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

I just had a crazy idea for your next musical mad science experiment (and it potentially includes 50edo):  See if it is possible to retune some of the quarter-tone ([[24edo]], "diatonicized chromatic")11L 2s (L/s = 2) scale works of Ivan Wyschnegradsky into other tuning systems that support 11L 2s and have a good approximation and single circle of 11/8 (or 16/11).  Plausible candidate tuning systems on the soft side are [[37edo]] (L/s = 3/2, and has a super-good 11/8), [[61edo]] (L/s = 5/3, but 61edo is big enough to be pushing the limits of plausibility), and [[50edo]] (L/s = 4/3 -- might be too soft).  Plausible candidate tuning systems on the hard side are [[35edo]] (L/s = 3), [[59edo]] (L/s = 7/3, but 59edo is big enough to be pushing the limits of plausibility), and [[46edo]] (L/s = 4/1 -- might be too hard).

Last modified: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:00, 9 April 2025 (UTC)

=== Table of odd harmonics for various EDO values supporting 17L 2s ===

{{Harmonics in equal|13|intervals=odd|prec=2|columns=28|title=[[13edo]] (L=1, s=1, [[16/11]] is 7) — Equalized 11L 2s}}

{{Harmonics in equal|76|intervals=odd|prec=2|columns=28|title=[[76edo]] (L=6, s=5, 16/11 is 41)}}

{{Harmonics in equal|63|intervals=odd|prec=2|columns=28|title=[[63edo]] (L=5, s=4, 16/11 is 34)}}

{{Harmonics in equal|113|intervals=odd|prec=2|columns=28|title=[[113edo]] (L=9, s=7, 16/11 is 61)}}

{{Harmonics in equal|50|intervals=odd|prec=2|columns=28|title=[[50edo]] (L=4, s=3, 16/11 is 27) — Supersoft 11L 2s}}

{{Harmonics in equal|137|intervals=odd|prec=2|columns=28|title=[[137edo]] (L=11, s=8, 16/11 is 74)}}

{{Harmonics in equal|87|intervals=odd|prec=2|columns=28|title=[[87edo]] (L=7, s=5, 16/11 is 47)}}

{{Harmonics in equal|124|intervals=odd|prec=2|columns=28|title=[[124edo]] (L=10, s=7, 16/11 is 67)}}

{{Harmonics in equal|37|intervals=odd|prec=2|columns=28|title=[[37edo]] (L=3, s=2, 16/11 is 20) — Soft 11L 2s}}

{{Harmonics in equal|135|intervals=odd|prec=2|columns=28|title=[[135edo]] (L=11, s=7, 16/11 is 73)}}

{{Harmonics in equal|98|intervals=odd|prec=2|columns=28|title=[[98edo]] (L=8, s=5, 16/11 is 53)}}

{{Harmonics in equal|159|intervals=odd|prec=2|columns=28|title=[[159edo]] (L=11, s=8, 16/11 is 86)}}

{{Harmonics in equal|61|intervals=odd|prec=2|columns=28|title=[[61edo]] (L=5, s=3, 16/11 is 33) — Semisoft 11L 2s}}

{{Harmonics in equal|146|intervals=odd|prec=2|columns=28|title=[[146edo]] (L=12, s=7, 16/11 is 79)}}

{{Harmonics in equal|85|intervals=odd|prec=2|columns=28|title=[[85edo]] (L=7, s=4, 16/11 is 46)}}

{{Harmonics in equal|109|intervals=odd|prec=2|columns=28|title=[[109edo]] (L=9, s=5, 16/11 is 59)}}

{{Harmonics in equal|24|intervals=odd|prec=2|columns=28|title=[[24edo]] (L=2, s=1, 16/11 is 13) — Basic 11L 2s}}

{{Harmonics in equal|107|intervals=odd|prec=2|columns=28|title=[[107edo]] (L=9, s=4, 16/11 is 58)}}

{{Harmonics in equal|83|intervals=odd|prec=2|columns=28|title=[[83edo]] (L=7, s=3, 16/11 is 45)}}

{{Harmonics in equal|142|intervals=odd|prec=2|columns=28|title=[[142edo]] (L=12, s=5, 16/11 is 77)}}

{{Harmonics in equal|59|intervals=odd|prec=2|columns=28|title=[[59edo]] (L=5, s=2, 16/11 is 32) — Semihard 11L 2s}}

{{Harmonics in equal|153|intervals=odd|prec=2|columns=28|title=[[153edo]] (L=13, s=5, 16/11 is 83)}}

{{Harmonics in equal|94|intervals=odd|prec=2|columns=28|title=[[94edo]] (L=8, s=3, 16/11 is 51)}}

{{Harmonics in equal|129|intervals=odd|prec=2|columns=28|title=[[129edo]] (L=11, s=4, 16/11 is 70)}}

{{Harmonics in equal|35|intervals=odd|prec=2|columns=28|title=[[35edo]] (L=3, s=1, 16/11 is 19) — Hard 11L 2s}}

{{Harmonics in equal|116|intervals=odd|prec=2|columns=28|title=[[116edo]] (L=10, s=3, 16/11 is 63)}}

{{Harmonics in equal|81|intervals=odd|prec=2|columns=28|title=[[81edo]] (L=7, s=2, 16/11 is 44)}}

{{Harmonics in equal|127|intervals=odd|prec=2|columns=28|title=[[127edo]] (L=11, s=3, 16/11 is 69)}}

{{Harmonics in equal|46|intervals=odd|prec=2|columns=28|title=[[46edo]] (L=4, s=1, 16/11 is 25) — Superhard 11L 2s}}

{{Harmonics in equal|103|intervals=odd|prec=2|columns=28|title=[[103edo]] (L=9, s=2, 16/11 is 56)}}

{{Harmonics in equal|57|intervals=odd|prec=2|columns=28|title=[[57edo]] (L=5, s=1, 16/11 is 31)}}

{{Harmonics in equal|68|intervals=odd|prec=2|columns=28|title=[[68edo]] (L=6, s=1, 16/11 is 37)}}

{{Harmonics in equal|11|intervals=odd|prec=2|columns=28|title=[[11edo]] (L=1, s=0, 16/11 is 6) — Collapsed 11L 2s}}

Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:00, 9 April 2025 (UTC)

== Musical Mad Science Musings on Diatonicized Sixth-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:

#  As the number of L intervals in a nL 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 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 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 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.

