Pitch Multiplication

In this post I’m going to talk about pitch multiplication - a topic related to pitch organization that for a very long time I found to be equally fascinating and perplexing. I first came across the term around 2011 or 12 when I first started studying Pierre Boulez’s music intensely, but I never took much time to really research the topic to understand it. To be fair, I was more concerned with Boulez’s use of instrument color, orchestration, gestural language and his writings on music. Though pitch structure was (arguably) a primary compositional element of Boulez’s music (and overall philosophical approach), I was never really interested in the theoretical underpinnings of how he created pitch structures. As a result, I never fully understood how pitch was organized in some of my favorite works (Eclat, Pli Selon Pli, Third Piano Sonata).

This isn’t to say I wasn’t interested in integral serialism or systematic ordering of pitches, it’s actually quite the opposite. I was just more interested in approaches by Xenakis, Berio, Robert Morris, and (my own teacher) Mikel Kuehn. Around 2014, my focus and interests in Boulez’s music started to shift and I refocused my attention on understanding his methods of inventing and working within rigorous mathematical pitch structures, but this was still mostly related to his method of integral serialism.

However, it wasn’t until late 2016 that I really began looking into pitch multiplication, as previous attempts were always unfruitful. I guess my background in math just wasn’t what it needed to be, or maybe I was just having issues wrapping my brain about the concept, because what I ultimately discovered is that the math is quite simple, I was just making it harder than it needed to be.

All of that aside, this leads to the main question at hand - what is pitch multiplication? Not only that, but how can it be used in a meaningful way that doesn’t render it an academic or theoretical exercise? The simplest explanation is that pitch multiplication is the act of multiplying one pitch class set with another, with the resultant product being a superset of pitch classes. This can be broken into simple multiplication and complex multiplication. In this post I will only be looking at simple multiplication and provide some examples, but I will also provide links for further reading related to complex pitch multiplication. I will also discuss some of the history of pitch multiplication and where it came from before diving straight into the mathematics behind the process.

Sidebar: the following information is taken primarily from Stephen Heinemann’s articles on pitch multipliation and Lev Kablyakov’s writings on multiplication in Boulez’s Le Marteau sans Maitre.

First a little bit of terminology:

Multiplicand - element x in the equation x * y = z
Multiplier - element y in the equation x * y = z
Product - element z in the equation x * y = z; the result of multiplying two elements together
Normal form - ordered pitch class set in which the total interval content is the most compressed
Initially ordered pc set (IP) - a pc set in which the first pitch is ordered, but the pcs that follow are not
Initial pitch class (r) - the first pc in an initially ordered pc set (important for determining OIS)
Ordered pitch-class interval structure (OIS) - interval set of an ordered pc set, determined by finding interval of initial class to other pcs in the set: given set (5,9,0), OIS = (<5,5>,<5,9>,<5,0>) = (047)


Pitch multiplication is a technique that was invented by Pierre Boulez as a means of creating variations of ordered and unordered pitch class content beyond the rules and guidelines of integral serialism. By the mid 1950s, many serial composers began moving into new territory and added individualized procedures to their approaches to serialism (aleatory, electronics, graphic notation, new approaches to ordering procedures). Pitch multiplication was one of Boulez’s primary compositional techniques that individualized his style that was characteristic of pieces from the 1950s and 60s, particularly his masterwork Le Marteau Sans Maitre. For Boulez, the practice of pitch multiplication allowed the composer to create ordered pitch class sets according to some kind of logic or rules, multiply the two together, resulting in a numerous supersets that could undergo other ordering procedures; the main point being that Boulez was able to create interrelated supersets from a small amount of pitch class set material, all of which could be ordered further by whatever scheme he chose. This approach could allow a composer to generate a massive amount of pitch material from a relatively limited starting point. It could potentially allow for repetition of pitches, depending of how the composer chose to filter the results of the superset (Boulez would remove repeated pitches, but one could leave repeated pitches in the final superset).

