Re: Fine points on Erlich meantone list -- Aaron, Vicentino

(Please note that an earlier copy of this message bounced -- I'm not sure
why, but am resubmitting. I got a curious message about "no user here by
that name, and would welcome an explanation if there's an appropriate
place to inquire, or if someone here would like to discuss this with me
via e-mail.)

Hello, there, and this is a quick note on Paul Erlich's table of
meantones, maybe a bit of "fine print" which may illustrate why any
such brief table must necessarily either list some items inviting
lengthy qualifications or annotations, or else make some omissions
which could be viewed as unsatisfying.

In another article, I hope to address the interesting question of
sonorities around 1600 involving minor sevenths and their intonation
in meantone, which I find quite appropriate. Of course, my attitudes
toward meantone may be influenced by that the fact that for me it _is_
my "common practice" for Renaissance and Manneristic music, along with
Pythagorean for medieval music. People who mostly compose or play in
other styles may have different views of what is "natural" or
"pleasing," something only to be expected on a Tuning List as diverse
as this one.

Here I'd like specifically to comment on the items in Paul's table
listing Pietro Aaron (1523) as the first to describe 1/4-comma
meantone, and Nicola Vicentino (1555) as the first to describe 31-tone
equal temperament (31-tet). In the first case, I agree with Paul and
various other scholars, although noting Mark Lindley's divergent view
on this point. With Vicentino and 31-tet, to borrow from the title of
a famous book by Peter Abelard, one might reply _Sic et non_, "Yes and
no." Drawing the Vicentino/31-tet connection involves a bit of
"kludging," but simply to omit it could be taken as at least equally
misleading.

In defense of Paul, I'd like to begin by recognizing that two leading
scholars of musical Mannerism in the 16th century, Maria Rika Maniates
and Claude Palisca, _have_ identified Vicentino's first tuning of his
36-note archicembalo with 31-tet, as did Lemme Rossi in 1666.

Similarly, various books on Renaissance music and tuning (including
Owen Jorgensen's) _have_ equated Aaron's meantone temperament
described in his _Toscanello in Musica_ (1523) specifically with a
1/4-comma tuning.

Let's look at both cases, and see the complications that arise with
Vicentino, and also Mark Lindley's argument that Aaron's tuning of
1523 does not necessarily specify a regular 1/4-comma temperament
for all intervals, although he is ready to agree that it describes
this tuning for the first five notes C-G-D-A-E.

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1. Vicentino: 31-tone quasi-equal temperament (31-tqet)
-------------------------------------------------------

We find that Vicentino indeed describes a tuning dividing each
whole-tone into five "minor dieses" or fifthtones, as I call them,
which he apparently regards as equal or equivalent. This does
precisely fit the concept of 31-tet. At the same time, however, he
notes that the first 12 notes of the tuning (Eb-G#) are tuned as on
usual keyboard instruments. This suggests the likelihood that his
conceptually equal fifthtones were realized as an _almost_ equal
31-note meantone based on pure major thirds -- in other words,
1/4-comma temperament.

However -- and this is worth checking out further -- I seem to recall
that at at some point, Vicentino describes his alternative tuning of
19-note meantone (Gb-B#) plus the second manual of his instrument in
pure fifths with these notes as realizing Ptolemy's syntonic diatonic
-- in other words, 5-limit just intonation for vertical sonorities. If
so, he implies a basic meantone temperament with pure major thirds, or
1/4-comma.

Here a problem of interpretation for anyone drawing up a brief chart
is that in Vicentino's time, the very slight difference between a
meantone with precisely equal fifthtones (31-tet) and a meantone with
pure major thirds (1/4-comma) might not have been a consideration. If
we take a 17th-century or later mathematical approach which takes
account of this difference, then "31-tet" may imply a deliberate
albeit minute tempering of the major third in order to obtain
_precisely_ equal fifthtones.

One possible solution for a brief table might be to invent a new
acronym, and say that Vicentino introduced "31-tqet" -- 31-tone
quasi-equal temperament. Because he is so important in the history of
the 31-division of the octave (precisely or _almost_ equal), I
wouldn't want to omit him, and 31-tqet permits his "mathematically
correct" inclusion.

Again, I want to see if I can find that passage where Vicentino may
speak of his second tuning as producing the "syntonic diatonic." That
would indicate pure major thirds -- and might also, to anticipate the
next point, be taken as one of the first descriptions of the 1/4-comma
temperament.

---------------------------------------------
2. Aaron's 1/4-comma meantone -- how regular?
---------------------------------------------

As for Aaron, the most judicious statement might be that he is indeed
the first to describe a process for dividing an apparently pure major
third into four equally narrowed or tempered fifths which, if
continued regularly throughout a tuning, _will_ result in a 1/4-comma
temperament. For many people, that is enough. Mark Lindley, however,
prefers to emphasize the caution that Aaron does not in his tuning
instructions specify that all fifths and major thirds should be
tempered in the same way -- although he does not exclude such a choice
either. See his "Early Sixteenth-century Keyboard Temperaments,"
_Musica Disciplina_ 28:129-151 (1974), with pp. 139-144 on Aaron. Here
one might want to compare the translation in Pietro Aaron, _Toscanello
in Musica_ (vol. 3): Book II, Chapters XXXVII-XXXX; Supplement_, Peter
Bergquest, tr., Colorado College Music Translations 4 (Colorado
Springs, 1970), pp. 10-11.

