Re: Vicentino's adaptive JI in context

Hello, there, and this a quick first response to some exchanges about
Vicentino's system (1555) of adaptive just intonation (JI) for his
_archicembalo_ with 36 notes per octave -- or, ideally, 38 notes.

While recognizing that Vicentino's system certainly can accommodate
18th-19th century major/minor tonality, as Paul Erlich ably discusses,
I would like to emphasize that this system grows out of 16th-century
music, with its modes, vertical progressions, and expressive nuances
of accidentalism.

Here, as a prelude to some longer articles on 16th-century verticality
and tunings, I would like to suggest how Vicentino's adaptive JI might
apply to one characteristic progression.

First of all, I would like to emphasize that Vicentino's adaptive JI
tuning does make it possible on a fixed-pitch instrument to achieve
pure 5-limit concords for any 16th-century (or later) music falling
within the compass of the instrument. Let's briefly consider this
reservation, which could be overcome with a somewhat larger
instrument.

In presenting his 36-note (or ideally 38-note) archicembalo, Vicentino
actually describes two tunings. The first tuning uses the 31 notes of
the first five "ranks" -- the 19 notes of the lower manual, and the
first 12 notes of the second -- to divide the octave into roughly
equal dieses of about 1/5-tone. The remaining 5 notes on the sixth
rank of the instrument -- the last rank of the second manual --
provide a few pure fifths with the notes of the first rank for
adaptive JI.

In practice, Vicentino evidently found it difficult to accommodate
more than 17 notes on his second manual -- but in principle, he would
like to have 19 notes on the upper manual as well as the lower,
allowing 7 notes rather than 5 for the sixth rank.

Here I give these extra notes in parentheses; an asterisk (*) shows a
note raised by a diesis or fifthtone, while an apostrophe or raised
comma (') shows a note raised by a "comma" -- which, for Vicentino,
can mean among other things the amount by which the fifth is tempered
in his basic meantone tuning. If we assume 1/4-comma meantone, this
"comma" would actually be equal to 1/4 syntonic comma, or ~5.38
cents.

6 (F') G' A' B' (C') D' E' F'
5 Gb* Ab* Bb* Db* Eb*
4 F* G* A* B* C* D* E*
--------------------------------------------------------------------
3 Gb Ab A# B# Db D# E#
2 F# G# Bb C# Eb
1 F G A B C D E F

The second tuning likewise has the first manual in a 19-note meantone
with a compass of Gb-A#, but has all notes of the second manual tuned
in pure fifths with the first. If the first manual is tuned in
1/4-comma temperament, then this arrangement indeed provides precisely
the pure 5-limit consonances associated in the 16th century with
Ptolemy's syntonic diatonic:

6 Gb' Ab' A#' (B#') Db' D#' (E#')
5 F#' G#' Bb' C#' Eb'
4 F' G' A' B' C' D' E'
--------------------------------------------------------------------
3 Gb Ab A# B# Db D# E#
2 F# G# Bb C# Eb
1 F G A B C D E F

While Vicentino's first tuning provides a complete system for
enharmonic or fifthtone music, plus a few adaptive JI sonorities, his
second provides a more or less complete adaptive JI system with a
compass of Gb-A#, sufficient for just about any 16th-century music
except such fifthtone compositions.

Discussing the application of this adaptive JI system -- or classic
5-limit JI a la Zarlino, for that matter -- to this music raises the
question of 16th-century approaches to vertical sonorities and
progressions. Here I present a single progression for consideration,
in order to introduce a few concepts and also illustrate Vicentino's
tuning in action.

Let us consider this common and pleasant progression for four voices,
with C4 as middle C, and higher numbers showing higher octaves. Note
that I show vertical intervals as integer ratios and as values in
rounded cents, with rounding errors of up to one cent possible:

G4 -- +193 -- A4
(4:3/498) (5:4/386)
D'4 -- +305 -- F4
(6:5/316) (4:3/498)
B3 -- +122 -- C'4
(5:4/386) (3:2/702)
G3 -- -193 -- F3

8H 10H
(M3-5)

Let's first consider the "fine-tuning," and then some general musical
concepts which may explain the shorthand symbols below the example --
the 8H, 10H, and (M3-5).

In Vicentino's adaptive JI, a pure fifth is formed by a note on the
"regular" lower manual plus its "comma key" on the second manual: thus
we have G3-B3-D'4-G4 for the first sonority, and F3-C'4-F4-A4 for the
second. A pure fourth conversely has a comma key for its _lower_ note
and a regular key for its upper note: here D'4-G4 and C'4-F4.

