Invocatio (MIDI): Solution to Darreg "puzzle"

Hello, there, everyone, and after a bit of time offline having to do
with some server problems at my ISP, please let share the solution to
my question about a technique described by Ivor Darreg and also used
by a very famous earlier xenharmonicist which appears in my
composition _Invocatio in Quarto Tono_:

http://value.net/~mschulter/invoc4a.mid

This piece, largely inspired by the enharmonic or fifthtone technique
of the great xenharmonicist Nicola Vicentino (1511-1576), uses a
practice of a kind appearing in some of Vicentino's compositions and
examples, and advocated by Darreg in writings such as his _Xenharmonic
Bulletin_ 9 (1978) on 1/4-comma meantone and the almost identical
tuning of 31-EDO.

My piece is based in 1/4-comma meantone, where fifths are tempered
narrow by 1/4 syntonic comma (~5.38 cents) to achieve pure 5:4 major
thirds. This may well have been Vicentino's temperament for his
_archicembalo_ or "superharpsichord" dividing the octave into 31
conceptually equal parts or "minor dieses" each equal to about 1/5 of
a tone.

In this type of tuning, the usual diatonic semitone (e.g. E-F, B-C) is
considerably larger (~117.11 cents) than the chromatic semitone
(~76.05 cents) -- in 31-EDO, precisely 3/5-tone and 2/5-tone. These
large diatonic semitones are a usual feature of 16th-century music.

However, what Vicentino sometimes practices and what Darreg strongly
recommends is using the more narrow or compact chromatic semitones at
or around 2/5-tone in cadential progressions, making these directed
progressions more efficient and decisive.

As Darreg writes in XHB 9 in reference to 31-EDO:

Some demonstrations of meantone were not readily appreciated
because of the flatness of the leading-tone. That's easy--
just play the tone 1/31 of an octave higher-- never mind what
it's called! Never mind either, if this melody-note you are
sharpening will therefore clash with a note in the accompaniment
which harmonic sense demands you NOT sharpen. It will work
anyway! Have a little imagination. Try it; you'll like it.

Here's a link to the complete issue, which I understand Brian McLaren
has most generously made available via the World Wide Web:

<http://www.ixpres.com/interval/darreg/XHB9.htm>

What Vicentino does in some of his compositions and examples involving
fifthtones, and what happens in some cadences in my piece also, are
shifts of a fifthtone or diesis at a cadence permitting the use of
this narrower semitone together with the usual vertical concords of
meantone.

For those curious about how this is musically pulled off, a score of
my piece might help. Here I should explain that an asterisk (*) shows
a note raised by a fifthtone or diesis, in my tuning 128:125 (~41.06
cents); Vicentino used a dot above a note to show this inflection.
There are also apostrophe (') signs while I'll explain a bit later,
but here it's the fifthtone inflections that are most relevant.

The piece is notated in what I might describe as a kind of 2/2 meter,
and a comma (,) in the line of numbers and bars showing the rhythm
indicates the conclusion of a phrase:

7
1 2 | 1 2 | 1 2 | 1 2 | 1 2 + | 1 2 | 1 2 ,|
G#4 A4 A4 A4 A4 A4 A4 G#*4
E4 E'4 E'4 F4 E'4 D4 C4 D4 E*4
B'3 C'4 C'4 C'4 C'4 A3 A3 B*'3
E3 A3 A3 F3 A3 F3 F3 E*3

1 2 + | 1 2 + | 1 2 | 1 2 | 1 2 + | 1 2 |
G#*4. G#*4 A*4 B*4 B*4 C'5 C#5 D5
E*4. E*4 E*'4 G*4 G*4 A4 A4 G4 F4 G4
B*'3. B*'3 C*'4 D*'4 D*'4 F4 E'4 D4
E*3. E*3 A*3 G*3 G*3 F3 A3 Bb3

14
1 2 | 1 , 2 | 1 2 | 1 2 | 1 2 | 1 + 2 |
C#*5. C#*5 D*'5 D*'5 D*'5 B*4 C*5 B*4 A*'4 B*4
A*4. A*4 B*4 B*4 B*4 G*4 G'*4 G*4
E*'4. E*'4 G*4 G*4 G*4 D*'4 E*4 D*'4
A*3. A*3 G*3 G*3 G*3 G*3 C*4 G*3

20 25
1 2 ,| 1 2 | 1 2 | 1 2 | 1 2 | 1 2 | 1 2 |
C*5 C'5. A4 A'4 A'4 G#*4
G*'4 A4. E'4 D4 D4 E*4
E*4 E'4. C'4 A'3 A'3 B*'3
C*4 A3. A3 F'3 F'3 E*3

1 2 | 1 2 | 1 2 | 1 2 ||
A4 G#4
E'4 E4
C'4 B'3
A2 E3

Let's take a look, for example, at the cadence at measures 6-7:

A4 G#*4
D4 E*4
A3 B*3
F3 E*3

Here we have basically a standard 16th-century cadence of the kind
with ascending whole-tones and descending semitones -- especially
characteristic of the Phrygian mode, in which this piece is written.
The major third F3-A3 between the lowest pair of voices expands to a
fifth, while the major sixth F3-D4 expands to an octave.

The "trick" is that instead of the usual F3-A3-D4-A4 to E3-B3-E4-G#4,
we raise each note of the resolving sonority by a diesis or fifthtone,
leaving the regular vertical intervals of each sonority unchanged. Or,
more precisely, that's what happens in the version I play on a 24-note
meantone keyboard (see below for an extra "surprise").

With this diesis shift, the semitones F3-E*3 and D4-C#*4 are around
2/5-tone instead of 3/5-tone (F3-E3, D4-C#4) -- and the whole-tone
steps A3-B*3 and D4-E*4, interestingly, are 1/5-tone wider than usual,
at a size of around 6/5-tone, rather close to 8:7.

