I casually mentioned in part 2 that Wendy Carlos had invented some innovative tuning systems. I wasn't going to leave it at that, though. So this post explores α, β and γ in a little more detail.
What's wrong with ordinary tuning?
This is a musicological can of worms. I'll try to pick out some of the juicier specimens for those who aren't familiar with the subject. Basically, it's a compromise in which no interval but the octave is quite right.
Back in ancient Greece, Pythagoras (the triangle man) worked out that musical intervals were related to simple mathematical ratios, in particular 1:2 and 2:3. Take a taut string; if you halve its length, the pitch it makes when plucked goes up by an octave (for example, from C to the next C up). Its frequency doubles. If, instead, you reduce its length to ⅔, its pitch goes up by a fifth (from C to G). Reduce it to ¾ and its pitch goes up by a fourth (C to F). You can work out pitches for all 12 semitones in the octave by multiplying together 1:2 and 2:3 ratios, but the problem is that they don't quite line up. An octave is 12 semitones, and a fifth is seven semitones, but 12 fifths is significantly different to seven octaves. This difference is known as the Pythagorean Comma. For those who think the world should be described by beautiful mathematics, it's a big problem. Practically, it means that when you tune an instrument using Pythagoras's system, some combinations of notes sound terrible, and so you have to write melodies without using the third or seventh, for example.
These days, we mostly get around the Comma by using equal temperament. This means that the octave is divided into twelve equal intervals, the frequency ratio of one semitone to the next being the twelfth root of two (about 1.059). This system makes all of the intervals almost perfect, which allows you to use all of the notes in one tune without it sounding horribly dissonant, and change key at will. But only the octave itself is exactly a 1:2 ratio. All the other intervals are approximations.
There's no reason it has to be twelve, either. Some traditional music of Indonesia, Thailand and China uses equally distributed octave (EDO) tunings of five, seven or nine intervals (known as 5 EDO, 7 EDO, etc.). Modern composers have used finer-grained EDOs such as 19, 29, 31, 41, 53 and even 96.
But the one thing all EDO tunings, including our familiar equal temperament (AKA 12 EDO) have in common is the perfect octave. Few composers have dared to use tunings that ignore that most fundamental of musical intervals.
Almost all historical work on multiple divisions of the octave in tuning theory has focused on whole number integer divisions. That assures us that after the particular number of notes of a particular division is added together, we arrive at a note exactly one octave (frequency ratio 2:1) away from the pitch at which we started. The so-called equally tempered scale is one of these divisions, as are all the important tunings based on 19, 31, and 53 equal steps. This notion has been around for so long that it almost sounds impertinent to suggest there might be a useful alternative which has been systematically ignored.
— Wendy Carlos, Three Asymmetric Divisions of the Octave
Which brings us to Wendy Carlos. In the mid 1980s, she discovered that by dividing the perfect fifth, rather than the octave, three very interesting scales resulted. Alpha, Beta and Gamma divide the fifth (a 3:2 ratio, you may remember) into nine, eleven and twenty steps respectively. They have some beautifully pure harmonies, though they sound very exotic and can also produce some wild dissonances.
I suggest you go and read the whole of the short article quoted above, to get a flavour of Wendy's thinking on these tunings. But first, have a listen. These are my first quick tryouts, playing a scale and a few chords in each of α, β and γ.
Three new friends
To me, these sound fairly friendly; exotic, rather than totally alien. Perhaps it's a function of the particular chords I landed on for each, but α has a kind of circus feel: playful, but slightly creepy. β sounds colder, more serious and dignified. γ is excited, high-energy, almost hallucinatory.
I am going to have some fun playing with them.