Re: TD 395: Reply to Johnny Reinhard -- Ives & Pythagorean F-A#

Hello, there, and in Tuning Digest 395, Johnny Reinhard wrote:

> Years later, Ives scholar John Kirkpatrick said the following for
> Vivian Perlis and her oral history project. "But much later, after
> he died, it finally dawned on me that what he had in mind was a
> suggestion of an interval that wasn't really a perfect fourth. The
> A-sharp would be a little higher than a B-flat would be, and the F
> natural would be a little lower than an E-sharp would be. So it was
> really slightly more than a perfect fourth, and for the words "The
> most are gone now," "gone" would be a little under what you'd expect
> as the interval of a fourth, and would be correspondingly expressive
> in that way." (Charles Ives Remembered: An Oral History, Da Capo
> Press, p.221)

Indeed, F-A# seems almost like an old friend to me: it defines a
Pythagorean "Wolf fourth" or augmented third formed by 11 fifths up,
177147:131072 or ~522 cents (a Pythagorean comma wider than 4:3).

This rings a bell because F-A# defines the chain of fifths for a
12-note tuning discussing by some early 15th-century theorists, and
possibly used according to Mark Lindley for the music in one organ
tablature of the epoch, with all accidentals tuned as Pythagorean
sharps. Of course, the absence of Bb, an integral note of the normal
gamut of system of _musica recta_, would be a serious drawback, and
Lindley notes that the examples of the tablature in question might be
taken to finesse this point by using the third Bb-D (realized on this
hypothesis as A#-D, a diminished fourth or schisma third at ~384.36
cents, only ~1.95 cents from 5:4) and avoiding the fifth Bb-F (which
would sound as A#-F, a Wolf fifth a Pythagorean comma narrower than
just).

Anyway, the Ives example suggests an interesting historical free
association: and I certainly agree that a Pythagorean interval of F-A#
would be quite different from a perfect fourth (at any rate in the
usual kinds of timbres we're considering).

In a weird "sort-of-neo-medieval/20th-century" mood, I've played
around with a resolution like

A#-B
F -E

where the interval expands to a pure fifth, each voice moving by a
usual Pythagorean diatonic semitone of 256:243 (~90 cents) -- ~180
cents of expansion in all. Somehow it seems almost "expressionist" or
something.

Most appreciatively,

Margo Schulter
mschulter@value.net