Re: Pythagorean septimal schisma -- confession of ethnocentrism

Hello, there, and thanks to such contributors as Joe Monzo and Kraig
Grady for reminding me that one danger of an "historical disclaimer"
that one may possibly disclaim too much, and in an ethnocentric manner
to boot.

When responding to a question whether Pythagorean intervals such as a
major third a comma wider than the regular ditone at 81:64, and major
sixth a comma wider than the regular 27:16 (respectively ~9:7 and
~12:7) were used "historically," I unreflectingly took the question to
refer to medieval European practice.

While one can find these intervals, for example, in the 17-note
Pythagorean keyboard octave of Gb-A# proposed by Prosdocimus of
Beldemandis (1413) and Ugolino of Orvieto (c. 1425-1440?) --
e.g. G#-Gb (~7:4); D#-Gb (~7:6); Gb-A# (~9:7) -- I am not aware of any
proposals suggesting that these intervals could be resolved in
"superefficient" cadences (possibly of the kind advocated by
Marchettus of Padua in 1318) to stable 3-limit intervals.

Xeno-Gothic tuning, practice, and theory, might be viewed in a sense
as just such a project -- but as a further offshoot of actual Gothic
theory, not as a reporting or restatement of that theory.

Having often discussed in this forum Marchettus and my Xeno-Gothic
interpretation (only possible choice) of his cadential aesthetics,
I felt a special responsibility to make this distinction between known
medieval practice and theory, and "neo-Gothic" offshoots.

However, as I have now been very constructively reminded, "historical"
uses of intervals are not a monopoly of European composed music: and
the system of srutis in India, and 17-note systems in various Arabic
and Persian traditions, for example, may indeed offer examples of
7-limit intervals as well as 5-limit intervals derived from
Pythagorean tuning in their schisma forms and eventually modified
slightly to obtain simple 7-based ratios.

Since the 5-limit schisma (~1.95 cents) and septimal schisma (~3.80
cents) are so small, this kind of modification might have no major
musical impact, but could permit a "neater" theoretical scheme of
ratios.

While I want to be cautious not to speak for Erv Wilson, who speaks
most eloquently for himself, I might add that Erv has communicated to
me his fascination for extended Pythagorean tunings documented or
reasonably postulated for various world musical traditions. Thus I
must confess cultural myopia, rather than mere invincible ignorance,
as a flaw revealed by my overly sweeping disclaimer.

Please let me warmly send my greetings to Kraig Grady and the
Anaphorians, whose territory in "North America" (or ?estom k'awi, "the
Middle Land (or Continent," as I sometimes call it in an immigrant
dialect of Nisenan, the indigenous language of my own territory,
incidentally including "South America" also) I would take to include
not only the USA but Canada and Mexico, the latter being also one of
the nations of Erv Wilson.

Most respectfully,

Margo Schulter
mschulter@value.net

Margo Schulter wrote:

> Since the 5-limit schisma (~1.95 cents) and septimal schisma (~3.80
> cents) are so small, this kind of modification might have no major
> musical impact, but could permit a "neater" theoretical scheme of
> ratios.

Maybe this is a trivial point, but that's never stopped me before. The
septimal schisma of 3.8 cents is a fairly arbitrary interval. Simpler
ratios are 225/224 and 5120/5103, which are between 5 and 8 cents. So
7-limit intervals are not as well approximated by Pythagorean tuning as
5-limit intervals are. Although still well enough to be recognized and
distinguished.

Graham