Thursday, April 25, 2013

Nihilism and optimism

n. the end, purpose, or significance of something
NB: Emphasis added.
Consider the definition given above by We're told the meaning of something is its purpose.  If a thing has a purpose, that purpose must have been intended by someone.  If we're told something has "inherent meaning," we must ask from whence this intent comes.  There doesn't seem to be an obvious answer to this question.

Monday, April 15, 2013

The tragedy of music: part three

In part one, we discussed some issues with Pythagorean tuning, and in part two we continued to quarter-comma meantone.  The logical conclusion is equal temperament, a widely used (though not quite universal) modern system.

Equal temperament is, in a way, simpler than any of the earlier systems.  Instead of building an octave out of some fixed ratio, we start with an octave and subdivide it.  The most natural way to do this is to make a semitone the twelfth root of two.  Note that this is simply "a semitone" rather than a chromatic or diatonic semitone; in equal temperament, those are the same thing.  Everything else can be built rather easily out of this fundamental unit, and we don't have overlapping or separated octaves.  This, in turn, means no wolf fifth.

Still, this fundamental unit can be unwieldy.  Writing out "the twelfth root of two" all the time is annoying, and the mathematical notation for it is rather ugly.  For this purpose, the so-called cent was invented.  There are 1200 cents in an octave; 100 make up a semitone.  It is a logarithmic unit.  Increasing a tone by 1200 cents means doubling its frequency.  700 cents make up a perfect fifth, and 400 cents are a major third.  The cent is a unit of relative measure; it is not meaningful to equate a single note to a given number of cents, except in relation to another note.  That "other note" is often middle A, which, for convenience, is typically tuned to 440 Hz.

This system is very practical, which accounts for its widespread use.  However, this is not a happy story with a happy ending; there is a problem.  The twelfth root of two is an ugly, irrational number, and all of the other intervals are powers of it.  There are no simple integer ratios at all.  While the cent may make the system look nice and clean, it is an artificial unit created to hide the irrational semitone.  In effect, we've taken the sourness of the wolf fifth and extended it over the whole octave, spreading thinly to mask the dissonance.  Practical this may be, but ideal it is not.

This leads me to a stark realization: the Platonist's ideal of "perfect yet unattainable" forms is inapplicable to music.  These problems are not of a physical nature.  Any musical system will suffer from them, except for the trivial system which has one note per octave.  There is a fundamental disconnect between equal temperament and just intonation.  We cannot have both, even in theory.  And that is the tragedy I've been talking about.

Monday, April 8, 2013

The tragedy of music, part two

In part one, we discussed Pythagorean tuning and its failings.  Despite this crucial issue, Pythagorean tuning was highly influential on musical theory.

In the 1500's, a variant of Pythagorean tuning called quarter-comma meantone became popular.  The "comma" in this name is not the Pythagorean comma we saw last time, but an entirely different comma.

A major third is an interval spanning three staff positions and four semitones.  It is not considered "perfect" in the same way as the perfect fifth, but is still regarded as a consonant interval, at least in theory.  Under Pythagorean tuning, the major third was a rather dissonant 81:64.  Quarter-comma meantone flattened this to a nicer 5:4, at the expense of a more dissonant perfect fifth.

Interestingly, we now have irrational intervals: the perfect fifth is the fourth root of 5.  This way, if we move up by four perfect fifths, and down by two octaves, we end up at 5:4, the justly-intoned major third.

This sort of trade-off is debatable, of course, but it was an explicit design goal of quarter-comma meantone; the flatter fifth was viewed as an acceptable price for a just third.

Now, suppose we start at middle C, as we did last time, and move up a major third.  We arrive at middle E at a ratio of 5:4.  Next we move up again to G♯, at a ratio of 25:16.  Finally, we get to C5, at a ratio of 125:64.  This is not an octave, but unlike in Pythagorean tuning, the interval is too flat rather than too sharp.  This means there's a gap between the octaves, unlike the overlapping octaves of Pythagorean tuning.

Like in Pythagorean tuning, one of the fifths must span this gap.  That fifth is sharpened by 128:125, a much larger interval than the Pythagorean comma.  It sounds extremely dissonant, to the point that it became known as the "wolf fifth" because it sounds like a wolf howling at the moon.  This moniker is sometimes also applied to the diminished sixth produced by Pythagorean tuning under the equivalent problem, but note that quarter-comma meantone is much worse in this regard.

Updated: In part three, we discuss modern equal temperament.

Saturday, April 6, 2013

The tragedy of music, part one

Once upon a time, people noticed that certain sounds are pleasant to hear, while others are not.  They began to experiment with it.  The most important early discovery in this vein was the notion that sound is vibration.  This is the fundamental principle of all musical theory, modern or ancient.  The next discovery was that of the octave: If you make two vibrations, one at double the frequency of the other, they sound... the same, in some way.  The faster one is clearly higher pitched, but they're harmonically equivalent.

The Ancient Greeks built upon this discovery, and eventually produced what we now call Pythagorean tuning.  Pythagorean tuning is built around simple frequency ratios; systems like this are called "just intonations."  The most important ratio other than the octave is the so-called "perfect fifth," which describes a gap of five staff positions in modern musical notation.  It spans a gap of seven semitones; there are a total of twelve semitones per octave in "conventional" systems such as Pythagorean tuning.  In Pythagorean tuning, a perfect fifth is a ratio of 3:2, meaning the faster vibration oscillates thrice for every two oscillations of the slower vibration.  So far, this is all very nice, and fits in well with the Pythagorean mathematical ideals of rationality.  But there's a problem.  I told you that a perfect fifth has a ratio of 3:2, is composed of seven semitones, twelve of which make up an octave, and the octave is 2:1.  Suppose we start at middle C (or C4), and move a perfect fifth up.  We arrive at G4, at 3:2 times our original frequency.  We continue to D5, at 9:4.  Next comes A6, at 27:8.  Then E6, at 81:16, B7, at 243:32, F♯7, at 729:64, C♯8, at 2187:128, and finally G♯8 at 6561:256.  Now suppose we go down from C4.  We find ourselves at F3 at 2:3, B♭3 at 4:9, E♭2 at 8:27, and A♭1 at 16:81.  The ratio from C4 to A♭1 is 81:16, and the ratio from G♯8 to C4 is 6561:256.  Multiplying, we see that the ratio from G♯8 to A♭1 is 531441:4096.  But A♭ and G♯ are supposed to be the same note.  Going by octaves, we should get 128:1.  The other ratio is rather large and unwieldy, because we have seven octaves of space, so if we divide those out, we get a ratio from G♯ to A♭ of precisely 531441:524288.  This interval is called the Pythagorean comma.  It's the difference (remember, in music, we never add or subtract frequencies, so this is actually the ratio) between a chromatic and a diatonic semitone, among other things.

So what does this actually mean?  It means Pythagorean octaves overlap, if only a little, since G♯ is a little sharper than A♭.  That's a problem for perfect fifths.  One of the twelve "perfect" fifths is ruined by spanning this overlap, being flattened to a rather dissonant diminished sixth.  So Pythagorean tuning, for all its beauty and simplicity, is not perfect.

Updated: In part two, we discuss quarter-comma meantone, a derivative of Pythagorean tuning.