March 30, 2007
Understanding the Western Tempered Scale (Part 12 of 15)

Take a moment to consider the frequency ratios of the various intervals in the tempered "Do-re-mi..." scale. m is our good old friend 2^{1/12} = 1.059463

root	1.0000	=	m^0
2nd	1.1225	=	m^2
3rd	1.2599	=	m^4
4th	1.3348	=	m^5
5th	1.4983	=	m^7
6th	1.6818	=	m^9
7th	1.8877	=	m^11

We have already commented on the fact that the 5th, 7 half steps above the root, has a frequency ratio very close to 1.5000 (i.e., 3/2). Now, it turns out the 4th therefore also has a number very close to 1.3333 (i.e., the ratio of the frequencies of two other harmonics of a vibrating string, or 4/3). But we could predict this, as a 4th up is the same thing as a 5th down (i.e., 1/ 1.4983 = 0.6674), and then shifted up an octave ( 0.6674 * 2 = 1.3348).

SO, if the 5th is strong in some way, the 4th will also be strong in a similar way. (This is sort of the basis of the fundamental chord progression of western music, the I IV V progresison.) Also note the 3rd has a frequency ratio which is also very close to the ratio of frequencies of harmonics of a string, 5/4. Even the 2nd is close to 9/8.

What does all this mean? It means that our ears are made to sense the order of certain types of vibrating things, like strings and tubes of air, and that subdividing the octave into 12 equally tempered intervals (not all of which are used in any given scale) comes remarkably close to making several of the notes exactly as they should be based on the physics of those strings or tubes of air.

In my naive statement of faith, I said that it is as if God has given us the notes, and it is up to us to make something beautiful in response to the wonder of these notes God has given us. Upon deeper consideration (and some historical reading), it rather appears that humankind has given us the notes of our scale, but that it just so happens to work out wonderously close to what our ears were somehow evolved (or created) to hear.

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