T HE M ATHEMATICS OF M USIC Katherine Goulde

O UTLINE 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

L ISTENING S AMPLE 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

D EFINITIONS 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

B ASIC T ONAL T HEORY 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 …) What is the frequency of A#? 440 x = What is the frequency of B? x = What about a full octave higher? Double the frequency.

B ASIC T ONAL T HEORY - N OTE 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 beat You can increase the value of the note or rest by 1.5 by adding a ‘dot’

B ASIC T ONAL T HEORY - I NTERVALS 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-stepsInterval nameFrequency Ratio 0Unison (or prime)1:1 1Minor second16:15 2Major second9:8 3Minor third6:5 4Major third5:4 5Perfect fourth4:3 6Augmented 4 th or Diminished 5 th or tritone 45:32 64:45 7Perfect fifth3:2 8Minor sixth8:5 9Major sixth5:3 10Minor seventh16:9 11Major seventh15:8 12Octave2:1

S CALES 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)

S CALES - C HROMATIC S CALE A scale with 12 pitches Each 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 parts

C HROMATIC S CALE 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 5th

O VERTONES 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

H ARMONICS 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 hz, 3 rd hz, 4 th hz What if we start with A with frequency 880hz? 1760hz, 2640hz, 3520hz Many stringed instruments produce overtones that approximate the harmonic series

‘H ARMONIC ’ OR ‘O VERTONE ’ 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…

R HYTHM 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)

R HYTHM 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

N ON -W ESTERN R HYTHM 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

N ON -W ESTERN R HYTHM 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

M USICAL F ORM - F UGUE 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

M USICAL F ORM - F UGUE Exposition1 st Middl e Entry 2 nd Middle Entry Final Entries COD A Sop. SubjectC1C2A Epi sod e C1C2Epis ode S C1Free counter point Alto AnsC1C2SC1C2SC1 Bass SC1C2AC1C2S

M USICAL F ORM - F UGUE Bach’s Fugue #2 from The Well-Tempered Clavier

D ISCUSSION 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!!!