Hypertones, counterset & other weird microanimals

Been away for awhile, so I'll first print all of Paul Erlich's post that
I'm replying to. This is a long post, folks, but I promise you a very
strange twist if you stick it out...

[Soderberg:]
> >So to
> >malign 12tET (so I assume) is to malign its effects on the ear, not its
> >12-space compositional-theoretic properties, which are quite potent
> >regardless of how you tune it. Likewise, arguments about how the
> diatonic
> >scale should *really* be tuned (for psycho-acoustic or historical or
> any
> >other reason) are mostly irrelevant to its basic 7-space
> >(compositional-theoretic) properties which carry over to *any* tuning
> that
> >closes (or pretends to close) at the octave. In the end, things like
> >n-tone theory or ME theory or hyperdiatonic theory are abstract and
> have
> >nothing to do with tuning.

[Erlich:]
> And almost nothing to do with music. Stephen, I couldn't disagree with
> this position more, but I have only the highest respect for your
> intellectual rigor, and I appreciate the respect you've shown to mine.
> As a starting point for some discussion, perhaps you could tell us how
> far you'd go in defending the phrase "regardless of how you tune it" in
> the first sentence above. I infer that some would require a Rothenberg
> proper 12-tone scale; how about you, Stephen?

After re-reading my post after Paul's comment, I can answer his
question "how far you'd go in defending the phrase 'regardless of how you
tune it'" quite easily: not far at all. I have to admit that it came off
sounding like I don't believe that tuning "matters" at all, and this is
certainly not true. My position would have been better served if I would
have approached it from this angle: once you choose a tuning -- ANY tuning
-- what then? You're in a new universe -- are the "rules" the same here
or are they different? What does "counterpoint" mean here? How do you
want to define a "chord" and what does this definition imply about
voice-leading? What's "consonant"? "dissonant"? "tonal"? "atonal"? Does
this universe offer more "degrees of compositional freedom" or less? Are
there formations, structures, transformations here that can't be found
elsewhere, and, if so, how can you take advantage of them? Etc. etc. In
7-scale diatonic (the language of the "Western canon"), these are all
non-questions unless you want to sound like Bach or Bacharach, and I sense
(with the possible exception of those legitimately interested in
historical tunings) most on the list are headed in clearly different
directions.

What I'm trying to get across (and this is a follow up to my "why
triads?" question) might best be seen through an example. So let me use
Paul's 10-scale (specifically, the non-ME "standard pentachordal major,"
SPM). The reason for this choice as an example is that Paul has offered
one of the best attempts I've seen to confront _publicly_ many of the
above questions in a decatonic environment. And as Paul and a few others
seem to have concluded, triads are not _necessarily_ the best or most
logical choice for basic sonorities in synthetic mega-scales for a number
of good reasons.

While I will be pointing out what might be termed some
"difficulties" with Paul's SPM, the analysis to follow is NOT meant to
"dis" this scale which has many interesting properties pointed out in
Paul's article. Looking on scale theory as a kind of "abstract geometry"
compels me to hold that, first, no scale (like no geometry) is "wrong,"
though it might be (compositionally) misused or misapplied, and second,
scales (like geometries) can be related to one another via their invariant
properties. This is not to say that the non-invariants of a given
scale/tuning are unimportant (which is what my previous post may have
implied); to the contrary, non-invariants give a specific scale its
characteristic "flavor." Thus, among 7-scales, the usual vanilla diatonic
major and the paprika Hungarian minor share some important invariant
properties, but no one is likely to confuse their non-invariant
properties.

The SPM is based on the interval string <2232223222>. While it is
not maximally even, it is nevertheless symmetric, an important defining
quality in chord formation. So the SPM is: (0,2,4,7,9,11,13,16,18,20). To
define/generate tetrachords, Paul uses the string <3322> -- again,
symmetric but not ME -- which is to say that we are to form a chord by
taking (generic) notes 1,4,7,9. The result is the following set of 10
basic tetrachords (to the left of the vertical line):

scale
step tetra string quality tetra name

I 0 7 13 18 <7654> Maj | 0 3 6 8 0x
II 2 9 16 20 <7744> Aug | 1 4 7 9 1x
III 4 11 18 0 <7744> Aug | 2 5 8 0 2x
IV 7 13 20 2 <6745> Min | 3 6 9 1 3x
V 9 16 0 4 <7645> Ma-mi | 4 7 0 2 4x
VI 11 18 2 7 <7654> Maj | 5 8 1 3 5x
VII 13 20 4 9 <7654> Maj | 6 9 2 4 6x
VIII 16 0 7 11 <6745> Min | 7 0 3 5 7x
IX 18 2 9 13 <6745> Min | 8 1 4 6 8x
X 20 4 11 16 <6754> Mi-ma | 9 2 5 7 9x

Now let's "reduce" the SPM to a chromatic 10-scale (call it 10X
for reference) whose string is <1111111111> (this is pretty much the same
as naming the diatonic scale steps):

0 2 4 7 9 11 13 16 18 20
| | | | | | | | | |
V V V V V V V V V V
0 1 2 3 4 5 6 7 8 9

Using the same chord-defining string as before, <3322>, we can
generate 10 chords in 10X, each of which corresponds 1-to-1 to a chord in
SPM (see above to the right of the vertical line -- each 10X chord is
"named" for reference.) Since each 10X chord is based on the same string,
<3322>, there is no way (other than relative scale position) to
distinguish and group them as there is with the chords in SPM. Thus there
is no invariance between these two "systems" WITH RESPECT TO chord
"quality" -- and thus, since SPM and 10X are two ways to partition the
octave (two distinct tunings), tuning matters for specific chord quality
(but matters less, as we shall see, for _patterns_ of chord qualities).

