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A key signature is a collection of paths in pitch-class space. As such, it provides a powerful tool for thinking about voice leading, one that can be used to answer familiar objections to the very notion of a voice leading between pitch-class sets. I begin by generalizing the traditional Western system of musical notation, defining paths in pitch-class space, and showing that key signatures are collections of such paths. I then show that it is always possible to find a minimal voice leading that has no voice crossings; however, it is not always possible to avoid voice crossings while maximizing common-tone retention. I describe a general method for identifying a minimal voice leading between arbitrary chords, and show that our familiar system of diatonic key signatures implements this method. I conclude by deriving a set of key signatures for the acoustic scale. These key signatures represent minimal voice leadings from the diatonic to the acoustic collection, and are interestingly related to the familiar diatonic key signatures. Submission received July 2005 VOICE LEADINGS AS GENERALIZED KEY SIGNATURES A rational reconstruction of Western musical notation DMITRI TYMOCZKO 0.1 This paper argues that key signatures provide a powerful tool for understanding voice leading. It has five sections. The first generalizes the system of Western musical notation, defining the notion of a basic scale—a collection of pitch classes provided with letter names. The second section shows that any key signature determines a voice leading from the basic scale to a target scale, and, conversely, that any voice leading can be expressed as a key signature. The third section describes three virtues that a voice leading or key signature may have: avoiding voice crossings, maximizing common tones, and minimizing the overall “size” of the voice leading. I show that two of these virtues are always compatible, but that all three cannot always be exemplified simultaneously. The fourth section asks how to find a minimal voice leading between arbitrary chords. I provide a general solution to this problem, and show that this solution is implemented by the familiar system of diatonic key signatures. The final section uses the same technique to develop a system of key signatures for the acoustic scale. 0.2 This investigation will require that we think rigorously about the foundations of Western musical notation. This is inherently a pedantic and somewhat painful enterprise. Readers may initially resent the author, and may wonder if he is motivated by a fetish for obscurantism or an unhealthy love of formalism for its own sake. I am not. Instead, I hope to demonstrate that some elementary features of our system of musical notation are as yet imperfectly understood. Furthermore, these features are directly related to a subject of intense current theoretical concern: the theory of voice leading. Understanding our familiar notational system, I will show, leads to new insights about the nature of pitch classes, pitch-class intervals, and voice leading. 0.3 This is chiefly because key signatures turn out to be entities of considerable intrinsic interest: collections of paths in pitch-class space. A path in pitch-class space resembles, but is not reducible to, a pitch-class interval. Paths are more fine-grained than intervals, allowing us to distinguish the one-semitone descending route between C and B from the eleven-semitone ascending route between them. This flexibility is quite useful when we are thinking about voice leading between pitch-class sets. Thus the humble key signature, seemingly unworthy of theoretical investigation, provides Music Theory Online 11.4 – 3 an extremely powerful instrument for investigating voice leading. 0.4 Before proceeding, I would like to acknowledge my indebtedness to the important work of Julian (Jay) Hook. Jay’s 2003 paper on “[key] signature transformations” prompted my own investigation into the broader subject of voice leading. Furthermore, Jay was the first to realize that the rows of my “interscalar path matrices” can be interpreted as key signatures (Section IV). The current paper brings Hook’s approach together with my own, demonstrating that the theory of voice leading and the theory of key signatures are inextricably intertwined. I. Numbers, letters, clefs, and accidentals 1.1 In this section, I generalize the traditional Western system of musical notation. This process will lead me to draw distinctions that are not ordinarily drawn, largely because we take for granted special features of our familiar notational system. For example, I will distinguish scale-dependent and scale-independent measures of distance in pitch-class space, and paths in pitch-class space from pitch-class intervals. The result will be greater clarity about conventional musical notation. 