Construction and interpretation of equal-tempered scales using frequency ratios, maximally even sets, and P-cycles Richard J. Krantza) Department of Physics, Metropolitan State College of Denver, Denver, Colorado 80217-3362
Jack Douthett Department of Arts and Sciences, Albuquerque Community College and Technical-Vocational Institute, Albuquerque, New Mexico 87106
共Received 15 January 1999; revised 31 January 2000; accepted 1 February 2000兲 Using recent developments in music theory, which are generalizations of the well-known properties of the familiar 12-tone, equal-tempered musical scale, an approach is described for constructing equal-tempered musical scales 共with ‘‘diatonic’’ scales and the associated chord structure兲 based on good-fitting intervals and a generalization of the modulation properties of the circle of fifths. An analysis of the usual 12-tone equal-tempered system is provided as a vehicle to introduce the mathematical details of these recent music-theoretic developments and to articulate the approach for constructing musical scales. The formalism is extended to describe equal-tempered musical scales with nonoctave closure. Application of the formalism to a system with closure at an octave plus a perfect fifth generates the Bohlen–Pierce scale originally developed for harmonic properties similar to traditional chords but without the perceptual biases of these familiar chords. Subsequently, the formalism is applied to the group-theory-based 20-fold microtonal system of Balzano. It is shown that with an appropriate choice of nonoctave closure 共6:1 in this case兲, determined by the formalism combined with continued fraction analysis, that this group-theoretic-generated system may be interpreted in terms of the frequency ratios 21:56:88:126. Although contrary to the spirit of the group-theoretic approach to generating scales, this analysis may be applicable for discovering the ratio basis of unusual tunings common in non-Western music. © 2000 Acoustical Society of America. 关S0001-4966共00兲01305-9兴 PACS numbers: 43.75.Bc 关WJS兴
INTRODUCTION
Musical scales, of necessity, are built on compromise among three competing expectations. First and foremost, we have come to expect music to be based on harmonious sounds. With this in mind, historically, musical scales were constructed of intervals generated by the ratios of partial frequencies.1,2 This expectation was formally established when, in the 19th century, Helmholtz3 showed that our perception of consonance is strongly influenced by the presence of these ratios. Second, we expect music to have variety and versatility. This is usually accomplished in Western music by modulation to closely related keys.4 Third, we expect a certain amount of convenience regarding modulation and transposition. Equal-tempered scales, that preserve 共or close兲 the octave, in which all half-steps are equally spaced logarithmically, make modulation and transposition straightforward.5 It is well-known that the expectations of a harmonious scale and the ease of modulation and transposition afforded by equal-tempered scales are incompatible.1,2,5–11 As a result, many authors6,12–15 have discussed various mathematical approaches for approximating harmonious scales in the context of equal temperament. Contemporary music is usually written with 12 equaltempered semitones to the octave. The reasons for this system is directly related to the musical expectations referred to a兲
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above. First, the octave, which is considered the most harmonious interval,5 is preserved. Second, 12 equal-tempered notes to the octave are used because the just 共harmonious兲 intervals of the perfect fifth 共frequency ratio 3/2兲, the major third 共frequency ratio 5/4兲, and the minor third 共frequency ratio 6/5兲 are reasonably approximated by their closest equaltempered interval.6 Therefore, our expectation of harmonious sounds is met for the octave and approximately so for the other three most important musical intervals. Third, 12 is the first relatively small number of notes to the octave that allows for a reasonable approximation of these just intervals and allows for enough variety by modulation through 12 different keys. Fourth, because it is an equal-tempered system transposition is relatively easy. In the first three subsections 共I A, I B, and I C兲 of Sec. I we analyze the 12-tone equal-tempered system in use today in light of recent developments in music theory.6,16,17 We do this in order to introduce these mathematical developments in the context of a familiar example. In Sec. I D the chord structure of our usual 12-tone system is discussed in the context of the mathematical notation developed in the first three subsections. In Secs. I E and I F we articulate and summarize our approach for constructing equal-tempered musical scales based on these recent developments. In Sec. II A we extend our formalism to describe nonoctave musical systems and then apply the extended formalism to the Bohlen–Pierce scale in Sec. II B. As another example, the formalism is shown to generate all the details of the
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FIG. 1. Ten-point desirability function 共closure of the octave by the pure fourth兲.
group-theoretic 20-fold system of Balzano in Secs. II C 1 and 2. The interpretive power of the formalism is shown in Sec. II C 3, where we show that Balzano’s 20-fold scale may be analyzed in terms of frequency ratios in the appropriate nonoctave system. I. A MATHEMATICAL INTERPRETATION OF 12-TONE EQUAL-TEMPERED MUSIC A. Equal-temperament, approximation of just intervals, and the desirability function6
Recently, Krantz and Douthett6 developed a single interval ten-point desirability function, based on the concept of octave closure, to assess the ability of c-tone equal-tempered
musical scales to best approximate just 共harmonious兲 musical intervals. In this notation, c refers to the chromatic cardinality 共number of notes兲 of the scale. A generalization of this desirability function, based on the concept of a generalized comma, was developed to assess the ability of c-tone equaltempered scales to best approximate multiple just intervals simultaneously. Furthermore, an extension of this generalized desirability function was developed in which individual single intervals could be weighted according to the relative preference of one interval over another. The weighted, multiple-interval, ten-point desirability function is N
D 共 c,N 兲 ⫽10⫺20
兺
i⫽1
p i 兩 兵 c log2 共 R i 兲 ⫹ 21 其 ⫺ 21 兩 ,
共1兲
FIG. 2. Generalized ten-point desirability function 共closure of the octave by the pure fourth, major third, and minor third, simultaneously, with equal weighting兲.
