Re: quarter-comma meantone on keyboard

Hello, there, Alita Morrison, and everyone.

Please let me begin by saying that like many other tunings, 1/4-comma
meantone can be precisely defined in mathematical terms, but will vary
in "real world" realizations, whether on a 16th-century harpsichord
tuned by ear or on a modern synthesizer with limited pitch resolution.
Below I'm including a Scala file and a frequency table from Scala (a
fine and free tuning program by Manuel op de Coul available on the
Internet).

(Around 1600 in Naples, a 19-note instrument with split keys was
rather popular, the Scala scale archive includes a file for this
tuning also, meanquar_19.scl.)

In 1571, Gioseffo Zarlino published what may be the first
comprehensive mathematical description of this tuning, which Aaron
(1523) may have defined more informally, and which Vicentino (1555)
likely used for his archicembalo with 36 notes per octave (in his
first tuning, 31 of them dividing the octave into nearly equal parts
of a circulating system, and the other five being used to provide pure
fifths for a few common notes). Interestingly, the great Spanish
theorist Salinas (1577) reports that he was considered the "inventor"
of this tuning during his stay in Rome about 40 years earlier.

The special quality of 1/4-comma meantone is that it has pure major
thirds at 5:4. Whether many 16th-century tuners focused especially on
these thirds is an open question, and Aaron's instructions of 1523
present an interesting case.

He says to start by tuning the octave C-C and the third C-E "as
sonorous and just as possible," with "the greatest possible unity."
Then the major third C-E is used as the basis for a chain of four
fifths, each of which should be a bit narrower than pure by the same
amount: C-G-D-A-E. Then the rest of the tuning is completed by moving
in fifths in either direction.

Zarlino more precisely defines that each fifth should be narrow by an
amount equal to 1/4 of the syntonic comma (81:80, ~21.51 cents), the
amount by which four pure fifths minus two octaves would exceed a pure
5:4 third. Using the logarithmic measure of cents, logarithms and
their application to music developing in the 17th century, we can say
that such a fifth is approximately 5.38 cents narrower than pure, with
a size of about 696.5784 cents.

Here is a table of frequencies for a 17-note tuning in 1/4-comma meantone
(Gb-A#) using a pitch level with C equal to its value in a modern tuning
with A=440. Since 16th-century instruments suggest that absolute pitches
often varied widely on both sides of that level, with an average for a
sample of some preserved wind instruments of A=466, maybe the modern
default isn't such a bad choice as one possible starting point. To the
Scala frequency data output I have added note names, with C4 as middle C:

|
0: 261.6256 Hertz 8.00000 oct [C4]
1: 273.3743 Hertz 8.06337 oct [C#4]
2: 279.9355 Hertz 8.09759 oct [Db4]
3: 292.5063 Hertz 8.16096 oct [D4]
4: 305.6418 Hertz 8.22434 oct [D#4]
5: 312.9772 Hertz 8.25855 oct [Eb4]
6: 327.0320 Hertz 8.32193 oct [E4]
7: 349.9192 Hertz 8.41952 oct [F4]
8: 365.6329 Hertz 8.48289 oct [F#4]
9: 391.2214 Hertz 8.58048 oct [G4]
10: 408.7899 Hertz 8.64385 oct [G#4]
11: 418.6009 Hertz 8.67807 oct [Ab4]
12: 437.3988 Hertz 8.74144 oct [A4]
13: 457.0410 Hertz 8.80482 oct [A#4]
14: 468.0100 Hertz 8.83903 oct [Bb4]
15: 489.0268 Hertz 8.90241 oct [B4]
16: 523.2511 Hertz 9.00000 oct [C4]

Here's a Scala file for this same scale, which you could use with
Scala to tune some sythesizers or the like automatically, as I
understand, or to do various kinds of analysis:

! meanquar_16.scl
!
1/4-comma mean-tone scale with split C#/Db, D#/Eb, G#/Ab and A#/Bb
16
!
76.04900
117.10900
193.15700
269.20600
310.26500
5/4
503.42200
579.47100
696.57800
25/16
8/5
889.73500
965.78400
1006.84300
1082.89200
2/1

This scheme splits all the usual accidentals into two parts except for
F#/Gb (only the standard F# included), and I'd agree with the author
of this file that A# seems at least as common as Gb in the
experimental 16th-century music I've seen.

By the way, I've played 16th-century European music and improvised in
related styles on a synthesizer with two independently tuned keyboards
to obtain between 13 and 24 notes per octave in 1/4-comma meantone.
One arrangement popular in Naples around 1600 was a 19-note instrument
(Gb-A#), and Scala has a number of files in 1/4-comma meantone for
differing numbers of notes per octave.

With 24 notes on two keyboards, you get a subset of Vicentino's
archicembalo with its 31-note meantone cycle. The keyboards differ by
Vicentino's "diesis" or fifthtone, often 128:125 or ~41.06 cents in
1/4-comma meantone (the distance between G# on the lower keyboard and
Ab on the higher one, for example).

Note that while Vicentino considers his tuning to divide the
whole-tone into five equal parts, he also appears to consider the
major thirds "perfect" or pure -- two characteristics which it was
shown by the later 17th century actually define two different
tunings, 31-tone equal temperament (31-tET) and 1/4-comma meantone.
In practice, variations in tuning by ear might be greater than the
theoretical difference between these two tunings, both of which
circulate nicely in 31 notes.

Also, while Vicentino himself noted that every interval was available
in his 31-note cycle from every note on the instrument, circularity
wasn't an important factor in even experimental 16th-century keyboard
music, although a few pieces (e.g. a puzzle piece for voices by
Willaert) may suggest such a closed system. In 1618, Fabio Colonna did
publish an "example of circulation" going through a cycle of 31 fifths
on his superharpsichord, somewhat similar to Vicentino's but with a
different keyboard layout, the _Sambuca Lincea_.

May I invite any further questions either on the List or via e-mail,
and wish you happy tuning and playing, and Happy New Year -- now that
we can discuss "21st-century music" very concretely and unequivocally.

Most respectfully,

Margo Schulter
mschulter@value.net