Tonal consonance parameters expose a hidden order in music. Jorge Useche 1, Rafael Hurtado 2

Transcription

1 Tonal consonance parameters expose a hidden order in music Jorge Useche 1, Rafael Hurtado Departamento de Física, Universidad acional de Colombia, Bogotá, Colombia Abstract Consonance is related to the perception of pleasantness arising from the combination of sounds and has been approached quantitatively using mathematical relations, physics, information theory and psychoacoustics. Tonal consonance is present in timbre, musical tuning, harmony and melody, and it is used for conveying sensations and emotions in music. It involves the physical properties of sound waves and is used to study melody and harmony through musical intervals and chords. From the perspective of complexity, the macroscopic properties of a system with many parts frequently rely on statistical properties of its constituent elements. Here we show that melody in musical pieces can be described in terms of the physical properties of melodic intervals and the existence of an entropy extremalization principle in music with psychoacoustic macroscopic constraints given by conserved quantities with musical meaning. This result connects human perception with human creativity through the physical properties of the musical stimulus. Pythagoras found that two sounds emitted simultaneously by vibrating strings of equal tension and density produce a pleasant sensation when the ratio between their lengths corresponds to the ratio between two small natural numbers n m 1,. This sensation is formally defined as consonance and it is present in melody, harmony, timbre and musical tuning,3,4. Many authors relate consonance with conveying musical information as emotions and meaning 5,6,7,8. From the perspective of the nature of tonality, consonance and dissonance give rise to emotions through tension and relaxation 6 in passages from satisfaction to dissatisfaction and again to satisfaction 7,9. A starting point for studying consonance is the superposition of pure tones or harmonic sound waves with frequency f 1,10,11 i. Musical instruments produce complex tones that can be represented by a superposition of pure tones; the set of frequencies with their corresponding amplitudes is the spectrum and characterizes the timbre of the instrument 1. Pitch is a subjective quality of sound that judges its height and depends strongly on the lowest frequency of the spectrum, the fundamental 1. Reinier Plomp and Willem Levelt found that the consonance level of pairs of simultaneous pure tones is related to the beats produced by fluctuations in the peak intensity of the resulting sound waves,1 (Supplemental ote 1). This approach to consonance is known as tonal or sensory because it depends on the physical properties of the stimulus regardless of the cultural context of listeners 1,13. William Sethares assigned a level of tonal consonance to timbre using the spectrum of the emitted sound and connected timbre with musical tuning 4,14. Musical tuning refers to adjusting a set of pitches to a musical scale using a fixed pitch of reference. Usually, a musical scale is a set of mathematical relations among the fundamental frequencies of pitches. Pairs of pitches in a musical scale define musical intervals with length L 1 jeusecher@unal.edu.co rghurtadoh@unal.edu.co given by the number of pitches between them. In musical theory the level of consonance assigned to a musical interval usually depends on its length and is frequently used to analyze harmony 15. Since musicians tend to apply the same rules for judging the consonance level of simultaneous and successive pitches (harmonic and melodic intervals respectively), and the short-term persistence of pitch in auditoriums may give rise to consonance sensations for successive pitches 3, then tonal consonance is suitable for analyzing melody. The ew Grove Dictionary of Music and Musicians defines melody as pitched sounds arranged in musical time in accordance with given cultural conventions and constraints 16. A definition that encompasses music and speech was given by Aniruddh Patel 17 : an organized sequence of pitches that conveys a rich variety of information to a listener. George Kingsley Zipf reported that the frequency of occurrence of melodic intervals in masterpieces of Western tonal music is inversely proportional to their length 18. From the information theory perspective, Güngör Gündüz and Ufuk Gündüz measured the probability of occurrence of musical notes during the progress of melodies and found that entropy grows until a limiting value smaller than the entropy of a random melody 19. Psychoacoustic representation of consonance The approach of Plomp and Levelt to tonal consonance of complex tones is independent of musical scales. They found that an interval of a given length might be more or less consonant depending on its timbre and location in the register and that this variation through the register is continuous and soft 1. They used two quantities for parametrizing the tonal consonance level of complex tones: The lowest fundamental frequency of the pair of pitches and the ratio between the fundamental frequencies f j f 1 i. This set of parameters is equivalent