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FIG. 3. Circle of fifths.
where 兵x其 is the fractional part of x, N is the number of intervals to be approximated, and c, the chromatic cardinality, is the number of equal-tempered intervals 共i.e., number of notes兲 per octave. The R i ’s are the frequency ratios of the individual intervals, and the p i ’s are the respective normalized weights of the R i ’s. If the intervals are weighted equally 共i.e., p i ⫽1/N for all i兲, then the desirability of the intervals collectively is the same as the average of the desirability of the individual intervals. Shown in Fig. 1 are the results of applying Eq. 共1兲 to the single interval of the pure fourth (R⫽4/3), or equivalently, the inversion of the pure fourth—the perfect fifth (R⫽3/2), for chromatic cardinalities up through 55. A value near 10 on the vertical axis indicates that the pure fourth is well approximated by the chromatic cardinality on the horizontal axis. As shown in the figure, for relatively small chromatic cardinalities, 12 is the best equal-tempered system which best approximates the just interval of the pure fourth. Shown in Fig. 2 are the results of applying Eq. 共1兲 to the just intervals of the pure fourth (R 1 ⫽4/3), major third (R 2 ⫽5/4), and minor third (R 3 ⫽6/5) simultaneously with equal weighting (p 1 ⫽p 2 ⫽p 3 ⫽1/3). As shown, 12 notes is the best choice for an equal-tempered scale with relatively few notes to the octave which provides enough variety and versatility, as discussed above. Historically, 12 notes to the octave was the preferred choice due, in part, to the wellapproximated perfect fifth 共or, equivalently, the pure fourth兲. Other examples are discussed in Ref. 6. B. Modulation
1. Circle of fifths and unidirectional P-cycles
As pointed out in the New Harvard Dictionary of Music,4 modulation is ‘‘the process of changing from one key to another, or the result of such a change.’’ The impor2727
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FIG. 4. Progression through a portion of the circle of fifths 共see the text for details兲.
tance of modulation in Western music is stated thus, ‘‘The capability for modulation, even more than the establishment of key, is the most distinctive and powerful property of the tonal system in Western music, especially since the community of twelve major and twelve minor keys was made intonationally practical by equal temperment.’’ As pointed out in Ref. 4, the most common modulations are between closely related keys which are those keys adjacent to each other on the circle of fifths. A diagram of the circle of fifths is shown in Fig. 3. The keys on the outside represent major keys and keys on the inside represent minor keys. Starting on any key 共major or minor兲, the next key clockwise 共or counterclockwise兲 represents an interval of a fifth. For example, starting on F] major, C] major is a fifth above and these two keys are socalled closely related keys. Again, from the New Harvard Dictionary of Music; ‘‘Because of the way in which sharps and flats are added to key signatures along the circle, the number of pitches in common between the starting key and each successive key outward in either direction decreases by one.’’ This property is most easily seen by reference to Fig. 4. We progress through a portion of the circle of fifths starting with the key of C major. The filled circles represent the 7 notes, out of the 12, used in the major scale. The heavy arrows show the progression through the circle of fifths. The dotted arrows indicate which note is moved going to the next key in the circle of fifths. As seen in the figure, each succes-
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sive change of key has six of seven notes in common with the previous key. As we progress further and further from the starting key, each successive key has one less note in common with the starting key. In recent works in music theory, microtonal versions of the circle of fifths have become known as unidirectional P-cycles 共P for proximity兲.18 More technically, a P-cycle is a cycle of three or more pairwise distinct sets 共scales or chords兲 from the same set class 共sets equivalent under transposition and inversion兲 such that there is a map between each pair of adjacent sets that leaves all but one of its notes fixed. Moreover, the note that moves is changed by precisely a half-step. If the sets in the cycle exhaust the set class then the cycle is said to be unidirectional, otherwise it is a toggling P-cycle 共these names came about as a result of particular properties inherent in these two types of cycles兲. Thus, the circle of fifths is a unidirectional P-cycle, while the 6-cycle of consonant triads e, E, g], G], c, C, and back to e is a toggling P-cycle. Since we will restrict our attention to unidirectional P-cycles, in what follows it will be understood that all P-cycles are unidirectional for our purposes.
Over the last 65 years other scale structures that have the properties of or related to unidirectional P-cycles have been constructed by Yasser,19 Mendalbaum,12 Chalmers,20–22 Wilson,23 Gamer,24 Balzano,25 Clough and Myerson,26,27 Mathews, Pierce, Reeves, and Roberts,28 Agmon,29 Clampitt,16 Clough and Douthett,17 Clough, Cuciurean, and Douthett,30 and Zweifel.31 In this discussion we adopt much of the terminology from the seminal works of Clough and Myerson.26,27 These terms arise as a result of musical intervals being associated with two somewhat incompatible lengths as described below. In the usual diatonic 共major兲 scale, the musical third comes in one of two sizes; three half-steps 共the minor third兲 and four half-steps 共the major third兲, whereas six half-steps 共the tritone兲 can either be a fourth 共augmented兲 or a fifth 共diminished兲. To generalize and relate these measures, let c be the size of the chromatic universe or the chromatic cardinality, as before. Similarly, for a given scale, let d be the diatonic cardinality which is the number of pitch classes 共e.g., all C s belong to the same pitch class兲 that define the scale. We may denote a given scale by the set 共2兲
where the elements are organized so that s 0 ⬍s 1 ⬍s 2 ⬍..., ⬍s d⫺1 . In our usual 12-tone chromatic universe, if we make the assignments C⫽0, C] /D[ ⫽1, D⫽2, etc. the C major scale may be represented by S 12,7⫽ 兵 0,2,4,5,7,9,11其 . Then the chromatic length from s i to s j is (s j ⫺s i ) expressed as the smallest non-negative integer modulo c, and the diatonic length from s i to s j is ( j⫺i) expressed as the smallest nonnegative integer modulo d. For the diatonic scale the chromatic length from one note to another 共reduced to within the octave兲 is the length of the interval in half-steps, and the diatonic length is one less 共for mathematical convenience兲 than the usual musical interval 共unison, second, third, etc.兲. 2728
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C. Maximally even „ME… sets 1. Diatonic scales
All of the seven note diatonic scales shown in the circle of fifths may be generated by the maximally even 共ME兲 algorithm developed by Clough and Douthett: Let c and d be the chromatic and diatonic cardinalities, respectively, and let n be any fixed integer such that 0⭐n⭐c⫺1. Then, a ME set with these parameters is n ⫽ J c,d
2. Mathematical generalization of unidirectional P-cycles
S c,d ⫽ 兵 s 0 ,s 1 ,s 2 ,...,s d⫺1 其 ,
For example, the chromatic length from the note B to E is 5, and since this interval is a fourth, the diatonic length is 3. Although their investigation was primarily to generalize other properties of the diatonic scale, Clough and Myerson26,27 were the first to construct an algorithm that generates, up to transposition, the complete family of sets that induces P-cycles. Also, Clough and Douthett17 extended and generalized this algorithm now known as the maximally even 共ME兲 algorithm. This algorithm is discussed in the next subsection.
