Re: Paul Erlich's 3-chord coverage and Gothic music

Recently Paul Erlich contributed this interesting item as part of a
discussion about his "three-chord coverage" desideratum that three
consonant chords should suffice to include all the tones of a scale;
see also his "Tuning, Tonality, and Twenty-Two-Tone Temperament,"
_Xenharmonikon_ 17 (Spring 1998), pp. 12-40 at 19:

> I forgot to point out that Medieval Western practice ("Gothic?")
> also fails the three-chord-covering property. The consonant chords
> were 3-limit dyads, but three of those would cover at most 6 notes,
> while the diatonic scale has 7. The diatonic scale with 3-limit
> harmony is another system which I wanted to leave out of my paper
> for brevity, but clearly cannot be dismissed altogether, as it was a
> language in which great musical works were produced for a few
> centuries.

This apparent exception to the "three-chord-covering" property invites
a further discussion of Gothic harmony, maybe beginning with a few
important cautions. My purpose here is to consider how a "tonality"
might be conceptualized and defined in Gothic music, and more
specifically in 13th-century writing for three or four voices.

Rather than open with a series of caveats taking considerable space, I
would simply caution that the following is my attempt to follow an
approach not inconsistent with actual Gothic theory. The medieval
theory of polyphony seems more focused on vertical sonorities and on
some ornamental and formal graces than on questions of "mode" or
scale, so what follows may reflect mainly my own proclivities.

Also, it would be well to observe that Paul Erlich himself has here
emphasized that his "three-chord coverage" property is intended mainly
to caution on the consequences of certain complex modern tuning
schemes, rather than to present a universal standard.

The complete unit of three-voice Gothic harmony is what Johannes de
Grocheio (c. 1300) refers to as embodying a _trina harmoniae
perfectio_ or "threefold perfection of harmony," and to which I shall
refer in English as a trine. One might define a trine as the 3-limit
equivalent of a 5-limit triad; it consists of an outer octave, and an
adjacent fifth and fourth:

| d' | d
| 4 | 5
8 | a 8 | g
| 5 | 4
| d | d

Note that in medieval theory, the octave counts as a "real" interval,
so that we have indeed a sonority of three tones and intervals. With
the fifth below and the fourth above, we have the more smooth and
conclusive form; with the converse arrangement, the harmony is
relatively stable but less conclusive. We may refer to these forms
respectively as 8|5-4 and 8|4-5 -- i.e. outer|lower-upper. Partchian
theorists might see these forms as representing respectively otonality
(2:3:4) and utonality (1/2:1/3:1/4), these numbers representing
frequency ratios (string ratios being conversely 6:4:3 and 4:3:2).

We may take a trine as the vertical embodiment of an octave-species,
including for the present purposes _eight_ diatonic notes,
e.g. d-d'. Following medieval concepts, we may regard these eight
tones as basic, but tones _beyond_ the octave as essentially
replicates: thus a diatonic octave-species has eight tones, as opposed
to seven or more than eight.[1]

Definitely engaging in the participatory sports of neologism as well
as "neo-medieval" theorizing, I might propose the term _trinality_ to
describe a stable trinic center plus the various other sonorities
which might be formed from the material of its octave-species.

CAUTION: Note that while many 13th-century pieces use exclusively or
almost exclusively the tones of a single diatonic octave-species,
inflections such as Bb/B and F/F# are not uncommon. Indeed, both B and
Bb are part of the basic medieval gamut or system of _musica recta_,
and they may fluidly alternate in a single melody; or we may have a
signature of Bb in the lowest part only, with a default of B in the
others. However, for the sake of initial simplicity, let us assume an
octave-species of eight tones.

Focusing mainly on the more conclusive 8|5-4 trine, we find that this
might be said to include the scale degrees 1-5-8, here written as
^1-^5-^8 to avoid confusion with all-important vertical intervals. The
less conclusive 8|4-5 trine would be ^1-^4-^8 -- but let us here
concentrate on the more conclusive form.

While Erlich's "consonant chord" might be read to mean a stable
sonority, in fact much of the feeling for scale and "trinality" in
Gothic music, at least for me, centers on cadential action involving
the contrast between stable trines and _unstable_ sonorities. Here I
use _cadential_ in its broad medieval sense of a tension-relaxation
progression in which unstable intervals resolve to stable ones; in
some compositions, what one might call "microcadences" can occur at
almost any "change in harmony" (motion of the lowest voice).

From this viewpoint, given the pervasive role of unstable sonorities
and cadential progressions in this music (3-limit has the virtue of a
wealth of unstable sonorities), it is easy to find progressions of
three or fewer sonorities which can either explicitly or virtually
cover a complete trinality or octave-species. Taking first the case
where all eight tones are explicitly presented:

d' e' f' ^6 ^7 ^8
a b c' ^3 ^4 ^5
g f ^2 ^1

In this cadential formula for a trinality on F, we have a relatively
consonant but unstable combination of g-a-d' or 5|M2-4 followed by a
somewhat more tense g-b-e' or M6|M3-4 resolving to a trine of f-c'-f'
(M6-8 + M3-5). In terms of scale degrees, we have all eight
represented.

