Set Theory

[Mr. Big Noodles]

Some set theory to get you thinking. I'll pose a bunch of rhetorical questions and answer them for your benefit.

0. What is a pitch?

A pitch is a note that we can name. These consist of the twelve notes of the chromatic scale and their enharmonic equivalents. A# and B♭are the same sounding pitch, but they may be considered different pitches on a theoretical level depending on their function within a given piece of music.

1. What is a pitch class?

A pitch class (abbreviated "p.c.") is a pitch and all of its enharmonic equivalents, minus the information about function. We use integer notation to represent pitch classes with numbers. We start at C, which is 0, and work our way up the chromatic scale until we hit B, which is 11. B♭/10 and B/11 are two-digit numbers in base 10. It's cleaner to have everything be single digit however, so we use t and e in place of 10 and 11.

0 - C
1 - C#/D♭
2 - D
3 - D#/E♭
4 - E
5 - F
6 - F#/G♭
7 - G
8 - G#/A♭
9 - A
t - A#/B♭
e - B

We also have [intervals] (and [interval classes]; more later), which are the measure of half steps between p.c.'s. So from 5 to 6 is [1], 5 to 7 is [2], and so on. [These brackets are the notation for anything involving intervals.]

2. What is a set?

A set is a group of pitch classes that have structural significance in a piece of music. Let's assume we are dealing with an A minor pentatonic scale.

A C D E G = {90247}

{These brackets indicate an unordered set. Unordered sets are the p.c.'s as you would find them in the wild.}

Of course, we know that we can have minor pentatonic scales in many different keys. Here's a B♭ minor pentatonic scale: {t1358} (Notice that I transposed by adding 1 to each integer.)

3. A word about transposition.

In set theory, we may or may not be dealing with music that is in a key. As such, key can become a hindrance to the overall point we make. Instead, we are concerned with the pitch level, or transposition level, at which the set occurs. If a set is based off of p.c. 4, then we would use the shorthand "T4" to indicate that is... based off of p.c. 4. If we were to transpose it up a half step, we would indicate the transposition level as "T5", and so on. We'll come back to this.

4. A word about inversion.

Sets can be flipped upside down to get what we call an "inverted set." Inversion takes the intervals of a set and changes their direction. If you have the two-note set {03}, then you can think of it as 0+3. Inversion would change it to 0-3, which results in the p.c.'s {09}. Huh?! Because this is mod12, 0-1=e, 0-2=t, 0-3=9.

Let's do a big set. Take that A minor pentatonic scale again: {90247}

Here is that set with everything shown as a sum from 9:

9+0=9
9+3=0
9+5=2
9+7=4
9+t=7

To invert it, we would change all the addition to subtraction:

9-0=9
9-3=6
9-5=4
9-7=2
9-t=e

So the inversion of {90247} is {9642e}. (If we are inverting around 9, that is.)

5. What is a set class?

A set class is an abstraction of every single transposition and inversion of a set. Set classes are expressed as a (prime form). Prime forms all start with 0 and are the smallest, most packed rotation of the set. To get to a prime form, we first need to convert a set to <normal order>. Normal order is sort of halfway between an {unordered set} and (prime form). Let's use an easy set:

{632}

Steps to getting normal order

1. Put the notes in some kind of ascending order.
2. Rotate it until you get the smallest intervallic arrangement of the pitch classes, packed to the left. In other words, the largest interval should be on the right.

Let's try it on this set.

Ascending! {236}

Smallest intervals! {23} spans [interval class 1], {36} spans [3]. If we go around from 6 to 2, we get [8], which is a huge interval. It's safe to say that {236} is as small as we can make this thing.

Because {236} is the smallest arrangement of those p.c.'s, we can upgrade its status to that of a normal order. Congratulations, <236>.


Converting <normal order> to (prime form)

1. Subtract the first p.c. down to 0.
2. Subtract the same number from the other p.c.'s in the set.

<236> minus 2 from everything = (014)


We would say that <236> is T2(014). In other words, <236> is (014) transposed up [2].


If we have an inverted set, we need to uninvert it to get normal order.