再 b c冎 ck⫹n d
k⫽0
共3兲
, d⫺1
where b x c is the smallest integer greater than or equal to x.17,30,32 For our usual 12-tone system with a 7-note diatonic scale, c⫽12 and d⫽7. If we let the notes C, C] /D[ , D, D] /E[ etc., in Fig. 4 be represented by the numbers 0, 1, 2, 3, etc., respectively, Eq. 共3兲 generates all the scales of the circle of fifths. For example, the C major scale is represented by the ME set 5 ⫽ J 12,7
再b
12k⫹5 7
c冎
6
⫽ 兵 0,2,4,5,7,9,11其 .
共4兲
k⫽0
Note that the index of the C major scale is n⫽5. As the index increases incrementally 共mod 12兲 the associated scales rotate clockwise around the cycle of fifths in Fig. 3. Because the ME algorithm reproduces the scales in the circle of fifths, in order, it preserves the properties of the unidirectional P-cycles referred to in the previous subsection. 2. Maximally even sets and P-cycles Clampitt16 has shown that a set can induce a P-cycle if and only if the set is maximally even and the chromatic cardinality, c, and the diatonic cardinality, d, are coprime, or (c,d)⫽1 in mathematical parlance. Clough and Douthett17 have shown that such sets are also generated sets with two distinct generators 共e.g., the fourth and fifth generate the major scales in our familiar 12-tone system兲. Moreover, the chromatic and diatonic lengths of the generators can be determined by the linear Diophantine equation
cI g ⫺dg⫽⫾1,
共5兲
where g is the chromatic length of a generator and I g is the corresponding diatonic length.33 For example, consider the usual diatonic scale 共c⫽12 and d⫽7兲. Then, Eq. 共5兲 becomes 12I g ⫺7g⫽⫾1.
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The only chromatic and diatonic lengths that satisfy Eq. 共6兲 are g⫽5, I g ⫽3 共the fourth兲 and g⫽7, I g ⫽4 共the fifth兲. It is left to the reader to show that the fourth or the fifth also generates the pentatonic scale 共c⫽12 and d⫽5兲. Although the diatonic length of a generator plays an important role in the theory of scales, our subsequent analysis will consider only chromatic lengths of generators. Therefore, Eq. 共5兲 simplifies to dg⬅⫾1 共 mod c 兲 .
共7兲
We will have occasion to use these generalizations in Sec. II.
TABLE I. Individual desirabilities for the frequency ratios 4/3, 5/4, and 6/5 for a 53-note equal-tempered system with octave closure. Also shown are the chromatic generators and diatonic cardinalities for each ratio. For comparison, the equal-weight ten-point desirability for all three ratios is shown. c⫽53
Ratios
Desirability function base 2
Chromatic generator
Diatonic cardinalities
4/3 5/4 6/5 All
9.940 8.756 8.816 9.171
22, 31 17, 36 14, 39 ¯
12, 41 25, 28 19, 34 ¯
D. Intervals and chords
In our usual 12-tone system the intervals of the fourth with a frequency ratio of 4/3, the major third with a frequency ratio of 5/4, and the minor third with a frequency ratio of 6/5 are of prime importance. The sequence of these ratios is 3:4:5:6, and the chromatic lengths of these intervals are 5, 4, and 3, respectively. 关Note that the ratio of the first element in the sequence of ratios 共3兲 to the last element 共6兲 produces the frequency ratio 6:3 or 2:1, which is the ratio of the octave. More on this later.兴 The major and minor triad structure can now be determined from the chromatic lengths. The sequence of these lengths arranged from largest to smallest is the step-interval sequence of the major triads. In this case the sequence is 共5,4,3兲 and the class of major triads 共equivalent under transposition兲 is M ⫽ 兵兵 0,5,9 其 , 兵 1,6,10其 , 兵 2,7,11其 , 兵 0,3,8 其 ,... 其 .
共8兲
The sequence is cyclic; therefore 共4,3,5兲 and 共3,5,4兲 are in the same class. These sequences just generate chord inversions that belong to the same class. Thus, this family consists of major triads F, F] ,G,G] ,...,E. Similarly, the minor triads are the triads whose step-interval sequence is the sequence of the chromatic lengths arranged from smallest to largest. The sequence is 共3,4,5兲 and the class of minor triads is m⫽ 兵兵 0,3,7 其 , 兵 1,4,8 其 , 兵 2,5,9 其 ,... 其 .
