Re: TD 874: Vicentino and Colonna -- 1/4-comma or 31-tET?

Hello, there, and I'd briefly to comment on a question that comes up
from time to time: should Vicentino's tuning for his 31-note division
of the octave be taken as 1/4-comma meantone, or as 31-tone equal
temperament (31-tET)?

Given the very close affinity of these two solutions -- maybe a
theoretical distinction outweighed by usual variations in a tuning by
ear -- the question may be moot. However, it's possible to argue that
Vicentino may have favored in practice a form of 1/4-comma meantone
considered equivalent to the division of the whole-tone into five
equal tones.

Vicentino's instructions that the first 12 notes on the archicembalo
(Eb-G#) should be tuned as in common practice suggests a circulating
31-note system based on 1/4-comma meantone.

At the same time, his division of the whole-tone into five "minor
dieses" not distinguished in size makes his scheme equivalent at a
certain theoretical level to 31-tET, with which Lemme Rossi equated it
in his treatise of 1666 giving string ratios for this tuning to within
about a tenth of a cent.

Maybe this is a question of level of description. In figuring
intervals on a 24-note archicembalo arrangement in 1/4-comma meantone,
I often count in "fifthtones," recognizing but conveniently
overlooking the distinction between larger and smaller fifthtones at
around 41 and 35 cents (together making up the chromatic semitone of
around 76 cents).

One approach to this question might be: "Does 31-tET simply imply a
conceptual division of the whole-tone into five equal parts, or does it
imply deliberately tempering the major third to make it very slightly
impure and thus distinct from 1/4-comma?"

If the former, then Vicentino's system can certainly be described as
31-tET, as scholars such as Maria Rika Maniates and Claude Palisca
have recently done in their fine translation of Vicentino's _Ancient
Music Adopted to Modern Practice_ (1996). So can Fabio Colonna's
31-note tuning of 1618, which he likewise notes divides the whole-tone
into five equal parts.

If the latter, then Vicentino doesn't seem to make any distinction between
the tuning of the first 12 notes in the two systems (31-note division of
the octave; 19-note tuning in a similar temperament with pure fifths added
for a form of Paul Erlich and others have termed "adaptive 5-limit JI").
Similarly, Colonna says that the first 12 or 19 notes or his 31-note
system are tuned as on smaller keyboard instruments, again suggesting a
circulating 1/4-comma meantone.

Personally I tend to regard both Vicentino and Colonna as representing
"a circulating 1/4-comma meantone almost equivalent to a literal
31-tET" rather than vice-versa, but this may be matter of viewpoint.
Rossi may have been the first theorist to state the slight
mathematical difference.

Incidentally, I'm not aware of allusions in Vicentino to 19-tET;
Guillaume Costeley (1570) may be the first to describe this
temperament.

Most respectfully,

Margo Schulter
mschulter@value.net