# What is Musical Set Theory?



## millionrainbows

Set theory is simply a listing of all possible 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 note sets.

From WIK: *Musical set theory* provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by *Howard Hanson* (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as *Allen Forte* (1973), drawing on the work in twelve-tone theory of *Milton Babbitt.* The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that. One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and which can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.

Two-element sets are called dyads, three-element sets trichords (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"-e.g., Rahn 1980, 140), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.

This is the raw material many modern composers draw on; Hanson used it tonally, Roger Sessions and Elliott Carter are two examples of composers who apllied their own individual methods to it. They're not serial, although the music can sound like that, because it is highly chromatic, concerned with the complete 12-note set, and uses smaller sets as 'motivic' material, or use the sets as abstractions to determine other aspects of form.

These are the books being referred to:

















It's also interesting to note that both books use the term "atonal" in their titles; both men are respected authors, *Allen Forte* being the head of the theory department at Yale for many years. "Atonal" methods like this are "non-tonal" methods, which do not use the hierarchy of tonality as the basis of their structural principles.

Atonal music is simply music which has no tonal center.


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## 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.


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## Albert7

Cool but I think that one of the questions is whether any of this affects the listener.


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## millionrainbows

Mahlerian said:


> 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.


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## Che2007

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.


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## matsoljare

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

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


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## Che2007

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.


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## millionrainbows

Mahlerian 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.


Yes, that's true, if the terms are being used to define _stylistic era practices;_

...but if we are truly and sincerely trying to define the terms in relation to the _*listening experience *_(the perception of tonal centers),then atonality is is _perceived_ as music which has no tonal center.

You're going to have to make up your mind as to the way in which you use these terms, and what your ultimate goal is: either historical reference, or as terms which describe the experience and perception of tonal centers in listened-to music.

As far as perceived tonal centers, as Che2007 says in his post, there is a fundamental difference between serial and tonal music, which is evident if one compares the way notes are related on a chromatic circle.

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.

This is why serial music is more often modeled on a straight number line, not a circle. Although octave equivalence is fundamental to hearing, a circle model is recursive, better showing the hierarchical nature of tonality within repeating octave/scale relations.

As well, serial music is concerned with quantities (interval distances) rather than identities (a note's function in an hierarchy within octave relations, to a key note).


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## Mahlerian

millionrainbows said:


> Yes, that's true, if the terms are being used to define _stylistic era practices;_
> 
> ...but if we are truly and sincerely trying to define the terms in relation to the _*listening experience *_(the perception of tonal centers),then atonality is is _perceived_ as music which has no tonal center.
> 
> You're going to have to make up your mind as to the way in which you use these terms, and what your ultimate goal is: either historical reference, or as terms which describe the experience and perception of tonal centers in listened-to music.


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

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.

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*.


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## millionrainbows

Mahlerian said:


> 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.



Mahlerian said:


> 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?"



Mahlerian said:


> 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).


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## Che2007

millionrainbows said:


> 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.


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## Che2007

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.


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## millionrainbows

Che2007 said:


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


I'm using a _general _definition of tonality, as used in the Harvard Dictionary of Music, when it says that "almost all music is tonal," meaning that it is* perceived* as having a tonal center.

When you say "there is modal music, then tonal music, and then atonal music," you are using the terms to refer to historical practice periods, not general perception.



Che2007 said:


> 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.


I'm not quibbling any more. I clearly distinguish between the use of the terms "tonal/atonal" as referring to certain historic stylistic periods, *or* using the terms generally, to refer to the perception of tonal centers in the listening experience, or the perception of no tonal center.

I question the separation of "modal" from "diatonic," my reason being that both derive their material from "scale/indexes" of notes that are unordered sets; plus, the church modes are part of the major scale, and the major scale is itself the Ionian mode. Also, the term "diatonic" means "using the notes of the scale," which to me would include modes.


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## millionrainbows

Che2007 said:


> 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/millionrainbows/1007-root-movement.html

Also:
http://www.talkclassical.com/blogs/millionrainbows/1056-tonality-serial-thought-two.html


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## Che2007

millionrainbows said:


> 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.


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## millionrainbows

Che2007 said:


> 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.


That was also my point; set theory certainly has nothing to do with tonal theory; but to understand either one fully, you must compare them and understand the differences. This of necessity requires that we mention both approaches.

