Download The Mathematics of Music

THE MATHEMATICS OF MUSIC
Katherine Goulde
OUTLINE

Basic tonal theory
Sound and Hertz
 Note values and rhythm
 Intervals

Scales
 Overtones
 Harmonics
 Rhythm



Western, Indian music, African music
Musical Styles and Forms

Fugues
LISTENING SAMPLE
Can you find a rhythm?
 What emotions does it evoke?
 Is this a particular style of music?
 Example1: Symphony No.40 in G Minor- Mozart
 Example2: The Rite of Spring- Stravinsky
 Example3: Horchata: Vampire Weekend

DEFINITIONS

Note: a pitched sound

Rest: an interval of silence



Rhythm: movement characterized by regular
recurrence or change of different patterns
Beat: the basic time unit of music (a pulse)
Interval: the relationship between the pitches of
two notes
BASIC TONAL THEORY

Note- a sound whose pitch has a corresponding
frequency measure in hertz (cycles per second)
A below middle C has a frequency of 440 hz
 The ratio of frequency between half tones= the 12th
root of 2 (which is 1.05946309…)


What is the frequency of A#?


What is the frequency of B?


440 x 1.0594.. = 466.16376
466.1637x 1.0594.. = 493.8833
What about a full octave higher?

Double the frequency.
BASIC TONAL THEORY- NOTE VALUES
Note value- the duration of a note
 There are values for rests as well







Whole note- 4 beats
Half note- 2 beats
Quarter note- 1 beat
Eighth note- 0.5 beat
Sixteenth note- 0.25 beat
You can increase the value of the note or rest by 1.5 by
adding a ‘dot’
BASIC TONAL THEORY- INTERVALS
Interval: the relationship between the pitches of two
notes
 An interval can be vertical (or harmonic) as well as
horizontal (or melodic)
 An interval can be shown as the ratio of the
frequencies of the two pitches



Ex) Octave-> 2:1, Unison-> 1:1, Perfect Fifth-> 3:2
An interval can be labeled according to the number of
scale steps
Number of Half-steps Interval name
Frequency Ratio
0
Unison (or prime)
1:1
1
Minor second
16:15
2
Major second
9:8
3
Minor third
6:5
4
Major third
5:4
5
Perfect fourth
4:3
6
Augmented 4th or
Diminished 5th or
tritone
45:32
64:45
7
Perfect fifth
3:2
8
Minor sixth
8:5
9
Major sixth
5:3
10
Minor seventh
16:9
11
Major seventh
15:8
12
Octave
2:1
SCALES
Scale- a collection of ordered notes used to create a
musical piece
 Can be classified according to the types of intervals
(diatonic or chromatic for example)
 Can also be classified by the number of tones per
octave- (Ex: pentatonic, hexatonic, heptatonic)

SCALES- CHROMATIC SCALE
A scale with 12 pitches
 Each pitch is a half step (semitone) apart

Multiply the frequency by the 12th root of 2
 Tuned using equal temperament
 Dividing the octave into equal parts

CHROMATIC SCALE

Why divide the octave into 12
parts?




Take the consonant intervals:
octave, fifth, fourth, Major 6th,
Major 3rd, Minor 3rd, and
Minor 6th.
12 is the smallest division of
the octave that best
approximates all 7 basic
consonant intervals
Why?
Take the scale as a cyclic
group of order 12 ->
 ({1, …, 12}
 Note that 5 and 7 are two of
the generators, and these
correspond to the perfect 4th
and perfect 5th

OVERTONES
Overtone- any frequency higher than the
fundamental frequency
 The fundamental together with the other
frequencies are called partials
 Overtones can be harmonic or inharmonic
 Inharmonic overtones- partials that have
frequencies not in proportion to the fundamental
frequency
 How does this work?

Natural vibrations of oscillators= normal modes
 When excited, will oscillate at several frequencies at
once

HARMONICS

What are harmonics?
Types of overtones
Waves at proportional
frequencies, and at inversely
proportional amplitudes
 Take the case of playing A below
middle C with full harmonics- A
has a frequency of 440 hz.



What are the first 4 harmonics??



What if we start with A with
frequency 880hz?


1st- 880hz,
2nd- 1320hz, 3rd- 1760hz, 4th2200hz
1760hz, 2640hz, 3520hz
Many stringed instruments
produce overtones that
approximate the harmonic
series
‘HARMONIC’ OR ‘OVERTONE’ SINGING




What is this? And How is it done?
Given the fundamental tone the singer is singing, he is able to
amplify the overtones simultaneously
The result is more than one distinct tone being sung at the same time
Let’s listen to a few examples…
RHYTHM
Movement characterized by regular recurrence or
change or different patterns
 Beat- the speed of the underlying pulse
 Tempo- how quickly the pulse repeats



Measured in beats per minute (bpm)
Time signature
Tells the number of beats per measure of music (the upper
numeral)
 Tells which note value represents equal one beat (the lower
numeral)

RHYTHM

There are different time signatures are associated with
types of music.
4/4 Common time
 2/2 Duple- Cut time-> marches, or fast orchestral music
 2/4 Duple-> often used for polkas or marches
 3/4 Triple-> often used for waltzes


It is possible to mix rhythms within one piece

Stravinsky’s The Rite of Spring
NON-WESTERN RHYTHM

Focuses more on
additive rhythm
Balinese and
Javanese music

Interlocking rhythms
of gamelon ensemble


The numbers are
pitches, dots are rests,
overbars indicate to play
2x as fast, dots above
and below indicate
octave
NON-WESTERN RHYTHM

African music often makes use of polyrhythms


2 or more rhythms at the same time
Indian music often uses complex rhythmic
cycles (called tala)

Most common tala is called Teental- which is a cycle
of four measures of four beats each
MUSICAL FORM- FUGUE



Fugue- a composition technique for a set number of
‘voices’
The word fugue is derived from a wording that means
to ‘chase’ or ‘flee’
Makes use of imitative counterpoint
The first voice enters with the main theme or subject
 There are subsequent entries by other voices imitating the
subject
 This series of entries is called the exposition
 After the exposition, there may be a connecting passage, or
episode


A fugue can have 1, 2, or 3 subjects which can be
developed simultaneously or at different points
MUSICAL FORM- FUGUE
Exposition
Sop.
Alto
Bass
Subject
1st
Middl
e
Entry
C1
C2
A
Ans
C1
C2
S
C1
Epi
sod
e
C1
C2
S
C1
C2
A
2nd
Middle
Entry
Epis
ode
S
Final
Entries
Episo
de
C1
Free
counter
point
C2
S
C1
C1
C2
S
COD
A
MUSICAL FORM- FUGUE

Bach’s Fugue #2 from The Well-Tempered
Clavier
DISCUSSION
How do different rhythmical structures change
the character of a song
 What is the correspondence between a number’s
characteristics and the ‘feel’ gives, with

Rhythm
 Intervals


Can you think of other connections between
music and mathematics?
• Thanks so much!!!