PC Set Calculator A Brief Introduction to Atonal Music Theory for Scientists and Mathematicians Precomposition in Stravinsky’s 12-tone Music Invariances in the 16 Rotational Arrays from Requiem Canticles Boulez, Structures 1a Hypercube A Brief Introduction to Atonal Music Theory for Scientists and Mathematicians Paul Lombardi A note produced from an acoustic instrument has a rich series of overtones. For the duration of this introduction, we will ignore the overtones and only consider the fundamental pitch. Pitch has a frequency that is measured in hertz. Two pitches sounded simultaneously or played in succession produce an interval. The ratio of the two frequencies in an interval determines whether the interval is consonant or dissonant. Consonant intervals sound harmonious while dissonant intervals sound harsh. Consider the interval from two pitches that have a frequency ratio of 2:1 as shown in Figure 1. When those two pitches are sounded simultaneously, we hear their sum. Some other intervals are shown in Figure 2, all of which are periodic. As the resulting wave from the sum of the two pitches becomes more complex, the interval becomes more dissonant. The 2:1 interval is so consonant that we perceive the two pitches as the same note, and it is called an octave. In adjacent octaves, harmonic systems are related geometrically. For example, if the note A has a frequency of 440 Hz, the A an octave higher has a frequency of 880 Hz and the A an octave lower has a frequency of 220 Hz. In modern Western Art Music, the octave is partitioned into twelve notes that aurally sound equidistant from each other, which we call equal temperament. A harmonic system of twelve divisions of the octave developed early in the history of Western Art Music as a result of composers’ preference for consonance. If a pitch has a frequency of υ in equal temperament, the next highest pitch has a frequency of . The interval of is called a semitone. Consider the interval shown in Figure 3 that spans 7 semitones. The interval of 7 semitones in nearly equal to the ratio of 3:2. The interval of 3:2 is perfectly in tune, while the interval of is slightly dissonant. Pythagoras said that musical ratios were the same as the distances between the planets, all combined into a beautiful harmony that he called Music of the Spheres. Plato further elaborated on this concept where he said that the difference between the musical intervals 3:2 and is analogous to the orbits of the planets in that all of the planets are never simultaneously in the same places. Plato knew that as intervals of 3:2 are stacked to preserve consonance, the harmonic system abruptly loses the equal temperament quantization.[1] Music early in the history of Western Art Music used tunings that relied on consonant ratios such as 3:2, 4:3 and 5:4. As music incorporated more dissonances and modulations, a system of equal temperament became necessary.[2] In a temperament system based on consonant ratios, some of the intervals must be out of tune to compensate for other notes to be in tune. For the remainder of this introduction, we will assume equal temperament. With the advent of atonal music at the beginning of the Twentieth Century, composers drastically changed their thinking about pitch. This new thinking was soon coined as octave equivalence, which means that a pitch in any octave has the same function, or a group of pitches arranged in any order in any octave has the same function. To simplify the understanding of atonal music, theorists use integers to refer to pitches. In this introduction, we refer to the twelve pitches of an octave with the integers 0-9 and A and B. The integers as they correspond to the pitches on a keyboard are shown in Figure 4. A pitch class or PC refers to a pitch without regards to which octave it’s in. For example, the PC 0 refers to any or all Cs regardless of which octave they’re in. Pitch is linear, but since the equal tempered harmonic system repeats every octave, a PC is cyclical; so we can understand PC in mod 12 space like the face of a clock as shown in Figure 5. The picture in Figure 5 is called a pitch-class wheel. In this harmonic system, intervals are the differences between PCs. Assuming octave equivalence, there are two ways of examining intervals. An ordered pitch-class interval is always computed in a positive clockwise direction around a PC wheel. Some examples of ordered pitch-class intervals are shown in Figure 6. (6 + 3)mod 12 = 9 (2 + 9)mod 12 = B (A + 6)mod 12 = 4 An interval class, or