Re: Parallel fifths, 16th-century and other

Hello, there, and the topic of parallel fifths and tuning systems is
an interesting case of possible but problematic interactions.

First, please let me agree with a couple of the nice points already
made in this discussion. It bears emphasis, as Judith Conrad has
observed, that the prohibition of parallels in Renaissance
counterpoint is a rule relating to _part-writing_, not always to
sounds. Further, it might apply partially or not at all to certain
genres of keyboard music, as we shall see below.

In Gothic polyphony up to around 1300, there is no restriction on
parallel fifths in theory or practice, aside from a general precept
that one should prefer a variety of vertical intervals and
motions. While contrary motion is especially favored, parallel fifths
and octaves as well as fourths are a regular part of the texture, and
occur in a great proportion of favorite cadences for three and four
voices.

As Frank J. Oteri has remarked, this practice nicely fits both with a
musical style where fifths and fourths are the most rich and complex
stable intervals, and with a Pythagorean tuning presenting these
preferred intervals in their ideal ratios of 3:2 and 4:3. In such an
environment, parallel fifths seem as "natural" as parallel thirds in
the Renaissance or Romantic period.

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1. A conditional rule: Ars Nova theory (c. 1300-1420)
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Interestingly, however, this apparent stylistic equilibrium seems to
shift around 1300, when some "modern" discant and counterpoint
treatises begin to present a new rule against consecutive fifths or
octaves in basic two-voice writing. This rule, foreign to traditional
13th-century music, may be seen as one manifestation of the Ars Nova
movement placing itself in contrast to the older Ars Antiqua.

The new rule, in fact, is associated with the introduction of the term
_contrapunctus_ or "counterpoint," referring in a strict sense to the
technique of writing for two voices note-against-note (or literally
"point against point").

Parallel fifths and octaves are often excluded in these 14th-century
treatises specifically under the conditions of two-voice,
note-against-note, writing -- not if three or more voices are
sounding, and not necessarily in more ornamental writing.

This _partial_ prohibition of parallel fifths and octaves seems
connected with some other shifting guidelines of the Ars Nova
theorists:

(1) A positive approval of two or three parallel thirds or sixths --
in contrast to the 13th century, when parallels of all kinds can be
found, but fifths seem often the most favored;

(2) A change in the status of the fourth (4:3) above the lowest voice
from a concord ranking with-but-after the fifth to a discord, or at
best a "concord _per accidens_" when accompanied, for example, by a
fifth below; and

(3) A restriction of M2 (9:8), m7 (16:9), and M9 (9:4) to a
"non-essential" role as dissonances -- in contrast to a 13th-century
tradition in which these intervals had some "compatibility."

Both the partial prohibition of parallel fifths and octaves, and the
restriction of the M2/m7/M9 family of "9-based" intervals, could be
seen as a significant shift in the "balance of power" of the intervals
toward thirds and sixths.

However, fifths and (upper) fourths remain the most complex stable
intervals, and the goal of directed cadential progressions;
Pythagorean tuning remains the ideal for a theorist at the end of the
Ars Nova era such as Prosdocimus, who in his counterpoint treatise of
1409 explains that parallel fifths and octaves are avoided because the
purpose of counterpoint is that two voices should not sing the same
thing, but rather than they should sing different but concordant
melodies.

It bears emphasis that even in two-voice writing, or in simple
note-against-note textures, 14th-century composers do not necessarily
feel bound by the "modern" rule against parallels, any more than
Guillaume de Machaut feels obliged to avoid the bold use of
combinations involving M2, m7, and M9 (whether as the continuation of
an older tradition, or for his own artistic reasons).

Likewise, in some English styles of writing around the middle to late
14th century, a predilection for thirds and sixths does not exclude
cadences such as this common and very effective 13th-century formula:

e'-f'
d'-c'
g -f

(M6-8 + M2-4)

Both the bold M2 between the upper voices of the first sonority, and
the parallel fifths between the lower voices, demonstrate either that
the "modern" 14th-century rules are indeed to be taken as qualified,
or that composers feel free to write what sounds good to them.

