Re: TD 929 -- Best wishes to Monz

Hello, there, with special thanks to the Monz for that essay on
Aristoxenes.

Monz, I want to wish you a happy interim, and also to thank you and
your discussions of arithematic monochord divisions for interesting
curiosity about Marchettus of Padua.

If we take his statement that the extra-wide cadential major sixth is
equally far from the 3:2 fifth or the 2:1 octave, with a distance of
"six dieses" from either, then one obvious interpretation is an
adaptive tuning system combining the pure 2:1, 3:2, 4:3, and 9:8 of
Pythagorean intonation with a fivefold division of the whole-tone much
like that of 29-tone equal temperament (29-tET).

This gives us a cadential major sixth in the expansive M6-8 resolution
at around 26:15 (~951 cents), or the _geometric_ mean of the pure
octave and fifth, at a distance of around 6/5-tone (~15:13 or 248
cents) from either of these concords.

However, suppose we take "at an equal distance from the fifth or the
octave" in an arithmetic sense, describing a monochord division.

Then if an octave is 24:12, say, and a fifth is 24:16, then the
arithmetic mean would be 24:14 (the latter term midway of the
monochord between 12 and 16), or 12:7.

The geometric interpretation seems to me, at least at first blush, to
fit better with the equation "at a distance of six dieses from either
the fifth or the octave."

However, for people who like the hypotheses that singers lean toward
simple ratios, 12:7 would define the simplest ratio for a major sixth
fitting the ideal of "closest approach" to this interval's cadential
goal of the stable octave. In the M6-8 expansion, one voice would move
by the usual 9:8 whole-tone which Marchettus describes in accord with
standard Pythagorean tuning, the other by a semitone or "diesis" of
28:27.

Anyway, whether or not it has any relevance to Marchettus, the idea of
12:7 as the arithematic mean of 3:2 and 2:1 is a neat relationship I
hadn't considered.

Again, please let me wish you the best during your time away from the
list.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@egroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

> Hello, there, with special thanks to the Monz for that essay on
> Aristoxenes.
>
> Monz, I want to wish you a happy interim, and also to thank you
> and your discussions of arithematic monochord divisions for
> interesting curiosity about Marchettus of Padua.

Thanks so much for the kind words, Margo. I've been enjoying
your recent flurry of terrific articles on Zarlino, and especially
this one on Marchetto. I think you'll be *most* interested in
the scales resulting from my new interpretation of Aristoxenus!

(and I'm really sorry I can only tease with it now... I promise
to get it up onto my website as soon as I can after my return.)

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

(HA! - I've found a new slogan!)

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

> 'All roads lead to n^0'
>
> (HA! - I've found a new slogan!)

I love it!

David Finnamore