Yahoo Tuning Groups Ultimate Backup tuning Re: Gentle introduction to neo-Gothic (1/Part 1)

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A gentle introduction to neo-Gothic progressions (1):
Trines, quads, and intonational flavors
Part 1 of 2: Most proximal quads
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A theme often sounded on this Tuning List is that tunings take place
in a musical context, and that understanding this context can help in
understanding a tuning and its qualities.

The purpose of this series of articles is to introduce and explain the
most common cadential progressions in neo-Gothic music, and to explore
some of the different intonational "flavors" which these progressions
may take on in various neo-Gothic tuning systems.

For example, it may be interesting to read that a given temperament
offers "excellent approximations of 14:11, 13:11, 21:17, and 17:14,"
but much more meaningful if one has a sense of how these unstable
intervals typically resolve in directed cadential progressions.

A verbal discussion, of course, even with ASCII musical examples, can
only tell part of the story; a cassette presentation including some
musical excerpts would be far more effective. However, a text-based
article can, at least, communicate some of the musical concepts
involved, and invite readers to try the examples on suitably tuned
instruments, or maybe even to prepare digitized versions of some of
the progressions and tunings.

Please let me emphasize that while many of the concepts here have
precedents in 13th-14th century European practice and theory, this is
an explanation in the year 2000 of an approach at once old and new.

In what follows, I use the English term "quad" to refer to any
four-voice sonority, and likewise "triple" for any three-voice
sonority. My main intention is to avoid the possible implication of
stability which the terms "tetrad" and "triad" may carry for some
readers. In 13th-century parlance, a _triplum_ refers to a third voice
or three-voice piece, and likewise a _quadruplum_ to a fourth voice or
four-voice piece.

From a xenharmonic perspective, one notable feature of neo-Gothic
music is the wide range of sizes or "flavors" for certain unstable
intervals, sometimes within the same tuning system and piece,
especially in cadential progressions.

A "major third" expanding to a fifth, for example, may range from
around 363 cents (a bit narrower than 21:17) to around 455 cents
(~13:10), a range of over 90 cents. Likewise, a "minor third"
contracting to a unison may range from around 248 cents (~15:13) to
around 341 cents (somewhat larger than 17:14).[1]

This first article, posted in two part, introduces some of the main
unstable cadential sonorities, especially the family known as "most
proximate" quads and triples, and surveys some of the principal
intonational shadings or "flavors" these sonorities may assume.

Future installments will explore in more detail these flavors, offered
in various combinations and fine nuances by different neo-Gothic
tuning systems.

Concern for the important distinction between Gothic and neo-Gothic
practice and theory bids me add that while the following "modernistic"
presentation starts with four-voice progressions, a more "medievalist"
approach would start with two-voice resolutions, moving from there to
three-voice and four-voice cadences.

Also, while resolutions of four-voice "quads" as described below do
occur in 13th-century compositions, the three-voice subsets of these
sonorities and resolutions described in Part II of this article are
more typical of 13th-14th century practice, where three-voice writing
is the norm.

For an introduction to historical 13th-century practice proceeding
from two-voice intervals to multi-voice structures and progressions,
readers are invited to visit my presentation kindly hosted by Todd
McComb of the Medieval Music and Arts Foundation:

http://www.medieval.org/emfaq/harmony/13c.html

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1. Cadential action: Trines and most proximal quads
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In Gothic and neo-Gothic music, the complete stable sonority is the
three-voice trine consisting of an outer octave, lower fifth, and
upper fourth.[2] Using a MIDI-style notation with C4 as middle C and
higher numbers showing higher octaves, an example of a trine would be
D3-A3-D4 or F3-C4-F4.

As realized in 3-limit just intonation (JI), also known as Pythagorean
tuning, the trine has pure interval ratios of 2:3:4, with a 2:3 fifth
below and a 3:4 fourth above together forming a 2:4 (or 1:2) octave.
Other Neo-Gothic regular tunings typically feature trines quite close
to these ideal ratios, but with fifths slightly wider than 2:3 and
fourths slightly narrower than 3:4.[3]

Defining the standard of rich and euphonious stability, the complete
trine serves as an optimal goal for directed resolutions of various
unstable sonorities.

