Convergent evolution in a large cross-cultural database of musical scales John M. McBride1,* and Tsvi Tlusty1,2,* 1Center for Soft and Living Matter, Institute for Basic Science, Ulsan 44919, South Korea 2Departments of Physics and Chemistry, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea *[email protected], [email protected] August 3, 2021 Abstract We begin by clarifying some key terms and ideas. We first define a scale as a sequence of notes (Figure 1A). Scales, sets of discrete pitches used to generate Notes are pitch categories described by a single pitch, melodies, are thought to be one of the most uni- although in practice pitch is variable so a better descrip- versal features of music. Despite this, we know tion is that notes are regions of semi-stable pitch centered relatively little about how cross-cultural diversity, around a representative (e.g., mean, meadian) frequency or how scales have evolved. We remedy this, in [10]. Thus, a scale can also be thought of as a sequence of part, we assemble a cross-cultural database of em- mean frequencies of pitch categories. However, humans pirical scale data, collected over the past century process relative frequency much better than absolute fre- by various ethnomusicologists. We provide sta- quency, such that a scale is better described by the fre- tistical analyses to highlight that certain intervals quency of notes relative to some standard; this is typically (e.g., the octave) are used frequently across cul- taken to be the first note of the scale, which is called the tures. Despite some diversity among scales, it is tonic. We refer to relative pitch in two ways: we talk the similarities across societies which are most of intervals between notes in units of cents, which can striking. Most scales are found close to equidis- be obtained by a logarithmic transform of two frequen- tant 5- and 7-note scales; for 7-note scales this cies f1 and f2, cents = 1200 × log2 f1=f2; or simply as accounts for less than 1% of all possible scales. a frequency ratio f1 : f2. Also, rather than referring to In addition to providing these data and statisti- each note in a scale relative to the tonic, we can repre- cal analyses, we review how they may be used sent a scale in terms of its set of adjacent intervals, the to explore the causes for convergent evolution in frequency ratio (or cents) between adjacent notes in the scales. sequence. In the example given in Figure 1A, one can see that the frequency ratios with respect to the tonic Introduction are amongst the simplest integer ratios. Early scholars believed that these intervals are innately important, not Music, like language, is a generative grammar consisting necessarily for valid scientific reasons [11]. Nonetheless, of basic units, and rules on how to combine them [1]. In the concept of tonal fusion – where some complex dyads melodies, the basic units are described by two qualities: may appear difficult to distinguish as separate – lends pitch (frequency) and duration (time). We generally refer credibility to this old idea [12, 13]. One interval stands to this basic pitch unit as a note, and a set of notes as out amongst the crowd: the octave, an interval of fre- a scale. Thus, as far as pitch is concerned, a scale is to quency ratio 2 : 1 (1200 cents). In many cultures two a melody what an alphabet is to writing. Despite their notes related by an octave are considered perceptually centrality to music, and apparent ubiquity, we know sur- similar (octave equivalence), and scales are considered to prisingly little about scales. Most studies focus on scales repeat when they reach the octave [14]. As a result, in arXiv:2108.00842v1 [physics.soc-ph] 23 Jul 2021 from a limited number of musical traditions [2, 3], and Figure 1A the first and last note of the scale are repre- the only finding from a broad statistical approach is that sented by the same letter. Despite many claims of uni- scales have 7 or fewer notes [4, 5]; there are anecdotal re- versality, experiments have shown that octave equivalence ports that certain notes are widespread, but this has not may be a weak perceptual phenomenon which is culture- been examined statistically [2, 3, 6–9]. We lack concrete specific. In this work, we aim to provide some statistical understanding of why we use scales, how diverse they are, evidence to assess how widely the octave is used. or how they came to be that way. We suspect that the In what follows, we explain how we create a database reason for this is simply a lack of suitable resources. Here of scales and measured tunings. We analyse these tunings we address this issue by presenting a data set of musical to see what intervals repeatedly occur, verifying the com- scales from many societies, extant and extinct, built upon mon belief that the octave is used among many cultures; a century of