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

← Older edit Newer edit →

VisualWikitext

@@ Line 115: / Line 115: @@
 The [[36edo]] equivalent of Diatonicized Chromaticism is [[17L&nbsp;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&nbsp;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&nbsp;2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range &mdash; you have to have a ''bad'' fifth to land outside of its range.  The range of [[7L&nbsp;2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L&nbsp;2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L&nbsp;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&nbsp;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&nbsp;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&nbsp;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.  (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 &mdash; 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&nbsp;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&nbsp;2s]] (as in [[Meantone]], [[Superpyth]], and their relatives) is very wide range &mdash; you have to have a ''bad'' fifth to land outside of its range.  The range of [[7L&nbsp;2s]] is still fairly wide, going from barely over [[52/35]] down to somewhat under [[81/55]]; [[9L&nbsp;2s]] is narrower, going from barely over [[25/17]] down to somewhat under [[19/13]]; [[11L&nbsp;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&nbsp;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&nbsp;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&nbsp;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&nbsp;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 &mdash; 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&nbsp;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).
 #  Tried assigning the generator as 23/16 ~ 13/9, tempering out [[208/207]] (the vicetone comma).  But the problem is that &mdash; as can be seen in the table of harmonics below &mdash; the 13th harmonic is not stable enough for the entire 17L&nbsp;2s tuning spectrum, although it gets close in 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&nbsp;2s into 2 or more temperaments.  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, except not in the right-most column of the tuning spectrum table.  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 (however, see below for the meantone subset of the soft half, for which the 23rd harmonic is actually stable enough).  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.  (In a rigorous test of the mapping of ~384/265, it did not perform as well as expected, missing the boat for the entire right-most column of the tuning spectrum table and in some other parts of the table, with no opportunities anywhere in the table to fix the mapping by adding a wart other than 17c.  Need to rewrite this section to stop depending upon 384/265, since it only works for the very complicated tuned generator.)  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 &mdash; 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&nbsp;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&nbsp;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&nbsp;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 &mdash; [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 family]] temperaments, 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&nbsp;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&nbsp;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 &mdash; [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.  Once the temperaments other than 112b that have warts have been removed, the remaining temperaments have a 23rd harmonic with stable enough mapping to use as the bright generator, even though it always maps sharp and is often inconsistent with nearby (particularly lower) harmonics.  For the hard half (to be dealt with later), ~13/9 appears to be usable as the generator, only needing a wart in one instance:  125f (which just barely misses being the patent val, while the wart improves consistency with nearby harmonics).
 # For the Meantone set, since 4 fifths get the ~5/4, and 3 (unnamed-53-limit-comma-tempered) bright generators get the fifth, this means that 12 of these bright generators get the ~5/4.  (Originally this was going to be a 2.3.5.13.23 subgroup extension of Meantone, with the Vicetone comma already having a fitting name, leading "Vicetone" as the name for this extension; but the 13th harmonic mapping just wasn't stable enough.  Too bad.  For now, going to have to go with something weird like Fitho-vicesimotertial for the comma and temperament name.)  Dropping EDOs for which warts greatly degrade accuracy (91c, 127cci, and 142ci), the optimal ET sequence is:  17c, 19, 36, 55, 74, 93, 112b, 129 (this list is manually generated &mdash; see above about x31eq having trouble with the high prime limit of the subgroup.)  The comma for this is made from the [[81/80|syntonic comma]] by substituting each instance of ~3/2 with an octave-reduced stack of 3 of our bright generator, which produces a spectrum of commas from |52 0 1 0 0 0 0 0 -12⟩ ~ |-80 -12 13 0 0 0 0 0 0 0 0 0 0 0 0 12⟩, of which |19 -3 4 0 0 0 0 0 -9 0 0 0 0 0 0 3⟩ (made by substituting 9 instances of (3/2)<sup>(1/3)</sup> by 23/16 and the other 3 instances of (3/2)<sup>(1/3)</sup> by 384/265) has the closest 53-limit just intonation value to 0 (5.4343638749{{c}}).  Naturally, 81/80 itself is also tempered out.
@@ Line 128: / Line 128: @@
 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]]) 03:50, 28 April 2025 (UTC)
+Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:12, 30 April 2025 (UTC)
 === Table of odd harmonics for various EDO values supporting 17L&nbsp;2s ===
 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) &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'')}}
+{{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|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)}}
+{{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|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'')}}
+{{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|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) &mdash; Supersoft 17L&nbsp;2s}}
+{{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|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'')}}
