ou may or may not be aware that the system of tuning used in virtually all the music you've ever heard is a fudge. It's a pretty good fudge, made for good reasons, but it's not right. The trouble is that none of the smaller intervals fit exactly into an octave. All of the notes we use are out by a few hundredths of a semitone. The worst note is out by 11.73 cents compared to its harmonic resonance. I can easily hear when something is out of tune by about 5 cents, so it is a big deal. Almost all the music that has ever been recorded is out of tune.
These different tuning systems are called temperaments. We commonly use equal temperament, in which octaves are divided into 12 equally sized semitones. Other temperaments that have been used historically include meantone temperament and well temperament. What I'm interested in exploring are different varieties of just temperament, or "Pythagorean" tuning.
If you play a pure tone, it naturally resonates at an a number of different frequencies, with a number of overtones. (Actually the number of overtones is infinite, but as you get further away from the main tone they get quieter until they're insignificant.) There's no such thing as a tone without overtones; it's impossible, that's just the nature of waves.
The main frequency you hear is the note itself. The most prominent overtone is the octave of the main note. Equal temperament has been defined so as to preserve this ratio so the octave is the only interval which is not out of tune. The next overtone is the perfect fifth. It turns out that this is 2 cents different to 7/12 of an octave (the fifth is seven semitones above the root). 2 cents isn't a problem to my ears, but already we've compromised. Then we get a perfect fourth, which, again, is 2 cents out but flat instead of sharp. A fourth and a fifth together make an octave so we're now two octaves above the root note.
This is where medieval organum music stops. For the medieval church, anything beyond this point was dissonance. If you want to make something sound hauntingly ancient, stick to these three intervals. It's not a bad plan in terms of tuning either because you don't have to make any significant compromises.
Then comes the major third, 8 cents out of tune. 8 cents! That's easily perceptible to the human ear. Each of the intervals comes in turn, defined by the mathematics of waves. (Two curiosities are worth commenting on: the minor third is actually more in tune than the major third, which is one reason minor keys sound lusher than major keys. Secondly, the 11th harmonic, often fudged as an augmented fifth, is 49 cents out of tune. That's a quarter tone out. This is a note which does not exist in western music yet has a closer harmonic relationship to the fundamental than many notes which western music does use. It's a strong argument for Arabic style 24-tone systems.)
The trouble comes when you try to make an instrument that can play in any key. An instrument which only plays in, say, D, can be tuned in just temperament without any problems. But if you then try to play it in Bb or many other keys, it won't work. Each of the notes is tuned slightly differently until you get to Bb. If you go up from D to Bb, you get the diminished fifth, 12 cents sharper than in equal temperament. But if you go down from D to Bb, you get the augmented fourth, 12 cents flatter than in equal temperment. This makes a difference of 23 cents between the two notes. In traditional Pythagorean tuning, they used the mean of the two values, the interval was called "diabolus in musica" and it was forbidden to use it. We can do better than that by employing an extra note, and playing one or the other according to context.
With computer equipment, it should be trivial to change between tunings depending on the key or the chord being played. With electronic music, there is no need to maintain the ugly compromise of equal temperament any longer. Down with ET! We want music that's in tune. Maybe I'll make a T-Shirt promoting Just Temperament.
(part 2 to follow)
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