Re: Dieses or commas? (was: learning to sing...) for Ibo Ortgies

Hello, there, Ibo Ortgies and everyone.

Thank you for your fascinating translations from the 18th century on
singing and the art of correctly distinguishing between the large and
small semitones. Please let me confirm that your English translations
are not only understandable but very pleasant to read.

From the passages you quote, I might guess that there are actually at
least two different models of intonation involved here: a kind of
split-key meantone-oriented model where indeed the "lesser diesis" of
128:125 (~41.06 cents) or about 1/5-tone defines the difference
between the semitones; and a model evidently borrowed and adapted from
late medieval or very early Renaissance times which divided a
whole-tone into nine "commas."

The latter model resembles both 15th-century and later explanations of
Pythagorean intonation where the small and large semitones
(respectively the diatonic _limma_ and chromatic _apotome_ from a
conventional Gothic point of view) are estimated at about 4/9-tone and
5/9-tone, and the system of 53 tones or "commas" to an octave
attributed to the younger Mercator and expounded by Kircher (1650).

In a 16th-century or later setting, the latter model is often modified
so that the usual diatonic semitone is the _large_ semitone of "five
commas," and the chromatic semitone the _small_ semitone of "four
commas." This reversal, in effect, produces a close approximation of
just intonation with pure ratios of 3 and 5, since the 1/9-tone
"comma" is quite close to the syntonic comma of 81:80 (~21.51 cents).

In 53-tone equal temperament (53-tET), where the ninefold division of
the tone precisely obtains, this 1/9-tone is about 22.64 cents --
quite distinct from the 1/4-comma meantone diesis of almost twice this
size often more or less closely approximated on split-key instruments.

However, in 5-based just intonation as in a 1/4-comma meantone tuning
with pure major thirds (sometimes known in Germanic traditions as the
"Praetorian" tuning because of its advocacy by Michael Praetorius in
his _Syntagma Musica_ of 1619), the lesser diesis is equal to the
amount by which three pure major thirds at 5:4 (~386.31 cents) fall
short of a pure 2:1 octave, or 128:125.

It may also be defined as the difference between a major third and a
diminished fourth, with the latter interval (e.g. C#-F) equal either
in 5-based just intonation according to Ptolemy's syntonic diatonic as
adopted by Zarlino, or in 1/4-comma meantone, to 32:25 (~427.37
cents), again a difference of 128:125 or about 1/5-tone. This lesser
diesis also defines the difference between the diatonic semitone of
16:15 and the usual chromatic semitone of 25:24 in the syntonic
diatonic.

During the meantone era of the late 15th-17th centuries, theorists
sometimes refer to both the 1/5-tone (lesser diesis) and 1/9-tone
("comma") models. For example, in the introduction to his "example of
circulation" demonstrating how one make cadences in turn on all 31
steps of his enharmonic keyboard, the _Sambuca Lincea_, Fabio Colonna
(1618) asserts that this musical piece proves that the whole-tone
divides neatly into five tones, and not into nine commas as some have
maintained. (His piece has a chain of cadences progressing around the
circle of fifths or fourths, and as he notes demonstrates the use of
such routine idioms as suspended fourths resolving to thirds.)

Of course, we know that either the 1/5-tone or the 1/9-tone model may
hold depending on the tuning system in use: 1/5-tones in a meantone at
or near 1/4-comma; and 1/9-tones in Pythagorean or 53-tET.

Both your fascinating sampling of 18th-century opinions and a
discussion of intonation in Kirnberger (1771) suggest that both models
may have played a role in the theory of this era also. For Kirnberger,
the two "standard" models for vocal intonation are the "older"
Pythagorean scale and the "modern" scale of Zarlino. For keyboards, he
favors some form of unequal well-temperament, and specifically warns
against the levelling of the major and minor keys which would occur
were 12-tET to prevail.

Here are passages you translate where a model based on the division of
the tone into nine "commas" rather than five dieses seems indicated to
me:

[Agricola, 1757, after discussing split keys, now considered outmoded
with the art of 12-note well-temperaments favored instead]

> ...[follows a short discussion of the division of the whole tone in
> nine "commas"]

Here it's interesting that the same author would apparently recommend
split keys as a model of correct intonation (likely separated by about
1/5-tone, at least if we assume a "Praetorian" or similar temperament)
and then present the 1/9-tone or "nine comma" model.

