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 scale tree, 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 scale tree 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.)

#  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 scale tree, although it gets close in the hard half of the scale tree (closer to just 13/9, including having the best 3rd harmonic within the scale tree).  Need to split the scale tree 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 scale tree.  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 scale tree, 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 scale tree 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 scale tree.  Maybe the 7th and 17th harmonics are stable enough for the middle of the 17L 2s scale tree?)

#  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 scale tree (leaving out the far right column of the scale tree 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 scale tree:  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 scale tree:  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.

Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 08:20, 4 April 2025 (UTC)<br>

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

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

Added:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 07:42, 8 April 2025 (UTC)<br>