If you read the blog on root movement, with quotes from Schoenberg's Structural Functions of Harmony, it is all made clear. "A fifth up, identical with a fourth down," and the others.

The intervals are "paired" with their inversions, such as P4/P5, etc. These also have a 'directionality' around the circle; C to G is a P5 up (going clockwise), or a P4 down (going counter-clockwise), and this affects the gravity of the root movement. This is all explained very clearly in the blogs.

This is because of the way we hear these intervals as vertical simultanities, not just horizontal movements.

A perfect fourth (G-C), because of our "harmonic hearing," will be heard as having the top (highest-pitched) note (C) as the 'root' or gravitational note of attraction; a fifth (C-G) will be heard as having the bottom note as "root". This is easily obvious by trying it out on a piano and listening.


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## Che2007

Hmm. I already know plenty about tonal root motion... You might be interested in Robert Hasegawa's work on a kind of rootedness for atonal sets: https://books.google.ca/books/about...e_Representation_i.html?id=2ce05XCL7hQC&hl=en

He is a very bright guy and profoundly musical.


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## millionrainbows

Che2007 said:


> You might be interested in Robert Hasegawa's work on a kind of rootedness for atonal sets:


That looks interesting, and it already clarified something for me when I read this:

"Historically, intervals between musical pitches have been understood through two distinct conceptual models: either as _distances_ in an imaginary space or as_ ratios_ between frequencies or string lengths. Each model has its own biases: the distance model is well-suited to constructing abstract pitch _geometries_, while the ratio model offers insight into an interval's sonic quality and stability."

I'd be interested in seeing whether he uses this as an analysis tool, or if he attempts to reconcile the two.

As I see it, the ratio model is flawed from the beginning, since the only reason Pythagoras stopped at 12 cycles was to preserve the octave and close the spiral. "Mean tone" schemes have been used ever since in attempts to salvage the major third.

Conversely, the "12" model of distances is arbitrary. There's no real reason to divide the octave that way, except for the interesting geometries it provides as a number divisible by both 3 and 4 (like a carpenter's ruler).

Same old dilemma: should music be a primarily sensual experience, or a cerebral one?


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## Che2007

millionrainbows said:


> A perfect fourth (G-C), because of our "harmonic hearing," will be heard as having the top (highest-pitched) note (C) as the 'root' or gravitational note of attraction; a fifth (C-G) will be heard as having the bottom note as "root". This is easily obvious by trying it out on a piano and listening.


This is dependent on context. If you have an arpeggiation of a bass running through a triad, then the 4th will sound like the root of the chord. However, in a cadential 6/4 chord, it is the note on the bottom of the 4th which sounds like the root.

In a more general point which relates to both tonal theory and set theory, we should remember that context is the key to understanding the way that you hear a set of pitch relations. For example, you can have the notes of a tonic chord and it sound like a dominant as in a cadential 6/4. In set theory you might have two different sets contextually closely related in one piece, but in another piece sound very distant.


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## millionrainbows

Che2007 said:


> This is dependent on context. If you have an arpeggiation of a bass running through a triad, then the 4th will sound like the root of the chord. However, in a cadential 6/4 chord, it is the note on the bottom of the 4th which sounds like the root.
> 
> In a more general point which relates to both tonal theory and set theory, we should remember that context is the key to understanding the way that you hear a set of pitch relations. For example, you can have the notes of a tonic chord and it sound like a dominant as in a cadential 6/4. In set theory you might have two different sets contextually closely related in one piece, but in another piece sound very distant.


In a cadential 6/4 chord sequence, say, F to C, the bottom note note (C) of the F chord (6/4) is heard as the *fifth* of the f, not the root of anything.

But these ideas are Schoenberg's, from his book Structural Functions of Harmony, and I accept them as givens, and they also make sense to my ears and brain. I'm not really interested in debating such things.


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## Che2007

> In a cadential 6/4 chord sequence, say, F to C, the bottom note note (C) of the F chord (6/4) is heard as the fifth of the f, not the root of anything.


