[Smt-talk] The Ubiquitous Triad

Ildar Khannanov etudetableau at gmail.com
Sun Jul 20 23:53:01 PDT 2014

Dear Stephen,

thank you for starting a fascinating discussion. My two cents, as always,
come from respect for 2500 years of tradition. I do not think that
Pythagoras was up for jokes. I know very well where from the critique of
the overtonal concept comes from but this is not the point.

The triad has been embedded in culture and history. Beyond its "music
theory today" aspect, it is related to major cultural constants. Rameau
calls it l'accord parfait for a reason. I disagree with Stephen Martin who
tried to dissociate Rameau and Descartes. I feel the breath of Cartesian
thought in Rameau. And Zarlino did provide the divine numeric proportions
for both major and minor triads (these numbers have been criticized as
well, but it is arguable and not finished yet). Triad is also related to
the morphologic forms of numerals that we have lost since Greeks.

Now, let us look at the triad; it is as much a concept of music theory as
it is a theological, philosophical, mathematical concept and a large part
of western worldview. What did we offer instead? A pitch class set, unlike
triad, has no roots in history, culture and national tradition. It is what
it is and nothing more, just the set of cardinal numbers.  I agree with Dr.
Forte's position: he intentionally created a tool that is independent of
the context. This is the goal of any scientific experiment. However, in
analysis we deal with music, which is par excellence a part of human
nature, humanities. Yes, it is annoyingly disorganized, illogical,
eclectic, etc. Does not meet rigorous criteria of scientificity. And yet,
it is what it is.

My suggestion would be to write a book on pitch class set in a wide
philosophical, theological, deconstructive context.

Is pitch class set a form of perfection? Is it capable of connecting
microcosmos with macrocosmos? Is it organic, natural, corporeal? Does it
allow for polysemy? How pitch class set is related to the concept of
harmony? What is the ontological status of pitch class set? It pitch class
set a teleological object (does it carry a goal in itself)? What is pitch
class set as a pure phenomenon? Does pitch class set relate to any
theological concept (Trinity, Transfiguration, Pneuma, etc.)?

These questions are overdue: the pc set has already become a tool of
composition, so it is not for theorists to doubt. As with the Dreyklang of
the figured bass tradition, we, theorists, deal with the status quo that
needs theoretization on a higher level.


Ildar Khannanov
Peabody Institute
etudetabeau at gmail.com

P.S. I really liked your reference to Borges! He indeed created new rules
of the game! I assigned my students to read his texts over the summer.
Expect an avalanche of ideas.

2014-07-20 18:01 GMT-04:00 Stephen Soderberg <hyperchord at me.com>:

