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MATHS IN MUSIC
Mathematics is involved in some way in every
field of study known to mankind. Any possible or
imagined situation that has any relationship
with space, time, or thought would also involve
mathematics.
Music is a field of study that has an obvious
relationship to mathematics.
Actually, music is first and most of all a
phenomena of nature, a result of the principles
of physics and mathematics.
Let’s see which are the elements that link Maths
to Music.
Definition of music
Mozart said: ‘Music is rhythm produced through sound.’
Sounds are vibrations transmitted through an elastic solid or a
liquid or gas, with frequencies in the approximate range of 20 to
20,000 hertz, capable of being detected by human organs of
hearing.
Consonance and dissonance
It is actually very hard to define the feeling of consonance
and dissonance, and Maths surely helps with it.
Ever wonder why some note combinations sound pleasing
to our ears, while others make us cringe? To understand
the answer to this question, you’ll first need to understand
the wave patterns created by a musical instrument. When
you pluck a string on a guitar, it vibrates back and forth.
This causes mechanical energy to travel through the air, in
waves. The number of times per second these waves hit
our ear is called the ‘frequency’. This is measured in Hertz
(abbreviated Hz). The more waves per second the higher
the pitch.
Now, to understand why some note combinations sound better, let’s first look
at the wave patterns of 2 notes that sound good together.
Let’s use middle C and the G just above it as an example:
Now let’s look at two notes that sound terrible together, C and F#:
Do you notice the difference between these two? Why is the first ‘consonant’ and the
second ‘dissonant’? Notice how in the first graphic there is a repeating pattern: every 3rd
wave of the G matches up with every 2nd wave of the C (and in the second graphic how
there is no pattern). This is the secret for creating pleasing sounding note combinations:
Frequencies that match up at regular intervals
A musical scale
A musical scale is the sequence of the sounds
included in one or more octaves. The sounds in a
scale are defined as ‘notes’, which are then used to
create melodies and harmonies. For our purposes,
we will only consider the basic diatonic scales used
in western music.
(the diatonic scale is made up of 7 notes from the
Chromatic scale, that go according to a precise
sequence of seven interludes, 5 tones and 2
semitones).
Tuning systems based on
Mathematics ratios
In the history of music the musical scale
developed from the Pythagorean tuning, to the
Natural tuning, ending up with the Equal
Temperament. Each one of this musical system
used a specific scientific ratio to get all the
frequencies which correspond to the notes in the
scale. We will go through the method used to
form this tunings and we will explain why the
Equal Temperament is used nowadays, while the
others are considered unsuitable.
To explain the scientific method used to form the scales it is
necessary to introduce the Harmonic series.
When we hear or produce any sound, this musically is never
by itself alone, but it always goes with other sounds, higher
and weaker, that are generated simultaneously from the
fundamental sound. These sounds are called Harmonics and
they are not perceptible by the human ear, unless you don’t
use scientific devices like the Helmholtz resonators. This is an
illustration of Harmonic series in musical notation:
• In the table we can notice that the Octave ratio is
the ratio between the second note and the first one
(ratio 2:1), while the perfect Fifth ratio is the ratio
between the third sound and the second one (ratio
3:2) and so on. Analyzing all the other frequency
ratios, compared to the musical interludes we get
this table:
Frequency ratio
Interlude
2/1
3/2
4/3
5/4
6/5
7/6
9/8
16/15
Perfect Octave
Perfect Fifth
Perfect Fourth
Major Third
Minor Third
Minor Third
Major Second
Minor Second
Pythagorean and Natural Tunings
Basing on the former table, the Pythagorean tuning
system consists in prefixing an interlude of a perfect fifth
(e.g. C – G) and then getting all the other frequencies
(corresponding to the notes in the scale) through
multiplications and divisions by 3/2 (the ratio
corrensponding to the perfect fifth).
The Natural Tuning, instead, uses all the ratios in
the former table to create the musical scale.
In both of the systems, Pythagorean and Natural,
the tuning depends on the fundamental
frequency from whom we start the calculation of
the other frequencies.
With time, musicians began to look for a new
musical system because in the Pythagorean one
some interludes were dissonant, in the Natural
one there were several notes which were
impossible to play practically.
The Equal Temperament
The Equal Temperament is the musical
system used nowadays. It derives from
the division of the octave in 12 equal
parts, through a geometric progression,
which is a sequence of numbers where
each term after the first is found by
multiplying the previous one by a fixed
non-zero number called the ‘common
ratio’.
So we actually have to calculate the
common ratio q in a geometric
progression which is made up of 13 terms
(the notes), whose first and last terms are
the value fo and the value 2 fo.
• Let’s have a look to this table, with all the terms and their values:
Term
Correspondence
Ex.
a1
f0
do3
a2
d0#3
a3
re3
a4
re#3
a5
mi3
a6
fa3
a7
fa#3
a8
sol3
a9
sol#3
a10
la3
a11
la#3
a11
si3
a13
2 f0
do4
Value
You can easily check that the value of ak is:
ak = fo ∙ q (k-1)
where (k-1) is the number of semitones
that exist between the frequency fo and
the frequency associated to the term ak .
So we get the reason q of the progression
with the relation:
q=
• That is, in case the 13° term (2fo ) is equal to the
double of the value of the first term (f0 ) :
q=
This value is exactly the coefficient that we have to
use in order to get all the frequencies of the Equal
Temperament. In fact, multiplying or dividing a
prefixed frequency (normally the frequency of 440
Hz, corresponding to the note A3 ) by this value q
(
) we get all the other frequencies in the scale.
Fibonacci Series in Music
The Fibonacci Series is a sequence of numbers in which 1 appears
twice as the first two numbers, and every subsequent number is the
sum of two preceding numbers:
1, 1, 2, 3, 5, 8, 13 ... and so on.
As it continues, the ratio between any number and its successor
approaches the ratio of golden section (1:1.618).
Various composers have used the
Fibonacci numbers when composing
music, for instance Debussy, Schubert,
Bach, but Bartòk in particular, who used
this Series when forming the chords or a
scale. The most typical chord is shaped on
the projection 8:5:3, where these
numbers are referred to the number of
semitones between the notes in the
chord.
This is exactly the base chord that is
used in ‘Music for Strings,
Percussion and Celesta’ that we are
listening in the background music.
The effect is a bit creepy, because it
is unusual, that is why they used it
as the main soundtrack in the
famous move The Shining, starring
Jack Nicholson.
But Fibonacci series is used in modern songs as
well, not only in classical music.
Lately Rock music, expecially Progressive Rock,
got involved more and more with composing
using Fibonacci numbers and the golden ratio.
For instance Genesis (with ‘Firth of Fifth’), the
Deep Purple (with ‘Child in Time’-the one we are
listening now) and the Dream Theater (with the
whole album ‘Octavarium’).
But the clearest use of Fibonacci Series in a
song can be actually seen in the song
‘Lateralus’, by the american rock band
Tool. They make a perfect use of the first
terms of the progression, and the song
continuosly refers to the spiral of the
golden ratio.
To close we are going to watch a video that
explains all the particulars of Fibonacci
series used in the song Lateralus.