{1t2}

Getting normal order from inverted sets

1. Arrange the set in ascending order. {12t}
2. Rotate it as best you can to get the smallest interval spans. {12}=[1], {2t}=[8], {t1}=[3]. 8 is the largest, so move it to the right: {t12}
3. The intervals are [3, 1], but we want [1, 3], so let's invert (or uninvert, if you will) this sucker. We have to go from inverted to uninverted, so we'll start with subtraction and change to addition. To get the right transposition level, start from the smallest interval and descend. (Start at 2, then go down to 1, etc.)

2-0=2

2-1=1

2-4=t

...

2+0=2
2+1=3
2+4=6

{236}

5. Success! <236>

Do the prime form!

<236> minus 2 = T2I(014)

Because this was an inverted set, we indicate the transposition as T2I. ("I" means "inverted.")

That's all for now. I'll take questions and hopefully get to interval vectors and Forte codes next time.

Summary of brackets:
{unordered set}
<normal order>
(prime form)
[intervals]
[interval vector]

 

[ElRay]

1) Thank you for posting.
2) Damn you for posting.

I always got these confused.

So, for a non-inverted Major chord, the intervals would be: [0,4,7], but when do you have to "minimize" the intervals? Rotating it to 3rd, 5th, root would give you [0,3,5] as the intervals, which would mean that the normal order for all major chords, regardless of the inversion, would be <0,3,5>, correct?

I started looking into this years ago, because I found out about all-interval-tetrachords, and wanted to develop an ear-training exercise, but got lost in [0,1,4,6] and [0,1,3,7] and trying to lay it out on a fretboard. Another example of a little bit of knowledge being a bad thing.

 

[Mr. Big Noodles]

Yeah, inversion is tricky. Hopefully I can answer this with a picture.

Notice that {047} contains the ascending intervals [4] and [3]. [3] is smaller, so if we want normal order, we need to make it ascend [3] and then [4]. We do this by flipping the intervals of the set around C (AKA "inversion") so that instead of going up [4], we're going down [4], and so on. This results in a DESCENDING succession of those original intervals. We then put the set in normal order by rotating the p.c.'s around to get them in ascending order. Getting prime form from there is a simple matter of subtracting to T0. (Although you can and probably should add up to 0, because it gives you the TxI number; <580>+7=(037); {047} can be notated T7I(037). Technically, the set is inverted around G, not C.)

You can also invert by subtracting from 12:

12
- 0 4 7
0 8 5

Flip it!

5 8 0

(+7/-5 gets you (037).)

 

[Mr. Big Noodles]

Here's a video that covers pitch class, prime form, and inversion in a way that might make more sense for some:

 

[bostjan]

...but what if A# is not the same as Bb?!

A# and Bb are only enharmonic equivalents because of the artifacts of 12-tone equal divided octave temperament. For that matter, the A in the key of A is not necessarily the same as the A in the key of F#m.

But I'm just splitting hairs. This is an excellent thread.

 

[Mr. Big Noodles]

They aren't the same pitch; they are the same pitch class. Same as C&#9837;&#9837; or Gx#. (Assuming 12tet, which this system does.)

 

[Mr. Big Noodles]

Interval vs. Interval Class

An interval class (i.c.) is like prime form for intervals. You take the smallest version of an interval, which may require inversion. Here's a list of intervals and corresponding interval classes:

m2 = 1
M2 = 2
m3 = 3
M3 = 4
P4 = 5
A4 = 6
P5 = 7 = 5
m6 = 8 = 4
M6 = 9 = 3
m7 = 10 = 2
M7 = 11 = 1

All the stuff in yellow inverts to a smaller interval class.

Interval Vectors

An interval vector is a tally of all the interval classes in a set. Let's start with (014).

01 = [1]
04 = [4]
14 = [3]

So we have one i.c. [1], one [4], and one [3]. We have a standardized form to indicate interval vectors, and it looks like this:

[000000]

Notice that there are six places (here filled with zeroes). The first place represents the number of i.c. 1's, the second place is 2's, and so on to 6's at the end.

