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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?

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The Mathematics of Music

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  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

  8. 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)

  9. 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

  10. 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

  11. 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

  12. 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

  13. ‘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…

  14. 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)

  15. 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

  16. 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

  17. 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

  18. 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

  19. Musical Form- Fugue

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

  21. 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!!!

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