Re: TD 738 -- Vicentino's cadential use of ~11:9

Hello, there, and in a recent article the question was raised of
whether Nicola Vicentino (1511-1576) used ratios involving the 11th
partial in practice as well as theory.

In fact, Vicentino does use his "proximate minor third," which he
associates with the approximate ratio of "5-1/2:4-1/2" or 11:9, in an
example of a cadential figure for four voices. The example appears in Nicola
Vicentino, _Ancient Music Adapted to Modern Practice_, tr. Maria Rika
Maniates, ed. Claude V. Palisca (New Haven: Yale University Press,
1996), ISBN 0-300-06601-5.

This example, from his treatise of 1555 which Maniates and Rika have
translated into English, is here shown using a notation where C4 is
middle C, higher note numbers show higher octaves, and the symbol "r"
shows a rest. The ASCII asterisk (*) indicates a note raised by a
diesis, roughly 1/5-tone, or about half of a meantone chromatic
semitone. Brackets indicate accidentals not indicated in Vicentino's
original edition. Note the crossing of the lower two parts:

1 & 2 & | 1 & 2 ||
C4 C[*]4 C4
Ab3 G*4 Ab3 Ab*3 A3
F3 C*3 F3
r F3 E*3 F3

In this example, the next-to-highest voice moves in the second measure
from the minor third above the bass through Vicentino's "proximate
minor third" to a concluding major third.

Vicentino describes the proximate minor third, at ~11:9, as relatively
concordant, in contrast to the "proximate major third" a diesis larger
than the usual major third, for which he gives an approximate ratio of
"4-1/2:3-1/2" or 9:7; he finds this latter interval to tend toward
dissonance.

More specifically, the proximate minor third on Vicentino's _archicembalo_
or "superharpsichord" is defined as an interval of a usual minor third
plus a diesis or fifthtone (e.g. C-Eb*). The size of this interval may
vary slightly depending on exactly how one interprets Vicentino's
temperament, with a circulating 31-note version of 1/4-comma meantone a
likely choice, but 31-tone equal temperament (31-tet) an interpretation
favored in the 17th century by Lemme Rossi in his treatise of 1666.

In 1/4-comma meantone, depending on the order of tuning, Vicentino's
proximate minor third may be equal to either a regular minor third
(~310.26 cents) plus a 128:125 diesis (~41.06 cents), ~351.32 cents;
or the pure 5:4 major third of this tuning (~386.31 cents) minus the
same diesis, ~345.25 cents. Either tuning is a close approximation of
11:9, ~347.41 cents. Note that the difference between these two
"flavors" of near-11:9 thirds, ~6.07 cents, is equal to the 31-note
meantone "comma" by which 31 meantone fifths fall short of 18 pure
octaves.

In 31-tet, which would exactly fit Vicentino's theoretical model of a
whole-tone divided into five "minor dieses," the proximate minor third
would have a precisely uniform size of 9/31-octave or 9/5-tone -- a
regular minor third of 8/5-tone (~309.67 cents) plus a diesis of
1/5-tone or 1/31-octave (~38.71 cents) -- or ~348.38 cents, within one
cent of 11:9.

Both Vicentino's direction that the first 12 notes of his instrument
(Eb-G#) should be tuned as on usual keyboards, and the limitations of
known 16th-century tuning methods, suggest 1/4-comma meantone as a more
likely realization than a precise 31-tet. Very possibly the slight
inaccuracies of an actual tuning around 1555 might be greater than the
small discrepancy between these two mathematical models.

In the cadence quoted above, the next-to-highest voice or alto
concludes by progressing through two "fifthtones" or minor dieses, in
emulation of the ancient Greek enharmonic genus: Ab-Ab*-A. This is a
kind of ornament which might be improvised to decorate various cadences.

Most respectfully,

Margo Schulter
mschulter@value.net