User:Lucius Chiaraviglio/Musical Mad Science: Difference between revisions
→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): More thoughts, with a more detailed check on stability of various harmonics across the 17L 2s tuning spectrum. |
→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): Added table of odd harmonics for various EDO values supporting 17L 2s |
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# 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. | # 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.) | # 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 the 13th harmonic is not stable enough for the entire 17L 2s tuning spectrum, | # 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.) | ||
=== Table of odd harmonics for various EDO values supporting 17L 2s === | |||
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) — Equalized 17L 2s}} | |||
{{Harmonics in equal|112|intervals=odd|prec=2|columns=28|title=[[112edo]] (L=6, s=5)}} | |||
{{Harmonics in equal|93|intervals=odd|prec=2|columns=28|title=[[93edo]] (L=5, s=4)}} | |||
{{Harmonics in equal|74|intervals=odd|prec=2|columns=28|title=[[74edo]] (L=4, s=3) — Supersoft 17L 2s}} | |||
{{Harmonics in equal|129|intervals=odd|prec=2|columns=28|title=[[129edo]] (L=7, s=5)}} | |||
{{Harmonics in equal|55|intervals=odd|prec=2|columns=28|title=[[55edo]] (L=3, s=2) — Soft 17L 2s}} | |||
{{Harmonics in equal|146|intervals=odd|prec=2|columns=28|title=[[146edo]] (L=8, s=5)}} | |||
{{Harmonics in equal|91|intervals=odd|prec=2|columns=28|title=[[91edo]] (L=5, s=3) — Semisoft 17L 2s}} | |||
{{Harmonics in equal|127|intervals=odd|prec=2|columns=28|title=[[127edo]] (L=7, s=4)}} | |||
{{Harmonics in equal|36|intervals=odd|prec=2|columns=28|title=[[36edo]] (L=2, s=1) — Basic 17L 2s}} | |||
{{Harmonics in equal|125|intervals=odd|prec=2|columns=28|title=[[125edo]] (L=7, s=3)}} | |||
{{Harmonics in equal|89|intervals=odd|prec=2|columns=28|title=[[89edo]] (L=5, s=2) — Semihard 17L 2s}} | |||
{{Harmonics in equal|142|intervals=odd|prec=2|columns=28|title=[[142edo]] (L=8, s=3)}} | |||
{{Harmonics in equal|53|intervals=odd|prec=2|columns=28|title=[[53edo]] (L=3, s=1) — Hard 17L 2s}} | |||
{{Harmonics in equal|123|intervals=odd|prec=2|columns=28|title=[[123edo]] (L=7, s=2)}} | |||
{{Harmonics in equal|70|intervals=odd|prec=2|columns=28|title=[[70edo]] (L=4, s=1) — Superhard 17L 2s}} | |||
{{Harmonics in equal|87|intervals=odd|prec=2|columns=28|title=[[87edo]] (L=5, s=1)}} | |||
{{Harmonics in equal|104|intervals=odd|prec=2|columns=28|title=[[104edo]] (L=6, s=1)}} | |||
{{Harmonics in equal|17|intervals=odd|prec=2|columns=28|title=[[17edo]] (L=1, s=0) — Collapsed 17L 2s}} | |||
(Need a way to combine the collection of tables into a single table for better readability.) | |||
Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:20, 4 April 2025 (UTC)<br> | Added: [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:20, 4 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]]) 07:42, 8 April 2025 (UTC) | ||