fractional exponents of prime-factors (was: Consistent notation, was: katapyknosis)

Hi Dave,

Glad you found my input useful!

About this:

> So I can consistently write things like
> 32/55 oct + 4.8 c
> whereas
> 2^(32/55) + 4.8c
> is not consistent.
> You'd have to write
> 2^(32/55) * 2^(4.8/1200)
> which is a bit verbose (numerose?).

I find this really interesting right at this time,
because I've been digging pretty deeply into factorizations
of tempered pitches, using fractional exponents for the
primes, and have been using an Excel spreadsheet I created
to do this.

I'm intrigued by this because Paul Erlich has said that
prime-factoring tempered ratios isn't useful, because
the fractional exponents do not give a unique factorization.

With my limited math ability, I don't really grasp all
the implications of this. But I find that it works and
can specify intervals correctly all the time.

For a very simple example, if we use vector subtraction to
calculate what note results from tempering the Pythagorean
"perfect 5th" by 1/12 of a Pythagorean comma, we get:

2^x 3^y

|-12/12 12/12| = 2^-1 * 3^1 = 3/2 = "perfect 5th"
- |-19/12 12/12| = ((2^-19)*(3^12))^(1/12) = 1/12 Pythagorean comma
-----------------
| 7/12 0 | = 2^(7/12) = 12-EDO "5th"

I've been doing a lot of calculations like this lately.
The main use I have for it is that I can create lattices
of tempered tunings which plot on top of my JI lattices.
Comments?

-monz
http://www.monz.org
"All roads lead to n^0"

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