Xenharmonic Wiki

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

← Older editNewer edit →
VisualWikitext
→Musical Mad Science Musings on Diatonicized Sixth-Tone Sub-Chromaticism(?): Add provision for performance checks of ~49/34 and ~75/52, but have still need to enter the values
Line 115: Line 115:
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]].  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 ''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 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.  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.)
#  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).
Line 128: Line 128:


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]]) 07:12, 30 April 2025 (UTC)
Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:02, 3 May 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 ===
Line 134: Line 134:
The following 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]].  For each EDO, it includes a list of plausible candidates for the 17L&nbsp;2s bright generator (BrightGen), with ''candidates failing to map to the bright generator in italics, along with plausible wart fixes''.
The following 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]].  For each EDO, it includes a list of plausible candidates for the 17L&nbsp;2s bright generator (BrightGen), with ''candidates failing to map to the bright generator in italics, along with plausible wart fixes''.


{{Harmonics in equal|19|intervals=odd|prec=2|columns=28|title=[[19edo]] (L=1, s=1, BrightGen is 10; patent ~[[13/9]] = 10; patent ~[[23/16]] = 10; patent ~[[62/43]] = 10; patent ~[[384/265]] = 10; ''patent ~[[59049/40960]] = 9'') &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 ~[[13/9]] = 10; patent ~[[23/16]] = 10; patent ~[[49/34]] = _; patent ~[[62/43]] = 10; patent ~[[75/52]] = _; patent ~[[384/265]] = 10; ''patent ~[[59049/40960]] = 9'') &mdash; Equalized 17L&nbsp;2s}}
{{Harmonics in equal|112|intervals=odd|prec=2|columns=28|title=[[112edo]] (L=6, s=5, BrightGen is 59; ''patent ~13/9 = 58; b val ~13/9 = 60''; patent ~23/16 = 59; patent ~62/43 = 59; ''patent ~384/265 = 60; b val ~384/255 = 61''; ''patent ~59049/40960 = 64; b val ~59049/40960 = 54'')}}
{{Harmonics in equal|112|intervals=odd|prec=2|columns=28|title=[[112edo]] (L=6, s=5, BrightGen is 59; ''patent ~13/9 = 58; b val ~13/9 = 60''; patent ~23/16 = 59; patent ~49/34 = _; patent ~62/43 = 59; patent 75/52 = _; ''patent ~384/265 = 60; b val ~384/255 = 61''; ''patent ~59049/40960 = 64; b val ~59049/40960 = 54'')}}
{{Harmonics in equal|93|intervals=odd|prec=2|columns=28|title=[[93edo]] (L=5, s=4, BrightGen is 49; ''patent ~13/9 = 50''; patent ~23/16 = 49; patent ~62/43 = 49; patent ~384/265 = 49; ''patent ~59049/40960 = 45'')}}
{{Harmonics in equal|93|intervals=odd|prec=2|columns=28|title=[[93edo]] (L=5, s=4, BrightGen is 49; ''patent ~13/9 = 50''; patent ~23/16 = 49; patent ~49/34 = _; patent ~62/43 = 49; patent 75/52 = _; patent ~384/265 = 49; ''patent ~59049/40960 = 45'')}}
