RE: Balzano's generalized diatonic

>Balzano is a much better mathematician than you seem to be giving him credit
>for. The product property _is_ required to get his three group-theoretic
>representations of the pitch set -- melodic, key, and harmonic. Since the
>title of his paper is _The Group-theoretic Description of 12-fold and
>Microtonal Pitch Systems_, and he gives these representations a huge amount
>of emphasis in his paper, there's no way you can exclude them from the list
>of "all the things he says he wants".

If I say coffee is all I want, it shouldn't be surprising that only coffee
will do. Balzano's paper is full of this sort of 'if A, then A' stuff; I
wish I had the time to quote it all here. At the beginning of the paper he
promises musical justification for his criteria. I re-read the paper, and
I still didn't see it. When it comes time to prove the product property,
he gives an excuse -- depends on how you formulate the properties.

It wasn't my intent to summarize the paper, it was my intent to summarize
everything I felt he justified. I did miss the bit about the generator
transformations causing the one note to change a _small amount_ with
respect to the 2nds of the scale; so we only consider chains which are 1D
periodicity blocks of the type you're looking for. But then, all of a
sudden, the small amount has to be one unit of the master scale, to get
homomorphism with the semitone-generated group. Product-space spanning
groups are thrown out when their semitone versions aren't isomers. Nonsense.

The paper was interesting, benevolent, and pointed out some important
properties of the diatonic scale. But its conclusions are hogwash.

-C.

Carl Lumma wrote,

>The paper was interesting, benevolent, and pointed out some important
>properties of the diatonic scale. But its conclusions are hogwash.

Agreed! Too bad theorists like Zweifel have been seduced by it.