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From A_numerical_theory_of_rhythm:


A.1 Euler Sonance

This great mathematician was interested (among many other areas) in the problem of musical consonance, and he elaborated an interesting theory on it, based on the decomposition, in prime factors, of the numbers that express the ratios of sound frequencies.

If a natural number A is expressed as product of primes, Euler defines its degree of complexity C(A), as:

                             P                                                                 P
                   A = P pi ai                   then              C(A) = 1 + S ai . (pi -1)                                                (1)

being pi each one of the P prime factors of A, ai its exponent, and P and S the product and sum signs. For example the complexity of 8 (23) is 1+3.(2-1) = 4 while that of 9 is 1+2(3-1) = 5; that is to say, 8 is simpler than 9.

The degree (of dissonance) of an interval, expressed as a fraction reduced to minimum numbers, is simply the degree of the product of numerator and denominator. So the fifth 3/2 has the degree 4, the same as that of 2x3=6 (1+2+1), the fourth 4/3 has 5 (=1+2+2), etc.

In general the dissonance of a simultaneous group of notes, as a chord, can be found to be equal the degree of complexity of the quotient between the minimum common multiple and the maximum common divider of their frequencies:

EuC (A,B,C,..) = C (N)           being          N =    m.c.m (A,B,C,..) / m.c.d (A,B,C,..)                              (2) 

For example, the frequencies 220:330:440, C-G-C’, have a dissonance as a chord equal to the degree of complexity of the quotient (3×4×110)/110 = 12, that it is 1+2×(2-1)+1×(3-1) = 5; and the same for the chords C-G-C’ or E-B-E’. The natural major chord in fundamental form, as C-E-G, reduced to simple figures, will always have frequencies proportional to 4-5-6, which takes us to a dissonance similar to the complexity of 60, that is 9, while the minor, A-C’-E’, proportional at 5/6-1-5/4, that is to say, 10-12-15, has also a degree of 60, again 9.

Other inversions of these chords can vary their dissonance: for example, the case G-C'-E', proportional at 3/4-1-5/4, or 3-4-5, gives 9, as before (a complexity of 60) but E-G-C', 5/4-3/2-2, or 5-6-8, gives 10 (that of 120, 1+3+2+4 = 10).

The theory of Euler seems to coincide quite well with the common feeling of the musician, and on the other hand, it offers a measure of a quality, seemingly so elusive to number, like the consonance. It is an interesting theory, and it is so even more for its simplicity and elegance (even transcendence, since it connects with the old Pythagorean theories).

It is possible to find a relationship between this estimate of consonance, seemingly only arithmetic, and the perception based on neurophysiology. Indeed, a pitched sound has a wave frequency of repetition whose duration is one period. The simultaneous audition of two sounds with prime periods will create a periodic wave form whose period will be the product of the two. That is to say, there will be a time of simultaneous coincidence, and the more complicated the ratio between those primes, the longer the period of the wave. And the complexity –dissonance– will be all the more so the more prime sounds we add. Figures 4 and 5, commented as rhythms, are equally valid to illustrate these tonal relationships: the tonal case is identical, but in much smaller temporal scale (1 to 100), to that of a rhythm, which indeed is more difficult to understand and to conceive, the more beats are contained in its measure.

Helmholtz extended these Sonance relations –based on the fundamental frequencies of several sounds– to their harmonic content. Now the complexity results from the interactions between the components of the harmonic families of each sound; nevertheless, he treats the subject in the domain of the frequencies, taking into consideration the whining that harmonics of neighboring frequency cause in the ear. Helmholtz centers rather on the neuro-physiological aspect, and thus moves away from the mainly numerical vision of Euler.

Fig. 21 shows the dissonances EuD (according to Euler) for the main supernumerary intervals –that is to say, of the type (n+1)/n– that form the perceptive base of our scales. Other dissonance values –which we will define in the following paragraph– are also included (all of them calculated by means of our program Euler97). The reader will judge if these numbers correspond to his perceptual estimation of dissonance, remembering, of course, that his musical experience has influenced his natural or primitive estimation [Sánchez, 92a, 93a, 94].

interval            ratio      cents     EuD       ED1      EDo2    ED2

octave               2:1         1200     2          1.00     1.00      1.00
fifth                  3:2           702      4          3.00    2.51      2.24
fourth               4:3           498      5          4.00    3.27      2.83
major third       5:4           386      7          6.00    5.01      4.47
minor third       6:5           316      8          7.00    5.37      4.58
major tone        9:8           204      8          7.00    5.74      5.00
minor tone      10:9           182     10        9.00     6.85      5.74
semitone         16:15         112     11       10.00    7.37      6.00
Fig.21. Euler y Essential r-Dissonances for some usual intervals, and variable r (1, 1.41,2) , with the interval ratios and sizes expressed in cents of semitone.

