Thursday, April 22, 2010

In Tune

My sister and I took piano lessons when we were children.  She continued to play after the lessons ended; I didn’t.  But although she likes to play the piano she really always wanted to play the violin.  So for her New Year’s resolution, she decided she would take some lessons.  She rented a violin and found a teacher and has been enjoying herself.   A few weeks ago she was playing some of the tunes she had learned for me and she remarked that the violin sounded out of tune.  So she started trying to tune it and ended up very frustrated.  I was no help.  I know nothing about violins and although I could hear that she was in tune on the first note I didn’t think she was out of tune later while she was sure that she was.  

The violin, unlike a guitar, has no frets.  So a violinist must tune only by ear without any help from the instrument itself.  Recently Sam, at Inside the Classics, wrote an interesting post that may explain why it was so hard.  He wrote it as an adjunct to a post by Jan Swafford at Slate on the History of Tuning.   Sam tells us that there is no such thing as Perfectly in Tune, it is all a matter of taste.

… what sort of tuning sounds best to your ear depends in large part on what kind of music you like to listen to. If you’re a big fan of renaissance music, for instance, you absolutely need all the fourths and fifths to be perfectly in tune, which means that some of the thirds won’t be, but that won’t bother you so much, since there aren’t very many of them in renaissance music. On the other hand, if you love big Romantic symphonies, those thirds are just crucial, since triads are the basic building blocks of that era, and your brain won’t even notice the occasional slightly out of tune fifth.

What is he talking about – thirds and fifths?  Swafford explains, and explains the problem:

In dealing with tuning, there are two main terms to know. One is interval. It means the distance between notes. The basic science of intervals was laid out in ancient Greece, perhaps first by the mathematician Pythagoras. The first notes of the C major scale are C, D, E, F, and G. The note E is the third note up from C, so the interval C-E is a third. The note G is five notes up, so C-G is a fifth. So musical intervals run second, third, fourth, fifth, and so on. (Some intervals can be major, like F to A, or minor, like F to A flat.)

OK? Now, as Pythagoras discovered, intervals are also mathematical ratios. If you take an open guitar string sounding E, stop it with your finger in the middle and pluck, you get E an octave above. The octave ratio, then, is 2:1. If you stop the string in the ratio 3:2, you get a fifth higher than the open string, the note B. The other intervals have progressive ratios; 4:3 is a fourth, and so on.

So far, all very tidy. But this is where things get hilarious. As Pythagoras also realized in mathematical terms, if you start with a C at the bottom of a piano keyboard and tune a series of 12 perfect 3:2 fifths up to the top, you discover that where you expect to have returned to a perfect high C, that C is overshot, intolerably out of tune. In other words, nature's math doesn't add up. A series of perfect intervals doesn't end at a perfect interval from where you started. If you tune three perfect 5:4 major thirds, it should logically add up to an octave, but it doesn't; the result is egregiously flat. It is this innate irreconcilability of pitch that, through the centuries, has driven men mad.  …

What all this means in practice is that in tuning keyboards and fretted instruments, you have to screw around with the intervals in order to fit the necessary notes into an octave. In other words, as we say, you have to temper pure intervals, nudge them up or down a hair in some systematic way. Otherwise, you get chaos. So there's the second word you need to remember: The business of adapting tuning to nature's messy math is called temperament.

How crazy is that?   Maybe this is why I liked playing the piano – I never had to tune it, there were professionals who did that.  I never had to make the choice that tuners have to make:

There is no perfection, only varying tastes in corruption. If you want your fifths nicely in tune, the thirds can't be; if you want pure thirds, you have to put up with impure fifths. And no scale on a keyboard, not even good old C major, can be perfectly in tune. Medieval tunings voted for pure fifths. By the late Renaissance the tuning systems favored better thirds. The latter were various kinds of meantone temperament. In meantone, most of the accumulated fudges were dumped onto two notes, usually G# (aka A flat) and E flat. The shivery effect of those two notes played together in meantone temperaments earned it the name "wolf," which, like its namesake, was regarded with a certain holy fear.

All of this is difficult for tuning fixed tuned instruments like the piano or harpsichords.  But violins aren’t fixed tuned instruments.  They can be tuned by the musician “on the fly”.  The voice is also not a fixed tune instrument and can be tuned on the fly.

Meanwhile, an orchestra is made of a bunch of instruments, some of which tune naturally by ear—strings, woodwinds, brass—but also instruments in fixed, equal temperament: harp, marimbas and xylophones, harpsichord and piano, etc. What do orchestras do to harmonize all those conflicting demands? They do the best they can and try not to think about it too much. It can make you crazy.

Swafford’s piece in Slate is well worth reading.  In the meantime, Sam plays the viola for the Minnesota Orchestra, so he understands string instruments.  And maybe understands why Sr. Emily really didn’t want to teach my sister the violin when she was a small child:

My viola has only four fixed pitches – my open strings – and I can even change those within a few seconds if I need to. I have no frets, either, to control where my fingers land for any particular pitch, so my intonation is entirely within my control (or lack thereof.) This is the major reason why a pianist who’s only been taking lessons for a few years will almost always sound miles ahead of a violinist with the same amount of training. There’s very little that bothers the ear more than out-of-tune music, and in a particularly cruel twist for parents of young musicians, correct intonation is one of the very hardest things to master on string, wind, and brass instruments.