Pitch multiplication... how does it work?

[Sharkissimo 1,973]

I get inversions, retrogrades, retrograde inversions, inverted retrogrades, transpositions and rotations, but not multiplication. Could someone, explain it to me? I want to be able to analyse Boulez's stuff.

And please don't say say 2 x 2 = 4.

[Wojo 2,453]

2 + 2 = 4 too. So I'm already confused.

[RedBard 71]

So you piqued my interest in pitch multiplication, Sharky. I'm now doing some light research into it. It certainly has its place in twelve-tone music.

From Bartok's Third Quartet:

Apparently, this is what happens when you "multiply" a chromatic tetrachord. I'm still trying to wrap my mind around what each number above each note signifies.

[BLUMENKOHL 1,068]

I get inversions, retrogrades, retrograde inversions, inverted retrogrades, transpositions and rotations, but not multiplication. Could someone, explain it to me? I want to be able to analyse Boulez's stuff.

And please don't say say 2 x 2 = 4.

Do you have an understanding of pitch-class set theory?

[Dixon Hill 4,232]

What are you not getting? The method itself, or the application?

[Sharkissimo 1,973]

More the application in the context of say Le marteau sans maître.

I get inversions, retrogrades, retrograde inversions, inverted retrogrades, transpositions and rotations, but not multiplication. Could someone, explain it to me? I want to be able to analyse Boulez's stuff.

And please don't say say 2 x 2 = 4.

Do you have an understanding of pitch-class set theory?

Yes.

So you piqued my interest in pitch multiplication, Sharky. I'm now doing some light research into it. It certainly has its place in twelve-tone music.

From Bartok's Third Quartet:

Apparently, this is what happens when you "multiply" a chromatic tetrachord. I'm still trying to wrap my mind around what each number above each note signifies.

That's pitch-class multiplication, which apparently is a bit different to pitch multiplication.

To explain it, 7 * 1 = 7, 7 * 2 = 2 and 7 * 3 = 9, and so on. Remember it's Modulo 12, so 14 =/= 2 and 21 =/= 9.

[BLUMENKOHL 1,068]

Then this ought to be a good read for the next couple of weeks:

https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/11257/9401433.pdf?sequence=1

[Glóin the Dark 1,218]

Is it this notion of multiplication you mean?

[Sharkissimo 1,973]

Yep.

[BLUMENKOHL 1,068]

I'm confused as to what you're confused about!

[Dixon Hill 4,232]

Ok. You've got some actual resources now. But I'd read this too if I were you. It features the piece you had in mind and the comments have a very interesting and level-headed discussion on what exactly the values of applying such processes are, if any.

http://www.artsjournal.com/postclassic/2010/06/how_to_care_how_it_was_made.html

I'm with those who feel it takes things to a point of absurdity - that it's probably little more than a slightly controlled academic excuse for writing whatever the hell you want.

[BLUMENKOHL 1,068]

Take two sets of notes, let's say:

{G} and {G#,A}

We cannot do math on letters, so we put them on a number line from 0 to 11, starting with C.

Our sets become

{7} and {8,9}

(remember 0=C, 1=C#, and so on)

Take their Cartesian Product (combine everything from one set with everything from the other):

{7} X {8,9}

and you get

{(7,8),(7,9)}

Boulez says, add the pairs mod 12. Why mod 12? because 7+8=15, and that doesn't correspond to a number on our number line going from 0 to 11 (it has 12 numbers on it!). So we wrap it around the octave with the mod 12 operation. A modulo operation just gives you the remainder after division.

12 goes into 15 once, with a remainder of 3. 12 goes into (7+9)=16, once, with a remainder of 4. So our final result is:

{3,4}

On our number line that means {D#,E}

[Glóin the Dark 1,218]

Here's how I understand the description. I'd say it might equally well be called "collective addition" of pitches.