1.2 I begin by providing numerical names for pitches and pitch classes. Somewhat unusually, I do so without presupposing a chromatic scale that divides pitchand pitch-class space into discrete “steps.” Instead, I develop a single, consistent set of numerical labels that can be applied to any tuning system and any chromatic universe. This system allows us to identify the diatonic scale (or more generally, a “basic scale” as defined below) prior to specifying how it is to be embedded in a larger, “chromatic” collection. This 1 Hook 2004. 2 Hook also read and commented on drafts of this paper. I would also like to thank Noam Elkies, who read a draft of this paper, and who has been extremely generous in teaching me mathematics over the past year. Similarly, conversations with Cliff Callender and Ian Quinn on a variety of topics have been quite stimulating. Two of Callender’s papers (2004, 2005) also sparked ideas in the present paper. Finally, my wife Elisabeth Camp provided moral support and inspiration—not to mention a very incisive set of comments on this paper. 3 The chromatic scale becomes relevant only when defining the symbols “s” and “f”; see §1.14. Music Theory Online 11.4 – 4 nicely reflects the fact that the diatonic scale was used long before it came to be interpreted as a subset of the chromatic scale. 1.3 The fundamental frequency f of a pitch can be associated with a real number p according to the equation: p = 69 + 12log2 (f/440) (1) This extends the standard system of MIDI note numbers to the microtones, associating any conceivable pitch with a unique real number and any real number with a unique pitch. In this continuous, linear pitch space, middle C corresponds to the number 60; the semitone is equal to a distance of one unit; the octave has size 12; and ascending motion in pitch corresponds to ascending motion along the real line R. 1.4 We form pitch-class space by identifying, or “gluing together,” all points p and p + 12 in pitch space. The result is the circular quotient space that mathematicians call R/12Z. We can visualize this space as shown in Figure 1. Note that Figure 1 is continuous: although I have labeled only the familiar pitch classes of twelvetone equal temperament, every point on the figure represents a distinct pitch class. 4 Curiously, music theory has standard numerical names for pitch classes but not pitches: theorists typically use spelling-specific designations like “C4” and “Cs6” to refer to pitches. I use MIDI note numbers here because they are reasonably well-known, and because they are consistent with the standard numerical labels for pitch classes. 5 According to the standard way of measuring distance in pitch space, the distance between two pitches p and q is equal to the absolute value of their difference, |p – q|. This is the familiar undirected interval between the pitches. 6 The description of pitch-class space as a circle is not simply a metaphor and does not depend on any specific visual representation of pitch-class space. Pitch-class space is topologically equivalent to a circle, and thus shares with it a well-defined mathematical structure. Consequently, we can translate many true statements about pitch-class space into true statements about circles, and vice versa. For example, any method of measuring distances on any circle (that is, any mathematical metric for the circle) defines a metric for pitch classes. Music Theory Online 11.4 – 5 Figure 1. Circular pitch-class space 1.5 We can label pitch classes using real numbers in the range 0 ≤ x < 12. These can be interpreted as clockwise arc lengths from the pitch class labeled 0. Ascending motion in pitch-class space corresponds to clockwise motion on the circle; descending motion corresponds to counterclockwise motion. This system generalizes the familiar integer-based system for notating pitch classes in 12tone equal temperament, in which C = 0, Cs = 1, and so on. In continuous pitch-class space these familiar pitch classes retain their familiar names; however, they are joined by microtones such as “C quarter tone sharp,” which is assigned the number 0.5, and “the 7 Real numbers in the range 0 ≤ x < 12 form a group under addition modulo 12Z. This is the quotient group R/12Z. Two real numbers x and y are congruent modulo 12Z if there exists some integer i such that x = y + 12i. The quantity “x + y modulo 12Z” is the number z, congruent to (x + y) modulo 12Z, and lying in the range 0 ≤ z < 12. Addition modulo 12Z resembles integer addition modulo 12, except that non-integral values are permitted. For example, 10 + 2.5 ≡ .5 modulo 12Z. 8 It is purely a matter of convention that we associate ascending motion in pitch-class space with clockwise motion on a circle. We could just as well associate ascending motion in pitch-class space with counterclockwise motion on an (appropriately labeled) circle. However, the distinction between ascending and descending motion in pitch-class space is not itself conventional; see note 23. Music Theory Online 11.4 – 6 pitch class 17 cents above D,” which is assigned the number 2.17. 