In an ideally efficient form of four-voice cadence, a trine provides a
restful goal for a moving four-voice sonority known as a "proximal
quad" with four unstable intervals all resolving by stepwise contrary
motion:

E4 F4
D4 C4
B3 C4
G3 F3

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

Here the outer major sixth of the quad expands to the octave of a
complete trine; the lower major third to the fifth of the trine; and
the upper major second to the fourth of the trine. The minor third
between the two middle voices contracts to a unison, both these voices
arriving at the fifth of the trine.

It's often convenient to identify a Gothic or neo-Gothic cadence as a
union of such directed two-voice progressions by contrary motion, thus
the shorthand below the example of (M6-8 + M3-5 + m3-1 + M2-4).

We call this quad "proximal" (i.e. close or neighboring) because its
voices and unstable intervals can all progress to their stable trinic
goal by means of ideally economic stepwise motion. Each voice moves by
step, and each unstable interval resolves by stepwise contrary motion.

It is further a "_most_ proximal" quad because each of the four
interval resolutions has one voice moving by a whole-tone and the
other by a diatonic semitone: the unstable interval need expand or
contract by only the total distance of a minor third to arrive at a
stable trinic interval (M6-8, M3-5, m3-1, or M2-4).

Using regular diatonic motions, as opposed to "unusual" intervals such
as the chromatic semitone, these are the most compact and efficient
resolutions possible by contrary motion from an unstable interval to a
stable one.

This last Gothic and neo-Gothic concept of resolutions to "the nearest
consonance" may become clearer as we look at most proximal quads in
more detail.

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1.1. Most proximal quads, expansive and contractive
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Most proximal quads come in two structural varieties: the _expansive_
variety we have just met, with an outer major sixth expanding to the
octave of a complete trine; and a _contractive_ variety with an outer
minor seventh contracting to a simple fifth, the prime trinic concord.
Both varieties feature four voices and four ideally efficient
two-voice resolutions by stepwise contrary motion:

Expansive Contractive

E4 F4 D4 C4
D4 C4 B3 C4
B3 C4 G3 F3
G3 F3 E3 F3

(M6-8 + M3-5 + m3-1 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

Here both quads cadence to the trine F3-C4-F4 or its fifth F3-C4.
While the terms "expansive" and "contractive" refer especially to the
behavior of the _outer_ major sixth or minor seventh (M6-8, m7-5),
more generally they describe the pattern followed by the majority of
intervals in each form.

Thus the resolution of the expansive most proximal quad features three
two-voice resolutions by expansion (M6-8, M3-5, M2-4), and one by
contraction (m3-1):

Expansive resolutions Contractive resolution

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

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

The resolution of the contractive most proximal quad likewise involves
three two-voice resolutions by contraction (m7-5, m3-1, m3-1) -- the
minor thirds E3-G3 and B3-D4 both progressing to unisons -- and one by
expansion (M3-5):

Contractive resolutions Expansive resolution

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

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

People who enjoy symmetrical patterns may take pleasure in noting that
the middle voices of an expansive most proximate quad form the
unstable interval which contracts (m3-1), while these same voices in
our contractive quad form the unstable interval which expands (M3-5).

Each quad features four of the five most important two-voice
resolutions to "the nearest consonance," or "closest approach"
progressions as they are sometimes termed in medieval and neo-Gothic
theory: M2-4, M3-5, and M6-8 (by expansion) and m3-1 and m7-5 (by
contraction).

From the viewpoint of vertical color, most proximate quads are
impressively active and dynamic but not intensely "dissonant," since
all of their intervals are to some degree "compatible." Major and
minor thirds are regarded in 13th-century theory as relatively
"concordant" or blending, although unstable; major seconds, minor
sevenths, and major sixths, while regarded as relatively tense, are
also accorded some "compatibility."

As we'll see (Section 3), there are also "crunchier" flavors of
proximate quads featuring strong dissonances such as minor seconds or
major sevenths, or augmented fourths or diminished fifths. For the
moment, however, keeping to the theme of most proximal quads and their
resolutions, let's focus on another and more contextual distinction
and introduce the vital topic of _intonation_.

