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→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): A plausible alternative to the generator spectrum (still needs to be vetted)
→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): More consideration of 62/43 as the bright generator; add patent val mappings of ~62/43
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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.  (But see later parts of this analysis, in which it is actually necessary to assign the 2.3.5... subgroup mapping first, at least for the hard half of the tuning spectrum.)
#  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.  (But see later parts of this analysis, in which it is actually necessary to assign the 2.3.5... subgroup mapping first, at least for the hard half of the tuning spectrum.)
#  It is noteworthy 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, except for needing a wart at 112b), 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 in the hard half of the spectrum, but is fairly stable in the soft half (although warts are needed for a few of the larger EDOs).
#  It is noteworthy 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, except for needing a wart at 112b), 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 in the hard half of the spectrum, but is fairly stable in the soft half (although warts are needed for a few of the larger EDOs).
#  Tried assigning the generator as 23/16 ~ 13/9, tempering out [[208/207]] (the vicetone comma).  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).  Need to split the tuning spectrum of 17L 2s into 2 or more temperaments.  Still have not found the right mix of harmonics for the hard half (36edo and harder), but for the soft half, the 5th and 53rd harmonics are stable enough to team up with the 3rd and 23rd harmonics to get a usable generator.  Nobody in their right mind is going to want to actually use the 53rd harmonic for constructing intervals, but for tuning the generator, it will have to do.  The bright generator (basic 19\36, spectrum from 10\19 soft to 9\17 hard) is therefore constituted as [[23/16]] (|-4 0 0 0 0 0 0 0 1⟩, 628.274347{{c}}) ~ [[384/265]] (|7 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1⟩, 642.1367415{{c}}), tempered together using the unnamed comma [[6144/6095]] (|11 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1⟩, 13.8623942563{{c}}); this comma is in fact tempered out in most of the EDOs on the soft half of the tuning spectrum, plus 17edo constituted as the often-used (and barely further from just) 17c val.  It follows that the dark generator is constituted as [[32/23]] (|5 0 0 0 0 0 0 0 -1⟩, 571.725653{{c}}) ~ [[265/192]] (|-6 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 1⟩, 557.863258{{c}}), tempered together using the same comma.  A plausible alternative to the generator spectrum is ~[[62/43]] (but this needs to be vetted); on the one hand, this would have the advantage of simplifying the description (if not the mathematics) of extensions; on the other hand, it has the disadvantage of not only replacing one moderately high harmonic having fairly stable mapping and one extremely high harmonic having rock-solid stable mapping with 2 very high harmonics having very unstable mapping, so the subgroup would have to map this ratio by direct approximation, as in 2.3...62/43.  (Coming in the future:  More work on the hard half of the tuning spectrum.  Maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s tuning spectrum?)
#  Tried assigning the generator as 23/16 ~ 13/9, tempering out [[208/207]] (the vicetone comma).  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).  Need to split the tuning spectrum of 17L 2s into 2 or more temperaments.  Still have not found the right mix of harmonics for the hard half (36edo and harder), but for the soft half, the 5th and 53rd harmonics are stable enough to team up with the 3rd and 23rd harmonics to get a usable generator.  Nobody in their right mind is going to want to actually use the 53rd harmonic for constructing intervals, but for tuning the generator, it will have to do.  The bright generator (basic 19\36, spectrum from 10\19 soft to 9\17 hard) is therefore constituted as [[23/16]] (|-4 0 0 0 0 0 0 0 1⟩, 628.274347{{c}}) ~ [[384/265]] (|7 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1⟩, 642.1367415{{c}}), tempered together using the unnamed comma [[6144/6095]] (|11 1 -1 0 0 0 0 0 -1 0 0 0 0 0 0 -1⟩, 13.8623942563{{c}}); this comma is in fact tempered out in most of the EDOs on the soft half of the tuning spectrum, plus 17edo constituted as the often-used (and barely further from just) 17c val.  It follows that the dark generator is constituted as [[32/23]] (|5 0 0 0 0 0 0 0 -1⟩, 571.725653{{c}}) ~ [[265/192]] (|-6 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 1⟩, 557.863258{{c}}), tempered together using the same comma.  A plausible alternative to the generator spectrum is ~[[62/43]] ; on the one hand, this would have the advantage of simplifying the description (if not the mathematics) of extensions; on the other hand, it has the disadvantage of replacing one moderately high harmonic having fairly stable mapping and one extremely high harmonic having rock-solid stable mapping with 2 very high harmonics having very unstable mappings, so the subgroup would have to map this ratio by direct approximation, as in 2.3...43/31 (direct-approximated 43/31 being the dark generator), or it would be necessary to add 'k' and 'n' warts as needed to achieve the same effect (amazingly, these harmonics covary sofficiently well that a wart is needed for only one EDO in the whole set:  142k — need to rework this section to take advantage of this near-rock-solid mapping stability of the ratio despite instability of the mappings of the individual harmonics, and should do the corresponding due diligence for ~13/9).  (Coming in the future:  More work on the hard half of the tuning spectrum.  Maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s tuning spectrum?)
