Re: Vicentino, meantone, 31-tet

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Vicentino, Meantone, and 31-tet:
A Renaissance Perspective
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Hello, everyone, and I would like to comment on a Renaissance
perspective regarding the relationship between what are now often
called "meantone" tunings and the system of 31-tone equal temperament
(31-tet). In 1555, as others have already noted, Nicola Vicentino
(1511-1576) proposed his _archicembalo_ based on 31-tet as a standard
not only for new styles of keyboard music, but also for the intervals
and consonances of vocal music.

Here I would like first to consider a generic concept equivalent to
"meantone" in the 16th century (the term "meantone" itself being
later), and then to consider how Vicentino deems 31-tet as synonymous
with such a scheme, or more precisely, one manifestation of it.

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1. Renaissance meantone -- or _participatio_
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Because tertian just intonation (JI) is generally the Renaissance
ideal for vocal music, and because it happens that two of the
Renaissance temperaments now known as "meantone" (1/4-comma,
1/3-comma) feature just major thirds at 5:4 or minor thirds at 6:5
respectively, it may not be so surprising that meantone is sometimes
conceived as a kind of subset of tertian JI. Specifically, the term
"meantone" often tends to imply the 1/4-comma tuning with pure major
thirds, if no other specification is given.

However, Mark Lindley has noted that early descriptions of what we
would call "meantone" do not specify any mathematical quantity of
temperament for the fifths, but rather sometimes mention more
generally a property of _participatio_ or "participation" by which
fifths are made slightly narrower than pure. Gaffurius (1496), for
example, says that fifths are narrowed on the organ "by a small and
hidden amount" -- that is, by an amount so small as not to be easily
measured under a theory in which the comma tends to be the smallest
recognized interval.

By 1555, this _participatio_ has become so accustomed -- having come
into vogue perhaps a century or so earlier, around 1450, in the epoch
of Conrad Paumann[1] -- that Vicentino treats is as one of the two basic
attributes of "modern" practice. He describes the music of the time as
"mixed and tempered music," that is, music mixing the intervals of the
different genera (diatonic, chromatic, enharmonic), and featuring a
slight tempering or "blunting" of the fifth in order to make the
thirds more concordant, in contrast with earlier Pythagorean tunings.

Vicentinoo remarks that the fifth is such an excellent concord that it
pleases even when "blunted" or "truncated" in this way to improve the
quality of the thirds, treated in "modern" music as full concords.

Thus if we are to seek a Renaissance equivalent of the meantone
concept, it would be this concept of "blunting" the fifth. Indeed,
Zarlino presents his archicembalo as a realization and further
extension of this practice.

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2. Vicentino's meantone: 31-tet
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As Lindley discusses, various shades of meantone were likely tuned in
the 16th century, not necessarily involving pure major or minor
thirds. Early instructions may give nonmathematical directions: "Tune
the fifths, narrowing them until they do not completely please you."
Zarlino (1558) takes a more mathematical approach, discussing for
example his 2/7-comma temperament (a compromise which "splits the
difference" from tertian JI among major and minor thirds).

What these tunings share in common is, of course, what Vicentino calls
the "blunting" of the fifth -- and he sees his own 31-tet as
synonymous with this practice.

Thus in discussing the tuning of his archicembalo, he specifies that
the first 12 notes should be tuned "as the good tuners do it," duly
blunting the fifths. Then the remaining chromatic and enharmonic notes
are added by reference to these standard tones (the usual Eb-G#).

While Vicentino does not identify this usual 12-note tuning as
1/4-comma meantone, in fact such a tuning (with pure major thirds)
would be very close to 31-tet. In Vicentino terms, the identifying
mark of his tuning is that a major semitone is equal to 3/5-tone, and
a minor semitone to 2/5-tone, with 31 "dieses" of 1/5-tone being equal
to an octave.

Thus Vicentino regards his tuning as both: (1) A tuning carried out
according to the best standard practice, with the fifths "blunted" (in
later terms, "meantone"); and (2) A tuning where 31 dieses of 1/5-tone
each are equal to an octave (in later terms, "31-tet").

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3. Conclusion
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One conclusion might be that at least to Vicentino, the categories of
a "blunted-fifths" tuning and a tuning with semitones as precise
fractional parts of the tone (major semitone as 3/5-tone, minor
semitone as 2/5-tone) are not mutually exclusive.

Some modern tuning theorists might consider "meantones" and "n-tet's"
two distinct categories, and in fact this approach need not clash with
Vicentino's, as long as we recognize that some tunings may fit in both
categories. Thus 1/4-comma meantone is a meantone tuning which is also
not (exactly) any n-tet; 12-tet is an n-tet which is outside the range
of characteristic meantone (where thirds are within about 1/3-comma of
pure); and 31-tet is also a meantone tuning.

Of course, the very close proximity of 1/4-comma and 31-tet may help
in explaining why Vicentino directs that the first 12 steps be tuned
in accordance with usual practice. Then, as now, the difference
between a pure 5:4 at ~386.31 cents, and a 10/31 octave major third at
~387.10 cents, may have been difficult to notice.

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Note
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1. This date of around 1450 marks the point at which a Pythagorean
tuning was felt no longer to meet the needs of a vertical style
treating thirds as more and more restful. It should be added that
Vicentino and other 16th-century theorists were apparently familiar
with Ancient Greek teachings regarding the Pythagorean concords
(e.g. as transmitted by Boethius), but not necessarily with Gothic
practice or theory (say 1200-1420), in which intervals such as the
Pythagorean major third (81:64) are treated as pleasing but unstable
"semi-concords."

Most respectfully,

Margo Schulter
mschulter@value.net