--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
Paul Erlich wrote:
> > That's understandable, since 31-tET conflates pairs of ratios
> > in his diamond.

This is useful terminology.
"<temperament> conflates ratios in <JI structure>"
means the same as
"<JI structure> overloads <temperament>".

> This whole thread about a possible MIRACLE intuition guiding
> Partch has made it abundantly clear to me that the literal
> structures embedded in the Tonality Diamond were of paramount
> importance to him.
>
> Since arguably the thing the Diamond shows best is the
> at-least-dual nature of each ratio, which is a property
> Partch emphasized repeatedly was inherent in ratios (quite
> obvious to my mind, since they're a relationship described
> by two numbers, duh!), then it seems to me to follow that
> this dual property was perhaps the primary conceptual focus
> of his tuning system.
>
> If this is the case, then I find that to be a very valuable
> insight into Partch's _modus operandi_.
>
>
> It's also fascinating that Partch was more interested in
> expanding his harmonic resources along Pythagorean lines
> (pun intended) rather than the higher-prime relationships
> approximated by MIRACLE.

What do you mean here by "higher-prime". I hope you only mean 5, 7 and
11.

But it seems that he went Miracle at first and then later changed only
four notes for Pythagorean. He changed only 49/48 to 81/80 and 27/20
to 15/11 (and their inversions).

So I can postulate 3 forces in historical order: First Diamondic, then
Miracle (which simply means that he wanted to fill in the diamond gaps
while minimising the number of extra notes and maximising the number
of 11-limit consonances, both strict and with small errors) and
finally the old Pythagorean/Diatonic reasserted itself sightly.

> I'm interested now more than ever in knowing some of Daniel
> Wolf's knowledge and opinions on this subject. A full-scale
> analysis of the *non*-JI harmonies in Partch's compositions
> would reveal a ton of information.

Yes indeed. We might be able to better answer the "schismic vs.
miracle" question based on that.

-- Dave Keenan