Re: Adaptive JI/RI tuning and neo-Gothic valleys (Part 3)

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Adaptive rational intonation and neo-Gothic valleys
A 24-note scheme a la Vicentino
(Essay in honor of John deLaubenfels)
Part III: Comma shifts, nuances, special sonorities (1)
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4. Comma shifts, nuances, and special sonorities: a sampler
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One feature as well as complication of a JI system combining pure
ratios of 2, 3, and 7 is the septimal comma, the interval separating
our two 12-note manuals in Pythagorean tuning (Eb-G#).

Here we consider some ways of approaching this comma of 64:63 (~27.26
cents) from the viewpoint: "If this be melodic unevenness, let us make
the most of it."[13]

The following discussion attempts merely to sample some of the
artistic possibilities of a system yet to be fully explored. In
addition to focusing on progressions involving direct melodic comma
shifts, we consider some alternative melodic nuances and related
special sonorities and idioms.

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4.1. Comma shifts and changes of flavor
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Direct comma shifts may occur as a calculated artistic effort to
maximize the subjective euphony or "consonance" of a cadential
progression, or to make an already special progression yet more
special.

They may also occur as a feature of "musical geometry" in certain
standard sequences if we seek the simplest possible ratios for all
vertical and melodic intervals.

Here we survey a few situations and idioms illustrating these diverse
themes.

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4.1.1. Cadential euphony and the _tertia automotiva_
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One striking kind of direct septimal comma shift involves a change in
cadential midstream, as it were, from a Pythagorean or 3-flavor major
third (81:64, ~407.82 cents) to a pure 7-flavor version (9:7, ~435.08
cents).[14] Here a carat sign (^) shows a note raised by a septimal
comma above its usual Pythagorean position, and thus located on the
upper keyboard manual of our 24-note tuning:

E4 F#^4 G4
C#4 C#^4 D4
A3 G3

This progression opens with a usual Pythagorean or 3-flavor sonority,
the mildly unstable _quinta fissa_ or "split fifth" combining an outer
fifth with a major and minor third, A3-C#4-E4, 64:81:96 or a rounded
0-408-702 cents.

Above the unchanged lowest note A3, the two upper parts proceed to the
pure 7-flavor version of the sixth sonority A3-C#^4-F#^4, 7:9:12 or
0-435-933 cents, the major third and sixth then resolving in the usual
expansive fashion to the fifth and octave of a complete trine
G3-D4-G4.

This change of flavor involves, most notably, a direct 27-cent comma
shift in the middle voice, C#4-C#^4. While this melodic shift is
striking in itself, the aesthetic logic of the progression may reflect
the somewhat ambivalent status of the pure 9:7 major third vis-a-vis
the usual Pythagorean major third at 81:64. Mixing both flavors of
this interval in the same cadential progression may serve to maximize
perceived euphony or contexual consonance.

Although the 9:7 or 7-flavor major third has a purer and "smoother"
ratio than the more complex 81:64, it may in many timbres and contexts
actually seem more tense or urgent, or less subjectively "concordant"
or independently euphonious.

The 9:7 is known in neo-Gothic theory as the _tertia automotiva_ or
"automotive third," both because of its especially dynamic and
forward-going qualities, and because of its proverbial resemblance to
an automobile horn -- the "car horn third" of popular speech, or more
poetically the _tertia clarionis_ or "clarion third."

More specifically, the 9:7 may have a strident quality in a split
fifth sonority where it is placed below a 7:6 minor third, for example
A3-C#^4-E4, 14:18:21 or 0-435-702 cents -- likewise known as the
_quinta fissa automotiva_ or "automotive split fifth." The traditional
Pythagorean 64:81:96, despite its complex ratios, often suggests a
quality better fitting the medieval ideal of an "imperfect concord" at
once unstable but _relatively_ blending.

In contrast, the 9:7 seems to have a more mellow or suave effect when
placed conversely above the 7:6, for example A^3-C3-E^4, 6:7:9 or
0-267-702 cents, the _quinta fissa suavis_.[15]

An expansive cadential sixth sonority such as A3-C#^4-F#^4 at 7:9:12
may have a quality somewhere between that of these two forms, but with
a certain "streamlined" kinship to the milder 6:7:9.[16] At the same
time, the directed tension of the outer major sixth with its
"striving" toward expansion to the octave lends impetus to a superb
cadential resolution (M6-8 + M3-5), with both two-voice resolutions
involving superincisive melodic semitones of 28:27.

Our hybrid progression with its septimal comma shift may therefore be
seen as an artful strategy for maximizing both vertical euphony and
efficient cadential action by mixing elements from standard 3-flavor
and 7-flavor versions of the same progression. Opening with the milder
64:81:96 of the 3-flavor version, we move to the streamlined 7:9:12 of
the 7-flavor version -- with the comma shift marking the colorful
rather than seamless point of juxtaposition.

