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 Korea2Departments of Physics and Chemistry, Ulsan National Institute of Science and Technology, Ulsan 44919,

South Korea*jmmcbride@protonmail.com, tsvitlusty@gmail.com

August 3, 2021

Abstract

Scales, sets of discrete pitches used to generatemelodies, are thought to be one of the most uni-versal features of music. Despite this, we knowrelatively little about how cross-cultural diversity,or how scales have evolved. We remedy this, inpart, we assemble a cross-cultural database of em-pirical scale data, collected over the past centuryby various ethnomusicologists. We provide sta-tistical analyses to highlight that certain intervals(e.g., the octave) are used frequently across cul-tures. Despite some diversity among scales, it isthe similarities across societies which are moststriking. Most scales are found close to equidis-tant 5- and 7-note scales; for 7-note scales thisaccounts for less than 1% of all possible scales.In addition to providing these data and statisti-cal analyses, we review how they may be usedto explore the causes for convergent evolution inscales.

Introduction

Music, like language, is a generative grammar consistingof basic units, and rules on how to combine them [1]. Inmelodies, the basic units are described by two qualities:pitch (frequency) and duration (time). We generally referto this basic pitch unit as a note, and a set of notes asa scale. Thus, as far as pitch is concerned, a scale is toa melody what an alphabet is to writing. Despite theircentrality to music, and apparent ubiquity, we know sur-prisingly little about scales. Most studies focus on scalesfrom a limited number of musical traditions [2, 3], andthe only finding from a broad statistical approach is thatscales have 7 or fewer notes [4, 5]; there are anecdotal re-ports that certain notes are widespread, but this has notbeen examined statistically [2, 3, 6–9]. We lack concreteunderstanding of why we use scales, how diverse they are,or how they came to be that way. We suspect that thereason for this is simply a lack of suitable resources. Herewe address this issue by presenting a data set of musicalscales from many societies, extant and extinct, built upona century of ethnomusicological enterprise.

We begin by clarifying some key terms and ideas. Wefirst define a scale as a sequence of notes (Figure 1A).Notes are pitch categories described by a single pitch,although in practice pitch is variable so a better descrip-tion is that notes are regions of semi-stable pitch centeredaround a representative (e.g., mean, meadian) frequency[10]. Thus, a scale can also be thought of as a sequence ofmean frequencies of pitch categories. However, humansprocess relative frequency much better than absolute fre-quency, such that a scale is better described by the fre-quency of notes relative to some standard; this is typicallytaken to be the first note of the scale, which is called thetonic. We refer to relative pitch in two ways: we talkof intervals between notes in units of cents, which canbe obtained by a logarithmic transform of two frequen-cies f1 and f2, cents = 1200 × log2 f1/f2; or simply asa frequency ratio f1 : f2. Also, rather than referring toeach note in a scale relative to the tonic, we can repre-sent a scale in terms of its set of adjacent intervals, thefrequency ratio (or cents) between adjacent notes in thesequence. In the example given in Figure 1A, one cansee that the frequency ratios with respect to the tonicare amongst the simplest integer ratios. Early scholarsbelieved that these intervals are innately important, notnecessarily for valid scientific reasons [11]. Nonetheless,the concept of tonal fusion – where some complex dyadsmay appear difficult to distinguish as separate – lendscredibility to this old idea [12, 13]. One interval standsout amongst the crowd: the octave, an interval of fre-quency ratio 2 : 1 (1200 cents). In many cultures twonotes related by an octave are considered perceptuallysimilar (octave equivalence), and scales are considered torepeat when they reach the octave [14]. As a result, inFigure 1A the first and last note of the scale are repre-sented by the same letter. Despite many claims of uni-versality, experiments have shown that octave equivalencemay be a weak perceptual phenomenon which is culture-specific. In this work, we aim to provide some statisticalevidence to assess how widely the octave is used.

In what follows, we explain how we create a databaseof scales and measured tunings. We analyse these tuningsto see what intervals repeatedly occur, verifying the com-mon belief that the octave is used among many cultures;however we also note the limited nature of this study on

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Figure 1: A: Illustration of relevant terms: as an example we show the Western major scale in 5-limit just intonationtuning, starting from middle C and spanning one octave. Scales can be represented symbolically (e.g. letters torepresent 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 howscale and tuning data can be classified.

evaluating universality. We then provide a comprehen-sive, statistical view of musical scales across cultures. Byconsidering the vast number of possible scales that are notused anywhere, we show that scales are more similar thandifferent: scales are overwhelmingly clustered around 5-and 7-note equidistant scales (scales where adjacent inter-vals are similar in size). Finally, we discuss the potentialmechanisms of how scales change over time, discuss thechallenges in understanding how scales have evolved, andpropose credible future directions.

Database

Database curation

A total of 55 books, journals, and other ethnomusicolog-ical sources were found to have relevant data on scales[15–69] (SI Table 1). We acknowledge that a previousattempt has been made to document scales using ethno-musicological records [70]. However this database lacksa defined methodology and does not link scales to refer-ences, so it cannot be independently verified.

We can define scales either prescriptively ("these arethe notes you can use in a melody") or descriptively("these are the notes that were used in the melody"),and we define ways of categorizing empirical scale data(Figure 1B). Theory scales consist of intervals with exact,mathematical frequency ratios. These are mainly foundin a limited set of cultures that exist along the old SilkRoad route, and they are not necessarily played as speci-fied. These are by definition prescriptive scales, althoughdescriptive scales can be found which closely match these.Measured scales are obtained where measurements have

been made of the notes on an instrument (instrument tun-ing), or a recording of a song has been analysed withcomputational tools to extract a scale. Instrument tun-ings are by default prescriptive, but can be descriptivein the case where all of the notes are used in a melody.Despite the fact that there is some error in these measure-ments (tools range from tuning forks, to the Stroboconn,to modern computational approaches; in all cases the er-ror is reported as less than 10 cents), this type of dataoffers the most objective insight into the scales used inmelodies. Measured scales taken from song recordingsare exclusively descriptive, and they make up the small-est part of the database. This is because it is yet quite achallenge to reliably infer scales from a recording of a per-formance using algorithms [56], and thus it requires a lotof manual effort and time. Despite this, we believe thatthe future of studies on musical scales lies in tackling thesechallenges, due to the potential of extracting descriptivescales from archives of ethnographic recordings [71, 72].However, it is clear that we currently lack the appropriatecomputational tools to perform such large-scale analyses[73, 74].

