Musical Pitch and Temperament

by R. Grothmann

This notebook tests pure versus equal tempered or approximated harmonics.

Create the time values for 4 seconds of sound.

>t=soundsec(4);

Let 440 Hz be our basic frequency.

>f=440;

New listen to a 220 Hz sound.

>s=sin(f*t/2);
>playwave(s);

The following is a pure fifth.

>s=sin(f*t)+sin(f*3/2*t);
>playwave(s);

Now we add the lower octave and vary the fifth in pitch. First the pure quint.

>s=sin(f*t)+sin(f*3/2*t)+sin(f*t/2);
>playwave(s);

Now the equal tempered fifth. The difference is small. But with sine curves the beat (oscillating amplitude) is audible.

>s=sin(f*t)+sin(f*2^(7/12)*t)+sin(f*t/2);
>playwave(s);

Now a fifth which is too much off.

>s=sin(f*t)+sin(f*1.49*t)+sin(f*t/2);
>playwave(s);

Here are the corresponding numbers

>3/2, 2^(7/12), 1.49

1.5
1.49830707688
1.49

Difference in tones is measured in cents, where 100 cents is a halftone, and the scale is logarithmic.

Here is the error in cents. First the tempered fifth.

>1200*logbase(2^(7/12)/(3/2),2)

-1.95500086539

The error is -2 cents. The error for the 1.49 approximation is already -11 cents. This is too much to be OK.

>1200*logbase(1.49/(3/2),2)

-11.5802040405

The difference in frequency explains the oscillation, since

Since the pure fifth has no beat, we can hear a beat with a following frequency of 4.4 Hz.

>f*(3/2-2^(7/12)), f*(3/2-1.49)

0.74488617426
4.4

Repeat the same with the major third.

>s=sin(f*t)+sin(f*5/4*t)+sin(f*t/4);
>playwave(s); // pure
>s=sin(f*t)+sin(f*2^(4/12)*t)+sin(f*t/4);
>playwave(s); // tempered
>s=sin(f*t)+sin(f*1.27*t)+sin(f*t/4);
>playwave(s); // way off
>5/4, 2^(4/12), 1.24

1.25
1.25992104989
1.24

Examples