Re: Rothenberg q

Kraig,

>looking at you might be interested In Walter O'Connell's All interval
sets. These are sets of
>four tones that each contain all 12 intervals ( by inversion). These are C
D F Gb, C Db E F#,
>C Db Eb G, C E F# G. These would be of use if you were looking for
efficient material.

This is interesting. These sets would be very efficient taking efficiency
to mean: number of distinct subsets given the total number of subsets (n!),
but fairly inefficient by Rothenberg's calculation (it usually takes only 2
members of the set to distinguish it from a transposition of itself). In
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.
Wilson's diagrams of scales generated by series of equal intervals
elegantly show, I think, that the series of fifths (approximate 3/2's) are
special in generating the most stable (in Rothenberg's sense) scales. I
haven't quite pinpointed the mathematical reason for this but I think it
follows from the fact that the fifth (actually the fourth) arithmetically
divides the octave (in the Greek sense).

jason

>of defining scales you might find interesting is Moments of Symmetry .
>http://www.anaphoria.com/mos.html

Hello again, Jason,

>This is interesting. These sets would be very efficient taking efficiency
>to mean: number of distinct subsets given the total number of subsets (n!)

That does sound interesting. Could you explain how you're using "distinct
subsets" here? And let's use a term other than efficiency, to keep this
new concept straight?

>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.

Eh? The diatonic scale doesn't contain all of the intervals of 19-tET.

>Wilson's diagrams of scales generated by series of equal intervals
>elegantly show, I think, that the series of fifths (approximate 3/2's) are
>special in generating the most stable (in Rothenberg's sense) scales.

Can you demonstrate this? I would be surprised if there are any special
generators here -- all generators should turn out their fair share of both
proper and improper scales, if the chaining is carried out far enough.

-Carl