{{Harmonics in equal|167|intervals=odd|prec=2|columns=28|title=[[167edo]] (L=9, s=7, BrightGen is 88; patent ~13/9 = 88; ''patent ~23/16 = 87''; patent ~62/43 = 88; ''patent ~384/265 = 89''; ''patent ~59049/40960 = 91'')}}
{{Harmonics in equal|167|intervals=odd|prec=2|columns=28|title=[[167edo]] (L=9, s=7, BrightGen is 88; patent ~13/9 = 88; ''patent ~23/16 = 87''; patent ~49/34 = _; patent ~62/43 = 88; patent 75/52 = _; ''patent ~384/265 = 89''; ''patent ~59049/40960 = 91'')}}
{{Harmonics in equal|74|intervals=odd|prec=2|columns=28|title=[[74edo]] (L=4, s=3, BrightGen is 39; ''patent ~13/9 = 40''; patent ~23/16 = 39; patent ~62/43 = 39; patent ~384/265 = 39; ''patent ~59049/40960 = 36'') &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 ~13/9 = 40''; patent ~23/16 = 39; patent ~49/34 = _; patent ~62/43 = 39; patent 75/52 = _; patent ~384/265 = 39; ''patent ~59049/40960 = 36'') &mdash; Supersoft 17L&nbsp;2s}}
{{Harmonics in equal|203|intervals=odd|prec=2|columns=28|title=[[203edo]] (L=11, s=8, BrightGen is 107; patent ~13/9 = 107; ''patent ~23/16 = 106''; patent ~62/43 = 107; ''patent ~384/265 = 109''; ''patent ~59049/40960 = 110'')}}
{{Harmonics in equal|203|intervals=odd|prec=2|columns=28|title=[[203edo]] (L=11, s=8, BrightGen is 107; patent ~13/9 = 107; ''patent ~23/16 = 106''; patent ~49/34 = _; patent ~62/43 = 107; patent 75/52 = _; ''patent ~384/265 = 109''; ''patent ~59049/40960 = 110'')}}
{{Harmonics in equal|129|intervals=odd|prec=2|columns=28|title=[[129edo]] (L=7, s=5, BrightGen is 68; ''patent ~13/9 = 69''; patent ~23/16 = 68; patent ~62/43 = 68; patent ~384/265 = 68; ''patent ~59049/40960 = 63'')}}
{{Harmonics in equal|129|intervals=odd|prec=2|columns=28|title=[[129edo]] (L=7, s=5, BrightGen is 68; ''patent ~13/9 = 69''; patent ~23/16 = 68; patent ~49/34 = _; patent ~62/43 = 68; patent 75/52 = _; patent ~384/265 = 68; ''patent ~59049/40960 = 63'')}}
{{Harmonics in equal|184|intervals=odd|prec=2|columns=28|title=[[184edo]] (L=10, s=7, BrightGen is 97; patent ~13/9 = 97; ''patent ~23/16 = 96, 'i' wart usable''; patent ~62/43 = 97; ''patent ~384/265 = 99''; ''patent ~59049/40960 = 101'')}}
{{Harmonics in equal|184|intervals=odd|prec=2|columns=28|title=[[184edo]] (L=10, s=7, BrightGen is 97; patent ~13/9 = 97; ''patent ~23/16 = 96, 'i' wart usable''; patent ~49/34 = _; patent ~62/43 = 97; patent 75/52 = _; ''patent ~384/265 = 99''; ''patent ~59049/40960 = 101'')}}
{{Harmonics in equal|55|intervals=odd|prec=2|columns=28|title=[[55edo]] (L=3, s=2, BrightGen is 29; ''patent ~13/9 = 30''; patent ~23/16 =  29; patent ~62/43 = 29; patent ~384/265 = 29; ''patent ~59049/40960 = 27'') &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 ~13/9 = 30''; patent ~23/16 =  29; patent ~49/34 = _; patent ~62/43 = 29; patent 75/52 = _; patent ~384/265 = 29; ''patent ~59049/40960 = 27'') &mdash; Soft 17L&nbsp;2s}}