Sorry but Schoenberg is too old fashioned in that. I would even go as far as to say he is wrong. Let me explain:

You approach a cadence in C major. Let's say you use some kind of chromatic pre-dominant like Gr+6 (Ab,C,Eb,F#). To avoid parallel 5ths you will follow this with a cadential 6/4.
The notes of this chord are (CEG) with G in the bass. This G is held while C and E resolve downwards to B and D. The key word here is *resolve*. They are resolving because they are behaving as a double suspension. When the cadential 6/4 arrived, it was in fact that low G that was being expressed. The other two notes were contextual dissonances that delayed the arrival of the chord tones B and D.
Therefore we would label this progression Gr+6 V6/4 V5/3 I.

There are 2 very important reasons for thinking like this:
1) That is the way most people hear this progression. When the first dominant chord (cadential 6/4) arrives, we don't hear a return to the tonic. We hear an arrival of the dominant - it is cliched to say but we all know what is coming next in this case. The dominant is being extended or intensified before the tonic arrival. 
2) There are other examples of this type of thinking that I am sure you wouldn't disagree with. For example, look at this:


http://imgur.com/Fbb19C6

You can see that the tonic bass has arrived on the downbeat of the new measure. However there are a load of notes in the right hand that don't fit. It looks like a dominant chord still. Yet, we can see that all of these notes resolve to the tonic harmony. In fact, all that has really happened is the notes of the tonic have been delayed. I don't think it would capture the sound of this snippet to say that the tonic doesn't arrive until the notes resolve. That would imply some kind of V11 chord. I don't think that is what is sounds like, and I certainly don't think we hear that bass C as the 11th of the chord! This is the reason for taking some intervals which are not innately dissonant as contextual dissonances. It allows us to tell the difference between a chord-tone and and non-chord-tone.

So yeah, in summation I would recommend you read a book like Aldwell and Sachter's _Harmony and Voice-leading_. Although they have their foibles (not using Weber roman numerals for one) they are much more reliable than Schoenberg's text.


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## millionrainbows

Che2007 said:


> Sorry but Schoenberg is too old fashioned in that. I would even go as far as to say he is wrong. Let me explain:
> 
> You approach a cadence in C major. Let's say you use some kind of chromatic pre-dominant like Gr+6 (Ab,C,Eb,F#). To avoid parallel 5ths you will follow this with a cadential 6/4.
> The notes of this chord are (CEG) with G in the bass. This G is held while C and E resolve downwards to B and D. The key word here is *resolve*. They are resolving because they are behaving as a double suspension. When the cadential 6/4 arrived, it was in fact that low G that was being expressed. The other two notes were contextual dissonances that delayed the arrival of the chord tones B and D.
> Therefore we would label this progression Gr+6 V6/4 V5/3 I.
> 
> There are 2 very important reasons for thinking like this:
> 1) That is the way most people hear this progression. When the first dominant chord (cadential 6/4) arrives, we don't hear a return to the tonic. We hear an arrival of the dominant - it is cliched to say but we all know what is coming next in this case. The dominant is being extended or intensified before the tonic arrival.
> 2) There are other examples of this type of thinking that I am sure you wouldn't disagree with. For example, look at this:
> 
> 
> http://imgur.com/Fbb19C6
> 
> You can see that the tonic bass has arrived on the downbeat of the new measure. However there are a load of notes in the right hand that don't fit. It looks like a dominant chord still. Yet, we can see that all of these notes resolve to the tonic harmony. In fact, all that has really happened is the notes of the tonic have been delayed. I don't think it would capture the sound of this snippet to say that the tonic doesn't arrive until the notes resolve. That would imply some kind of V11 chord. I don't think that is what is sounds like, and I certainly don't think we hear that bass C as the 11th of the chord! This is the reason for taking some intervals which are not innately dissonant as contextual dissonances. It allows us to tell the difference between a chord-tone and and non-chord-tone.
> 
> So yeah, in summation I would recommend you read a book like Aldwell and Sachter's _Harmony and Voice-leading_. Although they have their foibles (not using Weber roman numerals for one) they are much more reliable than Schoenberg's text.


All of this is irrelevant to what I was saying. I was talking about root movement, not voice leading. The intervals I mentioned (in detail) refer to root movement only, so only an interval is involved; the movement from one scale-step to another. The intervals, in this sense, are to be considered as harmonic entities, to be looked at in isolation, with no context. I see no reason for your response other than to demonstrate your knowledge.

