Re: Gentle introduction to neo-Gothic (1, Part 2B)

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A gentle introduction to neo-Gothic progressions (1):
Trines, quads, and intonational flavors
Part 2B of 2: Proximal quads and triples
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[Please note that this is Part 2B, to be followed by Part 2C, a
division motivated by a desire to keep these installments to a length
of about 500 lines each.]

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3. Omnitonal and mixed proximal quads: Flavors of crunchiness
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To this point, we have focused on most proximal quads involving
"closest approach" resolutions where one voice moves by a whole-tone
and the other by a semitone (expansive M2-4, M3-5, M6-8; contractive
m3-1, m7-5).

These quads are of special interest from an intonational point of
view, because they are apt to attract cadential shifts within a given
tuning system to "superefficient" cadential flavors. Thus a 24-note
Pythagorean tuning provides a choice between usual 3-flavor cadences
and accentuated 7-flavor ones, and 29-tET a choice between the usual
11-flavor and the striking 13-flavor.

Other varieties of proximal quads also have standard resolutions in
which all unstable intervals proceed to stable ones by stepwise
contrary motion, but which seem less prone to special intonational
treatment. These cadential sonorities tend to follow the usual or
"native" tuning in a given system: 3-flavor in Pythagorean, 11-flavor
in 29-tET, 7-flavor in 22-tET, etc.

The _omnitonal_ proximal quad features resolutions by stepwise
contrary motion in which every voice moves by a whole-tone, and each
unstable interval expands or contracts by a major third or ditone in
order to reach its stable goal (expansive m2-4, m3-5, m6-8;
contractive M3-1, M7-5). Here are examples of such standard
resolutions for the expansive and contractive omnitonal quads:

Expansive omnitonal quad Contractive omnitonal quad

F4 G4 E4 D4
E4 D4 C4 D4
C4 D4 A3 G3
A3 G3 F3 G3

(m6-8 + m3-5 + M3-1 + m2-4) (M7-5 + M3-1 + m3-5 + M3-1)

The expansive omnitonal quad features an outer minor sixth expanding
to the octave of a complete trine, while its contractive counterpart
has an outer major seventh contracting to a simple fifth.

In addition to their characteristic whole-tone motion in all voices,
omnitonal quads differ from most proximal quads in their inclusion of
strong dissonances as judged by Gothic or neo-Gothic standards: the
upper minor second of the expansive quad (here E4-F4) or major seventh
of the contractive quad (here F3-E4). Additionally, the expansive quad
has an outer minor sixth (here A3-F4), often regarded in 13th-century
theory as comparably discordant.

To borrow a term of the modern theorist Keenan Pepper, these intervals
introduce an element of Gothic "crunchiness," much relished in the
music of around 1200 and through the 13th century, and also treasured
in neo-Gothic music.

With most proximal quads, in contrast, all unstable intervals have
some degree of "compatibility," and the element of semitonal melodic
motion (intensive or remissive) provides an important ingredient of
directed cadential action. For omnitonal quads, the "crunchy" vertical
tension and release of the m2-4 or M7-5 progression, and also to at
least a degree the m6-8 progression of the expansive quad, provides a
different but equally compelling form of cadential amplification.

Cadences involving omnitonal quads tend to follow the usual flavor for
a given tuning, for example the regular 3-flavor of Pythagorean:

Expansive Contractive

F4 ----- +204 ----- G4 E4 ----- -204 ----- D4
(90) (498) (408) (0)
E4 ------ -204 ----- D4 C4 ----- +204 ----- D4
(498,408) (498,0) (702,294) (702,702)
C4 ----- +204 ----- D4 A3 ----- -204 ----- G3
(792,702,294) (1200,702,702) (1110,702,408) (702,702,0)
A3 ------ -204 ----- G3 F3 ----- +204 ----- G3

Each voice moves by a generously large 9:8 whole-tone (~204 cents),
and each unstable interval expands or contracts by a total of around
408 cents, equal to an 81:64 major third, in reaching stability.
The rather keen-edged quality of the vertical minor second at 256:243
(~90 cents) or major seventh at 243:128 (~1110 cents) may add to the
tension or "crunchiness" of these intervals.

