Re: Bob Valentine on n-limit counterpoint

> From: Robert C Valentine <bval@ihp009.iil.intel.com>

> Obviously this idea is supported by many sorts of 'world music' and
> going back to contrapuntal writing before the Baroque era (I
> believe) when the codification of harmony really was
> solidified. (tonic dominant dominant tonic subdominant tonic
> dominant tonic).

Hello, there, and thank you for a very interesting post. Above all,
expressing my gratitude for your kind words, I would like to applaud
your concept of systems of counterpoint for 7-limit or higher
tunings. In a sense, Nicola Vicentino took this step in 1555 when he
recognized the "proximate minor third" (which he defined as around
11:9) as a practical consonance. However, it is a road largely still
untaken.

Further, I would agree that if we define "harmony" in terms of a
system of chords with key functions, then this development falls
somewhere in the Baroque era, maybe roughly around 1660-1680, the
epoch of Stradella and Corelli. This is also, interestingly, around
the time that Werckmeister (1681 and later) marks the beginning of
documented unequal well-temperaments, associated with major/minor
tonality, in contrast to the meantone temperaments associated with
5-limit modal polyphony.

One point about medieval through early 17th-century European polyphony:
I would describe it as an art of simultaneous melodies and
consonances. While it is often said that vertical sonorities are
"incidental" to the melodies, what I perceive is an exquisite balance.
The central difference between modal and tonal polyphony, as I see it,
is not that the vertical sonorities and progressions are "incidental"
in the first, but that they follow patterns and "rules" different than
those of the major/minor key system.

Such compositions have regular vertical sonorities, progressions, and
cadences -- but different, and I might argue in some ways more diverse
and fluid, than those of 18th-19th century tonality. One might say
either than the diversity of these vertical progressions permits the
free development of melody, or that these progressions serve as a
splendid amplification of melody.

> In investigating melody and simultaneous melody, it is sufficient
> (and proper) to do it in the subset of a temperment that may be
> described as "modal", which again, makes sense since it can evolve
> the rules of tonality in the new tuning. 12 tones should be able to
> contain a particular 'scale' and its 'rotations' that may be used in
> a new tuning.

Maybe one way of putting this is to say that where tonal music often
focuses on transposing the same scale patterns (major or minor) to
various pitches, modal music often shifts among different centers or
scales using the same pitch materials, with appropriate cadential
accidentals, chromatic touches, etc. For example, a piece in D Dorian
may often have internal cadences on F and/or A. Also, a piece may mix
various modes, a technique discussed by Vicentino in 1555 and the
Monteverdi brothers in 1607, for example.

The availability of 12 modes provides resources for variety and
contrast analogous to circular modulation in tonal music. Each system
has its own charms, so that, as Ed Foote has pointed out, 1/4-comma
meantone may be ideal for Cabezon but a bit "bland" for Mozart.

For the most part, a conventional 12-note meantone tuning (Eb-G#)
nicely covers the range of standard accidentals in the 14th-16th
centuries. I should add that experimental pieces generally extend this
range without requiring a closed system. Thus if we take 1/4-comma
meantone as roughly equivalent to 31-tet, then avant garde
16th-century pieces often call for 14 out of 31, or 17 out of 31, or
(in the case of Vicentino and Bertrand) up to 24 out of 31 -- but not,
to my best knowledge, all 31 out of 31 for a single piece.

The typical Renaissance solution for such pieces is to use "split
keys" on a meantone keyboard for G#/Ab and Eb/D#, etc. Vicentino
describes his two-manual keyboard with split keys providing 36 notes
per octave in all (a quasi-31-tet plus a few extra notes, likely for
some pure fifths). One modern solution is to use two standard 12-note
keyboards which can be tuned independently, permitting anything from
12 to 24 notes per octave.

Thank you again for a post which I hope may lead to much future
dialogue.

Most appreciatively,

Margo Schulter
mschulter@value.net