INTEGER PROGRAMMING FORMULATION OF THE PROBLEM OF GENERATING MILTON BABBITT’S ALL-PARTITION ARRAYS Tsubasa Tanaka, Brian Bemman, David Meredith To cite this version: Tsubasa Tanaka, Brian Bemman, David Meredith. INTEGER PROGRAMMING FORMULATION OF THE PROBLEM OF GENERATING MILTON BABBITT’S ALL-PARTITION ARRAYS. The 17th International Society for Music Information Retrieval Conference, Aug 2016, New York, United States. hal-01467883 HAL Id: hal-01467883 https://hal.archives-ouvertes.fr/hal-01467883 Submitted on 14 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. INTEGER PROGRAMMING FORMULATION OF THE PROBLEM OF GENERATING MILTON BABBITT’S ALL-PARTITION ARRAYS Tsubasa Tanaka Brian Bemman David Meredith STMS Lab : IRCAM, CNRS, UPMC Aalborg University Aalborg University Paris, France Aalborg, Denmark Aalborg, Denmark [email protected] [email protected] [email protected] ABSTRACT cur is constrained. It turns out that this is a hard combina- torial problem. That this problem was solved by Babbitt Milton Babbitt (1916–2011) was a composer of twelve- and one of his students, David Smalley, without the use tone serial music noted for creating the all-partition array. of a computer is therefore interesting in itself. Moreover, it The problem of generating an all-partition array involves suggests that there exists an effective procedure for solving finding a rectangular array of pitch-class integers that can the problem. be partitioned into regions, each of which represents a dis- Construction of an all-partition array begins with an tinct integer partition of 12. Integer programming (IP) has I × J matrix, A, of pitch-classes, 0; 1;:::; 11, where each proven to be effective for solving such combinatorial prob- row contains J=12 twelve-tone rows. In this paper, we only lems, however, it has never before been applied to the prob- consider matrices where I = 6 and J = 96, as matri- lem addressed in this paper. We introduce a new way of ces of this size figure prominently in Babbitt’s music [13]. viewing this problem as one in which restricted overlaps This results in a 6 × 96 matrix of pitch classes, contain- between integer partition regions are allowed. This permits ing 48 twelve-tone rows. In other words, A will contain an us to describe the problem using a set of linear constraints approximately uniform distribution of 48 occurrences of necessary for IP. In particular, we show that this problem each of the integers from 0 to 11. On the musical surface, can be defined as a special case of the well-known prob- rows of this matrix become expressed as ‘musical voices’, lem of set-covering (SCP), modified with additional con- typically distinguished from one another by instrumental straints. Due to the difficulty of the problem, we have yet register [13]. A complete all-partition array is a matrix, to discover a solution. However, we assess the potential A, partitioned into K regions, each of which must contain practicality of our method by running it on smaller similar each of the 12 pitch classes exactly once. Moreover, each problems. of these regions must have a distinct “shape”, determined by a distinct integer partition of 12 (e.g., 2 + 2 + 2 + 3 + 3 1. INTRODUCTION or 1+2+3+1+2+3) that contains I or fewer summands greater than zero [7]. We denote an integer partition of an Milton Babbitt (1916–2011) was a composer of twelve- integer, L, by IntPart (s ; s ; : : : ; s ) and define it to be tone serial music noted for developing complex and highly L 1 2 I an ordered set of non-negative integers, hs ; s ; : : : ; s i, constrained music. The structures of many of his pieces 1 2 I L = PI s s ≥ s ≥ · · · ≥ s are governed by a structure known as the all-partition ar- where i=1 i and 1 2 I . For exam- ray, which consists of a rectangular array of pitch-class ple, possible integer partitions of 12 when I = 6, include integers, partitioned into regions of distinct “shapes”, each IntPart12(3; 3; 2; 2; 1; 1) and IntPart12(3; 3; 3; 3; 0; 0). corresponding to a distinct integer partition of 12. This We define an integer composition of a positive integer, structure helped Babbitt to achieve maximal diversity in L, denoted by IntCompL(s1; s2; : : : ; sI ), to also be an his works, that is, the presentation of as many musical pa- ordered set of I non-negative integers, hs1; s2; : : : ; sI i, PI rameters in as many different variants as possible [13]. where L = i=1 si, however, unlike an integer partition, In this paper, we formalize the problem of generating an the summands are not