The Area Set Value Relationships in Atonal Music

Year: 2018 Authors: Nikita Mamedov

Core claim

ASV reveals mathematical relationships among unrelated atonal set-classes and produces consistent patterns in Schoenberg’s Suite for Piano, Op. 25, and String Quartet No. 4, Op. 37.

Topics

atonal music analysis, pitch-class set theory, maximally even sets, twelve-tone technique

Domains

geometry, modular arithmetic, graph theory, trigonometry, music theory, composition analysis, performance interpretation

Methods

set-class classification, polygon area calculation, unit circle mapping, tone-row segmentation

Media

pitch-class sets, Schoenberg scores, Cartesian plane, unit circle

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2018 Conference Proceedings

The Area Set Value Relationships in Atonal Music

Nikita Mamedov

Department of Music, Louisiana State University, Baton Rouge; nmamed1@lsu.edu

Abstract

The Area Set Value analysis is based on works of Rappaport (2005 and 2007), allowing one to understand the mathematical connections that exist between set-classes in atonal music. While traditional music theory distinguishes between 216 prime forms in sets with cardinality of 3 or larger, the ASV analysis allows for 48 values. This technique groups unrelated prime forms according to mathematical properties. The application of ASV is employed in Arnold Schoenberg’s Suite for Piano, Op. 25, and String Quartet No. 4, Op. 37.

The ASV Groups

Using mathematics in music analysis allows one to examine a variety of repertoire. For instance, set theory uses graph theory, categorizes musical pitch-class sets, and defines their relationships in non-tonal context, while Neo-Riemannian analysis applies the notion of group theory to model intervals and chords [8]. This paper, based on the research of Rappaport (2005 and 2007), combines set and Neo-Riemannian analyses and examines the Area Set Values (ASVs), which are denoted with geometric representations on the unit circle in the Cartesian plane $({\mathbb{R}}^2)$ with a center at the origin and with radius of 1 [6]. The ASVs define chords of any cardinality as convex polygons. In his works, Rappaport uses integer partitions to generate scales [5]. As in Rappaport’s research, the analysis in this paper will map pitch-classes onto polygons constructed from points equally spaced on the unit circle and calculate the area of the constructed polygons. Unlike Rappaport’s research, this study groups all sets into unique categories and applies them to the musical analysis of Schoenberg’s 12-tone technique.

The ASV employs integer notation, where 0 represents pitch-class C, 1 represents pitch-class $\mathrm{C}\#$ or $\mathrm{D}\&gt{;}_{\mathrm{b}}$ , and so on. Likewise, the ASV incorporates enharmonic $(\mathrm{F}\#=\mathrm{G}\&gt{;}_{\mathrm{b}})$ and octave equivalences, which allow for cyclic representation of the pitch-classes in a closed musical cycle [4]. Therefore, the notion of modular arithmetic is applied, where 12 maps onto 0, 13 maps onto 1, and so on. There are likewise three nomenclatures (prime form, forte-number, and interval-class vector) under which the ASV analysis operates. Prime form is the most compact way to define a set that always begins with integer 0. For example, set $\{456\}$ has a prime form of (012). Each prime form is defined by a forte-number $(x-y)$ , where $x$ represents the cardinality of the set and $y$ represents the index of a certain prime form among all prime forms of a given cardinality. For instance, the forte-number of all major and minor chords in musical pitch-space (037) is 3-11. Finally, each forte-number contains a unique interval-class vector $(icv)$ - the intervallic make up of a pitch-class set, such as $<x_1$x_2="" $x_3=""$x_4="" $x_5=""$x_6="" $x_7$="" $x_8=""$x_9$=""$x_{1}$=""$x_{2}$=""$x_{3}$=""$x_{4}="" (012).="" -="" 11.="" 12,="" 12.="" 13,="" 14.="" 2k\pi="" 3-11.="" 4.="" 5.="" a="" and="" any="" are="" as="" at="" be="" better="" by="" calculations="" calculations="" class="" components="" connected,="" connected.="" components,="" components.="" connected,="" corresponding="" defined="" different="" distribution="" e}="" each="" equal="" every="" except="" for="" fortnord="" function="" functions="" general="" generated="" has="" have="" however,="" if="" in="" is="" it="" its="" j="" j),="" k$=""lsu.edu=""m=""m$="" m$<=""m$⇐"" m$<=""m$).="" m$<sup=""m$).="" m$<sup=""m$<sup="" n\leq="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le$="" n\leq$=""n\le 12.=""12.1)=""12.1).=""12.1).=""13.=""14.=""15.=""16.=""17.=""2.=""2.1=""2.1.=""3=""3-11.=""4.=""407=""5.=""a=""aav=""aavs.=""all=""also=""analysis=""analysis,=""and=""any=""are=""area=""area.=""as=""atonal=""b=""b)$="" b)$),=""b)$).="" b)$<sup=""b)$<sup="" b)$<sup=""b)$<sup="" b)$<sup=""b)$<sup="" b)$<sup=""b)$<sup="" b)$<sup=""b)$<sup="" b)$<sup=""b)$<sup="" b)$<sup=""b)$<sup="" b)$<sup=""b)$</x_1>

