Re: TD 924: Neo-Gothic and 11-limit (Paul Erlich)

Hello, there, and it's a great pleasure to respond to this question
from Paul Erlich:

> Playing around in 22-equal, I noticed that one can get a 4:6:7:9:11
> chord in hyper-Pythagorean, e.g., A-E-G-B-Eb. Margo, can you think
> of a way of incorporating a chord like this into the neo-Gothic
> language?

Please let me emphasize, Paul, how much I love this kind of creative
question which at once engages the musical imagination and raises
issues about the "stylistic bounds" of a given genre. Maybe if we
compare usual Gothic/neo-Gothic styles to tertian music of the 18th
and 19th centuries, your sonority places us in a situation a bit
analogous to modern jazz with its complex sonorities of the ninth and
eleventh, etc.

How can a new sonority be integrated into an established musical
vocabulary and cadential "grammar" -- or, at what point does the
process really lead us into a somewhat different langauge?

An important point: the "neo-Gothic language" as I tend to define it
is only one possible viewpoint on what "neo-Gothic" what be; and the
fact that a given idiom might be outside the usual "neo-Gothic"
category as I or anyone else may define it doesn't make that idiom
less beautiful. It may or may not be "cricket," but if not, maybe it's
championship baseball or soccer.

Something like A3-E4-G4-B4-Eb5 (using MIDI notation with C4 as middle
C), when I first saw it in your message, looked more complex than what
I would associate with "neo-Gothic," which usually involves up to four
voices (three or four being standard in the 13th and 14th centuries),
and actually hearing it in 22-tET confirmed that it is more "complex"
or "dissonant" than anything I'm accustomed to. That, of course,
doesn't make it "wrong," only "very strange for this idiom as I know
it."

Before considering a possible way of handling all five notes, I might
share a very intriguing progression which gets as far as including the
lowest four, A3-E4-G4-B4, and which I find a strikingly "xenharmonic"
and beautiful variation on a usual cadence:

B4 Bb4
G4 F4
E4 F4
A3 Bb3

(m7-5 + m3-1 + M9-8)

Thank you so much for setting a problem which gave me the opportunity
to discover this progression, beautiful not only in 22-tET but also in
usual Pythagorean, and I suspect likely in just about any neo-Gothic
tuning. Here the parenthetical notation, for people who may be
unfamiliar with it, shows two-voice resolutions by contrary motion
from unstable intervals to stable ones.

The lower three voices are standard: a minor seventh sonority, or
three voices of a complete contractive quad (A3-C4-E4-G4), resolving
here in an intensive manner (ascending semitonal motion) to the fifth
Bb3-F4. In 22-tET, as you note, this is a nice approximation of 4:6:7,
and I discuss this kind of cadential intonation more in my upcoming
article on "flavors."

We might describe the added B4, the major ninth above the lowest
voice, as what some jazz musicians at least call a "tension": it adds
to the complexity of the seventh sonority without being a "usual" part
of it.

Note that if we omit the seventh above the lowest voice G3, then we
have A3-E4-B4, a usual and relatively concordant 4:6:9. Thus _either_
4:6:7 or 4:6:9 is routine neo-Gothic; it's the superimposition of the
two, 4:6:7:9, which is "xeno" for this style.

However, that ninth A3-B4 _does_ permit a resolution which fits in
with the usual cadence in the lowest three voices: like the minor
seventh (m7-5) and minor third (m3-1), it can also resolve by a kind
of stepwise contrary motion: to the outer octave of the complete trine
Bb3-F4-Bb4, with one of its voices moving by a diatonic semitone or
limma A3-Bb3 and the other by a chromatic semitone or apotome B4-Bb4!

In 22-tET, where the limma is 1/4-tone and the apotome 3/4-tone, the
contrast of these motions may be especially striking; I like it also
in Pythagorean.

Note that with A3-E4-B4 alone, we could have a typical resolution by
contrary motion like this, with usual diatonic intervals:

B4 G4 B4 G4 B4 C4
E4 G4 E4 C4 E4 F4
A3 C4 A3 C4 A3 F3

(M9-5) (M9-5) (M9-12)

The first two options are identical except for the partwriting in the
middle voice, with the outer ninth contracting to a fifth; in the
third option, this interval expands to a twelfth by "near-conjunct"
contrary motion (stepwise in one voice, thirdwise in the other).

Another medieval alternative, even more common, is an oblique
resolution in which the ninth simply moves by step to the octave of a
complete trine:

B4 A4
E3
A3

M9-8
5

However, as the resolution of A3-E4-G4-B4 shows, a major ninth _can_
resolve by stepwise contrary motion to an octave as well, and this is
the kind of chromatic progression that Marchettus of Padua might have
come up with (he has a major sixth similarly resolving to a fifth).

Really, thank you for the gift of such a wonderful four-voice cadence:
all I had to do was resolve the lower three voices as usual, and let
the fourth voice move so as to complete the trine.

Now for the full five voices. While this goes beyond what I would call
"neo-Gothic," one solution would be like this, with the same
progression in the lower four voices, with the Eb5 held over and then
resolving from eleventh to twelfth above the lowest part:

Eb5 F5
B4 Bb4
G4 F4
E4 F4
A3 Bb3

Of course, if we agree that in a "neo-neo-Gothic" style with
sonorities this complex, something like Bb3-F4-F4-Bb4-Eb5 or 6:9:12:16
might be considered stable, then we could consider this sonority as
the resolution without a need for further movement of the highest
voice.

Having voices remain stationary at a cadence (other than a sustained
note or "pedal point" in certain styles) is less typical in Gothic or
neo-Gothic music than stepwise or thirdwise motion in all voices, but
it does happen in some conventional progressions, so why not here,
too?

By the way, with unstable sonorities more conventionally neo-Gothic or
less complex than this, I would say that if we want to have stable
sonorities going beyond the complete trine (2:3:4), then after 1:3:9
or 1:6:9, my choice would be 6:8:9, 8:9:12, 4:6:9, or 9:12:16.

A small aside: it's interesting that in 22-tET, your 4:6:7:9:11 indeed
has an outer interval very close to 4:11 (~1745 cents, around 6 cents
narrow); the 9:11, in contrast, is somewhat ambiguous at around 327
cents (B4-Eb5), a diminished fourth which I might describe as
somewhere between 6:5 (a kind of "schisma third") and 17:14. With
something like 29-tET, where an augmented second of 8/29 octave is
around 331 cents, I'd call it a gentle 17:14.

Anyway, thanks for setting a problem which, in its lower four-voice
subset, practically "solves itself" in a really beautiful way, and
enriches the neo-Gothic vocabulary in a fashion I might not have
considered.

Most appreciatively,

Margo Schulter
mschulter@value.net