Re: Response to Paul Erlich and Monz

Hello, there, and I'd like to reply to some comments by Paul Erlich
and by Joe Monz (whom I'll refer to, following his request for
informality, as "Monz").

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1. Medieval polyphony and "5-limit"
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First of all, not so surprisingly, I'd like to agree with Paul's
eloquent statements about the stylistic issues raised by any
proposal for 5-limit tuning of medieval polyphony based on a
quintal/quartal as opposed to tertian system of harmony. It does seem
to me a great deal like proposing that Monteverdi or Beethoven were
actually performed in 7-limit or 9-limit because these tunings are
preferred by some 20th-century just intonation (JI) enthusiasts.

As Paul remarks, there may not be very much to add on this question,
but I might mention an analysis of the opening of Machaut's Mass in
Easley Blackwood's _The Structure of Recognizable Diatonic Tunings_
which concludes that Pythagorean is much preferble to 5-limit.

Also, I'm tempted to suggest that if one _must_ propose an alternative
to Pythagorean (3-limit JI) for such music, why not something like
12-tet, which would still permit relatively compact diatonic
semitones, and leave thirds and sixths relatively active? If asked to
perform Machaut in something other than 3-limit JI, I'd certainly
prefer 12-tet to 5-limit, as does Blackwood.

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2. Marchettus: the Musical Context
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Having alluded to the controversy between Marchettus of Padua (1318)
and his critic Prosdocimus, who wrote about a century later to condemn
his "errors," I'd like to make it clear that I do not share the
negative judgment of the latter theorist, although I agree that the
mathematics of Marchettus are not always clear.

In fact, I'd like to urge that Marchettus is a stellar theorist and
interpreter of a world of early 14th-century music, and that this
mathematics be interpreted -- however uncertainly -- in the context of
his general theory of concord and discord.

From this viewpoint, I would urge that far from advocating "5-limit"
tuning, Marchettus is leaning toward something like a "7-limit" tuning
for cadential thirds and sixths.

First of all, Marchettus emphasizes the instability of thirds as well
as sixths by referring to these intervals as "tolerable discords" --
in contrast to the typical 13th-14th century term "imperfect
concords." This may be a rather fine semantic distinction, but it
hardly suggests a "5-limit" outlook which would seek more blending
thirds and sixths than Pythagorean at the expense of wider
leading-tones.

In fact, Marchettus deserves recognition and admiration for not only
formulating the principle of "close approach," but applying it as it
were on a microtonal level. This principle requires that a third
contracting to a unison be minor, and third expanding to a fifth, and
likewise a sixth to an octave, be major. Marchettus carries this
standard 14th-century rule further by calling for a cadential
leading-tone as narrow as possible, in theory only a diesis equal to
something like 1/5-tone or 2/9-tone (about half of a normal
Pythagorean semitone at 256:243 or ~90.22 cents).

However one chooses to realize the precise mathematics -- and a
leading-tone of around 28:27, associated with "superwide" cadential
major thirds at around 9:7 and major sixths at around 12:7 is one
possibility -- Marchettus is calling for _narrower_ leading-tones and
_wider_ major thirds and sixths at cadences, the opposite of 5-limit.

As a curious aside, I might note for Paul's amusement that a literal
reading of Marchettus's diesis might not be too far from a 22-tet step
of ~54.55 cents.

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3. Medieval English polyphony
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As we have discussed, there is clear evidence for tertian harmony in
some regions of 13th-century England, about two centuries before the
transition to such a harmonic system (and to 5-limit or meantone) on
the Continent. Monz raises the question of possible origins for this
predilection.

A possible clue is given by Giraldus Cambrensis, a philosopher and
traveller who records around 1200 that the people of certain regions
in Britain have a unique style of part-singing. He suggests that they
may have acquired this technique from the Danes, who had often invaded
and occupied these same regions. If one couples these statements with
those around 1275 from Coussemaker's Anonymous IV that thirds are
counted "the best concords" in the "Westcountry" of England, then a
tertian or "5-limit" interpretation of Giraldus is certainly possible.

Also, a Hymn to St. Magnus from the Orkney Islands for two voices
mostly in parallel thirds may date from around this same era, and come
from a seat of Norwegian culture. The Lydian mode of this piece might
possibly also suggest a Scandinavian provenance, since this mode is a
favorite, for example, in traditional Icelandic music.

