Father–3 equivalence continuum/Godtone's approach: Difference between revisions
m explain mutt and the 3 & 118 microtemp's mapping |
m explain significance of half-integer n in title |
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{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ style="font-size: 105%;" | Temperaments with half-integer ''n'' | |+ style="font-size: 105%;" | Temperaments with half-integer ''n'' aka<br />temperaments with a gen of ~sqrt(8/5)* | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
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| 5/2 | | 5/2 | ||
| [[Novamajor]]* ({{nowrap|77 & 80}}) | | [[Novamajor]]** ({{nowrap|77 & 80}}) | ||
| 19791209299968/19073486328125 | | 19791209299968/19073486328125 | ||
| {{ monzo| 41 2 -19 }} | | {{ monzo| 41 2 -19 }} | ||
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| {{ monzo| 55 2 -25 }} | | {{ monzo| 55 2 -25 }} | ||
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'''<nowiki>*</nowiki> This corresponds to the denominator of 2 implying that 3 must be reached in a half-integer number of ~5/4's. | |||
<nowiki>*</nowiki> Note that "novamajor" ([[User:Godtone]]'s name) is also called "isnes"; both names are based on the size of the generator being around 405 cents, but "isnes" was discovered as a point in the continuum while "novamajor" was discovered as one temperament in the [[fifth-chroma temperaments]]. | <nowiki>**</nowiki> Note that "novamajor" ([[User:Godtone]]'s name) is also called "isnes"; both names are based on the size of the generator being around 405 cents, but "isnes" was discovered as a point in the continuum while "novamajor" was discovered as one temperament in the [[fifth-chroma temperaments]]. | ||
If we approximate the [[JIP]] with increasing accuracy, (that is, using ''n'' a rational that is an increasingly good approximation of 1.72125...) we find these high-accuracy temperaments: | If we approximate the [[JIP]] with increasing accuracy, (that is, using ''n'' a rational that is an increasingly good approximation of 1.72125...) we find these high-accuracy temperaments: | ||
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The simplest of these 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. | The simplest of these 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. | ||
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. | ||
Also note that at {{nowrap|''n'' {{=}} −{{frac|2|3}}}}, we find the exotemperament tempering out [[32/27]]. | Also note that at {{nowrap|''n'' {{=}} −{{frac|2|3}}}}, we find the exotemperament tempering out [[32/27]]. | ||