Musique & Mathématique

11 septembre 2009

The music scale

Filed under: Non classé — admin @ 13 h 33 min

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The music scale – a little theory

Summary

In the literature concerning the music theory, the division of the octave in 12 intervals is often presented as a cultural receipt which was refined during centuries.
However this division in 12 semitones or the value retained for the ‘comma’ can be synthesized easily by a simple calculation…

Recall of physics

A sound is the propagation of an acoustic wave in the ambient air, of a fast variation of pressure generated by a source.
This variation is very weak in amplitude:

  • A variation of 20 Pascals only for one atmospheric pressure from approximately 100000 Pascals reaches the ‘threshold of obstruct’ for our ear (120 decibels).

These waves are known as spherical because they are propagated at the same speed in all the directions. A traditional image of this propagation is that of the circles generated on the surface of a calm water level when a stone there is thrown.
Of course the water waves are two-dimensional, they are propagated on a surface, while the acoustic waves are three-dimensional and they are progagent in a volume of air. The analogy is complete if we identify the height of the waves of the water waves to the variation of pressure of the acoustic waves.
In both cases we realize that the speed of the wave does not depend on the disturbance applied but only on the medium in which it is propagated:

  • approximately 15 km per hour for the watery waves, it depends in fact the height on liquid and its viscosity.
  • approximately 340 meters per second for the acoustic waves, it depends on the atmospheric pressure and the speed of wind.

By observing the height of water in a particular point of the water level, we can see that it passes by a minimum and a maximum while oscillating in a periodic way. The difference between the maximum and the minimum is called ‘amplitude’.
The number of oscillations per second is called the Hertz (Hz), it measures the ‘frequency’ of a wave.

Physiology of the ear

Our ear makes it possible to perceive a whole field of sound waves which it is possible to represent in the frequency-amplitudes plan in the following way:

domaine

(no..no… it’s not the map of the United States!)

The ear has a greater sensitivity towards a frequency of 4000 Hertz.
It can, towards this frequency, to detect a variation of pressure of 0,00001 Pascal (i.e. -10 decibels) for an atmospheric pressure of 100000 Pascals!!

Conversely the maximum amplitudes of 120 decibels correspond to a threshold of pain for the ear and beyond the irreversible damage can be caused.

The frequencies perceptible by the human ear go from 20Hz (bass sound) to 20000Hz approximately (treble sound). It is the acoustic frequency of wave which corresponds in music to a height of note.

Nature made that we have a logarithmic perception of the acoustic frequencies i.e. we are sensitive to the ratio of the frequencies between two sounds and not to their difference.

Example

The A3 (440Hz) and the A4 (880Hz) are separated by an interval which musicians call octave, whose characteristic is a ratio of frequency = 2 (880/440).

We perceive in an identical way the interval between the A1 note (110Hz) and the A2 note (220Hz) although their difference in Hertz is smaller.

Harmonics et sounds

The French mathematician Joseph Fourier showed that any periodic signal of frequency F can be broken up all things considered of sinusoidal signals of frequencies F, 2F, 3F, 4F, 5F, etc, called ‘harmonics’.

From a mathematical point of view this sum is infinite, but applied to the sounds we can ignore the harmonics starting from a certain row because they are inaudible for us beyond 20000 Hertz. The amplitudes of the various harmonics determine the ‘stamp’ of a sound. Generally, we considere that the sounds agree well if they have many common harmonics.
In particular the following intervals play a part determining:

  • an octave: it is the interval between the base frequency and the harmonique2, the ratio of frequency is 2.
  • a fifth: it is the interval between the harmonique3 and the harmonique2, the ratio of frequency is 3/2.
  • a tone: it is defined like a combination of the two preceding intervals; it is equal to two fifths minus an octave, the ratio of frequency of the tone is (3/2)*(3/2)/2 = 9/8; it is also the interval between the harmonique9 and the harmonique8.

Problem of the scale

How many fifths are there in an octave?

We see that two fifths correspond to a ratio of frequencies of (3/2)*(3/2) = 9/4 = 2,25 exceeding the octave of a tone. The number of fifths in the octave thus lies between 1 and 2.

In fact, we need S such as (3/2)^S = 2, i.e. S = log(2)/log(3/2) = 1,70951129…

The problem of this number S, is that it is irrational: it cannot be written in the form of fraction P/Q, and it is thus impossible to find a graduation (a scale) which contains at the same time the fifths, the octaves and their combinations.

Continued fractions

Fortunately a mathematical process called ‘continued fractions’ makes it possible to solve this problem, all at least by successive approximations.
This process makes it is possible to find the continuation of the best approximations of an irrational number by fractions P/Q.
The decomposition of our number S in continued fraction will enable us to find the rational P/Q giving us a scale in which P fifths equal Q octaves.

The continuation of the fractions for the S approximation is (see procedure here) :

1/1, 2/1, 5/3, 12/7, 41/24, 53/31, 306/179, 665/389, 15601/9126, …

The first two approximations are too coarse to have a musical translation:

  • 1/1: 1 octave = 1 fifth (the error is 71%), in this scale we can evaluate an interval only in whole number of octaves… rather limited…
  • 2/1: 1 octave = 2 fifths (the error is 29%), this scale is twice more precise than the preceding one, the intervals are graduated in half-octaves… but we cannot define the ‘ton’ because two fifths minus an octave = zero.