Here it should be emphasized that English music of the 13th and 14th
centuries varies a great deal in style, ranging from pieces actually
ending with sonorities including thirds, to those more or less similar
to Continental 3-limit music of the same era.

Exactly what intonation may have been used for the music of Dunstable
(c.1370-1453) is an interesting question, although some approximation
of 5-limit wouldn't be surprising.

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4. Hoppin and the Gothic/Renaissance transition
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As Paul notes, Hoppin's description of the "paradigm shift" from
Gothic to Renaissance style during the epoch starting around 1420
(e.g. the early Dufay) nicely captures the 3-limit to 5-limit
transition. Here I'd just like add one point which ties into my
concluding remarks to follow about the relative "sophistication" of
theory in 1300 and 1500.

While Hoppin rightly focuses on the much more restricted use of
"dissonance" in Renaissance vis-a-vis Gothic style, I would like to
add that from another viewpoint, the Gothic in practice and theory
often treats major seconds or ninths and minor sevenths as to some
degree "concordant" or "compatible," while the Renaissance treats them
as unequivocal discords.

This important change reflects two factors: the shift from 3-limit to
5-limit in theory and practice, and the shift more specifically to a
rather _homogenous_ 5-limit texture as the Renaissance ideal. While
Hoppin nicely communicates the latter point, the first point may be of
special interest from the viewpoint of tuning systems.

In a system like that of medieval Continental Europe, based on fifths
and fourths as primary concords, the major second (9:8) and minor
seventh (16:9) have some "respectability." The first interval is the
difference between a 3:2 fifth and a 4:3 fourth, while the second is
the sum of two pure fourths. Likewise the major ninth (9:4) represents
the sum of two pure fifths.

Even as bare intervals, M2 or M9 and m7 are held to be somewhat
"compatible" by various medieval theorists of polyphony ranging from
Guido d'Arezzo (c. 1030) to Jacobus of Liege (c. 1325). Around 1200,
when Perotin and other composers begin writing regularly for three or
four voices, these intervals can actually have a relatively blending
effect when combined with two fifths or fourths.

Thus Jacobus much likes the "concordant" effect of the major ninth
when presented in a sonority with two fifths (e.g. G3-D4-A4, 4:6:9),
or a minor seventh in a sonority with two fourths (e.g. G3-C4-F4,
16:12:9). In 13th-century practice, a combination of fifth, fourth,
and major second is also popular (e.g. G3-C4-D4, 6:8:9, or G3-A3-D4,
8:9:12).

Such sonorities, also found in many other world polyphonies based on
fifths and fourths as primary concords, give Gothic music an element
of color and contrast. These sonorities have a kind of "energetic
blend" which fits the overall quintal/quartal flavor but at the same
time brings into play the considerable excitement and tension of the
major second or ninth or minor seventh. From Perotin to Machaut, these
relatively blending 3-limit sonorities (along with the unstable but
relatively concordant thirds) are an attractive feature of the music.

In fact, Jacobus advocates what might be called an ideal of
"pancompatibility": _all_ the regular Pyhagorean intervals, except for
outright "discords" (i.e. m2, M7, A4, d5, m6), may be used pleasingly
in a wide range of vertical combinations, many of which he
catalogues. Notably included in these essential combinations are M2,
m7, and M9.

In contrast, the 15th-century style of "panconsonance" described by
Hoppin relegates major seconds or ninths and minor sevenths to the
status of restricted discords. This shift, advocated by "modern"
14th-century theorists such as Johannes de Muris (or a student
writing "according to" his teaching around 1320), gets more and more
consistently implemented in practice starting around 1420, Hoppin's
transitional epoch.

Also, during the epoch of 1420-1450 there seems to be an increasing
tendency to avoid parallel fifths more consistently in practice as
well as theory. Thirds and sixths more and more emerge as the
privileged intervals: both the traditionally preferred fifths, and the
"9-based" intervals of 3-limit (M2, m7, M9), lose their medieval
liberties.

The result, as Hoppin eloquently describes, is a Renaissance ideal of
rather homogenous textures flowing smoothly between tertian (5-limit)
harmonies, meaning "the third plus fifth or sixth" (Zarlino, 1558).

Accompanying this shift in styles is a shift in tunings: from 3-limit
JI as a medieval ideal for voices and a very practical keyboard
scheme, to 5-limit JI as a vocal ideal and meantone for keyboards.