IC, is always the shortest distance between 2 PCs regardless of the direction around the PC wheel. Some examples of ICs are shown in Figure 7. (6 + 3)mod 12 = 9 (B + 3)mod 12 = 2 (A ± 6)mod 12 = 4 (9 - 3)mod 12 = 6 (2 - 3)mod 12 = B A set of 3 or more PCs is called a pitch-class set. Consider the PC set 236A. Here, the 4 PCs are sorted in increasing order, which is called ascending form. An interval vector counts all of the interval classes between the pairs of PCs in a set, as shown in Figure 8. The interval vector for 236A is <101310> because there is one IC 1, zero IC 2, one IC 3, three IC 4, one IC 5, and zero IC 6 between the pairs of PCs in the set. 236A: (3-2)mod 12=1 (6-2)mod 12=4 (2-A)mod 12=4 1 2 3 4 5 6 (6-3)mod 12=3 (3-A)mod 12=5 < 1 0 1 3 1 0 > (A-6)mod 12=4 PC sets can be transposed by adding a constant integer mod 12 to each PC. 2368 is transposed as shown by the equations in Figure 9, where T indicates transposition and the subscript indicates the distance of transposition. T2(2368) = 458A T7(2368) = 9A13 = 139A The first line shows a transposition by 2 semitones, and the second line shows a transposition by 7 semitones. After each transposition, the PCs are resorted into ascending form. PC sets can be inverted by reflecting the PCs around an axis as shown in Figure 10. I(2368) = A964 = 469A Here, I indicates an axis that passes through PCs 0 and 6 in the PC wheel. 2368 inverted around this axis gives the PC set 469A, where 2 inverts to A, 3 inverts to 9, 6 inverts to 6, and 8 inverts to 4, and then the resulting set is sorted into ascending form. Notice that the pairs of PCs are complementary in base 12. PC sets can be inverted around other axes by combining inversion with transposition. 2368 inverted around the axis that passes through 1 and 7 is given by the equation shown in Figure 11. T2I(2368) = 0B86 = 068B> Here, the PC set is transposed by 2 after being inverted around the 0/6 axis. Notice that the inverted set has 2 PCs in common with the original set. Figure 12 shows the TnI transformations for all axes of inversion. Notice that the axis of inversion may be between PCs rather than passing through them. PC sets are symmetrical when they have all common PCs under some Tn or TnI transformation. An example of a common PC is illustrated by the PC sets 247 and 348 that have PC 4 in common. There are two types of symmetry: inversional and transpositional. Inversionally symmetrical PC sets have all common PCs at some TnI transformation. For example, 2457 is symmetrical around the axis that passes between 4 and 5 as shown in Figure 13. T9I(2457) = 7542 = 2457 T2I(159) = 195 = 159 T6I(159) = 519 = 159 TAI(159) = 951 = 159 Also shown in Figure 13, some PC sets, such as 159, have more than one axis of symmetry. PC sets are transpositionally symmetrical if they have all common PCs at some transposition level other than T0. 159, shown in Figure 14, is transpositionally symmetrical because it has all common tones at T4 and T8. T0(159) = 159 T4(159) = 591 = 159 T8(159) = 915 = 159 The ascending form of PC sets can be rotated so that any of their PCs can be listed first. The three rotations of 23B are 23B, 3B2, and B23 as shown in Figure 15. 23B 3B2 [B23] (2 + 9)mod 12 = B (3 + B)mod 12 = 2 (B + 4)mod 12 = 3 In each of the rotations, the PCs remain in a clockwise order around the PC wheel. For each of these rotations, the interval between the first and last PCs is 9, B and 4 respectively. B23 has the smallest interval between its first and last PCs, so B23 is the most compact rotation of the set. Therefore, B23 is in normal form. When two rotations have the same interval between their first and last PCs, the smaller interval between the first and second-to-last PCs is used to determine normal form. For any PC set, the most compact rotation of the ascending form is called normal form. PC sets are given in square brackets when they are in normal form. A set class, or SC, includes all transpositions and inversions of a PC set. Figure 16 shows all of the transpositions and inversions for the PC set [013], all of which are members of the same SC. The 24 sets in this SC are all listed in normal form. SC 3-2 (013): T0[013] = [013] T6[013] = [679] T0I[013] = [9B0] T6I[013] = [356] T1[013] = [124] T7[013] = [78A] T1I[013] = [A01] T7I[013] = [467] T2[013] = [235] T8[013] = [89B] T2I[013] = [B12] T8I[013] = [578] T3[013] = [346] T9[013] = [9A0] T3I[013] = [023] T9I[013] = [689] T4[013] = [457] TA[013] = [AB1] T4I[013] = [134] TAI[013] = [79A] T5[013] = [568] TB[013] = [B02] T5I[013] = [245] TBI[013] = [8AB] Of