Vicentino 24 tuning for two keyboards

Nicola Vicentino (1511-1576) is a figure of great interest to
xenharmonicists, and in this post I'd like to focus especially on one
approach to putting his theory of the _archicembalo_ and _archiorgano_
into practice on 20th-century microtunable synthesizers.


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1. Practice and theory
----------------------

Like the sophisticated Gothic theorists of the era 1200-1435 or so who
adapted Pythagorean traditions to complex techniques of polyphony,
Vicentino sought to adapt the traditions of Ptolemy and Aristoxenos to
the requirements of 16th-century polyphony. Also, like some of his
Gothic counterparts (e.g. Prosdocimus, 1413), he found that 12 notes
per octave were not enough.

Recognizing a keyboard temperament of 1/4-comma meantone with its pure
major thirds, or some approximation, as established "modern" practice,
Vicentino in fact defines the prevalent contemporary style as "mixed
and tempered music." Building on the usual 12-note meantone scale, and
seeking to implement the Greek diatonic, chromatic, and enharmonic
genera on an _archicembalo_ or "superharpsichord" as we might now say,
he proposes an instrument with 36 or 38 notes per octave.

For this instrument he proposes two alternate tunings, one of them
providing a complete system of all three genera which seems
essentially the same as 31-tone equal temperament (31-tet), plus
either five or seven extra notes supplying pure fifths for some or all
of the basic diatonic notes.

The second tuning ideally uses 38 notes per octave: 19 notes to
implement the diatonic and chromatic genera, and another 19 to provide
pure fifths for these notes. Thus either system, while premised on
1/4-comma meantone or 31-tet with its somewhat narrow or "blunted"
fifths, can provide just fifths (and minor thirds) for at least some
commonly used sonorities which Zarlino (1558) describes as the
_harmonia perfetta_ of third and fifth above the bass, and Lippius
(1612) calls the _trias harmonica_ or "triad."

Bill Alves has done a fine article on Vicentino's modified just
intonation system for extended keyboards.[1] Additionally, a complete
English edition of his famous treatise of 1555 by Maria Rika Maniates
and Claude Palisca has now been published.[2] These sources may lend
some perspective to what follows.


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2. Vicentino 24: a subset
-------------------------

At the end of the 20th century, one approach to a partial
implementation of Vicentino's system is to use two standard 12-note
keyboards, yielding a 24-note subset of his archicembalo tunings.

As it happens, his two alternative tunings for 36 notes seem to share
24 notes in common: the 19 diatonic and chromatic notes of 1/4-comma
meantone from Gb to B#; and five additional notes providing pure
rather than usual meantone fifths above G, A, C, D, and E. These five
notes are finals of the five natural modes most commonly used in
Renaissance music (D Dorian, E Phrygian, G Mixolydian, A Aeolian, and
C Ionian). Thus the availability of the just fifths D, E, G, A, and B
makes possible a pure _harmonia perfetta_ on the finals of these
modes, or as Lippius might say, a pure triad.

With Vicentino's instrument, the first 19 notes are arranged into
three "ranks." The first two ranks are equivalent to the white and
black keys of a usual 12-note meantone keyboard, while the third rank
supplies seven additional accidentals:

Rank 3: b# db d# e# gb ab a# b#
Rank 2: c# eb f# g# bb
Rank 1: c d e f g a b c

The additional five notes providing pure fifths to c, d, e, g, and a
might be notated g5, a5, b5, d5, and e5. These notes apparently make
up the "sixth rank" of Vicentino's first tuning[3] (where the fourth
and fifth ranks provide the notes of the enharmonic genus, comprising
with the first three ranks a 31-note set essentially equivalent to
31-tet).

Thus our "Vicentino 24" subset looks like this:

Rank 6: c5 d5 e5 g5 a5 b5 c5
Rank 3: b# db d# e# gb ab a# b#
Rank 2: c# eb f# g# bb
Rank 1: c d e f g a b c

The tones of Rank 6 (e.g. c5) are higher than their equivalents in
Rank 1 (e.g. c) by about 5.38 cents, the amount by which the fifth is
tempered in 1/4-comma meantone to obtain pure major thirds. This
difference is one of the senses in which Vicentino uses the term
"comma," a term with other interpretations to make life a bit more
interesting (and complicated) for modern readers and analysts.


