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Mean tricks to play on your mother-in-law #27
2nd-Mar-2004 07:20 pm

Tune her piano to Kirnberger III (**).

Of course, this being the 2nd time in my life that I've attempted to tune a piano, I imagine getting it to sound even remotely correct will be something of a triumph. (Last time around, people were saying "Gee, you ought to get that tuned" after I was done). On the other hand, last time around was 20 years ago and back then I didn't have clue #1 about temperament issues (*).

Suffice it to say, having a sense of absolute pitch --- and yes, I know people call it "perfect" pitch, but it just isn't, okay? --- is almost completely useless for this sort of job, just in case you were wondering.

On the other hand, given the state that her piano is in now (hasn't been tuned for maybe 20 years), almost anything will be an improvement, and since she's mostly tone-deaf anyway, I could probably tune it completely randomly and she wouldn't notice a thing (though and I almost certainly would...).

Also, I have this theory that Kirnberger III, with all of its perfect fifths, will be a lot easier to do than Equal Temperament. We'll see.

27  ≠  (3/2)12
They're very close (128 vs. 129.746337890625) but still no cigar. It's one of the many sad tragedies of number theory.

What this ultimately means is that that "circle of fifths" they always used to tell you about in music theory class (Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B) is more of an endless spiral rather than a closed circle; 12 perfect fifths do not an octave make and the leftover bits just keep adding up the further you go.

A lesser-known piece of the puzzle has to do with the fact that most of the familiar intervals people care about sound good exactly because they involve low-integer frequency ratios. I've already mentioned the octave (frequency ratio = 2/1) and the perfect fifth (frequency ratio = 3/2), but there are others that people necessarily care about as well. Which brings up the next set of number-theoretic tragedies
(5/4)3  ≠  2
(3/2)4  ≠  5
5/4, in case you were wondering, is the pure major third. In other words,
1. Three major thirds do not make an octave, as you might otherwise have been led to believe from a piano keyboard --- and in this case it's not even close (1.253 = 1.953125, not 2.0) and
2. The (Pythagorean) major third that you get by composing a bunch of perfect fifths is (much) wider than the pure major third (1.265626 vs. 1.25), meaning that even if when you do manage to get the fifths right, you still lose on the thirds and it's a difference that's big enough to hear.
And this has consequences all over the place. Most singers, e.g., have a natural tendency to adjust the pitches they're singing to be the pure intervals, because that's what'll sound most in tune, and while the violinists and trombonists could conceivably do the same, the other members of the orchestra unfortunately have much more fixed notion of what the intervals are due to the physical limitations of valves, holes, frets, and keyboard design. Which makes for all kinds of fun when you're trying to coordinate an orchestra and a chorus and there's this huge a capella section with a mess of key changes or chromatic weirdness. Good luck still being on pitch when the orchestra comes back in.

This "temperament" problem has no exact solution; no matter what you do, you have to compromise somewhere. Organ/harp/keyboard-builders et al from about 1600 onwards lost lots of sleep over this small matter of just what was the Right Way to divide up the octave. The Pythagorean system (yes, ancient Greek who invented right triangles, same guy) based everything on the perfect fifths, i.e., 11 perfect intervals and if you were stupid enough to write something in B major (and thus had to contend with the B-F# "wolf" interval) that was your funeral. But somewhere in the Middle Ages they started caring more about getting the thirds to sound right, too, which then led to the Meantone system, which vaguely boils down to, "Let's make C major sound good, and to hell with everything else." (yes, I know it wasn't quite like that.) Which worked okay for a few hundred years, but eventually folks like J. S. Bach got really, really tired of C major, so they started playing around with other systems of temperament.

Somewhere in the middle of the 19th century, the piano/organ tuners all got together and said, "Oh, for fuck's sake!" and settled on Equal Temperament, which is where you decree the half-step to be 21/12 = 1.0594630943592953..., which means you get exactly twelve of them in one octave, and then you can say, "All right, we are done now."

And if those little dots at the end of 1.0594630943592953... are screaming "Hello? IRRATIONAL NUMBER!" at you, or if this happens to make all of the larger intervals wrong --- and, in particular, all of the major thirds end up really wide, though not as wide as they were in the Pythagorean system --- well, we just don't care anymore. Wagner, Liszt or whoever else can now go off and do whatever strange chromatic shit they like with nary a worry because it'll now sound equally bad whether it's in C major or G# major...

... well okay, that's not quite true since, while it's pretty easy (at least in the sense of it being Somebody Else's Problem; bwahahahahaha!) to equal-temper a keyboard instrument, doing the same for, say, brass instruments is quite another matter, seeing as a 3-valve instrument only gives you 3 degrees of freedom in setting relative pipe-lengths, 4 if you count the extendable slide that trumpets/cornets typically have on valve #3. Even the (double) French Horn only gives you 7, and since there's no reasonable relationship between any of the scale degrees, you just lose on the other five (i.e., time for Stupid Lip and Hand Tricks).

... just in case you were wondering why all those pieces you did in high school band --- oops, I'm sorry, I meant Wind Ensemble --- were all in keys like F, Bb, Eb, or Ab.

(**) And if anybody ever told you that Bach used Equal Temperament and that's what The Well-Tempered Clavichord (two sets of 24 preludes and fugues written in each of the major and minor keys) was about, well, they were lying --- but it's a rather widespread misconception so they're in good company.

Bach was, in fact, trying out one of the many systems of "well-tempering" in use at the time, the idea behind said systems was to make all of the keys usable, not to make them all the same (hint: If they were all the same, you'd only need one prelude and fugue. The point of writing 24 sets was to exploit the various different features of each possible key).

Anyway, Kirnberger III is the system They (i.e., some historians who have websites) think he was using.

We'll see how it goes.
2nd-Mar-2004 08:59 pm (UTC)
Do you have a stroboscope with a Hz readout? If not, a computer with some kind of RTA software will be useful. Many stroboscopes have 12 note readouts, but of course they assume equal temperament. And yes, having perfect pitch (absolute, relative, or even so-called microtonal) won't do you a lick of good. Hell, I have very good relative pitch, but the A I can pick out of the air is always a few cents flat.

Yeah, yeah, et the viola jokes roll.

2nd-Mar-2004 11:04 pm (UTC)
At the moment he appears to be doing it by ear and listening for beats. He also has most of the multiple strings muted out with felt, which in and of itself makes the thing sound a WHOLE HELL OF A LOT BETTER.

2nd-Mar-2004 11:08 pm (UTC)
But can you tuna fish?
2nd-Mar-2004 11:10 pm (UTC)
"God gave wRog a gift. wRog geeks. wRog geeks well."

^_^
19th-Aug-2004 11:52 am (UTC)
I would have been a trombonist much, much longer if someone(*) had sat down with me in 8th or 9th grade and explained all that to me, rather than trying to muck with my rather literal-minded idea of where slide positions were without explanation. ("You have to sharpen up the leading tone!" "Why?" "Because otherwise it's too flat!")

(*) That someone being my otherwise-excellent trombone teacher, who was principal horn for the Mainzer Oper.