Regularly spelled major thirds or tenths on each keyboard are already
pure: e.g. G3-B3 and F3-A4 -- and likewise minor sixths or thirteenths.
Minor thirds or tenths have a regular key for their lower note and a
comma key for their upper note: e.g. B3-D'4. Minor sixths or
thirteenths conversely have a comma key for their lower note and a
regular key for their upper note: e.g. C'4-A4.

How does this kind of "adaptive JI" affect melodic progressions?
As long as a melodic line stays on a single manual, we have the usual
tempered intervals of 1/4-comma meantone: for example, usual
whole-tones of around 193.16 cents in the two outer voices, G3-F3 and
G4-A4. When a voice shifts between manuals, however, there occurs a
variation of 1/4 syntonic comma, or ~5.38 cents, from the usual
tempered interval.

Thus the tenor or second to lowest voice of this example progresses
B3-C'4, an interval of ~5.38 cents larger than the usual diatonic
semitone in 1/4-comma of ~117.11 cents, or ~122.49 cents. Similarly,
the alto or second to highest voice progresses D'4-F4, a minor third
at ~5.38 cents smaller than the usual meantone interval of ~310.26
cents, or ~304.88 cents.

A critic might observe that an altered whole-tone such as B'3-C4 is
more than 10 cents wider than the already amply wide 16:15 of classic
5-limit JI -- although it is still narrower than the whole-tone of
19-tet or 1/3-comma meantone. This melodic stretching might be a
tolerable compromise for the sake of pure concords.

Having considered the tuning, let's consider the progression itself.

In the 16th century, composers in practice and theory seek to
maintain what Gioseffo Zarlino (1558) calls _harmonia perfetta_ or
"perfect harmony," defined as a complete sonority consisting of "the
third plus the fifth or sixth" above the bass. In terms of later
tuning theory, this might be described as a "saturated 5-limit
sonority."

Another theorist especially interested in keyboard composition and
improvisation, Tomas de Santa Maria (1565), more specifically
classifies various four-voice sonorities according to the interval
between the two outer voices and the _differencias_ or adjacent
intervals added by the middle voices. Among his most preferred
sonorities of the "first grade" are three arrangements which would fit
into Zarlino's "third-plus-fifth" category, having outer intervals of
an octave, tenth, and twelfth:

G4 A4 A4
D4 F4 F4
B3 C3 D4
G3 F3 D3

8 10 12

Santa Maria defines these _differencias_ in terms of the ascending
adjacent intervals: thus we have here the octave "differentiated" into
third, third, and fourth; the tenth into fifth, fourth, and third; and
the twelfth into octave, third, and third.

Since these three "differences" of the octave, tenth, and twelve are
much favored in practice, we can let the symbols "8", "10", and "12"
stand for these specific arrangements unless otherwise qualified.

Zarlino, in discussing _harmonia perfetta_, discusses a distinction
between the "harmonic division" of the fifth with a major third below
and a minor third above (e.g. G3-B3-D4), and the "arithmetic division"
with the minor third below and the major third above (e.g. D3-F3-A3).
More generally, in any "third-plus-fifth" variety of sonority, the
harmonic division has the major third above the bass, and the
arithmetic division the minor third above the bass.

Combining the approaches of Zarlino and Santa Maria, we can classify
these sonorities both by their outer intervals and "differences," and
by their use of the harmonic or arithmetic division, thus

G4 A4 A4
D4 F4 F4
B3 C3 D4
G3 F3 D3

8H 10H 12A

From this viewpoint, our progression may be described as:

G4 A4
D4 F3
B3 C3
G3 F3

8H 10H

This shorthand tells us that we are progressing from the most common
difference of the octave to that of the tenth, both sonorities being
harmonic divisions of the fifth (major third or tenth above bass).

Additionally, the lower two voices define a progression from major
third to fifth by stepwise contrary motion, one of the standard
two-voice resolutions of 3-limit Gothic music (e.g. m3-1, M3-5, m3-5,
M6-8) now serving to guide progressions between the new stable 5-limit
sonorities. To show both the vertical sonorities, and this guiding
resolution, we can use the following shorthand, as in the example
above:

G4 A4
D4 F4
B3 C3
G3 F3

8H 10H

(M3-5)

While Vicentino's second or adaptive JI tuning covers a compass of
Gb-A#, it would be quite possible to implement such a tuning for
Vicentino's complete enharmonic system of 31 notes per octave if we
build an instrument with 62 keys per octave: the 31 notes of the lower
five ranks of the instrument in Vicentino's first or fifthtone tuning,
plus another 31 notes each a "comma" (i.e. 1/4 syntonic comma) higher.
This tuning permits both fifthtone music with pure 5-limit concords,
and a basic circulating temperament very close to 31-tet.

Most respectfully,

Margo Schulter
mschulter@value.net