Science of Sound Week 2


Previously, we talked about wave length and frequency. We measure frequency in Herz, abbreviated as Hz. A 1Hz sine wave goes through a complete cycle one time per second. A 440Hz sound wave goes through a complete cycle 440 times per second. The frequency is the reciprocal of the duration. A single cycle of a 440 Hz sine wave is  \frac {1} {440}th of a second.

We also talked about the speed of sound, which is 340 m/s at 20 degrees celsius. If we have a 1 Hz wave, travelling at 340m/s, it takes one full second to get through the complete cycle. Which means that the front of the sound wave is 340 metres away from the back. The wavelength is 340 metres.

A 2 Hz sine wave also travels 340m/s. The time it takes to get through each cycle is half a second. In half a second, the front has travelled 170 metres, which is to say that’s the wave length.

A 10 Hz sine wave lasts \frac{1}{10}th of a second, so the wave length is \frac{340}{10}, which is to say 34 metres.

A 100 Hz sine wave is \frac{340}{100} = 3.4 metres. The octave higher, 200 Hz, is 1.7 metres. The wave length is the speed of sound (c in the formula) divided by the frequency. \lambda = \frac{c}{f}


We mentioned 440 Hz in the first paragraph. If that sounds familiar, it’s because it’s also the frequency of most tuning forks. It’s the defined frequency for A.

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We also know that if we double the frequency to 880, that’s also and A. Or if we halve it to 220.

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110 Hz, 55 Hz and 27.5 Hz are also As. As we get lower the frequencies get closer together and as we get higher they’re farther apart. 7040 Hz and 14080 Hz are also As.

We know that all As are 440 multiplied or divided by a power of 2. We also know that doubling any frequency gives us an octave of that frequency. We can generalise from this to come up with a formula for a one note scale based on the octave. Where f is frequency, f \times x = 2f. It’s obvious here that x is 2.

What if we want a two note scale that uses Equal Temperament? This is a system where all the notes are equally distant from each other perceptually. We know that this has to be based on multiplication. We want an equal ratio between all the notes. Therefore to get from the bottom note to the next one, we need to multiply by some number x. And then to get from the middle note to the octave, we multiply by x again. f \times x \times x = 2f We can simplify those two xs.  \therefore f \times x^2 = 2f And divide both sides by f.  \therefore x^2 = 2 Solving for x:  \therefore x = \sqrt{2}. Our two note scale is 440, 622.25, 880. This is because  440 \times \sqrt{2} = 622.25 and  622.25 \times \sqrt{2} = 880

What about a three note scale? f \times x \times x \times x = 2f Which means  \therefore x^3 = 2 and so  \therefore x = \sqrt[3]{2} To work out this scale,  440 \times \sqrt[3]{2} = 554.37,  554.37 \times \sqrt[3]{2} = 698.46, and 698.46 \times \sqrt[3]{2} = 880.

If we want a 4 note scale, we can use \sqrt[4]{2} or for a five note scale \sqrt[5]{2}. But for a piano, we want 12 notes, including all the white and black keys.

Therefore, the tuning used by the piano, called “12 Tone Equal Temperament” (or 12tet) uses \sqrt[12]{2}.

We know that the frequencies are exponential, but perceptually, the difference between a C and and A is the same in any octave. Our scales and keyboards and the musical concept of pitch is linear. Every octave may double in frequency, but it’s always only 12 semitones.

Figure 5: “Logarithmic plot of frequency in hertz versus pitch of a chromatic scale starting on middle C.” via Image by Jono4174, public domain via Wikimedia Commons.

You now know enough to work out the frequency for every single note on the piano. (Or, you can look it up on wikipedia.) You can also work out the wavelength for every frequency on the keyboard. If the lowest note is A0, the frequency is 27.5 Hz, so the wavelength \lambda = \frac {340}{27.5} = 12.4 metres. And the highest note, C8 is 4186 Hz, so \lambda = \frac {340}{4186} = 0.081 metres. What a range! And that’s not even the highest note we can hear!

Going Further

Not all scales are based on octaves! The Bohlen-Pierce scale is based on multiplying frequencies by 3. How could you compute an equally tempered scale for Bohlen Pierce? If you wanted the scale steps to be roughly the same size as 12tet, how many scale steps would you use?

Published by

Charles Céleste Hutchins

Supercolliding since 2003

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