Now that we have a basic understanding of how sound acts in air, let’s talk about how these physics relate to music.
Earlier we mentioned in passing a couple of cases in which the relationships between the vibrations in the air, the relationships of our “innertones”, coincide, perfectly or nearly so, with modern Western 12-tone equal temperament.
Here’s a neat thing about 12-tone equal temperament. Play any tone. From there, you will always have other tones within 12-tone equal temperament (“12-tET”) which vibrate at twice the rate (2:1), three times the rate (3:1), and tones which coincide with the vibration relationships we find in our “innertones” of 3:2 4:3, and the octaves (half, twice, four times, one eighth, etc. the rates) of these such as 8:3, 2:3, etc.
Now, while the 2:1 (and 1:2, 4:1, etc.) relationships are exact in 12-tET, right in tune with the vibrations in the air, the 3:2 and 4:3 (the perfect fifth and perfect fourth) and their octaves (a fourth an octave higher, 8:3, for example) are just a wee hair off. This is called “tempering”, and it’s why we call it 12-tone equal temperament.
It’s a very clever bit of math, first documented by a Chinese prince half a century before appearing in Europe. It’s clever because if you try to make the vibrations of twelve tones all match the vibrations of 3:2 and 4:3, the pure fifths and fourths, happening in the air, you get eleven pure fifths (and corresponding pure fourths) and one real stinker, the famous “wolf”.
This wolf fifth is “lower” (slower, larger) than a fifth vibrating at a rate of 3:2. (The corresponding fourth is higher/faster/smaller).
So what, you might say. Well, think back to our vibrations within vibrations. When we play a single tone on our cello, or voice, or horn, the vibrations in the air all fit nicely within each other. Let’s say we have another instrument playing along, and the fundamental of the tone it is playing a perfect fifth “higher” than, that is, at a relative rate of 3:2 to, our tone of 100 Hz. The vibrations created by this sound will also fit nicely within those of our first sound. The fundamental of this sound is 150 Hz of course, it’s second harmonic at 300 hz, then 450 Hz, etc.
tone 1 tone 2 (in vibrations per second)
…and so on.
You can see how these vibrations, these, all match up or nestle inside each other harmoniously.
If we tuned that second instrument to that “wolf fifth”, however, it would vibrate at 147.9810552817716 times a second. You see that the harmonics of this tone will *not* fit in. If you imagine the vibrations in the air, you can “see” that the vibrations are moving at almost, but not quite the same rates. What this does is create new vibrations.
The second harmonic vibration of our wolf fifth tone is going to be conflicting with the third vibration of our original tone. It will be vibrating at 295.9621105635432 times a second, while the third harmonic of our original tone at 100 Hz. will be vibrating at 300 Hz.
These vibrations clash in the air and create what are called “beats”. In this case, the difference between the frequencies at that first point of clashing (second harmonic of “higher” tone with third harmonic of “lower” tone) is 4 vibrations per second, 4 Hz.
If you snap your fingers slowly and steadily, once a second, and sing evenly “january, february…”, that’s about exactly the rate of the beating. What it sounds like with musical instruments is a kind of throbbing, and with the other clashes within all the other “innertones” vibrating in the air (at the sixth harmonic of the first tone, for example), the effect taken all together is kind of a “howling” sound. And that is why it is called a wolf fifth.
But as I said earlier, the fifth of 12-tET is tempered and therefore a little “off”. Why doesn’t it howl as well? Well, if we played the 12-tET fifth along with our 100 Hz tone, we’d get our first and usually loudest clash between about 299.66 Hz and 300 Hz. The beat rate is only about 1/3 of a Hertz. That means it would take a full 3 seconds in this case (at these pitches)to complete a full beat, and the difference, the throbbing sound, is going to be almost completely unnoticed except in long smoothly sustained tones.
So here is where we find the acoustic connections between the 12-tET tuning system and the vibrations in the air: the vibrations that happen between all the second and third partials, and their simple multiples, are all very well in tune. Not perfect, but perfect enough in almost all cases, especially when you take into consideration that musicians with feeling and skill continually make little adjustments whenever possible- some vibrato, some more pressure on the guitar string playing the fifth (which raises the pitch to a pure 3:2) in a “power chord”, and so on.
For those of you into numerology and mysticism, you can see that the 12-tET system, with just a touch of compromise, fulfills the ancient Pythagorean ideal of everything, including music, being described by the numbers 1, 2 and 3.
But what about all the other tones, not just fourths, fifths and octaves, and all those resulting vibrations in the air?
Well, the price of ingeniously tweaking everything so that the vibrations between the first few (four, really) harmonic vibrations of musical tones
are all vibrating very well together is that the rest of the vibrations clash a lot.
If our two instruments were to play a 12-tET major third in harmony together, for example, the clash between the fifth harmonic component of the “lower” tone and the fourth harmonic component of the “higher” tone is very strong. This is the main reason you’ll hear some musicians complain that 12-tET is so “out of tune”. It is discordant if you value the “pure major third” which vibrates at the rate of 5:4. For several centuries in Western music, until about the time of Beethoven, the pure major third was the star of the show, so much so that people were willing to temper the fifth quite a bit and tolerate, not a wolf, but a rather wobbly fifth, in order to keep the 5:4 major thirds as pure as possible.
Surely there’s a way to keep your fourths, fifths and major thirds all pretty much perfectly in tune with the vibrations in the air, all at the same time, someone protests.
Indeed there is, and it’s been known for centuries, advocated by people from Isaac Newton to Schoenberg, and still in use to this day in the Turkish koma system. Instead of 12 perfect fifths, all you need is 53 pure fifths!
And here is where things get complicated, and paths diverge, and a myriad of approaches arise. We’ll talk about the main kinds of approaches in the next blog entry.