Such a mixing of models, however, may have interesting precedents.
Thus around 1500, theorists such as Gaffurius and Aaron present
traditional Pythagorean tuning with its small diatonic and large
chromatic semitones as the favored theoretical model, although both
describe some form of meantone tuning as what is used in practice on
keyboard instruments. Here the nature of the Pythagorean-meantone
transition makes such theoretical mixing of the two approaches not so
surprising.

[From Quantz, 1752]

> The major semitone has 5 commata, the minor has 4 [commata].
> Therefore the E-flat must be a comma [=small diesis] higher than D#.

Here, we have the theoretical model of the "nine commas" to a tone, as
would literally obtain in 53-tET as suggested by Kircher, here
following an interpretation of interval spellings based on a major
third close to 5:4 (17/53 octave), with a large diatonic semitone of 5
"commas" (close to 16:15). Here the difference between Eb and D# would
be roughly equal to either a syntonic comma of 81:80, or to the
Pythagorean comma of 531441:524288 (~23.46 cents).

The "five diesis" model is especially characteristic of enharmonicists
such as Vicentino (1555) and Colonna, and the former specifically
recommended the archicembalo with its 31-note division of the octave
as a model for singers. This model, of course, would also fit split
key instruments of from 13 to 16 notes per octave as long as they are
tuned in a meantone temperament not too far from 1/4-comma.

However, there is one possible hypothesis that might roughly reconcile
the "split-key diesis" and "comma" models: a split-key instrument
tuned not at or near 1/4-comma, but rather in the kind of meantone
with major thirds somewhat wider than pure favored around 1700, say
1/5-1/6 comma (the latter temperament being identified with the famed
organ designer Silbermann).

While the meantone diesis at 1/4-comma is indeed identical to the
lesser diesis at about 1/5-tone or ~41.06 cents, as the amount of
temperament is decreased this diesis becomes smaller. At 1/5-comma, it
is about 28.16 cents, say 1/7-tone; while at 1/6-comma it shrinks to
about 19.55 cents, say 1/10-tone, the latter quite close to the
"comma" of the traditional ninefold division.

By a curious coincidence, 2/11-comma produces a diesis of about 23.46
cents, virtually identical to the Pythagorean comma which inspired the
approximate "nine comma" division of the tone proposed in the 15th
century. If later split-key instruments were often tempered by about
this amount, then the "nine comma" model _would_ quite nicely fit the
actual intonation, and difference between semitones, on these
keyboards.

This raises a question, however: what would be the typical amount of
meantone temperament on late split-key instruments, or on those known
to the 18th-century authors you quote?

There is at least some anecdotal evidence that split-keys would have
been useful for 1/6-comma organs such as Silbermann's; a humorous
story has it that Bach would delight in playing in remote keys on such
instruments in order to demonstrate the limits of satisfactory
transpositions, and once even induced Silbermann himself to leave the
scene in order to escape the dissonance of his own organ's Wolf!

Was the taste for lesser amounts of temperament in meantones around
1700 inspired in part by a desire to make diminished fourths at least
marginally "playable" as major thirds by the standards of the times --
or were other considerations such as smoother fifths or narrower
diatonic semitones the main motivation?

Would there be a tendency to temper split-key meantone instrument
somewhat more heavily, since correctly-spelled thirds would be
available at more locations, while somewhat lessening the temperament
of 12-note meantone instruments in order to make diminished fourths
more nearly equivalent to usual major thirds?

Please let me warn that here I am getting outside my main era of
familiarity, and should be especially subject to correction.

However, the interplay between the "five dieses" and "nine commas"
models for dividing the whole-tone is apparently a notable factor in
18th-century theory, as your quotes suggest, and as in 16th-17th
century theory, this interplay might merit much further study.

Thank you again for your translations, at once informative and
stirring the imagination to consider more closely the interactions of
singers, instruments, and musical concepts adapted to new settings.

Most appreciatively,

Margo Schulter
mschulter@value.net