Talk:11L 2s

Musical Mad Science Musings on Diatonicized Chromaticism

The ramble (originally from YouTube comments) which gave the impetus for the content below has been moved here. Lucius Chiaraviglio (talk) 09:31, 18 March 2025 (UTC)

However, some of this does have a rationale for being here, at least until a proper temperament page is developed for it. Ivan Wyschnegradsky came up with a temperament that he called "diatonicized chromaticism", and made some really impressive compositions in it. So I thought it would be fitting to look into the properties of this temperament.

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 that could have reasonably dealt with it 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.

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.

Todo: complete section Need comma for back-extension to 5th harmonic and maybe back extension to 7th harmonic

Even after the above gets its own space on a temperament page (if that ever happens), something I would like to see (but have no way to generate myself) on the 11L 2s page would be a musical samples akin to what the 5L 3s page has for musical samples. Ivan Wyschnegradsky wrote a considerable volume of diatonicized chromatic music, including Préludes dans tous les tons de l'échelle chromatique diatonisée à 13 sons (24), for 2 pianos in quarter tones, Op. 22 (1934, rev. 1960). If a short one of these works (such as one or two of the Preludes) could be gotten in MIDI form and processed through a synthesizer with a high-quality piano emulation in various tuning systems (as noted in the temperament discussion above), this would provide an excellent Musical Samples section for this page.

Rewritten Lucius Chiaraviglio (talk) 09:31, 18 March 2025 (UTC)