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| 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 org… |
| 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 aboveThe notated result would look like thisExample 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 numbersExample 2The 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 rOrdered 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 processExample 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. 4/15/2020 11:13:40 pm Thanks for going to the effort of writing this all out, cool stuff! Reply Jon Fielder 4/15/2020 11:16:25 pm Hey! Thanks for reading. I had a math error in there at some point but I think I fixed it... Feel free to reach out if you have any questions, comments, etc. Reply JazzWarrior 1/21/2021 12:58:51 am Hi Jon - From one theorist to another : thanks for your post here - a nice simplification of Heinemann’s thesis work, and contextualisation of Slonimsky’s famous thesaurus of scales and patterns. It’s good to see more discussion of this concept. However, I’d like to offer some corrections on a couple of things, for the sake of accuracy/clarity, and for ‘pooling’ knowledge. As you mentioned in your above reply to Samuel Killick, you have made a calculation error, but I don’t see that it has been ‘fixed’ yet. Example 3, step 2 gives the following result for (Set X @ Set Y) : T5 {0,1,4,5} = {5,6,B,0} Presuming that ‘B’ = 11 in your notation system, and knowing that ‘0’ equals either ‘0’ or ‘12’, the above is incorrect, and should read: T5 {0,1,4,5} = {5,6,9,A*} [ * presuming ‘A’ = 10 in your notation system] This error is worth fixing because it is early in your article, and could lead to confusion for many readers who are not already a bit ‘hip’ to this stuff. [I have not ‘cased’ the whole article for further errors, but you might want to double-check]. Further on this point, I truly believe that notation systems which mix numerals and alphabet characters create unnecessary confusion because they are inconsistent sets. I understand that the rationale behind using letters such as ‘t’ & ‘e’ (or in your case, ‘A’ & ‘B’) as substitutes for the integers ‘10’ and ‘11’, respectively, is to avoid the potential visual confusion of mixing single numeral integers with double-numeral integers. However, I feel that this is a trivial concern compared to the conceptual and mathematical confusion which can arise from mixing letters with numbers (which may have even contributed to the above miscalculation). This is particularly so where a legend is not offered at the outset of any series of examples, or included in any discussion about them, which would clarify the details of representation for the reader (as is the case with your article). This is especially problematic when offering information to people unfamiliar with these variations of what is specifically known as, “Integer Notation”. The easiest way to avoid these opportunities for confusion is to avoid mixing character substitutes for integers with actual integers, and to just offer a consistent system of integer presentation, reserving letter characters exclusively for algebraic representation of operations on the pitch-class sets and their elements. This also avoids cross-confusion with characters employed in any associated algebraic operations, which may be the lower or upper case counterpart of the ‘integer substitutes’ (as with ‘T’ for ‘transposition’ and ‘t’ for ‘ten/10’). If integers are consistently presented with commas between them, (as in : {5,6,9,10}; or {0,11,10,4}) what really is the problem with reading the numbers ‘10’ or ‘11’? Please give this some consideration (it’s an issue I feel compelled to help resolve in striving for consistent communication of theory). Lastly, a historical correction : Slonimsky’s work was categorically NOT the “earliest recorded example of pitch multiplication”. That honor falls to Swiss mathematician and music theorist Leonhard Euler (of ‘Euler’s number’ fame), and his ‘Tonnetz’ (German for ‘tone-network’) lattice diagram, which was published 1st January 1739, in the treatise, “Tentamen Novae Theoriae Musicae”. The ‘Tonnetz’ is an example of what has been called chord/pitch-set ‘squaring’, a basic form of pitch multiplication whereby the intervallic makeup of a harmonic entity is reproduced upon its constituent pitches as transpositions of the original shape. In other words, the pitch-set is multiplied by itself. This also produces a larger set of unique tones. In the case of the ‘Tonnetz’, an initial root position major triad, gives birth to two more (root position) major triad transpositions, generated at its 3rd and 5th respectively. Euler’s original diagram, although basic compared with modern Tonnetz designs, clearly demonstrates the ‘squaring’ concept, showing how an F major triad generates an A major triad and a C major triad, and how they in turn generate further triads by the same process, and so on. Each ‘squaring’ of a triad gives a total of three discrete triads, and a total set of six unique tones: Intervals {M3,m3} @ {C,E,G} = {C,E,G} + {E,G#,B} + {G,B,D} Total unique pitches = {C,D,E,G,G#,B}; [ or: {C Maj. 9 (add #12)} ] Euler did not conceive the Tonnetz as a tool for creating pitch-class sets, however, but as a tool for visualising and conceptualising the relationships between triads and key centres, within the context of major/minor tonality, as it was being defined in the 18th century. He did not use any terminology such as ‘pitch-multiplication’ or ‘squaring’, but as a music theorist he conveyed the idea clearly, and as a ground-breaking mathematician, he would have recognised the mathematical parallels. Incidentally, I may be wrong, but I do not recall Slonimsky’s thesaurus d Reply Jon 1/21/2021 09:12:44 am Hello! Thanks for the detailed and thoughtful comment and for pointing out that I had not corrected that error. I thought I’d changed it at the time I made that last comment in April of 2020 but it looks like I never saved the updates. It’s all good now, and a