The earliest recorded example of pitch multiplication resulting in a superset from multiplying two pitch class subsets is Nicholas Slonimsky’s book Thesaurus of Scales and Melodic Patterns, which contained 1300 scales and patterns constructed with a type of pitch multiplication. Slonimsky created his patterns by taking a pitch class set, for example set X = (0134), and multiplying it by another, let’s say set Y = (015). The end result would look like this:

                        (0134) (015)  =  <0,1,3,4,1,2,4,5,5,6,8,9>

Notice resulting superset is an ordered set that allows for repeated pitches. The superset is created by transposing set X by each element of set Y. This would be shown as the following:

                               A = T0 (0134)  =  {0,1,3,4}
                               B = T1 (0134)  =  {1,2,4,5}
​                               C = T5 (0134)  =  {5,6,8,9}

The results of the transposition are then joined as A B C to create the ordered superset:

                         {0,1,3,4,1,2,4,5,5,6,8,9}  as shown above

The notated result would look like this

Example 1

It’s important to notice that Slonimsky allows for repeated pitches and creates his scales/patterns by ordering set X by transpositions of set Y. While the end result can be found through a process of pitch multiplication, it is not the same application as applied by Boulez, thought it is successful at creating a sequence of pitches related by interval interval with transpositional variation within the sequence. It is an elegant method for creating scalar pitch collections.

Other examples of pitch multiplication can be found in Stravinsky and Lutoslawski (as examined by Heinemann in his articles, linked below), although these examples outline the theoretical concept of multiplicative relationships among pitch class sets, and were not intentional practiced by the composers. Boulez was the first composer to intentionally theorize, invent and apply his approach to pitch multiplication. We won’t look at the examples in Stravinsky and Lutoslawski, but if this topic is interesting to you beyond Boulez or other simple applications I strongly suggest looking into Heinemann’s article.

Lets take a closer look at calculating supersets with pitch multiplication. The method employed by Slonimsky works as a method, but again it generates ordered transpositions of a single set. Another method more closely related to Boulez’s method is to create a Cartesian product of the two pc sets. A Cartesian Product is the product of two sets made up of each pair of the elements. For ease of reference we’ll call (0134) set X and (015) set Y. A Cartesian product of set X containing elements (a,b,c,d) and set Y containing elements (x,y,z) is {(a,x),(a,y),(a,z),(b,x),(b,y),(b,z),(c,x),(c,y),(c,z)}. The example below puts this into context with numbers

Example 2

​The equation for creating a Cartesian Product:  Set X Set Y = {(x+y) mod 12 | x X y Y}


                                  Set X = (3479)
                                  Set Y = (035)
                                  Set X * Set Y =
        {(3,0),(3,3),(3,5),(4,0),(4,3),(4,5),(7,0),(7,3),(7,5),(9,0),(9,3),(9,5)}

Each pair is then summed at mod12, resulting in the following set:
                                    (3,0) = 3
                                    (3,3) = 6
                                    (3,5) = 8 ...
                                    Total =  {3,6,8,4,7,9,7,A,0,9,0,2}

This can now be used as an ordered set, as Slonimsky would, which would look like this:
​

or we can remove duplicate pitches, and put the set in normal order:  =  {6,7,9,A,0,2,3,4}


The result of this multiplication process provides an 8-note collection that can be used in numerous ways; 8 notes of a scale, the superset broken into subsets, undergo a process of transposition to extend it’s use, etc. Additionally, it could be used as an intermediary superset between two moments in a piece. The product of the multiplication contains pitch classes 2, 6 and A, none of which are contained in either set X or Y, thus the augmented chord <2,6,A> could be used as an arrival point from the a process of moving through sets X and Y.

