Music And Math

Pythagoras was important to both mathematicians and musicians.

Pythagoras

Greek, 6th century B.C.E. Monday, March 26, 12

2:1 = octave 3:2 = fifth

Pythagoras figured out that certain musical intervals are related to each other by simple whole number ratios.

4:3 = fourth simple, whole number ratios Monday, March 26, 12

music of the spheres “celestial bodies move in proportions equivalent to pure musical intervals” It was tempting to think that everything in the universe could be summed up neatly. This view persisted for centuries.

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physical ratios 695 mm

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Let’s look at physical ratios, such as string length on a cello. A string is measured in mm’s.

695 = 347.5 2

695 = 231.66 3 1/2 the string & 1/3 the string Those numbers are getting awkward.

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we measure with touch and sound Luckily, when we play, we don’t use those kinds of measurements, we go by what we hear and feel.

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string (or pipe) length By playing the note that occurs at 1/2 the string length, we find that Pythagoras was right.

1 2

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L = one pitch

L = one octave higher

“flutes” of different lengths I made a set of “flutes” from PVC pipe. They don’t sound great, but they illustrate the Pythagorean ratios.

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2:1

flute = 27” long

Aaron’s flute is 27” long, so his piccolo, which plays an octave higher, should be 13.5” long. It isn’t. Can you guess why?

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piccolo = 12.5” long

flute =

The t wo instruments aren’t quite the same shape on the inside, which affects their pitch/length ratio.

cylinder

cone piccolo

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measure of pitch = frequency vibrations per second Each pitch has its own frequency, which is the number of times it vibrates per second, named after Heinrich Hertz.

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or Hertz

One example of a tuning fork. If you strike it and look at it closely, you can see that it looks blurry--it’s vibrating very fast.

440Hz

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A440 vibrates 440 times per second

frequency ratios: if X = one pitch 2 X = one octave higher

X

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2X

string (or pipe) length and

frequency are

inversely proportional Monday, March 26, 12

2:1 = octave Pythagoras used these t wo ratios to find all the notes within one octave. He was very clever.

2X frequency

3:2 = fifth 1.5 X frequency

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2. It’s easy to find an octave lower than D, just divide its frequency in half. From D he went up to A, then up to E, which again had to drop down an octave. Repeat....

1. From the starting note of C, he found G, a fifth higher. Then he found a fifth above G, which was D, but it was outside the octave.

a fifth

a fifth

an octave

C D

E

G

A

3. Now you’re ready for a math problem.

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C

D E

math problem: calculator allowed first note’s frequency X 1.5 = 2nd note (fifth) 2nd note X 1.5 = 3rd note (fifth) 3rd note ÷ 2 = octave lower that frequency X 1.5 = 4th note, etc do this 12 times (“circle of 5ths”)

last is NOT an exact 2:1 ratio to first hence, all pianos are slightly “out of tune” Monday, March 26, 12

the complicated world of “tuning” 12th root of 2 (1.0594630943593...) Most pianos are tuned using “frequency X 12th root of 2” (a truly magical number!) to find the next semi-tone, so the octave comes out even. This is called “equal tempered” tuning.

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