Talk:11L 2s
Musical Mad Science Musings on Diatonicized Chromaticism
These ramblings are copied from my comments (minus the aggravation of YouTube eating posts, and plus Xenharmonic Wikification of links, including sadly decapitalizing EDO to get said links to work) on Claudi Meneghin's YouTube instance of his arrangement of John Dowland's Pavana «Lachrimae» (from «Seven Tears») in 50edo, the hearing of which led me to realize that 50edo can actually support Ivan Wyschnegradsky's Diatonicized Chromaticism (11L 2s) scale.
My YouTube comments start here
I just had a crazy idea for your next musical mad science experiment (and it potentially includes 50edo): See if it is possible to retune some of the quarter-tone (24edo, "diatonicized chromatic")11L 2s (L/s = 2) scale works of Ivan Wyschnegradsky into other tuning systems that support 11L 2s and have a good approximation and single circle of 11/8 (or 16/11). Plausible candidate tuning systems on the soft side are 37edo (L/s = 3/2, and has a super-good 11/8), 61edo (L/s = 5/3, but 61edo is big enough to be pushing the limits of plausibility), and 50edo (L/s = 4/3 -- might be too soft). Plausible candidate tuning systems on the hard side are 35edo (L/s = 3), 59edo (L/s = 7/3, but 59edo is big enough to be pushing the limits of plausibility), and 46edo (L/s = 4/1 -- might be too hard).
Most of Ivan Wyschnegradsky's quarter-tone pieces are for 2 pianos tuned a quarter tone apart (in a few cases with other instruments); he did have a couple of quarter-tone pianos and even a quarter-tone harmonium built, but was not very satisfied with them (based on quarter-tone piano photos and video footage, I am going to hazard a guess that this was for ergonomic reasons); I think that with the way he wrote this music, it really does need the resonance and timbre of pianos.
The easiest pieces to deal with in this way would probably be a couple of his 24 Preludes (my favorites are III, VII, and VIII, but that is not an exclusive list of good choices).
If you look up a certain music organization named after Greek mythological figures responsible for inspiring artists that also has sheet music but that YouTube's censoring algorithm seems to think is a terrorist organization or something MuseScore, and there you look up Ivan Wyschnegradsky, they have several of his compositions. I looked at the 24 Preludes, and they offer several formats, including MIDI as well as some formats displayable as sheet music, although I haven't tested their output myself.
It is also worth going to Wikipedia and looking him up to get a list of compositions, and then searching for them and listening to them on YouTube. These compositions (other than the early ones in 12edo or 12WT) are not just somewhat xenharmonic like something that was originally written in quarter-comma meantone or some well-temperament -- they are seriously xenharmonic out of the box.
For the 24 Preludes, I would recommend Ivan Wyschnegradsky - 24 Quarter-Tone Preludes for two pianos Op. 22 (audio + sheet music) for an introduction that has the English translation of his own writing on diatonicized chromaticism followed by a complete set of the 24 Preludes well-performed with sheet music so that you can see the dynamic and tempo markings (not sure how much of that makes it through in the MIDI files or even in the MuseScore sheet music). The only downside of that one is that reading the music as 2 12edo piano parts for instruments tuned a quarter-tone apart might be problematic. In that case, a subset of the 24 Preludes are available (after you scroll down some ways) as individual videos on the YouTube channel of musicaignotus which have sheet music written for an actual quarter-tone piano. Ivan Wyschnegradsky used a non-standard semiflat symbol -- instead of looking like a (possibly narrower) backwards flat, it looks like a normal flat that has the bottom peeled open so that it looks like a cross between a normal flat and an 'h'.
Got a chance to look into this a bit more. For the fifths in the basic 11L 2s (Wyschnegradsky) diatonicized chromatic scale (the version that 24edo yields), the 11/8-span of a patent fifth is a stack of 10 intervals of 11/8, octave-reduced.
On the soft side, this still works for 37edo and 61edo, but if you go further afield, with 50edo you instead get the 5edo fifth, so that is too far on the soft end of the scale tree (although might still be good to include it for instructional purposes).
On the hard side, this still works for 35edo (the flat fifth ends up being the patent fifth by a hair), and for 59edo you get the not the patent fifth but the alternate flat fifth, which is barely further away from just and is still in the range of flattone, so we can call that still sort of working; but with 46edo you instead get the 23edo flat fifth, so that is too far on the hard end of the scale tree (although might still be good to include it for instructional purposes).
I'll do this separately for the major third, although the 24edo major third is at best mediocre in terms of relative error, being as sharp as that of 12edo, but in the context of increments of half the size.
Doing this for the 5/4 major third: The 16/11-span (goes the other way around the circle of 11/8) of this is in 24edo is 8.
On the soft side, this works very well for 37edo, for which the 5/4 major third is nearly just, as well as for 50edo, for which the 5/4 major third is just slightly flat, and for 61edo, for which the 5/4 major third is mildly sharp (although in terms of relative error it ends up being even worse than for 24edo).
On the hard side, this already quits working for 35edo, for which the 5/4 major third is fairly flat (you instead get a very sharp alternative major third, a bit sharp of a Pythagorean major third, and almost up to 33/26). For 46edo, it gets even worse, giving an alternate-alternate sharp major third (the patent 5/4 (sub-)major third is already slightly sharp), actually more like 14/11. For 59edo, the situation is similar, even though the 59edo 5/4 (sub-)maor third is just barely sharp of just, instead giving 19/15. So even the mildly hard side of the 11L 2s tuning spectrum doesn't work for the 5th harmonic, despite working for the third harmonic.
I didn't do the 7th harmonic, because Ivan Wyschnegradsky himself wrote in the text at the beginning of the 24 Quarter-Tone Preludes for two pianos linked above that you really need something more than 24edo to get the 7th harmonic (his choice for this in almost all of his compositions was 36edo or 72edo, although he also wrote a single 31edo composition for the Fokker organ; but none of those support his diatonicized chromatic scale, other than 72edo in redundant form (being 3 * 24edo). It would be interesting to do as a future back-extension to the 24 & 37 temperament, but it is understandable why he didn't use it, since 24edo has a bad 7th harmonic, and you have to go to either the superhard (46edo) region or the soft region (37edo through 50edo) to get a good 7th harmonic within the 11L 2s tuning spectrum.
That's all for now
Lucius Chiaraviglio (talk) 10:18, 25 January 2025 (UTC)
(Fixed some links) Lucius Chiaraviglio (talk) 11:04, 25 January 2025 (UTC)
Comma for getting the fifth on the circle of 11/8 or 16/11 in the middle of the 11L 2s tuning spectrum
The comma |-33 -1 0 0 10⟩ (11.224¢) equates a stack of ten 11/8 (octave-reduced) to 3/2. However, this only gives the patent fifth in more or less the range 35EDO to 37EDO. For 50EDO (as noted above) it gives the Blackwood (pentatonic) fifth; while for 46EDO it gives the 23EDO flat fifth. Lucius Chiaraviglio (talk) 11:04, 16 February 2025 (UTC) Edited Lucius Chiaraviglio (talk) 11:06, 16 February 2025 (UTC)