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1.2. Intensive and remissive manners in Pythagorean tuning
----------------------------------------------------------

Most proximate quads of the expansive and contractive varieties can be
recognized by their intrinsic structure: G3-B3-D4-E4 is an expansive
quad inviting (M6-8 + M3-5 + m3-1 + M2-4), while E3-G3-B3-D4 is a
contractive quad inviting (m7-5 + m3-1 + M3-5 + m3-1).

Since this is a "gentle introduction," and many readers may be used to
reckoning intervals in terms of the lowest voice, we might
conveniently describe the expansive quad as having a major third,
fifth, and major sixth above this voice. The contractive quad has a
minor third, fifth, and minor seventh above the lowest voice.

For many readers, another "user-friendly" way of expressing these same
sonorities is in terms of intervals in cents above the lowest voice.
In a standard medieval Pythagorean tuning, we have an expansive quad
at a rounded 0-408-702-906, and a contractive quad at 0-294-702-996.

While the distinction between the expansive and contractive quads is
one of intrinsic structure, either type of quad may resolve in two
_manners_ depending on the musical context.

In the _intensive_ manner, all four unstable intervals resolve by
progressions involving descending whole-tone motion in one voice and
ascending semitonal motion in the other. In the _remissive_ manner,
conversely, all four of these resolutions involve ascending whole-tone
motion and descending semitonal motion.

Comparing these resolutions for our quads G3-B3-D4-E4 and E3-G3-B3-D4
may make the distinction more clear:

Intensive manner

Expansive Contractive

E4 F4 D4 C4
D4 C4 B3 C4
B3 C4 G3 F3
G3 F3 E3 F3

(M6-8 + M3-5 + m3-1 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

Remissive manner

Expansive Contractive

E4 F#4 D4 C#4
D4 C#4 B3 C#4
B3 C#4 G3 F#3
G3 F#3 E3 F#3

(M6-8 + M3-5 + m3-1 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

Our two intensive resolutions to a trine or fifth on F are in fact
identical to those of Section 1.1, and the reader may refer to the
listings of their two-voice progressions in that section in order to
confirm that each of these progressions involves ascending semitonal
motion in one voice and descending whole-tone motion in the other.

For our two remissive resolutions of these same quads to a trine or
fifth on F#, a similar listing will confirm the converse pattern of
descending semitonal motion and ascending whole-tone motion:

Expansive quad/Remissive manner Contractive quad/Remissive manner

E4 F#4 B3 F#3 E4 F#4 D4 C#4 D4 F#4 G3 F#3 D4 C#4 B3 C#4
G3 F#3 G3 F#3 D4 C#4 B3 C#4 E3 F#3 E3 F#3 B4 C#4 G3 F#3

(M6-8) (M3-5) (M2-4) (m3-1) (m7-5) (m3-1) (m3-1) (M3-5)

The terms "intensive" and "remissive" derive from medieval Latin
descriptions of ascending and descending motion, and these terms may
nicely convey some of the musical qualities of these two manners of
resolution.

The intensive manner with its ascending semitonal motion tends to be
more decisive and definitive, while the remissive manner with its
descending semitonal motion may have a more "relaxed" and open-ended
quality, more "laid back" as one might say in colloquial English.

As illustrated in our last example, a given most proximal quad offers
two alternative resolutions by ideally efficient cadential motion, one
in the intensive manner and one in the remissive manner.

Here is another example illustrating this choice for the quad
F3-A3-C4-D4 or D3-F3-A3-C4, this time showing first the more common
remissive resolution to a trine or fifth on E, and then the more
"remote" intensive resolution to a trine or fifth on Eb:

Expansive quad Contractive quad

Remissive Intensive Remissive Intensive

D4 E3 D4 Eb3 C4 B3 C4 Bb3
C4 B3 C4 Bb3 A3 B3 A3 Bb3
A3 B3 A3 Bb3 F3 E3 F3 Eb3
F3 E3 F3 Eb3 D3 E3 D3 Eb3

Just as a given most proximal quad may regularly resolve to either of
two stable trines or fifths, so a given trine or fifth may be
approached by regular cadences from either of two most proximal quads,
one resolving intensively and the other remissively. Let us first
consider the example of the complete trine D3-A3-D4 or its fifth
D3-A3:

