Here is an excerpt from Schönberg's letter to Busoni of 24 August 1909 (just a few days before Schönberg began the composition of Erwartung), in which he describes his experimentation with quarter-tones.
Your "Outline of a New Aesthetic of Music" gave me uncommon pleasure, above all on account of its audacity. Particularly at the beginning, there are a few powerful sentences, of compulsive logic and superlative acuteness of observation.
I have also thought a lot about your idea of thirds of tones, though in a different way. But I had been thinking of quarter-tones, am however now of the opinion that it will depend less on the construction than on other things.
Moreover, one of my pupils calculated, at my suggestion, that the next division of the octave with similar properties to our twelve semitone division would have to introduce 53 notes.
[from Monzo: see my analysis of Robert Neumann's suggestion from Schönberg's Harmonielehre. The "similar properties" to which Schönberg refers are the good approximations which 53edo has to the "perfect 5th" of ratio 3:2 and the "major 3rd" of ratio 5:4, as explained by Neumann and shown in my analysis.]
If you adopt 18 thirds of tones, that would be approximately equal, for 3 x 18 = 54. But then the semitones would disappear completely.
[from Monzo: This is an odd pronouncement. Schönberg seems here to be trying to say that since 53 and and 18*3=54 are so close, 53edo contains an 18-tone subset which approximates Busoni's 18edo "sixth-tone" scale. Since Schönberg mentions 53edo as having "similar properties to our twelve semitone division", perhaps he was aware that 53edo contains an excellent approximation to Pythagorean tuning, which in turn is not too dissimilar from 12edo; see the graph below.
Schönberg thus seems to be proposing to Busoni that Neumann's 53edo achieves both of the same goals as Busoni's 36edo -- a larger universe of pitch resources, and the division of each whole-tone into both Semitones and 1/3-tones -- and, in addition, offers a closer proximity to the ratios of just-intonation.]
Earlier I thought out the following method of notating quarter-tones:
< and > are
However, I scarcely think that such attempts at notation will catch on; for I confidently hope that the notation of the future will be - how can I say : "wirelesser".
I also think differently about tonality - my music shows that. I believe: everything one can do with 113 keys can also be done with 2 or 3 or 4: major-minor, whole-tone and chromatic. Anyway, I have long been occupied with the removal of all shackles of tonality. And my harmony allows no chords or melodies with tonal implications any more.
(see A Century of New Music in Vienna for further background info)
2002.3.31 -- page created
2002.12.14 -- added graphic and further commentary