Yahoo Tuning Groups Ultimate Backup tuning Dufay and schisma thirds -- notation question

Hello, there, everyone, and I'd like to address an excellent question
that someone raises within the last week or so about the notation of
intervals in early 15th-century European pieces by composers such as
Dufay where the use of Pythagorean schisma thirds has recently been
suggested by Mark Lindley and others.

First of all, as far as I know, the intervals in question are notated
in the manuscript with regular note spellings, e.g. A3-C#4-E4 rather
than A3-Db4-E4. Thus the hypothesis that sonorities of this type were
performed with schisma thirds on many fixed-pitch keyboards of the
epoch, and may have often been sung or played by flexible-pitch
ensembles with similar smooth thirds, is based on evidence other than
explicit note spellings.

As to notation, one significant hint does come from Prosdocimus de
Beldemandis writing in 1413, and discussing the distinction between a
"fully perfected" Pythagorean major third or sixth such as A3-C#4 or
A3-F#4, and a diminished fourth or seventh such as A3-Db4 or A3-Gb4.

Prosdocimus notes that a written notation such as A3-C#3 might
represent either the usual Pythagorean form (which he prefers) or the
diminished fourth A3-Db4 a Pythagorean comma smaller. He leaves it an
open question to what degree the human ear might make a distinction
between these forms.

Thus as his modern editor and commentator Jan Herlinger observes,
Prosdocimus lends support to the theory of Lindley and others than a
notation such as A3-C#4-E4 could represent either the traditional
Pythagorean intervals or the schisma intonation A3-Db4-E4.

The line of argument followed by Lindley is that there is a large
amount evidence, both theoretical and stylistic, to support the use of
the Pythagorean Gb-B tuning for 12-note keyboards.

Discussions of keyboard tunings and designs addressing the
arrangements of minor and major Pythagorean semitones (the smaller
diatonic semitone or limma at 256:243 or ~90.22 cents, and the larger
apotome or chromatic semitone at 2187:2048 or ~113.69 cents), provide
rich documentation for the Gb-B tuning.

The use of prolonged noncadential sonorities with thirds in keyboard
sources of the time, and the tendency of these sonorities to appear
mainly on degrees where schisma thirds would occur in the Gb-B tuning,
provides an important argument that actual musical style reflected the
intonational distinctions of this keyboard tuning.

Thus Lindley suggests that much of the early vocal music of Dufay, and
some other composers of the era around 1400-1440 (or a bit earlier in
some parts of Italy), suggests similar patterns of intonation -- or,
at least, that the composers may have been influenced by the tuning of
a keyboard instrument in a Gb-B scheme or the like.

The remark of Prosdocimus that the written A3-C#4 might also represent
A3-Db4 supports the view that performers might realize regularly
spelled intervals and sonorities in either form.

Such flexibility also would seem to fit a situation of performer
discretion where a written E3-G3-C4 before D3-A3-D4 might be sung or
played without inflection, as Eb3-G3-C4, or as E3-G#3-C#4.

Note that on a 12-note keyboard instrument in Gb-B, schisma thirds
would occur automatically, since the performer would perforce press
the key actually tuned at a Pythagorean Ab in order to play a written
note G#; and likewise with Db for written C#, and Gb for written F#.

Here it is interesting that Prosdocimus himself raises this issue in
order to champion the traditional Pythagorean thirds and sixths at
81:64 and 27:16 -- around 408 cents and 906 cents. He describes these
intervals as "fully perfected," striving in the most efficient way to
attain the "perfection" of expansion to the 3:2 fifth and 2:1 octave
respectively. When given their full size, these intervals more
"closely approach" these fifths and octaves, resolving by way of
efficient 90-cent diatonic semitones.

His goal is to have these full Pythagorean thirds and sixths available
throughout an ample gamut, which leads to his proposal for a 17-note
system (Gb-A#) supporting truly intoned closest approach progressions
even in remote transpositions.

Reading between the lines, we might take his critique of a system with
five flats only -- the Gb-B gamut -- as evidence that by 1413, this
type of 12-note keyboard tuning was already coming into vogue, as the
Faenza Codex documenting Italian keyboard music of around this era
also suggests, with Lindley providing some analyses focusing on
prominent schisma thirds.

Considering both the apparent popularity of the Gb-B tuning for
keyboards and the traditional "closest approach" ethos of Prosdocimus,
a possible synthesis is available: the use of schisma thirds for
prolonged noncadential sonorities involving written sharps, but
regular Pythagorean thirds for sonorities leading to directed
resolutions, especially those of the traditional 14th-century variety
uniting the major third to fifth and major sixth to octave
progressions (e.g. A3-C#4-F#4 to G3-D4-G4).

Whether flexible pitch ensembles -- or possibly players on keyboard
instruments with more than 12 notes -- may have done this remains a
guess, but such a "discretionary intonation" would reflect the
cadential ethos of Prosdocimus as well as some of the notational
flexibility he may be documenting, and also the evidence that Lindley
has brought forward for the Gb-B tuning and its possible influences on
Dufay and other composers of vocal music.

Most appreciatively,

Margo Schulter
mschulter@value.net

Dufay and schisma thirds -- notation question

🔗mschulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com