User:Godtone/Augmented-chromatic equivalence continuum: Difference between revisions
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m 3 & 118 is named squarschmidt |
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| 7/4 | | 7/4 | ||
| {{nowrap|3 & 118}} | | [[squarschmidt]] ({{nowrap|3 & 118}}) | ||
| [[186773283746309210112/186264514923095703125|(42 digits)]] | | [[186773283746309210112/186264514923095703125|(42 digits)]] | ||
| {{ monzo| 61 4 -29 }} | | {{ monzo| 61 4 -29 }} | ||
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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''. (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 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 [[squarschmidt]] 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. | ||
Finally, the 3 & 612 microtemperament at ''n'' = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament [[612edo]]. The denominator of 7 indicates that 128/125 is split into 7 equal parts. | Finally, the 3 & 612 microtemperament at ''n'' = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament [[612edo]]. The denominator of 7 indicates that 128/125 is split into 7 equal parts. | ||