Cross-cultural data shows musical scales evolved to maximise imperfect fifths John M. McBride1,* and Tsvi Tlusty1,2,* 1Center for Soft and Living Matter, Institute for Basic Science, Ulsan 44919, South Korea 2Department of Physics, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea *[email protected], [email protected] June 1, 2020 1 Musical scales are used throughout the world, but harmonicity follows from the idea that exposure to harmonic 43 2 the question of how they evolved remains open. Some sounds in animal vocalisations may have conditioned humans 44 3 suggest that scales based on the harmonic series are to respond positively to them [9, 10]. Musical features related 45 4 inherently pleasant, while others propose that scales to harmonicity { harmonic intervals [11{13], octave equivalence 46 5 are chosen that are easy to communicate. How- [14, 15], a link between consonance and harmonicity [16{18] { 47 6 ever, testing these theories has been hindered by the are indeed widespread, although their universality is disputed 48 7 sparseness of empirical evidence. Here, we assimilate [19, 20]. One study attempted to explain the origin of scales 49 8 data from diverse ethnomusicological sources into a as maximization of harmonicity [8], however the universality 50 9 cross-cultural database of scales. We generate pop- of their findings is limited by the scope of cultures considered 51 10 ulations of scales based on multiple theories and as- [21]. 52 11 sess their similarity to empirical distributions from the The vocal mistuning theory states that scales, and intervals, 53 12 database. Most scales tend to include intervals which were chosen not due to harmonicity, but because they were 54 13 are close in size to perfect fifths (\imperfect fifths”), easy to communicate [22, 23]. We perceive intervals as cat- 55 14 and packing arguments explain the salient features of egories [24{27], and due to errors in producing [22, 28, 29], 56 15 the distributions. Scales are also preferred if their and perceiving notes [30{34], musical intervals are not exact 57 16 intervals are compressible, which may facilitate effi- frequency ratios, but rather they span a range of acceptable in- 58 17 cient communication and memory of melodies. While terval sizes. Any overlap between interval categories will then 59 18 scales appear to evolve according to various selection result in errors in transmission. This theory appears promis- 60 19 pressures, the simplest, imperfect-fifths packing model ing, but it has not yet been rigorously investigated. 61 20 best fits the empirical data. These two theories were proposed separately, yet they are 62 21 How and why did humans evolve to create and appreciate not mutually exclusive. In this paper, we modify, integrate 63 22 music? The question arose at the dawn of evolutionary the- and expand upon these ideas to construct a general, stochastic 64 23 ory and been asked ever since [1{4]. Studying musical fea- model which generates populations of scales. Our aim is to test 65 24 tures that are conserved across cultures may lead to possible which model best mimics scales created by humans. To this 66 25 answers. [5, 6]. Two such universal features are the use of dis- end, we assembled the most diverse and extensive database of 67 26 crete pitches and the octave, defined as an interval between two scales from ethnomusicological records. By comparing model- 68 27 notes where one note is double the frequency of the other [7]. generated theoretical distributions with the empirical distri- 69 28 Taken together, these form the musical scale, defined as a set butions, we find that the theory that best fits the data is 70 29 of intervals spanning an octave (Fig. 1A). Musical scales can the simplest. Most scales are arranged to maximise inclusion 71 30 therefore be considered solutions to the problem of partitioning of imperfect fifths { perfect fifths with a tolerance for error. 72 31 an octave into intervals, and thus can be treated mathemat- Scales are often found to be compressible, which may make 73 32 ically. Examination of scales from different cultures can help them easier to transmit. Adding more detail, beyond fifths, to 74 33 elucidate the basic perception and production mechanisms that harmonicity-based theories decreased their performance, which 75 34 humans share and shed light on this evolutionary puzzle. suggests that only the first few harmonics are significant in this 76 35 One theory on the origin of scales suggests that the fre- context. 