The Mathematics of Music

1. The Mathematics of Music Katherine Goulde

2. 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

3. 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

4. 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

5. 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#? • 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.

6. 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’

7. 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

9. 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)

10. 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

11. 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

12. 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

13. 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?? • 1st- 880hz, • 2nd- 1320hz, 3rd- 1760hz, 4th- 2200hz • What if we start with A with frequency 880hz? • 1760hz, 2640hz, 3520hz • Many stringed instruments produce overtones that approximate the harmonic series

14. ‘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…

15. 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)

16. 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

17. 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

18. 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

19. 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

20. Musical Form- Fugue

21. Musical Form- Fugue • Bach’s Fugue #2 from The Well-Tempered Clavier

22. 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!!!