共9兲
The union of M and m constitutes the set class of consonant triads. In each diatonic scale there are six major/minor triads 共three of each兲. For example, the triads G, C, D, a, b, and e are all embedded in the G major scale. The only embedded 共third-generated兲 triad not of major or minor quality is the F] dim triad. Therefore, using the chromatic lengths of the frequency ratios, 3:4:5:6 which are best approximated by our usual 12-tone equal-tempered system, we have determined the chord structure of the system. E. An approach to constructing musical scales
We may now formalize an approach for constructing equal-tempered musical scales. Given a sequence of frequency ratios, which may be chosen for a variety of reasons 共e.g., because they are ‘‘harmonious,’’ they convey some ‘‘mood,’’ or they meet some other mathematical, acoustical, or perceptual criteria兲, an equal-tempered system based on the desirability function, Eq. 共1兲, that best approximates these ratios may be determined. The chromatic lengths of 2729
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these ratios can then be determined and used as generators to identify scales that induce P-cycles; Eqs. 共7兲 and 共3兲. These scales then have the modulation properties associated with the familiar circle of fifths. Once the chromatic lengths of the chosen frequency ratios are known, the triad structure of the scale can be determined from the step-interval sequence; e.g., Eqs. 共8兲 and 共9兲 for the usual diatonic scale. We invite the reader to apply the formalism, using the fourth 共or fifth兲 as the ‘‘chosen’’ interval and verify that the usual pentatonic scale (c⫽12,d⫽5) is generated. The results should be the complement of the diatonic set and produce the open circles shown in Fig. 4. It is also left to the reader to verify that the ME algorithm 关Eq. 共3兲兴 generates the P-cycle of pentatonic scales. For another example, it is well-known that the 53-tone, equal-tempered system closely approximates the ratios of the fourth, major third, and minor third. The desirability function shown in Fig. 2 reflects this result and shows that the 53note-to-the-octave, equal-tempered scale is considerably better at approximating these intervals simultaneously. In Table I we summarize some results, using Eqs. 共1兲 and 共7兲, for a chromatic cardinality of c⫽53. Since we are weighting the intervals equally, the overall desirability is the same as the average of the desirabilities of the individual intervals. It is left to the interested reader to construct all the scales that induce P-cycles 共There are six set classes, two for each ratio, whose sets induce P-cycles.兲 and determine the embedded major and minor triads in each. F. Summary
In light of recent developments in music theory we have articulated an approach for constructing equal-tempered musical scales that: 共1兲 closes the octave, 共2兲 approximates multiple chosen intervals simultaneously, 共3兲 preserves modulation properties generalized from those of the circle of fifths, and 共4兲 determines the scale and chord structure for the system. For historical and pedagogical reasons we have shown that the mathematical patterns inherent in the familiar 12tone, equal-tempered system may be generalized. Alternatively, we could have taken an axiomatic approach by describing the formalism, applying it to the sequence of harmonious ratios 共3:4:5:6兲, and generating our usual 12tone, equal-tempered system. In the next section, we extend the formalism to account for nonoctave closure and show that some unusual scales
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FIG. 5. GDF for the ratios 5/3, 7/5, and 9/7 共closure at the tritave; b⫽3兲.
generated by other workers25,28,31 based on very different criteria, that range from group-theoretic arguments to acoustical and perceptual reasons, are described by our approach. II. ANALYSIS OF OTHER EQUAL-TEMPERED SYSTEMS A. Extension to nonoctave systems
1. Nonoctave desirability function
The ten-point desirability function need not be restricted to octave 共base 2兲 closure. A straightforward generalization of the weighted, multiple-interval desirability function yields the following: N
D b 共 c,N 兲 ⫽10⫺20
兺
i⫽1
p i 兩 兵 c logb 共 R i 兲 ⫹ 21 其 ⫺ 21 兩 ,
共10兲
where c, N, p i , and R i are defined as above. In the following, we refer to D b (c,N) as the generalized desirability function 共GDF兲. The base, b, of the logarithm represents the interval of closure. For octave closure, the usual case, the base is 2; D(c,N)⫽D 2 (c,N).
until c⫽271 with a GDF of 9.90. This desirability exceeds even that of the sequence of ratios 3:4:5:6 at 53 divisions to the octave 共compare with Table I兲. The desirability function, chromatic lengths, and diatonic cardinalities associated with this sequence of ratios are given in Table II. There are six scales 共up to transposition兲, with diatonic cardinalities of 2, 3, 4, 9, 10, and 11, that induce P-cycles. Diatonic cardinalities d⫽2, 3, and 4 are too small to be of much interest 共i.e., the scales would have too few pitch classes for much variation or variety兲. Therefore, the diatonic cardinalities d⫽9, 10, or 11 are better choices. Bohlen,35 based on combination tones, and somewhat later Mathews, Pierce, Reeves, and Roberts,28 based on the intonation sensitivity measure of Roberts and Mathews,36 chose a diatonic cardinality d⫽9. This scale has become known as the Bohlen–Pierce scale. With this in mind we will use d ⫽9 for our example and discuss these results in the next subsection. B. The Bohlen–Pierce scale28
Having chosen a chromatic cardinality of c⫽13 共according to our generalized desirability function兲 and a diatonic
2. Nonoctave example
In general, if the sequence of chosen ratios is r 1 :r 2 :r 3 :r 4 the base for the desirability function is just b ⫽r 4 /r 1 , as was the case for the pure fourth, major third, and minor third described above where the sequence of ratios was 3:4:5:6. In anticipation of the next section we chose the sequence of ratios 3:5:7:9 for our example. In this case, the base of the desirability function is 3 共⫽9/3兲, which means that our equal-tempered scale will close 共repeat兲 every octave plus a fifth. Shown in Fig. 5 is the GDF with equal weighting for these ratios. It is clear from the figure that the best, reasonably small, chromatic cardinality is c⫽13 with a GDF of 9.40.34 In fact, a better chromatic cardinality does not occur for these three ratios simultaneously, with equal weighting, 2730
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TABLE II. Individual generalized desirabilities for the frequency ratios 5/3, 7/5, and 9/7 for a 13-note equal-tempered system with tritave 共base 3兲 closure. Also shown are the chromatic generators and diatonic cardinalities for each ratio. For comparison, the equal-weight generalized desirability for all three ratios is shown. c⫽13
Ratios
Desirability function base 3
Chromatic generators
5/3 7/5 9/7 All
9.107 9.630 9.477 9.405
6, 7 4, 9 3, 10 ¯
Diatonic cardinalities 2, 11 3, 10 4, 9 ¯
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from largest to smallest 共6, 4, 3兲 and smallest to largest 共3, 4, 6兲, respectively, as before. The class of Bohlen–Pierce major triads is M BP⫽ 兵兵 0,6,10其 , 兵 1,7,11其 , 兵 2,8,12其 , 兵 0,3,9 其 ,... 其 .