{{Harmonics in equal|201|intervals=odd|prec=2|columns=28|title=[[201edo]] (L=11, s=7, BrightGen is 106; patent ~13/9 = 106; ''patent ~23/16 = 105''; patent ~62/43 = 106; ''patent ~384/265 = 108''; ''patent ~59049/40960 = 110'')}}
{{Harmonics in equal|201|intervals=odd|prec=2|columns=28|title=[[201edo]] (L=11, s=7, BrightGen is 106; patent ~13/9 = 106; ''patent ~23/16 = 105''; patent ~49/34 = _; patent ~62/43 = 106; patent 75/52 = _; ''patent ~384/265 = 108''; ''patent ~59049/40960 = 110'')}}
{{Harmonics in equal|146|intervals=odd|prec=2|columns=28|title=[[146edo]] (L=8, s=5, BrightGen is 77; ''patent ~13/9 = 78''; ''patent ~23/16 = 76''; patent ~62/43 = 77; ''patent ~384/265 = 78''; ''patent ~59049/40960 = 73'')}}
{{Harmonics in equal|146|intervals=odd|prec=2|columns=28|title=[[146edo]] (L=8, s=5, BrightGen is 77; ''patent ~13/9 = 78''; ''patent ~23/16 = 76''; patent ~49/34 = _; patent ~62/43 = 77; patent 75/52 = _; ''patent ~384/265 = 78''; ''patent ~59049/40960 = 73'')}}
{{Harmonics in equal|237|intervals=odd|prec=2|columns=28|title=[[237edo]] (L=13, s=8, BrightGen is 125; patent ~13/9 = 125; patent ~23/16 = 124; patent ~62/43 = 125; ''patent ~384/265 = 127''; ''patent ~59049/40960 = 129'')}}
{{Harmonics in equal|237|intervals=odd|prec=2|columns=28|title=[[237edo]] (L=13, s=8, BrightGen is 125; patent ~13/9 = 125; patent ~23/16 = 124; patent ~49/34 = _; patent ~62/43 = 125; patent 75/52 = _; ''patent ~384/265 = 127''; ''patent ~59049/40960 = 129'')}}
{{Harmonics in equal|91|intervals=odd|prec=2|columns=28|title=[[91edo]] (L=5, s=3, BrightGen is 48; ''patent ~13/9 = 49''; patent ~23/16 = 48; patent ~62/43 = 48; ''patent ~384/265 = 49''; ''patent ~59049/40960 = 46'') &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 ~13/9 = 49''; patent ~23/16 = 48; patent ~49/34 = _; patent ~62/43 = 48; patent 75/52 = _; ''patent ~384/265 = 49''; ''patent ~59049/40960 = 46'') &mdash; Semisoft 17L&nbsp;2s}}
{{Harmonics in equal|218|intervals=odd|prec=2|columns=28|title=[[218edo]] (L=12, s=7, BrightGen is 115; patent ~13/9 = 115; ''patent ~23/16 = 114''; patent ~62/43 = 115; ''patent ~384/265 = 117''; ''patent ~59049/40960 = 120'')}}
{{Harmonics in equal|218|intervals=odd|prec=2|columns=28|title=[[218edo]] (L=12, s=7, BrightGen is 115; patent ~13/9 = 115; ''patent ~23/16 = 114''; patent ~49/34 = _; patent ~62/43 = 115; patent 75/52 = _; ''patent ~384/265 = 117''; ''patent ~59049/40960 = 120'')}}
{{Harmonics in equal|127|intervals=odd|prec=2|columns=28|title=[[127edo]] (L=7, s=4, BrightGen is 67; ''patent ~13/9 = 68''; ''patent ~23/16 = 66, 'i' wart usable''; patent ~62/43 = 67; ''patent ~384/265 = 68''; ''patent ~59049/40960 = 64'')}}
{{Harmonics in equal|127|intervals=odd|prec=2|columns=28|title=[[127edo]] (L=7, s=4, BrightGen is 67; ''patent ~13/9 = 68''; ''patent ~23/16 = 66, 'i' wart usable''; patent ~49/34 = _; patent ~62/43 = 67; patent 75/52 = _; ''patent ~384/265 = 68''; ''patent ~59049/40960 = 64'')}}
{{Harmonics in equal|163|intervals=odd|prec=2|columns=28|title=[[163edo]] (L=9, s=5, BrightGen is 86; ''patent ~13/9 = 87''; ''patent ~23/16 = 85''; ''patent ~62/43 = 87 'k' wart usable''; ''patent ~384/265 = 87''; ''patent ~59049/40960 = 83'')}}
