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From A_numerical_theory_of_rhythm:
APPENDIX. TWO NUMERICAL THEORIES OF SONANCE
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:
(6)
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:
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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