To enable a range of potential analyses we collectedinformation pertaining to, where applicable, the societythe scales / tunings came from (country, language or eth-nic group), geography (country, region), instrument type,tonic note, and modes (for lack of a better word, we definemodes as the new scale that one gets if you pick a newnote as the tonic so that the scale is circularly permuted).Some of these (society / geography) we considered so im-portant that we declined to include data if these werenot identifiable in some capacity. Others were found in-frequently; tonic was only identified in 126 out of 413

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Figure 2: Scales in the database either come from amathematical theory (theory) or from measurements frominstruments or recordings (measured). The map showsthe geographic origin of the scales (theory: regions; mea-sured: dark shaded countries), with sample size, S, indi-cated by the marker size.

instrument tunings; only two sources precisely identifiedsets of modes that could be drawn from an instrumenttuning [58, 68].

Extracting scales from raw data

The database exists in two forms: the raw data, and aset of scales that is generated automatically from the rawdata according to a set of choices. (i) Theory scales aretypically given in symbolic notation such as letters / sylla-bles (e.g., solfège, swara), which can then be converted toa numerical sequence by matching the symbols to a tuningscheme such as just intonation, or 12-tone equal temper-ament (12-TET). For each theory scale in the database,we provide a set of tuning schemes that were plausiblyused, so a single theory scale may have several versionsin different tunings (it is possible to easily change the as-sumptions used to create a different set of scales). (ii)The tonic is the first note in all theory scales, and inmany cases modes are also found in musical traditions.However, if one wants completely ignore tonality, one canalso create all possible modes from all theory scales. (iii)Tonality is rarely noted in studies of instrument tunings,which means it is impossible to uniquely identify the orderof notes as intended by the performer. For these scales,one can: include only those that denote the tonic; in-clude all plausible scales (all modes); or include more thanone scale if there are multiple octaves without includingmodes. We define a plausible scale on an instrument tun-ing as a sequence of notes that span an octave. (iv) Inpractice, intervals are not perfect so we accept scales ifthe intervals sum to 1200±O cents, where O is the errortolerance. (v) In some cases, sources indicated that theoctave was used in performances, but reported the scaleswithout the octave. For these cases, we can choose to in-clude the missing interval that is needed to complete the

octave (affects 41 out of 413 samples). Measured scalesinferred from recordings typically do not span more thanan octave; the only relevant choice for these scales is (v).A complete workflow from source to database, includingexamples, is given in SI Fig. 1.

While it is possible to create different versions of thescale database (and details are given on the github repos-itory), we report statistics for one created according tothe following choices: (i) We match theory scales fromeach musical tradition to a set of tuning systems given inSI Table 2. (ii) We do not include all possible modes oftheory scales. (iii) We do not exclude scales inferred frominstrument tunings if we do not know the tonic, but wedo not include all possible modes. (iv) We include scalesthat require an extra interval to complete the octave. (v)We use a tolerance of O = 50, finding that few of the ex-tracted scales have such a large octave deviation (SI Fig2). This results in a total of 896 scales (462 theory scales,383 scales from instrument tunings, 51 scales from songrecordings). The theory scales span 5 regions, while themeasured scales span 8 regions, and 39 countries (Figure2). However, we must note that an assumption central tothe above methodology is that scales add up to a octave.Since we are unsure about how valid this assumption is,we first study the statistics of the raw instrument tuningsbefore studying the extracted scales.

Statistics of Instrument Tunings

Studying the statistics of instrument tunings allows theleast biased view of scales / intervals, since we do not alterthe raw data. This does not mean they are completely freefrom bias, and we refer the reader to the discussion sectionfor more detail. With this data, we can ask what notes arefound more or less than others, and estimate how signif-icant these results are according to different assumptionsabout how scales are generated. There is, of course, noinnately correct null distribution for instrument tunings;hence, our approach is to try several statistical models.

Outstanding tonic intervals: 200, 700, 1200 cents

In this section we limit our analysis to tonic intervals, as-suming that the lowest note is the tonic. Our first modelassumes that intervals, I, (as measured from the firstnote) are drawn independently from a lognormal distri-bution, P (I) = lnN(µ, σ2). Lognormal is an appropriatechoice on two accounts: (i) very small intervals should beunlikely due to limits in pitch perception, so P (I) → 0as I → 0; (ii) for clear physical reasons (human anatomyconstraints; instrument material cost, etc.) large intervalsare be less likely, so P (I) → 0 as I → ∞. This null dis-tribution states that intervals are entirely independent ofeach other, and the probability only depends on the sizeof the interval. While this is inherently wrong (notes areunlikely to be repeated within a given instrument tuning,which contradicts the idea of independence), it allows usto compare different I to say which are especially frequent/ rare. We compare the real distribution with the null

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Figure 3: A: Distribution of intervals relative to the tonic for instrument tunings. We show distributions for 6different samples: all samples (All); a subset balanced by region (Reg-Samp); a subset balanced by culture (Cult-Samp); subsets that only include data taken from idiophones, aerophones, and chordophones. For Reg-Samp andCult-Samp, we show distributions from three independent samples. We show a lognormal fit to each distribution(dotted line), and indicate the 99% bootstrapped confidence interval (shaded region). We truncate the x-axis at3000 cents for clarity. B: We repeated the same analyses on an artificial set of tunings resampled using the posteriordistribution of adjacent intervals sizes and exact same numbers of notes in a tuning.

distribution for the following samples (Figure 3A): all ofthe instrument tunings (‘All’, 404 tunings), a region sub-sample (‘Reg-Samp’, 8 contintent groups, at most 10 tun-ings chosen per region, 78 tunings), a culture sub-sample(‘Cult-Samp’, 72 cultures, at most 5 tunings per culture,200 tunings), and three sub-samples based on the typeof instrument the tuning is observed on (Idiophone, 195;Aerophone, 76; Chordophone, 55).

