source file: m1606.txt Date: Thu, 10 Dec 1998 15:54:37 -0500 (EST) Subject: Hypertones, counterset & other weird microanimals From: Stephen Soderberg 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