Re: Vicentino's piece in Vicentino's adaptive JI (MIDI)

Hello, there, Bob and Paul, and thanks for your encouragement on the
Vicentino adaptive tuning, and for your comments, which give me some
ideas.

For now, I'll try quickly to address the question of how a "virtually"
closed system such as 1/4-comma meantone, a circulating temperament
with some minor asymmetries (at least in theory, with practical tuning
variations maybe greater than differences in theoretical models), led
to the first precise formulations of 31-tET.

First, curiously, neither Vicentino (1555) nor Colonna (1618)
specifically says: "The way to realize this system of an octave
divided into 31 dieses or fifthtones, with all intervals available
from all steps, is to tune each regular major third at a pure ratio of
5:4, or to temper each regular fifth by 1/4 syntonic comma."

With Vicentino, the use of his instrument for adaptive JI with
"perfect fifths and perfect thirds," based in his second tuning on the
same "usual practice" meantone temperament of the first 19 notes as
his fifthtone cycle, suggests that both tunings might feature pure
major thirds. That gives us, with quartercomma shifts, precise
vertical JI; thus my 62-note system is based on 1/4-comma.

Colonna, who's concerned (at least in his treatise) with tuning his
instrument in a circulating system of 31 fifthtones, but not with
Vicentino-style adaptive JI for this instrument, says both that the
tuning is based on common practice for the first 19 notes, and that
each tone is neatly divided into five dieses, without any odd "commas"
or other complications.

If we view 31-tET as the deliberate slight tempering of major thirds
for perfect theoretical symmetry, then neither Vicentino nor Colonna
specify 31-tET as opposed to 1/4-comma meantone, with the latter
seeming especially to fit Vicentino's adaptive JI tuning.

With Colonna, while he doesn't specify 31-tET in this kind of
definition (a deliberate tempering of each major third), one could
take his assertion that each tone divides neatly into five dieses
without any commas or complications as expressing an ideal conception
precisely fitting the 31-tET model emerging, to my best knowledge,
somewhat later in the 17th century.

The "pragmatic" argument of what either Colonna or Vicentino were
_likely_ to have tuned might focus on the question of how likely one
would be to take 1/4-comma as a practical guide in tuning by ear, or
to make the kind of adjustments which "31-tET," as opposed to
1/4-comma, tends to imply.

Then, again, this also raises the question: how was 1/4-comma
typically tuned in this era? Aaron (1523) describes a process of
tuning the major third C-E as "sonorous and just" as possible, with
the greatest possible "utility," and then equally tempering the fifths
C-G-D-A-E -- as you've described, Bob, in recent threads.

However, Aaron then describes a process of tempering the other fifths
without expressly mentioning the possibility of simply tuning the rest
of the gamut in terms of pure major thirds. He does urge the tuner to
listen for good thirds as well as appropriately tempered fifths.

Maybe looking at the writings of people like Zarlino or Salinas in
terms of what they say about the practical tuning process would be
interesting: Zarlino does comment in 1571 that 1/4-comma "is not very
difficult to do," with the pure major thirds likely playing some role
in this conclusion.

Mainly, when discussing this question, especially as someone who leans
toward 1/4-comma meantone as a choice for this music, I should
emphasize that Vicentino doesn't actually say, "My tuning systems are
based on a regular chain of fifths with the major thirds pure." While
I consider it a reasonable inference to draw from his adaptive JI
system, some leading scholars (Maria Rika Maniates and Claude Palisca,
for example), have taken 31-tET as the basis for the system.

Scala, the outstanding scale definition and analysis program by Manuel
Op de Coul, also has files in the related archive of scales giving
versions of Vicentino's tunings based on 31-tET (vicentino1.scl,
vicentino2.scl).

I do feel a responsibility now and then to state this side of the
question.

Now we come to the first systematic presentation of 31-tET of which I
am aware, by Lemme Rossi (1666), which specifically presents this
equal division as the tuning of Vicentino's archicembalo, and gives
string ratios accurate to about a tenth of a cent. He also describes
1/4-comma meantone, calling it a _systema participata_, the term
_participatio_ for a temperament, i.e. the slight narrowing of the
fifths, going back to Gaffurius (1496).

It was Christiaan Huygens, near the end of this same century, who
compared 1/4-comma meantone and 31-tET, defining the latter
mathematically and noting the very small difference in the tempering
of the fifth, suggesting that the very slightly smaller impurity in
the fifth was an advantage of 31-tET.

Anyway, please feel free to ask more about this sometimes rather
intricate question, and thanks for giving me some ideas, to say the
least.

Most appreciatively,

Margo Schulter
mschulter@value.net