We find the most prominent deviations from the lognor-mal distribution at 1200, 700 and 200 cents (more likely),and 600 cents (less likely); these are consistently outsideof the 99% confidence intervals across almost all samples.Of the rest, we see 2400 cents three times (All, Cult-Samp,Aerophone), and ∼500 cents twice (All, Idiophone).

Our second model assumes that scales are constructedby combination of adjacent intervals, IA. We create a testdata set (TEST1) by matching the number of tunings,and the number of notes per tuning, and then samplingadjacent intervals with replacement from all adjacent in-tervals observed in instrument tunings. We then comparethis test set against a lognormal distribution (Figure 3B),finding that the only significant interval is ∼ 200 cents,which is also the mean size of an adjacent interval (198cents). Thus the reason that 200 cents appears to be sig-nificant in Figure 3A is simply due to its prevalence asan adjacent interval, while the the other significant re-

sults cannot be said to be simply due to the combinationof the available adjacent intervals. The adjacent inter-vals appear to be arranged in some specific order whichresults in some intervals occuring more frequenctly thanexpected by chance.

We then check whether we get the same result if we pre-serve the original sets of adjacent intervals for each tuning:we create a second test data set (TEST2) by matching thenumber of tunings, and the number of notes per tunings;however, instead of sampling from all observed adjacentintervals, we simply shuffled the adjacent intervals in eachtuning (SI Fig 3). In this case we find that the octave ismuch more frequenctly than expected from a lognormaldistribution. However, this is because there are many in-strument tunings where the highest interval is 1200± 15;when we control for this, we find that only 200 cents is sig-nificant again (SI Fig 4). Thus we can say that the peakat 200 cents is due to this being the mean size of adjacentintervals, but the peaks at 500, 700 and 1200 cannot beexplained by any of the suggested null distributions.

Further tests support outstanding intervals

The previous method was able to establish whether in-tervals are found more than expected by chance acrossa collection of instrument tunings. We can extend thisapproach by using the full range of intervals that can be

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Figure 4: A: Mean deviation of original intervals from the octave compared to shuffled intervals for 147 instrumenttunings; * p < 0.05, ** p < 0.005, *** p < 0.0005, or not significant (NS). Black line: x = y. B: Fraction of resultsthat are significant, and show that an interval size is found either more or less frequently than expected by chance.We show results for the whole dataset (All), region-balanced set (Reg-Samp), a culture-balanced set (Cult-Samp),and for a sampling scheme (TEST1) that only maintains the overall distributions of adjacent interval sizes and notesper tuning (Null). C: Fraction of significant results indicating that the octave is found more or less frequently thanchance against tuning devation in a test set which maximises the number of octave relationships. Shaded regionshows 95% bootstrapped CI.

made on an instrument, not just the tonic intervals. Foran instrument with N notes, this results in N×(N−1)/2intervals. Since this samples many more intervals, thistest is powerful enough to sometimes detect significantresults within single instrument tunings, not just acrossthe entire collection.

This test compares each instrument tuning with alter-native tunings that can be made by shuffling the order ofthe adjacent intervals (TEST2). This allows us to ask,for a given interval I, does this interval occur more oftenin the instrument tuning than one would expect if theyknow the adjacent intervals but not the order. To answerthis, for each tuning we find all intervals within some dis-tance w of I and calculate how far they are from I, andthen repeat the process for 50 shuffled versions of the tun-ing. We then use a Mann-Whitney U test to test whethereither set of intervals (original or shuffled) is significantlycloser to the octave than the other. We then repeat this100 times to get a converged average.

To demonstrate this analysis we show the mean devia-tion from the octave for the sets of intervals taken fromthe orignal tuning and the shuffled sets (Figure 4A). Formost tunings the results are not significant since the sam-ple size of intervals that are within w = 100 of 1200 centstends to be small. Another reason for non-significant re-

sults is that there are many scales where the adjacentintervals are of similar size, and can add up easily to aparticular value (in this case the octave); in these cases,the shuffled scales are just as likely to have many octaveintervals as the original scales.

We then extend this analysis to intervals over the range200 ≤ I ≤ 2600 cents, for the whole set of tunings, andfor two sub-samples (Figure 4B). We see that, out of allintervals, the first and second octaves are those most oftenfound in tunings at a rate that is significantly greater thanchance (40% of the time). They are followed by intervalsof 500, 700, 1000, and 1400 (and corresponding intervalsan octave above), in agreement with the data in Figure 3.These results are weaker for ‘Reg-Samp’, but nonethelessthere is good agreement across sub-samples. Similarly, wefind that 600 cents, 1150 and 1250 cents are often missingfrom tunings at a rate that is significantly greater thanchance. To put these results in perspective, we once againcreate test data as in TEST1, and re-run the analyses(Figure 4B, Null), which shows the range of significantresults expected by chance. This clearly demonstratesthat the previous high fractions of significant results werenot an artefact due to testing multiple hypotheses. Also,we show that the results do no depend on our choice ofw (SI Fig 5). These results reinforce the conclusion that

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these intervals are not simply found due to a preference forcertain sets of adjacent intervals – the adjacent intervalsare clearly arranged to achieve some preferred intervalswith respect to the lowest note.

