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

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 #  It is noteworthy (more detail needed) 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, 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&nbsp;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 with 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.  (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). 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).
+#  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).
 #  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.
 # 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 is manually generated &mdash; see above about x31eq having trouble with the high prime limit of the subgroup.)
 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]]) 10:02, 22 April 2025 (UTC)
+Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 20:56, 22 April 2025 (UTC)
 === Table of odd harmonics for various EDO values supporting 17L&nbsp;2s ===
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 (Need a way to combine the collection of tables into a single table for better readability.)
-In preliminary 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 presumably its mapping would break if I put in the rest of the right-most column of the MOS spectrum table.  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 meantone extension; it goes 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 just barely misses being rock-solid in this zone (just barely breaks on 125edo, for which 125f would be not bad); still need to check what the 13th harmonic does in the soft half of the spectrum.  Although those harmonics would also appear less solid if I included the rest of the MOS tuning spectrum.
+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 would break if I put in the rest of the right-most column of the MOS spectrum table.  And the mapping of the 53rd harmonic is similarly rock-solid.  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 just barely misses being rock-solid in this zone (just barely breaks on 125edo, for which 125f would be not bad); still need to check what the 13th harmonic does in the soft half of the spectrum.  Although those harmonics would also appear less solid if I included the rest of the MOS tuning spectrum.
 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]]) 10:03, 21 April 2025 (UTC)
+Last modified:  [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 20:56, 22 April 2025 (UTC)