77 36 quency ratios of intervals in a scale ought to consist of simple 37 integers [8]. After the octave (2:1), the simplest ratio is 3:2, re- Results 78 38 ferred to in Western musical theory as a perfect fifth. Frequen- 39 cies related by simple integer ratios naturally occur in the har- Harmonicity Models 79 40 monic series { a plucked string will produce a complex sound 41 with a fundamental frequency accompanied by integer multi- The main assumption underpinning the harmonicity theory is 80 42 ples of the fundamental. The theory that scales are related to that human pitch processing evolved to take advantage of nat- 81 1 82 ural harmonic sounds [37]. For example, harmonic amplitudes A 83 typically decay with harmonic number [9, 15], and correspond- 84 ingly, lower harmonics tend to be more dominant in pitch per- 85 ception [38]. Thus many proxy measures of harmonicity con- 86 tain parameters to account for harmonic decay [39]. However, CDEFGAB C 87 to minimize a priori assumptions and model parameters, we 88 avoid explicitly modelling harmonic decay. Instead, we test 89 two simple harmonicity theories which differ in how they treat 90 higher order harmonics. 91 The first harmonicity model (HAR), assumes that there is 92 no harmonic decay, for which the model of reference [8] is ap- 0 200 400 600 800 1000 1200 Interval size / cents 93 propriate. This model scores an interval f2=f1 defined by two 94 frequencies, f1 and f2, based on the fraction of harmonics of f2 B 95 that are matched with the harmonics of f1 in an infinite series. 96 Humans do not notice small deviations from simple ratios [40{ 97 42], and the model accounts for this by considering intervals 98 as categories of width w cents; intervals are measured in cents score Harmonicity f2 99 such that an octave is 1200 log = 1200 cents. Intervals are 2 f1 0 200 400 600 800 1000 1200 100 assigned to a category according to the highest scoring interval Interval size/ cents 101 within w=2 cents. The resulting template (Fig. 1B) is used to C 102 calculate the average harmonicity score for each scale across i ii iii 103 all N × (N − 1) possible intervals, apart from the octave; N 104 is the number of notes in a scale. We make no assumptions 105 about tonality, and thus all intervals are weighted equally. The 0 100 200 0 100 200 0 100 200 106 HAR model assumes that scales evolved to maximise this har- 107 monicity score. 108 The second harmonicity model (FIF) considers the limiting Figure 1: A: The major scale starting on a C shown on a piano 109 case of high harmonic decay. As harmonic decay increases, (equal temperament), and as interval sizes (in cents) of each 110 eventually a few intervals in a harmonic series become dom- note measured from the first note. Intervals between adjacent 111 inant (SI Table 1), in the order of unison, octave, fifth, etc. pairs of notes are denoted by IA. B: We use a harmonicity tem- 112 only Thus, this model assumes that due to harmonic decay, plate to assign harmonicity scores to intervals [8]. The highest 113 the octave and the fifth significantly affected the evolution of scoring intervals (frequency ratios are shown for the top five) 114 scales. Given this, we simply count the fraction of intervals act as windows of attraction, whereby any interval in this win- 115 × − that are fifths, out of all N (N 1) possible intervals { we dow is assigned its score. In this case the maximum window 116 do not count the octave, and we make no assumptions about size is w = 40 cents. C: Adjacent interval, IA, sets for the 117 tonality. We allow a tolerance for errors, w, and thus define major scale in equal temperament (i) and just intonation (ii & 118 ± \imperfect fifths” as intervals of size 702 w=2 cents. The iii). Boxes indicate interval categories, where the width repre- 119 FIF model assumes that scales evolved to maximise the num- sents the error, and boxes with similar colours are related by 120 ber of imperfect fifths that can be formed in a scale. a common denominator: (i) is losslessly compressed; a similar compression of (ii) is lossy; (iii) is uncompressed and therefore 121 Transmittability Model costs more to transmit. 122 The transmittability theory assumes that intervals are per- 123 ceived as broad categories, and the humans make errors in both 138 124 production and perception of intervals. Intervals must thus be munication of melodies. An example of a compressible scale 139 125 large enough to avoid errors in transmission due overlapping is the equal temperament major scale (Fig. 1A). We can rep- 140 126 interval categories (SI Fig. 1). This is not sufficient, how- resent this scale using its notes (C, D, etc.) or as a sequence 141 127 ever, to explain the considerable convergence in scales across of adjacent intervals, IA: 200; 200; 100; 200; 200; 200; 100. The 142 128 cultures. We can further consider that scales are optimized most compressed representation uses an alphabet size of one 143 129 for minimizing errors in transmission by favouring the use of by encoding the large interval (200) in terms of the small one 144 130 large intervals, however this bias exclusively favours equidis- (100) (Fig.