Tuning variability hinders search for universal in-tervals

What fraction of significant results should we expect tofind if those tuning instruments do indeed aim to tuneoctaves (or any interval)? To answer this we create a testset of scales (TEST3) with maximal possible octave inter-vals. We match the number of tunings, and the numberof notes per tuning, but we alter the intervals so that forevery interval where I ≥ 1200, there is a correspondinginterval an octave below. In reality humans (as any mea-surement device) have imperfect pitch resolution, so wealter this test set by adding normally distributed noise,N(µ = 0, σ2) to see how much this reduces the chance ofdetecting significant results. We find that even withoutnoise (σ = 0), this test can only find significant results90% of the time (Figure 4C). A realistic estimate of σcould start by considering that tunings in the databaseare usually measured to within 10 cents, but this is notthe main source of variation. In humans, interval discrim-ination drops below about 100 cents [75–78], and humanstypically sing with a standard deviation of at least about10 - 20 cents [10, 69, 79–84]. Thus we can expect a rea-sonable upper bound to the fraction of significant resultsthat can be detected of about 0.3 to 0.7. For comparison,we find the octave to occur significantly more than chanceabout 40% of the time. Thus, due to and empirical mea-surement (human or otherwise) error, there is a limit tohow much one can infer the significance of the prevalenceor lack of a particular interval in a single scale, melody ortuning. Despite this limitation, we find that our analy-sis finds a fraction of significant results supporting use ofthe octave which is within the range of this hypotheticallimit.

Ultimately, we cannot say how scales are chosen. Arethere some important intervals that are fixed first (e.g.octave), and then the rest are chosen to fill the gaps?Or are adjacent intervals of a certain size chosen, andthen arranged in some preferred order? All this analysishas shown is that whichever way scales are created, theyshow a significant, consistent bias towards including somenotes, and avoiding others.

Assessing evidence for the use of the octave bysource

While the octave is often reported as one of the most uni-versal features of music [2, 3, 7–9, 85], this status has beendisputed [69, 86–89]. To understand why, we review therelevant physical, statistical and perceptual ways in whichthe octave is considered special. Physically, the octave isproduced naturally as part of the harmonic spectra. Itis the most common and salient interval to appear in theharmonic spectra (occurs most frequently, occurs in rela-tion to most salient harmonics). Mathematically it is the

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Figure 5: Scale degree distributions for different samplesof the scale database: all scales, theory scales, measuredscales, sample balanced by region, sample balanced byculture.

simplest frequency ratio (2:1). In terms of statistics, it iswidely described as a universal feature of music and scales[2, 3, 7–9, 85]. Some state that it is better described asa statistical universal: present in many, but not all musi-cal traditions [87, 88]. It is difficult to judge which viewis correct, since most studies refer to only a handful ofspecific examples or to anecdotal evidence from experts.As of yet, however, there is no study that systematicallyexamines the prevalence of octave usage across cultures.The evidence provided in this work has so far lent supportto the notion that it is widespread, but it is still unclearto what extent. In terms of perception, octaves are oftenperceived as similar, and scale structure is often seen ascircular (or helical), with a periodicity of one octave [14].This can be seen most clearly in cultures where notes arenamed, and notes that are related by an octave have thesame name. Or when instruments or singers that havedifferent frequency ranges play / sing together, they of-ten play / sing in parallel octaves. Despite these culturalphenomena, percepual experiments on the octave tend todemonstrate rather weak effects [90–93], which are oftenconditioned on musical training [94–96]. Nonetheless, re-cent work has shown strong effects for harmonic sounds ingeneral, with regard to tonal fusion [13, 93], the ability tohear in noise [97], and memory of complex tones [98]. Al-together, the evidence points to a scenario where humansmay be able to take advantage of the mathematical regu-larities, and statistical abundance of harmonic sounds, insuch a way that this faculty can be strengthened throughcultural traditions and experience. It suggests that hu-mans can learn to appreciate the octave in a special way,but they need to be taught to first. To further supplementour earlier statistical analysis of instrument tunings, wenext consider what descriptive evidence exists in the stud-ies cited in the scale database.

After a first general reading of the sources, it seemedappropriate to classify sources in terms of evidence for:use of the octave when tuning instruments (8 sources),

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Figure 6: Distributions of adjacent intervals (left) and notes in scales (right) for different samples of the scaledatabase: (top) all scales, theory scales, measured scales; (middle) sample balanced by region, sample balanced byculture; (bottom) measured scales according to each instrument type. Shading indicates 95% CI. Histogram bins are20 (left) and 30 (right) cents.

using the same name for notes an octave apart (8sources), performing melodic harmony in parallel octaves(11 sources). Overall, we found descriptive evidence insupport of some sort of awareness of the octave as a spe-cial interval in 24 sources. Some studies lack any discus-sion of whether the octave is used in the musical tradi-tion in question, but separate studies indicate use of theoctave (7 sources). Instruments were sometimes tunedwithout reference to the octave, but instead: in scalarorder [18, 29, 31], or in fifths [32], or by playing tunesand seeing if the tuning fits [24]. No instances were foundof unique names for notes outside the range of an oc-tave. Authors rarely discussed the absence of the octave,except in terms of intervallic range of melodies [21] Inseveral cases authors discuss the octave in detail with-out any description of whether the octave is appreciatedin any way by the original performers [22, 61, 65]. Over-all, when including both descriptive evidence and the sta-tistical evidence from the previous section, a total of 36sources exhibit some evidence for specific appreciation ofthe octave, 2 show evidence to suggest that the octave isactively avoided [65, 69], 11 do not show any evidence (outof which 3 are reporting on ancient instruments obtainedthrough archeological sites [28, 37, 43]). These details arecovered in more detail in SI Table 3. A limitation of thisanalysis is that the sources used are only those that in-clude quantitative information on scales. An independentinvestigation of ethnographic texts (such as [99]) into thestatistics of octave use is recommended for a more com-plete picture. Nonetheless, these findings strongly sup-port the view that octaves are widespread, but they arenot as universal as once supposed.

Statistics of Scales

Scale degree

We find that, in agreement with previous work [5], mostscales have 7 or fewer degrees (Figure 5). There is a cu-rious lack of 6 note scales, that persists across samplingschemes, but is minimized when controlling for Culture.

Adjacent interval and scale note distributions

We next look at distributions of two important represen-tations of scales. Adjacent scales are particularly relevantfor scalar motion in melodies. Scale notes are the inter-vals with respect to the tonic. We show three differentsets of distributions: all scales, theory scales and mea-sured scales; sub-sets of scales sampled by region and byculture; sub-sets of measured scales according to the typeof instrument they come from.