2 to one with the same ratio f j f i and the absolute value of the difference between the fundamental frequencies f j f i (Supplemental ote ). For scales based on the Pythagorean rule the difference of fundamental frequencies is related with their sum f j f i = [(n m)/(n + m)](f i + f j ). (1) For the Just and the Pythagorean scales the quantity (n m)/(n + m) depends on the length L L(f i, f j ) of the corresponding interval and for f j f i (i.e. L 0) this relation can be expressed as (Figure 1) f j + f i = ( 1) h a L b (f j f i ) () with h = 0 for f j > f i and h = 1 for f j < f i. Up to three octaves, as melodic intervals in musical pieces usually do not exceed this length 15,0, the fit parameters to a power law for the Pythagorean scale are a = ± and b = ± 0.006, with R = , and for the Just scale are a = ± and b = 0.95 ± 0.005, with R = Equation () does not hold for tempered scales, however since pitches in the Twelve-Tone Equal-Tempered scale are 1 given by f i = f 1 i, where f 1 is a reference frequency, then 1 f j f i = (j i) 1 = L, with f j > f i (3) and for f j f i f j + f i = L/1 + 1 L/1 1 (f j f i ) = ( 1) h a L b (f j f i ) (4) In this case the fit parameters are a = ± and b = ± 0.004, with R = , see Figure 1. equations with two variables, therefore relating the use of tonal consonance in a musical piece with the selection of pitches made by the composer. A quantity that combines the two parameters selected for describing tonal consonance for complex tones is f j f i = (f i + f j )(f j f i ). For scales with unique values f j f i for each pair of pitches, such as the Twelve-Tone Equal-Tempered scale and in practically all cases of the Just and the Pythagorean scales (Supplemental ote 3 and Supplemental Figure 1), this quantity allows reconstructing both parameters, (f i + f j ) and (f j f i ), as well as assigning a tonal consonance level to each pair of pitches. If both sound waves propagate in the same medium (with density ρ) and their amplitudes T and phases are assumed to be equal then the quantity f j f i is proportional to the difference of the average density of the total energy carried by the two waves ϵ j ϵ i = π ρt(f j f i ) 1. This relation holds for pure tones and for complex tones it corresponds to the difference of the average of the energy density carried by the fundamental components. The magnitudes of f j f i and f j f i distinguish intervals of equal length played in different parts of the register and between intervals of different length, especially those between one and thirty-six semitones except for unisons f j = f i with degenerated value 0, see Figure for f j f i and Supplemental Figure for f j f i. Figure 1 Relation between musical scale parameters and the interval length for Just, Pythagorean, and Twelve-Tone Equal- Tempered scales. Interval length from one to thirty-six semitones. n > m and L > 0. Musical scale parameters are presented in Supplemental Table 1. Hence, for these scales, the sum of the fundamental frequencies contains information about the height of pitches and, from equations () and (4), about the tonal consonance parameter f j f i per unit of interval length, as b 1. Since each ratio f j f i corresponds to a length L and it depends on the ratio (f j f i )/(f j + f i ), then a complete description of tonal consonance can be made using the sum and the difference of the fundamental frequencies. Additionally, the set of these two quantities distinguishes each pair of pitches as there are two Figure Relation between the quantity f j f i and the interval length in semitones for an eighty-eight key piano. The quantity f j f i distinguishes intervals of equal length played in different parts of the register and between intervals of different length, especially those between one and thirty-six semitones. The upper branch comes from j = 88 (highest pitch) and i varies from 88 to 1. The tuning comes from the frequency relation for the Twelve-Tone Equal-Tempered scale with A = 440Hz. Finally, this model uses the fundamental for parametrizing pitch and timbre that can be included through consonance curves for complex tones, such as those produced by Plomp and Levelt 1. Use of tonal consonance in real melodic lines We study the use of tonal consonance regarding the selection made by the composer of melodic intervals characterized by their length and position in the register.

3 For this purpose, we study the probability distribution of the physical quantities f j f i and f j f i. If i indicates the chronological order of appearance of pitches in a melody, then the quantities f i+1 f i and f i+1 f i can be used to study tonal consonance with the sign distinguishing between ascending (f i+1 > f i ) and descending (f i+1 < f i ) transitions 17. We analyze vocal and instrumental pieces of the Baroque and Classical periods played in the Twelve-Tone Equal- Tempered scale with A = 440Hz. The selected pieces contain melodic lines characterized by their long length, internal coherence and rich variety of instruments and registers 15. Brandenburg Concerto o. 3 in G Major BWV Johann Sebastian Bach: The polyphonic material in this concert for eleven musical instruments makes it possible to assume that each instrument has a melodic motion. Missa Super Dixit Maria. Hans Leo Hassler: Polyphonic composition for four voices (soprano, contralto, tenor and bass). First movement of the Partita in A Minor BWV Johann Sebastian Bach: This piece has just one melodic line for flute. Piccolo Concerto RV444. Antonio Vivaldi (arrangement by Gustav Anderson): We selected the piccolo melodic line because of its rich melodic content. Sonata KV 545. Wolfgang Amadeus Mozart: We selected the melodic line for the right hand of this piano sonata assuming that it drives the melodic content. Suite o. 1 in G Major BWV 1007 and Suite o. in D Minor BWV Johann Sebastian Bach: The melodic lines of these pieces written for cello have mainly successive pitches. In the case of the few simultaneous pitches, the conduction of the melodic lines was made to the highest pitch. First, we extract from the scores the probability distributions of the quantities f i+1 f i and f i+1 f i and the complementary cumulative distribution function (CCDF) for f i+1 f i and f i+1 f i, which give information about the functional form of the probability distributions. Since the concept of Chroma states that the consonance level of the unison is equivalent to the corresponding octave and the octave can be an ascending (f i+1 f i = 1) or a descending (f i+1 f i = 1 ) transition, then for our analysis we consider the unison as an ascending transition as well as a descending one (Supplemental ote 4). In the cases of ascending transitions (f i+1 > f i ), descending transitions (f i+1 < f i using f i+1 f i and f i+1 f i ) and the joint set of them (using f i+1 f i and f i+1 f i for all transitions), the CCDF fit to exponential functions for all sets with average determination coefficient R ~0.99 and Standard deviation SD~0.01 (Supplemental Table ). Since the number of transitions in the studied melodic lines is maximum one order of magnitude larger than the total number of possible transitions between pairs of successive pitches in the same ambitus (range between the lowest pitch and the highest one) and the number of possible transitions for any melodic line is finite independently of its length, then the probability distributions must be represented in histograms with bin width moderately dependent on the number of transitions. These conditions are satisfied by the Sturges criterion. Histograms for both quantities fit to exponential functions with the highest R for f i+1 f i with ascending and descending transitions taken separately: For ascending transitions R = with SD = and for descending ones R = with SD = 0.016; see Supplemental Table. From now on we drop in the analysis the quantity f i+1 f i as f i+1 f i removes possible degenerations more efficiently (Supplemental Figure 1), produces better fits to exponential functions (Supplemental Table ) and might be expressed in units of energy. To present ascending and descending transitions in the same histogram, we merge left and right branches of the distribution functions by defining the bin width as the average of the two bin widths. Then, the probability distribution of f i+1 f i can be written as H P(ε) = { F + H e ε G + F H e ε G H for ε > 0 for ε < 0, (5) where the notation ε emphasizes that these distributions are constructed over bins, and the functional form of the CCDF is P(f i+1 f i ) = { F + C e (f i+1 F C e (f i+1 fi ) G+ C fi ) G C, f i+1 f i > 0, f i+1 f i < 0. (6) Supplemental Table 4 contains the values of F + H, F H, G + H, G H, F + C, F C, G + C, G C of fits and the determination coefficient R. These probability distributions resemble the Asymmetric Laplace Probability Distribution with different amplitudes for positive and negative branches and a discontinuity in the origin 3. Figure 3. Figure 3 General form of the asymmetric probability distributions P(ε) and P(f i+1 f i ) and their Complementary Cumulative Distribution Function for the right branches and the Cumulative Distribution Function for the left branches. In the symmetric case P 1 = P and α 1 = α. Figure 4 shows the histogram of the probability distribution of ascending and descending transitions