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3. Implementation on two keyboards
----------------------------------

The "Vicentino 24" tuning which follows might be implemented either on
an acoustical instrument with two manuals (e.g. a harpsichord), or on
a synthesizer system of some kind supporting independent tunings for
two keyboards. Each keyboard might be supported by its own
synthesizer, or a multitimbral synthesizer such as the Yamaha TX-802
might support both keyboards through its "part-tuning" feature.

While 31-tet is the ideal tuning for Vicentino's complete scheme,
since it permits a closed system with the equal fivefold division of
the whole-tone he describes, 1/4-comma meantone should give
essentially the same results with our Vicentino 24 subset.

Generating the first 19 notes of this subset, corresponding to
Vicentino's first three ranks, can be accomplished on synthesizers
such as the TX-802 by selecting two transpositions of 1/4-comma
meantone (Yamaha's "MeanTone" preset). The first keyboard, in a
standard Eb-G# tuning with a "Wolf" fourth or fifth between these
notes, represents the 12 notes of his first two ranks, the seven
diatonic notes plus the five usual accidentals (c#, eb, f#, g#, bb).

The second keyboard, in a G-B# tuning, adds the seven accidentals of
the third rank -- five of these accidentals mapped to the black keys
(db, d#, gb, ab, a#), plus e# and b# on the usual f and c keys.

On the TX-802, for example, the first Eb-G# keyboard would be selected
on the front panel as "MeanTone" with Yamaha's "key of C" specified;
the second G-B# keyboard is Yamaha's "key of E." A more general hint
for the TX-802's preset Pythagorean and meantone tunings: take the
Yamaha "key" and add an ascending minor third to find the lower note
of the Wolf fourth. Thus "key of F" means a Wolf at Ab-C#, and "key of
G" a Wolf at Bb-D#, etc.

While simply selecting these two preset 1/4-comma meantone
temperaments supplies 19 notes of our Vicentino 24 subset, we need to
custom-tune the remaining five notes of the second keyboard to get our
desired pure fifths. With synthesizer keyboards, the best approach
might be to tune these notes (c5', e5', g5', a5', b5') to the nearest
approximation of a perfect fifth above f, a, c', d', and e' on the
lower keyboard. This would call for intervals of 599 tuning steps on a
1024-tet synthesizer, 449 steps in 768-tet, etc.[4]

Here is a diagram showing the notes on the two keyboards in cents,
with c' as the point of reference:

117.1 269.21 620.53 813.69 965.78
db' d#' gb' ab' a#'
b# d5' e5' e#' g5' a5' b5' b#'
-41.06 198.53 391.69 462.36 701.96 895.11 1088.27 1158.94

Keyboard 2: G-B#, 1/4-comma (Yamaha "key of E") with modified "5" notes
Keyboard 1: Eb-G#, 1/4-comma (Yamaha "key of C")

76.04 310.26 579.47 772.63 1006.84
c#' eb' f#' g#' bb'
_76.0|117.1_117.1|76.0_ _76.0|117.1_76.0|117.1_117.1|76.0_
c' d' e' f' g' a' b' c''
0 193.16 386.31 503.42 696.58 889.74 1082.89 1200
193.16 193.16 117.11 193.16 193.16 193.16 117.11

It may be noted that in this 1/4-comma meantone implementation (with
five notes modified to provide pure fifths), accidental pairs such as
c#/db and d#/eb are a lesser diesis (128:125 or about 41.06 cents)
apart. In Vicentino's ideal scheme with five equal dieses to a
whole-tone, these accidentals would be separated by 1/5-tone, or half
of his minor semitone of 2/5-tone -- a "diesis" equal to one step in
31-tet, or about 38.71 cents.


--------------------------
4. Concluding observations
--------------------------

While Vicentino's full 31-tet system is required to obtain all of his
genera and intervals, even a 19-note or 24-note implementation can
reveal some interesting possibilities. For example, between db and e#
on the second keyboard (Vicentino's third rank), we find an interval
of about 345.25 cents in 1/4-comma tuning -- or 348.39 cents in
31-tet (9/31 octave).

