Re: Digest Number 716

Christopher Bailey wrote,

>>terms of propriety, these sets are off the board. Balzano, in his 1980
>>paper in Computer Music Journal, showed that the smallest proper set of
>>12tET to contain all twelve intervals is the diatonic scale. In fact, I
>>think this is true of any series-of-fifths-gerenated scale tuned in the
>>next highest ET in the series 3, 5, 7, 12, 19, 31.
>
>Actually, the smallest subset of 12TET containing all 12 (or 6,
>disregarding inversions) intervals IS the C-Db-E-F#, or C-Db-Eb-G set and
>their inversions.

But they are not proper subsets.

>I think what Balzano was pointing out was a even wieirder property of the
>diatonic scale. If you look at the intervals between: each note of the
>scale, and all of the other notes in the scale; then, you will see that
>each kind of interval appears a unique number of times.

I don't remember him pointing this out, but I wouldn't put it past him --
what possible significance could this property have?

>This helps "efficiency" I guess, as if you get one semitone (say, B-C)
>and then part of the other semitone (F#) then you know what transposition
>of diatonicity you're in (BCDEF#GA).

Sorry, I don't follow. B-C-F# may be a sufficient set of the diatonic
scale, but I don't see how the unique-number-of-times property helps
lower efficiency. And since Balzano was interested in duplicating every
property of the diatonic scale he happened to notice, he would want high
efficiency, anyway.

-Carl