User:Godtone/Augmented-chromatic equivalence continuum: Difference between revisions

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 : * 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.
 * 16/15 = 25/24 * 128/125, so that tempering out 16/15 (father) is found at ''n'' = -1. The reason it shouldn't be found at ''n'' = 0 instead is because ''n'' = -1 has an absurdly sharp tuning of ~5/4 because of being equated with ~4/3, which breaks the pattern from dicot at ''n'' = 0 having an absurdly flat tuning of ~5/4 because of being equated with ~6/5, and from ''n'' = 0 onwards, ~5/4 is tuned increasingly sharp. This observation is important enough for its own point:
-* 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.)
 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.