Say we take A as the root note, so A=0 in the 0,1,...,11 numbering of pitch classes. Then, in the standard concept of "pitch addition", we would have:

B+D=E (since 2+5=7)

B+F=G (since 2+8=10)

Now suppose instead of wanting to add one single note to another, we want to "add" one chord to another (or, more generally, one collection of notes to another). Suppose one of our two chords is the two-note chord {D,F} and, to be as simple as possible, the other is a one-note chord {B}. Then the "collective sum" is:

{ B } + { D, F } = { B+D, B+F } = { E, G }.

Next, let's replace the one-note chord {B} with a two-note chord {B,C}. Then, having done the calculations C+D=F and C+F=Ab, we can form:

{ B, C } + { D, F } = { B+D, B+F, C+D, C+F } = { E, G, F, Ab } = { E, F, G, Ab }.

Finally, let's replace {B,C} with {B,C,E}. as E+D=A and E+F=C, we get:

{ B, C, E } + { D, F } = { B+D, B+F, C+D, C+F, E+D, E+F } = { E, G, F, Ab, A, C} = { A, C, E, F, G, Ab }.

Just keep going until you've included as many notes as you want on the red side and the blue side!

And instead of "{ B, C, E } + { D, F }", the Wikipedia article suggests we should write "{ B, C, E } x { D, F }", and call the operation multiplication rather than collective addition, but that's nothing more than a superficiality.

(It's quite possible that I've completely misunderstood the description...)

[BLUMENKOHL 1,068]

There you go. The algorithm explained mathematically and musically between me and Gloin. The modulo is implicitwhen you think of it musically, you have an infinite piano!

Note that I set C=0 and Gloin set A=0, just to confuse you.

[RedBard 71]

It's a sea of Skittles and math! Aaaaaaaaagh!

[Glóin the Dark 1,218]

Note that I set C=0...

Ha! How perverse!

Sharky said it was more the application of the idea that he was inquiring about, though, so I guess we're telling him what he already knows. Except in different keys...

you have an infinite piano!

Infinite in one sense and yet so very finite in another, having, as it does, a lamentable twelve notes per octave!

I refuse to play on any piano which isn't tuned to a 224-tone equal temperament system, at the bare minimum.

[Uni 306]

Looks like I should've picked up that copy of Music Expressed as Warp Field Equations for Dummies.

[Sharkissimo 1,973]

Take two sets of notes, let's say:

{G} and {G#,A}

We cannot do math on letters, so we put them on a number line from 0 to 11, starting with C.

Our sets become

{7} and {8,9}

(remember 0=C, 1=C#, and so on)

Take their Cartesian Product (combine everything from one set with everything from the other):

{7} X {8,9}

and you get

{(7,8),(7,9)}

Boulez says, add the pairs mod 12. Why mod 12? because 7+8=15, and that doesn't correspond to a number on our number line going from 0 to 11 (it has 12 numbers on it!). So we wrap it around the octave with the mod 12 operation. A modulo operation just gives you the remainder after division.

12 goes into 15 once, with a remainder of 3. 12 goes into (7+9)=16, once, with a remainder of 4. So our final result is:

{3,4}

On our number line that means {D#,E}

Here's how I understand the description. I'd say it might equally well be called "collective addition" of pitches.

Say we take A as the root note, so A=0 in the 0,1,...,11 numbering of pitch classes. Then, in the standard concept of "pitch addition", we would have:

B+D=E (since 2+5=7)

B+F=G (since 2+8=10)

Now suppose instead of wanting to add one single note to another, we want to "add" one chord to another (or, more generally, one collection of notes to another). Suppose one of our two chords is the two-note chord {D,F} and, to be as simple as possible, the other is a one-note chord {B}. Then the "collective sum" is:

{ B } + { D, F } = { B+D, B+F } = { E, G }.

Next, let's replace the one-note chord {B} with a two-note chord {B,C}. Then, having done the calculations C+D=F and C+F=Ab, we can form:

{ B, C } + { D, F } = { B+D, B+F, C+D, C+F } = { E, G, F, Ab } = { E, F, G, Ab }.

Finally, let's replace {B,C} with {B,C,E}. as E+D=A and E+F=C, we get:

{ B, C, E } + { D, F } = { B+D, B+F, C+D, C+F, E+D, E+F } = { E, G, F, Ab, A, C} = { A, C, E, F, G, Ab }.