It is only later, around the middle of the 15th century, that two
events appear to occur at roughly the same time: the _consistent_
avoidance of parallel fifths and octaves in multi-voice textures; and
the move toward meantone temperaments for keyboards.

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2. Renaissance parallels: part-writing, keyboards, and meantone
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Both developments might reasonably be associated with a move from
3-limit to 5-limit systems of harmony and tuning: thirds, ranked as
"imperfect" (i.e. "partial") concords in 13th-14th century theory, are
approaching musical stability.

This association might lead one to wonder: might parallel fifths have
sounded especially displeasing, even in a multi-voice texture, on a
keyboard with narrowed or "blunted" (Vicentino, 1555) meantone fifths?
In 1/4-comma meantone with pure major thirds, for example, the fifth
is about 5.38 cents narrower than a just 3:2; might a succession of
two such fifths, as has been suggested here by Frank Oteri, call
attention to this compromise of temperament in an especially blatant
way?

Looking through some keyboard compositions likely intended for
meantone (e.g. ending on a sonority of c-g-e'), it becomes clear that
the _sound_ of such fifths can hardly be in great disfavor, e.g.

1 2 3 |
e' e' d'
c' b
a g

In vocal and instrumental music alike, consecutive fifths are by no
means excluded if they involve crossing of voices so that parts, to
borrow the earlier words of Prosdocimus, do not "sing the same thing."
A very common three-voice cadence of the 15th century, although not
necessarily devised specifically to avoid violating the rule against
parallel fifths (which may only have come into consistent observance
after this cadence had become popular), illustrates this point:

e'-f'
g -f
c -c'

Here the upper two voices have a standard M6-8 resolution still
recognized in the Renaissance as a leading two-voice cadence; the
lowest voice leaps an octave to arrive at the fifth of the final
sonority. In sound, we have the two consecutive fifths c-g and f-c'.

Such progressions with "parallel fifths in sound" occur often in
vocal and keyboard music alike, and Vicentino (1555) explicitly notes
that consecutive fifths are not against the rule if the voices cross.

One radical theorist, Vincenzo Galilei (1581), goes so far as to
challenge the rule against parallel fifths as a principle of
part-writing on this basis: if the fifths continually played on
instruments do not offend the ear, why try to avoid parallelism on
paper? Uncannily mirroring the actual history, he suggests that the
rule originally applied to a texture of two voices only; but unlike
conventional Renaissance theorists, he sees the extension to
multi-voice writing not as a refinement but as a pedantic
extravagance.

As Kraig Grady rightly points out, Zarlino as an enthuastic exponent
of the rule objects not only to parallel fifths, but also to
consecutive thirds or sixths of the same proportion, for the same
reasons: there is no variety either in the consonances or in the
motions of the voices. However, he adds that the unequal whole-tones
of tertian just intonation for voices (9:8, 10:9) may provide some
variety to mitigate this effect if both voices move conjunctly.

A curious corollary of this analysis would be that parallel thirds or
sixths of the same size should be more acceptable for voices than for
a keyboard instrument, where a meantone tuning will result in the same
ratios for all regular whole-steps.

At any rate, Viadana (1602) reflects the situation at the end of the
16th century when he states in his explanatory notes to his collection
of church music for voices and continuo after the new manner: "The
organ part is never under an obligation to avoid two fifths or two
octaves, but those parts which are sung by the voices are."

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3. Conclusion
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As people have noted, there are some interesting tuning-related
reference points in the history of parallel fifths. They are
consistently avoided _on paper in conventional part-writing_ starting
around 1450, the same epoch evidently as the Pythagorean-meantone
transition; and they are "re-admitted" to "serious" practice and
theory on a regular basis at the end of the 19th century
(e.g. Debussy), about the same epoch as the increasing standardization
of 12-tone equal temperament with its almost-pure fifths.

Yet the acceptance of parallel fifths in sound during the Renaissance,
and their frequent occurrence in keyboard music, suggests that
interactions of this kind are more complex than they might seem,
involving something more than a case of "You compose what you tune."

Most respectfully,

Margo Schulter
mschulter@value.net