The Mathematics of Music - PowerPoint Presentation

Slide1

The Mathematics of Music

Katherine

GouldeSlide2
Outline

Basic tonal theorySound and Hertz

Note values and rhythm

Intervals

Scales

Overtones

Harmonics

Rhythm

Western, Indian music, African music

Musical Styles and Forms

FuguesSlide3
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 WeekendSlide4
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 notesSlide5
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 12

th

root of 2 (which is 1.05946309…)

What is the frequency of A#?

440 x 1.0594.. = 466.16376

What is the frequency of B?

466.1637x 1.0594.. = 493.8833

What about a full octave higher?

Double the frequency.Slide6
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’Slide7
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 stepsSlide8

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 fourth4:3

6Augmented

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

2:1Slide9
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

)Slide10
Scales- Chromatic Scale

A scale with 12 pitchesEach pitch is a half step (semitone) apart

Multiply the frequency by the 12

th

root of 2

Tuned using equal temperament

Dividing the octave into equal partsSlide11
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 4

th

and perfect 5thSlide12
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 modesWhen excited, will oscillate at several frequencies at onceSlide13
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??

1

st

- 880hz,

2

nd

- 1320hz, 3

rd- 1760hz, 4th- 2200hzWhat if we start with A with frequency 880hz?

1760hz, 2640hz, 3520hzMany stringed instruments produce overtones that approximate the harmonic seriesSlide14
‘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…Slide15
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)Slide16
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 SpringSlide17
Non-Western Rhythm

Focuses more on additive rhythmBalinese 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 octaveSlide18
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 eachSlide19
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 pointsSlide20
Musical Form- Fugue

Exposition

1

st

Middle Entry

2

nd

Middle Entry

Final Entries

CODA

Sop.

Subject

C1

C2

A

Episode

C1

C2

Episode

S

Episode

C1

Free counterpoint

Alto

Ans

C1

C2

S

C1

C2

S

C1

Bass

S

C1

C2

A

C1

C2

SSlide21
Musical Form- Fugue

Bach’s Fugue #2 from The Well-Tempered ClavierSlide22
Discussion

How do different rhythmical structures change the character of a songWhat 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!!!