1.6 NB. The system of pitch-class labels described here has been defined to be consistent with the system of labeling pitch classes using scale degrees of the familiar 12-tone equal-tempered “chromatic scale.” However, these labels do not depend on the existence of this scale, and have been defined without reference to it. The consistency of the two systems is purely a matter of notational convenience. Convenience, however, is purchased at the potential cost of confusion, and it is important to distinguish the two systems. I will use the term semitone to refer to a unit of length in pitch-class space that is defined without reference to any chromatic scale. A semitone is 1/12 of an octave, regardless of the chromatic scale. A “chromatic scale step” (or “chromatic step”) refers to a length defined in terms of some chromatic scale (not necessarily the familiar one). Only in twelve-tone equal temperament does one chromatic scale step always equal one semitone. 1.7 We can now provide pitch classes with letter names. We choose some multiset of pitch classes to serve as the basic scale. We order this scale by choosing some element as the first scale degree and arranging the remaining elements in “scalar order”—that is, so that the absolute sum of the intervals between successive pitch classes totals 12 or less. Finally, we label the successive scale degrees of the basic scale with the letter names A, B, C, D, ... 1.8 I will typically list basic scales in letter-name order. For example, I will notate the familiar white note scale as (9, 11, 0, 2, 9 The distance between any two pitch classes a and b is usually taken to be the smallest distance between two pitches belonging to those pitch classes. This is the quotient metric, corresponding to the interval class (or undirected pitch-class interval) between two notes. This metric allows us to use pitch-class distances to make general statements about pitch distances. For example, “pitch class E is four semitones away from pitch class C” implies “for every pitch belonging to pitch class C there is a pitch belonging to pitch class E four semitones away from it.” 10 A multiset is a collection that may contain multiple instances of a single object. Footnote 35 motivates the use of multisets. 11 Here we consider the intervals as real numbers in the range 0 ≤ x < 12, and we add them in the normal way, rather than modulo 12Z. Given the circular model shown in Figure 1, we require that it be possible to traverse the ordering by starting at the first element, moving exclusively clockwise on the circle, and traveling no more than one circumference in the process. Music Theory Online 11.4 – 7 4, 5, 7). This indicates that letter name “A” corresponds to pitch class 9, “B” corresponds to pitch class 11, “C” corresponds to pitch class 0, and so on. If the basic scale is the pitch-class series (0, 4, 7, 0), corresponding to the C major triad with doubled root, then the letter name “A” corresponds to pitch class 0, “B” corresponds to pitch class 4, “C” corresponds to pitch class 7, and “D” corresponds to pitch class 0. Here, the letter names “A” and “D” refer to the same pitch class. Finally, if the basic scale is the equal-tempered pentatonic scale (1, 3.4, 5.8, 8.2, 10.6) then “A” corresponds to pitch class 1, “B” corresponds to pitch class 3.4, and so on. Note that the procedure described in the preceding paragraph ensures that it is always possible to rotate the basic scale so that it is in ascending numerical order. Thus (0, 0, 4, 7), (0, 4, 7, 0), (4, 7, 0, 0), and (7, 0, 0, 4) are acceptable basic scales, while (0, 4, 0, 7) is not. 1.9 We can provide pitches with letter names by appending an octave number to the letter name of the pitch class containing that pitch. Thus letter name C4 corresponds to pitch number 60 (“middle C”), while letter name B3 corresponds to pitch number 59, a semitone below it. 1.10 Pitch classes in the basic scale can be identified using the familiar staff-and-clef system. A clef indicates that a certain staff line (or staff space) corresponds to a letter name; the next space (or line) above this corresponds to the next letter name, and so on. Thus, without knowing anything about the basic scale, we can say that the pitch classes in Figure 2 are called “C,” “B,” “C,” and “A.” However, in order to translate these letter names into numerical names we must know what the basic scale is: in the conventional system, the letter names indicate pitch classes (0, 11, 0, 9). When the basic scale is (0, 4, 7, 0), then the letter names indicate the pitch classes (7, 4, 7, 0). When the basic scale is (1, 3.4, 5.8, 8.2, 10.6), then the letter names indicate the pitch classes (5.8, 3.4, 5.8, 1). 12 In this paper, regular parentheses denote ordered lists, and curly braces denote unordered collections. Thus (a, b, c) is ordered, whereas {a, b, c} is not. 13 Let a be a numerical label for a pitch as defined in §1.3. The octave number of a is ⎣a/12⎦ – 1, the greatest integer less than or equal to (a/12) – 1. Thus “middle C” has octave number 4, as do all pitches between middle C and the next-highest C. 14 Note that letter names “wrap around” from the end of the ordering to