E4 F#4 G4
C#4 D4
standard 3-flavor version A3 G3

E4 F#^4 G4
C#4 C#^4 D4
change-of-flavor version A3 G3

E4 F#^4 G4
C#^4 D4
standard 7-flavor version A3 G4

Of course, we might choose any of these three versions: the standard
Pythagorean version with its active-but-not-too-strident unstable
sonorities; the standard 7-flavor version with its characteristic
opening 14:18:21, a sonority with a smooth as well as a strident side;
or the mixed version with its calculated juxtaposition.[17]

Each choice has its own artistic attractions, reminding us that
musical perception is a matter not only of acoustical complexity or
harmonic entropy, but also of stylistic context or harmonic inertia,
with the comma shift as one artful form of "acceleration."[18]

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4.1.2. The "commatic genus" commatically varied
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In 1357, the theorist Johannes Boen[19] reports a taste for a new kind
of "commatic" melody involving two successive diatonic semitones, a
pattern distinct from that of the recognized diatonic, chromatic, or
enharmonic genus, e.g.:

F4-E4-D#4

This kind of progression occurs in some early 15th-century music, for
example that of Guillaume le Grant. In a usual Pythagorean 3-flavor,
this style involves cadences such as the following:

Eb4 D4 C#4 D4
Bb3 A3 G#3 A3
G3 F3 E3 D3

Here we have a series of sixth sonorities leading to a usual expansive
resolution of (M6-8 + M3-5). Both upper voices feature Boen's
consecutive descending diatonic semitones: Eb4-D4-C#4, or Bb3-A3-G#3.
These are the usual Pythagorean semitones at 256:243 (~90.22 cents).

To make this progression yet more distinctive, we can use the 7-flavor
for the last sixth sonority leading to the stable trine:

Eb4 D4 C#^4 D4
Bb3 A3 G#^3 A3
G3 F3 E3 D3

This nuance results in a joining of two _unequal_ semitones, a 90-cent
Pythagorean semitone followed by a supercompact 67-cent 28:27 semitone
(Eb4-D4-C#^4, or Bb3-A3-G#^3) bringing us to the penultimate cadential
sonority, in turn resolved by ascending 28:27 semitones in the upper
voices.

Here the shift from the "inertial frame" of the 3-flavor to the
7-flavor underscores the contrast between the closely juxtaposed flats
and sharps, joined by the two consecutive descending semitones: the
vertical and horizontal dimensions share in this added touch of drama.

As an alternative to either variety of "special" progression, of
course, we could conclude in a more usual late medieval fashion with a
remissive cadence featuring the descending semitone Eb3-D3 in the
lowest voice, nicely fulfilling a scenario suggested by the opening
flats and descending semitones in the upper voices:

Eb4 D4 C4 D4
Bb3 A3 G3 A3
G3 F3 Eb3 D3

As a variation on this more routine and very pleasant progression in
the usual Pythagorean 3-flavor, we could conclude with a 7-flavor
cadence, here starting out on the upper manual:

Eb^4 D^4 C^4 D^4
Bb^3 A^3 G^3 A^3
G^3 F^3 Eb3 D^3

In addition to the vertical nuance of the 7-flavor resolution, this
version introduces a subtle melodic contrast between the upper parts
with their standard Pythagorean steps and the lowest voice with its
tetrachord of 9:8-8:7-28:27 in the manner of Archytas and Al-Farabi.

As these examples may suggest, such choices may present the musical
equivalent of a kind of multidimensional chess, with many moves and
lines of variation open.

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4.1.3. Vertical sequences and septimal comma pumps
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To this point, we have focused on progressions using the septimal
comma shift as a calculated "special effect." Additionally, certain
common vertical progressions may require such direct comma shifts as a
simple matter of "musical geometry" if we seek consistently to observe
two conditions:

(1) Unstable vertical intervals should have the simplest
possible ratios; and

(2) Melodic cadential semitones should be tuned at 28:27,
the supercompact 7-flavor size of ~62.96 cents.

In practice, the "necessity" for direct septimal comma shifts becomes
a matter of artistic choice, since we can if desired modify either of
these conditions, for example by sometimes resolving pure 7-based
sonorities using variant sizes of melodic whole-tones and semitones
(Section 4.2).

However, the comma shift often lends a very pleasant and distinctive
air to these standard sequences, adding an element of drama and
underscoring cadential action, so that if it were not "necessary" we
might wish to invent it.