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Figure 1: The twelve pitch-class locations, ASV of 3-1, and ASV of 12-1.

Table 1 provides the ASV analysis for chords with cardinality between 3 and 12, incorporating a total of 216 forte-numbers. These include 12 trichords, 29 tetrachords, 38 pentachords, 50 hexachords, 38 heptachords, 29 octachords, 12 nonachords, 6 decachords, 1 undecachord, and 1 dodecachord.

Table 1: The ASV data for trichords through dodecachord.

The general trend of decreasing number of unique areas can be observed, as the cardinality of sets increases (with the exceptions of tetrachords). Also, as the cardinality increases, the smallest, largest, and average areas of each cardinality likewise increase, which leads to the notion of maximally even distributions that allow for the equal division of an octave, based on the interval content [3]. Examples of maximally even sets are tritones (2 tritones = 1 octave), augmented triads (3 major thirds = 1 octave), diminished seventh chords (4 minor thirds = 1 octave), and whole tone collections (6 major seconds = 1 octave). Furthermore, within each cardinality, the chromatic sets (3-1, 4-1, etc.) have the smallest ASVs, while the maximally even distributions (3-12, 4-28, etc.) have the largest ASVs. For each cardinality, the largest ASV comes from the polygon which is “closest” in total area to a regular polygon of the given order. Also, the “further” from a regular polygon the generated polygon is (based on its area), the smaller the ASV is, so that the smallest ASV is associated with the “least regular” of the polygons, which is the reason for $y=1$ in forte numbers of smallest ASV sets in each cardinality group. Table 2 presents detailed analysis of all trichords, including its prime forms, $icv$ , interval arrays (the directed interval distance between elements of the corresponding prime form), ASVs, as well as side lengths and angle values [2]. Three distinct observations can be made. First, the ASV increases as the forte number increases, except for 3-6, which is the subset of a whole tone collection. Second, the angles correspond to the values in the

The Area Set Value Relationships in Atonal Music

interval array and move in the increments of $15^{\circ }$ , where an interval array of 1 represents $15^{\circ }$ , an interval array of 2 represents $30^{\circ }$ , and so on. Third, while different in its prime form and $icv$ , 3-4 and 3-6 share identical ASVs. Furthermore, the uniqueness property of ASV analysis allows one to group two or more forte-numbers based on the mathematical area of their respective shapes. This demonstrates that the cardinality and the interval content does not reflect the area of the polygon that is being constructed. Therefore, while the total number of areas is tantamount to total chords analyzed, the number of unique areas is not and certain forte-numbers will generate identical areas. 37 of 48 ASVs contain more than 1 forte-number. The most common ASV is 2.366, occurring in 6-27, 6-Z28, 6-Z29, 6-30, 6-31, 6-Z45, 6-Z46, 6-Z47, 6-Z48, 6-Z49, 6-Z50, 8-2, 8-4, 8-5, and 8-6.

Table 2: The ASV analysis of trichords

Application to Schoenberg

It is possible to interconnect the mathematical and musical theories behind the ways composers use sets in serial music. One of such connections can be seen in Schoenberg’s use of 12-tone approach - a compositional technique that Schoenberg establishes to ensure the presentation of all 12 pitch-classes before any of them can undergo repetition [1]. An example, shown in Figure 2, can be seen in the second movement of Schoenberg’s Suite for Piano, Op. 25, where the composer introduces the initial tone row (row $x$ ) in the third measure: {4-5-7-1-11-0-9-10-6-3-8-2}.