Vicentino himself describes this interval as having an effect
somewhere between a minor and a major third, and states its
"irrational ratio" as approximately 5 1/2:4 1/2 -- or, converting to
an integer ratio, 11:9. He finds this a concordant and useful vertical
interval on the archicembalo, with its nature tending somewhat toward
the major third, a sonorous consonance lending a certain kindred
character to such an intermediate form.

In addition to being of interest from the viewpoint of Vicentino's own
theory and practice, a 19-note or 24-note subset of his archicembalo
can be also be helpful in playing some of the most adventurous works
of composers such as Gesualdo, who goes as far as E# and B#.

For Renaissance and early 17th-century music which goes beyond the
range of a 12-note meantone keyboard, but not quite this far, a more
moderate "split key" arrangement of two manuals might be easier
because it permits more unisons between the keyboards. Certain
chromatic works of Orlando di Lasso and Gesualdo, for example, might
have a range of Eb-A# or Ab-D# or the like, requiring only a couple of
nonunisonal notes between the two keyboards (e.g. Eb/D# and Bb/A#; or
Ab/G# and Eb/D#). Implementing this kind of arrangement for
synthesizer(s) might only require selecting two transpositions of a
preset 1/4-comma meantone option.

Incidentally, if one wants a _closed_ Renaissance meantone system with
19 or 24 notes per octave, then 19-tet (almost identical to 1/3-comma
meantone) is a logical choice. Here, one might choose 19-tet plus five
extra notes to provide pure fifths _below_ the virtually pure _minor_
thirds of this tuning.[5]


---------------
Notes
---------------

1. This article by Bill Alves, originally appearing in _1/1: Journal
of the Just Intonation Network 5(No.2):8-13 (Spring 1989), is
available at http://www2.hmc.edu/~alves/vicentino.html. It includes a
very helpful bibliography of material such as the extensive studies of
Henry Kaufmann.

2. Nicola Vicentino, _Ancient Music Adapted to Modern Practice_,
tr. Maria Rika Maniates, ed. Claude V. Palisca (New Haven: Yale
University Press, 1996).

3. One point of ambiguity is that Vicentino describes the keys of the
sixth rank as a "comma" higher than corresponding keys of the first
rank, a term which can have various meanings in his usage. Here it
seems reasonable to take the comma in question as the difference
between a 31-tet or 1/4-comma meantone fifth and a pure 3:2 fifth --
in other words, actually 1/4 of a syntonic comma. This reading nicely
fits in with Vicentino's description of the comma, cited by Alves, as
an interval which can "help a consonance." Alternatively, a "comma"
for Vicentino can also mean an interval of half an enharmonic diesis
or 31-tet step -- roughly 19 cents. Since adding a comma in this sense
to the usual tempered fifth of his system would yield a fifth of
about 716 cents, or 14 cents wider than just, the first interpretation
seems to me more likely.

4. Such a custom synthesizer retuning of the five notes on the second
keyboard supplying pure fifths would obviously be easier using a
computer-based program such as the brilliant Scala by Manuel Op de
Coul, or setting the tuning once on the synthesizer's front panel and
then saving it to a RAM cartridge or the like for repeated use.

5. Such pure fifths for 19-tet or 1/3-comma meantone are located a bit
less conveniently than in 31-tet or 1/4-comma meantone from the
viewpoint of usual Renaissance practice. In the latter case, for
example, b5' (about 5.38 cents above b' in 1/4-comma, 5.18 cents in
31-tet) serves as a pure fifth for the common sonority E-G#-B, often
sounded on the final of E Phrygian. In contrast, the tone we might
call b-5' in 19-tet (about 7.17 cents _below_ b' in 1/3-comma, or 7.22
cents in 19-tet) supplies the lowest tone of the somewhat unusual
sonority B-D#-F#, outside the most typical 16th-century meantone range
(Eb-G#) although it occurs in various adventurous works. The pure
sonorities supported by the other "just fifth" keys in 1/3-comma or
19-tet are more prevalent: d-5' (D-F#-A); e-5' (E-G#-B); g-5' (G-B-D);
and a-5' (A-C#-E).

Most respectfully,

Margo Schulter
mschulter@value.net