One such "septimal comma pump" sequence involves a descending series
of major sixth sonorities expanding to complete trines, at points
where a remissive resolution (descending semitonal motion) is followed
by an intensive resolution (ascending semitonal motion). Here I show
an example in duple meter, although various phrasings in duple or
triple (or other) meters are possible, and invite exploration:

1 2 | 1 2 & | 1
F4 G^4 A^4 G^4 F#^4 G4
F4 D^4 E^4 D^4 C#^4 D4
Bb3 A^3 A3 G3

For the first remissive resolution, we move from B3-D^4-G^4 (7:9:12)
to the trine A^3-E^4-A^4, with the descending 28:27 cadential semitone
Bb3-A^3 in the lowest voice. This is in keeping with the usual pattern
that remissive resolutions in the 7-flavor arrive at a stable trine or
fifth on the upper keyboard manual, with descending semitones moving
from the lower to the upper manual (e.g. Bb3-A^3).

Now that this resolution has "pumped us up" onto the upper manual at
A^3-E^4-A^4, we proceed to the next progression in our chain, the
intensive resolution from A3-C#^4-F#^4 (7:9:12) to G3-D4-G4, here by
way of a beautiful quintal-quartal sonority A^3-D^4-G^4 (9:12:16).

With or without this optional connecting sonority, our sequence from
A^3-E^4-A^4 to A3-C#^4-F#^4 at a pure 7:9:12 inevitably involves the
comma shift A^3-A3 in the lowest voice. On our 24-note keyboard, we do
not have available the alternative solution of building a pure 7:9:12
on the unchanged lowest tone A^3, calling for A^3-C#^^4-F#^^4, with
the upper voices raised in level by another septimal comma.

Within our 24-note system, the "geometric necessity" and high artistic
opportunity of the downward A^3-A3 shift makes it possible to obtain
A3-C#^4-F#^4 at a pure 7:9:12, resolving in a standard intensive
manner with ascending 28:27 semitones in the upper voices moving from
the upper to the lower manual (C#^4-D4, F#^4-G4). At the same time, it
provides what I consider a felicitous touch of color and emphasis.

With human voices or other nonfixed-pitch instruments hypothetically
adhering to our classic 2-3-7 JI model using integer ratios only[20],
such a "comma pump" sequence might result in "comma drift" rather than
a comma shift, with the pitch level raised to permit successive major
sixth sonorities at 7:9:12 and consistent 28:27 cadential semitones:

1 2 | 1 2 & | 1
F4 G^4 A^4 G^4 F#^^4 G^4
F4 D^4 E^4 D^4 C#^^4 D^4
Bb3 A^3 G^3

The cumulative upward drift involved in such adjustments may be
illustrated more dramatically if we continue the sequence further,
with each remissive cadence (Bb3-A^3 and F^3-E^^3 in the lowest voice)
"pumping up" the pitch level by a septimal comma:

1 2 | 1 2 & | 1 2 | 1 2 | 1 2 | 1
F4 G^4 A^4 G^4 F#^^4 G^4 E^^4 F^4 D^^4 E^^4 C#^^^4 D^^4
F4 D^4 E^4 D^4 C#^^4 D^4 B^^3 C^4 A^^3 B^^3 G#^^^3 A^^3
Bb3 A^3 G^3 F^3 E^^3 D^^3

In a 24-note keyboard version, these comma drifts are balanced by
downward comma shifts whenever a remissive cadence is followed by an
intensive cadence (A^3-A3-G3 or E^3-E3-D3 in the lowest voice), thus
keeping within the range of the gamut:

1 2 | 1 2 & | 1 2 | 1 2 | 1 2 | 1
F4 G^4 A^4 G^4 F#^4 G4 E^4 F4 D^4 E^4 C#^4 D4
F4 D^4 E^4 D^4 C#^4 D4 B^3 C4 A^3 B^3 G#^3 A3
Bb3 A^3 A3 G3 F3 E^3 E3 D3

Another kind of septimal comma shift sequence involves a progression
from a trine to a fifth by way of a contractive minor seventh sonority
(remissive resolution), and then back to the original trine by way of
an expansive sixth sonority (intensive resolution). Here I apply an
iambic pattern of triple meter often favored in a medieval European
setting, placing stable sonorities in stressed positions while giving
some duration and emphasis to unstable sonorities also:

1 2 3 | 1 2 3 | 1
G^4 F4 E^4 F#^4 G4
D^4 Bb3 A^3 C#^4 D4
G^3 A^3 A3 G3

Here the first contractive/remissive resolution is from G^3-Bb3-F4
(12:14:21) to the fifth A^3-E^4 (m7-5 + m3-1), with descending 28:27
semitones (Bb3-A^3, F4-E^4). As with our previous sequence, we must
now follow with the comma shift A^3-A3 in the lowest voice in order to
permit an expansive sixth sonority A3-C#^4-F#^4 at a pure 7:9:12
resolving intensively to G3-D4-G4. Again, the shift can lend extra
color to the cadence as well as facilitating the desired intonation.

