Stuart Isacoff, Temperament: How Music Became a Battlefield for the Great Minds of Western Civilization (2001, 2003). $13.95. [Amazon]
Go over to your piano and play a middle C. Now play the next higher C. That's an octave difference; the string for the lower C should be twice as long as the higher one. That's a 2:1 ratio. Another ratio is to play the C, and then play the higher G. That's a fifth, and the ratio of the lengths should be 3:2. Pythagoras -- or his school -- figured all that out in the 6th century B.C.E. But here's the bad news. Produce a series of octaves where you keep halving the length of the string, and see what it sounds like after you've done it seven times. Now do the same thing with the fifths. This time, you will need to do twelve times. Pythagoras believed that you would arrive at the same final tone (i.e., tone with the same name) at the end. But guess what? You don't:
The tones sounded by his two instruments were, however, almost the same, yet slightly -- disturbingly -- out of tune. The fact is, octaves and fifths, when created with Pythagoras's pure mathematical ratios, are incommensurate: The further they move away from a common starting point, the more the structures built from these "perfect" intervals diverge. (40) Thus ensues centuries of argument between those who seek to keep the standard octaves and fifths, and those who advocated something that came to be called "equal temperament," where the distance between the tones are fixed, and, apparently, you can't hear the slight dissonaces that are introduced. It's this equal temperament that your modern piano uses. In this great argument are major players such as Descartes, Mersenne, Vincenzo Galilei (Galileo's father), Rousseau, Rameau, and many others. They really go at it. I know these figures from the history of science and philosophy, and it was amazing to me to see how invested they were in music; this probably says something about how intellectual disciplines isolate their objects of study like so many individual bees -- you forget the variety of flowers they serviced. On the one hand, it would seem, are those who might be called the Pythagoreans, who believe that there is some mystical force in the "just" intervals. And there are the moderns, who advocate equal temperament.
The first half of this book is great, as you see the Pythagorean ideal crumble when faced with the prospect of constructing a playable keyboard-- the illustrations are outrageous (the key ones are woodcuts or engravings from Marin Mersenne's Harmonie Universalle [1636-1637]; try as I might, I couldn't find reproductions on the mighty Internet). The 2nd half gets a bit tedious, with many digressions, but still worthy if you can stick it out. There are nuggets all the way through, and having finished it, I can say that I'm just about dying to listen to a couple of things: Adrian Willaert's Quid non ebrietas (16th century), which was apparently designed to show the pain of just intervals, and then some new pieces by Michael Harrison, a sometime collaborator of La Monte Young, which strives to evoke the mysticism and physical sensation of the early systems.
I'll close this with a juicy quote from d'Alembert:
comments powered by Disqus
All freedoms are bound together and are equally dangerous. Freedom in music implies freedom to feel, freedom to feel implies freedom to think, freedom to think implies freedom to act. (223-224) Amen.