Connections between Math and Music
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Connections between Math and Music Laura Harlow – HSPVA Rhodora Maligad – Austin HS PowerPoint PPT Presentation

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

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Math and music

Connections between Math and Music

Laura Harlow – HSPVA

Rhodora Maligad – Austin HS


Math and music

A village without music is a dead place.

African proverb


Math and music

  • 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 musically-inclined.


Math and music

  • Overview

  • Historical Connections

  • Obvious Connections

  • Functional Connections

  • Geometric Connections


Math and music

  • 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


Math and music

  • 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


Math and music

  • 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


Math and music

  • 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


Math and music

  • 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.


Math and music

  • The Obvious Connection: Rhythm

  • Rhythm is the basis upon which music is built just as the concept of number is the basis of mathematics.


Math and music

  • 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.


Math and music

  • 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.


Math and music

  • 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


Math and music

  • 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.


Math and music

What makes music different than noise?

The answer is in the mathematics.


Math and music

  • 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.


Math and music

  • 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)


Math and music

  • 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


Math and music

Functional Connections

y = f(x)

y = f(x) + 2

y = 2 f(x)

y = -f(x)

y = 2 f(x) + 4


Math and music

Composers use math in subtle ways to create musical compositions that are pleasing to hear.


Math and music

Geometric Connections

Many geometric transformations have musical counterpart

MusicMath

repeat horizontal translation

transposition horizontal and vertical translation


Math and music

Many geometric transformations have musical counterpart

MusicMath

inversion vertical reflection

retrogressionhorizontal reflection

retrograde inversion180° rotation


Math and music

  • 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


Math and music

“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)


Math and music

  • 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/>


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