Figure 2: Schoenberg, Suite for Piano, Op. 25, movement II, mm. 1-3.

Two important subsets are derived from row $x$ . The first four pitches of row $x$ ( $x_1$ ) are $\{4,5,7,1\}$ that belong to the set 4-12 with prime form of (0236). The third and the fourth elements of $x_1$ (G and $D_b$ ) generate an interval of a tritone – the most unstable interval in Western music. The next set extracted from row $x$ is 4-Z15 with prime form of (0146). This set initializes from the third and the fourth elements of $x_1$ and additionally includes the successive pitch-classes 6 and 3 in the upper staff, played by the right hand on the piano. Therefore, $x_2=\{7,1,6,3\}$ . Finally, a new subset can be extracted by amalgamating the

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third and the fourth elements of $x_2$ with successive pitch-classes 8 and 2 in the upper staff (the final two notes A $♭$ and D played by the right hand). Therefore, $x_3=\{6,3,8,2\}$ . The generation of one 4-12 and two 4-Z15 sets out of row $x$ is not a coincidence and while $x_2$ begins with the tritone, $x_1$ and $x_3$ end on one. Both 4-12 and 4-Z15 have identical ASV of 1.183. The first movement of Arnold Schoenberg’s String Quartet No. 4 presents another example of ASV connection to the twelve-tone technique. Schoenberg uses the tone row {2-1-9-10-5-3-4-0-8-7-6-11} in the opening six measures of Op. 37 (row $y$ ), shown in Figure 3 [9]. Sets in this work have discrete properties and appear in the tone row on more than one occasion. 3-4 (015) occurs in row $y$ twice as {2,1,9} and {7,6,11}. 4-4 (0125) occurs in row $y$ twice as {5,3,4,0} and {8,7,6,11}. The third tetrachord of row $y$ is 4-7 (0145). While the notion of discreet tetrachords is evident, the ASV analysis demonstrates that the tetrachordal segmentation of row $y$ generates three separate tetrachords with identical ASV value, as sets 4-4 and 4-7 have the ASV of 0.75.

Figure 3: Schoenberg, String Quartet No. 4, Op. 37, movement I, mm. 1-6.

The ASV can be used by music theorists, mathematicians, and performers. For theorists and mathematicians, the ASV allows for an alternate arrangement of pitch-class sets into 48 unique categories and generates significant patterns that can be related to atonal music. Outside of set and transformational theories, the ASV analysis incorporates trigonometric properties that allow one to understand how the intervallic content of each set reflects the side lengths, the angles, and the total areas of the polygons. More importantly, the ASV explains the similarities that exist in mathematical properties of two or more sets that are unrelated, based on their intervallic content. For performers, the ASV can be used to generate and shape their own musical interpretation. An identical ASV in two or more pitch-class sets is a unique property and can be emphasized musically with dynamics, tempo, and articulation.

References

[1] J. Boss. “Schoenberg on Ornamentation and Structural Levels.” Journal of Music Theory, vol. 38, no. 2, 1994, pp. 187–216. [2] R. Chrisman. “Describing Structural Aspects of Pitch-Sets Using Successive-interval Arrays.” Journal of Music Theory, vol. 21, no. 1, 1977, pp. 1-28. [3] J. Clough and J. Douthett. “Maximally Even Sets.” Journal of Music Theory, vol. 35, no. 1, 1991, pp. 93–173. [4] A. Popoff. Towards A Categorical Approach of Transformational Music Theory. 2012. [5] D. Rappaport. “Geonetry and Harmony.” Bridges Conference Proceedings, Alberta, Canada, 2005, pp. 67–72. [6] D. Rappaport. “Maximal Area Sets and Harmony.” Akiyama-Chvatal Festschrift, Graphs and Combinatorics, vol. 23, no. 1, 2007, pp. 321–329. [7] S. Rings. Tonality and Transformation. Oxford University Press, 2011. [8] M. Russ. “Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts.” Music and Letters, vol. 90, no. 4, 2009, pp. 722–725. [9] J. N. Straus. Introduction to Post-Tonal Theory. Pearson Prentice Hall, 2005. [10] R. Sturman. “A Dynamical Systems Approach to Musical Tuning.” Dynamical Systems, vol. 27, no. 1, 2012, pp. 131–143.