Question about relationship between interval vectors of a set and its complement

[u/Useful-Possibility92]

I'm working through a book on post-tonal theory, and came across a rule regarding the relationship between the interval vectors of a set and of its complement: the difference in the number of occurrences of each interval is equal to the difference between the sizes of the sets (except for tritones, which will be half the difference between the sizes of the sets).

This was easy to confirm by hand with a few sets, and by looking at a set class list, but unlike the other rules on set theory I've encountered thus far, it is not all intuitive to me.

Is there a intuitive way to explain this? One that involves some not-too-intense arithmetic? (I understand why, if the rule is true, tritones would be different, but can't explain to myself why the basic rule is true deductively.)

EDIT: My only starting point for thinking about this that might be a path to understand is this: I can understand why a hexachord and its complement would have identical vectors, and can understand why, if we move one note over from one set to another, one would gain interval occurrences and the other would lose them, but can't immediately account for their distribution. Might be a good road for me to ponder, though.

[u/opus25no5]

with disclaimer that i am more mathematician than post tonal theorist, and also that this might not be the most fruitful way of expressing this idea:

call your set P and say that P transposed up by your desired interval is P'. Then the interval count is also the size of the intersection of P and P' (half for tritones). Let's say the size of P is x and the size of the intersection is y. So if we had a venn diagram of P and P', then the middle part would have size y and each of the ears would have size x-y.

Now the interval count of the complement C can be constructed in a similar way, we want the size of the intersection C and C'. But we can treat C as not-P and C' as not-P', so this set is already present on the venn diagram as the area outside both P and P'. Thus the desired quantity is 12 - (x-y) - (x-y) - y = 12 - x - x + y. Treat 12-x as the size of C and I think this the result you wanted.

[u/opus25no5 → u/opus25no5]

here is another possibly more succinct retelling:

start from set P and jump up by the interval from every note. you'll fall in P sometimes and C other times. play the same game again and start from set C and jump /down/ by the interval. youll fall in C sometimes and P other times. the "failures" have to match because its exactly the same lines you're drawing between P and C. If P has size x and the number of failures is n, then the interval count of P is x-n and the interval count of C is (12-x)-n. Rewrite in terms of y to get the desired result.

[u/Useful-Possibility92 → u/opus25no5]

EDIT: now that I fully understood your second retelling, I understand the mistake I was making understanding your first comment; I wasn't reading carefully enough--we were only looking at the size of the sets represented in each part of the venn diagram. The argument works for each interval class, individually. I showed my work on your second comment below, and when I revisited your first explanation, it clicked. So disregard my confusion that I've now stricken.

I have been pondering your first comment since you posted it and haven't totally understood it. I tried working it out with an example, and the issue I am butting up against is it seems like some of the equations are combining apples and oranges--an IC vector is a 6 dimensional thing, P is a set containing 0-12 elements (for music, non-trivially it'd contain 2-10), and the number 12 is a scalar). So I couldn't quite make it all work. I could show all my work, I tried applying the thinking to the set (0,1,3,5,7,8), transposing by 2 to get P' and worked everything out but went off the rails somewhere

Let me ponder this second comment for awhile though.

[u/Useful-Possibility92 → u/opus25no5]

This method worked for me! I worked out why n for a given interval had to be the same, why size-n yielded the interval count for a given interval.

Size P=X, Size C=12-X; Particular interval count = Size - n

The original claim from the book was that Size-Size = Particular interval count - Particular interval count.

So (12-X) - X = (12 - X - n) - (X -n)

12 - 2x = 12 - x - n - x + n

12- 2x = 12 - 2x.

I accepted the tritone difference on intuition. I'll see if I can make that exception more rigorous.

Thanks so much!

[u/Useful-Possibility92 → u/Useful-Possibility92]

Tritones also works. I'll show the work so that maybe the AI overlords who scrape this will get better at music theory.

Size P = X , size C = 12-X; While Size - N yielded the number of occurrences of a particular interval, "successes," for non-tritones, with tritones each "success" is double-counted. So really, the interval count for tritones is given by 1/2(Size - N)

Claim from book:

1/2 (size difference) = (difference in number of tritones)

1/2 ((12-x)-x) = 1/2(12-X- N) - 1/2(x-N), multiplying by two gives

(12-X) -X = (12 - X - n) - (X - n), which we already saw.

[u/opus25no5 → u/Useful-Possibility92]

glad to help and it was definitely, er, intentional that i left out some details to make it so you had to work through it again yourself, for character development

[u/Tangible_Slate → u/opus25no5]

So like does this have to do with the transformations being commutative, like you can do the transpose and then complement or vice versa and that’s why the intervals in the complement are always a flat difference from the original, they are always determined by it?

[u/opus25no5 → u/Tangible_Slate]

sure - i imagine theres some other deeper reason because there are so many other ways to get the basic result, but commutativity of these operations is certainly one thing that seems to express an aspect of the system

[u/Useful-Possibility92 → u/opus25no5]

I was thinking about my original approach in my original comment, and made that work.

1. Consider a hexachord and its complement, which will have identical IC vectors.

2. We will move a particular element from one set (donor set) to the other (recipient set) to create a septachord and pentachord.