Speaking of context, the only reason I brought up root movement is because "literal inversion of chords" was brought up, and I don't think this is applicable to tonality. Thus, I assumed the knowledge of this aspect was missing. Howard Hanson talks about inverting chords this way, but it is an abstract method not based on tonal principles in the old sense.

At any rate, I'm not here to debate, but to clarify, and the 'context' of this discussion seems to have veered off into territory I'm not interested in exploring.


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## Che2007

millionrainbows said:


> All of this is irrelevant to what I was saying. I was talking about root movement, not voice leading. The intervals I mentioned (in detail) refer to root movement only, so only an interval is involved; the movement from one scale-step to another. The intervals, in this sense, are to be considered as harmonic entities, to be looked at in isolation, with no context. I see no reason for your response other than to demonstrate your knowledge.
> 
> Speaking of context, the only reason I brought up root movement is because "literal inversion of chords" was brought up, and I don't think this is applicable to tonality. Thus, I assumed the knowledge of this aspect was missing. Howard Hanson talks about inverting chords this way, but it is an abstract method not based on tonal principles in the old sense.
> 
> At any rate, I'm not here to debate, but to clarify, and the 'context' of this discussion seems to have veered off into territory I'm not interested in exploring.


I was just clarifying that talking about root as a summation of intervals in isolation is not really that profitable a way of talking about tonal music.


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## Guest

Che2007 said:


> [...] You approach a cadence in C major. Let's say you use some kind of chromatic pre-dominant like Gr+6 (Ab,C,Eb,F#). *To avoid parallel 5ths* you will follow this with a cadential 6/4.


Just a point of interest to make here, Ché2007. It's true that the harmony textbooks alert us to possible parallel 5ths when resolving the German A6, urging us to take the cadential 6/4. This is certainly what Haydn does. However, Mozart allows himself these parallel fifths by resolving directly to the dominant (V). For this reason these seemingly "forbidden" parallels are referred to as "*Mozart fifths*". My own harmony book from school (Lovelock) says: "_The augmented 6ths on the flattened 6th of the scale resolve to Ic or V or V7 with the proviso that the German 6th cannot proceed to V without producing 5ths. It is therefore inadvisable to resolve it thus in examination work, *though there is no valid musical objection to the progression*_."


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## millionrainbows

Che2007 said:


> I was just clarifying that talking about root as a summation of intervals in isolation is not really that profitable a way of talking about tonal music.


Then, talking about "literal inversion of scales" is unprofitable too, when talking about tonality. While it is possible to invert a scale, why would you want to? It would change all the pitches.

Plus, you can't "retrograde" a scale. Since a scale has no order, that makes no sense. You can retrograde a melody, but not a scale.

It seems that there are some basic gaps in communication becoming apparent.

If you can't follow or justify what I've said so far about harmony/vertical and melodic/horizontal, and unordered scales vs. ordered rows, harmonic vs. melodic, then you will never understand the principles and reasons behind hexachords and combinatoriality in 12-tone and atonal music procedures.

The methods of dealing with the vertical/harmonic aspect of serial composition is a prime consideration.


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## millionrainbows

Che2007 said:


> 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.


No, I'm saying more than that, because I'm comparing "identity" with "quantity." In tonality, inversion is identity based (C up to G is the same as G up to C), but in set theory, +7 and -7 create different results.


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## Che2007

millionrainbows said:


> If you can't follow or justify what I've said so far about harmony/vertical and melodic/horizontal, and unordered scales vs. ordered rows, harmonic vs. melodic, then you will never understand the principles and reasons behind hexachords and combinatoriality in 12-tone and atonal music procedures.


Hmmm... I guess next time I am teach 20th Century theory at my university I'll just refer them to your blog since you think you grasp serial methods better than me.


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## Che2007

TalkingHead said:


> It is therefore inadvisable to resolve it thus in examination work, *though there is no valid musical objection to the progression*[/I]."


It does appear in some music but I generally teach that composers more often resolve via the V6/4. You are right though of course, they do appear in some music! I was really just trying to have a musical example where you might find a V6/4


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## millionrainbows

Che2007 said:


> Hmmm... I guess next time I am teach (sic) 20th Century theory at my university I'll just refer them to your blog since you think you grasp serial methods better than me.


 Oh, is that what I think?

I never said that I "grasp serial methods better than anyone else here."

I'm happy with what I know, and I will keep studying and adding to my knowledge.