deeper perusal didn’t return any other calculation errors that I could see. Regarding the use of A and B (or t and e in other systems), I don’t really have a problem using that method, mostly because one of those two systems is taught in essentially all post-tonal music theory classes (collegiate or otherwise) in the United States and in music theory textbooks. While I also don’t always care for the alphabetical representations of pitches, I think it is the best option for when commas are not used - prime form, for example. I understand not caring for it, but using A and B or T and E is, at least for now, much more prevalent than using numeric 10 and 11. For the sake of consistency and cross-referencing I’ll stick with what I have. I’ve read extensively about Euler, Tonnetz, and further writing by Lewin, Hyer, and Dmitri Tymoczko’s further writing on the subject. I think it’s a very interesting way of looking and the interrelatedness of triadic structures, but I’m hesitant to include it here as a historical precursor of pitch class multiplication, because it simply isn’t. I included Slonimsky because his method is a combination of subsets to create an ordered superset. Tonnetz deals with multiplication in terms of how pitches are related by frequency and ratio to a fundamental, not as a means of establishing pitch class sets. For example, moving “right” on a Tonnetz lattice represents multiplication by 3, or up by perfect 5th. This is because the relationship of 3x a fundamental frequency is an octave+5th higher than a fundamental in the overtones series. Moving left on the Tonnetz is multiplication by 5, or a major 3rd. The same applies, in that 5x a fundamental is 2 octaves and a major 3rd above the fundamental. Like you mentioned in your comment, this is not pitch multiplication in the same sense that I’m outlining in my post, and I think it’s very important for the sake of clarity to not merge the two. I’m not making the claim that this is the first example of multiplication or mathematics in any form being used in music theory, only that Slonimsky is the first example pitch multiplication as it is outlined here - as a means of generating pitch class sets. Reply JazzWarrior 2/13/2021 12:39:47 am G’day Jon, Thanks for responding to my comment on this post - being the first comment I have ever posted on a music blog! (@ 50 y/o mind you!) - nice to have a dialogue with you. I actually had more to the comment, but evidently exceeded the available word/space limit, so I’ll try to keep it shorter! [Incidentally I did try to add another comment several times but to no avail - hopefully this time will work]. I’m glad you got the correction sorted, since this is a very useful article for people new to the concept. What you have said in response to my criticisms of the mix of letter symbols with numbers in integer notation is perfectly reasonable - we each have our perspectives and practical purposes to nurture, and your rationale is totally fair. Re Euler & the Tonnetz, I do agree with what you have put forth, in the sense that Euler was not using ‘pitch-class multiplication’ deliberately as a compositional tool for creating hybrid pitch sets. However, while I understand the kind of ‘multiplication’ you are discussing in regard to frequencies etc (and the original tonnetz being an infinite expansion of calculations using harmonic ratios), I cannot agree that the Tonnetz does not constitute an example of ‘simple multiplication’, since it is categorically an example of ‘chord (pitch set) squaring’ - ie. the interval (class) set by itself, upon the pitches of a given transposition. Was it designed as an example of p.c. set multiplication? - No. Is it nevertheless an example of a fundamental p.c. set multiplication operation? - Yes. That is why I did assert that it is a prior case of the phenomenon to Slonimsky. However, you are right about Slonimsky if we are talking about a conscious effort to make multiplication ‘a thing’. Howard Hanson’s “Harmonic Materials of Modern Music” (1960) does in fact utilize ‘squaring’ triads to create larger pitch sets (See Ch II - III), only he calls the process ‘projection’. He uses the ‘projection’ of the major triad unit to create 3 unique pentads, and an ultimate hexad. He also demonstrates projections of other ‘triad’ types. The major triad projection is no different to the first two (and all subsequent) branches in Euler’s Tonnetz. Hanson’s extension of “complementary ‘involution’ [inversion]” is equivalent to squaring the minor triad, and, if continued, results in the same lattice of minor triads as can be derived from the Tonnetz (in ‘reverse’). A few years ago I began using this concept as a harmonic basis to musical material incorporating serialism (not 12-tone) and algorithmic prescriptive processes. I began with the major triad as an experiment. That experiment turned me on to the Tonnetz and Boulezian multiplications. However, it is mainly pitch-set ‘squaring’ which I have pursued, and I haven’t looked back yet! I call my method ‘Matrix Harmony’ and have written a uni thesis which discusses that. I use mirrored ‘squarings’ of various chords/scales/p.c. sets to create harmonic and melodic material. If you are interested in what I’m up to with that, please feel free to contact me on my email (presuming you can access that as an admin here - if not, reply to me here & I’ll get in touch). Thanks again for the article and the opportunity to discuss music together. All the best, Jon. Regards, Kyle Watson (AKA the ‘jazz warrior’). 8/2/2022 09:13:50 am Sitemize giriş yap ve tiktok düşmeyen türk takipçi satın al. Tiktok ucuz türk takipçi satın almak hiç bu kadar kolay ve zahmetsiz olmamıştı. Web sitemizden tiktok düşmeyen türk takipçi al ve hesabının keyfini çıkart. Reply Scott Lindroth 9/15/2024 04:06:03 pm The errors pointed out by earlier commenters have not been fixed. They are still present in Sept. 2024! Reply ## Leave a Reply. | The “Direct Sound” Page is dedicated to general blog posts and discussions. Various topics are covered here. |