{{Harmonics in equal|163|intervals=odd|prec=2|columns=28|title=[[163edo]] (L=9, s=5, BrightGen is 86; ''patent ~13/9 = 87''; ''patent ~23/16 = 85''; patent ~49/34 = _; ''patent ~62/43 = 87 'k' wart usable''; patent 75/52 = _; ''patent ~384/265 = 87''; ''patent ~59049/40960 = 83'')}}
{{Harmonics in equal|36|intervals=odd|prec=2|columns=28|title=[[36edo]] (L=2, s=1, BrightGen is 19; patent ~13/9 = 19; patent ~23/16 = 19; patent ~62/43 = 19; patent ~384/265 = 19; ''patent ~59049/40960 = 18'') &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 ~13/9 = 19; patent ~23/16 = 19; patent ~49/34 = _; patent ~62/43 = 19; patent 75/52 = _; patent ~384/265 = 19; ''patent ~59049/40960 = 18'') &mdash; Basic 17L&nbsp;2s}}
{{Harmonics in equal|161|intervals=odd|prec=2|columns=28|title=[[161edo]] (L=9, s=4, BrightGen is 85; ''patent ~13/9 = 86'' ; ''patent ~23/16 = 84''; patent ~62/43 = 85; ''patent ~384/265 = 86''; ''patent ~59049/40960 = 83'')}}
{{Harmonics in equal|161|intervals=odd|prec=2|columns=28|title=[[161edo]] (L=9, s=4, BrightGen is 85; ''patent ~13/9 = 86'' ; ''patent ~23/16 = 84''; patent ~49/34 = _; patent ~62/43 = 85; patent 75/52 = _; ''patent ~384/265 = 86''; ''patent ~59049/40960 = 83'')}}
{{Harmonics in equal|125|intervals=odd|prec=2|columns=28|title=[[125edo]] (L=7, s=3, BrightGen is 66; ''patent ~13/9 = 67, 'f' wart usable''; ''patent ~23/16 = 65''; patent ~62/43 = 66; ''patent ~384/265 = 67''; ''patent ~59049/40960 = 65'')}}
{{Harmonics in equal|125|intervals=odd|prec=2|columns=28|title=[[125edo]] (L=7, s=3, BrightGen is 66; ''patent ~13/9 = 67, 'f' wart usable''; ''patent ~23/16 = 65''; patent ~49/34 = _; patent ~62/43 = 66; patent 75/52 = _; ''patent ~384/265 = 67''; ''patent ~59049/40960 = 65'')}}
{{Harmonics in equal|214|intervals=odd|prec=2|columns=28|title=[[214edo]] (L=12, s=5, BrightGen is 113; ''patent ~13/9 = 114''; ''patent ~23/16 = 112''; patent ~62/43 = 113; ''patent ~384/265 = 114''; ''patent ~59049/40960 = 111'')}}
{{Harmonics in equal|214|intervals=odd|prec=2|columns=28|title=[[214edo]] (L=12, s=5, BrightGen is 113; ''patent ~13/9 = 114''; ''patent ~23/16 = 112''; patent ~49/34 = _; patent ~62/43 = 113; patent 75/52 = _; ''patent ~384/265 = 114''; ''patent ~59049/40960 = 111'')}}
{{Harmonics in equal|89|intervals=odd|prec=2|columns=28|title=[[89edo]] (L=5, s=2, BrightGen is 47; patent ~13/9 = 47; patent ~23/16 = 47; patent ~62/43 = 47; patent ~384/265 = 47; ''patent ~59049/40960 = 46'') &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 ~13/9 = 47; patent ~23/16 = 47; patent ~49/34 = _; patent ~62/43 = 47; patent 75/52 = _; patent ~384/265 = 47; ''patent ~59049/40960 = 46'') &mdash; Semihard 17L&nbsp;2s}}
{{Harmonics in equal|231|intervals=odd|prec=2|columns=28|title=[[231edo]] (L=13, s=5, BrightGen is 122; ''patent ~13/9 = 123''; ''patent ~23/16 = 121''; patent ~62/43 = 122; ''patent ~384/265 = 124''; ''patent ~59049/40960 = 121'')}}