> I want to thank Nicolas for his thoughtful response (copied below).  I
> have read his comments over several times and must say that there is
> nothing in them  that I find truly dissonant with points I have made in my
> blog entry –– which is not to imply a consonance with his conclusion. And
> so ....
> I must admit that I hadn't thought about it all starting as a gentlemen's
> agreement ("Western music or its musicians decided" –– consciously? Maybe
> "decided" is not the right word, but I know what Nicolas is getting at)
> that was periodically renewed across the centuries ("'our' choice"? ––
> Really?? I don't remember being asked and I doubt Nicolas was ––
> unconscious acceptance - the unexamined life - is one of my primary
> issues). Nevertheless, I like the audacity of this justification. An
> argumentum ab auctoritate stretched over 1,000 years is nothing to sneeze
> at. But neither is it anything to genuflect for.
> Re my identification of the culprit as the "tonal triad": I'm trying to
> critique the "perdurability" (as one anonymous reader put it –– I'll add
> that to the growing list) and relatively recent near fetishist attachment
> associated with the usual/traditional/common-practice-period/over-determined/ubiquitous/tonal/consonant/diatonic/3-11/<3,4,5>/(p,sign)/pretty-sounding
> triad. I had to pick one and stick with it. So please pardon me –– to avoid
> a confusion not unlike that of trying to keep Tolstoy's characters straight
> through the first five chapters, a scholar's accuracy was not as important
> to me here. In particular, I consciously avoided using "consonant" for
> several  reasons, among which are those hinted at by Nicolas.
> But for the record: of course I know the triad *itself* is innocent; and
> as to that triad itself, I'm an admirer. In fact, as will soon be seen, I
> believe the object in question *may* have a future role to play *beyond*
> its essential backward-facing role in support of "music theory today" whose
> goals are embedded in analysis. But perhaps its new role will be taken down
> a notch or two or three. As a preview of thoughts I don't know that anyone
> has even had yet, we might consider yet another modifier to describe a
> future role: the Janus-faced triad –– but not a Janus like the one we have
> today where one face is looking into the past and the other face is looking
> into a nostalgic mirror of the past. This will not be easy & maybe not even
> possible for a very long time.
> One final note re Nicolas' statement, "Musical analysis, that I know,
> never claimed to state how music should be." Not directly perhaps, but
> this claim is less convincing now and has become much more difficult to
> defend in the past century or so. The sheer weight of accumulating analysis
> and the curriculum to support it and the oversupply of teachers to teach it
> all clearly favor an approach that strongly urges, without necessarily
> meaning to say such an impolite thing out loud, that "this music we're
> teaching is superior" –– and I'm not only referring to ethnologically
> different musics "of other traditions" as Nicolas seemed to be suggesting
> as the only other options (we've managed to infiltrate a lot of them at any
> rate), but also to musics that have emerged over the past century from
> the larger liberal Western intellectual tradition, whether as evolutionary
> developing variations on that tradition or reactions and experiments going
> in radically new paths.
> Re Isaac Malitz' comments on, shall we say, the folly of getting all
> feverish over the notes: I agree, but I'll be taking it in a different
> direction that, again, doesn't involve considering the note to be a
> "relic." A minor triad, for example, is *both* three individual notes in
> a certain relationship (however they got there)* and* a gestalt one
> doesn't have to analyze into parts  to recognize immediately. But another
> time on that.
> Stephen Soderberg
> Keswick, VA
> On Jul 18, 2014, at 05:29 PM, Nicolas Meeùs <nicolas.meeus at scarlet.be>
> wrote:
> It seems to me too easy and futile to render "the tonal triad" responsible
> of so many confusions.
> Western (or Occidental, or European, as you like) music or its musicians
> decided, about a thousand years ago, to base the musical game on an
> opposition between consonance and dissonance. Many other musics made other
> choices, e.g. oppositions of rhythms, or of timbres, etc. But
> consonance/dissonance was 'our' choice (you may belong to another
> tradition, but I don't think Stephen does).
> Now, basing the musical game on consonance/dissonance entails defining
> these terms. This took time, several categorizations have been tried, and
> several justifications; these would deserve a discussion here, but as this
> isn't the question raised by Stephen, I'll refrain entering it now. Our
> forefathers came to a general agreement on the intervals that could be
> considered consonances and those considered dissonances. Once again, I
> won't discuss now their conscious or unconscious reasons. We all know the
> result: unison, third, fourth, fifth, sixth and octave are consonances,
> seconds and sevenths dissonances, and augmented fourths/diminished fifths
> somehow between the two.
> The logical consequences are simple, even if not immediately evident:
> – The maximum number of pitch classes all 'consonant' between themselves
> is three.
> – Granting the principle of inversion, they all and exclusively form
> "triads" (properly speaking, those that Stephen calls "tonal triads").
> – All other combinations of intervals include 'dissonances' and therefore
> do not form "triads" properly speaking. (One argument for saying this is
> that in most definitions of "chords", dissonant combinations of three notes
> imply some other, absent note. This boils down to the theory of piled up
> thirds, but again this is not the question raised by Stephen: he should
> perhaps begin realize that his question is not the good one...).
> – "Tonality" by no means is implied in this, so that the notion of "tonal"
> triad is irrelevant.
> Neo-Riemannian Tonnetze (which describe triads) invite us to consider
> other possible combinations. I think to remember that Rick Cohn shew that
> the richest Tonnetz is that of which the coordinates are arranged in fifths
> and thirds.
> All this evidences specific properties of triads, and explains why they
> have played such a prominent role in our music for the last thousand years.
> Triads by no means are a norm for ANY music. They certainly were a norm for
> our music for these last centuries, especially (but not exclusively) in its
> tonal period, and I see no problem in acknowledging that. Musical analysis,
> that I know, never claimed to state how music should be. It does try to
> show how the music analyzed has been: our Western (etc.) music has been
> triadic for many centuries, and I see no problem in this.
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