Tempered, Pythagorean, and Just Tuning


There are a number of ways to tune the intervals which make up the chromatic scale (the scale including all sharps and flats). Modern keyboard instruments almost always use the "tempered" tuning. This page, and the mp3 music files it contains, will show why. The goal is, given the frequency of the note "c", to decide the frequencies of all of the notes, namely, We want to do this in a way such that the intervals one actually plays, in music, will be close to the ideally consonant intervals; specifically, when notes separated by 7 half-steps are played we want the ratio of frequencies to be 3/2; notes separated by 5 half-steps should have frequencies in ratio 4/3; 4 half steps should be 5/4, and 3 half-steps should be 6/5. The latter are somewhat less musically important, and much harder to achieve.
The "traditional" way to tune, called the Pythagorean scale, is to

1) Tune one note, say, "c", to some value;

2) Tune the octaves to be precisely a factor of 2 in pitch, and

3) Tune the other notes by the rule that "5'ths are perfect," meaning that two notes separated by 7 half-steps are made to have a frequency ratio of exactly 3/2.

The great advantage of this technique is that it is "easy to do," since the octaves and fifths can be achieved by avoiding "beats". The problem is that, somewhere, there is a 5'th which is mis-tuned; because 12 factors of (3/2) is not the same as 7 factors of 2 (though it is close).
The frequencies in the Pythagorean scale are the frequency of c times,

Note that, for the "black Piano keys," I had to choose whether to write them as sharp or flat. When I write one as sharp, it means I determine the frequency by going up a fifth from the note below (F sharp is determined from B); the flats are determined by coming down a fifth from the note above (B flat is determined from F). The answers would be different if we approached from the opposite side.
What does the Pythagorean scale sound like? In some keys it sounds fine; but in other keys, one of the 5'ths is imperfect, and it sounds out of tune. Here are two tunes: Frere Jaque and a bit from Samuel Barber's Adajio for Strings, played in each key--but using the fixed intonation determined above. Judge for yourself.
An attempt to improve on is scale is the Just scale. The idea is to force all thirds to be "perfect" as well, that is, the major thirds should have the notes in the ratio 5/4 and the minor thirds should have the ratio 6/5. But whereas we could make all but one 5'th be ideal, it is only possible to make 8 of the 12 major 3'rds ideal. In many keys, there will be musically important 3'rds and 5'ths which are quite far off. Therefore, this tuning sounds better in the intended scale, but even worse when we play in some unusual key.

The frequencies, with respect to the frequency of c, are

Now the music: Here are the same tunes played in each key--but using the fixed intonation determined above. I think you will agree that, while the "tonic" C sounds fine, some of the more unusual keys are almost painfully out of tune.
The tempered scale is a compromise. All half-steps are made of equal size (that is, equal logarithmic range of frequency). Therefore, every third and every fifth is off from the ideal relation, but the amount they are off is the same in every key and does not get bad for keys with lots of sharps and flats. The ratio of half-step separated notes is the twelfth root of 2, which is 1.059463.

Therefore, the relation of the note frequencies to C are

Music in this key sounds fine in every key; again we demonstrate using Frere Jaque. Part of our preference for the tempered scale is that we are used to it. Our ears expect the amount that it misses certain musical intervals. Nevertheless, it is a real advantage that it does not sound bad in unusual keys.

Note: The music files here are completely synthesized. If you want to know how I made them, e-mail me at (guymoore at physics.mcgill.ca)