [[quinmite]] temperament.


400edo doubles [[200edo]], which holds a record for the best 3/2 fifth approximation. 400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the Sym454 calendar scale with 231\400 as the generator, which can be treated as 2/5 syntonic comma meantone.  
400edo doubles [[200edo]], which holds a record for the best 3/2 fifth approximation. 400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the Sym454 calendar scale with 231\400 as the generator, which can be treated as 5/12 syntonic comma meantone, which is the first meantone in the continued fraction that offers good precision. Other items like 1/3 and 2/5 eventually become inconsistent with the edo.  


=== Prime harmonics ===
=== Prime harmonics ===
Line 20: Line 20:
|unison
|unison
|1/1 exact
|1/1 exact
|
|-
|28
|5/12-meantone semitone
|6561/6250
|
|
|-
|-
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|231
|231
|Gregorian leap week fifth
|Gregorian leap week fifth
|118/79
|118/79, twelfth root of 800000/6561
|
|
|-
|-
Line 35: Line 40:
|perfect fifth
|perfect fifth
|[[3/2]]
|[[3/2]]
|
|-
|372
|5/12-meantone seventh
|12500/6561
|
|
|-
|-

Revision as of 09:50, 4 December 2021

The 400 equal divisions of the octave (400edo) is the equal division of the octave into 400 parts of exact 3 cents each.

Theory

400edo is consistent in the 21-odd-limit. It tempers out the unidecma, [-7 22 -12⟩, and the qintosec comma, [47 -15 -10⟩, in the 5-limit; 2401/2400, 1959552/1953125, and 14348907/14336000 in the 7-limit; 5632/5625, 9801/9800, 117649/117612, and 131072/130977 in the 11-limit; 676/675, 1001/1000, 1716/1715, 2080/2079, 4096/4095, 4225/4224 and 39366/39325 in the 13-limit, supporting the decoid temperament and the quinmite temperament.

400edo doubles 200edo, which holds a record for the best 3/2 fifth approximation. 400 is also the number of years in the Gregorian calendar's leap cycle. 400edo supports the Sym454 calendar scale with 231\400 as the generator, which can be treated as 5/12 syntonic comma meantone, which is the first meantone in the continued fraction that offers good precision. Other items like 1/3 and 2/5 eventually become inconsistent with the edo.

Prime harmonics

Script error: No such module "primes_in_edo".

Table of intervals

Step Name Associated ratio Notes
0 unison 1/1 exact
28 5/12-meantone semitone 6561/6250
35 septendecimal semitone 17/16
231 Gregorian leap week fifth 118/79, twelfth root of 800000/6561
234 perfect fifth 3/2
372 5/12-meantone seventh 12500/6561
400 octave 1/1 exact

Scales

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