Complex Harmonies
The study of music offers a very good perspective on the struggle between simplicity and complexity.
We humans find certain combinations of sounds pleasing, or harmonious. We often find music beautiful. Beauty is subjective, so harmony should be subjective too. Or is it?
The first theory of harmony is due to Pythagoras, who discovered that the juxtapositions of sounds that sound harmonious follow some simple principles. For instance, if you strike two strings, one of which is twice as short as the other, they will produce a very harmonious perfect octave. The same experiment with the ratio of 2/3 produces another very harmonious interval--a perfect fifth. Those intervals are called perfect, because they sound perfect.
Pythagoras found out that if the ratios of the lengths of strings are equal to simple fractions, they sound harmonious. The simpler the fraction (smaller numerator and denominator) the more perfect the interval. The simplest fraction, 1/2, produces an octave, which sounds almost like the unison (which is 1/1). Next comes 1/3 (an octave + a fifth) and 2/3 (a fifth). 1/4 corresponds to two octaves, and then we have fourths, thirds, etc., which are still harmonious, but progressively less so. So, according to Pythagoras, harmony is not subjective--it has its roots in mathematics.
Much later it was discovered that harmony has something to do with frequencies, which are parameters in the Fourier transform of a sounds. Shorten the string by half and it will double the frequency of the sound it produces. The ratios of frequencies (which are more commonly used) are the inverse of the ratios of string lengths.
Why is Fourier transform so important? First, because this is what our ear does with all sounds before sending them to the brain. But there is also a physical aspect to Fourier transform--sine waves, which are the basis of Fourier analysis, are closely tied to harmonic oscillators. And harmonic oscillators are some of the simplest physical systems. So harmony has its roots in physics, too.
Well, to be fair, there are no harmonic oscillators in nature; just like there are no straight lines or circles. A harmonic oscillator is a Platonic idea, an abstraction. And abstractions are subjective--they are products of our brains. Se we are back to the starting point--harmony is subjective, but at a higher plane.
But that's not the end of the story. Since harmony is based on mathematical considerations, it should be possible to codify it--derive it from first principles or axioms. Using simple intervals, one can build a simple scale--the Pythagorean pentatonic. It has the following intervals 1/1, 9/8, 5/4, 3/2, 5/3 (it can also be built by stacking up perfect fifths modulo an octave). It loosely corresponds to the sequence C, D, E, G, A.
The pentatonic scale is used all over the world. It has one shortcoming though--it has large gaps. To fix this problem two more steps were added, corresponding to F and B in Western musical notation. This was an arbitrary decision, since one could as well have added F-flat and B-flat. In fact, the blues scale plays more freely with those steps, not to mention the minor scale.
This was the origin of the diatonic scale, that we all know and love. In its basic form it consists of the steps we call C, D, E, F, G, A, B (and back to C)--the white keys on the piano. You may notice that the Western notation sort of follows the alphabet, but not really. That's because the choice of the starting note (the tonic) is to some extent arbitrary. You can, for instance, start the scale with A (A, B, C, D, E, F, G) and obtain a variation which is now called the minor scale (or Aeolian mode). In the middle ages, the accepted canon contained all seven transpositions of those steps (these are now called church modes).
For people with perfect pitch, all this makes perfect sense. As for the rest of us--we don't really care what sound starts the particular mode. All we care about is the sequence of intervals. So if you take all those church modes and transpose them in such a way that they all start on the same note, you sort of get 11 possible notes. They fall in the vicinity of our chromatic scale (C, C#, D, D#, E, F, G, G#, A, A#, B). I say "sort of" because things are not as perfect in music as one would hope.
The problem is that "transpositional invariance" (my term) is incompatible with perfect intervals. Transpositional invariance in music is something that most of us find quite natural. If somebody plays a melody starting at C, and, some time later, plays the same melody a semitone higher (starting at C#), most of us won't notice the difference. It will be the same melody--all the intervals will be the same. Only people with perfect pitch may notice the difference.
So if you play all the church modes starting on the same note, but preserving the original intervals, you'll get all kinds of small discrepancies. In essence, intervals are defined using rational numbers, whereas transpositions require real numbers (in particular roots of rational numbers). These two don't jive.
The equivalent of the Goedel's incompleteness theorem in music is the existence of the Pythagorean comma. If you take a leap of a fifth 12 times in a row (for instance, starting with C, go to G, then to d, a, etc...), you'll reach a tone that is almost exactly the same as if you made 7 leaps of an octave. The tiny difference is called the Pythagorean comma.
To have a cake and eat it too, a new scheme was adopted, called equal temperament, in which all intervals, except the octave, became irrational. That's how modern pianos are tuned. When you play a fifth on the piano, for instance by striking C and G, you won't hear a perfect fifth. In fact most people couldn't tell a perfect fifth from a tempered fifth--the difference is so small. On the other hand, because of equal temperament, the interval between C and G is exactly the same as between D and A, and so on. Also twelve tempered fifth do add up to seven octaves.
A lot of people are heartbroken when they discover that music is not perfect. Others enjoy the richness of complexity brought by little imperfections. J. S. Bach was so enthusiastic that he composed 24 musical pieces in 12 major and 12 minor scales under the common title "The Well-Tempered Clavier." Equal temperament allowed clean transitions (modulations) from any key to any other key--something we now take for granted.