ethnomusicological enterprise. however we also note the limited nature of this study on 1 A B Piano Absolute Relative Freq. Adjacent Prescriptive Descriptive Freq. Freq. Ratio Intervals C 112 cents Theory Measured B 204 cents Song A 182 cents Recordings G 204 cents F 112 cents Instrument Tunings E 182 cents D 204 cents C Figure 1: A: Illustration of relevant terms: as an example we show the Western major scale in 5-limit just intonation tuning, starting from middle C and spanning one octave. Scales can be represented symbolically (e.g. letters to represent notes); as a set of absolute frequencies; as a set of relative frequencies in cents or as frequency ratios, relative to the first note in the scale (tonic); as a sequence of adjacent intervals. B: Venn diagram indicating how scale and tuning data can be classified. evaluating universality. We then provide a comprehen- been made of the notes on an instrument (instrument tun- sive, statistical view of musical scales across cultures. By ing), or a recording of a song has been analysed with considering the vast number of possible scales that are not computational tools to extract a scale. Instrument tun- used anywhere, we show that scales are more similar than ings are by default prescriptive, but can be descriptive different: scales are overwhelmingly clustered around 5- in the case where all of the notes are used in a melody. and 7-note equidistant scales (scales where adjacent inter- Despite the fact that there is some error in these measure- vals are similar in size). Finally, we discuss the potential ments (tools range from tuning forks, to the Stroboconn, mechanisms of how scales change over time, discuss the to modern computational approaches; in all cases the er- challenges in understanding how scales have evolved, and ror is reported as less than 10 cents), this type of data propose credible future directions. offers the most objective insight into the scales used in melodies. Measured scales taken from song recordings are exclusively descriptive, and they make up the small- Database est part of the database. This is because it is yet quite a challenge to reliably infer scales from a recording of a per- Database curation formance using algorithms [56], and thus it requires a lot A total of 55 books, journals, and other ethnomusicolog- of manual effort and time. Despite this, we believe that ical sources were found to have relevant data on scales the future of studies on musical scales lies in tackling these [15–69] (SI Table 1). We acknowledge that a previous challenges, due to the potential of extracting descriptive attempt has been made to document scales using ethno- scales from archives of ethnographic recordings [71, 72]. musicological records [70]. However this database lacks However, it is clear that we currently lack the appropriate a defined methodology and does not link scales to refer- computational tools to perform such large-scale analyses ences, so it cannot be independently verified. [73, 74]. We can define scales either prescriptively ("these are To enable a range of potential analyses we collected the notes you can use in a melody") or descriptively information pertaining to, where applicable, the society ("these are the notes that were used in the melody"), the scales / tunings came from (country, language or eth- and we define ways of categorizing empirical scale data nic group), geography (country, region), instrument type, (Figure 1B). Theory scales consist of intervals with exact, tonic note, and modes (for lack of a better word, we define mathematical frequency ratios. These are mainly found modes as the new scale that one gets if you pick a new in a limited set of cultures that exist along the old Silk note as the tonic so that the scale is circularly permuted). Road route, and they are not necessarily played as speci- Some of these (society / geography) we considered so im- fied. These are by definition prescriptive scales, although portant that we declined to include data if these were descriptive scales can be found which closely match these. not identifiable in some capacity. Others were found in- Measured scales are obtained where measurements have frequently; tonic was only identified in 126 out of 413 2 octave (affects 41 out of 413 samples). Measured scales inferred from recordings typically do not span more than an octave; the only relevant choice for these scales is (v). A complete workflow from source to database, including examples, is given in SI Fig. 1. While it is possible to create different versions of the scale database (and details are given on the github repos- itory), we report statistics for one created according to 200 176 171 175 Theory the following choices: (i) We match theory scales from Measured 110 each musical tradition to a set of tuning systems given in 100 91 69 SI Table 2.