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Connections between Math and Music
Laura Harlow – HSPVA
Rhodora Maligad – Austin HS
A village without music is a dead place.
African proverb
Our Goal
•To understand some of the mathematics
found in music
•To make mathematical functions and
geometric transformations better understood
through music
•To be able to differentiate the curriculum for
the auditory learners and/or the musicallyinclined.
Overview
•Historical Connections
•Obvious Connections
•Functional Connections
•Geometric Connections
Connections between the two
disciplines have been studied since
ancient times.
• Pythagoras (580 BC)
• Plato (424 – 347 BC) elaborated on
“music of the spheres”
• Archytas (estimated 430-350) On Music
• Nicomachus (100 AD) Introduction to Music
• Ptolemy (100-165 AD) Harmonics
More Historical Connections…
• Boethius (500 AD) Principles of Music
• Kepler (1571 – 1630 AD) refined
“music of the spheres”
• Galileo (1600) some combinations of tones are
more pleasing than others
• Euler (1707 - 1785) A New Theory of Music
• Bernoulli (1700 – 1782) extended Euler’s work
Pythagorean music
•Identified music with numbers
•Music was defined and restricted by the math that
dictated its theory
•Pythagoras used only whole number ratios of
string length and the frequencies of notes
•If you divide an octave into 12 equal parts, we get
the irrational number 2 1/12
Pythagorean music
•Proved the existence of irrational numbers but
chose to ignore numbers that could not be written
as a fraction
•The omission of irrational numbers resulted in
scale known as a minor scale
•Speculations arise about the effect on Greek play
(tragedies) since the music is much more sinister
Cultural Differences
•Cultures have developed their music in various ways,
among them differences in the ways they divided an
octave into notes.
•Western music uses a pattern of 5 - 7 notes in a scale.
•African cultures also use 7 notes with the 3rd and 7th
notes slightly flattened, these are now known as “blue
notes”
•Most Asian music uses a pattern of 12 notes in a
scale.
The Obvious Connection: Rhythm
Rhythm is the basis upon which music is built
just as the concept of number is the basis of
mathematics.
Measures of Time
Time signature is a fraction whose numerator tells us
how many beats make up a measure and whose
denominator tells what note is assigned to that beat.
GCD in Music
The concept of Greatest Common Denominator
and Addition of Fractions can be used to
determine if a musician is working within the
given time signature or rhythm.
Note Combinations That Work
• ½ note + ¼ note + ¼ note = 4/4 = 1
• ¼ note + ½ note + 1/8 note + 1/8 note = 8/8 =1
Note Combinations That DON’T WORK
• ¼ note + ½ note = ¾ < 1
• ¼ note + ⅛ note + ⅛ note + ½ note + ⅛ note = 1⅛ > 1
LCM in Music
The math concept of Least Common Multiple can be
used to determine where the second note will fall in
relation to the three-note rhythmic scale.
What makes music different than noise?
The answer is in the mathematics.
We need some definitions
•Frequency – number of vibrations per second
•Pitch – a listener’s evaluation of frequency
•Tone – a sound that lasts long enough and is
steady enough to have pitch, quality and
loudness
•Octave – same note (tone), frequency doubled
•Scale – The pattern used to travel an octave.
Some other interesting definitions
used in music:
•Amplitude – distance between max and min
•Wavelength – distance traveled in a cycle
•Period – time to complete a wavelength
•Loudness – listener’s evaluation of amplitude
•Pure tone – constant frequency and amplitude
(creates the sine wave)
So, what is music and what is noise
•Music is an organization of sounds with some
degree of rhythm, melody, and harmony
•Music is said to be an art and often defined by
contrast with noise
•Noise is a mixture of different frequencies
•White noise – equal amounts of sound power from
each spectrum of available frequencies
Functional Connections
y = f(x)
y = f(x) + 2
y = 2 f(x)
y = -f(x)
y = 2 f(x) + 4
Composers use math in subtle ways
to create musical compositions that
are pleasing to hear.
Geometric Connections
Many geometric transformations have musical
counterpart
Music
Math
repeat
horizontal translation
transposition
horizontal and vertical
translation
Many geometric transformations
have musical counterpart
Music
Math
inversion
vertical reflection
retrogression
horizontal reflection
retrograde inversion
180° rotation
Some Miscellaneous Information
•Golden proportion
•Fractal music
•More child prodigies in math and music than any
other disciplines
•Music and math do not require much experience
and interpretation on manipulation of symbols is
significant
•Mozart’s Melody Dice – Use 2 6 sided dice rolled
to determine what was played in each of 16 bars of
music to create a waltz
“That person is a musician, who, through
careful rational contemplation, has
gained the knowledge of making music,
not through the slavery of labor, but
through the sovereignty of reason.”
Boethius (A.D. 480)
Bibliography
•Garland, Trudi Hammel and Kahn, Charity Vaughan. Math and Music: Harmonious
Connections. Palo Alto: Dale Seymour Pulbications, 1995.
•Beall, Scott. Functional Melodies. Key Curriculum Press, 2000.
•Peterson, Ivars. “Circles of Dissonance" MAA Online. November 24, 1997. June
15, 2005 <http://www.maa.org/mathland/mathtrek_11_24.html>.
•Peterson, Ivars. “Medieval Harmony" MAA Online. 1999. June 15, 2005
<http://www.maa.org/mathland/mathtrek_1_25_99.html>.
•Mathematics and Music. Rusin, David.2004. June 15, 2005.
<http://www.math.niu.edu/~rusin/uses-math/music/>