The _total_ expansion from a major third to a fifth, or from a major
sixth to an octave, remains about 8/5-tone, or the sum of these two
motions -- but here produced by combining narrow semitone steps of
2/5-tone and wide whole-tone steps of 6/5-tone, rather than the usual
steps of 3/5-tone and 5/5-tone.

Anyway, that's the solution to the "puzzle."

Here I should add that those apostrophe symbols in the score show a
different and smaller kind of inflection also described by Vicentino:
raising certain notes by a quartercomma (~5.38 cents) to obtain pure
fifths (3:2) and minor thirds (6:5) as well as major thirds.

In modern terms, this would be called "adaptive JI." As it happens,
Vicentino's own keyboard with 36 notes per octave could support
_either_ a 31-note cycle of fifthtones plus a few extra notes for
these pure adaptive JI sonorities, _or_ a system focused on pure
sonorities based on a 19-note meantone range (Gb-B#).

However, using 21st-century technology, it's possible to model a
version of Vicentino's germinal ideas combining both tuning systems
for a single piece, permitting fifthtone inflections and pure
sonorities. A player harpsichord or organ a la Nancarrow could do it
also, with enough notes per octave.

Anyway, it's a great pleasure to honor Nicola Vicentino and Ivor
Darreg while maybe providing a bit of diversion with my "puzzle."

Most appreciatively,

Margo Schulter
mschulter@...

Hello, everyone, and recent discussions about chromaticism and
adaptive tuning have suggested to me a "test" presenting some
chromatic music in adaptive JI -- and also making available some MIDI
files of historical pieces which people might retune in various ways.

Here I want to focus specifically on 16th-century European music in an
authentic 16th-century adaptive JI system -- Nicola Vicentino's, based
on a regular meantone tuning with very small melodic shifts to get
purely concordant sonorities as defined in a Renaissance setting: pure
fifths at 3:2 and minor thirds at 6:5, as well as major thirds at 5:4.

This system evidently uses very small melodic adjustments involving
two versions of the same note about 5 cents apart -- more on this
below for the curious.

One striking feature of meantone chromaticism is the unequal size of
the semitones: the large diatonic semitone (e.g. E-F, B-C) is around
117 cents, while the small chromatic semitone (e.g. F-F#, Bb-B) is
around 76 cents. That's a difference of a full 41 cents, about
1/5-tone, an interval known as an enharmonic diesis (e.g. G#-Ab).

While Vicentino also uses this diesis or "fifthtone" as a melodic
interval -- one of his distinctive contributions as a xenharmonicist
-- I've decided for these "tests" to stick to the (relatively) more
"usual" chromaticism. One reason is that fifthtone progressions tend
to take center stage, and people sometimes confuse them with the much
smaller adaptive JI adjustments.

Here's the first "test," or sample, a chromatic setting of _Alleluia:
Haec dies_ which Vicentino included in his treatise of 1555. My
adaptive JI version is something that could be played in theory on
Vicentino's harpsichord or organ with 36 notes per octave, but is much
easier to do with MIDI:

MIDI example: <http://value.net/~mschulter/qcmaval1.mid>

For the curious, I should explain that to tune his instrument for
adaptive JI, Vicentino first places the 19 notes of the lower manual
(Gb-B#) in a usual meantone -- likely 1/4-comma, with the fifths and
minor thirds 1/4 syntonic comma (~5.38 cents) narrower than pure, so
that four tempered fifths form a pure 5:4 major third.

Next each note of the upper manual -- ideally 19, but Vicentino could
only fit in 17 on his actually instruments -- is tuned at a "just
fifth" to a note on the lower manual. For example, we might take C4
(middle C) on the lower manual, and tune G4 on the upper manual to its
pure fifth.

This note would be 1/4 comma, or 5.38 cents, higher than G4 on the
lower manual. Vicentino uses a "comma sign" looking like an apostrophe
(') to show this difference -- at least as one of its meanings, so we
can refer to the pure fifth C4-G'4 -- C4 on the lower manual, G'4 on
the upper manual.

The idea of this system is for the player to mix notes from the two
manuals so as to get pure concords -- e.g. C4-E4-G'4-C4, with the
upper note of the fifth C4-G'4 and of the minor third E4-G'4 raised by
1/4-comma. Let's call this a "quartercomma" adjustment.

Similarly, in a sonority like F'3-A'3-D4-A'4, the lower note of a
fourth (A'3-D4) or major sixth (F'3-D4) is raised by a quartercomma to
give a pure ratio of 4:3 or 5:3.

One further technological innovation might have made Vicentino's JI
system a practical as well as theoretical triumph: some kind of
"player harpsichord" or "player organ" that could automatically choose
the right notes. For a human player, in real time, mixing the right
notes from the two keyboards with any consistency would be difficult
at best. However, using _some_ just sonorities could have a very
striking effect, with usual meantone intervals at other points.

With MIDI files and pitchbends, it's a great system for Renaissance
music; I've felt free to avoid quartercomma shifts in places where
they might cause complications, for example where a shift would be
required in the middle of a single sustained note.

Also, I'd want to emphasize that Vicentino's system is based on
Renaissance music and styles; with music based on major thirds at 9:7
rather than 5:4, for example, an adaptive JI system would obviously
have different patterns.

Maybe my main point is that "chromaticism" can mean different things
in different cultures and eras. For the European Renaissance variety,
Vicentino's adaptive JI is a neat solution, and it's a pleasure to
give people a chance to hear it.

Again, I welcome anyone who would like to use this MIDI file as a
basis for other tunings or adaptive retunings, for example a variable
adaptive system like that of John deLaubenfels.

In peace, inviting some exciting and friendly dialogue,

Margo