By Eric Garneau
The Final Movement
Here we are at the series finale of lecturer John Crooks' 'Introduction to Pitch Systems in Tonal Music' - or maybe it's just the season finale? In true television fashion, all the mysteries from the past ten weeks of video lectures are now revealed for our learning pleasure. Key among them: just how can we come up with a reasonable system of tuning our instruments? We know a bunch of ways that don't work, so how about finding one that does?
As it happens, our answer's found in a system known as equal temperament. It's based totally on mathematics that disregards a lot of the basics of harmony we've learned over the past few weeks (such as fifths and triads) and aims only to keep our octaves in perfect harmony. That makes sense, because as we discussed last week (and many weeks before that) octaves are really the key to organizing our system of pitches, thanks to a phenomenon known as octave equivalence.
Never Mind the Triads, Here's the Octaves
Once we settle on only really caring whether our octaves are in-tune, calculating the proper frequencies for the other notes in our 12-tone set becomes pretty simple - we just need to have a system whereby every 13th value perfectly doubles our initial pitch. By using the 12th root of two, or 1.059463, we can obtain something of a magic number that will produce the desired results. Once we've got that, we can multiply it by whatever our starting frequency is to get our pitch a half step up, then again for the next half-step, ad naseum to construct a scale with pure octaves.
Of course, as we already discussed that does leave our fifths and triads (which Crooks calls 'a central aspect of tonal harmony') slightly out of tune. But Crooks points out that somehow our ears compensate for this minor discord - we hear a triad in equal temperament and we equate it with a pure triad, at least until the two are stacked side-by-side. As Crooks notes, that says some interesting things about our cognition - that we're willing to ignore minor disharmonies in order to find an overall tuning system that works. Go us.
Crooks also briefly addresses some other topics that might prove interesting to us, including just why it is that we value a 12-pitch scale. The reason: we need it in order to have a completely modular pitch system, meaning we can play a composition in any key and keep its intervals intact. We also learn a bit about the historical context of tuning development; it turns out that equal temperament has really only been in vogue for the last hundred or so years, and that formerly musicians tended to favor a Pythagorean system of tuning, which made key modulation more difficult.
Perhaps all of this still sounds like trivia - interesting, but impractical. Certainly there were lessons in this course that made me think so, but now that we've reached the end, I'm not so sure. Probably the most key lesson we can take away here is this: tuning can be, as Crooks states, 'anything we want it to be.' There's nothing more primal about equal temperament tuning than any other system, it's just what we enjoy right now. The more enterprising musicians among us are free to create whatever system we feel is most useful, and thanks to Crooks' lectures we have a grasp of basic mathematical and harmonic concepts necessary to navigate the compromises we'll need to do so. Lots of working musicians (like myself) probably won't feel the need to experiment with alternate tuning methods any time soon, but doesn't it make our control of the craft that much more potent if we can? I think so, and I think it's made Crooks' course quite worthwhile.
Apply the concepts you learned here to some great tunes by checking out these music blogs.