So, we will begin the study starting from fraction 5/3.

1st approximation S = 5/3

We obtain a graduation where 5 fifths are worth 3 octaves (with an error of 4,3%).
The octave is then divided into 5 equal parts, and the fifth in 3 equal parts.

gam5

C D+ F- G+ B- C

We see that the smallest interval of this scale is precisely the tone defined higher as 2 fifths minus 1 octave. For this reason, this scale is called pentatonic scale (with 5 tones). It is the scale characteristic of the blues.

2nd approximation S = 12/7

We obtain a graduation more precise than the preceding one where 12 fifths are worth 7 octaves (with an error of 0,48%).
It is the traditional moderate scale which everyone learned at the school.

gam12

C C# D D# E F F# G G# A A# B C

In this scale, the octave is divided into 12 units, the fifth is divided into 7 units.
The tone (2 fifths minus 1 octave) is thus worth 7+7-12 = 2 units of this graduation. The unit of the moderate scale is thus called with reason a semitone!!

The extent of this scale is exactly 6 tones.

… Let us continue on our impetus…

3rd approximation S = 41/24

In this scale 41 fifths are worth 24 octaves (the error is 0,12%).
Eh yes… more S is precisely approximate, more the numbers P and Q grows and more the basic unit becomes small (we cannot have butter and the money of butter ; -)

If this scale were used, the octave would be divided into 41 units, the fifth of 24 units and the tone into 24+24-41 = 7 units.
This scale would thus be cut out into 41/7 = 5,8571… tones.

If this scale is not used it is because with few expenses additional, we obtain the following approximation much more precise…

4th approximation S = 53/31

In this scale 53 fifths are worth 31 octaves (the error is 0,017%).
It means that the octave is divided into 53 units, the fifth in 31 units, and the tone into 31+31-53 = 9 units.

This scale is thus cut out into 53 ninth of tone, that is to say 5,8888… tones.
Just as it is, this scale is impracticable bus to locate a note, it would have to be written on a scale making it possible to distinguish the 53 levels of the octave. But let us pass in addition to, we can reason with multiple intervals of this unit.

Let us break up the smaller fifth and the octave in interval (in tones):

  • 1 fifth (31 units) = 9+9+9+4 units = 3 tones + 4 units.
  • 1 octave (53 units) = (9+9+9+4)+9+9+4 units = 5 tones + 2*4 units.

Let us call D the interval of 4 units which we find in the two expressions, we have:

  • 1 fifth = 3 tones + D
  • 1 octave = 5 tones + 2 D = 1 fifth + 2 tones + D

If you learned the music theory, you know this D: it is the diatonic semitone. This graduation related to S=53/31 thus makes it is possible to build the diatonic scale!!

Still a small effort…

The difference between a tone and a semitone diatonic is called in music theory ‘chromatic semitone’.
In this graduation the chromatic semitone is thus worth 9-4 = 5 units.

The difference between chromatic semitone and a diatonic semitone is also called ‘comma’, it is worth 5-4 = 1 unit.
Thus, the comma is the basic unit of this graduation!!


So… it means that the value of the comma, which in the music theory course was presented to us like the smallest discernible interval at the ear, rises in fact directly from the 4th approximation from S = log(2)/log(3/2) in continued fractions!

You can involve yourselves to distinguish the comma which is between C (523,25Hz) and B# (530,39Hz):

C B#


Let us refine the value of the comma…

5th approximation S = 306/179

In this scale 306 fifths are worth 179 octaves (the error is 0,0014%).
I would not extend on this approximation because with few expenses additional, we obtain the following approximation 50 times more precise…

In this scale, the octave is divided into 306 units, the fifth in 179 units.

By way of exercise, to calculate the number of units of the following intervals:

  • tone
  • diatonic semitone
  • chromatic semitone
  • comma

6th approximation S = 665/389

In this scale 665 fifths are worth 389 octaves (the error is 0,00003%).
It means that:

  • the octave is divided into 665 units
  • the fifth in 389 units
  • the tone in 389+389-665 = 113 units
  • the diatonic semitone (D = 1 fifth – 3 tones) = 389-3*113 = 50 units
  • the chromatic semitone (1 tone – D) = 113 – 50 = 63 units
  • the comma (chromatic semitone – D) = 63-50 = 13 units.

It is found that the distribution of the semitones diatonic and chromatic in a tone is légérement different that in the case from the diatonic scale; while being reduced to the traditional cases where 1 ton is worth 9 commas, we obtain:

  • a semitone diatonic = 9*50/113 = 3,9823 commas (instead of 4)
  • a chromatic semitone = 9*63/113 = 5,0177 commas (instead of 5).

The octaves and the fifths of this scale will be then righter (600 times righter) than in the traditional case. The advantage of such a precision is that it makes it possible to control the phenomenon of beats in certain agreements, often unpleasant to the ear…

♦ ♦


Un commentaire

  1. Typically the circle of fifths is used in the analysis of classical music whereas the circle of fourths is used in the analysis of Jazz music but this distinction is not exclusive. The numbers on the inside of the circle show how many sharps or flats the for this scale has.

    Commentaire by Christina Carabini — 16 juin 2011 @ 19 h 22 min

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