Intermittent Archytan touches in late medieval polyphony?
Part I: Some first two-voice examples

In order to show how the 8:7 tone, here considered first as a melodic step,
could have arisen in late 13th-14th century European music, we might best
begin with some simple two-voice progressions that are themselves familiar.
Here is one where the voices progress from a stable fifth through an
unstable third to a unison -- 5-3-1 -- with the "closest approach" principle
coming into vogue by around 1300 calling for a third about to contract to a
unison to be minor, if necessary by inflection. In closest approach
resolutions by stepwise contrary motion, one voice moves by a tone, and the
other by a diatonic semitone.

..-204.....-204...
...8:9......8:9...
B4------A4------G4
3/2.....4/3....32/27
702.....498.....294

...204......90...
...9:8....256:243
E4------F#4-----G4
1/1.....9/8....32/27
.0......204.....294

.3.......32......1
.2.......27......1

In this diagram, I arbitrarily the starting note of the lower voice, E4, as
the 1/1 reference for note locations, and thus note B4, on which the upper
voice starts, as 3/2. The melodic intervals for each voice are shown as
ratios, and in cents. Thus the lowest voice moves E4-F#4-G4, ascending by
steps of 9:8 or a regular tone, and then 256:243 or a regular limma or
diatonic semitone. Here the melodic ratios are shown with the larger number
first, 9:8 and 256:243, to indicate ascending motion. These numbers may be
taken as string lengths, as on a monochord, so that smaller numbers mean
higher notes.

The upper voice descends by two 9:8 steps, B4-A4-G4, with these melodic
steps shown as 8:9, with the smaller number or string length first, which
indicates descending intervals, as do also the notations in cents showing
negative values for these steps at "-204" cents.

At the bottom of the diagram, I show ratios for the vertical or simultaneous
intervals: E4-B4 at a stable 3:2 fifth; then F#4-A4 at an unstable but
relatively concordant or blending 32:27 minor third; and finally G4-G4, a
stable unison to which the minor third resolves. We thus have an alternation
of stable and unstable sonorities, or "perfect-imperfect-perfect" as we
might sum up this 5-min3-1 idiom.

Before focusing on the min3-1 resolution itself, we should briefly note that
all melodic steps are 9:8 tones or 256:243 semitones. The lower voice
ascends E4-F#4-G4 or 1/1-9/8-32/27 through a regular minor third (294
cents), moving by a tone and then a 256:243 limma; the upper voice descends
B4-A4-G4 through a regular major third or ditone at 81:64 (408 cents) made
up of two identical 9:8 tones. The ascent of the lower voice by a 32:27
minor third, combined with the descent of the upper voice by an 81:64 major
third, together bring us vertically from the fifth to the unison.

We now come to the resolution from F#4-A4 to G4-G4, and the inflection of
the lower voice. As one of the patterns of closest approach becoming
standard around 1300, a minor third often seeks to contract to a stable
unison, with one voice moving by a 9:8 tone, and the other by a compact
256:243 limma. The minor third at a usual 32:27 ratio is itself at once
somewhat blending and sweet, and at the same time has considerable tension
or complexity by comparison with a stable interval (1:1 unison, 4:3 fourth,
3:2 fifth, 2:1 octave). The resolution releases this tension in a satisfying
way, and also involves the decisive melodic motion of a 256:243 limma
(F#4-G4), here made to stand out because of the inflection.

In describing such standard resolutions, a few terms may be helpful. We may
term this closest approach resolution "contractive," because it involves a
minor interval contracting to a smaller stable or perfect interval (here a
unison). It is in the "intensive manner," meaning that it involves an
ascending semitonal motion ("intensive" in a medieval European context sometimes
meaning "ascending") and descending motion by a tone.

In usual Pythagorean intonation, of the kind that most theorists describe
and would likely take place on a 14th-century keyboard instrument, the step
E4-F#4, or 1/1-9/8, would be by a standard 9:8 tone, with the # sign (more
generally, in standard intonation, a "mi-sign" often resembling a modern
natural, but sometimes taking forms more like a modern sharp) raising the
step F4 in the regular or _musica recta_, at 256/243 (90 cents) or a limma
above E3, by an apotome of F4-F#4 at 2187:2048 (114 cents), the difference
between the limma and the 9:8 tone.

