Intermittent Archytan touches in late medieval polyphony?
A prologue to some examples and speculations

In response to your comment, Roger Wibberley, about the 8:7, I'd like to
explore how it may well have arisen as a melodic interval, and sometimes as
a vertical interval, in 13th-14th century European polyphony. Here I would
emphasize that the uses I address of 8:7, 28:27, and other steps of Archytas
involve a momentary "stretching" or "bending" of the usual Pythagorean
fabric, which then returns to its usual shape, so that the kinds of comma
shifting or drifting which arise with the Syntonic in the 16th century are
not an issue here.

If we take Pythagorean intonation as the starting point, then the Syntonic
and Archytan in some ways are mirror images of each other, with comma
alterations of Pythagorean tuning in opposite directions. The Syntonic as
applied in the 16th century (and approximated for fixed-pitch instruments by
meantone temperament), of course, aims at smooth 5:4 and 6:5 thirds, and
results in wider diatonic semitones or mi-fa steps at 16:15 rather than the
concise Pythagorean 256:243. This style, Roger, as you have eloquently
summed it up, "wallows in consonance," that is Zarlino's _armonia perfetta_
produced by maintaining wherever possible, in pieces for three or more
voices, the saturated stable sonority of "the third plus the fifth or
sixth."

A critical point about the Syntonic as desired and applied or approximated
in 16th-century Europe is the imperative in this style for _consistent_
maintenance of Syntonic thirds and sixths. From another view, this involved
by the time of Zarlino what I have called the "invalidation" of Pythagorean
major thirds at 81:64 as "dissonant" in the sense of unpleasant and out of
place in textures pervaded by sonorities including 5:4 and 6:5 thirds. By
invalidation, I mean the gradual process by which a standard and acceptable
tuning such as the medieval 81:64 for a major third becomes undesirable and
eventually is felt as inappropriate "dissonance."

In a 13th-14th century context, in contrast, the use of some Archytan steps
and intervals would contrast with these later Syntonic developments not only
as to the direction in which comma alterations of melodic steps and vertical
intervals occur, but in the absence of any invalidation effect for regular
Pythagorean intervals such as thirds and sixths. Rather, the Archytan could
have and may well have arisen, in some dialects and performance styles for
13th-14th century music, as a momentary accentuation or intensification of
tendencies already present in a Pythagorean ethos of complex polyphony.

Thus the "closest approach" ethos, involving frequent inflections in
directed progressions which may have come into vogue in the later 13th
century and become a standard part of early 14th-century theory
(interestingly including Jacobus as a champion of the late 13th-century Ars
Antiqua or Ars Vetus as well as Philippe de Vitry and Marcheto of Padua in
the Ars Nova camp), fits nicely with a usual Pythagorean tuning. 

Complex and active major thirds at 81:64 and minor thirds at 32:27 (408/294
cents), resolving respectively by stepwise contrary motion to a fifth or
unison in directed progressions involving melodic steps of a 9:8 tone and a
compact 256:243 limma, nicely express a vertical aesthetic based on what I
will call a contrast between imperfect and perfect sonorities.

If we describe some basic elements of this medieval Pythagorean polyphoic
aesthetic as wide and active major thirds and sixths "striving" to expand to
stable fifths and octaves, narrow minor thirds likewise seeking contraction
to stable unisons, and directed progressions involving narrow and efficient
semitones to underscore resolutions from imperfection to perfection, then
intermittent use of Archytan steps and intervals represents an accentuation
of this this aesthetic rather than its blurring or compromise.

Thus the Archytan system offers extra-wide major thirds at 9:7 and
extra-narrow minor thirds at 7:6 (435/267 cents), a 64:63 comma larger than
81:64 or smaller than 32:27, which resolve in directed progressions with
melodic motions of a usual 9:8 tone plus a superincisive 28:27 thirdtone
(204 and 63 cents). Marcheto associates alteration of the sizes of thirds
and sixths in this general direction, and extra-narrow semitones or dieses,
with inflected directed progressions involving mi-signs outside the _musica
recta_ gamut defined by a chain of fifths Bb-F-C-G-D-A-E-B -- or, in other
words, sharps. Here both the inflection and the momentary alteration of the
intonational fabric heighten our sense of excitement and anticipation of a
resolution to perfect consonance.

For Marcheto himself, based on many plausible readings of his mathematics,
an Archytan realization of his accentuated directed progressions involving
sharps might actually seem rather conservative and cautious. The Archytan
thirdtone at 28:27 or 63 cents is still easily accepted by my ear as a
diatonic semitone; diesis steps in the range of 42-48 cents, as suggested by
these plausible readings, involve for me a more radical warping of the
intonational fabric.

If Christopher Page is correct, tendencies in French and Italian music of
the late 13th and 14th centuries toward what I will generally term a
"Marchetan" (or "Marchettan," depending on which Latin or Italian spelling
of his name one follows) accentuation of Pythagorean tuning may have been
widespread. Page in fact considers such tendencies toward wider major thirds
and sixths than Pythagorean, narrower minor thirds, and extra-narrow mi-fa
semitones in directed progressions to be likely in various _musica recta_ as
well as _musica ficta_ contexts.

Based on this assumption, I will demonstrate how momentary Archytan
modifications of Pythagorean tuning could nicely service such progressions
both in two-voice and multivoice polyphony. A unifying theme will be the
tendency or desire for narrower mi-fa semitone steps in directed
progressions, which could lead singers to Archytan steps (or close
approximations) without any necessarily knowledge of or adherence to the
theory of the relevant ratios (e.g. the large 8:7 tone at 231 cents and
narrow 28:27 semitone at 63 cents).

Here I will start with conventional Marchetan inflections in two-voice
contexts, and then move to three-voice progressions, possible vertical uses
of the 8:7 tone and 7:4 minor seventh, and direct chromaticism.

What I seek to sketch out is a possible world often quite different than
that of regular Pythagorean intonation, and featuring such sonorities as the
extra-active 7:9:12 (e.g. E3-G#3-C#4) or 14:18:21 (e.g. A3-C#4-E4), and also
the alluringly smooth 6:7:9 (e.g. C#4-E4-G#4) -- and also, in styles
favoring a robust use of the vertical seventh, 4:6:7 (e.g. E3-B3-D4 in an
progression to F4-C4 of the kind that occurs in Machaut). Whatever may have
happened in 14th-century Europe, exploring this world of sound can expand
our musical universe in the 21st century.

Margo Schulter
19 June 2017
Reviewed 5 July 2017