repost: Erlich's contest

>>Those of you who have read my paper or followed my posts know that I >>suggest replacing the 7-out-of-12 scale, which has defined most Western >>music for centuries if not millenia, with a 10-out-of-22 scale.

I think it's rather inaccurate to say that the 7-out-of-12 scale has "defined" Western music for centuries. What's "Western" music? The 7 tone MOS is one of the most commonly used scales in the world. Always has been.

If we do recognize a "Western Music", then we'll notice that it's been using to awesome effect the 12 tone MOS for over 100 years, and I do not mean serialism. It's also made ample use 5, 6, and 8 tone scales.

While the argument for the 10-of-22 scale is thorough, well-presented, and very compelling, it remains to be proven or disproven only through a body of music, since that is what the theory is *for*. Unfortunately, the only reasonable instrument for decatonic music that exists at the moment is the guitar, which is simply not my can of worms...

>>I haven't PROVEN that something significantly different from 10 of 22 can't >>work, but I doubt it.

Can't work for what? The specific set of rules you chose to generalize diatonicity? There must be other sets of rules that capture the essence of G.D. just as well:

I think that as long as we keep propriety in mind, and make so that the set of intervals (scale steps) can rotate through the set of acoustic magnitudes in some systematic way with the scale's circular permutations, and keep everything to a digestable yet challenging size (see discussion of cognitive limits below), we have "got it".

>>I know of no 9-limit or 11-limit generalized-diatonic scales, but they >>might exist (I don't know how important that would be, since the 9-limit >>and especially 11-limit analogues of the minor chord sound pretty dissonant >>to me, despite Partch's excellent use of them).

The 9-limit utonalities sound good to me. The 11's work with the right (especially electronic) timbres, and/or tasteful amplitude balance and voicing. And I don't think that we have to rotate through major and minor to achieve diatonicity. We could rotate through higher and lower identities, or all sorts of things.

But if we insist on complete chords, and I don't think we have to, higher-limit generalized diatonic scales run into another problem: many of the desirable effects of diatonicity drop off as the number of tones in the scale increases. Some will drop off because of the Miller limit (which has to do with tracking events over time), and some will drop off due to the Subitizing limit (which has to do with tracking multiple, simultaneous events).

1. Miller limit

(a) I don't believe in just one point-of-no-return Miller limit, at least not in the application of how listeners experience melodic symmetries. Rather, I think that there may be several types of memory effects that fall in and out as the number of tones in a melody changes. Exact numbers would depend to some extent on how much practice the subject has had at this stuff, but here's a rough idea of what I'm thinking (we assume octave equivalence)...

tones effect propriety music
-----------------------------------------------------------------------------
2-4 too easy little importance chant
less interesting ritual song
more "join in" potential

5-12 tracking starts to slip most important polyphony
mind has fun trying to parallel harmony
keep its place melody over chords

11-22 tracking the entire scale some importance parallel harmony
impossible: mind "chunks" melody over chords
scale into proper subsets melody over drone
and tracks within/between
those

23-34 inability to focus or no importance conceptualism
and up mind begins to fuse
individual simuli and
re-interprets as if
hearing 5-9 tone scale

(b) I'd say that the Miller limit has claimed all it will from generalized diatonicity by 12 notes. This would seemingly nix anything higher than the 9 limit, whose smallest possible G.D. scale has 11 members. But I suspect that most listeners will need quite a bit of practice (and maybe a few Millers...) before getting the most out of even a 9 tone G.D. scale.

(c) Miller complained that he couldn't explain the performance of those subjects with absolute pitch. There are some very good reasons to believe that almost everyone is capable of very accurate absolute pitch. But I do not believe that the ability to remember the tones (as measured in Miller's experiments) using absolute pitch means that we are not experiencing a loss of some type of experience. For example, someone with a well-developed sense of absolute pitch may not have a problem correctly tracking 34 tones/oct. However, I believe she would suffer the same loss of ability at tracking melodic symmetries at this number of tones as someone without absolute pitch. With this I admit to some difficulty defining and measuring "melodic symmetries".

(d) I list "mind begins to fuse individual tones and re-interprets as if hearing a 5-9 tone scale" as one of the effects of a melody with over 23-34 tones. I list "conceptualism" as the kind of music you'd make with it. Here, I am insulting "conceptualist" music (the idea behind a work of music is extremely important to me as a listener and composer, and "conceptualism" belittles this). But there is a way to profit from the brain's tendency to fuse tones when they are this close in size and this many in number -- the performance of generalized diatonic music in just intonation! Choirs have been doing it for centuries.

2. Subitizing limit

(a) It's been shown that average dudes from all over can count how many stones you toss on the ground almost instantly- so long as you don't toss more than six stones at a time. Since a good deal of the interest of G.D. scales comes from the interaction between parts in polyphonic composition, it seems that we'll lose something if we go above the 11-limit.

(b) While this ability should be more easily improved with training than the Miller limits discussed above (remember Rainman and the toothpicks?), its carry-over to the tracking of simultaneous parts in a polyphonic music is not entire. This is due to the fact that our psychoacoustic bandwidth (keeping notes with their respective parts) is not as great as our visual bandwidth (as used for counting stones) -- especially when listening to music produced on speakers, which lacks the spatial cues of acoustic performance. I think six parts is a good practical upper limit for polyphony. Parallel harmony shouldn't have a limit so long as we stick to otonalites.

-Carl