Just keep going until you've included as many notes as you want on the red side and the blue side!

And instead of "{ B, C, E } + { D, F }", the Wikipedia article suggests we should write "{ B, C, E } x { D, F }", and call the operation multiplication rather than collective addition, but that's nothing more than a superficiality.

(It's quite possible that I've completely misunderstood the description...)

Thanks for that!

I sort of understood it, but now (i think) I do. There's just one niggle. Blume combined both sets (took their Cartesian blah blah) but Gnome didn't. If he did, it would look like:

{ B, D } + { B, F }

Would that make much of a difference to the product?

... I'll find out.

I'm with those who feel it takes things to a point of absurdity - that it's probably little more than a slightly controlled academic excuse for writing whatever the hell you want.

Oh I know, it's pretty arbitrary. I just don't like the idea of Pierre Boulez understanding something and not me!

Ok. You've got some actual resources now. But I'd read this too if I were you. It features the piece you had in mind and the comments have a very interesting and level-headed discussion on what exactly the values of applying such processes are, if any.

http://www.artsjournal.com/postclassic/2010/06/how_to_care_how_it_was_made.html

Nice Blog (I'm a big fan of Stravinsky's rotational arrays and row columns--seems much more practical)--but the writer glosses over the whole pitch multiplication process.

Nevertheless, I get that Boulez divides up his row into five segments in five possible ways based on a rotating number series:

That makes sense. The five rotated segments being single pitch classes, dichords, trichords and tetrachords...

And so on. I also get that he “multiplies” each of those five segments by all five of them to build up derived unordered pitch sets – the process of “chord multiplication” being to transpose one chord to all the pitches of the other chord and add all the pitches together. And you can see how (if you take the trouble) each gesture is drawn from the pitches of these chord-multiplication products:

... wait what?

[Dixon Hill 4,232]

Hahaha fair enough. (to the Boulez understanding comment)

[Glóin the Dark 1,218]

If Blume had done my sum, it would have gone essentially like this:

• we want to calculate { B } + { D, F }
• first take the Cartesian product: { B } X { D, F } = { ( B , D ) , ( B , F ) }
• now perform addition within each pair of round parentheses: { B+D, B+F }
• finally, doing the actual sums this becomes { E , G }

So, really, I was implicitly taking the Cartesian product in the step

{ B } + { D, F } = { B+D, B+F }

[Sharkissimo 1,973]

So would the Cartesian product for the second example (the two note chord) look like this?

{ ( B , D } , ( B , F ) , ( C , D ) , ( C , F ) }

[Glóin the Dark 1,218]

Exactly.

[BLUMENKOHL 1,068]

I'm confused. What's labeled B is

Assuming C=0...

{11, 2, 10, 1}

and what's labeled C is

{9,0}

Cartesian product of the two is

{8, 11, 2, 7, 10, 1} (I removed duplicate pitches)

which corresponds to {G#, B, D, G, A#, C#}

Those are not the six notes he's labeled under BC. Or am I that bad at reading music?

[Glóin the Dark 1,218]

Assuming C=0...

Therein lies your problem!

The given chord is a translate (by five semitones) of your one. Try F=0.

[BLUMENKOHL 1,068]

Bloody hell!

[Glóin the Dark 1,218]

F in hell!

[KK 3,307]

Treating notes as Cartesian vectors? Seems like a fascinatingly absurd and pointlessly excessive system by which to go by.

Love it.

[BLUMENKOHL 1,068]

It's actually kind of elegant how remarkably well the products of the operation work! They fit nicely with their component sets

[KK 3,307]

Sure, and it creates a tightly woven interconnectivity in seemingly "chance" music. But musically, the procedure is a bit pointless isn't it?

As a system, I can see it being an aid in writing serialist music. But I guess one must ask just how diverse its applications can really be.

