Re: [tuning] Digest Number 716

> 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.

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.

thus in the diatonic scale you find there are 2 semitones, 5 whole-tones,
4 minor thirds, 3 major thirds, 6 perfect fourths, and one tri-tone.
So, a Unique amount of each interval[-class (i.e.--not counting
inversional equivalents (5th =4th)].

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).

Actually this "unique-amount-of-each-interval" property is shared by a
6-note chromatic "wad", but, that's not as exciting (at least according
to most of the world.)

bye bye

Christopher Bailey