In 17-tET, with its usual 23-flavor (major thirds ~23:18, minor thirds
here a bit larger than 27:23), these same resolutions look like this:

Expansive Contractive

F4 ----- +212 ----- G4 E4 ----- -212 ----- D4
(71) (494) (424) (0)
E4 ------ -212 ----- D4 C4 ----- +212 ----- D4
(494,424) (494,0) (706,282) (706,706)
C4 ----- +212 ----- D4 A3 ----- -212 ----- G3
(776,706,282) (1200,706,706) (1129,706,424) (706,706,0)
A3 ------ -212 ----- G3 F3 ----- +212 ----- G3

Here the melodic whole-tones at around 212 cents, and total expansive
or contractive motions of a major third (~423.53 cents) in the
resolutions of unstable intervals, have grown yet more spacious; the
urgently tense minor second of ~70.59 cents, or major seventh of
~1129.41 cents, has grown yet keener.

In the native 7-flavor of 22-tET, near the far end of the spectrum of
regular neo-Gothic tunings, these trends are carried yet further:

Expansive Contractive

F4 ----- +218 ----- G4 E4 ----- -218 ----- D4
(55) (491) (436) (0)
E4 ------ -218 ----- D4 C4 ----- +218 ----- D4
(491,436) (491,0) (709,273) (709,709)
C4 ----- +218 ----- D4 A3 ----- -218 ----- G3
(776,709,273) (1200,709,709) (1145,709,436) (709,709,0)
A3 ------ -218 ----- G3 F3 ----- +218 ----- G3

Here melodic whole-steps are around 218 cents, very close to the mean
between 9:8 and 8:7, and two-voice resolutions involve expansion or
contraction by a major third of ~436.36 cents, very close to 9:7.
Minor seconds or major sevenths at ~54.55 cents or ~1145.45 cents are
yet keener, packing a considerable "crunch."

In mixed proximal quads, some unstable intervals resolve in an
omnitonal manner (whole-tone motion in both voices), but others in an
intensive or remissive manner (whole-tone motion in one voice,
ascending or descending semitonal motion in the other). This situation
is associated with the strong Gothic dissonance of the augmented
fourth or tritone, or the diminished fifth.

Here are some of the possible permutations of mixed proximal quads,
with the tritone or diminished fifth characteristically behaving much
like a usual fourth or fifth and proceeding by parallel motion while
other resolutions by contrary motion guide the progression:

Expansive omnitonal/intensive Contractive omnitonal/intensive

E4 F4 A4 G4
D4 C4 F4 G4
Bb3 C4 D4 C4
G3 F3 B3 C4

(M6-8 + m3-5 + M3-1 + m2-4) (m7-5 + m3-1 + m3-5 + M3-1)

Expansive omnitonal/remissive Contractive omnitonal/remissive

C4 D4 F4 E4
Bb3 A3 D4 E4
G3 A3 B3 A4
E3 D3 G3 A4

(m6-8 + m3-5 + m3-1 + M2-4) (m7-5 + M3-1 + m3-5 + m3-1)

Like omnitonal quads, these mixed quads tend to take on whatever the
native or usual flavor may be in a given tuning, and the following
examples show each of the above four progressions in a different
regular tuning:

Pythagorean (3-flavor) 29-tET (11-flavor)

E4 ------ +90 ----- F4 A4 ----- -207 ----- G4
(204) (498) (414) (0)
D4 ------ -204 ----- C4 F4 ----- +207 ----- G4
(612,408) (498,0) (703,290) (703,703)
Bb3 ------ +204 ----- C4 D4 ----- -207 ----- C4
(906,702,294) (1200,702,702) (993,579,290) (703,703,0)
G3 ------ -204 ----- F3 B3 ----- +83 ----- C4