We humans find certain combinations of sounds pleasing, or harmonious. We often find music beautiful. Beauty is subjective, so harmony should be subjective too. Or is it?
The first theory of harmony is due to Pythagoras, who discovered that the juxtapositions of sounds that sound harmonious follow some simple principles. For instance, if you strike two strings, one of which is twice as short as the other, they will produce a very harmonious perfect octave. The same experiment with the ratio of 2/3 produces another very harmonious interval--a perfect fifth. Those intervals are called perfect, because they sound perfect.
Pythagoras found out that if the ratios of the lengths of strings are equal to simple fractions, they sound harmonious. The simpler the fraction (smaller numerator and denominator) the more perfect the interval. The simplest fraction, 1/2, produces an octave, which sounds almost like the unison (which is 1/1). Next comes 1/3 (an octave + a fifth) and 2/3 (a fifth). 1/4 corresponds to two octaves, and then we have fourths, thirds, etc., which are still harmonious, but progressively less so. So, according to Pythagoras, harmony is not subjective--it has its roots in mathematics.
Much later it was discovered that harmony has something to do with frequencies, which are parameters in the Fourier transform of a sounds. Shorten the string by half and it will double the frequency of the sound it produces. The ratios of frequencies (which are more commonly used) are the inverse of the ratios of string lengths.
Why is Fourier transform so important? First, because this is what our ear does with all sounds before sending them to the brain. But there is also a physical aspect to Fourier transform--sine waves, which are the basis of Fourier analysis, are closely tied to harmonic oscillators. And harmonic oscillators are some of the simplest physical systems. So harmony has its roots in physics, too.
Well, to be fair, there are no harmonic oscillators in nature; just like there are no straight lines or circles. A harmonic oscillator is a Platonic idea, an abstraction. And abstractions are subjective--they are products of our brains. Se we are back to the starting point--harmony is subjective, but at a higher plane.
But that's not the end of the story. Since harmony is based on mathematical considerations, it should be possible to codify it--derive it from first principles or axioms. Using simple intervals, one can build a simple scale--the Pythagorean pentatonic. It has the following intervals 1/1, 9/8, 5/4, 3/2, 5/3 (it can also be built by stacking up perfect fifths modulo an octave). It loosely corresponds to the sequence C, D, E, G, A.
The pentatonic scale is used all over the world. It has one shortcoming though--it has large gaps. To fix this problem two more steps were added, corresponding to F and B in Western musical notation. This was an arbitrary decision, since one could as well have added F-flat and B-flat. In fact, the blues scale plays more freely with those steps, not to mention the minor scale.
This was the origin of the diatonic scale, that we all know and love. In its basic form it consists of the steps we call C, D, E, F, G, A, B (and back to C)--the white keys on the piano. You may notice that the Western notation sort of follows the alphabet, but not really. That's because the choice of the starting note (the tonic) is to some extent arbitrary. You can, for instance, start the scale with A (A, B, C, D, E, F, G) and obtain a variation which is now called the minor scale (or Aeolian mode). In the middle ages, the accepted canon contained all seven transpositions of those steps (these are now called church modes).
For people with perfect pitch, all this makes perfect sense. As for the rest of us--we don't really care what sound starts the particular mode. All we care about is the sequence of intervals. So if you take all those church modes and transpose them in such a way that they all start on the same note, you sort of get 11 possible notes. They fall in the vicinity of our chromatic scale (C, C#, D, D#, E, F, G, G#, A, A#, B). I say "sort of" because things are not as perfect in music as one would hope.
The problem is that "transpositional invariance" (my term) is incompatible with perfect intervals. Transpositional invariance in music is something that most of us find quite natural. If somebody plays a melody starting at C, and, some time later, plays the same melody a semitone higher (starting at C#), most of us won't notice the difference. It will be the same melody--all the intervals will be the same. Only people with perfect pitch may notice the difference.
So if you play all the church modes starting on the same note, but preserving the original intervals, you'll get all kinds of small discrepancies. In essence, intervals are defined using rational numbers, whereas transpositions require real numbers (in particular roots of rational numbers). These two don't jive.
The equivalent of the Goedel's incompleteness theorem in music is the existence of the Pythagorean comma. If you take a leap of a fifth 12 times in a row (for instance, starting with C, go to G, then to d, a, etc...), you'll reach a tone that is almost exactly the same as if you made 7 leaps of an octave. The tiny difference is called the Pythagorean comma.
To have a cake and eat it too, a new scheme was adopted, called equal temperament, in which all intervals, except the octave, became irrational. That's how modern pianos are tuned. When you play a fifth on the piano, for instance by striking C and G, you won't hear a perfect fifth. In fact most people couldn't tell a perfect fifth from a tempered fifth--the difference is so small. On the other hand, because of equal temperament, the interval between C and G is exactly the same as between D and A, and so on. Also twelve tempered fifth do add up to seven octaves.
A lot of people are heartbroken when they discover that music is not perfect. Others enjoy the richness of complexity brought by little imperfections. J. S. Bach was so enthusiastic that he composed 24 musical pieces in 12 major and 12 minor scales under the common title "The Well-Tempered Clavier." Equal temperament allowed clean transitions (modulations) from any key to any other key--something we now take for granted.