these 24 sets, only [013] and [023] begin with PC 0. [013] is more normal than [023] (i.e. the interval between the first and second-to-last PCs is smaller), so we say that (013) is in prime form. Sets in prime form are given in parentheses. Each SC has only one prime form. SCs have been cataloged differently by several theorists, but the catalog by Allen Forte has become the standard. The Forte names for SCs include a number that indicates the cardinality of the set, followed by the catalog number. The PC sets shown in Figure 16 are all members of SC 3-2. The three indicates that the set has 3 PCs, and the 2 indicates that the set is the second set in the catalog of sets with a cardinality of 3. As shown in Figure 17, the complement of [013] in base 12 is [2456789AB]. Complementary SCs have the same catalog number, so the complement of SC 3-2 is SC 9-2. [013] [2456789AB] 3-2 (013) 9-2 (012345679) All PC sets in a SC have the same interval vector. There are some sets that have the same interval vector that are not members of the same SC. That is, even though they have the same interval vector, they are not related by transposition or inversion. Different SCs that have the same interval vector are said to be Z-related, so a Z is included in their Forte name. As shown in Figure 18, SCs 5-Z12 and 5-Z36 are Z-related because they both have an interval vector of <5222121>. 5-Z12 (01356) 5-Z36 (01247) <5222121> <5222121> So far in this introduction, we have arranged sets in ascending, normal and prime forms. In serial music, however, the order of the PCs is important. Curly braces are used to indicate a series in a fixed order. Serialism is not just limited to PCs, but any musical element may be serialized such as duration, instrumentation, intensity, and articulation. Twelve-tone music is a specific type of serialism that makes use of a series that has one occurrence of each of the twelve PCs. Figure 19 shows a twelve-tone series by Arnold Schoenberg labeled P for prime. P {45716382B09A} R {A90B28361754} T8I I {4317250698BA} RI {AB8960527134} The retrograde of the twelve-tone series is labeled R. The inversion of the series is labeled I. Here, the series is inverted around the axis that passes through PC 4 (T8I) so that P and I begin with the same PC. RI is the retrograde of the inversion. There are twelve transpositions of each of P, R, I and RI. As shown in Figure 20, a twelve-tone matrix is a handy way of compactly listing all of these transpositions. P is listed in the top row of the matrix, and I is listed in the left column of the matrix. The twelve transpositions of P are listed in the twelve rows of the matrix reading from left to right. The twelve transpositions of I are listed in the twelve columns reading from top to bottom. Likewise the transpositions of R and RI are listed in the matrix as the retrogrades of P and I. We can refer to any of the 48 forms of the series as follows. The transposition of P beginning with PC 4 is labeled P4, which is in the top row of the matrix. The transposition of P beginning with PC 3 is labeled P3, which is in the second row of the matrix. The retrograde of P4 begins with PC A, and is labeled R4 and not RA. This way, we can know that P4 and R4 are retrogrades of each other without having to know the last note in P4. The transpositions of I and RI are handled similarly. For example, the third column of the matrix is I7 and its retrograde is RI7. I7 begins with PC 7, while RI7 begins with PC 1. Frequently, it is necessary to refer to elements within a series based on their order position. So, we refer to a PC’s placement within the series by using a coordinate system where the top left PC in the matrix is coordinate 0,0. For example, the first PC of P4 is 4, which has an order position of 0; while the first PC of R4 is A, which has an order position of B. This introduction rudimentarily discusses some basic topics of atonal music theory. For a more detailed discussion of these topics, see Joseph Straus’s Introduction to Post-Tonal Theory.[3] © 2007 Paul Lombardi [1] James Haar: ‘Music of the spheres’, Grove Music Online ed. L. Macy (Accessed [15 January 2007]), <http://www.grovemusic.com>. [2] Mark Lindley: ‘Equal temperament’, Grove Music Online ed. L. Macy (Accessed [15 January 2007]), <http://www.grovemusic.com>. [3] Joseph Straus, Introduction to Post-Tonal Theory, 3rd Ed. Upper Saddle River, NY: Pearson Prentice Hall, 2005. This page has been accessed times since August 11, 2009.