Re: TD 764 -- near-31-tet (1/4-comma meantone)

Hello, there, and in response to the question about 31-tet, I'm one of
those people mentioned by Paul Erlich who uses the almost identical
1/4-comma meantone tuning -- in sets ranging from 12 to 24 notes per
octave -- for Renaissance and Manneristic music from around the late 15th
to early 17th centuries.

In articles posted around late last October, I discuss some of my
experiences with a 24-note keyboard arrangement in 1/4-comma meantone, a
subset of Nicola Vicentino's _archicembalo_ or "superharpsichord" (1555)
with a complete 31-note circulating version of what is very likely this
same tuning with pure major thirds. Also, in 1618, Fabio Colonna published
a treatise about a similar instrument with a circulating 31-note meantone
temperament, including an example of a piece moving through cadences on
all 31 steps of the instrument.

The interval of a meantone diesis (128:125) -- or the very similar
1/31-octave in 31-tet -- can be very expressive both melodically and in
yielding certain altered vertical intervals. This might be called a
"fifthtone," and Vicentino uses it in some of his compositions
(unfortunately, only a few sample sections of madrigals using this style
and one complete Latin motet honoring a patron have come down to us).

Really we might describe 31-tet as just a mathematically symmetrical
version of circulating 31-note 1/4-comma meantone. In the latter tuning,
we need to make the last odd fifth of the cycle about 6.07 cents wider
than the others in order to arrive at a cycle of an even 18 octaves --
interestingly making this last fifth almost pure, since the others are
narrowed by 1/4 syntonic comma (~5.38 cents).

While my articles of last year cover mainly Renaissance/Manneristic
music and derivative styles, people like Paul Erlich and Dave Keenan are
more familiar with some of the many other uses of 31-tet (or its close
meantone relative), and the Huygens-Fokker site is a great place to begin
also.

Incidentally, since Paul has mentioned adaptive tuning, I should mention
that both he and Vicentino have apparently independently come up with
schemes for just intonation of all 5-limit vertical intervals (the
standard concords of Renaissance music) using 1/4-comma meantone as a base
gamut, but making modifications where required. One theory is that this is
what singers and players of non-fixed-pitch instruments may do in
practice.

One historical note: while 1/4-comma meantone and 31-tet are
mathematically distinct, and some people have reported being able to
distinguish the two tunings, there's not necessarily a clear distinction
in the Renaissance. Thus Vicentino figures his intervals in terms of
fifths of a tone, and do I very often using 1/4-comma meantone, although
as it turns out these five parts of the tone are slightly unequal -- just
as this planet is not quite a perfect sphere, but the concept of
circumnavigation is nevertheless quite practical. I tend to suspect that
when tuning an instrument by ear over 31 fifths, the variations in
practice might be greater than the mathematical distinction in theory.

Most respectfully,

Margo Schulter
mschulter@value.net