Mathematics of Equal Temperament

My nephew Sam is a bassist. He showed me this video by Adam Neely about how polyrhythms are the same thing as harmonies. Really spectacular information, really well explained.

Sam then asked me to explain the equation of Equal Temperament. Here goes:

What are musical notes?

Musical notes, like all sounds, are vibrations in the pressure of the air. Those vibrations arrive at your ear, shake your ear drum back and forth, which shakes the fluid in your inner ear, which stimulates nerve cells. Your brain interprets the signals from those nerves as sounds, and, if the vibrations arrive at regular intervals, as notes. As Adam Neely points out, notes are the same thing as beats, except that notes are fast enough for you to hear them as a constant tone, whereas beats are slow enough that you hear them as discrete events at regular intervals in time.

We describe a beat in “beats per minute”, which is a count of how many pulses of sound arrive at your ear in one minute of time. We describe a tone or note in “beats per second”, otherwise known as “cycles per second”, abbreviated ‘cps’, or, in modern units, Hertz, abbreviated ‘Hz’. 1 Hz = 1 cycle per second = 1 pulse per second. The tick of a second hand of a mechanical clock or watch happens at 1 Hz. 1 Hz = 1 cycle per second = 60 beats per minute = 60 bpm. These units measure the same thing, the frequency of arrival of a pulse of pressure in the air at your ear drum. So we say that bpm and Hz are both units of Frequency. We also use the term Pitch interchangeably with Frequency.

Absolute versus relative pitch for a single note:

Higher frequency = higher pitch. Lower frequency = lower pitch. The precise absolute frequency of a note is not very important. When a musician tunes her instrument, she adjusts the absolute pitch of her instrument to match that of the others in the ensemble. So long as the instruments are adjusted to the same set of absolute frequencies as the other instruments, we say that the ensemble is “in tune”, and everything sounds good. A frequency of 440 cycles per second = 440 Hz is the note A, and is a pretty standard absolute tuning. A little lower, using an A note of 436 Hz, and the orchestra will still sound good, even though it will sound flat (lower in frequency) compared to another orchestra tuned to an A of 440 Hz. A little higher, an A note of 443 Hz, and the orchestra will still sound good, even though it will sound sharp (higher frequency) compared to the 440 Hz tuned orchestra. There is more information on the fundamental frequency of A here.

Octave:

So we tune up, with everybody setting their instruments so that the A note makes the air vibrate 440 times a second, at 440 Hz. That’s only one note. What about all the other notes?

If a person plays a note that is double that pitch, at 880 Hz, twice as many vibrations per second as 440 Hz, that note is also an A, but one octave (eight whole notes: A, B, C, D, E, F, G, A) higher. 220 Hz gives you an A as well, but one octave lower than the original 440 Hz note. (If you were playing in a group tuned to 443 Hz for A, then the A an octave up would be 886 Hz, and the A an octave down would be at 221.5 Hz.)

Octaves are physically significant. Physical objects, like musical instrument strings or your inner ear, wiggle in similar ways to frequencies that are powers of 2 apart. There is a lot of interesting physics to this “harmonic motion”, which I am going to skip over, for now. I mention the physical significance of octaves to point out that, while octaves are determined by physics, the differences in frequencies between notes within an octave is something that is set by the musicians and the culture the musicians are playing in.

12 Tones:

In much of Western music, we split the octave up into 12 tones: A, B-flat, B, C, D-flat, D, E-flat, E, F, G-flat, G, and A-flat. Vi Hart covers variations on Twelve Tones in a brilliant video. It doesn’t have to be 12 tones. There are other ways in other systems of music, using other numbers to break up the octave.

As far as I know, all musical systems break the octave (doubling of pitch) into intervals. If you know of a system that uses a fundamental interval different from an octave, please let me know. You can contact me at the Ask button at the bottom of the homepage.

There is a lot of really musically and physically and mathematically interesting and useful information about intervals between notes that I am going to skip over. Perhaps I will come back to explain more about it at a later time. Adam Neely does a good job of explaining some of it in the video I link to at the top of this page.

What is Equal Temperament?

We need some way of breaking up an octave into 12 notes, and we need some systematic way to do that so that musicians can understand, and play with, each other. We can break the interval into musically equal differences of frequency between notes, or into unequal-but-possibly-nicer-sounding differences of frequency between notes. The latter we call “Just Temperament” and the former we call “Equal Temperament.” The comparison between Just Temperament and Equal Temperament, and the history of the development of Equal Temperament is fascinating. Find it here, at Wikipedia, as an entry point into further study.

You want to break an octave (doubling of frequency) into musically even divisions of frequency. How do we do that?

We can’t make each note twice the frequency of the previous note, because then it would be an octave higher, and we would get only one note per octave, a “One Tone” scale. Pretty boring.

We can’t make each note one times the frequency of the previous note, because we would never get to an octave if all the notes are exactly the same.

So we need a number between 1 and 2 to multiply by the frequency of the first note to get the next note. Let’s call that number “p”.

Also, I’ll clean up the equations later, so you need to know what I mean by the symbol ^, which we use to mean “to the power of”.  2 x 2 = 4, and 2 x 2 x 2 = 8, and 5 x 5 x 5 = 125, and so on. 5 cubed = 5 x 5 x 5 = 5^3 = 125. 2^2 = 4, 2^3 = 8, and anything^0 = 1.

Let’s start with A = 440 Hz. In terms of frequency,

A = A x p^0 = A

B-flat = A x p^1 = A * p

B = A x p^2

C = A x p^3

D-flat = A x p^4

…and so on…

G = A x p^10

A-flat = A x p^11

A (an octave up) = A x p^12

So, p needs to be a number such that when we raise it to the power of 12, we get 2. That’s easy! It’s the 12th root of 2. (12th root of 2)^12 = 2, by definition. So that’s our interval multiplier. It’s a number between 1 and 2, specifically, it’s 1.05946309436. Google has a built in calculator, so you can get the 12th root of 2 here.

Let’s try this out. Here’s part of a table of the frequencies of notes from Michigan Tech. A4 refers to the A in the 4th octave on a piano, which covers 8 octaves. The changeover in the octaves happens at the C note, so the C above A4 is C5.

Frequencies of Notes

Frequencies of Notes

If we start at A 440 Hz, and we want to get C, we use:

C = A x p^2 = 440 Hz x (12th root of 2)^3 = 440 Hz x 1.189207115 = 523.251130601 Hz = C.

The Equation of Equal Temperament

The general form of the equation is

f = f0 x p^n = f0 x (12th root of 2) to the nth power

where n can range from 0 to 12, f0 is the frequency of the starting note, and f is the frequency of the nth note in that octave.

Let’s try it for some other note. B4 to G5.

B = 493.88 Hz

G = B x p^n = 493.88 Hz x (12th root of 2)^8 = 493.88 Hz x 1.58740105197 = 783.985631546 = G.

Since the multiplicative interval between each note and the next is the 12th root of 2, regardless of what note you start on, the musical scale is uniform. This lets you play any tune in any key.


As always, if you see something that is unclear or incorrect, please let me know. You can contact me at the Ask button at the bottom of the homepage.

(Aside: blue light is an octave higher in frequency than red light, which is why violet and purple (mix of red and blue) look similar.)

(Aside: axial motion in a tuning fork stem is caused by the axial displacement of the arms, even though you give the arms radial displacement to make them go. mechanical advantage is why you can get the tone out of the stem with very little damping.)

 

Skills

Posted on

2016-08-14

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