By 1477, Tinctoris is reporting the fashionable opinion that there is
no music worth hearing save that written in the previous 40 years; and
by 1482, Ramos is describing 5-limit JI and very likely some kind of
meantone tuning for keyboards.

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5. Theory between 1300 and 1500
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From the above, it might be inferred that the concord/discord theory
of a musician such as Jacobus around 1300 could actually be regarded
as more "sophisticated" than that of the paradigm prevailing around
1500. Jacobus, like many 20th-century theorists, finds that M2 or m7
has a notable degree of "concord," in contrast to m2 or M7. The
Renaissance simply regards all seconds and sevenths as "discords."

Here I do not wish to make invidious comparisons, since each theory
nicely fits the style of a beautiful music. However, I do wish to
counteract the common view of "progress" in which the 18th-century
harmonic system becomes a kind of directive goal for all previous
practice and theory. The 3-limit harmony of the 13th century is very
beautiful, and very non-18th-century.

Monz raises the question as to whether medieval theorists described
Pythagorean tuning because they did not have available the mathematics
to describe anything else, theory's tools "improving" between 1300 and
1500.

Here one might raise the question of why English theorists such as
Theinred of Dover (13th c.?) and Walter Odington (c. 1300) write of
thirds at or near ratios of 5:4 and 6:5, while the advocacy of such a
tuning on the Continent only occurs around the time of Ramos
(1482). To me, an attractive hypothesis would be that tertian or
"5-limit" styles of harmony are recorded in regions of 13th-century
England, but only became common in Continental composition during the
15th century.

However, we are fortunate to have direct evidence that a Continental
theorist around 1300 favoring Pythagorean tuning _was_ quite capable
of describing 5-limit and 7-limit ratios.

Jacobus of Liege, in the Fourth Book of his _Speculum musicae_
("MIrror of Music") comparing the various concords, remarks that it
would be quite possible to build instruments using such interval
ratios as 5:1 and 7:1, but that these intervals would not be properly
formed in terms of whole-tones and semitones.

He goes on to demonstrate what we call the syntonic and septimal
commas: 5:1 or 80:16 is smaller than the regular major third plus two
octaves at 81:16 by a factor of 81:80; and 7:1 or 63:9 is smaller than
the regular minor seventh plus two octaves at 64:9 by a factor of
64:63.

The inquiring Jacobus also notes the gap between the classic 3-limit
concords of the fifth (3:2) and fourth (4:3), and the next "useable"
superparticular ratio (i.e. n+1:n) of the major second (9:8),
suggesting that the quite "imperfect" quality of the latter concord
may reflect this gap. Thus he recognizes that the intervening
superparticular ratios (5:4, 6:5, 7:6, 8:7) are conceivable in theory,
but don't fit the tuning system used in practice.

To me, this approach seems much like that of 20th-century authors who
observe that the harmonic series indeed includes a seventh partial,
but that it doesn't fit the tuning of classical European music. Those
who appreciate the musical value of intervals at or near 7:4 may
rightly emphasize that such authors are hardly exhausting the
possibilities. However, the issue is one of style rather than of
mathematics.

Around 1500, a mathematical change in music theory is indeed afoot:
the recognition of the _irrational_ ratios needed to describe the new
temperaments such as keyboard meantones or the coming 12-tet for
lutes. This trend does indeed correlate with an interest in Euclidean
geometry as a method for calculating organ pipes dividing a 9:8
whole-tone into two equal parts, and eventually also with the
16th-century admiration of Aristoxenes.

In my view, the shift in 15th-century theory toward 5-limit ratios
nicely fits the changing practice of the times as described both by
contemporary authors such as Tinctoris and by modern writers such as
Hoppin.

Since a 5-limit system is indeed more complicated than a 3-limit
system, theory inevitably became more complex; and at least one
16th-century theorist argues that some vocal temperament is required
in order to avoid undesired pitch shifts. Vicentino, by neatly
proposing a 31-tet keyboard as a standard for vocalists, avoids such
"comma shift" problems while opening up new microtonal possibilities
with his 1/5-tones of the "enharmonic genus."

One point not so often noted is that the Renaissance lost touch with
Gothic theory and practice, and that it is fun to imagine how music
history and theory might have been different, for example, if
Monteverdi had cited the bold 3-limit minor sevenths of Machaut in
justifying his own "modern" 5-limit liberties.

Most respectfully,

Margo Schulter
mschulter@value.net