Intensive cadences to D Remissive cadences to D

Expansive Contractive Expansive Contractive

C#4 D4 B3 A3 C4 D4 Bb3 A3
B3 A3 G#3 A3 Bb3 A3 G3 A3
G#3 A3 E3 D3 G3 A3 Eb3 D3
E3 D3 C#3 D3 Eb3 D3 C3 D3

In this case, the accidentals called for by both the intensive and
remissive choices are available within a typical 12-note tuning of
Eb-G#. For an example involving more remote accidentals, let us
consider the usual intensive cadence and more "remote" remissive
cadence to the trine F3-C4-F4 or fifth F3-C4:

Intensive cadences to F Remissive cadences to F

Expansive Contractive Expansive Contractive

E4 F4 D3 C4 Eb4 F4 Db3 C4
D4 C4 B3 C4 Db4 C4 Bb3 C4
B3 C4 G3 F3 Bb3 C4 Gb3 F3
G3 F3 E3 F3 Gb3 F3 Eb3 F3

A 17-note Pythagorean tuning of Gb-A#, or an equivalent tuning in a
neo-Gothic temperament, would provide all the accidentals needed for a
regular remissive cadence to F, as would smaller tunings such as the
15-note subset Gb-G# or the 12-note subset Gb-B.

More generally, _any_ trine or fifth may be regularly approached in
either an intensive or a remissive manner, as long as our gamut
provides the requisite accidentals.[4]

Having become acquainted with these intensive and remissive
progressions, let us consider their intonation in Pythagorean tuning,
known in neo-Gothic theory as the "3-flavor."

As we shall see in the next section, the concept of "flavor" applies
to _unstable_ sonorities and intervals, as opposed to stable intervals
and trines which quite closely approximate the ideal ratios of 2:3:4
over the usual range of neo-Gothic tunings.

More specifically, "flavor" focuses mainly on different nuances and
shadings for the unstable intervals of the expansive and contractive
most proximal quads: M2, m3, M3, M6, and m7. The concept tends to
center especially on such intonational variations for M3, m3, and M6,
somewhat more sensitive to small changes in fifth size than M2 and m7,
but with the latter intervals also playing a vital role in larger
changes of flavor.

In the "3-flavor" ideally realized by Pythagorean tuning, these
intervals have or approximate integer ratios based on powers of three:
M3 at 81:64 (~408 cents); m3 at 32:27 (~294 cents); M6 at 27:16 (~906
cents); M2 at 9:8 (~204 cents); and m7 at 16:9 (~996 cents).

Additionally, the "flavor" concept focuses on the melodic dimension of
tuning, and especially on variations in the size of whole-tones and
semitones in cadential and other settings. In Pythagorean, the melodic
whole-tone or major second has the same pure 9:8 ratio we just noted
in a vertical context (~204 cents), while the diatonic semitone or
limma has a size of 256:243 (~90 cents).

The following diagrams show both vertical and melodic intervals in the
Pythagorean "3-flavor" for intensive and remissive progressions from a
most proximate quad to a trine or fifth. Numbers in parentheses above
a given voice show vertical intervals in rounded cents formed between
it and each higher voice, while signed numbers show melodic motions in
the ascending (positive) or descending (negative) direction:

Expansive/Intensive Contractive/Intensive

E4 ----- +90 ----- F4 D4 ----- -204 ----- C4
(204) (498) (294) (0)
D4 ----- -204 ----- C4 B3 ----- +90 ----- C4
(498,294) (498,0) (702,408) (702,702)
B3 ----- +90 ----- C4 G3 ----- -204 ----- F3
(906,702,408) (1200,702,702) (996,702,294) (702,702,0)
G3 ----- -204 ----- F3 E3 ----- +90 ----- F3

Expansive/Remissive Contractive/Remissive

E4 ----- +204 ----- F#4 D4 ----- -90 ----- C#4
(204) (498) (294) (0)
D4 ----- -90 ----- C#4 B3 ----- +204 ----- C#4
(498,294) (498,0) (702,408) (702,702)
B3 ----- +204 ----- C#4 G3 ----- -90 ----- F#3
(906,702,408) (1200,702,702) (996,702,294) (702,702,0)
G3 ----- -90 ----- F#3 E3 ----- +204 ----- F#3

