User:Lucius Chiaraviglio/Musical Mad Science: Difference between revisions

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 The [[36edo]] equivalent of Diatonicized Chromaticism is [[17L&nbsp;2s]].  So I've been giving a bit of thought to how to start constructing a temperament (or set thereof) that uses this scale.  This is in a very rudimentary stage, but so far I have observed:
-#  As the number of L intervals in a nL&nbsp;2s scale grows, the range of qualifying generator sizes shrinks, and so the scale becomes more brittle to tempering of the generator, and it becomes hard to find good ratios for specifying the generator.  Considering the wider of each pair of generators, the range of [[5L&nbsp;2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range &mdash; you have to have a ''bad'' fifth to land outside of its range.  The range of [[7L&nbsp;2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L&nbsp;2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L&bsp;2s]] ([[Ivan Wyschnegradsky]]'s original Diatonicized Chromatic scale) brackets [[16/11]]; and the ranges get progressively narrower and the ratios more complicated until by the time we get to 19L&nbsp;2s, the range falls between two ratios, the second of which is not even all that simple:  [[13/9]] and [[36/25]].  The first is too sharp by somewhat over 1{{c}}, and the second is barely too flat; although since it is near-just as 10 steps of [[19edo]], which is equalized 19L&nbsp;2s, we can count it as snapping to the lower end.  It is possible to come up with more complicated ratios by mediation between these slightly out-of-bounds endpoints, such as [[75/52]] and [[49/34]], or even [[62/43]] in the middle, but the latter uses unacceptably large primes, while the previous ratios and even 36/25 itself fail to map properly in the patent [[val]]s of some of the equal temperaments within the range of 17L&nbsp;2s (this flaw of 36/25 making it tempting to use the slightly flatter [[23/16]], so before considering the next point, it seems better to specify the generator as a tempered 36/25 ~ 13/9, or perhaps even 23/16 ~ 13/9, either way with the proviso that the generator can never reach the just value of either endpoint without going out of range.  But the choice of generator tempering comma will need to depend upon which subgroup(s) counts as the core of this temperament, so let's not throw out any of the above intervals just yet.
+#  As the number of L intervals in a nL&nbsp;2s scale grows, the range of qualifying generator sizes shrinks, and so the scale becomes more brittle to tempering of the generator, and it becomes hard to find good ratios for specifying the generator.  Considering the wider of each pair of generators, the range of [[5L&nbsp;2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range &mdash; you have to have a ''bad'' fifth to land outside of its range.  The range of [[7L&nbsp;2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L&nbsp;2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L&nbsp;2s]] ([[Ivan Wyschnegradsky]]'s original Diatonicized Chromatic scale) brackets [[16/11]]; and the ranges get progressively narrower and the ratios more complicated until by the time we get to 19L&nbsp;2s, the range falls between two ratios, the second of which is not even all that simple:  [[13/9]] and [[36/25]].  The first is too sharp by somewhat over 1{{c}}, and the second is barely too flat; although since it is near-just as 10 steps of [[19edo]], which is equalized 19L&nbsp;2s, we can count it as snapping to the lower end.  It is possible to come up with more complicated ratios by mediation between these slightly out-of-bounds endpoints, such as [[75/52]] and [[49/34]], or even [[62/43]] in the middle, but the latter uses unacceptably large primes, while the previous ratios and even 36/25 itself fail to map properly in the patent [[val]]s of some of the equal temperaments within the range of 17L&nbsp;2s (this flaw of 36/25 making it tempting to use the slightly flatter [[23/16]], so before considering the next point, it seems better to specify the generator as a tempered 36/25 ~ 13/9, or perhaps even 23/16 ~ 13/9, either way with the proviso that the generator can never reach the just value of either endpoint without going out of range.  But the choice of generator tempering comma will need to depend upon which subgroup(s) counts as the core of this temperament, so let's not throw out any of the above intervals just yet.
 #  In 36edo, the original inspiration for this attempt at a temperament, 19L&nbsp;2s lends itself to making good use of 36edo as a 2.3.7... subgroup temperament, with the generator 19\36.  With this scale, it is possible to choose a mode of this scale (UDP 11|7, cyclic order 14, LLLLLsLLLLLLLLLsLLL, no mode name assigned yet) that includes the following key 2.3.7 intervals:  root (0\36), [[9/8]] (6\36), [[7/6]] (8\36), both flavors of split neutral third (10\36 and 11\36), [[9/7]] (13\36), [[4/3]] (15\36), [[3/2]] (21\36), [[7/4]] (29\36), [[16/9]] (30\36), and on to the root, all the while filling in the scale with 2\36 stacked to various extents.  It also includes the generator interval 19\36, but let's not assign the generator a (tempered) ratio just yet.  The choice of other modes enables use of other intervals relative to the root, while a decent subset of them still support both the 3-limit fourth and fifth.
 #  Have noticed (more detail needed) that harmonics 3 and 23 are very stable over the tuning spectrum of this scale, although the 23rd harmonic is guaranteed to be sharp; the 7th harmonic is also reasonably stable, although it changes enough over the tuning spectrum to get rather bad at the extremes; the 5th harmonic is definitely not stable, and would need different extensions for at least the hard and soft halves of the tuning spectrum; commas including powers of 5 should be avoided in the core of the associated temperament, while commas including powers of 3 and 23 (and probably 13 &mdash; need further checking to be sure) seem like they would be good choices for the core.  (Coming in the future:  Checking this further.)