Thread: What is Musical Set Theory?

From the preface to The Structure of Atonal Music:

"In 1908 a profound change in music was initiated when Arnold Schoenberg began composing his "George Lieder" Op. 15. In this work he deliberately relinquished the traditional system of tonality, which had been the basis of musical syntax for the previous two hundred and fifty years."

In other words, atonality is not defined in relation to your "broad conception of tonality" that includes modal musics, but rather in relationship to common practice tonality only.

Originally Posted by Mahlerian

From the preface to The Structure of Atonal Music:

"In 1908 a profound change in music was initiated when Arnold Schoenberg began composing his "George Lieder" Op. 15. In this work he deliberately relinquished the traditional system of tonality, which had been the basis of musical syntax for the previous two hundred and fifty years."

In other words, atonality is not defined in relation to your "broad conception of tonality" that includes modal musics, but rather in relationship to common practice tonality only.

I think you are interpreting that statement too rigidly. I do think that atonality can be defined generally, as "music which has no tonal center." That would include common-practice, as well as any generally tonal/modal /scalar system.

The only difference in common practice, as a form of tonality, is its specific scale choices (major/minor) and specific harmonic functions and practices, as part of that tradition;

...but essentially, "tonality is tonality," meaning that there is an hierarchical system in place based on harmonic principles of a fundamental (or key) note, and its constituent parts, all based on scales which span an octave, using the octave as "1" or starting point; and each step can have a function, with triads built on it, which gives it harmonic function. That's not the exclusive realm of common-practice tonality, by any means.

It's very simple, really, and there's no need to over-complicate it. And I agree that none of this is going to affect the listener; the listener will intuitively hear whether or not there is a tonal center.

My point is further reinforced by Howard Hanson's use of set theory, which he has modified somewhat to fit the requirements of tonality. Hanson's ideas expand the idea of tonality until it becomes a much more flexible version of tonality, with many more possibilities than common practice tonality.

These respected professionals, like Hanson and Forte, have no need for a limiting, strictly academic view of tonality as exemplified by "common practice;" these men are interested in creating music which is free to access all possible forms of scales and harmonic practices, and are responsible for creating the "expanded" view of tonality that I have espoused.

Hanson's book Harmonic Materials of Modern Music. The word "harmonic" means that Hanson approaches the set material always by listening, with consideration to the ear and tonality.

One way of thinking is to consider set theory as a kind of figured bass but where the relationships you are labeling aren't figured from the bottom of the chord but by collapsing the set into the smallest possible interval via octave equivalence. Your mileage may vary as to how musically rewarding that is. As it stands, there are really very few ways of talking about harmony when tonal function/behaviour is ruled out. To put it simply, if we don't know how a chord should behave we don't really have good methods for categorizing it perceptually and setting up a method for hearing it.

Set theory is the creation of categories for relating together sets of notes. There are several methods for relating together sets:

1) By calling them the same harmony. For example if I had Bb, E and C I would call that (026). If I had C, F# and D I would also call that (026). By this measure I would say they are equivalent harmonies. We might also look at how close one set is to another mathematically. For example (01356) and (02356) might be claimed to be strongly related sets.
2) By indexing their interval vectors. If I had (0137) and (0146) I would note that each has the same number of each possible interval within the set. Therefore they have the same interval vector and might be said to sound alike. We could also make observations between sets that had similar but different interval vectors and claim there is some similarity there.
3) By saying there is a strong relationship to one set and to its compliment (i.e. all the notes left out of the set). So we might say that (0148) and (01245689) are closely related sets.
4) We might look at union sets where one chord is made up of the combination of smaller sets. An example of this might be (014) having (023) added to the top of it to make (01467).

However, the way we decide to interpret these relationships of perceptually, compositional or mathematical closeness is entirely up to the theorist. This is the basic problem of set theory, it doesn't theorize musical action at all. It just gives an apparatus to start talking about a piece.

I'm still looking for someone who can explain this better:

http://en.wikipedia.org/wiki/Klumpenhouwer_network

So a Klumpenhouwer Network (or K-Net) is a musical network where the relationship between the different pitches is invariant even though the absolute pitch interval changes. I think it is best explained through example (the best explanation is Lewin on Schoenberg Opus 11 http://tinyurl.com/p7e3jwa) but there a few things we need to understand before we get there.

Firstly we need to be at home with transposition and inversion. So we all hear octave equivalence. This is that we name notes an octave apart the same note. This means, whatever group of notes we have at whatever pitch level, we can imagine them within one octave without disturbing the relationships between them. We can take this one step further and imagine them as forming a circle as show here: http://tinyurl.com/ow8ss3j. We assign each note a number 0 thru 11.