#  It is noteworthy (more detail needed) that harmonics 3 and 23 are very stable over the tuning spectrum of this scale (at least for EDO values up into the mid double digits), although the 23rd harmonic is guaranteed to be sharp, meaning that at larger EDO values, increasingly fine divisions of the octave will cause the mapping to disagree with 10\19 and 9\17 (and thereby with 19\36), thus requiring an 'i' [[wart]].  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 possibly 13 — need further checking to be sure, but this is looking less good than 3 and 23) seem like they would be good choices for the core.  (Coming in the future:  Checking this further.)

#  Tentatively assigning the generator as 23/16 ~ 13/9, tempering out [[208/207]].  But the problem is that — as can be seen in the table of harmonics below — the 13th harmonic is not stable enough for the entire 17L 2s tuning spectrum, even for the for the hard half of the tuning spectrum (closer to just 13/9, including having the best 3rd harmonic within the tuning spectrum). Maybe splitting the tuning spectrum of 17L 2s into 2 or more temperaments is in order?  Maybe the 5th harmonic is stable enough for the soft half of the 17L 2s tuning spectrum (closer to just 23/16, but even closer to the just barely out-of-reach 36/25)?  And maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s tuning spectrum?  (Coming in the future:  Checking this further; may need to insert some more supporting material above.)

Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:42, 8 April 2025 (UTC)

This table (actually a collection of tables for now) is for tracking trends in odd harmonics along the tuning spectrum of [[17L 2s]]; it is intended to match the organization of [[17L_2s#Scale_tree|the corresponding scale tree]], except for omitting the right-most column other than the top and bottom extremes:

{{Harmonics in equal|19|intervals=odd|prec=2|columns=28|title=[[19edo]] (L=1, s=1, BrightGen is 10) — Equalized 17L 2s}}

{{Harmonics in equal|112|intervals=odd|prec=2|columns=28|title=[[112edo]] (L=6, s=5, BrightGen is 59)}}

{{Harmonics in equal|93|intervals=odd|prec=2|columns=28|title=[[93edo]] (L=5, s=4, BrightGen is 49)}}

{{Harmonics in equal|74|intervals=odd|prec=2|columns=28|title=[[74edo]] (L=4, s=3, BrightGen is 39) — Supersoft 17L 2s}}

{{Harmonics in equal|129|intervals=odd|prec=2|columns=28|title=[[129edo]] (L=7, s=5, BrightGen is 68)}}

{{Harmonics in equal|55|intervals=odd|prec=2|columns=28|title=[[55edo]] (L=3, s=2, BrightGen is 29) — Soft 17L 2s}}

{{Harmonics in equal|146|intervals=odd|prec=2|columns=28|title=[[146edo]] (L=8, s=5, BrightGen is 77)}}

{{Harmonics in equal|91|intervals=odd|prec=2|columns=28|title=[[91edo]] (L=5, s=3, BrightGen is 48) — Semisoft 17L 2s}}

{{Harmonics in equal|127|intervals=odd|prec=2|columns=28|title=[[127edo]] (L=7, s=4, BrightGen is 67)}}

{{Harmonics in equal|36|intervals=odd|prec=2|columns=28|title=[[36edo]] (L=2, s=1, BrightGen is 19) — Basic 17L 2s}}

{{Harmonics in equal|125|intervals=odd|prec=2|columns=28|title=[[125edo]] (L=7, s=3, BrightGen is 66)}}

{{Harmonics in equal|89|intervals=odd|prec=2|columns=28|title=[[89edo]] (L=5, s=2, BrightGen is 89) — Semihard 17L 2s}}

{{Harmonics in equal|142|intervals=odd|prec=2|columns=28|title=[[142edo]] (L=8, s=3, BrightGen is 75)}}

{{Harmonics in equal|53|intervals=odd|prec=2|columns=28|title=[[53edo]] (L=3, s=1, BrightGen is 28) — Hard 17L 2s}}

{{Harmonics in equal|123|intervals=odd|prec=2|columns=28|title=[[123edo]] (L=7, s=2, BrightGen is 65)}}

{{Harmonics in equal|70|intervals=odd|prec=2|columns=28|title=[[70edo]] (L=4, s=1, BrightGen is 37) — Superhard 17L 2s}}

{{Harmonics in equal|87|intervals=odd|prec=2|columns=28|title=[[87edo]] (L=5, s=1, BrightGen is 46)}}

{{Harmonics in equal|104|intervals=odd|prec=2|columns=28|title=[[104edo]] (L=6, s=1, BrightGen is 55)}}

{{Harmonics in equal|17|intervals=odd|prec=2|columns=28|title=[[17edo]] (L=1, s=0, BrightGen is 9) — Collapsed 17L 2s}}

Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:42, 8 April 2025 (UTC)

Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:10, 9 April 2025 (UTC)