The adjacent intervals show markedly different distri-butions for theory and measured scales (Figure 6). Onone level this is to be expected, as the theory scales aregiven as exact theoretical values, and as such the naturalvariation in instrument tuning of vocal intonation is omit-ted. As such, the peaks (100, 200, 300 cents) in theoryscales are very sharp. In comparison, distributions formeasured scales are much broader. Taking subsamplescan either smooth the distributions (Region, Culture), orexaggerate differences (see peak at ∼ 350 cents for Chor-dophones). Some features of adjacent interals, however,are general across all groups: they are rarely less than100 cents, and rarely greater than 400 cents (in Carnaticmusic, rules explicitly forbid larger intervals [100]); theyare mainly concentrated between 100 and 300 cents, with

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Figure 7: A: Dendrogram showing the relations between scales, grouped into 6 clusters. Scale names are shown asthey are given in the database. For each cluster we show: (B) the scale note distributions, with dotted lines indicatingthe average scale; (C) the frequency and relative frequency of occurrence within each region.

a salient peak at about 170− 200 cents.

The scale note distributions show remarkable conver-gence across the different samples (Figure 6). There is aclear peak in all cases at 200, 500 and 700 cents. In all dis-tributions the peak at 700 cents is sharp. The next mostcommon peak is at 200 cents; although 500 cents is veryprominent in some distributions, it is not as prominent inaerophones and idiophones. There are then broad regionsthat tend to be well-populated: 300-400 cents, 800-900cents, 1000-1100 cents. We also see regions that are notpopulated. Particularly notable are the regions border-ing 0, 700 and 1200 cents (also 280 cents, 450 cents, 950cents). The tritone (600 cents) is rarely found in mea-sured scales, but is found in theory scales.

The shape of these distributions depends heavily on thenumber of notes in a scale (SI Fig 7). This means thatthe graphs shown here depend on the over-representationof 7-note scales. Despite this, we still find salient peaks

at 200, 500 and 700 cents, regardless of the scale degree.

Scale clusters

To get a sense of the variation between scales we clusterthem according to their scale notes (hierarchical cluster-ing, Ward’s method, euclidean distance [101]) to create 6clusters of scales. We divide the scales into only 6 clus-ters, simply to facilitate discussion of the results. Usingthe euclidean distance restricts us to studying scales ofthe same length, so we present results for 7-note scales asan example since they are most numerous. We show theclusters in Fig 7, indicating some of the names of scalesor the instruments / cultures they were taken from (fullset is given in Supplementary Data).

The largest cluster (a, purple, 220 scales) contains thescales that are closest to an equiheptatonic scale (e.g.“Asena 27”), and it is very well represented across geo-graphical regions (except Oceania, which only has 3 7-

8

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Figure 8: A: Distribution of shortest euclidean distance between each real scale and its nearest neighbour (Real-Real), and between each scale enumerated on a grid (Real-Grid) and its closest real scale. Dashed lines indicateboundaries used to establish subsets of Grid-scales as Close to (within 50 cents) or Far from (greater than 100 cents)real scales. B: Distribution of adjacent intervals for real scales, Close, and Far. C: Distribution of distance from theequiheptatonic scale for real scales, Close, and Far. D: Distributions of scale notes for notes 2-7 (missing tonic andoctave) for real scales, Close and Far. Dotted lines show corresponding position of notes in the equiheptatonic scale.

note scales in total). The second largest cluster (e, blue,100 scales) is distinguished by being the most evenlyspread across geography in terms of relative frequency,and also by the lack of fifths compared to the tritone.This is followed by a cluster (d, orange, 90 scales) whichcontains the Major scale, which is also known as Bi-lawal (North India), Maqam Ajam Ushayran (Arabic),and Qing Yue (Chinese). The smallest three clustersshow progressively less global geographical distributions(b, brown, 57 scales; c, green, 42 scales; f, red, 35 scales).It is possible that these clusters are found in fewer placessimply due to the smaller size of these clusters. Anotherinterpretation is that the use of certain scales was spreadvia migration and trade: for cluster f, the scales are al-most entirely from regions in Asia; for cluster c, most ofthe scales are found between Europe, Middle East andSouth Asia.

Some regions show much greater diversity than others.South Asia shows the greatest diversity (as measured bythe entropy of cluster distributions; SI Fig 8), which isprimarily due to the 72 melakarta of the Carnatic tra-dition. These are 72 scales, each with their own name,which were originally derived via combinatorial enumer-ation of intervals within constraints. The diversity fromSouth East Asian scales, on the other hand, stems almostexclusively from two named scales: the Thai 7-note scale,and the Pelog scale from the Gamelan tradition. Despitebeing named the same, there are countless versions of thesame scale, to the point that there are versions of thePelog scale in 4 out of the 6 clusters (the majority, how-

ever, being in cluster f ).

Comparison with all possible scales

To put the diversity of scales in a broader context, wecan consider the hypothetical world of possible scales byenumerating them on a grid. For brevity, we only presentresults for 7-note scales, since they are most numerous.We take 20 cents as a basic grid size, for several reasons(limits on singing accuracy and pitch perception percep-tion; historical smallest intervals – Greek comma, sruti).By limiting the scales to those that include adjacent in-terval sizes between 100 and 350 cents we already removeover 97% of possible scales (using a grid size of 50 centsresults in a reduction of 94%). We enumerate all remain-ing possible scales on this grid (grid scales), and comparereal and grid scales by calculating the distances betweenthem.