4 (including unisons) in the case of the first movement of the Partita in A minor BWV 1013 as well as the bin degeneration for the corresponding ambitus. To explain the effect of bin degeneration notice that the distance in Hz between pairs of differences f j f i for the Twelve-Tone Equal-Tempered scale varies in such a way that the number of differences inside an arbitrary bin ε, it is its degeneration, reduces when f j f i increases and the probability distribution is equivalent to the one of a random melodic line (Supplemental Figure 3). The comparison between the distributions of real melodic lines with those from bin degeneration for the corresponding ambitus indicates that the scale contributes to the observed results but does not explain them. Additionally, the probability distribution for bin degeneration fits better to a Power law function (R =0.963) than to an exponential function (R = 0.934) (Supplemental Table 3). Figure 4 Twelve-Tone Equal-Tempered scale effect. Comparison between the probability distribution for the real melodic line of the first movement of the Partita in A minor BWV 1013, Johann Sebastian Bach, and the corresponding bin degeneration for the same ambitus. The quantitative difference between the probability distribution for a real melodic line and its corresponding random one gives information about the order introduced by the composer coming from the selection of successive pairs of pitches. A mathematical tool for comparing two probability distributions is the Kullback- Leibler divergence or relative entropy 3 D KL = p k ln ( p k ), q k (7) where p k is the probability distribution for the real melodic line to be compared with the a priori distribution q k coming from the degeneration of the k th bin and that we associated with a random melodic line with the same ambitus of the real one. is the number of bins coming from the ambitus with bins for each branch. Statistical model For modeling the system in analogy to equilibrium statistical physics we use the definitions of melody 16,17 and the results obtained by Gündüz and Gündüz 19, assuming that the composer would have to create a melodic line among the richest ones in terms of the combination of successive pitches. This procedure is equivalent to minimize the relative entropy (7) with mathematical constraints containing relevant musical information. The first constraint of the model comes from normalization. Since there are three kinds of different melodic intervals (ascending, descending and unisons), then the normalization constraint is given by p a + p d + p u = 1, where p a is the probability of ascending intervals, p d is the probability of the descending ones and p u the probability of unisons. If p u is known then p a and p d can be found from the normalization constraint and the asymmetry between ascending and descending intervals p a p d. An equivalent alternative is to use the normalization constraint and to measure from histograms the two observables / p d + p u = p k and p a + p u = p k. k=(/)+1 (8) The difference between the probability distributions of real melodic lines and their corresponding random ones leads to the second constraint. It comes from the selection made by the composer of the length and position in registry of musical intervals, independently if they are ascending or descending ones, and can be associated with the use of tonal consonance as follows: From equation () with b 1 and assuming that each bin k can be represented by the median of its extremes ε k (Supplemental Figure 4), since ε k is linked with the interval length L and the tonal consonance parameter f j f i, then the expected value of ε is = a [ p k σ L k ε = p k ε k + p k f j f i /L k ] (9) where σ is the variance for the length of intervals inside a bin (Supplemental ote 5). Using histograms this is the best estimation that we can make of ε. The first term in the second line of equation (9) is related with the dispersion of musical interval length inside each bin and the second term with the location of intervals in the register for each bin. The location of musical intervals is measured in terms of the average of the frequency differences, which is formally related to tonal consonance (Supplemental ote 6). In this model the expected value ε is conserved just as the expected value of energy in the canonical ensemble model of Statistical physics 4. To explore the nature of ε we consider the case in which each bin contains maximum one possible transition. Assuming that the amplitudes of both sound