A.2 Essential Sonance

Due to the necessity of obtaining certain mathematical properties (musical intervals as vectors in a linear space, and dissonance as a distance in that space, defining in this way a metric space) we introduced (Sánchez, 92,93) a variation of the Euler’s measure of dissonance: we suppress the added ‘1', which does not affect the relative hierarchy of the interval dissonances; furthermore, we generalize the notion for variable exponents of the coefficients (ai) and the prime factors weights (pi-1) –whose product are the vector coordinates. We define therefore the Essential Complexity of a positive integer fraction A:

                   P                                                                              P
      A =    P pi ai          as:        ECr(A) =  ( S (pi -1) r  ai  r ) 1 / r       1 <=r < infinite                      (3)

being r a real exponent that weighs differently each exponent ai according to its value (for r = 4, only the greater of them counts); pi (p sub i) the prime factors of A, ai (a sub i) their exponents, and P, S the product and sum signs. Also, similarly:

The degree of Essential r-dissonance of a group of notes, as a chord, will be given by the Essential r-complexity of the quotient between the minimum common multiple and the maximum common divider of their (integer) frequencies:

EDr (A,B,C,..) = ECr (N)           being          N =    m.c.m (A,B,C,..) / m.c.d (A,B,C,..)           (4)

For example, the frequencies 220:330:440, C-G-C’, have a chord dissonance equal to the degree of dissonance of the quotient (3×4×110)/110 = 12, that it is 2×(2-1)+1×(3-1) = 4 (it was 5 for Euler); and the same for the chords C-G-C’ or E-B-E’. The major chord in fundamental form, as C-E-G, reduced to simple figures, will always have frequencies proportional to 4-5-6, which takes us to a dissonance similar to the complexity of 60, that is 8 [(2×(2-1)+1×(3-1)+1×(5-1)], while the minor, A-C’-E’, proportional at 5/6-1-5/4, that is to say, 10-12-15, has also a degree of 60, again 8: it can be observed that both chords are equally consonant –or dissonant–, which coincides with the general appreciation.

Other inversions of these chords can vary their dissonance: for example, the case G-C'-E', proportional at 3/4-1-5/4, or 3-4-5, gives 8, as before (a complexity of 60) but E-G-C', 5/4-3/2-2, or 5-6-8, gives 9 (that of 120, 3+2+4 = 9): it is a slightly more dissonant inversion that the fundamental form.

In Fig.21 the EDr appears for r =1, square root of 2, and 2; we will usually use the first one..

A.3 musical interpretation of the number N

The number N, defined in the previous paragraph as the quotient among maximum common multiple and minimum common divider of the frequencies of the notes heard simultaneously, has an interesting sound and musical interpretation. Clearly, by dividing the m.c.d. of the frequencies by their m.c.d., we eliminate the influence of the chord tessitura –general pitch height–, reducing the chord notes to a common scale: in this way chords are equaled in different octaves. For example: N would be same for the chords C3-E3-G3, with frequencies: 262 : 262×5/4 : 262×3/2 than C5-E5-G5, (1047 : 1047×5/4 : 1047×3/2): indeed, both, N1 = (262×4×3×5) / (262) = 3×4×5; and N2 = (1047×4×3×5) / (1047) = 3×4×5: are the same, the tessitura does not have any influence in the essential dissonance; in this way, all triads proportional to 4-5-6 have the same N and therefore the same EDr.

A.4 Sonance of a duration rhythm

Situ             1                                                   n
           CP 2)2)2@2)2)2)2@)2)2@2)2)2
Order           n       o1                  o2             o3     2n-1
Fig.22. Notations in a rhythm inside a measure of n CP

Let us consider a duration rhythm (DR) like a group of strokes inserted in a measure (Fig.22) of at least 3 CP or chronos protos (2 would not allow the perception of the pattern formed by the blows, whether they be 2 or 1, because both strokes will have the same duration, 1 CP, and could not be differentiated). We will have then from 1 to n-1 blows, since n blows in a n-measure constitutes a uniform tempo: there will be still no duration pattern, for at least one CP cell must be empty. The group of strokes that compose the DR forms what we can call a rhythmic chord, a group of events with a collective Sonance that depends on the Sonance of each, like it happens with a chord of tones or notes. Therefore, adapting the expression (4) to the rhythmic-temporal domain we find:

Let a DR be defined as the presence (stroke) or absence (non stroke or silence) in each of the n CPs that compose the measure: if we have m strokes with successive orders o1 o2,..., om, the DR stands for:

               DR = ( n, (o1,o2,..., om ) )  being      2<n,              0 < m <=n ,        0<= oi < n             (5)

and then we define its rhythmic essential dissonance (RED) as the Essential r-dissonance of the rhythmic frequencies of these strokes; themselves –the frequencies– defined as the quotient of their rhythmic order divided by n, number of CPs in the measure; that is to say:


The rhythmic order of a stroke in situation i in a measure of n CPs is defined as its situation regarding the beginning of the previous measure, beginning to count in 0, that is to say, it value is n+i-1. See Fig. 7 and 22, and paragraph 1.5, where these points are illustrated and these definitions applied.