We find that the real scales are much more similar toeach other than the grid scales (Fig 8A). We then de-fine subsets of grid scales either far from real scales (Far;distance greater than 100 cents) or close to real scales(Close; distance less than 50 cents). We find again thatreal scales, and Close scales, tend to have adjacent inter-vals of about 200 cents, and rarely have large or smallintervals; compared to Far scales, which exhibit a moreuniform distribution (Fig 8B). By comparing the distribu-tions of each note separately, we can see that real scalestend to have notes near to 7-TET tuning systems (Fig8D); this is especially salient for notes near 500 and 700

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cents. We then look at the average distance between notesin real / grid scales and the equidistant heptatonic scale(Fig 8C). We find that in 80% of real scales, notes arewithin on average 50 cents of 7-TET. In contrast, only20% of the sampled grid scales are within this region. Ifwe take into account the entire set of grid scales (not onlythose with 100 ≥ IA ≥ 350), the probability of randomlyselecting a scale within 50 cents of 7-TET is less than 1%.Repeating the analysis for 5-note scales leads to the samequalitative conclusion (SI Fig X).

Discussion

How different are scales?

A key finding in this work is that despite some differ-ences, scales across cultures are remarkably similar. Outof all the scales we found, most can be thought of asvariations on the 5- and 7-note equidistant scales. Thisis at odds with some recent literature that claims thatequidistant scales are rare [5, 9, 102–104], which seemto be aided by some hypothesised benefits of unequalstep sizes [105, 106]. It seems that equidistant scales canonly be considered rare if you are extremely conserva-tive about what level of precision must be reached to becalled equidistant. Out of all possible scales that one canmake, equidistant scales are exceedingly rare (Figure 8);yet 80% of heptatonic scales found in this work are closerto equiheptitonic than 99% of possible heptatonic scales.Thus, the results unequivocally show that scales tend tobe approximately equidistant, with 5 or 7 notes (Figure5); and variation in scales can be described in terms ofhow they deviate from equidistance.

It is important to consider that some differences maybe artefacts arising from comparing theory and mea-sured scales. Theory scales are ideal representations, butwhen performed they must exhibit some tuning variabil-ity. When this is studied, it typically contains questionsof which ideal tuning system is intended – e.g., 12-TETor just intonation (JI) – [83, 84, 107] despite the factthat differences in intervals from different tuning systemsare undetectable due to limits on singing stable pitchsand pitch perception [11]. In contrast, when discussingintonation in societies without theoretical scales, tun-ing variation is often highlighted as an intrinsic part ofthat culture [44, 48, 108–110]. There is indeed evidencethat variability in tuning is intentional in some societies[107, 109, 111], and in other societies musicians may tol-erate large deviations in their tunings [112], but it feelsat times like there is an underlying assumption in the lit-erature of differences between societies with establishedmusic theory, and those without. Unfortunately, we lacksufficient examples of instrument tunings from societieswith theory scales for a thorough comparison, and micro-tonal deviations may have been airbrushed out of histor-ical records due to the reliance on Western notation intranscription [113]. Despite this, we briefly compare theempirical tuning variability in cultures. We first calcu-lated standard deviations between notes in the slendro

scale for two cases [26]: comparing repeated measure-ments of the same orchestra at different times (8 cents),comparing notes across orchestras (16 cents). Pelog scalesshowed more variability (13 cents, and 37 cents). Analysisof all Thai scales in the database gives a standard devi-ation of 22 cents. To compare with Western music wecalculate the deviation between notes in 12-TET, just in-tonation (JI) and Pythagorean tuning of Greek modes (7cents), and the deviation between similar-sized intervalsin a Belgian carillon (15 cents) [114]. While these resultslack statistical power, they suggest that the differencesin tuning variability between societies are not necessarilythat great – i.e., what scholars perceive as difference, maynot be perceived as such by others.

How did far-away societies come to use such similarscales? The explanation undoubtedly entails some combi-nation of cultural diffusion and convergent evolution, butdiscerning between these two is fiendishly difficult. Thefinding that most scales are almost equidistant seems toresult from convergent evolution (Figure 8), yet we alsofind evidence that geographical regions can also special-ize in certain types of scales (Figure 7). Several attemptshave been made to link musical traditions between SouthEast Asia and Africa [60, 115, 116] (and Central Americaand Africa [65]). However, lacking any rigorous analy-ses of scale content, they only focused on the apparentequiheptitonic nature of the scales. This appears to befurther evidence of bias in the perception of differencesbetween scales in the literature cited here. Many equi-heptatonic scales from across the world are similar to the-ory scales, but this is rarely stated. For example, thereare Thai scales that are differ from the JI Lydian modeby less than 43 cents (euclidean distance), which is thesame distance between the JI and the 12-TET version ofthe Lydian mode. Upon actually comparing the statisticsof how similar scales are from Africa, South East Asia,and Western Europe, we find substantial overlap betweenall regions (SI Fig 9). While there is undoubtedly sometransmission of information and scales across cultures, thelevel of similarity seen across such great geographical di-vides points mainly to convergent evolution, rather thancultural diffusion.

How do scales change over time?

Scales can change, or persist, over time in a variety ofways. On a short time-scale, vocal (descriptive) scalesare inherently statistical, due to a lack of precision in mo-tor control and pitch perception [117]. Performances areephemeral, so in between performances scales reside inmemory – this introduces another mechanism of change.Unlike vocal scales, instrument tunings physically persistthrough time, and the extent depends on how the instru-ment is constructed. Environmental changes (tempera-ture, humidity) will quickly affect the tuning of stringedinstruments, but effects will be slower for a metal xy-lophone. Through physical force, the metal reeds of anmbira may be knocked out of place [29], or a string maybe broken. To illustrate this point, we checked sources for

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examples of repeat tunings of the same instruments acrossa specific timeframe: Gamelan orchestras were found tohave a standard deviation of 8 (slendro) and 13 (pelog)cents over about 25 years [26]; a likembe (plucked metallamellophone) was found to vary by 18 cents over a fewweeks [29]; a kora was found to vary by 27 cents over oneweek [32]. Alternatively, scales can exist in collective hu-man memory through music theory; e.g., one can tracethe history of the major scale from the Ionian mode givenin Pythagorean tuning, up to the present-day commonuse of 12-TET. To counteract this change, musical theory(and other technological innovations) is perhaps the mostrobust method of preserving a tuning standard over time.Implicit awareness of tonal fusion would have been suffi-cient to establish an early method of maintaining a tuningstandard over time, by tuning simultaneous tones accord-ing to octaves and fifths. In more recent times this hasbeen reinforced though global tuning standards (concertpitch) and inventions such as fixed pitch instruments andelectric tuners. Thus, we can see that scales change; thetimescale of change can depend on technology (quality ofinstrument construction / materials, invention of digitalmusic technology).