The Cartesian Product method of multiplying pc sets is simple and straight-forward, but the application of the process as described by Boulez takes on another level of specificity, specifically ordering the sets. This is process that Heinemann describes in his dissertation on pitch multiplication in the music of Boulez. This method involves taking two sets, again X and Y, using set X as the multiplicand and set Y as the multiplier, and determining the product of the two based on an ordered interval series of set X. Let’s break all of that down, by first finding the ordered pitch-class interval structure (or OIS as defined by Heinemann) of a set:

Example 3
           Set X = {2,3,6,7}, put in ascending order for simplification of the explanation
           Set Y = {2,5}

For this example we’ll assume that {2,3,6,7} is the order of the set. We’ll look at different orderings in examples below

 
Step 1 - find the ordered pitch-class interval structure, and finding the interval relationship of each element to the first element in the set:

               Set X = {a,b,c,d}, OIS = (i<a,a>,<a,b>,<a,c>,<a,d>)
               Set X = {2,3,6,7}, OIS = (i<2,2>,<2,3>,<2,6>,<2,7>) = (0,1,4,5)
               NOTE: the first interval class will always be 0 in an OIS
    
Step 2 - Use elements of set Y as transposition levels for the OIS of set X
    
                            Set X  Set Y  =  {2,3,6,7} {2,5}
                               {2,3,6,7}  =  {0,1,4,5}
                            T2 {0,1,4,5}  =  {2,3,6,7}
                            T5 {0,1,4,5}  =  {5,6,9,A}

Step 3 - Join the results together, remove any duplicate pitch classes, put in ascending order

                        {2,3,6,7}  +  {5,6,B,0}  =  {B,0,2,3,5,6,7}
                  The result of step 3 is the product of  {2,3,6,7} * {2,5}


Let’s take this a step further. Because this method of pitch class multiplication involves ordering set X, we are able to get different results depending on the order of the multiplicand. For this we’ll need to take the ordered. It is important to remember that sets X and Y MUST BE IN NORMAL FORM, but beyond that they can be re-ordered for the process of multiplication. For this extension of the technique, concept of initially ordered pc set and initial pitch class (r) become important. Given sets X and Y, either can be the multiplicand and multiplier, but let’s stick with X as the multiplicand for now.


Example 4
                         Set X = {6,0,9,7}, in normal order is {6,7,9,0}
                         Set Y = {2,5}


We can use the normal order of the set for the multiplicand or rotate the set
In normal order, the value of r is 6, and the set is written as {6<7,9,0>}. When rotating and changing the value of r, the order of the pitch classes are also reordered. The following are the four permutations of Set X through rotating each pc to the value of r

Ordered sets:       6,<7,9,0>}    {7,<9,0,6>}    {9,<0,6,7>    {0,<6,7,9>}
         OIS:       {0,1,3,6}     {0,2,5,B}      {0,3,A,B}     {0,6,7,9}


Having these sets now allows us to create new products using set X as the multiplicand. If set Y remains the multiplier, we can use the OIS from each permutation of set X as the multiplicand and get the following:

    
Example 5
                                {6<7,9,0>} {2,5}
                                {6,<7,9,0>}   =  {0,1,3,6}
                                T2 {0,1,3,6}  =  {2,3,5,8}
                                T5 {0,1,3,6}  =  {5,6,8,B}
                                {B,2,3,5,6,8}

                                {7<9,0,6>} {2,5}
                                {7,<9,0,6>}   =  {0,2,5,B}
                                T2 {0,2,5,B}  =  {2,4,7,1}
                                T5 {0,2,5,B}  =  {5,7,A,4}
                                {A,1,2,4,5,7}

                                {9<0,6,7>} {2,5}
                                {9,<0,6,7>}   =  {0,3,A,B}
                                T2 {0,3,A,B}  =  {2,5,0,1}
                                T5 {0,3,A,B}  =  {5,8,3,4}
                                {0,1,2,3,4,5,8}