While 14th-15th century theorists use the concept of "coloration" or
"coloring" to describe different kinds of inflections, Ugolino of Orvieto
(writing around 1425-1440) uses an especially lucid terminology I will
borrow here. As he puts it, what we are doing with the F#4 inflection is
"coloring" the major third F4-A4 in the regular gamut at a usual 81:64
diatone (408 cents) so that it is reduced to a correct minor third at 32:27
(an apotome smaller, F#4-A4) which can efficiently attain its goal of the
unison with the desired melodic steps of 9:8 (A3-G3) and 256:243 (256:243).

Ugolino's coloration, where it is desired to reduce an unstable major
interval to a minor one for a contractive resolution, thus has intertwined
two dimensions. From a vertical perspective, the resulting F#4-A4 at 32:27
or 294 cents is an ideal size to resolve by opposite motions of a 9:8 tone
and 256:243 limma at 204 cents and 90 cents respectively, together producing
(204 + 90) or 294 cents total contraction, the amount needed to move from
this minor third to a unison.

From a melodic perspective, the inflection of the lowest voice reduces its
step in this directed progression from a 9:8 tone, F4-G4, to a semitone
F#4-G4 at a usual 90 cents. Both the inflection, and the following semitone
step, may add an element of decisiveness to the resolution. In the 13th
century, there are many beautiful directed progressions where all voices
move by steps of a tone; but by around the end of the 13th century,
resolutions involving semitonal motion were especially relished, a
motivation for many inflections.

For most writers of this era from around 1300 to 1440 or so, 9:8 tones and
256:243 limma are the appropriate melodic steps for the _musica recta_ gamut
or its extensions through _musica falsa_ (literally "false music") or
_musica ficta_ (which might be taken as "invented music" or "contrived
music"). The alternation of stability and apt instability, however the
latter is interpreted in a given style (with thirds and sixths as standard
"imperfect concords," and major seconds or ninths and minor sevenths also
regarded as having some degree of "concord" or "compatibility in certain
styles, e.g. Machaut), thus takes place within a classic and consistently
proportioned Pythagorean intonational fabric.

However, Christopher Page suggests that in practice, French and Italian
musicians of the 13th-14th centuries may have often followed tendencies
given a theoretical voice by Marcheto (or Marchetto or Marchettus) of Padua
in his _Lucidarium_ of 1317-1318. For Marcheto, what I have termed an
intensive resolution involving a note with a mi-inflection involving _musica
falsa_, as with F#4 in our example, should be sung higher than its usual
Pythagorean position. That is, the distinctive "sign of _musica falsa_"
which Marcheto advocates should raise it by more than a usual apotome. This
means that a minor third contracting to a unison like F#4-A4 will be yet
narrow than the Pythagorean 32:27 or 294 cents; and the directed semitone
step F#4-G4 (mi-fa) appreciably smaller than the already compact and
efficient 256:243 limma at 90 cents.

Let us suppose that singers take this general tact with our progression, and
that the performer of the lower voice happens to place F#4 at about a 64:63
comma (27 cents) higher than a canonical Pythagorean position. This tuning,
or something like it, would result:

..-204.....-204...
...8:9......8:9...
B4------A4------G4
3/2.....4/3....32/27
702.....498.....294

...231......63...
...8:7.....28:27
E4------F#4-----G4
1/1.....8/7....32/27
.0......231.....294

.3.......7.......1
.2.......6.......1

Here the usual descending 9:8 steps in the upper voice are unaltered, and
likewise the overall interval of a 32:27 minor third (E4-G4) by which the
lower voice ascends to meet the upper voice at a unison. However, the
melodic division in the lower voice of the 32:27 third is changed: we have
an 8:7 tone (E4-F#4) at 231 cents, larger than 9:8 by an Archytan or 64:63
comma; and an extra-compact 28:27 semitone or thirdtone step F#4-G4 at 63
cents. This melodic contrast between a generous 9:8 tone and an incisive
256;243 limma in Pythagorean intonation, noted by Mark Lindley, is yet
further enhanced.

Vertically, the opening 3:2 fifth leads to an imperfectly or relatively
concordant minor third with a different nuance: at a 64:63 comma smaller
than the usual 32:27 or 294 cents, this third at an extra-narrow 7:6 or 267
cents has a notably simple ratio, reflected in a distinctive acoustical
smoothness for some listeners, and at the same time can contract to its goal
of a unison in a superefficient manner, with steps of 9:8 and 28:27, or 204
and 63 cents (204 + 63 = 267 cents).