#  To get ~[[3/1]], we stack 3 bright generators (and then octave reduction gets us ~[[3/2]]), putting this temperament on the alpha-tricot part of the [[ploidacot]] system (not to be confused with the [[Alphatricot]] temperament, and not to be confused with [[Mothra]], which is tricot without the alpha, which only works for the subset of the following EDOs that also have octaves divisible by 3).  The comma for this is [[12288/12167]] (|12 1 0 0 0 0 0 0 -3⟩, 17.13195906{{c}}) in the 23-limit, and [[18874368/18609625]] (|21 2 -3 0 0 0 0 0 0 0 0 0 0 0 0 -3⟩, 24.45522370876{{c}}) in the 53-limit.  Except as noted with warts, this works for the patent vals of all of the smaller and mid-size EDOs in the 17L 2s tuning spectrum (leaving out the far right column of the tuning spectrum table apart from its top and bottom ends, since the EDO sizes in between the top and bottom ends get very large).  As noted above, things are different between the soft half (up to and including 36edo) and the hard half (36edo onwards) later on, so even though they work the same way for the 3rd harmonic, I split the EDO list in half, although oddly enough 17c works as if it was on the soft half despite being at the hard end (collapsed 17L 2s). EDO list for soft half of tuning spectrum:  19, 36, 55, 74, 91c, 93, 112b, 127cci, 129, 146i (and note that 112b just barely misses being the patent val of the dual-fifth 112edo). EDO list for hard half of tuning spectrum:  17, 36, 53, 70, 87, 89, 104, 123, 125, 142 (this list is manually generated — [http://x31eq.com/temper/ Graham Breed's x31eq temperament finder] has trouble with the high prime limit of the subgroup, and only finds part of the spectrum, and shows some other EDOs in addition).  Note that some of the commas in each comma spectrum listed below (corresponding to the generator spectrum above) have negative just intonation values, because each spectrum crosses through 0.
#  To get ~[[3/1]], we stack 3 bright generators (and then octave reduction gets us ~[[3/2]]), putting this temperament on the alpha-tricot part of the [[ploidacot]] system (not to be confused with the [[Alphatricot]] temperament, and not to be confused with [[Mothra]], which is tricot without the alpha, which only works for the subset of the following EDOs that also have octaves divisible by 3).  The comma for this is [[12288/12167]] (|12 1 0 0 0 0 0 0 -3⟩, 17.13195906{{c}}) in the 23-limit, and [[18874368/18609625]] (|21 2 -3 0 0 0 0 0 0 0 0 0 0 0 0 -3⟩, 24.45522370876{{c}}) in the 53-limit.  Except as noted with warts, this works for the patent vals of all of the smaller and mid-size EDOs in the 17L 2s tuning spectrum (leaving out the far right column of the tuning spectrum table apart from its top and bottom ends, since the EDO sizes in between the top and bottom ends get very large).  As noted above, things are different between the soft half (up to and including 36edo) and the hard half (36edo onwards) later on, so even though they work the same way for the 3rd harmonic, I split the EDO list in half, although oddly enough 17c works as if it was on the soft half despite being at the hard end (collapsed 17L 2s). EDO list for soft half of tuning spectrum:  19, 36, 55, 74, 91c, 93, 112b, 127cci, 129, 146i (and note that 112b just barely misses being the patent val of the dual-fifth 112edo). EDO list for hard half of tuning spectrum:  17, 36, 53, 70, 87, 89, 104, 123, 125, 142 (this list is manually generated — [http://x31eq.com/temper/ Graham Breed's x31eq temperament finder] has trouble with the high prime limit of the subgroup, and only finds part of the spectrum, and shows some other EDOs in addition).  Note that some of the commas in each comma spectrum listed below (corresponding to the generator spectrum above) have negative just intonation values, because each spectrum crosses through 0.