Aaron, writing an introduction to music in Italian (_Toscanello_ is a
title honoring his native Tuscany), advises the not necessarily
experienced reader to start by making the octave C-C "just," and then
the major third C-E "sonorous and just, as united as possible." While
Aaron does not specify a pure 5:4 for the major third, this seems to
me a natural reading of "sonorous and just." Then the fifths C-G-D-A-E
are tuned so that each is slightly "flat" or "lacking," and each by
the same amount. For example, A should be the same "distance" from D
as from E -- in other words, the fifths D-A and A-E should be tempered
from "perfection" by the same quantity.

As far as I'm concerned, this is enough to justify listing Aaron's
instructions as a description the procedure for obtaining 1/4-comma
meantone -- as various authors have done, and Paul does in his table.
Further, Aaron adds after his C-G-D-A-E series of fifths that F should
be tuned by a similar but opposite procedure, making F in the fifth
F-C "a little high, passing a bit beyond perfection." If one assumes
symmetry, then 1/4-comma meantone indeed results.

Correctly pointing out that Aaron's instructions do not give a
mathematical description of 1/4-comma meantone, or explicitly direct
that all major thirds be made pure, Lindley raises the question of
what Aaron means by his statement that "thirds and sixths are blunted
or diminished" in this temperament.

One interpretation which occurs to me is that Aaron is comparing
meantone _major_ thirds and sixths with the same intervals in a
Pythagorean tuning with pure fifths -- where they are indeed somewhat
larger. Lindley emphasizes such ambiguous statements to argue that
Aaron's instructions do specify 1/4-comma for the first notes
C-G-D-A-E, but might lend themselves to various slightly irregular
tunings for the other fifths and thirds.

Another point made by Lindley is that Aaron uses the adjective
_giusta_ ("just") to refer not only to his initial octave and major
third (where "pure" is an attractive reading), but also for the
temperament as a whole -- _participatione & acordo giusto & buono_, "a
just and good temperament and tuning." However, Lindley himself is
ready to accept Aaron's opening C-G-D-A-E with its "sonorous and just"
major third C-E as a description of 1/4-comma. His statement about the
octave C-C and major third C-E being as sonorous or "united" as
possible would support this conclusion even we assume that "just" may
mean simply "euphonious" or "pleasing."

In his instructions for the final stage of the temperament, the tuning
of the sharps, Aaron directs that C#, tuned in relation to the fifth
A-E, should be a major third from A and a minor third from E, and
likewise with F# in relation to the fifth D-A, etc.

From one viewpoint, this language may simply be reminding the student
of the locations of major and minor thirds involving accidentals.
Lindley, however, pursuing his argument that a regular 1/4-comma
temperament is not _necessarily_ implied, argues that one _could_ read
this language to suggest something like Zarlino's 2/7-comma meantone
for the temperament of the sharps, with major and minor thirds
compromised by about the same amount from pure.

In arguing that Aaron's tuning is not _necessarily_ a regular
1/4-comma meantone, Lindley may have two motivations. First, he wishes
to emphasize that Aaron's instructions are not mathematically precise;
and indeed, I would agree that Zarlino (1571) and Salinas (1577) may
be the first known theorists to give such mathematical definitions of
1/4-comma and other meantone temperaments.

Secondly, Lindley wants to correct the view that 1/4-comma was a
universal standard in the 16th century. Since Aaron's instructions are
often taken as the paradigm case of this tuning, showing that they are
actually open to more than one interpretation would fit with his
larger campaign against "1/4-comma hegemony."

However, I find it noteworthy that without invoking any complex
mathematical concepts or even defining a syntonic comma, Aaron has
described in what I find beautiful as well as musicianly terms the
idea of tuning a pure major third and then dividing it into four
equally tempered fifths. Aaron's remaining instructions, including his
suggestion of a similar but opposite tempering of fifths in the flat
direction (F-C, and then Bb-F and Eb-Bb), permit a regular 1/4-comma
temperament, even if they do not compel it or define it in
mathematical terms.

If I were making a table like Paul's, I might list Aaron for 1/4-comma
and add a footnote or annotation like this:

"Aaron evidently describes a temperament with a pure major third C-E
and equally narrowed fifths for his first five notes C-G-D-A-E, with
further instructions permitting but not explicitly specifying that
other major thirds are pure; Zarlino (1571) and Salinas (1577) give
mathematically precise descriptions of a regular 1/4-comma tuning."

Anyway, having shared my annotations, I would add that I might feel
uneasy about a table of meantone tunings avoiding these issues by
omitting Aaron and Vicentino.

Most respectively,

Margo Schulter
mschulter@value.net