Re: 2-3-7 optimizations (for Graham Breed & Paul Erlich)

Hello, there, Paul Erlich, and thank you for asking two excellent
questions, the first especially which helps me (hopefully) to correct
promptly a misunderstanding which a reader might very naturally draw
from my post.

Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM> wrote in Tuning Digest
1093, quoting my comments on 19-note meantone keyboards in a dialogue
with Graham Breed:

>> Costeley (1570) describes a similar 19-note keyboard for 19-tET,
>> and 19-note keyboards or _cembali chromatici_ ("chromatic
>> harpsichords) had some vogue in Naples around 1600, the environment
>> of the famed composer Gesualdo, where keyboard composers such as
>> Trabaci wrote pieces especially for this instrument.

> Are you saying Trabaci wrote specifically for 19-tET? Any other composers,
> with dates, that we might add to Joe Monzo's list in his dictionary entry
> for "Equal Temperament"?

Here the answer is "No," although a reader could easily draw such a
conclusion from my failure to distinguish the general theme of 19-note
keyboards from the more specific theme of 19-tET as advocated by
Costeley.

The 19-note _cembalo chromatico_ was tuned very likely to 1/4-comma
meantone (pure major thirds) or something similar; Zarlino's 2/7-comma
tuning with major and minor thirds equally tempered by 1/7 comma, for
example, could have been an alternative. It thus presented an _open_
19-note tuning (typically Gb-B#).

There are some revealing pieces of evidence confirming this
interpretation. First, Trabaci wrote a piece with a sonority of
D#-F##-A#, calling for a major third above D#, and going one fifth in
the sharp direction beyond the usual 19-note Gb-B# range. He advises
that this sonority may be played D#-F#-A# (with a minor third) on a
usual _cembalo chromatico_. This suggests that the piece may have
written especially for an _archicembalo_ or similar instrument with a
full 1/4-comma meantone cycle of 31 notes a la Vicentino (1555) and
Colonna (1618), with one such instrument (designed a bit differently
from the instruments of either of these authors) coming down to us
from around 1606.

Also, Colonna says that the first 19 notes of his 31-note _Sambuca
Lincea_ are tuned as on the "chromatic harpsichords" -- well-known in
Naples. Since his system (like Vicentino's) is based on a 31-note
cycle dividing the whole-tone into five parts, this points to
1/4-comma meantone as a likely solution -- or as described in later
17th-century theory, 31-tET (Rossi, Huygens).

Possibly one of the reasons I may not have made a due distinction
between 19-note keyboards and specifically 19-tET ones is that Graham
and I were mostly discussing 19-note meantones in the vicinity of
1/4-comma or 31-tET, so that the Costelely 19-tET application was a
sort of digression; but that is much clearer once stated!

Thank for your question, which gives an opportunity to suggest that
Trabaci might be placed under "likely 1/4-comma meantone with 19 or
more notes," also very close to "19-or-more-out-of-31-tET."

>> In contrast, a minor seventh at or near 9:5 (e.g. a Pythagorean
>> augmented sixth, to continue our "role reversal" theme) seems to me
>> often more tense than either a 16:9 or a 7:4, although the
>> 1020-cent interval of 20-tET is quite fine,

> 1020 cents is only 2 cents off 9:5 -- certainly "near", wouldn't you
> say?

Yes, and my intended interpretation was, "When I compare the tunings
of the minor seventh, I tend to find 9:5 more tense, but interestingly
I find the very close approximation in 20-tET not especially tense in
practice." Curiously, 1020 cents is also almost exactly the size of
the Pythagorean augmented sixth (e.g. Eb-C#) or _pentatonus_ of
Jacobus of Liege (five 9:8 whole-tones of ~204 cents each,
59049:32768) -- a ~1.95-cent schisma wider than 9:5.

As long as we're discussing this point, an interesting thing about
20-tET is that neo-Gothic minor seventh sonorities combining this
interval with a minor third and a fifth have alternative forms such as
0-720-960 cents (upper interval of 240 cents) or 0-720-1020 cents
(upper interval of 300 cents). The upper interval in either case
typically contracts to a unison, acting in a minor-third-like way,
with the outer minor seventh contracting to a fifth.

Anyway, in generalizing that 16:9 or 7:4 seems to me maybe a bit
milder than 9:5, I wanted to cite 20-tET as an example to qualify this
generalization; also, in Renaissance/Manneristic meantone, I find the
minor sevenths quite fine (midway between 16:9 and 9:5), and suspect
that a pure 9:5 might be similarly quite in keeping with the style.

Any of these sizes seems to me a variation in "modal color" or the
like on the basic theme of a minor seventh, and I would add that a
neat feature of 20-tET is the way it provides different sizes of
ordinary intervals in a usual neo-Gothic arrangement -- unlike 22-tET
for neo-Gothic, "usual" is here a bit different from "regular," as I
hope to discuss soon in an article on this tuning.

Maybe 20-tET to neo-Gothic is like 22-tET to Classic diatonic/triadic;
there are all kinds of incredible possibilities, but not exactly the
accustomed ones.

Most appreciatively,

Margo Schulter
mschulter@value.net