A related sequence featuring simultaneous septimal comma shifts in the
two lower voices is the following:

1 2 3 | 1 2 3 | 1
G^4 F4 E^4 F#^4 G4
D^4 E^4 E4 D4
G^3 A^3 A3 G3

Here our contractive/remissive seventh sonority is G^3-D^4-F4 (4:6:7),
resolving to A^3-E^4, and followed by the expansive/intensive sixth
sonority A3-E3-F#^4 (14:21:24), resolving in turn to G3-D4-G4. In
moving to the penultimate sixth sonority, both lower voices negotiate
direct comma shifts (A^3-A3, E^4-E4).

The "double comma shift" lends what I find a very pleasant ambiance to
the cadence, making the penultimate 14:21:24 sonority yet more
colorful, and amplifying the overall pattern of the two lower voices
moving together in fifths, and the upper voice in contrary motion.
For the lower voices to descend together by a septimal comma at the
same time as the upper voice ascends to the elevated F#^4 might be
said to reinforce this contrast on a fine intonational level.

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Notes to Part III
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13. For earlier parts of this article, please see:
http://www.egroups.com/message/tuning/16640 (Part I)
http://www.egroups.com/message/tuning/17034 (Part II)

14. For a full, step-by-step presentation on neo-Gothic sonorities,
cadences, and flavors, please see the series "A gentle introduction to
neo-Gothic progressions":
http://www.egroups.com/message/tuning/15038 (1/Pt 1)
http://www.egroups.com/message/tuning/15630 (1/Pt 2A)
http://www.egroups.com/message/tuning/15685 (1/Pt 2B)
http://www.egroups.com/message/tuning/16134 (1/Pt 2C)

15. The term _quinta fissa suavis_ is in tribute to the great
mathematician and theorist Leonhard Euler, who in 1764 advocated the
use of 7-based intervals including the 7:6 minor third. See Joe
Monzo's translation with commentary of an article by Patrice
Bailhache, http://www.ixpres.com/interval/monzo/euler/euler-en.htm.

16. In various theoretical approaches based on concepts of the 17th
century and later, the 6:7:9 and 7:9:12 are inversions sharing the
same pitch classes, e.g. F#^3-A3-C#^4 and A3-C#^4-F#^4.

17. A similar logic may inspire such late medieval idioms as the
direct chromatic semitones advocated by Marchettus of Padua in his
_Lucidarium_ of 1318, for example in the following progression here
shown with two possible alternative versions:

C4 C#4 D4 C#4 D4 C4 D4
F3 E3 D3 F3 E3 D3 F3 E3 D3
(Marchettus) (Variant 1) (Variant 2)

The version of Marchettus, with its direct chromatic progression of
C4-C#4 in the upper voice, makes it possible to open on the pure fifth
F3-C4 _and_ to obtain a major sixth E3-C#4 before the octave D3-D4
(standard 14th-century resolution of M6-8). At the same time, this
special melodic interval has a striking quality in its own right
evidently embraced by Marchettus as an artistic resource rather than a
reluctant necessity. Possible alternatives include an augmented fifth
F3-C#4 preceding the major sixth E3-C#4 (Variant 1, a not unlikely
interpretation in some 14th-century pieces); or a minor sixth E3-C4
before the resolving octave D3-D4 (Variant 2).

18. While "harmonic entropy" postulates an intrinsic degree of
"consonance/simplicity" or "dissonance/complexity" for a given
interval or sonority, "harmonic inertia" focuses on a listener's
expectations within a given tuning system or style. In a style where
the rather complex Pythagorean 81:64 represents the regular and
accustomed major third, for example, a cadential 9:7 may represent a
notable "acceleration" or _change_ in entropy level, albeit arguably a
decrease in entropy, just as in Newtonian physics a deceleration is
also an acceleration (i.e. a change of velocity).

19. Wolf Frobenius, _Johannes Boens Musica und Seine
Konsonanzenlehre_, Freiburger Schriften zur Musikwissenschaft
(Musikwissenschaftliche Verlags-Gesselschaft mbH Stuttgart, 1971),
Latin text at p. 78. Boen's "new genus" _quod commaticum dici potest_
("which may be called commatic") is illustrated by the descending
tetrachord F4-E4-D#4-C4, a division of 256:243-256:243-19683:16384
(two Pythagorean diatonic semitones followed by an augmented second at
~317.60 cents), or a rounded 90-90-318 cents.

20. In practice, some kind of "adaptive JI" (Paul Erlich) involving
very small pitch adjustments might obtain, as modelled in other
musical contexts, for example, by Vicentino's adaptive tuning (see
Section 3) and the variable adaptive tuning of John deLaubenfels.

Most respectfully,

Margo Schulter
mschulter@value.net