{{Harmonics in equal|231|intervals=odd|prec=2|columns=28|title=[[231edo]] (L=13, s=5, BrightGen is 122; ''patent ~13/9 = 123''; ''patent ~23/16 = 121''; patent ~49/34 = _; patent ~62/43 = 122; patent 75/52 = _; ''patent ~384/265 = 124''; ''patent ~59049/40960 = 121'')}}
{{Harmonics in equal|142|intervals=odd|prec=2|columns=28|title=[[142edo]] (L=8, s=3, BrightGen is 75; patent ~13/9 = 75; ''patent ~23/16 = 74''; ''patent ~62/43 = 74, 'k' wart usable''; ''patent ~384/265 = 76''; ''patent ~59049/40960 = 74'')}}
{{Harmonics in equal|142|intervals=odd|prec=2|columns=28|title=[[142edo]] (L=8, s=3, BrightGen is 75; patent ~13/9 = 75; ''patent ~23/16 = 74''; patent ~49/34 = _; ''patent ~62/43 = 74, 'k' wart usable''; patent 75/52 = _; ''patent ~384/265 = 76''; ''patent ~59049/40960 = 74'')}}
{{Harmonics in equal|195|intervals=odd|prec=2|columns=28|title=[[195edo]] (L=11, s=4, BrightGen is 103; ''patent ~13/9 = 104, 'f' wart usable but requires 'e' wart for 11th harmonic''; ''patent ~23/16 = 102''; patent ~62/43 = 103; ''patent ~384/265 = 104''; ''patent ~59049/40960 = 102'')}}
{{Harmonics in equal|195|intervals=odd|prec=2|columns=28|title=[[195edo]] (L=11, s=4, BrightGen is 103; ''patent ~13/9 = 104, 'f' wart usable but requires 'e' wart for 11th harmonic''; ''patent ~23/16 = 102''; patent ~49/34 = _; patent ~62/43 = 103; patent 75/52 = _; ''patent ~384/265 = 104''; ''patent ~59049/40960 = 102'')}}
{{Harmonics in equal|53|intervals=odd|prec=2|columns=28|title=[[53edo]] (L=3, s=1, BrightGen is 28; patent ~13/9 = 28; patent ~23/16 = 28; patent ~62/43 = 28; patent ~384/265 = 28; patent ~59049/40960 = 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 ~13/9 = 28; patent ~23/16 = 28; patent ~49/34 = _; patent ~62/43 = 28; patent 75/52 = _; patent ~384/265 = 28; patent ~59049/40960 = 28) &mdash; Hard 17L&nbsp;2s}}
{{Harmonics in equal|176|intervals=odd|prec=2|columns=28|title=[[176edo]] (L=10, s=3, BrightGen is 93; patent ~13/9 = 93; ''patent ~23/16 = 92''; patent ~62/43 = 93; ''patent ~384/265 = 94''; patent ~59049/40960 = 93)}}
{{Harmonics in equal|176|intervals=odd|prec=2|columns=28|title=[[176edo]] (L=10, s=3, BrightGen is 93; patent ~13/9 = 93; ''patent ~23/16 = 92''; patent ~49/34 = _; patent ~62/43 = 93; patent 75/52 = _; ''patent ~384/265 = 94''; patent ~59049/40960 = 93)}}
{{Harmonics in equal|123|intervals=odd|prec=2|columns=28|title=[[123edo]] (L=7, s=2, BrightGen is 65; patent ~13/9 = 65; ''patent ~23/16 = 64, 'i' wart is usable''; patent ~62/43 = 65; patent ~384/265 = 65; patent ~59049/40960 = 65)}}
{{Harmonics in equal|123|intervals=odd|prec=2|columns=28|title=[[123edo]] (L=7, s=2, BrightGen is 65; patent ~13/9 = 65; ''patent ~23/16 = 64, 'i' wart is usable''; patent ~49/34 = _; patent ~62/43 = 65; patent 75/52 = _; patent ~384/265 = 65; patent ~59049/40960 = 65)}}
{{Harmonics in equal|193|intervals=odd|prec=2|columns=28|title=[[193edo]] (L=11, s=3, BrightGen is 102; patent ~13/9 = 102; ''patent ~23/16 = 101''; patent ~62/43 = 102; ''patent ~384/265 = 104''; ''patent ~59049/40960 = 103'')}}
