400edo: Difference between revisions
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== Theory == | == Theory == | ||
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. It tempers out 4914/4913 in the 17-limit, and [[1729/1728]] in the 19-limit. | 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. It tempers out 4914/4913 and [[24576/24565]] in the 17-limit, and [[1729/1728]] with 93347/93312 in the 19-limit. | ||
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. | 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. | ||
The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to [[18/17]], making it an interpretation of [[18/17s equal temperament]]. Since it tempers out the 93347/93312, a stack of three 18/17s is equated with 19/16. | |||
=== Prime harmonics === | === Prime harmonics === | ||
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|5/12-meantone semitone | |5/12-meantone semitone | ||
|6561/6250 | |6561/6250 | ||
| | |||
|- | |||
|33 | |||
|small septendecimal semitone | |||
|[[18/17]] | |||
| | | | ||
|- | |- | ||
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|septendecimal semitone | |septendecimal semitone | ||
|[[17/16]] | |[[17/16]] | ||
| | |||
|- | |||
|37 | |||
|diatonic semitone | |||
|[[16/15]] | |||
| | |||
|- | |||
|99 | |||
|undevicesimal minor third | |||
|[[19/16]] | |||
| | |||
|- | |||
|100 | |||
|symmetric minor third | |||
| | |||
| | |||
|- | |||
|200 | |||
|symmetric tritone | |||
|[[99/70]], [[140/99]] | |||
| | | | ||
|- | |- | ||
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|perfect fifth | |perfect fifth | ||
|[[3/2]] | |[[3/2]] | ||
| | |||
|- | |||
|323 | |||
|harmonic seventh | |||
|[[7/4]] | |||
| | | | ||
|- | |- | ||
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|400 | |400 | ||
|octave | |octave | ||
| | |2/1 exact | ||
| | | | ||
|} | |} | ||