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

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 musically- inclined.

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.

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

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

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

Composers use math in subtle ways to create

musical compositions

that are pleasing to hear.

Geometric Connections

Many geometric transformations have musical counterpart

translation

Many geometric transformations have musical counterpart

Some Miscellaneous Information

•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