2.2 The problem
2.2.1 Specifying the problem
Since this chapter centers a history of a problem in musical combinatorics, recounting our current understanding of the problem and its solutions will be the first move. This retelling serves two functions. First, it fully specifies the problem and its solutions to those readers for whom it is unfamiliar. Second, it emphasizes how the solution to the all-interval problem can be expressed as an algorithm: an imperative procedure, often involving some repetition, for solving mathematical problems.
The all-interval series problem can be stated succinctly:
4. Contrast with Taruskin on the history of twentieth-century music: “The contradictory or ‘dialectical’ themes broached in these introductory paragraphs—triumph vs. insecurity, responsibility vs. escape, science-as-savior vs. science-as-destroyer, esotericism vs. utility, intellect vs. barbarism, faith in progress vs. omnibus suspicion—will be the cantus firmi of the next several chapters, along with the all-pervading image of rubble and waste, and the paralyzing (or inspiring) prospect of rebuilding. All of the bizarre and contradictory musical events and phenomena to be recounted must be understood as counterpoints against these intractable and irresolvable dilemmas that unbalanced the world’s mind.” Richard Taruskin, The Oxford History of Western Music (Oxford: Oxford University Press, 2005), vol. 5, 3.
Which twelve-tone series contain all (distinct and unique) eleven ordered pitch-class intervals between directly successive pairs of notes?5
Take the following series: <E
♭
, G♭
, D♭
, G, C, D, B, B♭
, A♭
, F♭
, F, A>. Listing the size in semitones of the intervals between directly successive pairs of notes in this series gives: <3, 7, 6, 5, 2, 9, 11, 10, 8, 1, 4>, an interval sequence in which no such ordered pitch-class interval is repeated. Now that we know that at least one such series exists, a composer interested in exploiting the features of an all-interval series in her own piece, would reasonably wonder if any others exist. By the natural application of a sufficient number of rounds of the four canonical serial operations (transposition, inversion, retrograde, and retrograde-inversion), we can obtain, at most, 47 other all-interval series.6Recall the effect of applying transposition to a twelve-tone series: the pitch level of the melody (the “row”) standing for the series changes, but the order of its intervals does not. Accordingly, it suffices
to summarize the effects of these operations on the interval sequence in Table 2.1.
Table 2.1: Forms of an all-interval series (
S
) under the canonical twelve-tone operators (transposition, inversion, retrograde, retrograde-inversion) showing the all-interval property is preserved.Form Pitches Intervals
T
0(S) <E♭
, G♭
, D♭
, G, C, D, B, B♭
, A♭
, F♭
, F, A> <3, 7, 6, 5, 2, 9, 11, 10, 8, 1, 4>I
0(S) <E♭
, C, F, B, G♭
, E, G, A♭
, B♭
, D, D♭
, A> <9, 5, 6, 7, 10, 3, 1, 2, 4, 11, 8>R
6(S) <E♭
, B, B♭
, D, E, F, A♭
, G♭
, D♭
, G, C, A> <8, 11, 4, 2, 1, 3, 10, 7, 6, 5, 9>RI
6(S) <E♭
, G, A♭
, E, D, D♭
, B♭
, C, F, B, G♭
, A> <4, 1, 8, 10, 11, 9, 2, 5, 6, 7, 3>.
If original series has the all-interval property, then so too do its canonical serial transformations. Pre-college mathematics tells us that there are no fewer (and no more!) than
12!
(12 x 11 x 10 … 2 x 1) twelve-tone series and we’ve only accounted for 48, a mere fraction of a percent. But the way5. This problem makes sense for other equal-tempered systems as well. Subsequently, unqualified references to the all-interval series problem should be understood to refer the framing in the twelve-tone equal temperament case.
forward is clear. We know how to measure intervals in a n-tone equal temperament system, and we can recognize when the set of successive intervals exhausts the (n-1) equal-tempered intervals possible in that system. Given these two competencies, it is trivial to specify a deterministic process, which when followed, guarantees a complete list of the all-interval twelve-tone series. This process can be summarized as a series of imperative statements, structured to facilitate computational implementation: as an algorithm.
Expressing the solution to a problem as an algorithm can lead to ready computational implementation, since programmable computers consume imperative instructions (in the form of program code) and excel at uncomplainingly completing repetitive calculations at high speed. Nathan Ensmenger and Paul Ceruzzi have noted the rising importance of algorithms around the time that the all-interval series catalogs were generated. The study of algorithms became the central concern of the inchoate subject of computer science.7 Algorithms may be analyzed and optimized, using a
combination of the tools of discrete mathematics, technical expertise with specific computer systems, and an understanding of the domain-specific affordances of the problem to which a given algorithm promises a solution.
If algorithms seem to be interlopers as objects of music theory, consider the process by which students are taught to identify the quality of chords in open position, or indeed, the process for determining the “prime form” of an arbitrary pitch-class set. Many readers will be aware that there are at least two popular definitions of “prime form” which diverge in that they prefer different prime-form representatives for a small number of pitch-class sets. Some might know that one is preferred by computer programmers: it requires, on average, fewer lines of code to implement (though this always depends on the exact programming language used). This is a concrete, and familiar, example of how computational criteria pervade the tools of music theory that we continue to use, even when we are away from the (computer) keyboard.