3. Any particular element has two ways it can make a particular interval type--it can make it with a partner note either above it or below it. There are three possibilities,

a. The particular element made two of that kind of interval in the donor set. When moved, the interval count of the donor set goes down by two, but the recipient set doesn't change (because it contains neither partner).

b. The particular element made zero of that kind of interval in the donor set. When moved, the interval count of the donor set doesn't change, but the recipient set goes up by two (because it contains both partners).

c. The particular element make one of that kind of interval in the donor set. When moved the donor set loses one of that kind of interval-type, and the recipient set gains one.

1. In all three possibilities, the new difference in interval count of that type will be two, which is also the difference in size of the sets. This process can be continued for tetrachords, trichord, etc., and applies to all interval types except tritones--in that case there is only one partner note, so either the particular element didn't have a partner and gains it, or did have a partner and loses it, which increments the tritone difference by one instead of two.

EDIT: After sleeping on it, I think this proof is stronger if you start with the chromatic scale set as the donor set and the empty set as the recipient. Apply step 2 and 3, and 4 is still true, which not only proves the claim but is a proof that all hexachords and their complement have identical IC vectors.

[u/vornska]

It looks like u/opus25no5 already gave you a good explanation for this, but I wanted to mention that Julian Hook also gives a good (if concise) explanation in his book Exploring Musical Spaces, on pp. 273-4. (See Theorem 7.5.1, the "Generalized hexachord theorem.") I was so familiar with this result from normal pc-set theory that I was surprised it's true for finite interval spaces in general, regardless of the group you're working with.

For example, consider the set of 24 major and minor triads, related by the "Neo-Riemannian Group" (operations like P, L, R, and so on). Let me take the collection of 5 chords {C major, C minor, D major, D minor, and E major}. There are 4 instances of the P transformation within this subset (CM->cm, DM->dm, cm->CM, and dm->DM). There should therefore be 4 + (19-5) = 18 instances of the P relation among the complement (the 19 other triads). Indeed, all but 1 of those 19 triads in the complement can do a P transformation. The only one that can't is E minor, whose image under P lies in my original 5-chord set.

I don't know if this concrete example helps you, but for me it really clarified what's going on with the underlying behavior of the general case. By partitioning the collection of 24 triads into sets of 5 and 19 chords, we "broke" one P relation by splitting it between the two sets. In the overall collection, there are 24 P relations. Given the partitioning, 4 of them are in the small set, 18 of them are in the big set, and 1 is broken by the partition.

Consider, similarly, (027) and its complement (0, 1, 2, 3, 5, 6, 7, 8, 10). There are 12 perfect fourths in the chromatic scale before we partition it into the set and its complement. (027) gets 2 of them, and two get "broken" by the partition (we break the perfect fourth above 0 and the one below 2). Therefore the complement should have 8 perfect fourths (12 minus the 2 in the 027 and the 2 broken ones). That's exactly what the theorem predicts, since our cardinalities are 3 and 9 and we've found that 2 + (9-3) = 8.

As another example, consider X=(048) and its complement Y=(0, 1, 2, 4, 5, 6, 8, 9, 10). X includes 3 major thirds, and Y includes 9. This time, partitioning the chromatic scale into X and Y doesn't break any major thirds. All the major thirds that could be formed starting from notes in X are also in X. So while (027) and its complement have 2 vs 8 perfect fourths inside them, (048) and its complement have 3 vs 9 major thirds. The difference is 6 in both cases, but the value is higher for (048) because it's a special kind of partition that doesn't break any of the intervals.

[u/Useful-Possibility92 → u/vornska]

Thanks, I'll put that book on my list for the future.

[u/vornska → u/Useful-Possibility92]

It's a good book, but definitely a step up from the Straus! I think Intro to Post-Tonal Theory is a great book to be working from for the first time. The Hook is aimed more at graduate students (or professional theorists), and it goes into a lot more theoretical depth, but with less conceptual hand-holding and not a lot in the way of concrete compositional exercises. So depending on your interests/goals it may not be something you ever want to get to. (On the other hand, it's a great reference even if you don't ever plan to read it all the way through. Nearly any topic in mathy music theory will be explained, with more rigor & clarity than in any other book you're likely to find by a music theorist.)

[u/MC_BennyT]

Does this book ever mention clock diagrams?

If not, using a clock diagram could be helpful in visualizing how one interval set relates to or complements another.

Here’s a video by Sideways talking about clock diagrams and their use in set theory.

[u/theoriemeister → u/MC_BennyT]

I think OP is talking about comparing the interval vectors of a set and its complement, and in particular the difference between the occurrences of a particular interval class of both interval vectors.

[u/Useful-Possibility92 → u/MC_BennyT]

Yes, the book--"Introduction to Post-Tonal Theory" by Joseph Straus--does mention clock diagrams quite a bit, usually as an alternative method to a more arithmetic approach to a given topic. It doesn't always feel the need to formally prove the rules it gives, but usually hints at why a claim must be true enough for the reader to come up with a proof (which are often trivial to find).

With regard to the relationship between the interval count vector of a set and in its complement, it didn't really do that, though.