5 waves are equal, an approximation frequently used in the theory of tonal consonance 11, then ε ~ f p+1 f p ϵ p+1 ϵ p /(π ρt), (10) where ϵ p+1 ϵ p is the expected value of the magnitude of the difference in the average density of the total energy carried by the two fundamental components of the sound waves: ϵ j ϵ i. Further consequences of this case are presented in Supplemental ote 7. We include a possible asymmetry in the selection of ascending and descending intervals as a third constraint. This asymmetry is present in the difference of the coefficients for left and right branches in equation (5), we first analyze the continuity equation resulting from the fact that the second pitch of a transition is the first pitch of a successive transition. For a continuous segment of M successive pitches, the quantity δ for this segment is defined as M 1 δ = f M f 1 = (f r+1 f r ), r=1 (11) where r is a natural number that indicates the chronological order of appearance of pitches. The magnitude of δ is smaller than or equal to the difference f max f min, given by the maximum and the minimum frequencies in the ambitus of the melodic line. If we have G transitions and U continuous segments separated by rests in a melodic line, the expected value of f r+1 f r is f r+1 f r = 1 G δ T. U T=1 (1) asymmetry between the use of ascending and descending intervals, ε must be much smaller than ε (Supplemental Table 4). Minimizing the relative entropy (7) subject to the constraints in equations (8), (9) and (13) we obtain the probability distribution 5 (Supplemental ote 8) ( p d + p u )q k e λ 1 ε k λ ε k, k [1, / [q m e λ 1 ε m λ ε m=1 m] ] p k = ( p a + p u )q k e λ 1 ε k λ ε k { [q m e λ, k [ (14) 1 ε m λ ε m] + 1, ] m=(/+1) where λ 1 and λ are the Lagrange multipliers for constraints (9) and (13) respectively. Using the expected values ε and ε obtained from the empirical distributions for the selected melodic lines, and allowing less than 1.0% of error between the results from the statistical model and those of real data, we obtain the values for λ 1 and λ (Supplemental Table 4), with λ 1 from one to two orders of magnitude larger than λ. While the values of λ 1 are positive those of λ can be positive or negative showing possible asymmetries in the use of ascending and descending intervals. Each fit parameter is of the same order of magnitude for real melodic lines and those from the statistical model, most of them are consistent within the error bars of the fits (Supplemental Table 4). Figure 5 shows the comparison between the statistical model and the empirical results in the case of the Suite o. BWV Some differences between the empirical data and the results from the statistical model are expected as there are patterns in real melodic lines that cannot be captured by this model. Melodic balance and the continuity of musical phrases are common properties of musical pieces, including those from the Baroque and Classical periods 15,0, leading to values of f p+1 f p much smaller than those of f p+1 f p. Using histograms the best estimation that we can make of the expected value of (1) is ε = p k ε k / = a [ p k σ L k / + p k f j f i L + a [ p k σ L f j f i k + p k L k ]. k=( +1) k=( +1) k ] (13) Equation (13) is our last constraint and shows that ε is due to the asymmetry in the use of the length and position in the register of ascending (positive terms) and descending (negative terms) intervals. Equations (9) and (13) imply that, unless there is an extremely unusual Figure 5 Comparison between the histograms for the statistical model and the real melodic line. Suite o. BWV 1008, Johann Sebastian Bach. To compare probability distributions of different melodic lines, independently if they come from empirical data or the statistical model, we use the CCDF. For the statistical model CCDF can be generated dividing the probability of a bin between all the possible transitions inside it from the ambitus of the corresponding melodic line; here we randomly distribute the probability inside each bin. Figure 6 shows the CCDF for empirical data and the corresponding results from the statistical model for the melodic lines of Missa Super Dixit Maria. Supplemental Figures 5 and 6 show the CCDF for empirical data and the