One factor that greatly complicates the study of evolu-tion of scales is that it is possible to invent an entirely newscale; this is completely at odds with typical ideas of evo-lution as incremental change. There are many examplesof the use and invention of microtonal tunings in West-ern music [16, 44, 118]. In fact, many theory scales bearthe hallmarks of mathematical generation. Greek modesare all circular permutants, based on simple integer ra-tios [119]. The current system of Carnatic melakarta is aresult of combinatorial enumeration of a set of intervals,constrained by a set of rules [100]. Similarly, other cul-tures that have a long history of mathematical scholarship[120] all use overlapping sets of scales based on theoreticaldivisions of the octave [121–126].

Scales naturally change over time, and they can be in-vented completely from scratch. Despite this, scales aresurprisingly similar across cultures, which suggests thereis some selection for certain types of scales.

How are scales selected?

What are the possible selection pressures acting on scales?We propose three categories of selection pressure. Cul-tural evolution biases are based on how many people, orwhich people in your group are using the scale [127]. Cog-nitive biases depend on how pitch is perceived by humans.Production biases depend on what is easy or difficult tosing, and physical constraints on instruments.

There are some notable examples of cultural selectionin scales. For some reason, humans tend to synchro-nize (multiple reasons have been suggested on the basisof evolutionary psychology, e.g., in-group signalling, so-cial bonding, signalling of coalition strength [128, 129]),which may be what leads to the apparent conformity bias(selection for popularity). In Western Europe, 12-toneequal temperament (12-TET) was selected for its ability

to allow musicians to synchronize tunings with less effort.Similarly, Hindustani music, which traditionally uses justintonation (JI) tuning, has shown a shift towards 12-TETsince the introduction of the harmonium, an imported in-strument with pitches fixed to 12-TET [83]. One studyfound that scales inferred from African recordings shiftedtowards 12-TET over time [130]. Novelty biases can befound in the numerous composers in the 20th centurywho experimented with microtonal tunings [118]; Game-lan tuners also indicate the use of tuning variability as aform of expression [109]. There is also some indicationof a prestige / success bias, where tunings are chosen be-cause they are associated with success, or good ability[67].

It is more difficult to talk of cognitive biases; they areless direct so it is difficult to verify their impact. Somehave suggested a bias towards harmonicity; while the sug-gested mechanism of vocal similarity leaves a lot to bedesired [131], the prevalence of octaves and fifths doeshint that harmonicity is important. Others have high-lighted the need to efficiently, and reliably, communicateand remember melodies [10, 132]. There is a trade offbetween the complexity of scales and the cost / error rateof these two processes, which may explain why adjacentintervals are rarely smaller than a semitone (∼ 100 cents)[77]. Some have also suggested that sensory dissonance,or beats, contributes to the evolution of scales [133]. Itis true that many cultures show an aversion to beats, butthe opposite trend is found in some Balkan and Lithua-nian singing traditions [81]. Despite the relevance of someestablished psychophysical phenomena, which, if any, af-fects the evolution of scales probably differs by society.

Production constraints can apply to both singing andinstruments [134]. When singing, large intervals are moredifficult to produce than small intervals, although due tolimits on motor control, it is impossible to reliably pro-duce intervals below a certain size [135]. Vocal rangeis typically constrained to about two octaves for bothmen and women, with women singing typically aboutan octave higher than men. For instruments, there areclear limits to how many notes one can fit (how manystrings, reeds, blocks, etc.), what size of interval rangesyou can achieve. Aerophones may naturally result in atendency to use harmonic intervals, since they are ofteneasily achieved by overblowing.

Due consideration should be given to the stories of peo-ple who tune instruments. Tuning instruments by com-paring harmonic intervals has a rich history. Using fifthsto match the sound of two strings goes back as far as an-cient Greek [119], and perhaps ancient Babylonian times[136], while in modern times the standard tuning on a gui-tar mainly requires matching of fourths. Similar accountsare found throughout the world on using fifths and oc-taves to tune instruments [30, 32, 39, 47, 68]. Musicianswill also tune their instruments according to the soundof adjacent intervals [18, 29, 31]. An underappreciatedform of tuning is to tune an instrument visually [67, 68];e.g., are the holes on a flute evenly spaced? Do the xy-lophone bars increase in size according to a smooth gra-

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dient? One can also tune according to some standard(outstanding Gamelan orchestra [26]; the best-soundingplayer in a group; using an electric tuner [67]). In ad-dition to the fundamental frequency, the spectral qualitycan be tuned as a secondary consideration [109]. Whileit may be difficult to reconcile the individual experiencesof tuning an instrument with some population-level cog-nitive bias, it would be imprudent to ignore it.

In addition to considering the possible selection pres-sures, we ought to consider the effect of group size onselection. Population genetics theory has shown that theeffect of selection is modulated by group size [137]. Ifthere is weak selection in a small population, the effectsmay not be noticeable. However, if the group size is large,even weak selection effects will be apparent. This mayhelp explain why in some of the regions where theoryscales are used, the same (or very similar) scales havebeen used for over a thousand years. This may be dueto their influential status in court and art music, whichwould have led to large group sizes.

How can we study the evolution of scales?

The challenge of studying the evolution of scales appearsquite formidible. Scales change over time, the rate atwhich they change may vary wildly depending on the in-strument / technology; moreover, they can be inventedfrom scratch. There are numerous possible selection pres-sures that are not mutually-exclusive, and their effectsmay be modulated by group size. Nonetheless we suggestthat the following two approaches are possible.