#  Dividing the EDO list into hard and soft halves is helpful for looking at these EDOs as 5-limit temperaments.  For the soft half, except as noted with warts, these are all [[Meantone]] temperaments, or conntortions under [[Meantone]] temperaments:  19, 36 (contorted under 12), 55, 74, 91c, 93 (contorted under 31), 112b, 127cc, 129 (contorted under 43), 146c.  Without the warts, 91edo and 127edo fall on [[Schismic–Pythagorean_equivalence_continuum#Python|Python]] (currently still named Lalagu by Graham Breed's x31eq temperament finder), for which 16 fifths (octave-reduced) are needed to reach [[5/4]]; while 146edo falls on the currently unnamed diploid temperament that flattens the fifth by (optimally) close to 1/10 of |-27 20 -2⟩ to get ~5/4.  For the hard half, the picture is more complicated (and the generator constitution is not even currently set), so we'll leave that for later.
#  Dividing the EDO list into hard and soft halves is helpful for looking at these EDOs as 5-limit temperaments.  For the soft half, except as noted with warts, these are all [[Meantone]] temperaments, or conntortions under [[Meantone]] temperaments:  19, 36 (contorted under 12), 55, 74, 91c, 93 (contorted under 31), 112b, 127cc, 129 (contorted under 43), 146c.  Without the warts, 91edo and 127edo fall on [[Schismic–Pythagorean_equivalence_continuum#Python|Python]] (currently still named Lalagu by Graham Breed's x31eq temperament finder), for which 16 fifths (octave-reduced) are needed to reach [[5/4]]; while 146edo falls on the currently unnamed diploid temperament that flattens the fifth by (optimally) close to 1/10 of |-27 20 -2⟩ to get ~5/4.  For the hard half, the picture is more complicated (and the generator constitution is not even currently set), so we'll leave that for later.
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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]]) 09:41, 23 April 2025 (UTC)
Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:10, 25 April 2025 (UTC)


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


This table (actually a collection of tables for now) is for tracking trends in odd harmonics along the tuning spectrum of [[17L&nbsp;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:
This table (actually a collection of tables for now) is for tracking trends in odd harmonics along the tuning spectrum of [[17L&nbsp;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) &mdash; Equalized 17L&nbsp;2s}}
{{Harmonics in equal|19|intervals=odd|prec=2|columns=28|title=[[19edo]] (L=1, s=1, BrightGen is 10; patent ~62/41 = 10) &mdash; Equalized 17L&nbsp;2s}}
{{Harmonics in equal|112|intervals=odd|prec=2|columns=28|title=[[112edo]] (L=6, s=5, BrightGen is 59)}}
{{Harmonics in equal|112|intervals=odd|prec=2|columns=28|title=[[112edo]] (L=6, s=5, BrightGen is 59; patent ~62/41 = 59)}}
{{Harmonics in equal|93|intervals=odd|prec=2|columns=28|title=[[93edo]] (L=5, s=4, BrightGen is 49)}}
{{Harmonics in equal|93|intervals=odd|prec=2|columns=28|title=[[93edo]] (L=5, s=4, BrightGen is 49; patent ~62/41 = 49)}}
{{Harmonics in equal|74|intervals=odd|prec=2|columns=28|title=[[74edo]] (L=4, s=3, BrightGen is 39) &mdash; Supersoft 17L&nbsp;2s}}
{{Harmonics in equal|74|intervals=odd|prec=2|columns=28|title=[[74edo]] (L=4, s=3, BrightGen is 39; patent ~62/41 = 39) &mdash; Supersoft 17L&nbsp;2s}}
{{Harmonics in equal|129|intervals=odd|prec=2|columns=28|title=[[129edo]] (L=7, s=5, BrightGen is 68)}}
{{Harmonics in equal|129|intervals=odd|prec=2|columns=28|title=[[129edo]] (L=7, s=5, BrightGen is 68; patent ~62/41 = 68)}}
{{Harmonics in equal|55|intervals=odd|prec=2|columns=28|title=[[55edo]] (L=3, s=2, BrightGen is 29) &mdash; Soft 17L&nbsp;2s}}
