Pitch Multiplication (2017)

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

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                                  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:
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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}

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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.