The Scarily-tempered Clavier

If you have Pythagoras and his entourage over for dinner, here are a couple of helpful tips: (a) don’t serve chicken, and (b) hide the piano.  The first is easy to explain: as Malvolio says, Pythagoras believed in reincarnation, and in particular that on’es grandmother might well currently be a chicken.  The second part is a little more complicated, but I’m pretty sure your guests would consider your harmless upright an engine pure evil.

The Pythagoreans actually loved music; they worshipped numbers, and music is where we can genuinely feel the beauty of numeric proportions, not just think about them.  Here is why (forgive me if this is old hat to you): If you pluck, say, a harp string, it will bounce back and forth, making wavelike shapes, but since the two ends of the string are held down, only certain sizes of wave will work, namely those that are just the right size so that a whole number of wave-lengths will take up exactly the length of the string.  The first few of these have simple mathematical relationships, and they also happen to be notes that sound good together.

Say we have a string where the wave that fits exactly once onto the string makes the note C.  The wave that is half as long will also live happily on the string: it will fit twice, and will sound a note exactly one octave above, also being C.  The wave that fits 3 times will make the note G; the jump from C to G is called a perfect fifth, and it sounds sweet to most people.  The wave that fits four times will be another, higher C, and the jump from G to this C is called a perfect fourth.  It also sounds harmonious.  (It is true that not everybody finds the same harmonies pleasing. There are people in the Balkans who think that having two singers a minor second apart is the coolest thing ever…but I’m sure they would still recognize the difference between consonance and dissonance, they just attach different values to them.)

It is pleasing that, even if you don’t know or even believe in the math, you can hear the difference between intervals that have a simple ratio (2:1, 3:2, 4:3) and some random interval.  This is super as long as you just want to play, say, nice intervals starting from C, but there’s trouble when you want to fill in the rest of the notes, and especially if you want to play in other keys.  You might think that you could just keep going up by fifths (and down an octave when necessary) until you hit all the notes and get back to C, but if you’re using perfect intervals, it won’t work; mathematically, you can keep multiplying by 3/2 forever and you will never get a whole number, much less a power of 2.

Musically, your perfect fifths will cause other intervals to be wrong; for example, by the time you get to F, instead of a perfect fourth above C you’ll be about halfway to F-sharp.  You can lower the F to make it work with C, but now anything you play in B-flat will sound like shit.  There’s no way to make all the intervals work perfectly.

One solution was to use a different instrument for playing in different keys, which works OK with penny-whistles but not great with Church organs or even harpsichords.  For those, the most common approach was to choose your favorite keys and intervals and make those either perfect or nearly so, and then just avoid the others.  Anyone learning to compose for the keyboard used to have to learn which chords would sound good and which had to be avoided unless you wanted the audience to throw things.

Not everybody was happy with this, and some instrument-makers adopted the approach of adding extra kludge keys: you might have one G that works as the dominant of C, and also tuck in an emergency stunt G to serve as the fourth above D.  You can imagine what most musicians thought of that.  Anyway, people got very worked up about these different systems of intervals, or “temperaments,”  and a stray comment on the virtues of Just Temperament was guaranteed to start a fight in any 17th-Century bar.

The system that finally won out was calculated to piss everybody off a little.  All 12 half-tones are exactly the same distance apart, so you can compose equally well in any key, and there are no chords that will accidentally produce that awful wow-wow-wow resonance that comes from a too-short interval.  On the other hand, none of the intervals are perfect—to anold-timey ear trained on perfect harmonies, everything on a modern keyboard would sound slightly out of tune.  The pure ratios of 3:2 and 4:3 are approximated.

But what would really freak out a Pythagorean is that the proportions between notes do not correspond to any ratio of whole numbers.  I wasn’t exaggerating when I said that they worshipped numbers, by which they meant whole numbers and the ratios of whole numbers.  They would not have accepted the intervals on a piano, which are various roots of 2, and they would have been especially horrified by the interval between C and F-sharp, a ratio that to them was a vertiginous mystery, about whose existence the initiates were sworn to secrecy.

They just couldn’t handle the fact that the square root of 2 is irrational—there are legends that this revelation unhinged some Pythagoreans and drove them to suicide (it must be admitted that some of them were maybe not completely hinged to begin with).  It was all the more scary that this hole in the beautiful universe was to be found right there in their favorite theorem: if you have a right triangle with sides of length 1, then the hypotenuse, by the Pythagorean Theorm, has length the square root of 2.    Harsh.

So like I said, hide the piano.


PS  it’s not hard to prove that squt(2) is irrational, and it is a good glimpse into the way that mathematicians think, but I’ll leave that for another time.

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