6 corresponding results from the statistical model for the other melodic lines. Cumulative Probability 1x10-1 1x10-1x10-3 a Soprano Contralto Tenor Bass -x10 5-1x10 5-5x x10 4 1x10 5 x10 5 f i+1 - f i (Hz ) -x10 5-1x10 5-5x x10 4 1x10 5 x10 5 f i+1 - f i (Hz ) Figure 6 Comparison between complementary cumulative distribution functions from the statistical model and real melodic lines. (a) Empirical data: Missa Super Dixit Maria, Hans Leo Hassler. (b) Statistical model. Conclusions Emergent properties in music, present in melodic lines, are portrayed by the occurrence of macroscopic quantities associated with the tonal consonance phenomenon. Melody is represented in terms of the Cumulative Probability 1x10-1 1x10-1x10-3 b Soprano Contralto Tenor Bass physical parameters of consonance and an entropy extremalization principle, with the selection made by composers of melodic intervals expressed by two macroscopic conserved quantities with musical meaning: ε measures the average preference about the length and the position in the register of melodic intervals, and ε measures the macroscopic asymmetry in the use of ascending and descending intervals. These results show that creativity in music involves rules and principles similar to those found in macroscopic physical systems, like energy conservation and entropy maximization. While many non-physical complex systems present similar emergent properties, here the variable used for describing the microscopic properties of the system is a physical quantity functionally similar to energy. References: 1. Rossing, T. The Science of sound. (Addison-Wesley Publishing Company, 1990). Roderer, J. The physics and psychophysics of music. An introduction. (Springer-Verlag, ew York, 009) 3. Regnault, P., Schön, D., Ystad, S. Sensory Consonance: An ERP Study. Music Perception 3, (004) 4. Sethares, W. Local consonance and the relation between timbre and scales. The Journal of the Acoustical Society of America 94, 118 (1993) 5. Copland, A. What to listen for in music. (Mentor Book, ew York, 1957) 6. Madell, G. Philosophy, Music and Emotion. (Edinburg University Press, Edinburg, 00) 7. Schopenhauer, A. The World as Will and Representation. Volume II. (Dover Publications, Inc., ew York, 1969) 8. Ball, P. Science & Music: Facing the music. ature 453, (008) 9. Budd, M. Music and the Emotions: The philosophical theories. (Routledge & Kegan Paul, London and ew York, 1985) 10. Helmholtz, H. On the sensations of tone as a physiological basis for the Theory of Music. (Longmans, Green and Co., London and ew York, 1895) 11. Heffernan, B., Longtin, A. Pulse-coupled neuron models as investigative tools for musical consonance. Journal of euroscience Methods 183, (009) 1. Plomp, R., Levelt, J. Tonal consonance and critical band width. Journal of the Acoustical Society of America 38, (1965) 13. Schellenberg, E., Trainor, L. Sensory consonance and the perceptual similarity of complex-tone harmonic intervals: Tests of adult and infant listeners. The Journal of the Acoustical Society of America 100, (1996) 14. Sethares, W. Tuning, timbre, spectrum, scale. (Springer-Verlag, London, 005) 15. Aldwell, E., Schachter C. Harmony and voice leading. (Harcourt Brace Jovanovich Publishers, 1988) 16. Apel, W. Harvard Dictionary of Music. (Harvard University Press, Cambridge MA, 1974) 17. Patel, A. Music, Language, and the Brain. (Oxford University Press, Inc., ew York, 008) 18. Zipf, G. Human Behavior and the Principle of Least Effort. (Addison-Wesley Press., Cambridge, 1949) 19. Gündüz, G., Gündüz, U. The mathematical analysis of the structure of some songs. Physica A 357, (005) 0. Rimsky-Korsakov,., Vitols, J., Shteinberg, M. Practical manual of harmony. (Carl Fischer, 1930) 1. Pain, H. The physics of vibrations and waves. (John Wiley & Sons Ltd, Chichester, 005). Scott, D. Multivariate density estimation: theory, practice, and visualization. (John Wiley & Sons, 199) 3. Cover, T., Thomas, J. Elements of information Theory. (John Wiley & Sons, Inc., Hoboken, 006) 4. Pathria, R. Statistical Mechanics. (Elsevier Butterworth-Heinemann, Oxford, 1996) 5. Kotz, S., Kozubowski, T., Podgorski, K. The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. (Birkhaüser, Boston, 001)