17-tET (23-flavor) 22-tET (7-flavor)

C4 ------ +212 ----- D4 F4 ----- -55 ----- E4
(212) (494) (273) (0)
Bb3 ------ -71 ----- A3 D4 ----- +218 ----- E4
(635,424) (494,0) (545,273) (709,709)
G3 ------ +212 ----- A3 B3 ----- -218 ----- A4
(918,565,282) (1200,706,706) (982,709,436) (709,709,0)
E3 ------ -212 ----- D3 G3 ----- +218 ----- A4

In addition to illustrating the variety of mixed proximal quads and
tunings, these examples may dramatize why the neo-Gothic 7-flavor, for
example, should not be confused with the 7-odd-limit or 7-prime-limit
of other theoretical systems.

In our 22-tET example of the 7-flavor, as advertised, major and minor
thirds are close to 9:7 and 7:6, while the outer minor seventh is not
too far from 7:4. The "flavor" concept applies to unstable intervals
such as these receiving primary directed resolutions (here m7-5, M3-1,
m3-5, m3-1).

However, the unstable diminished fifth B3-F4, not part of such a
primary directed resolution by contrary motion although it adds much
vertical color, has a size of 10/22 octave or ~545.45 cents, quite
close to 11:8. This routine neo-Gothic 7-flavor tuning of G3-B3-D4-F4
in 22-tET, 0-436-709-982 cents in a shorthand measuring intervals with
reference to the lowest voice, may be quite different from what a
"7-limit" tuning of this same sonority might imply.[11]

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4. Proximal triples: Sixth and seventh combinations
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In Gothic and neo-Gothic textures with three voices rather than four
-- the most typical number of parts in the 13th and 14th centuries --
cadences often involve three-voice subsets or _triples_ of full
unstable quads. With such _proximal triples_, as with proximal quads,
all unstable intervals can resolve by stepwise contrary motion.

We begin with triples featuring an outer sixth inviting expansion to
the octave of a complete trine (as in the full expansive quad), or an
outer seventh inviting contraction to a fifth (as in the full
contractive quad). These three-voice sonorities, and also the full
quads of which they are subsets, may be described respectively as
sixth combinations and seventh combinations.

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4.1. Most proximal triples
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Let us consider first the category of most proximal triples, with a
major sixth expanding to an octave or a minor seventh contracting to a
fifth. We can derive the most typical forms of these sonorities from
the corresponding most proximal quad by omitting either of the middle
voices.

For major sixth combinations, this approach yields two characteristic
three-voice triples, with resolutions here shown in the intensive
manner (ascending semitonal motion):

Full expansive quad Derivative triples of the sixth

E4 F4
D4 C4 E4 F4 E4 F4
B3 C4 B3 C4 D4 C4
G3 F3 G3 F3 G3 F3

(M6-8 + M3-5 + m3-1 + M2-4) (M6-8 + M3-5) (M6-8 + M2-4)

While the resolution of the full quad involves four directed two-voice
progressions, three expansive (M6-8, M3-5, M2-4) and one contractive
(m3-1), the two derivative triples have resolutions uniting two
expansive progressions: (M6-8 + M3-5) or (M6-8 + M2-4).

While both three-voice resolutions are compelling, the latter may have
an extra degree of dynamic intensity because both of its unstable
intervals (M6 and M2) are regarded as relatively tense by 13th-century
standards, in comparison with the relatively blending M3. We reach a
similar conclusion applying 14th-century standards, where either M3 or
M6 is regarded as somewhat milder than M2.

Similarly, from the contractive most proximal quad, we can derive two
characteristic triples of the minor seventh:

Full contractive quad Derivative triples of the seventh

D4 C4
B3 C4 D4 C4 D4 C4
G3 F4 G3 C4 B3 C4
E3 F3 E3 F3 E3 F3

(m7-5 + m3-1 + M3-5 + m3-1) (m7-5 + m3-1) (m7-5 + m3-1)

The resolution of the full contractive quad includes three contractive
two-voice progressions (m7-5, m3-1, m3-1) and one expansive
progression (M3-5); either derivative triple features the two
contractive progressions (m7-5 + m3-1). While both triples share these
two progressions, and have identical sets of intervals (m7 + 5 + m3),
their adjacent intervals are conversely arranged, with the minor third
below the fifth in E3-G3-D4 but above it in E3-B3-D4.