For our purposes transposition is the shortest distance around this 12 step circle. Imagine if you went from C (0) to Ab (8) - that would be 4 steps. We would therefore label this as T4 (Transposition 4).

Inversion is where map one set of pitches onto another around a particular axis. This sounds a bit mad but it relates to a genuine musical experience - you will sometimes have sets that are the same as another except it is turned upside-down. The best known example of this is major and minor chords. One way of thinking about a minor chord is to imagine it as an upside-down major chord. In a major chord you have a major 3rd at the bottom whereas in a minor chord it is at the top. It is time to get a little more formal to describe this. Basically turning things upside-down means exchanging an interval above a certain pitch with the same interval below it. This means that pitch is the axis around which the inversion takes place. To return to our circle diagram, it is important to note that if you draw a line across it you are always going to write a line through notes a tritone apart (06). If you picked a set of notes around the circle, then you pick a spot on the circle to invert around you might come up with something that looks like this: http://imgur.com/otwk99B. You will notice that each red note is linked to a blue note. These notes map to each other. The label I1 means inversion where the numbers sum to 1 mod 12. I know that might seem a bit arbitrary but since you can invert around the space between two notes it is pretty important to name it like that.

The last thing we need to know is what a network is. Put simply, it is a set of pitch relationships. You have a group of pitches, keys or whatever and they are related to each other by certain operations (ie. transposition, inversion etc.). This is often represented diagrammatically in order to ease understanding and often an analyst tries to get to some underlying symmetry by describing these networks (but that is not expressly what a network has to be).

So we now have the intellectual resources to talk about K-Nets. The name is after Henry Klumpenhouwer who it seems came up with the idea while under the supervision of David Lewin.

A K-Net is a set of pitch relations which can describe the relationship between sets which might otherwise look unrelated. I am going to give you an example of a K-Net but I am making it up as I go along so please don't be disappointed that it isn't that musically interesting! Let us imagine you had a chord made of C E F# G. That would be set (0137). Compare that to C# D F B which would be (0236). Pretty dissimilar eh? Well there is a relationship I could bring out a set of relationships which would be mathematically true.

In the first set C E F# G. Take C as your starting point. Now invert around the pitch Eb. Now go one semitone up. Now go a minor third down. There you go you have gotten all of the pitches of that set.
In the second set C# D F B. Take F as your starting point. Invert around the pitch Eb. Now go one semitone up. Now go a minor third down. There you have again gotten all the members of your set.
You follow the same path, the same set of relationships but you get out two very different sets. What I have constructed is one type of K-Net whereby you have just one path and you have to start in one particular place (i.e. there are two parts of the network that aren't connected). You can construct K-Nets that are entirely connected up.

That is just to start of what you can do with K-Nets however. As you might have realized, you can follow the network from any pitch and you will get out a set. That means each K-Net results in a family of sets that have the same path embedded within them.

Where it gets interesting though is in recursion. This should be totally mind-blowing btw so hold onto your hat!
You can imagine that K-Nets themselves might be related to one another. That is to say you might have a network that resembles another network. This might seem mad but it could be something where the inversion parts of the K-Net are themselves taking part in an inversion. So if you had one K-Net where you invert around Eb and you had another where you invert around F then you might say that there is a meta-iversion between the two (around E). Another relationship you might get is invariance of the the transposition intervals (this is a bit more musically salient since it guarantees particular intervals will be present in a set). You could also have an inversion relationship between the distance of transposition.

Let me say this flatly - that all sounds totally bonkers right? I mean it is hard enough to imagine that inversion around a pitch we don't even hear could inform the construction of two sets such that they are somehow related. That is hard enough to swallow. But now I am saying that you could have meta-operations happening on those unheard pitches! How is that musically important or valid?!?!

Well I don't really think it is that important but Ido know one pretty incredible thing. In Lewin's tutorial on K-Nets linked above, he found exactly this in Schoenberg's music. He not only found a set of K-Net relationships, but he then found a higher level network that resembled his original K-Net. He then found that this meta K-Net fed into an even higher K-Net. Honestly, that is pretty damn unbelievable! I can barely get my head around it!

So there you go, that is what K-Nets are! They are certainly arcane, they are hotly debated, and frankly they are not that musically useful or salient. However, they are quite interesting.

Originally Posted by Mahlerian

I am choosing my definitions on the basis of music as I perceive it.