{{Harmonics in equal|193|intervals=odd|prec=2|columns=28|title=[[193edo]] (L=11, s=3, BrightGen is 102; patent ~13/9 = 102; ''patent ~23/16 = 101''; patent ~49/34 = _; patent ~62/43 = 102; patent 75/52 = _; ''patent ~384/265 = 104''; ''patent ~59049/40960 = 103'')}}
{{Harmonics in equal|70|intervals=odd|prec=2|columns=28|title=[[70edo]] (L=4, s=1, BrightGen is 37; patent ~13/9 = 37; patent ~23/16 = 37; patent ~62/43 = 37; patent ~384/265 = 37; patent ~59049/40960 = 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 ~13/9 = 37; patent ~23/16 = 37; patent ~49/34 = _; patent ~62/43 = 37; patent 75/52 = _; patent ~384/265 = 37; patent ~59049/40960 = 37) &mdash; Superhard 17L&nbsp;2s}}
{{Harmonics in equal|157|intervals=odd|prec=2|columns=28|title=[[157edo]] (L=9, s=2, BrightGen is 83; patent ~13/9 = 83; ''patent ~23/16 = 82''; patent ~62/43 = 83; ''patent ~384/265 = 84''; ''patent ~59049/40960 = 84'')}}
{{Harmonics in equal|157|intervals=odd|prec=2|columns=28|title=[[157edo]] (L=9, s=2, BrightGen is 83; patent ~13/9 = 83; ''patent ~23/16 = 82''; patent ~49/34 = _; patent ~62/43 = 83; patent 75/52 = _; ''patent ~384/265 = 84''; ''patent ~59049/40960 = 84'')}}
{{Harmonics in equal|87|intervals=odd|prec=2|columns=28|title=[[87edo]] (L=5, s=1, BrightGen is 46; patent ~13/9 = 46 ; patent ~23/16 = 46; patent ~62/43 = 46; ''patent ~384/265 = 47''; ''patent ~59049/40960 = 47'')}}
{{Harmonics in equal|87|intervals=odd|prec=2|columns=28|title=[[87edo]] (L=5, s=1, BrightGen is 46; patent ~13/9 = 46 ; patent ~23/16 = 46; patent ~49/34 = _; patent ~62/43 = 46; patent 75/52 = _; ''patent ~384/265 = 47''; ''patent ~59049/40960 = 47'')}}
{{Harmonics in equal|104|intervals=odd|prec=2|columns=28|title=[[104edo]] (L=6, s=1, BrightGen is 55; patent ~13/9 = 55; ''patent ~23/16 = 54''; patent ~62/43 = 55; ''patent ~384/265 = 56''; ''patent ~59049/40960 = 57'')}}
{{Harmonics in equal|104|intervals=odd|prec=2|columns=28|title=[[104edo]] (L=6, s=1, BrightGen is 55; patent ~13/9 = 55; ''patent ~23/16 = 54''; patent ~49/34 = _; patent ~62/43 = 55; patent 75/52 = _; ''patent ~384/265 = 56''; ''patent ~59049/40960 = 57'')}}
{{Harmonics in equal|17|intervals=odd|prec=2|columns=28|title=[[17edo]] (L=1, s=0, BrightGen is 9; patent ~13/9 = 9; patent ~23/16 = 9; patent ~62/43 = 9; ''patent ~384/265 = 10''; c val ~384/265 = 9; ''patent ~59049/40960 = 10''; c val ~59049/40960 = 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 ~13/9 = 9; patent ~23/16 = 9; patent ~49/34 = _; patent ~62/43 = 9; patent 75/52 = _; ''patent ~384/265 = 10''; c val ~384/265 = 9; ''patent ~59049/40960 = 10''; c val ~59049/40960 = 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.)
Line 173: Line 173:


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]]) 07:12, 30 April 2025 (UTC)
Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 09:02, 3 May 2025 (UTC)
Retrieved from "https://en.xen.wiki/w/User:Lucius_Chiaraviglio/Musical_Mad_Science"