                                {0<6,7,9>} {2,5}
                                {0,<6,7,9>}   =  {0,6,7,9}
                                T2 {0,6,7,9}  =  {2,8,9,B}
                                T5 {0,6,7,9}  =  {5,B,0,2}
                                {8,9,B,0,2,5}


Additionally, we can also use set Y as the multiplicand and set X as the multiplier. However, in this instance set X must remain in normal form, and set Y must be converted into an OIS. The example below demonstrates this process


Example 6
Ordered Y Sets:  {2,<5>}    {5,<2>}
           OIS:  {0,3}        {0,9}

                                  {2,<5>} {6,7,9,0}
                                  {2,<5>}   =  {0,3}
                                  T6 {0,3}  =  {6,9}
                                  T7 {0,3}  =  {7,A}
                                  T9 {0,3}  =  {9,0}
                                  T0 {0,3}  =  {0,3}
                                  {6,7,9,0,3}

                                  {5,<2>} {6,7,9,0}
                                  {5,<2>}   = {0,9}
                                  T6 {0,9}  =  {6,3}
                                  T7 {0,9}  =  {7,4}
                                  T9 {0,9}  =  {9,6}
                                  T0 {0,9}  =  {0,9}
                                  {3,4,6,7,9,0}


At the end of the process we have a total of 6 new supersets from the result of multiplying two sets from one another. These can now be used as ordered sets for melodic writing, they can be used as unordered sets to determine harmonic structures. They could be used as unordered sets and be applied free to both melody and harmony to have a consistent pitch structure throughout a section. They could be used as blocked chords and with some working out one could find smooth voice leading to get from one chord to the next. If we were Boulez, we would probably establish some kind of localized ordering scheme and use each of these as unordered sets to be filtered through whatever ordering systems is applied to the overall piece, section of the piece, etc. The example below shows these chords in standard notation to show the sets in a musical context.

(Chord spelling used for visual assistance, to keep noteheads from stacking on top of one another; no ordering melodic or vertical ordering is implied by the voicing)

    While the above method uses simple math, it can be a little cumbersome and not always clear to see. Heinemann demonstrates another method for deriving supersets, that might be a preferred method for someone more used to looking at pitch matrices. The example below demonstrates how th matrix method works using the same pc sets and OISs:


Example 7
Ordered sets:     {6,<7,9,0>}    {7,<9,0,6>}    {9,<0,6,7>    {0,<6,7,9>}
         OIS:     {0,1,3,6}      {0,2,5,B}      {0,3,A,B}     {0,6,7,9}

                
                                       0    1    3    6
                                    2| 2    3    5    8
                                    5| 5    6    8    B
                                    Total = {B,2,3,5,6,8}

    
                                       0    2    5    B
                                    2| 2    4    7    1
                                    5| 5    7    A    4
                                    Total: {A,1,2,4,5,7}


                                       0    3    A    B
                                    2| 2    5    0     1
                                    5| 5    8    3     4
                                    Total: {0,1,2,3,4,5,8}


                                       0    6    7    9
                                    2| 2    8    9    B
                                    5| 5    B    0    2
                                    Total: {8,9,B,0,2,5}

​
As you can see, both methods yield the same results. It pretty much comes down to the method you prefer: equations or matrices, either way you’re doing basic addition and mod12.

The example below shows these used in a musical framework to demonstrate how I might go about using the results of the multiplication.

So that’s simple pitch multiplication. Hopefully this blog entry cleared up any confusion you had before, and if you’ve made it this far and still want to know moe I suggest you check out the Heinemann and Kablyakov articles below. There is plenty more to uncover with this technique and Heinemann and Kablyakov do it more justice than I can here. That said, I strongly encourage you to try this method. It doesn’t have to just be used for atonal collections, you could choose to utilize more consonant pitch collections, which can then be manipulated in various ways to sound more consonant when used melodically or harmonically. One could also utilize these for their ordered properties and filter out anything that doesn’t fit within a given superset. The possibilities are endless, and for that reason one could see why Boulez was so drawn to this technique.