Now, Paul gives some very logical functional names to SPM chords
based on the identification of Q (3:2) with 13 chromatic steps. We will
isolate here what he identifies as the tonic (I), the subdominant (V), and
the dominant (VII). It's at this point that we can begin to see some very
important differences between 10-scales and 7-scales. First, unlike
7-scales with <223> chords, 10-scales with <3322> chords only have one
triple of chords that covers the scale. In the SPM scale, a triple of
_consecutive_ tetras will form a cover, but note that, although we might
expect it of a quasi-diatonic system, the union of the tonic, dominant,
and subdominant that Paul identifies is _not_ a cover of the scale. In
fact, consecutive triples are the only sets which can form a cover for
SPM, and furthermore, due to "scale covariance" (described in my MTO
article), this characteristic is shared by _any_ 10-scale system based on
<3322> chords. This can be checked with the simple 10X system given
above, or by listing a random 10-string such as <1714539926> and listing
its <3322> chords. So the T-D-SD relationship doesn't provide "key
coherence" as we might expect.

A second characteristic of all 10-scale/<3322> systems is that,
unlike 7-scale/<223>&<2221> systems and others, they aren't saturated with
what Eytan Agmon calls "efficient linear transformations." In the usual
diatonic, for example, any triad can be linked to any other triad by
either a unison or a scale-step move. In SPM/<3322> there are two
fundamental progressions that don't connect smoothly: those whose roots
are 4 and 6 scale steps apart, respectively, e.g.:

I V I VII

18--------->16 18-------->20
13-->(11)--> 9 13-------->13
7---------> 4 7--------> 9
0---------> 0 0-->(2)--> 4

whereas, for example, I-VI describes a typical ELT (smooth voice-leading):

I VI

18---->18
13---->11
7----> 7
0----> 2

Once again, this is true of all 10-scale/<3322> systems due to scale
covariance.

Another interesting property that scale covariance carries through
all 10-scale/<3322> systems is the common-tone pattern. Starting with any
chord in the list above, chord pairs n scale steps apart will have x tones
in common:

n: 0 1 2 3 4 5 6 7 8 9
x: 4 0 2 2 1 2 1 2 2 0

Finally, the pattern of chord qualities is invariant for 10-scale
systems based on a scale string (where x ne y are any
integers) and generic string <3322> as above. To illustrate, let Maj = a,
Min= a', Ma-mi = b, Mi-ma = b', and Aug = c; then the quality pattern will
always be some rotation of: a c c a' b a a a' a' b' (a skew-symmetric
pattern generated by the interaction of the scale and generic strings).

The bottom line is, while scale tuning "matters" in many ways (not
the least of which is vertical & horizontal quality/modality, imparting a
defining "flavor" to a system), once you have chosen a scale, the real
work is just beginning since most scales offer multiple "geometric"
possibilities. Let's now see what happens if we use a triad as a basic
chord in SPM.

If we keep the demand for a 3:2 basis for a majority of chords (a
possibly psychologically reasonable but not logically necessary
condition), one plausible triad list uses the ME generic string <334>
which in effect simply deletes the top note of Paul's tetras yielding
triads:

I 0 7 13 Maj
II 2 9 16 Aug
III 4 11 18 Aug
IV 7 13 20 Min
V 9 16 0 Maj
VI 11 18 2 Maj
VII 13 20 4 Maj
VIII 16 0 7 Min
IX 18 2 9 Min
X 20 4 11 Min

This reduces the 5 tetrachord qualities to 3 trichord qualities: <769>
(Maj), <679> (Min), and <778> (Aug).

First, this tells us something interesting about minimal covers
because it focuses attention on triads of identical quality. We noted
above that the only cover (with the usual criterion for a triple of
chords) for tetras here was three consecutive tetras and therefore the
"tonic" could not participate with the "dominant" and "subdominant" to
form a cover. But the above list suggests that the cover principle works
differently in this scale's universe. Three consecutive triads won't
cover the scale, but four consecutive triads will. AND, by altering the
cover principle in this way, we find other, more interesting covers which
tend to alter the way we might envision the T-D relationship. If we
collect all the Maj chords (I,V,VI,VII), say, we disclose a second minimal
cover which can be expanded to any four chords in that (root)
relationship. Furthermore, while ELT (smooth voice leading) is fairly
depleted in this (triadic) interpretation, it is in full force for
quadruples of chords in a cover: the progression I-V-VI-VII-I is
ELT-saturated. All of this suggests the interesting possibility that, in
this universe, there are _three_ "dominant" functions -- "dominant" (VII),
"subdominant" (V), and an auxiliary(?) "spanning dominant" (VI) since
I-VII and I-V are naturals for voice-leading, but you can't easily lead V
to VII without VI.

