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

#  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.)

#  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, 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 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 — 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 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 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.

The following 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]] (but this needs to be updated for addition of another column on 2025-05-12 — will do this after filling in candidate generator mappings for EDOs already in this table).  For each EDO, it includes a list of plausible candidates for the 17L 2s bright generator (BrightGen), with ''candidates failing to map to the bright generator in italics, along with plausible wart fixes (if any)''.

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 — 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 2s tuning spectrum, plus 17c

Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 21:04, 14 June 2025 (UTC)