User:Godtone/Augmented-chromatic equivalence continuum: Difference between revisions
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* For ''n'' a nonnegative integer, half-integer or third-integer('''*'''), increasing ''n'' corresponds to increasingly sharp tunings of ~5/4. In the limit, as ''n'' goes to infinity, these all approach ~5/4 = 1\3, corresponding to [[augmented temperament]]. | * For ''n'' a nonnegative integer, half-integer or third-integer('''*'''), increasing ''n'' corresponds to increasingly sharp tunings of ~5/4. In the limit, as ''n'' goes to infinity, these all approach ~5/4 = 1\3, corresponding to [[augmented temperament]]. | ||
:: ('''*''' It is conjectured by [[User:Godtone]] that for a given choice of denominator ''b'' in ''n'' = ''a''/''b'', a larger value of ''a'' always corresponds to a sharper tuning of ~5/4, where the sharpness in a pure-octaves tuning is always strictly flat of 1\3, so that (more trivially) taking the limit as ''a'' goes to infinity, ~5/4 = 1\3. The intuition for why we might expect this to be true is that in a pure-2's pure-3's tuning, we are always constraining ~128/125's size to be equal to the appropriate relation to ~25/24, where as 2 and 3 are fixed, the ~5 is the only free variable and depending only on ''n'', with that ''n'' essentially indirectly specifying the degree of tempering.) | :: ('''*''' It is conjectured by [[User:Godtone]] that for a given choice of denominator ''b'' in ''n'' = ''a''/''b'', a larger value of ''a'' always corresponds to a sharper tuning of ~5/4, where the sharpness in a pure-octaves tuning is always strictly flat of 1\3, so that (more trivially) taking the limit as ''a'' goes to infinity, ~5/4 = 1\3. The intuition for why we might expect this to be true is that in a pure-2's pure-3's tuning, we are always constraining ~128/125's size to be equal to the appropriate relation to ~25/24, where as 2 and 3 are fixed, the ~5 is the only free variable and depending only on ''n'', with that ''n'' essentially indirectly specifying the degree of tempering.) | ||
* Also, if one is interested in what intervals are present between ~5/4 and ~4/3, it is simple to observe because (16/15)/(25/24) = 128/125, meaning for a nonnegative integer ''n'' there is exactly one more interval between ~5/4 and ~4/3 as between ~6/5 and ~5/4. As there is ''n'' - 1 intervals between ~6/5 and ~5/4 (because of splitting ~25/24 into ''n'' parts), that means that (for nonnegative integer ''n'') there is exactly ''n'' intervals between ~5/4 and ~4/3. More generally, for rational ''n'' = ''a''/''b'', we have ''a'' - 1 intervals between ~6/5 and ~5/4 and because there is another ~128/125 between ~5/4 and ~4/3 we have ''a''/''b'' + 1 = ''a''/''b'' + ''b''/''b'' for the translated coordinates so that we have ''a'' + ''b'' - 1 intervals between ~5/4 and ~4/3, corresponding to splitting ~16/15 into ''a'' + ''b'' equal parts. | |||
Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts. | Therefore, if ''n'' = ''a''/''b'' is a rational with ''b'' > 1 and ''b'' not a multiple of 3 (so that 3''a''/''b'' + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather ''b'' equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in ''b''(3''a''/''b'' + 2) = 3''a'' + 2''b'' generators, and also means that ~128/125 is split into ''b'' equal parts. | ||
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|- | |- | ||
| 5/3 | | 5/3 | ||
| [[Mutt]] ({{nowrap|84 & 87}}) | | [[Mutt]] ({{nowrap|84 & 87}}) {{nowrap|(generator {{=}} ~[[Würschmidt's comma]])}} | ||
| [[mutt comma]] | | [[476837158203125/474989023199232|mutt comma]] | ||
| {{ monzo| -44 -3 21 }} | | {{ monzo| -44 -3 21 }} | ||
|- | |- | ||
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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 a sequence of increasingly accurate temperaments ''n'' = 2, 5/3, 7/4, 12/7: | 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 a sequence of increasingly accurate temperaments ''n'' = 2, 5/3, 7/4, 12/7: | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ style="font-size: 105%;" | | |+ style="font-size: 105%;" | Temperaments closely approximating the just ''n'' | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
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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 | 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 (7) indicates that 128/125 is split into 7 equal parts, while the numerator indicates that each (128/125)<sup>1/7</sup> part represents (25/24)<sup>1/12</sup>, that is, a twelfth of 25/24. | ||
[[Category:3edo]] | [[Category:3edo]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||