共12兲
The class of Bohlen–Pierce minor triads is m BP⫽ 兵兵 0,3,7 其 , 兵 1,4,8 其 , 兵 2,5,9 其 ,... 其 .
共13兲
For this scale, the set class of ‘‘consonant triads’’ is M BP艛m BP . Although Bohlen35 and Mathews, Pierce, Reeves, and Roberts28 did not have the GDF, the ME algorithm, or the P-cycle algorithm, they constructed a scale consistent with our application of these tools for constructing scales given a chosen sequence of ratios. C. A ratio interpretation of Balzano’s c-fold microtonal system25
1. Introduction
FIG. 6. A 13-note chromatic scale with 9-note embedded diatonic scale generated by the ME algorithm compatible with unidirectional P-cycles. The Bohlen–Pierce scale is superimposed on the first diagram. Progression through a few closely related keys is also shown.
cardinality of d⫽9 共in accordance with the modulation properties of unidirectional P-cycles兲 for the sequence of ratios 3:5:7:9 共chosen by previous workers28,35,36 for acoustic and perceptual reasons兲, with chromatic generators of 6 共or 7兲, 4 共or 9兲, and 3 共or 10兲, for an equal-tempered system with base 3 closure, we are ready to construct the ‘‘diatonic’’ scale and chord structure for the system. The ME algorithm for c ⫽13, d⫽9, and n⫽4 yields the following for a typical scale:
再b
13k⫹4 4 ⫽ J 13,9 9
c冎
8
⫽ 兵 0,1,3,4,6,7,9,10,12其 .
共11兲
k⫽0
This scale is shown by the filled circles in the first diagram of Fig. 6. Superimposed on that diagram are the nine tones of the Bohlen–Pierce scale, denoted as I, II, III, etc. Also, shown in Fig. 6 are the first few closely related keys generated by the ME algorithm, with indices; n⫽4, 5, 6, 7, 8, and 9. Because the chromatic cardinality and the diatonic cardinality are coprime these sets induce P-cycles, and therefore have the modulation properties of the generalized circle of fifths. Also shown in Fig. 6, as a dotted arrow, is the note that moves to generate the next closely related key. As before, the heavy arrows indicate the progression through the generalized circle of fifths. The classes of major and minor triads are generated by arranging the step-interval sequence for the chosen ratios 2731
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Our next example of a nontraditional system is based on Balzano’s well-known c-fold microtonal system where c ⫽n(n⫹1) and n is an integer greater than or equal to 3. Balzano points out that ‘‘With the advent of the computer, the possibilities of exploring microtonal systems of octave division broadens considerably.’’ As an alternative, he bases his octave division constructs on mathematical grouptheoretic properties rather than constructing microtonal systems based on good fits to frequency ratios. The chromatic cardinalities considered by Balzano are the products of two consecutive integers 共a generalization of 12⫽3•4 in our familiar 12-tone system兲 and the diatonic cardinalities are the sum of those two integers 共a generalization of 7⫽3⫹4兲. His ‘‘F to F] property’’ is equivalent to requiring scales that induce P-cycles and his ‘‘major’’ and ‘‘minor’’ triads have the interval sequence (n 2 ⫺n⫺1,n ⫹1,n) and (n,n⫹1,n 2 ⫺n⫺1), respectively. Again, these sequences are the generalization of the usual n⫽3 case for the familiar 12-tone system. As a specific example, we will use Balzano’s 20-fold system for comparison. 2. P-cycles, ME sets, and triads
Balzano’s 20-fold system requires that d⫽4⫹5⫽9. With a diatonic cardinality of 9, the two generator lengths that satisfy Eq. 共7兲, and hence generate scales that induce P-cycles are g⫽9 and g⫽11. 共The length g⫽11 will also be important in the construction of the major and minor triads.兲 Since Balzano’s scale will induce a P-cycle, it can be determined via the ME algorithm 6 J 20,9 ⫽
再b
20k⫹6 9
c冎
8
⫽ 兵 0,2,5,7,9,11,14,16,18其 .
