Re: Neo-Gothic (response to Joseph Pehrson)

Hello, there, Joseph Pehrson and everyone.

In response to your questions, I would say that Paul Erlich has
offered good answers. Specifically, the idea of neo-Gothic
temperaments with fifths wider than pure is indeed a _neo_-Gothic
concept, not an element of medieval theory, which is based on
Pythagorean or 3-limit just intonation.

As Paul points out, medieval Pythagorean systems do extend the gamut
to up to 17 notes per octave, but this is different from 17-tone equal
temperament (17-tet), which combines accentuated versions of some
standard Pythagorean intervals (e.g. the wide major third and narrow
minor third) with other intervals quite different from those of known
medieval practice (e.g. the neutral thirds and sixths).

In discussing the medieval references to intervals such as
thirdtones, quartertones, or "fifth parts of a tone" which do occur
now and then, I feel it important to add that these are typically
hints to singers, not guides to keyboard tunings or to a systematic
definition of the various intervals of polyphonic music.

Also, especially through Boethius, medieval theorists might be aware
of the enharmonic genus of Greek theory with its dieses equal to half
of a semitone. Thus after endorsing the idea of a 17-note Pythagorean
keyboard (with five flats and five sharps) for the "intelligent
organist," Ugolino of Orvieto around 1435 or so suggests that an extra
two keys might be added dividing the semitones E-F and B-C into equal
parts in the ancient Greek manner, but notes that this is not used in
current music. It seems more of an "antiquarian" touch, so to speak.

In short, I would like to emphasize that either Pythagorean systems
with more than 17-notes per octave (e.g. Xeno-Gothic with 24), or
temperaments with fifths wider than pure, are 20th-century variations
on Gothic music and its system of Pythagorean just tuning with up to
17 notes per octave.

Also, the Xeno-Gothic system has a certain medievalist humor to it. In
the early 14th century, Marchettus of Padua was going outside the
bounds of Pythagorean mathematics by proposing a division of the
whole-tone into "five parts" -- especially if we take this as a
geometric division into equal ratios, as did such critics as
Prosdocimus about a century later.

The Xeno-Gothic tuning provides one _possible_ realization for the
cadences of Marchettus based on a 20th-century application of
Pythagorean mathematics: make cadential major thirds or sixths
precisely a Pythagorean comma wider than their usual forms. In a
typical three-voice cadence combining M6-8 and M3-5, this produces an
unstable sonority very close to 7:9:12 expanding to a 2:3:4 trine, and
diatonic semitones (or thirdtones) a bit larger than 28:27. Using C4
as middle C, and "^" to show a note raised by a Pythagorean comma, for
example:

E^4 F4
B^3 C4
G3 F3

In certain timbres, at least, this "solution" sound very "right" to
me, but the basic interpretation of the statements by Marchettus about
cadential intonation remain a matter for debate, let alone any
specific choice of interval sizes.

Most appreciatively,

Margo Schulter
mschulter@value.net