Remembering our interval breakdown of (014), we would fill in the interval vector like this:

[101100]

Let's try another one. Here's the breakdown of (0135):

01 = 1
03 = 3
05 = 5

13 = 2
15 = 4

35 = 2

So now we collect all the i.c.'s and plug them into the vector:

[121110]

Notice that there are two instances of [2] in that set (between {13} and {35}), so we would tally "2" in the 2's place in the vector. Conversely, there are no instances of [6], so that slot remains at 0.

ElRay brought up the all-interval tetrachord. Let's see what happens when we do the interval vector for that one.

(0146)

01 = 1
04 = 4
06 = 6

14 = 3
15 = 5

46 = 2

Do the tally...

[111111]

Heyyyy, it has all the interval classes. Makes sense that we call it the "all-interval tetrachord" now, doesn't it?

Z relation

We just saw that (0146) has the interval vector [111111]. There's another all-interval tetrachord: (0137)

01 = 1
03 = 3
07 = 7 = 5

13 = 2
17 = 6

37 = 4

[111111]

Alright, so (0146) and (0137) have the same interval vectors. Whenever two different set classes share the same interval vector, we say that they are Z related. You might exploit this in a piece of music by using certain intervals to create motivic coherence, but situating those intervals in different set classes to achieve variety. In much post-tonal music, there is a a tendency to seek ways to maximize both coherence and variety.

 

[Mr. Big Noodles]

Forte Codes

We have been referring to set classes as prime forms, for example (013) or (0125). This can get a bit hairy when you're discussing larger sets, such as (0134679t). Thankfully, there is a shorthand. I refer you to the set class table. This table contains every single prime form, their corresponding interval vectors (this table does not use the brackets I use, but I would encourage you to follow the conventions I specified, because it makes everything clearer), a transpositional symmetry number (not a number I find particularly useful), and a two-part catalog number that looks something like this: 3-1

This two-part number is what we refer to as a "Forte code," named after their inventor, Allen Forte. Every prime form has one of these. There is a bit of a pattern to them.

(012) = 3-1
(013) = 3-2
(014) = 3-3
(015) = 3-4

You see what's happening? the first number tells you how many p.c.'s are in the set. The second number seems to correspond with how small or large the intervals are in the set. After a while, that pattern breaks a bit.

4-22 = (0247)
4-24 = (0248)
4-23 = (0257)
4-27 = (0258)

I'm not entirely sure why it goes this way, but that's how it is.

Anyway, the nice thing about Forte codes is that instead of saying (0134679t), we can now refer to it as 8-28 (also known as the octatonic scale). Right off the bat, you can tell that the set has 8 p.c.'s, and if you know a bit about the ordering, you can tell that this comes relatively late on the table and contains a decent distribution of intervals. 8-28 happens to be a maximally even set; its p.c.'s are distributed regularly throughout an octave. 8-1, by contrast, is (01234567), which is a tight cluster of p.c.'s, and then a dead zone spanning a major third.

Z related Forte codes

Remember those Z related sets from last post? Mr. Forte was nice enough to point those out for us in his numbering system. Any set that has a Z related pairing looks like this:

5-z29

That one has the interval vector [555553]. If we look around a bit, we can find 5-z15, which is also [555553].

Complementary Sets

Another nice thing about the Forte codes is that they pair up to show you complementation. This means that for any given set, there is another set that fills in the gaps and creates a full chromatic scale (what we call an "aggregate"). For example, if I have the set 1-1 (not on the table, this is literally just one note), then there is an 11 note set out there that supplies the remaining pitches of the chromatic scale. Let's go back to 8-28:

(0134679t)

What's missing? 2, 5, 8, and e. Let's find the prime form of that set.

<258e> (This set is symmetric, so any rotation will do.)
-2222
(0369)

Now locate (0369) on the table. You should end up with 4-28. Hey, that's interesting: 8-28 and 4-28 have the same number in the second part, and the first numbers add up to 12. You'll find this to be the case for any set on the table: 5-13 is complemented by 7-13, 3-1 is complemented by 9-1. I think this has something to do with why the table has funky numbering in some places.