The first approach is to study the evolutionary dynam-ics in cases where it is appropriate to make simplifyingassumptions, and where one has sufficient data. We canthink of one such case. Gamelan orchestras are typicallytuned in reference to another orchestra [109]; the instru-ments go out of tune at a slow, steady rate compared toother instruments. Also, each Gamelan orchestra typi-cally has been tuned to the same orchestra for a while,such that there is a network of dependencies betweenGamelan orchestras. Gamelan tunings are also exten-sively documented; if these are supplemented by record-ings, there ought to be a considerable amount of datathat may extend back by several decades. Nonetheless,it is not clear that such a study on evolution of Gamelanorchestra tunings would tell us much about evolution ofscales in other settings.

Another approach is to disregard the mechanisms bywhich scales change, and to focus instead on the selectionpressures. Indeed, we have shown that there are likelysome strong selection pressures at play, given the substan-tial convergence in scales across cultures. This approachrequires tractable, relevant mathematical models; mul-tiple selection pressures must be considered in tandem;models ought to be able to not only predict the scalesin use, but predict which scales are not used. Finally,multiple models may have convergent predictions, whichmeans it may be impossible to disentangle the differenteffects. A major barrier to this approach is that any re-

sults will depend on the sample of scales studied, yet itis impossible to know a priori what is a suitable repre-sentative sample of scales. A representative sample couldmean one that equally weights different societies. Does itmatter if societies have different population sizes? Is thisappropriate if one society tends to use a single scale, whileanother has a richer repertoire? There are methodologicalchoices in drawing boundaries between societies, and oneought to control for transmission between societies; moreabstractions are not necessarily good. Is it appropriateto balance by frequency of use? Indeed, in some culturesthere are many scales and some are not necessarily usedmuch [138], however there is little data on this. Ulti-mately, this is a difficult problem, and it seems like theonly appropriate approach is to try multiple approaches.

Bias in the scale database

We are aware that the database itself has several poten-tial sources of bias. First, there is the separation betweentheory and measured scales; a naive user may wronglyassume that in cultures which use theory scales intervalshave always got exact values. Then there are problemsassociated on relying on data collected by only a few eth-nomusicologists. Being limited in number, they are lim-ited in the types of locations they visited; one can see inFigure 2 that some regions are totally absent. Some havesuggested that ethnomusicologists have a bias towards re-porting findings that are considered ‘interesting’, and thusinflating diversity in the database [88]. Some musical tra-ditions (Gamelan, Thai) were very popular research top-ics, so they are over-represented. Unfortunately, in somerare instances it seems that there are statistical irregular-ities in the reporting of tunings: Surjodiningrat et al. [26]note that Jaap Kunst [139] (not included in this database)reported gamelan tunings where all the higher notes wereexactly an octave above the lower ones, and that this isextremely unlikely; in a study of prehistoric bone fluteswhere the tunings are given to an accuracy of 1 cent, oneflute is recorded as having a series of equal tempered in-tervals: [200, 200, 200, 300] [43]. In all, these biases donot invalidate the data, but one must consider how biasesmay affect any analyses and subsequent conclusions.

Limitations to studying scale evolution

It may be the case that, as with languages, we are wit-nessing the death of diversity of scales due to the com-bined forces of globalization and technological change[140]. There is evidence of homogenization, resulting inwidespread adoption of 12-TET [83, 130]. Perhaps thisis due to the proliferation of mass-produced, fixed-pitchinstruments. Perhaps it is due to a full-strength confor-mity bias, as the internet effectively increases group size.Ultimately, this suggests if we want to understand howscales evolve, we must look to the past. However, thereinlies a different problem: the older the instrument, the lesscertain we can be about how they were played. For prehis-toric artefacts, we cannot be sure whether they (or theirreconstructions) faithfully resemble the instrument in its

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original condition, and the only instruments that remainsufficiently intact to play are aerophones. Depending onthe type of aerophone, it is possible to get a range of abouta semitone (100 cents) by varying air flow, flow direction,and embrouchure [141, 142]. Some instruments, like ocari-nas, have many viable cross-fingerings that can result ina dazzling range of scales [141–143]. Thus, we believethat the best source of scales is in ethnographic record-ings spanning the past century [71], so it is imperativethat methods be developed that can faithfully infer scalesfrom large samples of songs. Algorithms must be devel-oped that can handle low-quality recordings [144], back-ground noise, instrument / singing segmentation [145],polyphonic stream segmentation [146], note segmentation[147], and tonal drift [148].

In addition to improved computational methods, it isstrictly necessary to work with large samples. Due to (i)relatively small differences between tunings, (ii) error inmeasurements of tunings, and (iii) inherent variability intunings due to cognitive limitations, it will take a lot ofdata to make definitive claims. In addition, the challengeis further compounded by the wide range of potential se-lection pressures that we have identified. The strengthof selection may vary widely across cultures, as may theselection pressures themselves. Some societies may fo-cus on rhythmic complexity, to the point that weak /few selection pressures appear to act on scales. Culturesthat lack simultaneous singing may by unmoved by theoctave [89, 92]. It is possible to untangle these more sub-tle effects, but only with multiple recordings from eachsociety, and also detailed ethnographic texts that indi-cate how music was performed. Despite the limitationson studying the evolution of scales, we believe that withsufficient large-scale efforts, we may one day gain a deepunderstanding of why music sounds the way it does.

Conclusion

Scales are a cornerstone of music across the world, uponwhich endless combinations of melodies can be generated.Surprisingly, despite a wealth of ethnomusicological re-search on the subject, we lacked a comprehensive, diversesynthesis of scales of the world. Here we remedy this issue,with a focus on quantitative data that will enable detailedstatistical analyses about how scales evolve. Our own pre-liminary analyses have lent quantitative and qualitativesupport for the widespread (but not necessarily univer-sal) use of the octave in some special capacity. We haveshown that despite the rich diversity of scales, when put incontext of how many scales are possible, what stands outis how remarkably similar they are across the globe. Wecompose a treatise on the evolution of scales, and proposepromising avenues for future research.

Author Contributions

J.M. and T.T. designed research; J.M. performed re-search; J.M. analyzed data; J.M. and T.T. wrote the pa-per.

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