Diatonics

To take your points in turn.

1. THe first of these criteria, is simple to show graphically with 12TET or
C12,

5 0 7 2 9 4 11 reordered 0 2 4 5 7 9 11 (0, has no points at which 3
adjacent PCs lie together.

5 0 7 2 9 4 11 6 1 reordered 0 1 2 4 5 6 7 9 11 (0, has several places where
three adjacent PCs lie together, Viz, 0 1 2 or 11 0 1 or 4 5 6 or 5 6 7.

2. The F F-sharp rule was discussed in section III, and from that dicussion,
it is simply the result of transposing the diatonic generated by the sum of
the two 'generators' (or the interval that behaves structurally like the 3:2
fifth, or 'generalised fifth', as Paul Zweifel dubs it) It must be noted
that the new tone in he diatonic shall have the following property:

It should be the 'leading tone' of the new diatonic

It can also be equivalent to adding 1 to the PC that lies a 'generalized
fifth' below the old tonic.

If the raised tone lies in the 'supertonic' position then the scale is an
anhemitonic scale, in that it contains no tones that are separated by the
difference between the two generators of the scale.

I know I use the tterm generators differently, but in the context of the
group theoretic structures proposed initially by Balzano. I am merely
continuing to use the term, and did not invent it myself.

Incidentally, can anyone point me to Paul Ehrlich's decatonics, a web
address etc...?

Mark

> From: tuning-math@yahoogroups.com
> Reply-To: tuning-math@yahoogroups.com
> Date: 21 Mar 2002 23:57:54 -0000
> To: tuning-math@yahoogroups.com
> Subject: [tuning-math] Digest Number 323
>
> I don't understand either of the second criteria you use for choosing
> diatonics. (That is the second numbered point on both the lists.) Can
> you clarify them?
>
>
> Graham

Mark Gould wrote:

> 5 0 7 2 9 4 11 6 1 reordered 0 1 2 4 5 6 7 9 11 (0, has several places
> where
> three adjacent PCs lie together, Viz, 0 1 2 or 11 0 1 or 4 5 6 or 5 6 7.

Ah, so this is relative to that base scale. And the equivalent 9 note
scale from 19-equal would still pass. Doesn't this rule get taken care of
by the later one rejecting "pentatonics"? At least along with another one
rejecting "degenerate" scales which are simply an equal scale.

> 2. The F F-sharp rule was discussed in section III, and from that
> dicussion,
> it is simply the result of transposing the diatonic generated by the
> sum of
> the two 'generators' (or the interval that behaves structurally like
> the 3:2
> fifth, or 'generalised fifth', as Paul Zweifel dubs it) It must be noted
> that the new tone in he diatonic shall have the following property:
>
> It should be the 'leading tone' of the new diatonic
>
> It can also be equivalent to adding 1 to the PC that lies a 'generalized
> fifth' below the old tonic.

But doesn't this always work for an MOS? Or is it this rule that
specifies an MOS?

> If the raised tone lies in the 'supertonic' position then the scale is
> an
> anhemitonic scale, in that it contains no tones that are separated by
> the
> difference between the two generators of the scale.

And the 7 note diatonic in more than 12 notes would have this property.

> I know I use the tterm generators differently, but in the context of the
> group theoretic structures proposed initially by Balzano. I am merely
> continuing to use the term, and did not invent it myself.

It sounds consistent with Gene's ideas, so long as you can describe the
octave in terms of the generators.

> Incidentally, can anyone point me to Paul Ehrlich's decatonics, a web
> address etc...?

I have it as <http://www-math.cudenver.edu/~jstarret/Erlich.html>. Only
one h in Erlich. It's also in Xenharmonikon.

Graham

--- In tuning-math@y..., Mark Gould <mark.gould@a...> wrote:

> 5 0 7 2 9 4 11 reordered 0 2 4 5 7 9 11 (0, has no points at which 3
> adjacent PCs lie together.

What do you mean by a "PC"?

> I know I use the tterm generators differently, but in the context of the
> group theoretic structures proposed initially by Balzano.

"Generators" is a term taken from mathematics, and I hope when we say "generator" or "group" we are all on the same page, and using the mathematical definitions. What are the two uses of "generator" you see around here? The looseness I've sometimes observed is "generator"
as opposed to octaves or a division thereof, but that should be clear from context.

--- In tuning-math@y..., graham@m... wrote:

> I have it as <http://www-math.cudenver.edu/~jstarret/Erlich.html>.

from there you'll want to go to

http://www-math.cudenver.edu/~jstarret/22ALL.pdf

carl lumma or someone inserted the word 'equal' into the title -- at
no time did any version of this document residing on my computer
contain the word 'equal' in the title.

note that page 20 is all wrong -- just ignore it.

also, the table of key signatures at the end is wrong -- i'll put up
the corrected version if anyone cares to see it.

cheers,
paul