User:Godtone/Augmented-chromatic equivalence continuum: Difference between revisions
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! Ratio | ! Ratio | ||
! Monzo | ! Monzo | ||
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| 3/2 | |||
| [[Ditonic]] ({{nowrap|50 & 53}}) | |||
| [[1220703125/1207959552]] | |||
| {{ monzo| -27 -2 13 }} | |||
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| 5/3 | | 5/3 | ||
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| {{ monzo| 17 1 -8 }} | | {{ monzo| 17 1 -8 }} | ||
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The simplest of these other than [[Würschmidt]] is [[mutt]] which has interesting properties discussed there. In regards to mutt, the fact that the denominator of ''n'' is a multiple of 3 tells us that it has a 1\3 period because it's contained in 3edo. The fact that the numerator is 5 tells us that 25/24 is split into 5 parts. From {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} we can thus deduce that each part is thus equal to ~cbrt(128/125) = (128/125)<sup>1/3</sup>, so that ~5/4 is found at 1\3 minus a third of a diesis, so that ~125/64 is found at thrice that. This observation is more general, leading to consideration of temperaments of third-integer ''n''. | The simplest of these other than [[Würschmidt]] is [[mutt]] which has interesting properties discussed there. In regards to mutt, the fact that the denominator of ''n'' is a multiple of 3 tells us that it has a 1\3 period because it's contained in 3edo. The fact that the numerator is 5 tells us that 25/24 is split into 5 parts. From {{nowrap|(128/125)<sup>n</sup> {{=}} 25/24}} we can thus deduce that each part is thus equal to ~cbrt(128/125) = (128/125)<sup>1/3</sup>, so that ~5/4 is found at 1\3 minus a third of a diesis, so that ~125/64 is found at thrice that. This observation is more general, leading to consideration of temperaments of third-integer ''n''. (Note that [[ditonic]] at ''n'' = 3/2 is included as an alternative approximation of ''n'' = ~1.7... as it finds relevance in [[53edo]], whose 5-limit is exceptionally accurate for its note count, but also because its increased complexity relative to Würschmidt allows it to spread damage over more generators.) | ||
The 3 & 118 microtemperament is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators. | The 3 & 118 microtemperament is at ''n'' = 7/4. Its generator is approximately 397{{cent}} so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)<sup>1/4</sup> needed to find prime 3 is thus four times the result of plugging ''n'' = 7/4 into 3''n'' + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators. | ||