+{{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|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)}}
+{{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|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'')}}
+{{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|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) &mdash; Soft 17L&nbsp;2s}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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'') &mdash; Semisoft 17L&nbsp;2s}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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) &mdash; Basic 17L&nbsp;2s}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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) &mdash; Semihard 17L&nbsp;2s}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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) &mdash; Hard 17L&nbsp;2s}}
+{{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|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'')}}
+{{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|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)}}
+{{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|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'')}}
+{{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|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) &mdash; Superhard 17L&nbsp;2s}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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'')}}
+{{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|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) &mdash; Collapsed 17L&nbsp;2s}}
+{{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|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}}
 (Need a way to combine the collection of tables into a single table for better readability.)
-In detailed (but not complete) observations of scrolling through the above table group, I started noticing interesting things, like how the harmonic/subharmonics of the generator have unstable mapping (because no simple ratio with a reasonable sized numerator and denominator fits into this zone), but the 3rd harmonic is nearly rock-solid (and 112b is a respectable if overly-complex quarter-comma meantone approximation), although its mapping causes strange effects in the right-most column of the MOS spectrum table, causing ~13/9 to map to the bright generator in several EDOs in the soft half of the tuning spectrum and to map 1 step sharp of the bright generator in several EDOs in the hard half of the spectrum.  And the mapping of the 53rd harmonic is reasonably solid as long as the right-most column of the tuning spectrum table is entirely left out; ~384/265 does not do as well as expected as a generator.  And there the 5th harmonic seems very much usable in the soft end of the scale tuning spectrum as long as the EDO sizes don't get too large (and even then, sometimes it is still okay), which looks to me like enabling a 2.3.5.23.53 meantone extension.  The 5th and 53rd harmonics go all over the place in the hard end, but there the 25th harmonic shines and is rock-solid as long as you don't go softer than 36edo (basic), and the 13th harmonic jis fairly rock-solid in this zone (and some of the exceptions are candidates for rescue by applying an 'f' wart); in the soft half of the spectrum, the 13th harmonic always maps to 1 step too many for ~13/9 to be usable as the bright generator, and an 'f' wart would worsen consistency with nearby harmonics, except strangely in most of the right-most column of the tuning spectrum table.  Finally, the ratio ~62/43 (bright generator) or ~43/31 (dark generator) has amazing stability &mdash; even though the mapping of its component harmonics is pretty unstable, they co-vary sufficiently well that a wart is needed for proper mapping in only a few places, and in each case the wart does not appear to hurt consistency, at least at a brief inspection.
+In detailed observations of scrolling through the above table group, I started noticing interesting things, like how the harmonic/subharmonics of the generator have unstable mapping (because no simple ratio with a reasonable sized numerator and denominator fits into this zone), but the 3rd harmonic is nearly rock-solid (and 112b is a respectable if overly-complex quarter-comma meantone approximation), although its mapping causes strange effects in the right-most column of the MOS spectrum table, causing ~13/9 to map to the bright generator in several EDOs in the soft half of the tuning spectrum and to map 1 step sharp of the bright generator in several EDOs in the hard half of the spectrum.  And the mapping of the 53rd harmonic is reasonably solid as long as the right-most column of the tuning spectrum table is entirely left out; ~384/265 does not do as well as expected as a generator.  And there the 5th harmonic seems very much usable in the soft end of the scale tuning spectrum as long as the EDO sizes don't get too large (and even then, sometimes it is still okay), which looks to me like enabling a 2.3.5.23.53 meantone extension.  The 5th and 53rd harmonics go all over the place in the hard end, but there the 25th harmonic shines and is rock-solid as long as you don't go softer than 36edo (basic), and the 13th harmonic jis fairly rock-solid in this zone (and some of the exceptions are candidates for rescue by applying an 'f' wart); in the soft half of the spectrum, the 13th harmonic always maps to 1 step too many for ~13/9 to be usable as the bright generator, and an 'f' wart would worsen consistency with nearby harmonics, except strangely in most of the right-most column of the tuning spectrum table.  Finally, the ratio ~62/43 (bright generator) or ~43/31 (dark generator) has amazing stability &mdash; even though the mapping of its component harmonics is pretty unstable, they co-vary sufficiently well that a wart is needed for proper mapping in only a few places, and in each case the wart does not appear to hurt consistency, at least at a brief inspection.  The generator ~[[59049/40960]] = ~|-13 10 -1⟩ of the established [[Alphatricot family]] only works for a narrow band in the hard to super-hard region of the 17L&nbsp;2s tuning spectrum, plus 17c
 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]]) 03:50, 28 April 2025 (UTC)
+Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:12, 30 April 2025 (UTC)