We might say that the 7:6 minor third could be described, borrowing and
extending Ugolino's concepts, as "supercolored," a comma closer to its
directed goal of the unison, and able to reach this goal through an
especially incisive 28:27 step.

This progressive momentarily stretches or alters the intonational fabric,
but not does not cause any persistent and inconvenient disturbance: the
resolution to a unison on G4 at a usual 32:27 minor third above E4 restores
the fabric to its usual proportions. Interesting, the melodic steps in the
lowest voice for this progression at 1/1-8/7-32/27 are like that of one of
the common permutations or steps for a tetrachord of the Archytan diatonic,
with 1/1-8/7-32/27-4/3 (0-231-294-498 cents), with steps of 8:7-28:27-9:8.
From the notes used in this example, including the accentuated F#4 at 8/7,
we can form this tetrachord from the notes E4-F#4-G4-A4.

How 14th-century French or Italian singers might have heard a 7:6 minor
third remains an open question. Theorists wrote of this ratio, called
sesquisexta ("and again a sixth part"), and were aware of its role in Greek
theory. To my ears, it very nicely serves the role of an "imperfect
consonance," being at once pleasing in itself, and yet seeking a stable
resolution. While the simplicity of the 7:6 ratio may give this third a
special allure, its compact size of 267 cents (exceeding a 9:8 tone by only
28:27 or 63 cents) may for some modern theorists introduce a degree of
acoustical dissonance or tension, since some of the upper partials of the
two notes are within the "critical band" where they may cause noticeable
beating. Since the medieval concept of an unstable "imperfect concord"
embraces both relative euphony and a degree of tension, the 7:6 third thus
for me has a special aptness for these accentuated progressions, although we
cannot be sure if 14th-century performers often shared this taste.

Let us consider another pervasive two-voice progression of the 14th century
which might, for musicians seeking the general kind of accentuated
progressions described by Marcheto, result in an 8:7 melodic step. Let us
consider first the classic and then the accentuated version:

........204...........90......
........9:8.........256:243...
..B3------------C#4----------D4
27/16.........243/128.......2/1
.906............1110.......1200

...............-204...........
................8:9...........
..E3-------------------------D3
..9/8.......................1/1
..204........................0

...3............27...........2
...2............16...........1

Here the two voices begin at the fifth E3-B3 and ultimately expand to the
octave D3-D4, with the lower voice descending by a 9:8 tone (E3-D3), and the
upper voice ascending by an overall 32:27 minor third divided into the 9:8
tone B3-C#4 followed by the 256:243 limma C#4-D4. The descent of a 9:8 tone
in the lower voice, together with the ascent of a 32:27 minor third in the
upper voice, add up to a total expansion of a 4:3 fourth (204 + 294 = 498
cents), the difference between a 3:2 fifth and a 2:1 octave.

Between these stable or perfect intervals of the fifth and octave there
mediates an unstable major sixth E3-C#4, made possible by the inflection of
the upper voice, at 27:16 (906 cents), which in another favorite closest
approach progression expands to the stable 2:1 octave, with the lowest voice
descending by a 9:8 tone and the upper voice ascending by a 256:243 limma. A
27:16 major sixth before an octave, like a 32:27 minor third before a
unison, can efficiently reach its directed goal through these steps. And
again, we have a voice -- the lower in the last example (E4-F#4-G4), and the
upper in this (B3-C#4-D4) -- with an ascending motion through a 32:27 third,
divided into a 9:8 tone followed by a 256:243 limma.

This progression of E3-C#4 to D3-D4 is again in the intensive manner,
involving ascending semitonal motion (C#4-D4 in the upper voice) and a
descending 9:8 tone. While the min3-1 resolution in the last example was
contractive, however, the Maj6-8 resolution is expansive, with a major
interval seeking to expand to its stable goal. The 27:16 major sixth is
quite bright and complex, combining relative concord with considerable
energy and tension impelling the directed progression to the 2:1 octave.

We can also describe this 27:16 major sixth, which mediates between the
stable 3:2 fifth E3-B3 and 2:1 octave D3-D4, as a "mediating sonority," and
the step C#4 in the upper voice that produces it as a "mediating tone." Such
a mediating note adds both melodic smoothness and vertical contrast to a
simple motion between stable concords by contrary motion, at once dividing
what would be the leap of a minor third in the upper voice, B3-D4, into
smooth stepwise motion, B3-C#4-D4; and producing the imperfect or unstable
major sixth to adorn the progression with an element of directed tension, a
perfect-imperfect-perfect alternation of 5-Maj6-8. Note that the voice with
the mediating tone approaches and leaves it in the same direction,
B3-C#4-D4.

