Re: schismatic tuning -- and permission to reuse

Hello, there, Monz and everyone, and thank you for such a courteous
response to seek permission to use my post on schismatic tuning, which of
course you're welcome to do -- I'm much complimented by the request!

Please let me note more generally that my posts here are generally
available for use under something like the GNU Copyleft. This means that
as long as you credit the author and note any changes or revisions you
might have made, a text is available for circulation, distribution,
reposting, etc.

Of course, I'm especially delighted to have an item used in your Tuning
Dictionary, Monz -- a really great resource including both your own
approaches to the subject and a compilation of views and commentaries by
others. That's one impressive project!

A point on the 17-note Pythagorean described in 15th-century Europe
(or more specifically Italy, by Prosdocimus de Beldemandis in 1413 and by
Ugolino of Orvieto sometime around 1425-1440 or so): while this system
(Gb-A#) indeed includes some schismatic intervals, to describe it as a
"schismatic" system might imply that near-5-limit ratios are the preferred
ones. Actually it's a regular diatonic Pythagorean scale (F-B) plus ten
accidentals, with a flat and a sharp dividing each whole-tone into a
pattern of limma-comma-limma (e.g. F-Gb-F#-G).

Both Prosdocimus and Ugolino advocate such a system above all as a
solution to the problem of having _regular_ Pythagorean intervals and
cadences available at as many places as possible, especially progressions
by stepwise contrary motion with 32:27 minor thirds before unisons
(min3-1), 81:64 major thirds before fifths, and 27:16 major sixths before
octaves (Maj6-8). In each stepwise progression, one voice moves by a
regular 9:8 whole-tone and the other by a 256:243 diatonic semitone.

In fact, they advocate the 17-note system as a way of _avoiding_ certain
schismatic variations of these regular intervals -- although a keyboardist
conversant with both the old style and the new might mix them creatively,
using for example schismic thirds and sixths for sustained noncadential
sonorities and regular Pythagorean thirds and sixths for directed
cadences. From this point of view, the 17-note system (or a system with
13 or more notes including both Gb and F#) does include all the notes of
the modern 12-note Gb-B tuning plus a pure fifth B-F# (which Ugolino notes
would be absent in Gb-B where B-Gb does not "fully perfect" this fifth,
resulting in what is now termed a Pythagorean Wolf fifth or diminished
sixth).

It might be best to say that in earlier 15th-century Europe, the 17-note
tuning is proposed mainly as a method of making more regular Pythagorean
intervals and cadences available at remote locations, but could also serve
as a system for mixing regular and schismatic thirds and sixths according
to the taste of the player.

It might further be worth specifically emphasizing that the basic diatonic
scale here is regular Pythagorean (F-B), with C-E, for example, as a usual
81:64 third. Likewise F-A and G-B are the usual 81:64 -- in contrast to
"schismatic" systems with intervals like C-Fb, F-Bbb, and G-Cb. The
17-note Pythagorean Europe adds more accidentals to a usual diatonic
Pythagorean system, rather than altering the system to make near-5-limit
intervals a harmonic norm.

As I discussed in my earlier recent post, the 12-note Gb-B system of this
same era likewise has usual Pythagorean thirds within the regular gamut of
the diatonic notes plus Bb, so that "partially schismatic" might be one
possible description (i.e. sharps tuned as flats).

Of course, any regular 12-note Pythagorean tuning will have some
schismatic intervals, including the Eb-G# arrangement likely common in the
14th century. Here, for example, the diminished fourth C#-F is a
schismatic major third (resulting, for example, in certain ornamented
versions of the min3-1 cadence in Italian music), while F-C# (which can
come up in some progressions) is a schismatic minor sixth. However, it's
debatable whether the "5-limit-like" qualities of these intervals was
significant much before the era of 1380-1410, when the partially
schismatic Gb-B tuning may have come into fashion.

Thus it might be worthwhile to decide just when a Pythagorean tuning with
some schismatic intervals becomes a "schismatic" -- or "semi-schismatic"
(e.g. Gb-B) -- tuning.

Such distinctions might depend in part on the orientation of the tuning --
how "near" or "remote" are the schismatic intervals (e.g. Eb-G# vs. Gb-B
around 1400, vs. something with C-Fb or the like), and what is the
interpretation of the person describing the tuning (if this is available)?

Thus it is possible that in one culture or subculture/taste a 17-note
Pythagorean scheme might be favored specifically to approximate a 5-limit
system for the most prominent intervals; in another, to make _regular_
Pythagorean intervals available at more locations (as advocated by
Prosdocimus and Ugolino, with the latter advocating a 17-note organ); and
in yet another, to mix or contrast both types of sonorities in the same
piece (as might have happened in 15th-century Europe if a 17-note keyboard
was actually built, a question left open by Lindley).

Most appreciatively,

Margo Schulter
mschulter@value.net