Lastly, I'd like to point out a particular achievement of Allen Forte and set theory. Some of you may be aware of Nicholas Slonimsky's Thesaurus of Scales and Melodic Patterns. That book spends over 200 pages describing every conceivable iteration of pitches in 12tet tuning, and it's obscure as hell. Forte's system does the same thing in a couple of pages and is quite self-explanatory once you know the pattern. If I give you the code, "4-z29," you already know that it has a Z related pairing somewhere (though you'd have to reference the table to know it's 4-z15) and that its complement is 8-z29 (which is in turn Z related to 8-z15). This gives a crystalline compactness and lends predictability to music that operates outside of the boundaries of traditional tonality.

 

[prlgmnr]

Can I promise you in all honesty that I don't mean this in a snarky or dismissive way.

What can you do with all this? Preferably I'd like an answer I can listen to rather than read.

 

[Mr. Big Noodles]

You can technically use set theory to describe the pitch language of any music, but it works best for the music that it was developed for. Really simply, sets are motives. Music that heavily exploits set theory involves a lot of transpositions and transformations of a small number of motives.

This is maybe not the most orthodox example, but I think it has something to offer to our discussion:

The Room motion picture soundtrack - Lisa and Mark


The motives at the beginning are {0e3} and {672}. These are Te(014) and T7I(015) respectively. Note that they both contain the interval [1] (and [4], for that matter), and that their contours are each other's inversion. In the context of the film (which, if you haven't seen it, you really should), Lisa is cheating on her boyfriend Johnny with his best friend, Mark. I hear these two sets as representing their affair: two sets embracing each other through their contour. I believe that Lisa is represented by (014), and her transposition level is usually Te. Mark is (015), and he is usually T7I.

Later on, Johnny overhears Lisa talking about him with her mother behind his back and becomes suspicious. He sets up a tape recorder to spy on Lisa. This is the music that happens at that point:

"I will record everything!"


And the sets:

0:00 - {017} T1I(016)
0:10 - {871} T7(016)
0:20 - {0e6} T0I(016), and it extends down just a bit to make {0e65} T5(0167), which is a symmetric set.
0:38 - {1718} T7(016)
0:40 - {3193t} Tt(0136)
0:45 - {t91} T9(014) < Lisa's set, wrong transposition level. Gets one more note to become {t910} T9(0134), another symmetric set.
1:00 - {107} T7I(016)

Notice that in this cue, there's a lot of (016), which bears a similarity relation (more on that later; basically, it shares all but one note with another set) with both (014) and (015). Remember, Johnny suspects Lisa at this point, but he does not know what exactly is going on. The sets and contours are all messed up, which to me indicates that he has a hazy notion of wrongdoing and an unclear image that somehow involves Lisa. Lisa's set only appears once in the cue, and Mark's is nowhere to be found. The music reflects Johnny's thought process through a leitmotivic treatment of sets. You could think of (016) as the "searching" leitmotif, and the four-note symmetric sets as doubtful, unsure suspicions.

 

[prlgmnr]

Awesome, thanks.

 

[KMunDy1303]

MBN, what pieces would you recommend to look at to further study set theory and it's application?

It's been a while since I took any theory classes but I remember looking at some Bartok and late Stravinsky pieces. I don't remember which pieces but I'm sure I can dig out my text books easily enough. Just looking for more to eat up.

Do you know of any resources similar to IMSLP.org that would have piece to look at or are we getting to recent for copyright/fair use?

 

[Mr. Big Noodles]

Bartók is good, but you have to know what to look for. For example, he likes to use mirror inversion in his music. You don't usually need set class analysis to show you that though. The second Viennese school composers (Schoenberg, Berg, Webern) and some of the more rigorous serialists (Babbitt, Boulez) are good because they use pitch class and set class invariance extensively. I'll cover this more when I do a twelve-tone thread. Ruth Crawford-Seeger, Russian-era Stravinsky, and a bunch of other modernists are good for it too. Shostakovich is rich with this stuff as well, but I feel his music requires a more advanced treatment.

It all depends what you are trying to say/do. Set theory is good for analyzing Messiaen because it allows you to easily identify subsets of his modes of limited transposition. It's good for discussing non-tertian chords and characteristic sonorities that don't lend themselves to lead sheet symbols. That sort of subset/superset relation is different from the motivic treatment we find in Webern's music, where every single motive and chord is 3-3. It's different from the secondary harmonies and non-contiguous sets Schoenberg uses in his twelve-tone music to expand the harmonic and melodic language of his tone row. Metal/jazz folks might find referential collections and subset/superset relations more immediately useful, as that's what the music already does to a large degree. Set theory is fairly multifaceted.