It is possible to classify the notes of an octave-species into three
groups, from a cadential standpoint. Tones ^1-^5-^8 define the stable
trinic center and goal; adjacent tones ^2, ^4, ^6, and ^7 typically
occur in penultimate unstable sonorities and tend to resolve ^2-^1,
^4-^5, ^6-^5, and ^7-^8. "Mixing and matching" these progressions
generally produces cadences in which unstable intervals resolve by
conjunct contrary motion, as in the following impressive example in
four voices:

e' f' ^7-^8
d' c' ^6-^5
b c' ^4-^5
g f ^2-^1

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

Note that these two sonorities represent all tones except ^3, the only
tone neither included in ^1-^5-^8 nor adjacent to any of its
tones. However, this tone is "implicitly" defined by the expectation
that the major third g-b will be divided into two whole-tones g-a-b
rather than a minor second and augmented second, e.g. g-ab-b.

In fact, a recognizable octave-species or trinality can often be
conveyed by fewer than all eight tones, as this example in a trinality
of G:

f'-g' ^7-^8
b -d' ^3-^5
a -g ^2-^1

(m6-8 + M2-5)

Here, although ^4 and ^6 are not explicitly sounded, the
characteristic diminished fifth between ^3 and ^7 gives this
progression a "natural G-species" quality. Note that here ^3 moves
thirdwise to ^5, arriving at the fifth of the resolving trine.

Another example of "implicit scale definition" shows the alternative
^3-^1 progression:

e' f' ^7-^8
b c' ^4-^5
a f ^3-^1

(M2-4)

Here the ascending semitonal progression ^4-^5 conveys a "natural
F-species" quality. From another viewpoint, the known major third
relationships a-f and e'-c' imply a-g-f and e'-d'-c', thus allowing us
to recognize the octave-species despite the absence of ^2 and ^6.

Allowing for the vital and pervasive role of _unstable_ sonorities in
defining a Gothic trinality, and the factor of "implicit" scale
definition through such cues as major third relationships and tritonic
intervals, the musical feeling of "coverage" in this music may be less
problematic than one might guess on the basis of theories applying,
for example, primarily to 5-limit or 7-limit music.

An actual Gothic piece, of course, will likely present more than one
potential trinic center, just as a 5-limit piece typically features
more than one potential triadic center. Sometimes it's easy to say
"this piece is obviously in the trinality of D (e.g. d-a-d')," and in
other cases the judgment may be much less clear, e.g. "this piece
gently oscillates between centers of f-c'-f' and g-d'-g', finally
settling on the latter." Each trinic center might typically be
reinforced, at least temporarily, by cadencing to it from unstable
sonorities.

This raises a final point: Gothic harmony features mostly conjunct or
near-conjunct (thirdwise) melodic progressions in all voices -- or,
from a vertical point of view, the resolution of unstable sonorities
by conjunct or near-conjunct contrary motion. Thus while the system
might be taken to fulfill Paul Erlich's property that "[t]he majority
of consonant chords have a root that lies a Q [fifth or fourth] away
from the root of another consonant chord," conjunct or thirdwise
motion in the lowest voice is more typical than _direct_ progressions
between sonorities a fourth or fifth apart.

--------
Note
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1. A reader familiar with the medieval modes may recognize that this
vertical division of the octave into fifth and fourth is somewhat like
the theoretical division of an octave-species into an authentic mode
with lower fifth and upper fourth above the final (e.g. Mode I or
Dorian, d-d'), or a plagal mode with lower fourth below the final and
upper fifth above it (e.g. Mode II or Hypodorian, A-a' with final d).
However, because the term "mode" for various medieval theorists such
as Johannes de Grocheio implies not only an octave-species but a
specific pattern of beginning, middle, and end for a melody, I have
preferred in a polyphonic context as opposed to plainsong to speak of
"octave-species."

Most respectfully,

Margo Schulter
mschulter@value.net

Margo Schulter makes some excellent points about Gothic music which
demonstrate that the style is far from a dyadic, 3-limit analogue to
common practice tonal music. She also discusses the concept of "implicit
scale definition" as an alternative to "coverage". It should be pointed
out that only two tones, namely thos comprising the diminished fifth,
are needed to identify the diatonic scale in any regular tuning other
than 12-tET. In 12-tET, the diminished fifth is identical to the
augmented fourth, so any one additional note is needed to identify the
scale. My pentachordal decatonic scales in 22-tET contain a 10/22 oct.
interval which is unique in each scale and thus serves to identify the
scale unambiguously.