BETWEEN U S: A HyperHistory of American Microtonalists

You may have never been aware of it, but you’ve been listening to music in five-limit tuning all your life – or rather, an equal-tempered approximation of it. European music is based on the desire to get two intervals in tune: the perfect fifth (3/2) and the major third (5/4), as well as the minor third (6/5) between those two intervals. The basic problem of tuning is that any two prime-numbered intervals are incommensurate. That is, there is no number of major thirds that will equal any other number of perfect fifths. Major thirds are 386.3 cents wide, perfect fifths are 701.955 cents wide, and there just aren’t any reasonably small numbers those will both divide into evenly. [Another way to say it: Major thirds are based on 5, perfect fifths on 3, and no power of 5 (5, 25, 125, 625...) will ever equal a power of 3 (3, 9, 27, 81, 243...).] Therefore we have to make decisions about which pitches to have fifths on and which to have thirds on.

Let’s start by taking four pitches and tuning them to perfect fifths: F, C, G, and D, with C as our tonic, defined as 1/1.

F 4/3 = 498 cents
C 1/1 = 0 cents
G 3/2 = 702 cents
D 9/8 = 204 cents

If we continue tuning the circle of fifths to these perfect 3/2 fifths, we’ll end up with a 3-limit tuning known as Pythagorean, because it is limited to the intervals Pythagoras is alleged to have discovered. That gives us rather harsh major thirds of 81/64 (408 cents), though, and right now we’re looking for pure major thirds of 5/4 (386.3 cents). So let’s build a pure major third above and below each of our four established pitches. This gives us a pretty evenly-spaced 12-pitch scale:

C 1/1 = 0 cents
Db 16/15= 112 cents
D 9/8 = 204 cents
Eb 6/5= 316 cents
E 5/4 = 386 cents
F 4/3 = 498 cents
F# 45/32 = 590 cents
G 3/2 = 702 cents
Ab 8/5 = 814 cents
A 5/3 = 884 cents
Bb 9/5 = 1018 cents
B 15/8 = 1088 cents

This is a fine, perfectly in-tune scale with 12 pitches. Its only drawback is that it is only in tune for the key of C. For example, if you want to play a D chord, the interval between D 9/8 and A 5/3 isn’t 3/2, as a perfect fifth should be, but 40/27 (5/3 divided by 9/8 = 5/3 x 8/9 = 40/27). And at 680 cents instead of 702, that “wolf fifth” between D and A is going to howl. You can retune A to be in tune with D, but then your F chord is no longer in tune.

There is a fascinating recording of a piano work in five limit tuning: Terry Riley‘s The Harp of New Albion (Celestial Harmonies 14018). The work employs a piano tuned to the above scale on C#, except that the G is 64/45 instead of 45/32. And the movements of the piece run through several keys, including D and Bb, so that you get a powerful sense of what happens when you intentionally modulate within a limited five-limit system.

This is the basic problem with tuning Western music, with its need for fifths and thirds and its limitation of only 12 pitches. If we allowed ourselves more than 12 pitches per octave, the problem would have many easier solutions. In fact, there were experiments in 16th century Italy with constructing harpsichords with octaves of 19 and 31 pitches to the octave just to avoid this dilemma. The academics of their day, as academics always will, prevented such innovations from catching on. But all of our historical European tunings, from meantone to well temperament and even our present accursed equal temperament, are approximations of this five limit tuning. So you can explore those historical European solutions, or you can go on to the composers who, after being stalled for 330 years, finally plunged ahead into seven-limit tuning.

From BETWEEN U S: A HyperHistory of American Microtonalists
by Kyle Gann
© 2001 NewMusicBox