 

[Mr. Big Noodles]

I want to demonstrate some subset/superset relationships because they are nifty and fairly easy to grasp. The idea is that any set that contains 11 or fewer pitch classes (so all but one of them) is a subset of one of more larger sets. For example, (024) (Forte class 3-6) is a subset of the diatonic scale. Here's C major: {024579e}, and in prime form (013568t). It's in the diatonic collection two more times, actually: {024579e} {024579e}. It's also in the whole tone collection six times total. I'll highlight where the set is and give the Forte class and its transposition level (Tn).

(02468t) T0 3-6
(02468t) T2 3-6
(02468t) T4 3-6
(02468t) T6 3-6
(02468t) T8 3-6
(02468t) Tt 3-6

Of course, every set is ultimately a subset of 12-1, which is the entire chromatic scale. However, we tend to be more concerned with subset/superset relationships that are tangible in the music. You can relate this to tonal music pretty easily. A diatonic progression such as I vi IV V with a diatonic melody is all going to boil down to a bunch of subsets of the diatonic collection, 7-35. I think that's a fairly intuitive observation.

 

[Mr. Big Noodles]

Now I would like to show you some collectional thinking and subset/superset relationships in a piece by Béla Bartók. Avail yourself of this diddy and an analyzed score:

Béla Bartók - 44 Violin Duos - No.33, Song of the Harvest

(Clean copy here.)

I have listed all of the sets in normal order, prime form, and by Forte class. Spoiler: it's a lot of 4-10. This is the sort of thing we see in post-tonal music: a lot of similar and identical set classes making up a sizable chunk or even the entirety of a piece. I've summarized each tetrachord pairing in the following graphic and have given you some tabs in case standard notation isn't your thing.

One thing to notice is that the two violin parts are always a tritone apart (a distance of interval class 6). So we have T9 4-10 paired with T3 4-10. The difference between 9 and 3 is 6. You can follow that through all the way to the last pairing, where it finally changes: 3-8=-5, and -5=7 in mod12. You can interpret the transpositional difference going from 6 to 7 as a sort of resolution or reconciliation of a harmonic problem in the piece. At the end, the two tetrachords are brought together by fixing their transpositional difference. I'll show you what I mean by combining these tetrachords into supersets.

This is an even more condensed view of the tune. You can see the first four sections play around with T1 and T2 of 8-28, which is the octatonic scale. I have put each tetrachordal subset under a slur. Finally, at the end, we get out of this vacillation between octatonic collections by breaking the i.c. 6 relationship between tetrachords. This puts the final section in A♭ dorian. In a way, this A♭ dorian section is a summary of what happened before, because T8 4-10 is a subset of T1 8-28, and T3 4-10 is a subset of T2 8-28.

By the way, when searching for a recording, I found this rendition for violoncelli by Paul Olefsky and Hai Zheng. It's a great interpretation, except that whoever is playing the violin 2 part isn't reading the key signature correctly: they're doing <3468> instead of <3568> in the first section because they missed the E#. One criticism that I have heard from listeners and performers about post-tonal music is that you can't tell when there's a wrong note. Here's your chance! The wrong notes stick out like a sore thumb for me on an intervallic level, because literally everything else in the tune is 4-10.

 

[ElRay]

Finally having a chance to catch-up on the threads that require more brain power.

MBN, what pieces would you recommend to look at to further study set theory and it's application?

I started noodling around with set theory after hearing Michael Nicolella's version of Elliot Carter's 'Shard'. Carter's done a few pieces based on AIT's.

... Béla Bartók - 44 Violin Duos

I have a book with the 44 arranged for guitar. I've been meaning to get into it. I have listened to the 44 done with two violins, and they're great pedagogically done in order, but not very listenable. The collections that have them grouped better (like 1, 8, 6, 9) are actually listenable.

Without side-tracking this, is there any other Bartók you'd recommend along these lines? I've got "For Children" and two other Bartók Collections (one specifically arranged for a guitar duo) arranged for guitar.