[Glóin the Dark 1,218]

So pitch addition is dependent on what we choose as the root note, meaning that there are twelve different operations of this kind (and hence twelve derived operations of "pitch multiplication" for chords) rather than a single, canonical one. I was wondering if there is such a thing as a straightforward, well-behaved, and reasonably natural operation on the pitch classes which is not dependent on the choice of root note, and instead would give the same answer whether we put C=0 or A=0 or F=0 or whatever. The only non-trivial examples of such an operation I could easily come up with are the operation * described by the expression

x * y = 4x - 3y

(all operations performed modulo 12), and its "reverse" defined by switching x and y on the right side of the equation. This operation is actually very similar in form to the original addition operations, but has the rather striking "tone invariant" property in addition. I would conjecture that, if normal addition does indeed have any audible significance, then so would this one.

Any music theorists here ever encountered it?

[BLUMENKOHL 1,068]

As applicable as any other compositional tool! It won't work everywhere, but useful to have in the old toolset. Might help you dig yourself out of a tight spot one day.

[KK 3,307]

As applicable as any other compositional tool! It won't work everywhere, but useful to have in the old toolset. Might help you dig yourself out of a tight spot one day.

Oh I agree, it could be an effective tool in certain situations. Just not sure how many pieces I'd write primarily based on this method.

[Dixon Hill 4,232]

I've done it once, and once is all I'll do!

[Ludwig 1,120]

Pitch multiplication means transposing one subset of a twelve-tone series by the intervals of another subset. So in the graphic below...

the minor third of C is stated on each of the notes of B (though for some reason, B is transposed up T4). And in the lower staff, the minor 7th of A is restated on each of the notes of B. The result (disregarding duplicate notes) is a new set, the notes of which can appear in any order.

On another note - what is going on in this thread? I feel like we're trying to figure out the theme to the Enigma Variations.

[Dixon Hill 4,232]

Auld Lang Syne!

[KK 3,307]

I feel like we're trying to figure out the theme to the Enigma Variations.

Why don't we have a thread for that?!

[Glóin the Dark 1,218]

Pitch multiplication means transposing one subset of a twelve-tone series by the intervals of another subset.

That's not quite it: in addition to the process you've described, the entire result must then be transposed by a degree determined by the position of the second subset. This is why, in the example you referenced, figure B is also transposed upwards by four semitones: the root note underlying this example is F, while the lowest note of figure C is four semitones higher at A.

Edit: It's a pity they couldn't have denoted those two figures X and Y or something sensible like that!

[Sharkissimo 1,973]

Assuming C=0...

Therein lies your problem!

The given chord is a translate (by five semitones) of your one. Try F=0.

Tried it and I got:

[10, 1, 4, 9, 0, 3]

7 semitones too high!

So pitch addition is dependent on what we choose as the root note, meaning that there are twelve different operations of this kind (and hence twelve derived operations of "pitch multiplication" for chords) rather than a single, canonical one. I was wondering if there is such a thing as a straightforward, well-behaved, and reasonably natural operation on the pitch classes which is not dependent on the choice of root note, and instead would give the same answer whether we put C=0 or A=0 or F=0 or whatever. The only non-trivial examples of such an operation I could easily come up with are the operation * described by the expression

x * y = 4x - 3y

Could you explain this expression a bit? Where do the 3 and 4 come from?

[Glóin the Dark 1,218]

Tried it and I got:

[10, 1, 4, 9, 0, 3]

7 semitones too high!

Starting to doubt myself here, but I think that the answer you got (i.e, {10, 1, 4, 9, 0, 3}) is what's printed in the score. Remember that F=0 throughout. Did you revert to C=0 when interpreting the outcome? That would explain the seven semitone discrepancy...

Could you explain this expression a bit?

Yeah, if the numbers x and y are denoting two pitch classes, we just (a) multiply x by 4, (b) multiply y by 3, and then (c) subtract (b) from (a); lastly, we covert the answer into a number modulo 12.

Let's say x=2 and y=9. Then:

4x-3y = (4 times 2) - (3 times 9) = 8-27 = -19.

But, modulo 12, we find that -19 = -7 = 5 (just adding 12 repeatedly until we get into the range 0,1,...,11).

So 4x-3y = 5 would be the answer in this case (whatever pitch class 5 represents).

Equally well, we could avoid negative numbers by changing the formula to

x * y = 4x + 9y.

