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:

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: Let’s have a look to this table, with all the terms and their values: 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 ) : 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.