In these diagrams, vertical intervals are indicated in a "tree-like"
arrangement. The lowest note G3 of the expansive quad, for example,
has intervals of (906,702,408) cents, or (M6,5,M3), with the three
upper voices at E4, D4, and B3 respectively. The next voice at B3 has
intervals of (498,294) cents or (4,m3) with the remaining higher
voices at E4 and D4; and the next voice at D4 has an interval of (204)
cents or (M2) with the remaining highest voice at E4.[5]

These cadences in the 3-flavor may bring out some general qualities of
Gothic and neo-Gothic music. Thirds and sixths have a relatively
complex quality, one nicely fitting their role as unstable and often
cadentially directed intervals. At the same time, there is a
satisfying contrast between wide 204-cent whole-tones and compact
90-cent diatonic semitones.

A measure of cadential efficiency bringing both dimensions into play
is the total distance an unstable interval must expand or contract in
order to arrive at its stable goal. As the above example shows for
various cadential permutations, in the Pythagorean 3-flavor each
unstable interval resolves by motions of a 204-cent whole-tone in one
voice and a 90-cent diatonic semitone in the other -- a total
expansion or contraction, as the case may be, of 294 cents, or a 32:27
minor third.

Neo-Gothic tunings in the immediate neighborhood of Pythagorean, for
example 53-tone equal temperament or 53-tET (fifths ~0.07 cents
narrow), or 41-tET (fifths ~0.48 cents wide), present unstable
interval sizes and melodic steps very close to those shown above, and
so may be regarded as offering fine variations on this basic
"3-flavor."

Other neo-Gothic intonational regions or flavors offer different
musical qualities based on the interplay of such variables as the
degree of "complexity" or tension in unstable intervals, the sizes and
relative proportions of melodic whole-tones and semitones, and the
amount of expansion or contraction involved in two-voice resolutions.

Further, a single neo-Gothic tuning system typically may offer two or
three contrasting cadential flavors, one motivation for carrying a
tuning beyond 12 notes so as to enjoy a choice of these flavors at
various positions of the gamut.

To illustrate these points, we next compare the 3-flavor with some
other common flavors for most proximal quads.

---------------
Notes to Part I
---------------

1. This is by no means to exclude "neutral thirds," which occur for
example in 17-tone equal temperament (17-tET).

2. Johannes de Grocheio (c. 1300) describes this sonority as
manifesting the _trina harmoniae perfectio_ or "threefold perfection
of harmony," thus from this Latin the English "trine."

3. The most characteristic or "Central" zone of neo-Gothic regular
temperaments moves outward from Pythagorean to 17-tET, where fifths
are about 3.93 cents wide; the "Far" neo-Gothic zone ranges out from
there to around 22-tET (fifths ~7.14 cents wide), or for some
"Setharianized" (e.g. gamelan-like) timbres to around 27-tET (fifths
~9.16 cents wide).

4. This is by no means to exclude the additional option of altered
progressions: for example, substituting F#3-Bb3-C#4-D#4 for the
regular expansive quad Gb3-Bb3-Db4-Eb4 in a cadence to F3-C4-F4.
Such substitutions result in melodic motions replacing diatonic
semitones with chromatic ones (e.g. F#3-F3 in place of Gb3-F3, etc.),
and augmented or diminished vertical intervals in place of usual ones
(e.g. the augmented third Bb3-D#4 in place of the fourth Bb3-Eb4). The
musical effect can sometimes be delightful. In certain neo-Gothic
temperaments, for example, the augmented third Bb3-D#4 will
approximate 11:8 (~551.32 cents), aptly styled a "superfourth" by
David Keenan, giving cadential resolutions a special beauty.

5. Apart from intonational matters, our diagrams illustrate a general
point which has been left largely implicit. Shifting from an intensive
to a remissive resolution of the same most proximal quad leaves the
sizes of vertical intervals unchanged, but changes the arrangements
and directions of melodic motions, and also the sonority of resolution
(here from F3-C4-F4 or F3-C4 to F#3-C#4-F#4 or F#3-C#4).

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Gentle introduction to neo-Gothic (1/Part 1)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

🔗Joseph Pehrson <josephpehrson@compuserve.com>