This then brings us back to the status of Paul's original tetra
selection. There are several ways to include triads here. The obvious
way is to declare the triad to be the scale's basic sonority. This would
then turn Paul's tetra into a "seventh" chord, and, while things would
look somewhat strange, the building blocks would pretty much resemble our
inhereted notions of chords.

A second interpretation holds, I think, much more promise -- and
beside, it's more radical. Here's where I jump off the high-dive and hope
there's water in the pool when I reach bottom. First, a preliminary
exercise. Take any typical 4-part harmonization of a hymn. Play the
soprano & alto lines up an octave and the tenor & bass parts down an
octave. Concentrate on hearing it, not as 4 lines and not as a succession
of chords, but as 2 lines, each of which consists of 2 lines.

One of the questions I asked near the beginning concerned
"counterpoint" in megascale systems. Let's assume that the tetra
described above is the basic harmonic sonority in our system analogous to
the triad in common practice theory (CPT). In CPT there's a hierarchy,

pitch (unad)
interval (doad)
triad
(7th chord)

along with well-defined notions of consonance and dissonance, which taken
together help shape the "rules" of CPT counterpoint. Replacing the CPT
triad with the SPM tetrachord, we now posit the following hierarchy:

pitch (unad)
interval (doad)
hyperinterval (triad)
tetrachord
("7th chord")

Ignoring the 7-th chord to keep the illustration simple, focus on
the new structure, the triadic "hyperinterval" (HI). Define a consonant
HI as the intervallic complex formed by any three pitches in a (consonant)
tetrachord (just as, ahistorically, we can define a consonant interval
backward as formed by any two pitches in a consonant triad). Call this
set of 3 pitches a "hypertone" (HT). We now have the basis for a
generalization of traditional counterpoint which might be called
"counterset." Hypothetically, traditional counterpoint (in any historical
version) should turn out to be a category of counterset.

What are the "rules" of counterset? I haven't the foggiest idea.
The reason is that their definition, in my opinion, must hinge on
compositional praxis (if anyone decides to play with this idea -- and I
won't hold my breath). But a random "2-part" exercise (with no
"dissonance" allowed) might look like:

Counterset: { 7 13 18} { 9 16 4} {11 2 7} {13 20 4} { 0 7 13}

CF:-) { 0 7 18} { 9 16 0} {11 18 2} {20 4 9} { 0 7 18}

(I) (V) (VI) (VII) (I)

Set notation for HTs is retained to emphasize that the elements of any HT
can appear compositionally in any order. Cantus firmus and Counterset
lines can be distinguished by tessitura, instrumentation, rhythm,
repetition, leaps, etc. Any HT can appear as a simultaneity or an
arpeggiation. At times the "lines" may not be aurally distinguishable at
all. Dissonance at the HI level can be introduced by passing or other
non-chord HTs -- e.g., in VI above replace {11 2 7} with a HT belonging to
another chord, such as {20 4 11} from chord X (the totality, {2 4 11 18
20}, is not a consonant "chord" since it contains a foreign HT).

In the preparatory exercise above, the 4-part chorale setting was
re-interpreted (in our new language) as a 2-part hypertonal setting. If
the original tessitura is now reinstated, the hypertonal interpretation of
the setting, while not "incorrect," appears absurd -- it collapses into a
mere paper theory. In other words, there is generally not much use for
hypertonal theory in traditional musics, EVEN THOUGH they may be
considered as "first-order" hypertonal... you just can't hear it, nor
should you be able to, nor should you want to (IMHO). But when we get into
megascales, CPT can't handle the increase in volume -- of the individual
notes and of the new relationships that begin to appear (I know I'm into
opinion-mongering here). So the next thing to appear is (as above) a
second-order hypertonal set of structures. I've seen it occur time and
again, not just with Paul's scales. And each time, these second-order
(and higher) structures bring with them a semblance of a very new (and
strange) type of order.

(Two other lower-order hypertonal systems can also be posited --
chord clusters and tone rows, but I won't muddy the waters with those
right now (especially the latter which would lead to a deconstruction
that denies the existence of "atonality"))

Returning to the original point, then, does tuning matter? Yes,
but as I've been trying to demonstrate, megascales tend to gravitate into
"bundles" of megascales which are defined by the invariants shared by the
bundle. And one of the most universal invariants appears to me to be
hypertonal ordering... the "laws" of counterpoint don't really change in
other universes, they just get really weird. And this leads to the final
question which, for now, has to remain open: Can we hear it? (Any
composers out there like to take on the challenge?)


Steve Soderberg