Not when you quibble over terminology, as when you said "...atonality is not defined in relation to your "broad conception of tonality" that includes modal musics, but rather in relationship to common practice tonality only." Here, you are clearly using the term "atonality" in a restricted sense.

Originally Posted by Mahlerian

Like I've said numerous times, though you take no notice, there is no such thing as a perception of atonality to me. I do not perceive any lack of tonal center in such music.

That's fine; however, that does not invalidate the very existence of music which is perceived by others as being atonal (having no tonal center), as implied by such threads as "Does atonal music exist?"

Originally Posted by Mahlerian

If I perceived any such quality that makes an atonal piece "atonal," or if anyone could prove to me that anyone else can discern such a distinct and definable quality, I would agree that the term is meaningful, but I have never seen any agreement as to what such a quality would be, and composers almost entirely avoid and ignore the term.

Ahh, finally we are making some progress, and clarifying things.

If the term "tonal" is used in a general sense (Harvard Dictionary), then I use its converse term "atonal" in the general sense.

The experience of music is subjective, and no one can "prove" their perceptions. As general terms which describe perceptions, there is no need to "prove" the terms or for an agreement. That's irrelevant, and does nothing to diminish the usefulness of the terms in perceptual terms.

In terms of musical structure, however, I think a good case can be made that "music which is based on a harmonic model" will more likely be perceived as having a tonal center. Thus, generally speaking, almost all music (modal, ethnic, folk, popular, classical) is perceived as being tonal in the general sense. Conversely, music such as Japanese Noh music, or Tibetan ceremonial percussion/horn music, is perceived as not having a tonal center (if calling it "atonal" confuses you).

Originally Posted by millionrainbows

In serial terms, a major triad can be 'turned upside down' (inverted) and will produce a minor triad: C-E-G (clockwise) becomes C-Ab-F (going counter-clockwise from C, the "axis" of symmetry).

Inversion is different in tonality, because relations between pitches are considered hierarchically, not as quantities measured in semi-tonal distances; we hear C-E-G, E-G-C, and G-C-E as all being C major triads.

Just to mention a technicality. You are really describing two different meanings of the word inversion and perhaps that is because I wasn't clear in my description.

Axial inversion is the inverse relationship between major and minor. Tonal inversion comes in two different forms - tonal mirror inversion where you invert by the generic interval within the tonal gamut. So in C minor, Eb inverts to Ab around C. The other type of inversion is the inversion of chords. This is inversion at the 8ve and really just identifies the equivalence of those chords you mention. All you are saying is that a major third is equivalent to a minor sixth.

Also, common musical parlance is to say that there is modal music, then tonal music, and then atonal music.

You can quibble around the edges of the problem but honestly people talk about renaissance music as modal with some tonal inflections. They talk about late Mahler as tonal with atonal elements and they talk about Ferneyhough as atonal with some tonal inflections.

That is the common meaning of the terms. I would no more call Ockeghem tonal than I would call Schoenberg tonal.

Originally Posted by Che2007

Just to mention a technicality. You are really describing two different meanings of the word inversion and perhaps that is because I wasn't clear in my description.

Axial inversion is the inverse relationship between major and minor. Tonal inversion comes in two different forms - tonal mirror inversion where you invert by the generic interval within the tonal gamut. So in C minor, Eb inverts to Ab around C. The other type of inversion is the inversion of chords. This is inversion at the 8ve and really just identifies the equivalence of those chords you mention. All you are saying is that a major third is equivalent to a minor sixth.

What you say is technically true, but essentially, tonality and harmonic root movement are concerned with only six basic intervals, not their inversions: m2(M7), M2(m7), m3(M6), M3(m6), P4(P5), and tritone(aug4/dim5); thus, a minor sixth is simply a major third going the other direction around the circle. Thus, the "identity" (not quantity/interval distance) of the pitch is revealed, as a 'place holder' in the tonal hierarchy, always in relation to "1" or the root.

See my blog "Root Movement"
http://www.talkclassical.com/blogs/m...-movement.html

Also:
http://www.talkclassical.com/blogs/m...ought-two.html

Originally Posted by millionrainbows

What you say is technically true, but essentially, tonality and harmonic root movement are concerned with only six basic intervals, not their inversions: m2(M7), M2(m7), m3(M6), M3(m6), P4(P5), and tritone(aug4/dim5);

Quite a few interpretations of tonal music would actually restrict root motion to 3rds and 5ths. Root motion by a step caused a great deal of problems for theorists like Rameau, Kirnberger and Sechter. I am not sure I follow why root motion has anything to do with different types of inversion. More widely, set theory certainly has nothing to do with tonal theory.