This is because +9 is the same thing as -3 in the modulo 12 universe. We would likely have to subtract 12 one or more times to get the answer into the range 0,1,...,11 in this case, but the final answer will always be the same no matter which of the two ways we go about it. E.g., with x=2 and y=9 as before,

4x + 9y = (4 times 2) + (9 times 9) = 8 + 81 = 89 = 77 = 65 = ... = 17 = 5.

Where do the 3 and 4 come from?

That's less easy to justify! Feel free to ignore this pile of jargon:

My aim was to find a process comparable to pitch addition but independent of what note we label as zero (something that wouldn't be vulnerable to the sort of error that Blume discovered earlier when he was working under the assumption that C=0, while the example implicitly had a different root). I did a bit of scribbling, trying to interpret symbolically what I was after, and homed in on this expression as something which (rather uniquely) worked.

If m and n are any whole numbers, we can make the formula

x * y = mx + ny

(everything done modulo 12 as always). Let's call an operation of this form a linear operation. For my formula, m = 4 and n = -3 (or, equivalently, m=4 and n=9). On the other hand, if we take m=1 and n=1, it becomes x * y = x + y, and the operation reduces to that of pitch addition. So we have a generalised notion which incorporates the addition operation utilised in the "Boulez process" above as a special case, but, by altering the parameters m and n, has several other variants including the one that I arrived at.

I wanted an operation whose outcome would be (unlike that of addition) independent of one's choice of root note. It turns out that, for a linear operation as notated above, this independence occurs if (and only if) m+n=1 (modulo 12). That's why my formula works in this respect (because 4+(-3)=1, or 4+9=13 : both are 1 modulo 12), whereas ordinary addition doesn't (as 1+1=2 isn't 1 modulo 12).

We could also achieve this with, say, m=5 and n=8, and the resulting linear operation

x * y = 5x + 8y.

But this operation lacks a rather fundamental mathematical property which I rather wanted to retain, called associativity. This is the requirement that, whatever notes x, y and z might be, the sums x * (y * z) and (x * y) * z should give the same result. Addition is an associative operation, but most linear operations aren't. It can be worked out that a linear operation is associative if (and only if) both m and n belong to the set {0, 1, 4, 9}.

To have both associativity and root note invariance, then, there are really only two possibilities: we can have m=0 and n=1 (or vice versa), or else we can have m=4 and n=9 (or vice versa) The former case is trivial: the operation is simply x * y = y, disregarding x altogether. So we have only one non-trivial possibility: x * y = 4x + 9y (or its reverse with x and y swapped). That's where my 4 and -3 = 9 came from!

[Ludwig 1,120]

Pitch multiplication means transposing one subset of a twelve-tone series by the intervals of another subset.

That's not quite it: in addition to the process you've described, the entire result must then be transposed by a degree determined by the position of the second subset. This is why, in the example you referenced, figure B is also transposed upwards by four semitones: the root note underlying this example is F, while the lowest note of figure C is four semitones higher at A.

Edit: It's a pity they couldn't have denoted those two figures X and Y or something sensible like that!

The transposition by four semitones is just an extra thing Boulez does with this particular piece. It's not part of the regular procedure of multiplication. He just transposes all his multiplied products by the interval created between the constant of F and the first note of the multiplied subset, in this case meaning F to A, which is four semitones.

That's the answer, Sharky.

[Glóin the Dark 1,218]

But is that the answer if the question is how to analyse Boulez's stuff using pitch multiplication? The thesis that Blume linked to near the top of the thread considers this type of multiplication (Ludwig's version, under the label simple multiplication) and makes the case that Boulez's process, at least in Le marteau, can be understood by a multiplication which is both more elegant (being commutative and non-multiplier replicating) and accounts perfectly for features such as the above four semitone transposition. I've only just glanced through the chapter (Chapter 6) in which he defines his supposedly effective multiplication, but, on first sight, it seems to me as though it's just the operation that Blume and I described in this thread, but presented in a more heavy-handed way (with his "transposition-determining constant" performing the same task as our selection of note zero). I could be mistaken about this...