These most proximal triples of the major sixth or minor seventh have
resolutions very similar from an intonational viewpoint to those of
most proximal quads, often inviting the use of superefficient 7-flavor
or 13-flavor variations in tunings where the 3-flavor or 11-flavor is
the norm (see Part IIA, Sections 2.2 and 2.3).

For example, here are (M6-8 + M3-5) and (M6-8 + M2-4) resolutions in
the usual 3-flavor and accentuated 7-flavor of Pythagorean intonation.
In the 7-flavor, the symbols ^ and @ show a note raised or lowered by
a Pythagorean comma of ~23.46 cents:

Pythagorean 3-flavor

Intensive Remissive

E4 ------ +90 ----- F4 E4 ----- +204 ----- F#4
(498) (498) (498) (498)
B3 ------ +90 ----- C4 B3 ----- +204 ----- C#4
(906,408) (1200,702) (906,408) (1200,702)
G3 ------ -204 ----- F3 G3 ----- -90 ----- F#3

(M6-8 + M3-5) (M6-8 + M3-5)

E4 ------ +90 ----- F4 E4 ----- +204 ----- F#4
(204) (498) (204) (498)
D4 ------ -204 ----- C4 D3 ----- -90 ----- C#4
(906,702) (1200,702) (906,702) (1200,702)
G3 ------ -204 ----- F3 G3 ----- -90 ----- F#3

(M6-8 + M2-4) (M6-8 + M2-4)

Pythagorean 7-flavor

Intensive Remissive

E^4 ----- +67 ----- F4 E4 ----- +204 ----- F4
(498) (498) (498) (498)
B^3 ----- +67 ----- C4 B3 ----- +204 ----- C4
(929,431) (1200,702) (929,431) (1200,702)
G3 ------ -204 ----- F3 G@3 ----- -67 ----- F3

(M6-8 + M3-5) (M6-8 + M3-5)

E^4 ----- +67 ----- F4 E4 ----- +204 ----- F#4
(227) (498) (227) (498)
D4 ----- -204 ----- C4 D@3 ----- -67 ----- C#4
(929,702) (1200,702) (906,702) (1200,702)
G3 ----- -204 ----- F3 G@3 ----- -67 ----- F#3

(M6-8 + M2-4) (M6-8 + M2-4)

In their Pythagorean 7-flavor versions, these unstable sixth
combinations have ratios quite close to 7:9:12 and 14:21:24.

To illustrate the resolutions of the minor seventh triple with the
minor third below and fifth above (m7-5 + m3-1), let us consider the
regular 7-flavor of 22-tET:

Intensive Remissive

D4 ------ -218 ----- C4 D4 ----- -55 ----- C#4
(709) (709) (709) (709)
G3 ------ -218 ----- F3 G3 ----- -55 ----- F#4
(982,273) (709,0) (982,273) (709,0)
E3 ------ +55 ----- F3 E3 ----- +218 ----- F#3

For the minor seventh triple with fifth below and minor third above,
also (m7-5 + m3-1), here are resolutions in the usual 11-flavor and
very dramatic 13-flavor of 29-tET, with the "*" and "d" signs showing
a note raised or lowered by a diesis (1/29 octave, ~41.38 cents):

29-tET 11-flavor

Intensive Remissive

D4 ------ -207 ----- C4 D4 ----- -83 ----- C#4
(290) (0) (290) (0)
B3 ------ +83 ----- C4 B3 ----- +207 ----- C#4
(993,703) (703,703) (993,703) (703,703)
E3 ------ +83 ----- F3 E3 ----- +207 ----- F#3