In some cases, optimization can precipitate computational applications that were previously impractical. The naive formulation of an algorithm to compute an exhaustive catalog of all-interval
7. Nathan Ensmenger, The Computer Boys Take over: Computers, Programmers, and the Politics of Technical Expertise (Cambridge, MA: MIT Press, 2012), 131.
for each permutation of the twelve pitch classes:
compute the sequence of intervals between successive pitch classes in the permutation if the sequence contains exactly and only the eleven intervals of the system
...then add the permutation to the list of all-interval series otherwise
...consider the next permutation
Figure 2.1: Algorithm 1; Brute force, tone set test
series would not just take an inconveniently long time to run, it would take an impracticably long time to run, and perhaps voraciously consume finite computational resources, for example, the data storage capacity of short- or long-term memory.8 Naive, first-time-round solutions can be improved
upon by the application of optimization techniques that exploit domain expertise pertaining to the specific problem to be solved. The optimization of algorithmic solutions to music-theoretical problems—exemplified by the all-interval series problem—involves the encounter between two kinds of expertise: the computational and the musical.
This algorithm requires exactly
12!
tests to be performed; we check the sequence of intervals induced by every single permutation of the twelve pitch classes to see if it exhausts what might be called the interval aggregate (that is, the eleven equal-tempered directed intervals). Now, knowing that if we have a single all-interval series, we can just apply the canonical serial transformations and get a handful for free, why not cut down the number of tests we have to perform by switching our focus to the interval sequences themselves? Instead of generating over 479 million permutations of the twelve tones, we can consider the11!
permutations of the eleven ordered pitch-class intervals. Then, for each such permutation, we ask if those intervals can appear in a twelve-tone series that exhausts the aggregate. Algorithm 2 reflects this change of focus.8. Some music researchers were all too aware of these limitations: Herbert Eimert pointed out the technical impossibility (at that time) of generating and storing on magnetic drum memory all possible permutations of the twelve-tone aggregate. Eimert, Grundlagen der musikalischen Reihentechnik, 43.
for each permutation of the eleven intervals:
compute the pitch-class set that arises from applying the permuted intervals as successive transpositions of a fixed starting pitch class
if the pitch-class set contains exactly and only the twelve notes of the system ...then add the permutation to the list of all-interval series generators otherwise
...consider the next permutation
Figure 2.2: Algorithm 2; Brute force, interval set test
When Algorithm 2 is coded in a programming language (antiquated or modern, the result must be the same if the algorithm is correctly implemented), 3,856 interval patterns result. In other words, there are exactly 3,856 permutations of the eleven ordered pitch-class intervals that can generate a number of equivalent all-interval twelve-tone series. But whereof 1,928? Inspecting the table above, consider the relationship between the interval sequences of series related by inversion. Most who implemented a computational search for all-interval series noticed that if an interval sequence generated the aggregate as required, then so too would the sequence consisting of the complement (mod 12) of its constituents. Thus, the computer programs they used would stop precisely mid-way through the total list of valid interval sequences, since the second half of the list could be recovered by complementation, a computer operation that is trivial compared to a lengthy search through the remaining permutations. This chapter, in part, shows how this trick and others like it were used to optimize the algorithmic specification of the all-interval series problem.
Both of these algorithms can be described as implementing a brute-force or exhaustive search strategy, the most naive and computationally expensive approach to such a problem. A strength of the brute-force strategy is its relative ease of implementation: simply generate all possible permutations and test each one. A significant weakness of such brute-force strategies is that they do not scale well as the search space grows. Imagine we were interested in all-interval series in the 24-tone equal temperament universe. How long would such an algorithm take to run to completion, assuming the test at each iteration remained the same length? If a gut response suggests “twice as long,” the
correct response—it would be 1.3 quadrillion times as long—comes as a sobering correction. The computational complexity of the task is indeed a function of the number of distinct pitch classes in the tonal system, but it is not a linear function. As the size of the tonal universe under consideration grows, so too do the number of tests required, given by the factorial function (n! = n x n-1 x ... 2 x 1). The combinatorial explosion represented by this dependency can also be exploited to improve running time.
Despite the superficial similarities between Algorithm 1 and Algorithm 2, Algorithm 2 represents a significant “win” for the implementer, because it operates on permutations of intervals, rather than permutations of notes. Again, assuming a fixed duration for each test, it reduces the computer time required by a factor of 12 (that is, more than one order of magnitude). Such speed-ups could make a difference when it came to the allocation of finite computing resources: a twelve-week computation might dominate computer time in a university computing center for an entire academic semester, whereas a one-week computation would not. And, as we shall see, even before the
computer was introduced to the problem, Krenek recognized the pedagogical virtue in restricting his attention to a hypothetical 6-tone musical system, to help make the problem more manageable, more thinkable.9