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## millionrainbows

Set theory is distinguished from 12-tone or serial music as well. 

A series is a line, not a set, of pitch classes. A pitch-class retains its identity no matter how its pitch-classes are ordered. In a series, however, the pitch-classes occur in a linear (usually non-scalar) order; the identity of the series changes if the order changes.


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## isorhythm

I'm bumping this thread because I want to know if someone can actually answer millionrainbows' original question - what is musical set theory? Or at least point me toward some free resources. I still have no idea, and I've tried many times to read up on it.

For example, the Wikipedia entry on musical set theory does not describe anything that could be called a theory.

So what the heck is musical set theory? What does it say? What are the basic principles? Anyone?


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## Mahlerian

My impression is that musical set theory is a tool for analyzing post common practice music and classifying its harmonies by their relationship to a note, independently of their traditional functional analyses.

The problem I have with it is that it generally ignores voicing, which is if anything more important in post-common practice music for harmonic identity, not less.


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## isorhythm

Mahlerian said:


> My impression is that musical set theory is a tool for analyzing post common practice music and classifying its harmonies by their relationship to a note, independently of their traditional functional analyses.


But surely there is more to it than this, right? Because a classification system is not a theory, or even the beginning of a theory.


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## Mahlerian

isorhythm said:


> But surely there is more to it than this, right? Because a classification system is not a theory, or even the beginning of a theory.


Well, it does provide a method for more easily identifying non-triadic chords and their complements (hence the set aspect), and also a way of relating harmonies to each other, but as far as I know, it's not something that's been used for the creation of music. It's strictly a method for analysis, and doesn't provide any information about form the way, say, Schenkerian analysis does.


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## EdwardBast

isorhythm said:


> I'm bumping this thread because I want to know if someone can actually answer millionrainbows' original question - what is musical set theory? Or at least point me toward some free resources. I still have no idea, and I've tried many times to read up on it.
> 
> For example, the Wikipedia entry on musical set theory does not describe anything that could be called a theory.
> 
> So what the heck is musical set theory? What does it say? What are the basic principles? Anyone?


Mahlerian was pretty much on target; Set Theory isn't really a theory. It is a systematic way of classifying musical materials intended to make unifying factors easier to hear and discuss. A taxonomic system. I suppose it has a certain organicist mindset at its roots, the assumption that there will be a strong internal motivic unity or unity of materials in even the most intractable musical constructions.



Mahlerian said:


> It's strictly a method for analysis, and doesn't provide any information about form the way, say, Schenkerian analysis does.


I would say maybe it's more like a set of tools that could be put in the service of an analytical method rather than an actual method? But yeah, it doesn't offer specific concepts of organization like Schenker's Urlinie or Ursatz or prolongation.


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## millionrainbows

isorhythm said:


> I'm bumping this thread because I want to know if someone can actually answer millionrainbows' original question - what is musical set theory? Or at least point me toward some free resources. I still have no idea, and I've tried many times to read up on it.
> 
> For example, the Wikipedia entry on musical set theory does not describe anything that could be called a theory.
> 
> So what the heck is musical set theory? What does it say? What are the basic principles? Anyone?


Set theory is usually used to analyze music which is not composed tonally. It basically observes what notes are in the music, and puts them into sets consisting of from 2 to 12 notes.

There are some principles which arise out of this process which can tell us about the music not just in statistical terms, but in harmonic terms. Unordered sets can be seen as having intervals which give them a certain sonic character. This is called the interval vector.

Really, there is no short answer. Why explain what a crossword puzzle is? The easiest way to grasp its essence is to simply do one.


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## millionrainbows

Mahlerian said:


> Well, it does provide a method for more easily identifying non-triadic chords and their complements (hence the set aspect), and also a way of relating harmonies to each other, but as far as I know, it's not something that's been used for the creation of music. It's strictly a method for analysis, and doesn't provide any information about form the way, say, Schenkerian analysis does.


Well, if you look at Howard Hanson's text Harmonic Materials of Modern Music, you can see how set theory can be used to generate scales, chords, and other materials for composing. But, no, it's not a "method" for composing, any more than the tonal system is. Is it supposed to be?


