Re: Note names

Hello, there, and this is a first response to your comments; I
regret a delay due to some Internet problems over the weekend.

On the question of notations, I would emphasize that different
musicians may favor different styles; the variety of notations for
31-note meantone (e.g. Vicentino 1555; Colonna 1618) and 31-note equal
temperament (31-tet), as well as for 53-note Pythagorean or 53-tet,
reflect this diversity.

The use of double or multiple sharps or flats has the attraction of
being familiar and logical; some other systems, such as Vicentino's or
Colonna's, may more closely reflect keyboard mappings.

Here a significant distinction may be that with Pythagorean or
meantone (and related equal temperaments), these other systems
distinguish between notes which also have distinct names or spellings
in conventional notation (sometimes involving multiple flats or
sharps). With syntonic comma variations in 5-limit or higher just
intonation (JI), however, conventional notation does not distinguish
between the two flavors of D in the just fifths G-D and D-A, for
example, so new systems are not merely an option but a necessity.

Also, notational systems can reflect musical contexts: my Pythagorean
spelling for 53-tet, or 22-tet for that matter, might be radically
different than a spelling taking 5-limit intervals (e.g. C-E as a
near-5:4 rather than a near-81:64 or near-9:7) as the norm.

As far as Marchettus, for now I would like simply to emphasize that my
"Xeno-Gothic" using complex Pythagorean tuning is merely one modern
approach to the musical implementation of one _possible_ reading of
what Marchettus says about cadential aesthetics.

In this reading, Marchettus seems to call for very narrow cadential
semitones which he describes as "dieses" of around "one of the five
parts of a whole-tone" in cadential resolutions, for example, from
major third to fifth or from major sixth to octave.

This language _suggests_ that these cadential thirds and sixths may
have been considerably larger than Pythagorean. If so, and people such
as Joe Monzo here have offered alternative scholarly interpretations,
then such intervals might possibly be located somewhere in the range
of around 408-455 cents for major thirds before fifths, or 906-955
cents for major sixths before octaves.

The idea is simply "a major third or sixth somewhere between the usual
Pythagorean size and something verging on a narrow fourth or minor
seventh." This is itself a controversial interpretation, and any
specific value within the range is a guess as to what certain singers
or instrumentalists following Marchettus _might_ have done, very
likely by ear, without concerning themselves with precise ratios,
Pythagorean or otherwise.

(Here I would emphasize that Xeno-Gothic is a keyboard tuning, and
that Marchettus is discussing the intonation of cadences in general
terms, without any statements about the ratios of cadential thirds or
sixths, or about keyboard tunings or temperaments.)

Please let me emphasize that the use of complex Pythagorean ratios is,
as far as I know, strictly my modern means of obtaining values (around
431 cents and 929 cents respsectively) which can be implemented on a
keyboard making available conventional Pythagorean intervals also. A
modern interpreter might just as reasonably favor a major third of
6/17 octave in 17-tet; 8/22 octave in 22-tet; or 19/53 octave in
53-tet.

Similarly, someone seeking to play compositions of the organist Conrad
von Paumann from around 1450, and finding meantone appropriate, might
choose 1/4-comma temperament as one possible realization, although the
amount of temperament favored in the first decades of meantone remains
unrecorded, and Gaffurius in 1496 was content to say that the fifths
were narrowed by "a certain small and hidden quantity."

For the moment, I would conclude by noting a certain paradox:
Xeno-Gothic offers one possible realization of Marchettan cadences on
a keyboard using complex Pythagorean ratios, when Marchettus himself
was well known for his disregard of the niceties of Pythagorean
mathematics. Maybe this is itself a touch of very late 20th-century
mannerism.

Most appreciatively,

Margo Schulter
mschulter@value.net