CPS stuff (was RE: The CPS Blues [combination product sets])

>But if the factors are not consecutive odds beginning with 1, then
>you're not really doing either according to the definitions in the
>Tuning Dictionary.

I do both by a generalized version of the definitions, neither by a
literal application of them. I suggest our current exchange demands
we review our use the consecutive odd factor assumption (I would
never assume it for CPS).

>I'd still rather say you're completing the chords with respect to
>the initial factor set, rather than saturating the chords with
>respect to the initial factor set.

Sounds good. We'll have to replace 'Partchian tonalities' with 'factor
set' in the entry for complete.

Further, "basic" and "stellated" would probably have been optimal in my
original post.

Speaking of incomplete chords, the non m/n=2/1 CPSs are looking cool;
check out the utonal triads in the 2)6 pentadekany (which I hadn't
noticed until I found Co and Cu).

>> N! N! N!
>> ------- + -------------- + ------------
>> M!(N-M)! (M-2)!(1+N-M)! (N-M-2)!(1+M)!
>>
>> ...but it looks like we're facing some negative factorials. I also tried
>> expressing it as a single fraction, but got some nasty polynomials.
>
>The first negative factorial results from dividing by (M-1), which
>is dividing by zero if M=1, so you mustn't do that.

Ah yes, I'm aware. As I said, I did find the correct common denominator...

> 3 3
> N! (M + (N-M) - N) N!
>------- + ------------------
>M!(N-M)! (M+1)!(N-M+1)!

...but forgot how to distribute for cubes. I had to go to bed, planned on
looking it up tonight. Thanks! Now I have something to check my work
against. Like I said, I couldn't believe my basic algebra had slipped so
far.

Looks like we have a winner. John- what formula did you publish that was
wrong?

-Carl