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## isorhythm

You don't even need to go all the way to Schenker to find a real theory of tonal music, though. Just the traditional description of how tonal music works, i.e. tonic, dominant and subdominant functions, dissonance and consonance, modulations, passing tones, cadences, all that stuff, constitutes a theory with explanatory power.

I think it's odd that something called "set theory" came to be considered important in America when as far as I can tell it has no explanatory power and what observations it's capable of making about music are trivial.


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## Mahlerian

The real issue is that no existing theory covers any kind of post-tonal music, whether it uses triads or not. All people have come up with are explanations of how individual composers and works function, and there's nothing yet that's comparable to common practice tonal theory in explanatory power and applicability.


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## EdwardBast

isorhythm said:


> You don't even need to go all the way to Schenker to find a real theory of tonal music, though. Just the traditional description of how tonal music works, i.e. tonic, dominant and subdominant functions, dissonance and consonance, modulations, passing tones, cadences, all that stuff, constitutes a theory with explanatory power.
> 
> I think it's odd that something called "set theory" came to be considered important in America when as far as I can tell it has no explanatory power and what observations it's capable of making about music are trivial.


One has to supply the explanatory power oneself, and the tools set theory affords are only as good as the insight of the person wielding them. Music beyond the bounds of triadic harmony, especially that which tends to be called atonal, is characterized, like any other music, by recognizable components and salient configurations, including vertical, quasi-harmonic entities, and melodically relevant groups of notes. If one wants to talk about this music, one must have names for different sorts of configurations and relations among same. Suppose for example, one hears intuitively a clear affinity of sonority between the different sections of a work. Finding that the harmonic configurations in these sections tend to be identical or closely related sets, allows one to say what precisely is at the root of ones intuitions. (But even establishing that two sets are the same or related often requires being able to manipulate them in certain ways, like putting them into "normal order.") Or if one wished to explain the sense of melodic unity among the different songs of Pierriot lunaire, for example, it could be an important insight to recognize that the set (0, 1, 4) is prevalent among several of them.

Facility with sets, that is being able to recognize when one is dealing with groups of notes related, for example, by inversion or complementarity, is, according to those who rely on set theory, part of an essential skill set for understanding the workings of much modern music.


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## Woodduck

EdwardBast said:


> One has to supply the explanatory power oneself, and the tools set theory affords are only as good as the insight of the person wielding them. Music beyond the bounds of triadic harmony, especially that which tends to be called atonal, is characterized, like any other music, by *recognizable components and salient configurations, including vertical, quasi-harmonic entities, and melodically relevant groups of notes.* If one wants to talk about this music, one must have names for different sorts of configurations and relations among same. Suppose for example, one hears intuitively *a clear affinity of sonority between the different sections of a work.* Finding that *the harmonic configurations in these sections tend to be identical or closely related sets*, allows one to say what precisely is at the root of ones intuitions. (But even establishing that two sets are the same or related often requires being able to manipulate them in certain ways, like putting them into "normal order.") Or if one wished to explain the sense of melodic unity among the different songs of Pierriot lunaire, for example, it could be an important insight to recognize that the set (0, 1, 4) is prevalent among several of them.
> 
> Facility with sets, that is *being able to recognize when one is dealing with groups of notes related, for example, by inversion or complementarity*, is, according to those who rely on set theory, part of an essential skill set for understanding the workings of much modern music.


I speak from a position of utter ignorance on this subject, but it sounds as if set theory would be fully applicable to tonal music too. All the kinds of relationships I've bolded above might characterize a Baroque ouverture or a Romantic tone poem. Apparently my brain has been trying to do set theory all my musical life! Is there anything new about it? Weren't we always analyzing music in terms of the kinds of internal connections set theory proposes to name? Is it only the names that are novel? And are "set theorists" (whoever _they_ are) using these names to describe Haydn and Schumann?


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## EdwardBast

Woodduck said:


> I speak from a position of utter ignorance on this subject, but it sounds as if set theory would be fully applicable to tonal music too. All the kinds of relationships I've bolded above might characterize a Baroque ouverture or a Romantic tone poem. Apparently my brain has been trying to do set theory all my musical life! Is there anything new about it? Weren't we always analyzing music in terms of the kinds of internal connections set theory proposes to name? Is it only the names that are novel? And are "set theorists" (whoever _they_ are) using these names to describe Haydn and Schumann?


