Yahoo Tuning Groups Ultimate Backup tuning Re: Paul Erlich on 55-tET as possible 18th-century vocal model

Hello, there, everyone, and to my earlier comments on the very
valuable translations and remarks by Ibo Ortgies concerning guidelines
for vocal intonation in the 18th-century German tradition, I would
like to add a brilliant hypothesis by Paul Erlich concerning the
theory of dieses or commas in this era.

Although in my initial post I proposed that references to division of
the whole-tone into "nine commas" might draw on the tradition of
15th-century Pythagorean models, or the later Mercator/Kircher model
of 53-tone equal temperament (53-tET), Paul has masterfully suggested
an alternative model of 55-tET.

In this tuning system, which Paul mentions in connection with the
great Baroque composer Telemann, we have a whole-tone of 9 steps (as
in 53-tET also), but a regular diatonic semitone of 5 steps (rather
than the 4 steps of 53-tET, closely approximating the Pythagorean
limma at 256:243 or ~90.22 cents).

Thus in 55-tET, the large diatonic semitone of 5/9-tone (~109.09
cents) and the small chromatic semitone of 4/9-tone (~87.27 cents)
differ by a "comma" of 1/9-tone or ~21.82 cents, equal to the diesis
whether defined as the difference between such accidentals as D#-Eb
(Eb higher), or the difference between 12 pure fifths and 7 pure
octaves, or the difference between 3 pure major thirds (~392.72 cents
each) and a pure octave.

The temperament of the 55-tET fifth is around 3.77 cents, not too far
from 1/6-comma meantone (~3.58 cents), so that this tuning would
nicely fit period sensibilities as reflected by Silbermann's organs.

The diatonic semitone of 9/55 octave, at 109 cents, is quite close to
the interval of 16:15 prescribed by Zarlino's system (~111.73 cents),
while the diesis or comma of 21.82 cents is very close to the syntonic
comma at 81:80 (~21.51 cents). Given the recognition of Zarlino's
syntonic diatonic as the "modern scale" by authors such as Kirnberger
(1771), this resemblance might be an attractive feature.

Above all, unlike a Pythagorean or 53-tET model, 55-tET permits the
approximation of 18th-century intonational ideals with a _regular_
meantone tuning scheme: a major second at about 196.36 cents is equal
to two fifths up less an octave, a major third to two such
whole-tones, and so forth.

In contrast, although similarly regular for a typical 13th-14th
century Gothic setting, a Pythagorean or 53-tET model runs into major
complications if one is seeking major thirds near 5:4 rather than
81:64. While a regular diatonic semitone or limma of 4/9-tone is ideal
for Gothic music, the 55-tET diatonic semitone of 5/9-tone similarly
fits an 18th-century setting.

After reading Paul's personal communication to me, and picturing a
student indeed taking as a reference a split-key instrument tuned in
55-tET or its rather close approximation of 1/6-comma meantone, I find
this hypothesis most engaging. Here the diesis is synonymous with
1/9-tone, as the authors suggest, and at the same time can be
demonstrated on a keyboard in a neatly regular meantone temperament.

One lesson of this discussion, online and offline, may be the
importance of actually known a given period and its tunings as a basis
for interpreting statements such as the theory of "nine commas" in a
whole-tone.

As a medievalist, I was familiar with the Pythagorean approximation of
nine commas in a tone, but not with the 18th-century model of 55-tET
which Paul very helpfully called to my attention.

Also, my interest in 16th-century tunings has well accustomed me to
1/4-comma meantone with its diesis of 128:125 (~41.06 cents), about
1/5-tone -- but here the smaller, commalike, diesis of 55-tET or
1/6-comma meantone might be more relevant to 18th-century theory and
practice.

There is an interesting analogy which one might draw between the
"fifthtone" pair of 29-tET/31-tET, and the "ninthtone" pair of
53-tET/55-tET. While either 29-tET or 31-tET divides the tone into
five equal parts, the former tuning is nicely adapted to a Gothic or
neo-Gothic setting (and may approximate one possible interpretation of
the vocal intonation system advocated by Marchettus of Padua, 1318);
the latter to a Renaissance/Manneristic setting (closely approximating
Vicentino's 31-note meantone cycle of 1555).

The difference is that 29-tET divides the octave into 5 whole-tones
plus two _small_ diatonic semitones of 2/5-tone, while 31-tET divides
it into 5 whole-tones plus two _large_ diatonic semitones of
3/5-tone. The first division fits Gothic or neo-Gothic polyphony with
its active thirds near 127:100 and 13:11, and the second fits
16th-century style with its restful thirds near 5:4 and 6:5.

Similarly, 53-tET divides the octave into 5 whole-tones plus two small
diatonic semitones of 4/9-tone, and 55-tET into 5 whole-tones plus tow
large diatonic semitones of 5/9-tone.

The latter division fits the concord/discord scheme of 18th-century
music where thirds are full and stable consonances rather than partial
and unstable ones; at the same time, the milder tempering of the fifth
and the somewhat more compact diatonic semitone vis-a-vis 31-tET (or
the almost identical 1/4-comma meantone) may reflect a difference
between 16th-century and 18th-century styles.

Since some of the passages translated in this discussion point to
split-key instruments as a standard for correct vocal intonation, a
relevant question might be whether or how often such instruments may
have approximated a shade of meantone near 55-tET or 1/6-comma.

Incidentally, 2/11-comma meantone (fifths ~3.91 cents narrow) would
also give a close approximation of 55-tET, with a diesis curiously
almost exactly equal to the Pythagorean comma of 23.46 cents
(531441:524288).

To conclude, I find Paul's proposal a remarkable synthesis of
theoretical models and documented period practice, with the use of
temperaments of around 1/6-comma or 2/11-comma in split-key
instruments a question inviting more discussion.

Most appreciatively,

Margo Schulter
mschulter@value.net

Re: Paul Erlich on 55-tET as possible 18th-century vocal model

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