Math and music.ppt
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Transcript of Math and music.ppt
7/25/2019 Math and music.ppt
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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%