One could indeed use set theory to describe and analyze baroque or romantic music, just as one could use binary code to describe and analyze a literary text. However, both endeavors would be insane, given the more suitable and efficient alternatives available. Traditional modes of analysis based on tonal function and principles of counterpoint are uniquely suited to apprehending the subtle gradations of tension and elements of grammar and syntax that comprise the common-practice tonal system. All of this subtlety would elude a set theoretic approach. Those adept at set theory understand this and so, of course, they happily analyze common-practice tonal music just like everyone else. No one ever intended or wanted set theory to supersede or assimilate tonal analysis into a more general system. Essentially, set theory is a brute force method designed for music in which tonal grammar and syntax do not apply or are of minimal or attenuated explanatory power.

Set theory is certainly new because many groups of notes that have no name (or are gibberish) in tonal music are described as easily as any other group in set theoretic terms. No one in their right mind would analyze Haydn or Schumann in set theoretic terms.


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## Woodduck

EdwardBast said:


> One could indeed use set theory to describe and analyze baroque or romantic music, just as one could use binary code to describe and analyze a literary text. However, both endeavors would be insane, given the more suitable and efficient alternatives available. Traditional modes of analysis based on tonal function and principles of counterpoint are uniquely suited to apprehending the subtle gradations of tension and elements of grammar and syntax that comprise the common-practice tonal system. All of this subtlety would elude a set theoretic approach. Those adept at set theory understand this and so, of course, they happily analyze common-practice tonal music just like everyone else. No one ever intended or wanted set theory to supersede or assimilate tonal analysis into a more general system. Essentially, set theory is a brute force method designed for music in which tonal grammar and syntax do not apply or are of minimal or attenuated explanatory power.
> 
> Set theory is certainly new because many groups of notes that have no name (or are gibberish) in tonal music are described as easily as any other group in set theoretic terms. No one in their right mind would analyze Haydn or Schumann in set theoretic terms.


Thanks. So we don't use set theory to describe common practice (or other tonal) music because our ways of comprehending such music do a better job than set theory could - i.e., nothing would be gained by using the classificatory terms of set theory to describe features of tonal music.


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## EdwardBast

Woodduck said:


> Thanks. So we don't use set theory to describe common practice (or other tonal) music because our ways of comprehending such music do a better job than set theory could - i.e., nothing would be gained by using the classificatory terms of set theory to describe features of tonal music.


Precisely. Nothing gained and quite a bit lost.

Although … I wouldn't rule out the possibility that set theory might facilitate insights into transitional music some of us are inclined to hear as tonal.


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## millionrainbows

Woodduck said:


> Thanks. So we don't use set theory to describe common practice (or other tonal) music because our ways of comprehending such music do a better job than set theory could - i.e., nothing would be gained by using the classificatory terms of set theory to describe features of tonal music.


This makes it sound as if set theory is deficient in some way. Actually, tonal music can be seen as just another special case of set theory. The C maj scale is a 7-note unordered set.


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## millionrainbows

EdwardBast said:


> ....(But even establishing that two sets are the same or related often requires being able to manipulate them in certain ways, like putting them into "normal order.")


Putting sets into 'normal order' is simply a way of eliminating redundancies, to reduce the number of sets which one has to deal with for their essential characteristics.

The major triad C-E-G, in set theory, is (0, 4, 7) (C up to E up to G). The interval content is the same if inverted (7, 4, 0) as C-Ab-F (C down to Ab, also a 4, down to F, also a 7), which is an F minor triad. Normal order simplifies the relations, since they are the same.

But in tonality, C major and F minor are totally different. Tonality also has no symmetric directionality; relations are interpreted as identities, not as quantities. C 'up' (clockwise) to E in tonality is totally different than C down (counterclockwise) to Ab, although both intervals (C to E and C to Ab) are 4 semitones. The note name (Ab or E), in tonality, are _identities_ rather than quantities, because everything is related recursively (on a circle) to C, or "1," the key note.

This is totally different than tonal thinking; all this talk of 'analyzing tonal music with set theory' is rather like trying to stuff a horse into a suitcase.

Set theory can certainly be used to compose music, and is not just a method of analysis. It is a manifestation of a completely different way of thinking about music.


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## millionrainbows

Elliott Carter uses set theory in his music, and from what he has said, it would appear that it is integral to the composition process. He said he is interested in "all-interval sets."