7 Supplemental material Supplemental otes: Supplemental ote 1: The superposition x(t) of two pure tones with different amplitudes (A 1 and A ), different phases (φ 1 and φ ) and different frequencies (f 1 = ω 1 π and f = ω π), where ω i is an angular frequency, is given by where t is the time. Defining then x(t) = x 1 (t) + x (t) = A 1 cos(ω 1 t + φ 1 ) + A cos(ω t + φ ) ω + = ω 1 + ω ; φ + = φ 1 + φ ; ω = ω 1 ω ; φ = φ 1 + φ x(t) = (A 1 + A )cos(ω + t + φ + )cos(ω t + φ ) + (A A 1 )sin(ω + t + φ + )sin(ω t + φ ) This equation indicates that the frequency of the oscillations in time depends just of ω + and ω, independently of phases and amplitudes. Since ω = πf then the frequency of oscillations is given by the sum and the difference of the frequencies f 1 and f. In the case of sound waves, for f 1 f 0 Hz with f 1 f f 1 + f people hear a tone of frequency f R = (f 1 + f ) with beats (shocks waves produced by fluctuations in the peak intensity) at the frequency f B = f 1 f. This can be seen directly by plotting the final equation of the superposition and counting the number of fast oscillations and peaks in the amplitude per unit of time. Supplemental ote : Reinier Plomp and Willem Levelt parameterized the consonance level of pairs of complex tones using the relation between tonal consonance and the fundamental frequency of the lowest tone for musical intervals of equal interval length. Using the fact that for the fundamental frequencies of a pair of complex tones the interval relation f j = α (j i) f i, with α (j i) given by the interval length j i for an specific musical scale, the frequency difference f j f i can be written as f j f i = f i (α (j i) 1). Since α (j i) 1 is constant for intervals of equal length, then tonal consonance and the difference of the fundamental frequencies have the same functional relation. Supplemental ote 3: We use three musical scales: Pythagorean, Just and Twelve Tone Equal Tempered. For each of these scales we consider the consequences of degenerated values f j f i = f n f m, except for unisons (i = j, m = n), where i m and j n. This equality holds for the following conditions: f j > f i and f n > f m for ascending transitions or f j < f i and f n < f m for descending transitions. Since for descending transitions f i f j = f m f n the difference between the cases of ascending and descending transitions is just the subindices order, then we analyze the case of ascending transitions without loss of generality. For each scale we can express f j = α (j i) f i and f n = α (n m) f m, then f i [α (j i) 1] = f m [α (n m) 1] or f i f = [α (n m) 1] m 1] [α (j i) For i > m then f i f m = α (i m) and for i < m f i f m = 1 α (i m). Finally, the degeneration equations are [α (n m) 1] [α (j i) 1] = α (i m) for i > m and [α (n m) 1] 1] = 1 for i < m. [α (j i) α (i m) We also use this procedure to obtain the degenerations equations for the difference of frequencies f j f i = f n f m [α (n m) 1] [α (j i) 1] = α (i m) for i > m and [α (n m) 1] [α (j i) 1] = 1 α (i m) for i < m. We generate all possible combinations of the coefficients α (n m) to find the number of times that the degeneration equations are satisfied, which depends of the precision in the decimal places. Supplemental Figure 1 shows the percentage of times that the degeneration equations are satisfied, over a total range of = possibilities (corresponding to an 88 pitch musical instrument like a 88-key piano), as a function of the precision (measured in decimal places) used to calculate the left and the right part of the degeneration equations. The values of α for the Pythagorean, Just and Twelve Tone Equal Tempered scales are calculated from the values n m and f j f i presented in the Supplemental Table 1. For all scales, the α coefficients corresponding to intervals larger than one octave are obtained by multiplying the corresponding coefficient of the previous octave by. In our algorithm, the degeneration equations are satisfied when the absolute value of the difference between the left and the right parts of each equation is less than Supplemental ote 4: We begin the bin count considering the bin box [0 Hz, x Hz ) for ascending transitions and the bin box (y Hz, 0 Hz ] for descending transitions, where x and y are the bin width of the histogram generated by the Sturges criterion.

8 Supplemental ote 5: Since each bin is represented by the average of its extremes and this number corresponds to the average of all possible states in the bin [(f ]k j f i ) generated from the corresponding ambitus, then we can link the quantities ε and f j f i and measure the expected value as ε = [p k ([ f ]k j f i )]. The last equation depends on the quantity [ f ]k j f i that can be calculated for each bin from the relation f j + f i = ( 1) h a L b (f j f i ) using the fact that f j f i = (f j + f i )(f j f i ). Then n k [ f ]k j f i = 1 f n j f i u k u=1 n k = 1 n k a L b (f j f i ) u u=1 where the sum runs from u = 1 to the total number of possible transitions f j f i in the k th bin, u = n k. As there are different combinations of pairs of frequencies that generate the same interval length we can split the sum in as many terms as possible interval lengths inside the bin W [ f ]k j f i = a [P(x)Lb x (f x i f i ) ], x=1 where the sum over all possible transitions inside the bin is expressed as a sum over all possible interval lengths inside it. In the last equation the sum runs from x = 1 to the maximum interval length in the k th bin, x = W. P(x) is the probability of finding an interval of length x inside the k th bin and (f j f i ) x is the average of the quantity (f j f i ) for intervals with length x inside the bin. The quantity (f x j f i ) can be related to the tonal consonance parameter f i f i over a single interval length x expressing it as σ x + f j f i, where σx is a variance that measures the spreading of forms used to construct a single kind of interval of length x in the k th bin and f j f i depends of the average x of the absolute value for the frequency differences of a single interval length x. Then W [ f ]k j f i = a {P(x)Lb x [σ x + f j f i ]} = a [P(x)Lb x σ x ] + a [P(x)Lb x f j f i x=1 Taking b 1 and using the notation σ L k and f j f i /L k to represent the corresponding expected values inside each bin x W x=1 [ f ]k j f i = a σ L k + a f j f i /L k As the expected value operator E is linear and each bin is characterized by the sum of two quantities a σ L k X and a f j f i /L k Y, then the expected value over all bins is E(X + Y) = E(X) + E(Y) concluding that ε = a p k σ L k + a p k f j f i /L k Supplemental ote 6: The frequency difference f j f i is a soft function F of the tonal consonance C. Since inside a bin the frequency differences of the same interval length are near between each other, then we can approximate the relation between tonal consonance and frequency difference around a point q using the Taylor series f j f i = F(C) F(q) + F (q)c F (q)q. Taking the mean value of the M frequency differences of the same length x inside the bin M f 1 j f i = M f j f i U U=1 M = 1 M F(C) U F(q) 1 M 1 + F (q) 1 M C U F (q)q 1 M 1, U=1 and taking q as the mean value of the tonal consonance corresponding to the M frequency differences of the same length inside the bin: q = C = 1 M then M U=1 f j f i F(C ) + F (C )C F (C )C = F(C ). This result implies that the average of the frequency differences corresponds to the frequency difference that is representative of the corresponding average of tonal consonance. Supplemental ote 7: From equation (9) ε = a [ p k σ L k + p k f j f i M U=1 /L k ], W x=1 M U=1 ] x x M U=1 C U when each bin contains maximum one possible transition the first term vanishes and the average disappears

