Re: Reply to Joe Monzo: Artusi, Kirnberger, etc.

Hello, and in response to Joe Monzo's fascinating analysis of my post
last summer in reference to an "Artusi solution" just tuning of a
near-12-tet with fifths of 16384:10935, I'd mainly like to emphasize
that virtually identifical tunings have been proposed by various
theorists starting at least as early as Kirnberger in 1766.

Technically speaking, the Kirnberger line of tunings may be trivially
different because they tend to be based on a fourth of 10935:8192
rather than the fifth which I describe.

In other words, my not-so-new proposal was based on a ~700-cent fifth
built from a schisma major third at 8192:6561 (~384.36 cents) plus a
pure 6:5 minor third at 6:5 (~316.64 cents). The more common "JI
near-12-tet" proposals of the later 18th-century tend to use a
~500-cent fourth built from a Pythagorean apotome or chromatic
semitone at 2187:2048 (~113.69 cents) plus a pure 5:4 major third at
5:4 (~386.31 cents).

A psychological reason for my variant -- actually requiring an extra
just fifth to derive each tempered interval, and also requiring a pure
6:5 as opposed to 5:4, and thus even _less_ parsimonious -- is my
fascination with the 8192:6561 schisma third, a vital interval in
early 15th-century keyboard tunings. Thus I seem to recall reasoning:
"What interval plus a schisma third would add up to ~700 cents?"

For those who may not have seen my original posts, I might explain
that the association with Giovanni Maria Artusi, the famous critic of
Monteverdi, comes from a statement of this theorist that he accepted
12-tet for the lute, but would not admit it for voices until it could
be associated with known ratios of rational numbers. Either
Kirnberger's original 10935:8192, or the variant of 16384:10935,
indeed might be taken as a solution to this "riddle of Artusi."

Most appreciatively,

Margo Schulter
mschulter@value.net