Babbitt and George Perle have also used set theory, and Perle calls his approach "Twelve-tone tonality," and has written a book of the same name, in which he demonstrates how to use these ideas to compose.

In the end, set theory is an index of all possible 2,3,4,5,6,7,8,9,10, and 11 note sets, put into 'normal order' to reduce them to their essential interval characteristics.

Another advantage of set theory is that when analyzing music, especially twelve-tone or serial music, the plethora of different pitch-names becomes irrelevant, and we have a manageable way of recognizing interval constructs, by their "reduced" normal forms.

This approach of recognizing interval constructs is totally irrelevant to tonality. Tonality works with "identity/key note" relations, not "quantity/interval templates."

You might be able to analyze tonal music with set theory and derive some general abstract principles from this, but that's all. The rest is not a 'failing' on set theory's part, it's just not applicable.


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## EdwardBast

millionrainbows said:


> Putting sets into 'normal order' is simply a way of eliminating redundancies, to reduce the number of sets which one has to deal with for their essential characteristics.


^^^
This is clear as mud. Try: Putting sets into normal order allows one readily to see if they are the same or different.


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## isorhythm

millionrainbows said:


> This makes it sound as if set theory is deficient in some way. Actually, tonal music can be seen as just another special case of set theory. The C maj scale is a 7-note unordered set.


Well, I think the problem is that it can't be seen as a subset of set theory, because set theory doesn't explain how tonal music works.

I'm going to get one of the major books on it from the library at some point, but from where I'm sitting now, it still looks to me like set theory is not a theory at all. It's a kind of glorified shorthand notation.


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## SuperTonic

isorhythm said:


> Well, I think the problem is that it can't be seen as a subset of set theory, because set theory doesn't explain how tonal music works.
> 
> I'm going to get one of the major books on it from the library at some point, but from where I'm sitting now, it still looks to me like set theory is not a theory at all. It's a kind of glorified shorthand notation.


I would urge you to wait until after you have read more about the theory before you make such statements. It does exist at a higher level of generality than say tonal harmony or the 12 tone method, but that doesn't mean that it isn't useful as a compositional or analytical tool.


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## millionrainbows

EdwardBast said:


> ^^^
> This is clear as mud. Try: Putting sets into normal order allows one readily to see if they are the same or different.


Well, that is unclear as well. Putting a set into "normal order" simply arranges the intervals in their most compact form, and eliminates redundancies. The set {0, 7, 4} in normal order is {0, 4, 7} (zero up to 4 up to 7).

To put a set in normal form, begin by putting it in normal order, and then transpose it so that its first pitch class is 0.

Thus, we can see equivalent sets even when they are transposed or in different order, and see that they are essentially the same in terms of the intervals they contain.


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## EdwardBast

millionrainbows said:


> Well, that is unclear as well. Putting a set into "normal order" simply arranges the intervals in their most compact form, and eliminates redundancies. The set {0, 7, 4} in normal order is {0, 4, 7} (zero up to 4 up to 7).
> 
> To put a set in normal form, begin by putting it in normal order, and then transpose it so that its first pitch class is 0.
> 
> Thus, we can see equivalent sets even when they are transposed or in different order, and see that they are essentially the same in terms of the intervals they contain.


No, it is perfectly clear. If you put two sets in normal order and you get different numbers, the sets are different. Which is what I said: "It allows one readily to tell if two sets are the same or different." Your last two posts on the purpose of putting sets in normal order have rendered a simple concept that can be explained in 15 words or less, virtually indecipherable.


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## millionrainbows

EdwardBast said:


> No, it is perfectly clear. If you put two sets in normal order and you get different numbers, the sets are different. Which is what I said: "It allows one readily to tell if two sets are the same or different." Your last two posts on the purpose of putting sets in normal order have rendered a simple concept that can be explained in 15 words or less, virtually indecipherable.


Your view applies, if you are dealing with ordered sets. Yes, the order is different. But when you analyze a set in terms of its interval content, this is a constant which does not change; for example, C-E-G is a major third (C-E), a minor third (E-G) and a fifth (C-G). This is its interval content, or interval vector: (0, 4, 7), which is the same three intervals whether the set is stated as (0,7,4) or (4, 7, 0). That's three possible names for the set, and normal order eliminates these redundancies.


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