A.5 Approximate Sonance

All the previous concepts, based on integer numbers and their ratios, rational numbers, can be applied also to  numbers in the so-called ‘reality’ if we approach these numbers, the real numbers –real in mathematical sense–, by means of rational numbers of near value, as was indicated in section 1.8. In the following paragraphs we will look for methods to achieve this good approach.

But in the case that occupies us, an additional element intervenes: not only the quality of the approach counts, but also, the integers that form the fraction –numerator and denominator–, should be the simplest from the point of view of their complexity, or Sonance, since that means also simplicity from the perceptive point of view (better ‘hearing’). Such simple numbers should be preferred to other smaller, but more complex–we already saw how 7 is more complex than 8, 9 and 10. When there are several numbers, the good perceptive approach will be the one that produces simple numbers.

Thus, we will try to rationalize several quantities, to convert several real numbers in rational, or what is the same thing, to find an unit of measure, or module, of which those real numbers are multiples. The problem has sometimes an exact solution: let us think of w, 2w, 3w ... which obviously admit the real unit w, which provides us with integer exact measures, 1, 2, 3. But in general we will not find that simplicity, due to the reasons previously given, so, we must soften the conditions of the problem, which is outlined in this way:

                We want to find a real number of which several others are approximately multiples.

Now the problem is transferred to the adverb 'approximately' . Indeed, it is always possible to find a real number (in fact anyone) that approaches more or less multiples: but clearly, the remainder will take any value. The task is then to delimit the remainders or errors that separate those approximate multiples from the exact values. For example the numbers 10.1, 20.1, 29.8 admit the approximate divider 10, with errors of .1, .1, -.2, respectively. Then we can define a collective measurement for those remainders, and to delimit that quantity so that only when it is smaller than a previously adopted boundary, that approach will be accepted. Thus, we specify our problem as:

Let a1, a2,... , an, be real numbers. Let us look for a real divider common to all whose Collective Error of Approximation (CEA) is small; that is to say, smaller than a certain boundary, B.

We will take as that approximation error CEA = ||ei||r, that is to say as the r-norm of the vector of partial errors ei, being these in turn defined by means of bi = ci×d + ei (ci is the integral quotient obtained by dividing the real bi by d, and ei the real remainder of this division). The remainder ei is, naturally, smaller than the dividing d. In general it will be bound, between 0 and d; and, considered as a uniform random variable, its half value will be proportional to d, and will then be bound by d/2.

So that to compare several dividers it will be convenient to normalize the remainders ei dividing them by their corresponding bound d/2. It is also suitable to normalize them with respect to the number of real ai, to avoid an increasing error with this number. So that instead of taking the r-norm as our CEA, we will calculate the r-mean, being thus the index of quality:

Qr(e)= Er(e) / (d/2)     that is      Qr(e) = 2 ( S | ei | r / n ) 1 / r   /   d       with 1 <= r < infinite            (7)

where all the numbers appearing are real (so, only the quotients will be positive integers). The r-mean is calculated adding all the r powers of the absolute values of the approximation errors, then dividing the sum by their number, and extracting the r root. For r = 1, we find the usual arithmetic mean of the absolute errors. Notice that we have normalized that mean when dividing by d/2, and with this we have finally found the boundary we looked for, C, which takes a value of 1:

           Qr(e)<1                                                                             (8)

We define the Approximate Maximum Common Divider now, a.m.c.d. as:

The a.m.c.d. of several real numbers ai is that integer divider that, between two limits, presents smaller Qr.

It seems intuitively reasonable to expect the integer dividers of the a.m.c.d. are also good dividers, because in turn they approach well d some multiple of d can even be also good candidate. But now the parameter Q is decisive because it compares all the approximations. An empirical value for Q1 is .15, much less than the absolute bound 1.

Now then, how to look for those good values? will this quality Q be calculated for all the possible dividers?. It has been already seen that is impossible, since they are infinite, and, although we can take only some, equally spaced values, as in a grid, intending to ‘scan’ the chosen range, will it will take long. Well-known techniques for searching local minima can be applied to speed the search. Another reasoning and its technique are described in Sánchez and Beltrán (1998).


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