User:Lucius Chiaraviglio/Musical Mad Science: Difference between revisions
m →Table of odd harmonics for various EDO values supporting 17L 2s: Add need for update after 17L 2s tuning table was extended on 2025-05-12 |
→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): Rename to Diatonicized Third-Tone Sub-Chromaticism and add explanation of this in the introduction; recommend future temperament name Wonderland |
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Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:25, 26 April 2025 (UTC) | Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:25, 26 April 2025 (UTC) | ||
== Musical Mad Science Musings on Diatonicized | == Musical Mad Science Musings on Diatonicized Third-Tone Sub-Chromaticism == | ||
The [[36edo]] equivalent of Diatonicized Chromaticism is [[17L 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: | The [[36edo]] equivalent of Diatonicized Chromaticism is [[17L 2s]]. (Originally I had this down as Diatonicized Sixth-Tone Sub-Chromaticism, following after the example of [[Ivan Wyschnegradsky]], but Diatonicized Chromaticism is really named after the large step in its [[11L 2s]] scale, so this should liewise be named after the large step in its 17L 2s scale, which approximates a third-tone.) So I've been giving a bit of thought to how to start constructing a temperament (or set thereof) that uses this scale. (And it has turned out to be a real rabbit hole, which suggests the name Wonderland for the temperament if I ever get to that point before somebody else takes that name for some other purpose.) This is (unfortunately still) in a very rudimentary stage, but so far I have observed: | ||
# As the number of L intervals in a ''n''L 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 2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range — you have to have a ''bad'' fifth to land outside of its range. The range of [[7L 2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L 2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L 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 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 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 such large primes as to give difficulty (otherwise it would be very good), 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 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. Also worthy of consideration is the generator ~[[59049/40960]] = ~|-13 10 -1⟩ of the established [[Alphatricot family]], although this only works for a narrow band in the hard to super-hard region of the 17L 2s tuning spectrum, plus 17c. 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. (More painstaking inspection has revealed that ~23/16 or ~13/9 are usable after all in significant parts of the tuning spectrum with only minimal use of warts, while ~62/43 is usable throughout again with only minimal use of warts — therefore this needs to be rewritten to prefer ~23/16 or ~13/9, with ~62/43 as a fallback, and the more complex sliding generator with the 53rd harmonic component only used as a last resort.) | # As the number of L intervals in a ''n''L 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 2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range — you have to have a ''bad'' fifth to land outside of its range. The range of [[7L 2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L 2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L 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 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 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 such large primes as to give difficulty (otherwise it would be very good), 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 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. Also worthy of consideration is the generator ~[[59049/40960]] = ~|-13 10 -1⟩ of the established [[Alphatricot family]], although this only works for a narrow band in the hard to super-hard region of the 17L 2s tuning spectrum, plus 17c. 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. (More painstaking inspection has revealed that ~23/16 or ~13/9 are usable after all in significant parts of the tuning spectrum with only minimal use of warts, while ~62/43 is usable throughout again with only minimal use of warts — therefore this needs to be rewritten to prefer ~23/16 or ~13/9, with ~62/43 as a fallback, and the more complex sliding generator with the 53rd harmonic component only used as a last resort.) | ||
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Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:42, 8 April 2025 (UTC)<br> | Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:42, 8 April 2025 (UTC)<br> | ||
Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) | Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:04, 14 June 2025 (UTC) | ||