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 and [[24576/24565]] in the 17-limit, and [[1729/1728]] | 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 [[936/935]], [[1156/1155]], 2058/2057, [[2601/2600]], 4914/4913 and [[24576/24565]] in the 17-limit, and 969/968, [[1216/1215]], [[1521/1520]], and [[1729/1728]] in the 19-limit. | ||
400edo doubles [[200edo]], which holds a record for the best 3/2 fifth approximation | 400edo doubles [[200edo]], which holds a record for the best 3/2 fifth approximation. | ||
The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to [[18/17]], making it an | 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 is close to 5/12 syntonic comma meantone. The leap week scale offers an interest in that 1/7th of its generator, 33\400, is associated to [[18/17]], making it an approximation of [[18/17 equal-step tuning]]. Since it tempers out the 93347/93312, a stack of three 18/17's is equated with 19/16. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Primes in edo|400|columns=15}} | {{Primes in edo|400|columns=15}} | ||
== | == Selected intervals == | ||
{| class="wikitable" | {| class="wikitable center-1" | ||
|+ | |+ | ||
!Step | ! Step | ||
! | ! Eliora's Naming System | ||
!Associated ratio | ! Associated ratio | ||
|- | |- | ||
|0 | | 0 | ||
|unison | | unison | ||
|1/1 | | 1/1 | ||
|- | |- | ||
|28 | | 28 | ||
|5/12-meantone semitone | | 5/12-meantone semitone | ||
|6561/6250 | | 6561/6250 | ||
|- | |- | ||
|33 | | 33 | ||
|small septendecimal semitone | | small septendecimal semitone | ||
|[[18/17]] | | [[18/17]] | ||
|- | |- | ||
|35 | | 35 | ||
|septendecimal semitone | | septendecimal semitone | ||
|[[17/16]] | | [[17/16]] | ||
|- | |- | ||
|37 | | 37 | ||
|diatonic semitone | | diatonic semitone | ||
|[[16/15]] | | [[16/15]] | ||
|- | |- | ||
|99 | | 99 | ||
|undevicesimal minor third | | undevicesimal minor third | ||
|[[19/16]] | | [[19/16]] | ||
|- | |- | ||
|100 | | 100 | ||
|symmetric minor third | | symmetric minor third | ||
| | |||
| | |||
|- | |- | ||
|200 | | 200 | ||
|symmetric tritone | | symmetric tritone | ||
|[[99/70]], [[140/99]] | | [[99/70]], [[140/99]] | ||
|- | |- | ||
|231 | | 231 | ||
|Gregorian leap week fifth | | Gregorian leap week fifth | ||
|118/79 | | 118/79 | ||
|- | |- | ||
|234 | | 234 | ||
|perfect fifth | | perfect fifth | ||
|[[3/2]] | | [[3/2]] | ||
|- | |- | ||
|323 | | 323 | ||
|harmonic seventh | | harmonic seventh | ||
|[[7/4]] | | [[7/4]] | ||
|- | |- | ||
|372 | | 372 | ||
|5/12-meantone seventh | | 5/12-meantone seventh | ||
|12500/6561 | | 12500/6561 | ||
|- | |- | ||
|400 | | 400 | ||
|octave | | octave | ||
|2/1 | | 2/1 | ||
|} | |} | ||