9 (f j f i ) ε = a p k k = a 1 p L xh x L k W x=1 h (f j f i ), xh where x runs from 1 to the total number of possible interval lengths W in the corresponding ambitus, h refers to the possible locations of length x for a given musical interval in the register and runs from 1 to the total number of possible positions in the register, p xh is the probability of finding inside the melodic line a particular interval of length x in the position h of the register. Supplemental ote 8: In analogy to the Gibbs-Boltzmann entropy maximization principle in Physics, that produces the probability distribution p n by maximizing the quantity S = k B p n ln (p n ) subject to constraints, maximizing the quantity D KL = p n ln ( p n subject to constraints when there is an a priori distribution q n produces the probability distribution p n. The probability distribution q n contains information about the degeneration of the accessible states and satisfies that 0 < q n < 1. A maximum of the quantity D KL is equivalent to a minimum of the quantity D KL. According to the Lagrange Multipliers theory, in order to find an extreme value of the quantity p n ln ( p n subject to constraints we have to calculate the auxiliary function F Then F = p n ln ( p n ) + λ q d + p u p n n (p q n ) + λ + (p a + p u p n ) λ 1 (< ε > p n ε n ) λ (< ε > p n ε n ) ) n= +1 F = p n ln ( p n ) λ q p n λ + p n λ 1 p n ε n λ p n ε n + λ 1 < ε > n n= +1 + λ < ε > +λ (p d + p u ) + λ + (p a + p u ). df = [ln ( p n ) + q n p n + λ dp n q n p n q 1 ε n + λ ε n ] n An extreme value occurs when df dp n = 0 and d F dp n 0, then and using the property n i=m 1 = n + 1 m leading to and then A possible solution of this equation is λ 1 λ + 1 n= +1 [ln ( p n ) λ q 1 ε n + λ ε n ] λ 1 λ + 1 = 0 n n= +1 [ln ( p n ) λ q 1 ε n + λ ε n ] λ ( n ) λ + ( ) = 0 [ln ( p n ) λ q 1 ε n + λ ε n ] λ λ + = 0 n [ln ( p n ) λ q 1 ε n + λ ε n + λ n + λ + ] = 0. q n ) that defining ln ( p n q n ) λ 1 ε n + λ ε n + λ + λ + = 0 λ = 1 + λ and λ + = 1 + λ +

10 becomes ln ( p n q n ) + λ 1 ε n + λ ε n + λ + λ + = 0. Finally p n = q n exp( λ 1 ε n λ ε n λ λ + ) and then p n = q n exp( λ λ + ) exp( λ 1 ε n λ ε n ). ow we find the value of exp( λ λ + ) using the constraints p d + p u = p n and p a + p u = p n k= +1 : for n [1, ] q n exp ( λ λ + ) exp( λ 1 ε n λ ε n ) = p d + p u then exp( λ λ p d + p u + ) = q n exp( λ 1 ε n λ ε n ) and for n [ + 1, ] q n exp ( λ λ + ) exp( λ 1 ε n λ ε n ) = p a + p u n= +1 exp( λ λ p a + p u + ) = q n exp( λ 1 ε n λ ε n ) n= +1 Finally, the probability p n is q n (p d + p u ) for n [1, / q n exp( λ 1 ε n λ ε n ) ] p n = q n (p a + p u ) { for n [ + 1, ] n=(/+1) q n exp( λ 1 ε n λ ε n )

11 Supplemental Figures: Supplemental Figure 1 Probability of satisfying the degeneration equations vs. decimal places used to calculated the degeneration equations. The figure shows the probability of satisfying the degeneration equations in the case of an 88 pitch instrument (see Supplemental ote 3) for the quantities f j f i and f j f i in three different scales: Twelve Tone Equal Tempered, Just and Pythagorean. For the Twelve Tone Equal Tempered scale the degeneration equations are not satisfied beyond 5 decimal places for f j f i and beyond 8 decimal places for f j f i. For the Just and the Pythagorean scales and taking 13 decimal places, the number of times that the degeneration equations are satisfied for the quantity f j f i reduces to 7 and 3 respectively (corresponding to a an order of magnitude of 1E-5 in the probability), however for the quantity f j f i the number of times that the degeneration equations are satisfied reduces to 341 and 9 respectively (corresponding to a an order of magnitude of 1E-3 in the probability). In our algorithm, the degeneration equations are satisfied when the absolute value of the difference between the left and the right part of each equations is less than f j - f i 5.0E E E+03.0E E E E E E E E Interval length (semitones) Supplemental Figure Relation between the quantity f j f i and the interval length for an 88 key piano tuned using the frequency relation for 1 the Twelve Tone Equal Temperate scale (f i = f 1 i ) with A = 440 Hz. The quantity f j f j distinguishes intervals of equal length played in different parts of the registry and between intervals of different length. The upper branch comes from j = 88 (highest pitch) and i varying from 88 to 1.