Mediating sonorities can range from quite transient to dramatically
prolonged (e.g. in a final cadence); the 13th-14th century treatment of
vertical instability is often very flexible, and mediating sonorities are
one characteristic pattern in the interaction of vertical and melodic
dimensions.

Focusing on the resolution of Maj6-8, here E3-C#4 to D3-D4, we find an
example of what Ugolino calls "perfection": altering what would be a minor
sixth E4-C4 at 128:81 or 792 cents to make it a major sixth at a 114-cent
apotome (C4-C#4) larger, and thus bringing it to its full or "perfect" size
of 27:16 (792 + 114 = 906 cents), so that it can efficiency attain its goal
of the octave through the expected melodic steps of 9:8 and 256:243.

Ugolino's "perfection" more generally applies to expansive closest approach
resolutions where what would otherwise be a minor interval, here the 128:81
minor sixth E3-C4, is "perfected" by an inflection enlarging it to its full
and active major size, here 27:16, thus setting the stage for a satisfying
progression to the 2:1 octave. In this context, "perfecting" an unstable
major interval such as a sixth means bringing it to its full size, with a
vibrant and energetic quality that facilitates an economical and satisfying
resolution to a perfect interval.

Let us now consider an accentuated version of this 5-Maj6-8 pattern with an
inflection at C#4 to achieve the major sixth before the octave:

........231...........63......
........8:7..........28:27....
..B3------------C#4---------D4
.27/16.........27/14........2/1
..906..........1137........1200

...............-204...........
................8:9...........
..E3------------------------D3
..9/8.......................1/1
..204........................0

...3............12...........2
...2............7............1

As in the last example for the lower voice, here in the upper voice the
ascent through a 32:27 minor third, now B3-C#4-D4, is altered from the usual
division of 9:8-256:243 (204-90 cents) to 8:7-28:27 (231-90 cents), the
division found in the Archytan Diatonic. The 12:7 major sixth at 933 cents,
a 64:63 comma wider than 27:16, has a simpler ratio, and at the same time is
yet wider, and to me brighter and yet more dynamic in its expansive
"striving" toward the 2:1 octave. This striving is satisfied by melodic
motions of a descending 9:8 tone in the lowest voice, and an ascending 28:27
thirdtone step in the highest voice, thus (204 + 63 = 267) cents of overall
contraction, the amount needed to move from the wide 12:7 major sixth to the
octave.

Following and building on Ugolino's approach, we might describe the 12:7
major sixth as "superperfected," having in my judgment a yet more dynamic
quality that the usual 27:16, and resolving in a progression featuring the
narrow mi-fa step of 28:27. The accidental inflection at C#4, the extra
comma by which this note is raised (momentarily stretching the intonational
fabric), and cadential thirdtone step all add drama and interest, while
allow this fabric to return to its usual shape when the resolution is
accomplished.

While the vertical progressions and the interval aesthetics they reflect may
be specific to medieval Europe, there is a more general musical tendency
also seen in Near Eastern music with its often purely melodic focus that may
be operating here: a desire for narrower cadential steps in approaching a
musical goal. In a medieval European context in the late 13th and 14th
centuries, these steps are especially semitones involved in directed
vertical progressions, although some theoretical texts suggest similar
tendencies in monophonic chant and likely secular songs also. 

In a 14th-century European setting, this desire for yet narrower semitones
than Pythagorean for certain directed progressions is in harmony with a
vertical ethos contrasting imperfect and perfect intervals and sonorities,
where "superperfecting" a major sixth before an octave to make it an extra
comma wide (e.g. 12:7 at 933 cents as compared with 27:16 at 905 cents), or
"supercoloring" a minor third before a unison to make it an extra comma
narrow (e.g. 7:6 at 267 cents as compared with 32:27 at 294 cents) serves
expressive vertical as well as melodic goals.

While Marcheto associates intonationally accentuated progressions especially
with mi-inflections involving _musica falsa_ outside the regular gamut,
Christopher Page suggests that singers may found these nuances apt for
directed progressions within the regular gamut also. In the next part of
this article, I consider some of the possibilities this raises.

Margo Schulter
20 June 2017