constrained to being in descending all-partition array using an integer programming paradigm order of size. For example, if L = 12 and I = 6, then in which a solution requires solving a special case of the IntComp12(3; 3; 3; 3; 0; 0) and IntComp12(3; 0; 3; 3; 3; 0) set-covering problem (SCP), where the subsets in the cover are two distinct integer compositions of 12 defining the are allowed a restricted number of overlaps with one an- same integer partition, namely IntPart12(3; 3; 3; 3; 0; 0). other and where the ways in which these overlaps can oc- Figure 1 shows a 6×12 excerpt from a 6×96 pitch-class matrix, A, and a region determined by the integer composi- tion, IntComp12(3; 2; 1; 3; 1; 2), containing each possible c Tsubasa Tanaka, Brian Bemman, David Meredith. Li- pitch class exactly once. Note, in Figure 1, that each sum- censed under a Creative Commons Attribution 4.0 International License mand (from left to right) in IntComp (3; 2; 1; 3; 1; 2), (CC BY 4.0). Attribution: Tsubasa Tanaka, Brian Bemman, David 12 Meredith. “Integer Programming Formulation of the Problem of Gener- gives the number of elements in the corresponding row of ating Milton Babbitt’s All-partition Arrays”, 17th International Society the matrix (from top to bottom) in the region determined for Music Information Retrieval Conference, 2016. by the integer composition. We call this part of a region Figure 2, the three horizontal insertions of pitch-class inte- gers, 3 (in row 1), 7 (in row 2), and 10 (in row 4), required to have each pitch class occur exactly once in the second integer partition region. Not all of the 58 integer partitions must contain one or more of these insertions, however, the total number of insertions must equal the 120 additional Figure 1:A 6×12 excerpt from a 6×96 pitch-class matrix pitch classes required to satisfy the constraint that all 58 integer partitions are represented. Note that, in order for with the integer composition, IntComp12(3; 2; 1; 3; 1; 2) (in dark gray), containing each pitch class exactly once. each of the resulting integer partition regions to contain every pitch class exactly once, ten occurrences of each of the 12 pitch classes must be inserted into the matrix. This typically results in the resulting matrix being irregular (i.e., “ragged” along its right side). In this paper, we address the problem of generating an all-partition array by formulating it as a set of linear con- straints using the integer programming (IP) paradigm. In section 2, we review previous work on general IP problems Figure 2:A 6 × 12 excerpt from a 6 × 96 pitch-class and their use in the generation of musical structures. We matrix with a region whose shape is determined by the also review previous work on the problem of generating integer composition, IntComp12(3; 3; 3; 3; 0; 0) (in light all-partition arrays. In section 3, we introduce a way of gray), where three elements (in bold) are horizontal inser- viewing insertions of elements into the all-partition array tions of pitch classes from the previous integer partition as fixed locations in which overlaps occur between con- region. Note that the two indicated regions represent dis- tiguous integer partition regions. In this way, our matrix tinct integer partitions. remains regular and we can define the problem as a special case of the well-known IP problem of set-covering (SCP), modified so that certain overlaps are allowed between the in a given row of the matrix a summand segment. For subsets. In sections 4 and 5, we present our IP formula- example, in Figure 1, the summand segment in the first tion of this problem as a set of linear constraints. Due to row for the indicated integer partition region contains the the difficulty of the problem, we have yet to discover a so- pitch classes 11, 4 and 3. On the musical surface, the dis- lution using our formulation. Nevertheless, in section 6, tinct shape of each integer composition helps contribute we present the results of using our implementation to find to a progression of ‘musical voices’ that vary in textural solutions to smaller versions of the problem and in this density, allowing for relatively thick textures in, for ex- way explore the practicality of our proposed method. We ample, IntComp12(2; 2; 2; 2; 2; 2) (with six participating conclude in section 7 by mentioning possible extensions to parts) and comparatively sparse textures in, for example, our formulation that could potentially allow it to solve the IntComp12(11; 0; 1; 0; 0; 0) (with two participating parts).