{{Harmonics in equal|55|intervals=odd|prec=2|columns=28|title=[[55edo]] (L=3, s=2, BrightGen is 29; patent ~62/41 = 29) &mdash; Soft 17L&nbsp;2s}}
{{Harmonics in equal|146|intervals=odd|prec=2|columns=28|title=[[146edo]] (L=8, s=5, BrightGen is 77)}}
{{Harmonics in equal|146|intervals=odd|prec=2|columns=28|title=[[146edo]] (L=8, s=5, BrightGen is 77; patent ~62/41 = 77)}}
{{Harmonics in equal|91|intervals=odd|prec=2|columns=28|title=[[91edo]] (L=5, s=3, BrightGen is 48) &mdash; Semisoft 17L&nbsp;2s}}
{{Harmonics in equal|91|intervals=odd|prec=2|columns=28|title=[[91edo]] (L=5, s=3, BrightGen is 48; patent ~62/41 = 48) &mdash; Semisoft 17L&nbsp;2s}}
{{Harmonics in equal|127|intervals=odd|prec=2|columns=28|title=[[127edo]] (L=7, s=4, BrightGen is 67)}}
{{Harmonics in equal|127|intervals=odd|prec=2|columns=28|title=[[127edo]] (L=7, s=4, BrightGen is 67; patent ~62/41 = 67)}}
{{Harmonics in equal|36|intervals=odd|prec=2|columns=28|title=[[36edo]] (L=2, s=1, BrightGen is 19) &mdash; Basic 17L&nbsp;2s}}
{{Harmonics in equal|36|intervals=odd|prec=2|columns=28|title=[[36edo]] (L=2, s=1, BrightGen is 19; patent ~62/41 = 19) &mdash; Basic 17L&nbsp;2s}}
{{Harmonics in equal|125|intervals=odd|prec=2|columns=28|title=[[125edo]] (L=7, s=3, BrightGen is 66)}}
{{Harmonics in equal|125|intervals=odd|prec=2|columns=28|title=[[125edo]] (L=7, s=3, BrightGen is 66; patent ~62/41 = 66)}}
{{Harmonics in equal|89|intervals=odd|prec=2|columns=28|title=[[89edo]] (L=5, s=2, BrightGen is 47) &mdash; Semihard 17L&nbsp;2s}}
{{Harmonics in equal|89|intervals=odd|prec=2|columns=28|title=[[89edo]] (L=5, s=2, BrightGen is 47; patent ~62/41 = 47) &mdash; Semihard 17L&nbsp;2s}}
{{Harmonics in equal|142|intervals=odd|prec=2|columns=28|title=[[142edo]] (L=8, s=3, BrightGen is 75)}}
{{Harmonics in equal|142|intervals=odd|prec=2|columns=28|title=[[142edo]] (L=8, s=3, BrightGen is 75; patent ~62/41 = 74, indicating need for 'k' wart)}}
{{Harmonics in equal|53|intervals=odd|prec=2|columns=28|title=[[53edo]] (L=3, s=1, BrightGen is 28) &mdash; Hard 17L&nbsp;2s}}
{{Harmonics in equal|53|intervals=odd|prec=2|columns=28|title=[[53edo]] (L=3, s=1, BrightGen is 28; patent ~62/41 = 28) &mdash; Hard 17L&nbsp;2s}}
{{Harmonics in equal|123|intervals=odd|prec=2|columns=28|title=[[123edo]] (L=7, s=2, BrightGen is 65)}}
{{Harmonics in equal|123|intervals=odd|prec=2|columns=28|title=[[123edo]] (L=7, s=2, BrightGen is 65; patent ~62/41 = 65)}}
{{Harmonics in equal|70|intervals=odd|prec=2|columns=28|title=[[70edo]] (L=4, s=1, BrightGen is 37) &mdash; Superhard 17L&nbsp;2s}}
{{Harmonics in equal|70|intervals=odd|prec=2|columns=28|title=[[70edo]] (L=4, s=1, BrightGen is 37; patent ~62/41 = 37) &mdash; Superhard 17L&nbsp;2s}}
{{Harmonics in equal|87|intervals=odd|prec=2|columns=28|title=[[87edo]] (L=5, s=1, BrightGen is 46)}}
{{Harmonics in equal|87|intervals=odd|prec=2|columns=28|title=[[87edo]] (L=5, s=1, BrightGen is 46; patent ~62/41 = 46)}}
{{Harmonics in equal|104|intervals=odd|prec=2|columns=28|title=[[104edo]] (L=6, s=1, BrightGen is 55)}}
{{Harmonics in equal|104|intervals=odd|prec=2|columns=28|title=[[104edo]] (L=6, s=1, BrightGen is 55; patent ~62/41 = 55)}}
{{Harmonics in equal|17|intervals=odd|prec=2|columns=28|title=[[17edo]] (L=1, s=0, BrightGen is 9) &mdash; Collapsed 17L&nbsp;2s}}
{{Harmonics in equal|17|intervals=odd|prec=2|columns=28|title=[[17edo]] (L=1, s=0, BrightGen is 9; patent ~62/41 = 9) &mdash; Collapsed 17L&nbsp;2s}}


(Need a way to combine the collection of tables into a single table for better readability.)
(Need a way to combine the collection of tables into a single table for better readability.)
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