共14兲
k⫽0
This scale represents what Clough and Douthett17 call a hyperpentatonic scale, which is a ME set whose diatonic cardinality is 1 less than half the chromatic cardinality and the chromatic cardinality is divisible by 4. Note that if c⫽12 the hyperpentatonic scale reduces to the familiar pentatonic scale. Balzano uses the diatonic length of 2 共the ‘‘third’’兲 to generate triads. As seen in Eq. 共14兲 the chromatic lengths of
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TABLE III. For a small number of bases 共that define closure兲, the principalconvergents-continued-fraction approximations for the step intervals 共11, 5, and 4兲 in Balzano’s 20-fold system are calculated.
TABLE IV. Analysis of Balzano’s 20-fold system using candidate sequences compatible with principal-convergent-continued fractions 共see the text for details兲.
c⫽20 and d⫽9 b 5/20
c⫽20 and d⫽9
Base 2 3 4 5 6
b 11/20
3/2, 19/13, 41/28 2/1, 9/5, 11/6 2/1, 13/6, 15/7, 433/202 2/1, 5/2, 12/5, 17/7 2/1, 3/1, 8/3, 67/25
6/5, 19/16, 25/21 4/3, 25/19, 229/174 3/2, 7/5, 17/12 3/2, 160/107, 643/430 2/1, 3/2, 11/7, 25/16
b 4/20 7/6, 8/7, 23/20 5/4, 71/57, 147/118 4/3, 29/22, 33/25 3/2, 4/3, 7/5 3/2, 10/7, 83/58
the thirds are 4 and 5. Therefore, along with the chromatic length of the generator g⫽11, the step-interval size for the third is 5. Along with a step interval of 11 for the generator, the step-interval sequences for major and minor triads become 共11, 5, 4兲 and 共4, 5, 11兲, respectively. The classes of Balzano 20-fold major and minor triads are M B 20⫽ 兵兵 0,11,16其 , 兵 1,12,17其 , 兵 2,13,18其 ,... 其 ,
共15兲
m B 20⫽ 兵兵 0,4,9 其 , 兵 1,5,10其 , 兵 2,6,11其 ,... 其 .
共16兲
It is left to the reader to verify that there are four major triads and four minor triads embedded in Balzano’s scale. For a 20-fold system, sets with diatonic cardinality 11 and generators g⫽9 or g⫽11 also induce P-cycles. Recently, Zweifel31 has investigated the properties of these 11note scales. These scales are the complements of Balzano’s nine-note scales and belong to a larger class of scales first isolated by Agmon29 and later studied in depth by Clough and Douthett17 and Clough, Cuciurean, and Douthett.30 Scales from this class are embedded in chromatic universes where c⫽0 共mod 4兲, and the diatonic cardinality is 1 more than half the chromatic cardinality. It is left to the reader to generate this class of scales and the resulting chord structure. In the next subsection, we take an alternative, nonoctave approach to analyzing Balzano’s 20-fold system. 3. Ratios and continued fractions
Schechter37 and Douthett, Entringer, and Mullhaupt38 have shown the utility of using continued-fraction-principal convergents39 for generating equal-tempered musical scales with octave closures. We have taken a somewhat different approach in analyzing Balzano’s 20-fold system. The step intervals of 11, 5, and 4 correspond to ‘‘frequency ratios’’ of the form: b 11/20, b 5/20, and b 4/20; where b is the base of the logarithm that defines closure. For example, b⫽2 defines the usual case of octave closure and b⫽3 defines closure at an octave plus a fifth, which was the case for the Bohlen–Pierce scale. We proceeded to choose a number of bases, b⫽2, 3, 4, 5, and 6 and calculated the three frequency ratios given above, in each base. We then generated the first few principal-convergents-continued-fraction approximations for each frequency ratio in each base. The results of these calculations are summarized in Table III. For convenience, in Table III, we have excluded the musically trivial 1/1 ratio. We then chose a candidate sequence of ratios in each base consistent with the principal-convergents-continuedfraction approximations and compatible with that base. For 2732
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Base
Sequence
3 4 5 5 6 6
30:55:72:90 7:15:21:28 5:12:18:25 14:34:51:70 21:56:88:126 15:40:63:90
b 11/20 Fractions 11Õ6 15Õ7 12Õ5 17Õ7 8Õ3 8Õ3
b 5/20 Fractions
b 4/20 Fractions
D b (20,3) Equal-weight GDF
72/55 7Õ5 3Õ2 3Õ2 11Õ7 63/40
5Õ4 4Õ3 25/18 70/51 63/44 10Õ7
8.71 8.00 8.39 9.13 9.31 9.06
example, for base 2 we chose the ratios 10:15:18:共20兲 because 15:10⫽3:2, 18:15⫽6/5, and 20:18⫽10/9 because 20:10⫽2 共the base兲. In general, we cannot choose a base and a sequence of ratios in which all these ratios are consistent with the continued fraction convergents simultaneously. We chose the base for closure. The first ratio was chosen because it was the generator of the diatonic scale that induces P-cycles. In the cases examined, we chose either the second or third ratio to be compatible with a 共relatively small integer兲 principal-convergent-continued-fraction approximation for that ratio. We allowed the remaining ratio to be compromised, i.e., it was not necessarily a continued-fraction convergent. For base 3, it turned out that the candidate sequence 5:9:12:15 yields the principal-convergent-continued fractions 9/5, 4/3, and 5/4. The ratio 15:5 yields the base. Similar results are obtained for other bases. Each of these candidate ratio sequences was then used to calculate the equal-weight GDF. The candidate sequences, for bases through b⫽6, leading to a GDF greater than or equal to 8.00 are shown in Table IV. Fractions shown in bold are the resulting principal-convergent-continued fraction approximation to the given step-interval ratio. The fractions given in normal type are the resulting compromised ratios. For example, for the base 3 ratio sequence 30:55:72:90 we get a ratio of 11/6 共⫽55/30兲 for the generator of the diatonic scale, 5/4 共⫽90/72兲 is the continued fraction approximation for 3 4/20, and the middle ratio 共72/55兲 is compromised. These ratios result in an equal-weight GDF of 8.71. As shown, based on the GDF, the best choice for closure that yields small-integer-principal-convergent-continued fraction ratios is base 6. This represents closure at two octaves plus a fifth. Alternatively, the ratio sequence that would generate Balzano’s 20-fold system with a 9-note scale that induces P-cycles and yields a reasonable desirability function value is 共21:56:88:126兲. For comparison, Fig. 7 shows the equal-weight, multiple-interval desirability function, base 6, for these ratios. In the sequence of chromatic cardinalities c⫽20 is the best small number chromatic cardinality. Note that although c⫽4 has a relatively good value, it may be neglected as too small to be interesting. 4. Summary
We have extended Balzano’s approach somewhat by suggesting, based on continued fractions and the GDF, a
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FIG. 7. GDF for the ratios 56/21, 88/ 56, and 126/88 共closure at the sextave; b⫽6兲.