12 Supplemental Figure 3 Comparison among a random melodic line played in the Twelve Tone Equal Tempered scale (A=440 Hz) and the probability distribution generated by the same scale. The random melodic line contains pitches without rests (9999 transitions) and was generated in the same ambitus of the Suite o. 1 in G Major BWV 1007 of Johann Sebastian Bach ( Hz Hz). The bin width, 114 Hz, is given by Sturges criterion. Supplemental Figure 4 Relation between the bin center and the average of the possible transitions f j f i inside the bin. The fit shows a lineal relation of the form f j f i = A + Bε with A = ± , B = ± 0.00 and determination coefficient R = Since A and B are close to 0 and 1 respectively, then the number associated to each bin center is representative of the possible transitions f j f i inside the bin. Bin width 114 Hz, ambitus Hz to Hz. Cumulative Probability 1x10-1 1x10-1x10-3 Violin 1 Viola 1 Cello 1 Harpsichord Violone Cumulative Probability 1x10-1 1x10-1x10-3 Violin 1 Viola 1 Cello 1 Harpsichord Violone -x10 5-1x x10 5 x10 5 f i+1 - f i (Hz ) -x10 5-1x x10 5 x10 5 f i+1 - f i (Hz ) Supplemental Figure 5 Comparison between the statistical model and the corresponding complementary cumulative distribution function for the Brandenburg Concerto o. 3 in G Major BWV Left. Experimental results. Right. Statistical model results.

13 Cumulative Probability 1x10-1 1x10-1x10-3 Piccolo concerto First mov. Partita Mozart sonata Suite 1 Suite Cumulative Probability 1x10-1 1x10-1x10-3 Piccolo concerto First mov. Partita Mozart sonata Suite 1 Suite -x10 6 -x10 6-1x10 6-5x x10 5 1x10 6 x10 6 x10 6 f i+1 - f i (Hz ) -x10 6 -x10 6-1x10 6-5x x10 5 1x10 6 x10 6 x10 6 f i+1 - f i (Hz ) Supplemental Figure 6 Comparison between the statistical model and the corresponding complementary cumulative distribution function for the Suite o. 1 in G Major BWV 1007, Suite o. in D Minor BWV 1008, Sonata KV 545, First Movement of the Partita in A Minor BWV 1013 and Piccolo Concerto RV444. Left. Experimental results. Right. Statistical model results.

14 Supplemental Tables Just scale Pythagorean scale Tempered scale Interval lenght (Semitones) n m n/m n m n/m f j /f i (0/1) (1/1) (/1) (3/1) (4/1) (5/1) (6/1) (7/1) (8/1) (9/1) (10/1) (11/1) (1/1) Supplemental Table 1 Coefficients for generating the Pythagorean, Just and Twelve Tone Equal Tempered scales. Interval length runs up to one octave (1 semitones). CCDF Histograms f(i+1)-f(i) f(i+1) -f(i) f(i+1)-f(i) + f(i+1)-f(i) - f(i+1) -f(i) + f(i+1) -f(i) - f(i+1)-f(i) f(i+1) -f(i) f(i+1)-f(i) + f(i+1)-f(i) - f(i+1) -f(i) + f(i+1) -f(i) - Violin Violin Third Brandenbug concerto Violin Viola Viola Viola Cello Cello Cello Violone Harpsichord Missa Dixit Maria Soprano Contralto Tenor Bass Suite Piece or movement Suite Mozart sonata First mov. Partita Piccolo concerto Avarage Standard desviation Supplemental Table Determination coefficient R for the fit to an exponential function of the Complementary Cumulative Distribution Functions (CCDF) and histograms. For the CCDF the exponential functions are of the form: P(x) = A ± e x /B ± and for histograms P(X) = C ± e X D ±. Ascending transitions are identified by the sign +,descending transitions are identified by the sign - and their combination is left without sign. We have use the notation x and X to distinguish between CCDF and histograms (constructed over bins) respectively; in both cases this notation applies to the quantities f j f i and f j f i. A ±, B ±, C ± and D ± are reported in Supplemental Table 4.

15 Third Brandenbug concerto Missa Dixit Maria Piece or movement Exponential R R Violin ± ± Violin ± ± Violin ± ± Violin ± ± Violin ± ± Violin ± ± Viola ± ± Viola ± ± Viola ± ± Viola ± ± Viola ± ± Viola ± ± Cello ± ± Cello ± ± Cello ± ± Cello ± ± Cello ± ± Cello ± ± Violone ± ± Violone ± ± Harpsichord ± ± Harpsichord ± ± Soprano ± ± Soprano ± ± Contralto ± ± Contralto ± ± Tenor ± ± Tenor ± ± Bass ± ± Bass ± ± Suite ± ± Suite ± ± Suite ± ± Suite ± ± Mozart sonata ± ± Mozart sonata ± ± First mov. Partita ± ± First mov. Partita ± ± Piccolo concerto ± ± Piccolo concerto ± ± R Avarage Twleve-Tone Equal-Tempered scale Power law: P= a(ε b ) a b (Hz ) Supplemental Table 3 Functional form of the degeneration distribution for the Twelve-Tone Equal-Tempered scale. Determination coefficient R for the fit to exponential and power law functions of the degeneration distribution. Supplemental Table 4 Data, fit parameters, determination coefficients, expected values and Lagrange multipliers.

16