[Ludwig 1,120]

The explanation I cite comes from published work by a PhD in music theory at one of the world's top universities. I'd go with that over something only on the Internet.

[BLUMENKOHL 1,068]

The explanation I cite comes from published work by a PhD in music theory at one of the world's top universities. I'd go with that over something only on the Internet.

Care to cite your source?

[Glóin the Dark 1,218]

The analysis I was talking about is also from someone with a Ph.D. in music theory, and his thesis is specifically about pitch multiplication in relation to Boulez. But it wasn't my intention to appeal to authority; the work can be judged on its merits by anyone interested.

[Ludwig 1,120]

The explanation I cite comes from published work by a PhD in music theory at one of the world's top universities. I'd go with that over something only on the Internet.

Care to cite your source?

Joseph Straus, Introduction to Post-Tonal Theory (2005), 3rd ed., pp. 235-40.

The analysis I was talking about is also from someone with a Ph.D. in music theory, and his thesis is specifically about pitch multiplication in relation to Boulez. But it wasn't my intention to appeal to authority; the work can be judged on its merits by anyone interested.

Yes I know we can judge any work on its merits. I was just implying that Straus' is the best viewpoint on the issue because it is the most up-to-date in published form. He does acknowledge Heinemann's work, which was published in a top academic journal nearly twenty years ago. Also, I don't believe Straus contradicts what Heinemann says, he just distills it down to its essentials for the sake of pedagogical clarity.

So in short, I think we can agree on what Heinemann calls "simple multiplication," yes? He also cites "compound multiplication," but says it's not particularly important to Boulez's Marteau, then cites "complex multplication," which adds transposition based on aspects of the subsets in the series. Here's what Heinemann says exactly in his article:

In Le Marteau, a single constant governs each domain, chosen according to this criterion: the transposition-determining constant is the rightmost of the first three pcs of the row appearing in set VXA, the first V-set partitioned from the row in each domain. Thus (as can be inferred from Example 11) in Domains 1 and 3, k equals F; in Domains 2 and 5, k equals D; in Domain 4, k equals Eb.

In the example we're discussing with the T4 transposition, that's the result of F being the rightmost pc he's talking about.

[Glóin the Dark 1,218]

I was just implying that Straus' is the best viewpoint on the issue because it is the most up-to-date in published form. He does acknowledge Heinemann's work, which was published in a top academic journal nearly twenty years ago. Also, I don't believe Straus contradicts what Heinemann says, he just distills it down to its essentials for the sake of pedagogical clarity.

Ah, okay. I took your comment at the end of the previous page as suggesting that "simple multiplication" is the only operation that could sensibly or usefully be meant by "pitch multiplication", and that the transposition aspect was a separate matter altogether. If simple multiplication is found to be the easiest version to grasp initially, in the world of music theory, then I guess that's the appropriate place to start. From my own perspective, I think that "Bluhrmenkehrl multiplication" just about beats simple multiplication in terms of conceptual simplicity, so I feel more drawn towards it as a framework within which to view these processes.

[BLUMENKOHL 1,068]

In other news, this thread helped me realize doing arithmetic on musical notes is clock arithmetic.

It's 7:00 PM. Add 8:00 hours to 7:00 PM.

(7+8) mod 12

The time is now 3:00 AM!

12 hours? 12 notes? COINCIDENCE? CONSPIRACY?

[Glóin the Dark 1,218]

"Be there at 5, sharp!" they said, and were furious when I showed up at 6. How was I supposed to know they didn't speak the language of musical set theory?

[Marcus 390]

There is another, simpler, less mathematic approach to this procedure, which is simply to multiply a chord by itself, that is to say expand upon it using the structures of the original sonority.

Let's say you have a chord consisting of the notes C, Ab and B, giving us an intervallic make-up of a minor sixth, a minor third and a major seventh. Using each of these intervals, one could create expansions derived from the original chord.

Starting from the top B,the original structure could be repeated, giving us G and Bb, and again, continuing up, F# and A.

Or one could start from the Ab, giving us E and G. And so on.

A much simpler process, but one that can generate interesting results that will also sound sonically related to the original construct.