sequence of ratios 共21:56:88:126兲 in a nonoctave system that may be used to generate his 20-fold, 9-note system. The ratios found to generate Balzano’s system do not necessarily define ‘‘consonant’’ intervals. This is in the spirit of Balzano’s approach that justified the system solely on grouptheoretic grounds. If an a priori reason for choosing the above interval sequence could be justified, our formalism could be used to generate this system. Otherwise, the formalism can be used to interpret the system in terms of frequency ratios including nonoctave closure. III. DISCUSSION
Based on good-fitting intervals, consistent with the multiple-interval desirability function6 and the modulation properties of unidirectional P-cycles,18 along with the application of maximally even set theory,17 we have developed a formalism for constructing equal-tempered scales, of any chromatic cardinality, with ‘‘diatonic’’ scales, and the associated chord structure, that have the modulation properties of the closely related keys in the usual 12-tone system. We have applied the formalism to the 12-tone, equal-tempered system to show that this approach describes the details of this traditional system, as well as to introduce the mathematical details of these recent music-theoretic developments in the context of a familiar example. Also, the formalism has been extended to describe nontraditional scales that include nonoctave closure. As an example, the formalism was applied to the frequency ratios 3:5:7:9 of the Bohlen–Pierce scale. This scale was chosen originally for its similarity to the structural and acoustic properties of the usual diatonic scale. It is shown that our approach reproduces all the details of the Bohlen–Pierce scale. More generally, given a sequence of frequency ratios determined for their acoustical, structural, mathematical, or perceptual properties, our formalism can generate an equaltempered system that: 共1兲 has closure, 共2兲 reasonably approximates the ‘‘chosen’’ frequency ratios simultaneously, 2733
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and 共3兲 can yield a scale and chord structure that has the modulation properties of closely related keys. As an example of the applicability of our approach, we show that the nontraditional 20-fold microtonal system of Balzano25 can: 共1兲 be generated by our formalism, and 共2兲 can be interpreted in terms of the ratio sequence 21:56:88:126 in a 6:1 nonoctave system when the formalism is coupled with continued fraction analysis. The above examples show the power of this approach for generating nonstandard musical systems. First, one can choose an interval for closure and assess the ability of various equal-tempered systems to approximate chosen intervals by application of the GDF, thereby determining an appropriate chromatic cardinality for the scale. Second, to ensure musical variety and variation, a diatonic cardinality that induces P-cycles, and therefore retains the modulation properties of closely related keys, can be chosen. Third, the ME algorithm can then be used to generate all the appropriate diatonic scales of the system. Fourth, the step-interval sequence, based on the chosen intervals, can be used to generate the chord structure for the system. Alternatively, our approach can be used as an analysis tool in which a given scale may be interpreted in terms of ratios in an appropriate nonoctave system. Used in this way, we hope to discover a ratio basis of unusual tunings common in non-Western music.
1
J. M. Barbour, ‘‘The Persistence of Pythagorean Tuning Systems,’’ Scr. Math. 1, 286 共1933兲. 2 J. M. Barbour, ‘‘Musical Logarithms,’’ Scr. Math. 3, 21 共1940兲. 3 H. Helmholtz, in On Sensation of Tone 共Dover, New York, 1954兲, Chaps. I and XVI 共originally published in 1885兲. 4 ‘‘Modulation,’’ in The New Harvard Dictionary of Music, edited by D. M. Randell 共Belknap, Cambridge, MA, 1986兲, p. 503. 5 ‘‘Temperaments,’’ in The New Harvard Dictionary of Music 共Ref. 4兲, p. 837. 6 R. J. Krantz and J. Douthett, ‘‘A Measurement of the Reasonableness of Equal-Tempered Musical Scale,’’ J. Acoust. Soc. Am. 95, 3642–3650 共1994兲.