29-tET 13-flavor

D4 ------ -207 ----- C4 Dd4 ----- -41 ----- C#4
(248) (0) (248) (0)
B*3 ----- +41 ----- C4 B3 ----- +207 ----- C#4
(952,703) (703,703) (952,703) (703,703)
E*3 ----- +41 ----- F3 E3 ----- +207 ----- F#3

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4.2. Omnitonal and mixed proximal triples
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Among proximal triples, as among proximal quads, we also find
omnitonal forms where an outer minor sixth expands to the octave of a
complete trine or an outer major seventh contracts to a fifth; and
mixed forms featuring an augmented fourth or diminished fifth (see
Section 3).

These triples, like omnitonal and mixed quads, are typically intoned
in the usual or "native" flavor for a given tuning. Here are some
examples of omnitonal triples and resolutions in the 3-flavor of
Pythagorean tuning, and of mixed triples in the 11-flavor of 29-tET:

Omnitonal triples -- Pythagorean 3-flavor

F4 ------ +204 ----- G4 F4 ----- +204 ----- G4
(498) (498) (90) (498)
C4 ------ +204 ----- D4 E4 ----- -204 ----- D4
(792,294) (1200,702) (792,702) (1200,702)
A3 ------ -204 ----- G3 A3 ----- -204 ----- G3

(m6-8 + m3-5) (m6-8 + m2-4)

E4 ------ -204 ----- D4 E4 ----- -204 ----- D4
(702) (702) (408) (0)
A3 ------ -204 ----- G3 C4 ----- +204 ----- D4
(1110,408) (702,0) (1110,702) (702,702)
F3 ------ +204 ----- G3 F3 ----- +204 ----- G3

(M7-5 + M3-1) (M7-5 + M3-1)

Mixed triples -- 29-tET 11-flavor

E4 ----- +83 ----- F4 C4 ---- +207 ----- D4
(621) (497) (90) (497)
Bb3 ----- +207 ----- C4 Bb3 ---- -83 ----- A3
(910,290) (1200,703) (792,579) (1200,703)
G3 ----- -207 ----- F3 E3 ---- -207 ----- D3

(M6-8 + m3-5) (m6-8 + M2-4)

F4 ------ -83 ----- E4 A4 ----- -207 ----- G4
(579) (703) (414) (0)
B3 ------ -207 ----- A3 F4 ----- +207 ----- G4
(993,414) (703,0) (993,579) (703,703)
G3 ------ +207 ----- A3 B3 ----- +83 ----- C4

(m7-5 + M3-1) (m7-5 + M3-1)

Omnitonal and mixed triples, like their counterparts among full quads,
can have an impressively "crunchy" quality, especially the varieties
featuring minor seconds, major sevenths, or augmented fourths or
diminished fifths; the (m6-8 + m3-5) resolution might be regarded as
somewhat milder, although the minor sixth is also often rated in
13th-century theory as an acute discord.[12]

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Notes to Part 2B
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11. More specifically, a "7-limit" tuning of G3-B3-D4-F4 might imply
4:5:6:7, with a 5:4 major third -- while the neo-Gothic 7-flavor
specifies a major third at or fairly close to 9:7. Curiously, while a
regular diminished fifth such as B-F in the usual 7-flavor of 22-tET
has a ratio close to 11:8, this same interval in the usual 11-flavor
of 29-tET (14/29 octave, ~579.31 cents) is quite close to 7:5.

12. A mixed proximal quad such as G3-Bb3-D4-E4 has as subsets both a
mixed triple (G3-Bb3-E4), resolving (M6-8 + m3-5); and a most proximal
triple (G3-D4-E4), resolving (M6-8 + M2-4). Likewise the contractive
G3-B3-D4-F4 has as subsets G3-B3-F4 (mixed), resolving (m7-5 + M3-1);
and G3-D4-F4 (most proximal), resolving (m7-5 + m3-1).

Most respectfully,

Margo Schulter
mschulter@value.net