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Connections between Math

and Music

Laura Harlow – HSPVA

Rhodora Maligad – Austin HS

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A village without music is adead place. African proverb

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Our Goal

!o understand so"e of the "athe"atics

found in "usic

!o "a#e "athe"atical functions and

geo"etric transfor"ations better understood

through "usic

!o be able to differentiate the curriculu" for

the auditor$ learners and%or the "usicall$&

inclined'

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Overview

Historical (onnections

Obvious (onnections

)unctional (onnections

Geo"etric (onnections

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(onnections between the two

disciplines have been studied sinceancient ti"es'

 P$thagoras *+,- .(/

 Plato *010 – 203 .(/ elaborated on

4"usic of the spheres5

 Arch$tas *esti"ated 02-&2+-/ On Music 

 6ico"achus *7-- A8/ Introduction to Music 

 Ptole"$ *7--&79+ A8/ Harmonics

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More Historical (onnections:

 .oethius *+-- A8/ Principles of Music 

 ;epler *7+37 – 792- A8/ refined

4"usic of the spheres5

 Galileo *79--/ so"e co"binations of tones are

"ore pleasing than others

 <uler *73-3 & 73,+/ A New Theory of Music 

 .ernoulli *73-- – 73,1/ e=tended <uler>s wor#

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P$thagorean "usic

?dentified "usic with nu"bers

Music was defined and restricted b$ the "ath that

dictated its theor$

P$thagoras used onl$ whole nu"ber ratios of

string length and the fre@uencies of notes

?f $ou divide an octave into 71 e@ual parts we get

the irrational nu"ber 1 7%71

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P$thagorean "usic

Proved the e=istence of irrational nu"bers but

chose to ignore nu"bers that could not be written

as a fraction

!he o"ission of irrational nu"bers resulted in

scale #nown as a "inor scale

Speculations arise about the effect on Gree# pla$*tragedies/ since the "usic is "uch "ore sinister 

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(ultural 8ifferences

(ultures have developed their "usic in various wa$s

a"ong the" differences in the wa$s the$ divided an

octave into notes'

Bestern "usic uses a pattern of + & 3 notes in a scale'

 African cultures also use 3 notes with the 2rd and 3th

notes slightl$ flattened these are now #nown as 4blue

notes5

Most Asian "usic uses a pattern of 71 notes in a

scale'

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!he Obvious (onnectionC Rh$th"

Rh$th" is the basis upon which "usic is built

 Dust as the concept of nu"ber is the basis of

"athe"atics'

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Measures of !i"e

!i"e signature is a fraction whose nu"erator tells us

how "an$ beats "a#e up a "easure and whose

deno"inator tells what note is assigned to that beat'

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G(8 in Music

!he concept of Greatest (o""on 8eno"inator

and Addition of )ractions can be used to

deter"ine if a "usician is wor#ing within thegiven ti"e signature or rh$th"'

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6ote (o"binations !hat Bor# E note F note F note 0%0 7

  note F E note F 7%, note F 7%, note ,%, 7

6ote (o"binations !hat 8O6>! BOR;

  note F E note I J 7

  note F ⅛ note F K note F E note F K note 7K 7

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What makes music diferent

than noise? The answer is in themathematics.

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Be need so"e definitions

)re@uenc$ – nu"ber of vibrations per second

Pitch – a listener>s evaluation of fre@uenc$

!one – a sound that lasts long enough and is

stead$ enough to have pitch @ualit$ and

loudness

Octave – sa"e note *tone/ fre@uenc$ doubled

Scale – !he pattern used to travel an octave'

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So"e other interesting definitions

used in "usicC

 A"plitude – distance between "a= and "in

Bavelength – distance traveled in a c$cle

Period – ti"e to co"plete a wavelength

Loudness – listener>s evaluation of a"plitude

Pure tone – constant fre@uenc$ and a"plitude

*creates the sine wave/

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So what is "usic and what is noise

Music is an organiation of sounds with so"e

degree of rh$th" "elod$ and har"on$

Music is said to be an art and often defined b$contrast with noise

6oise is a "i=ture of different fre@uencies

Bhite noise – e@ual a"ounts of sound power fro"

each spectru" of available fre@uencies

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Composers use math in

subtle ways to createmusical compositionsthat are pleasing to hear.

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Geo"etric (onnections

Man$ geo"etric transfor"ations have "usicalcounterpart

  Music Math

  repeat horiontal translation

  transposition horiontal and vertical

translation

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Man$ geo"etric transfor"ations

have "usical counterpart

  Music Math

  inversion vertical reflection

  retrogression horiontal reflection

  retrograde inversion 7,-N rotation

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So"e Miscellaneous ?nfor"ation

Golden proportion

)ractal "usic

More child prodigies in "ath and "usic than an$

other disciplines

Music and "ath do not re@uire "uch e=perience

and interpretation on "anipulation of s$"bols is

significant

Moart>s Melod$ 8ice – se 1 9 sided dice rolled

to deter"ine what was pla$ed in each of 79 bars of

"usic to create a walt

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That person is a musician!who! through care"ul rational

contemplation! has gainedthe knowledge o" makingmusic! not through theslavery o" labor! but throughthe sovereignty o" reason.#

.oethius *A'8' 0,-/

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.ibliograph$

Garland !rudi Ha""el and ;ahn (harit$ Vaughan' Math and MusicC Har"onious

(onnections' Palo AltoC 8ale Se$"our Pulbications 7+'

.eall Scott' )unctional Melodies' ;e$ (urriculu" Press 1---'

Peterson ?vars' 4(ircles of 8issonanceQ MAA Online' 6ove"ber 10 73' une

7+ 1--+ JhttpC%%www'"aa'org%"athland%"athtre#7710'ht"l'

Peterson ?vars' 4Medieval Har"on$Q MAA Online' 7' une 7+ 1--+

JhttpC%%www'"aa'org%"athland%"athtre#71+'ht"l'

Mathe"atics and Music' Rusin 8avid'1--0' une 7+ 1--+' JhttpC%%

www'"ath'niu'edu%Trusin%uses&"ath%"usic%