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7
D. E. Hall, ‘‘The Objective Measure of Goodness-of-Fit for Tunings and Temperaments,’’ J. Music Theory 17, 274 共1973兲. 8 D. E. Hall, ‘‘Quantitative Evaluation of Musical Scale Tunings,’’ Am. J. Phys. 42共7兲, 543 共1974兲. 9 D. E. Hall, ‘‘Acoustical Numerology and Lucky Temperaments,’’ Am. J. Phys. 56共4兲, 329 共1988兲. 10 D. E. Hall, ‘‘A Systematic Evaluation of Equal Temperaments Through N⫽612,’’ Interface 共USA兲 14, 61–73 共1985兲. 11 G. C. Hartmann, ‘‘A Numerical Exercise in Musical Scales,’’ Am. J. Phys. 55共3兲, 223 共1987兲. 12 J. Mandelbaum, ‘‘Multiple Division of the Octave and Tonal Resources of 19-Tone Temperament,’’ Ph.D. dissertation 共Indiana University, Bloomington, IN, 1961兲. 13 W. Stoney, ‘‘Theoretical Possibilities for Equally Tempered Musical Systems,’’ in The Computer and Music, edited by H. B. Lincoln 共Cornell University Press, Ithaca, NY, 1970兲, pp. 163–171. 14 D. De Klerk, ‘‘Equal Temperament,’’ Acta Musicol. 51, 140 共1979兲. 15 M. Yunik and G. Swift, ‘‘Tempered Music Scales for Sound Synthesis,’’ Comput. Music J. 4共4兲, 60 共1980兲. 16 D. Clampitt, ‘‘Pair-Wise, Well-Formed Scales: Structured and Transformational Properties,’’ Ph.D. dissertation 共SUNY at Buffalo, Buffalo, NY, April 1997兲. 17 J. Clough and J. Douthett, ‘‘Maximally Even Sets,’’ J. Music Theory 35, 93–173 共1991兲. 18 In July of 1993 Clough 共SUNY at Buffalo兲 assembled a group—known as the ‘‘SUNY Buffalo Working Group’’—of music theorists, music psychologists, and mathematicians to investigate and expand on the ideas put forth by Richard Cohn 共University of Chicago兲 on late 19th century voice leading. The term P-cycles was introduced by Cohn and was among the topics explored at the meeting. This group met again in July of 1997 to explore a related topic, neo-Riemannian Transformations. For more information on the above, including a bibliography of related topics, see the special topics edition of the Journal of Music Theory 42共2兲 共1998兲. 19 J. Yasser, The Theory of Evolving Tonality 共American Library of Musicology, New York, 1932兲. 20 J. Chalmers, ‘‘Cycle Scales,’’ Xenharmonikon 4, 69–78 共1975兲. 21 J. Chalmers, ‘‘Polychordal Matrices and MOS Scales,’’ Xenharmonikon 7–8, 156–167 共1979兲. 22 J. Chalmers, ‘‘Construction and Harmonization of Microtonal Scales in Non-Twelve-Tone Equal Temperaments,’’ Proceedings of the 8th International Computer Music Conference, Venice, Italy, pp. 534–555, 1982 共unpublished兲. 23 E. Wilson, Private communication to J. Douthett 共1996兲. 24 C. Gamer, ‘‘Some Combinatorial Resources of Equal-Tempered Systems,’’ J. Music Theory 11共1兲, 32–59 共1967兲.
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25
G. Balzano, ‘‘The Group-Theoretic Description of 12-Fold Pitch Systems,’’ Comput. Music J. 4, 66–84 共1980兲. 26 J. Clough and G. Myerson, ‘‘Musical Scales and the Generalized Circle of Fifths,’’ Am. Math. Monthly 93共9兲, 695–701 共1986兲. 27 J. Clough and G. Myerson, ‘‘Variety and Multiplicity in Diatonic Systems,’’ J. Music Theory 29, 249–270 共1985兲. 28 M. V. Mathews, J. R. Pierce, A. Reeves, and L. A. Roberts, ‘‘Theoretical and Experimental Explorations of the Bohlen–Pierce Scale,’’ J. Acoust. Soc. Am. 84, 1214–1222 共1988兲. 29 E. Agmon, ‘‘A Mathematical Model of the Diatonic System,’’ J. Music Theory 33, 1–25 共1989兲. 30 J. Clough, J. Cuciurean, and Douthett, ‘‘Hyperscales and the Generalized Tetrachord,’’ J. Music Theory 41共2兲, 67–100 共1997兲. 31 P. Zweifel, ‘‘Generalized Diatonic and Pentatonic Scales: A Group Theoretic Approach,’’ Perspect. New Music 34共1兲, 140–161 共1996兲. 32 To get a sense of why these sets are called maximally even, S. Block and J. Douthett, J. Music Theory 38, 21 共1994兲 constructed a measure of the eveness of subsets in the same cardinal family 共e.g., same chromatic cardinality or same diatonic cardinality兲. This measure is consistent with the ME algorithm. 33 J. Douthett, ‘‘The Theory of Maximally and Minimally Even Sets, the One-Dimensional Antiferromagnetic Ising Model, and the Continued Fraction Compromise of Musical Scales,’’ Ph.D. dissertation 共University of New Mexico, Albuquerque, NM, May 1999兲. 34 When c is a denominator of a continued fraction convergent of the logb of a ratio, then the ratio’s desirability at c is better than at any smaller chromatic cardinality 共Ref. 6兲. Remarkably, for the sequence of ratios 3:5:7:9, c⫽13 is a denominator of continued fraction convergents to log3 of all three ratios: log3(5/3)⬇6/13, log3(7/5)⬇4/13, and log3(9/7)⬇3/13. This coincidence explains why this small chromatic cardinality has such an extraordinarily good desirability. 35 H. Bohlen, ‘‘13 Tonstufen in der Doudezeme,’’ Acustica 39, 76–86 共1978兲. 36 L. Roberts and M. Mathews, ‘‘Intonation Sensitivity for Traditional and Nontraditional Chords,’’ J. Acoust. Soc. Am. 75, 954–959 共1984兲. 37 M. Schechter, ‘‘Tempered Scales and Continued Fractions,’’ Am. Math. Monthly 87, 40–42 共1980兲. 38 J. Douthett, R. Entringer, and A. Mullhaupt, ‘‘Musical Scale Construction: The Continued Fraction Compromise,’’ Util. Mathemat. 42, 97–113 共1992兲. 39 A. Ya Khinchin, Continued Fractions 共The University of Chicago Press, Chicago, 1992兲.
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