ALMOST PERFECT PITCHby David IsleAny music lover would enjoy an afternoon in Vienna’s Sammlung
ALMOST PERFECT PITCHby David IsleAny music lover would enjoy an afternoon in Vienna’s Sammlung alter Musikinstrumente (Collection of old Musical Instruments). There are beautiful pianos adorned with inlays and paintings. 19th century narrow-waisted guitars. The accoutrements of one of history’s great musical scenes. But for me, the most interesting items in this bestiary are the strange, extinct instruments. The ecosystem of instruments we know today, varied though it may seem, is the result of centuries of evolution. Many mutants and missing links were left behind along the way. You can see some of them in the Sammlung alter Musikinstrumente.Co-evolving with the musical instruments of the day was the musical scale itself. You may think that the Western 12-note scale represents all the notes that are possible to play. But this exact set of notes is a compromise and compromised solution, negotiated over several centuries, to a math problem. Shown above is an 18th-century instrument from the Vienna collection called a Harmonie-Hammerflügel (it probably has an English name, but I don’t know it). It has 31 different notes instead of the 12 that a modern piano has. What were these different notes, what were they good for, and why have they disappeared?A song is not identified by the absolute notes so much as the relationship between them. The note “C” doesn’t sound much different than “A” to most people if they listen to them 15 minutes apart. But an interval is something people can recognize. The two most important intervals are an octave and a fifth. The interval at the beginning of “Somewhere Over the Rainbow” is an octave (the Some-Where part). The interval from the first to the second “Twinkle” in “Twinkle, Twinkle Little Star” is a fifth.So you’re building an instrument and you have to decide what notes it can play. Given the importance of octaves and fifths, it would be nice if, for any note that the instrument can play, it can play a fifth above that note and an octave above that note. How do we make that happen?Now we get to the math problem. Notes are defined by frequencies. Frequency is measured in “hertz” (Hz). Concert A, for instance, is 440 Hz. Suppose we’re starting with that one. Now, I want to include the A that is an octave above concert A. It turns out that if you want to raise the pitch by an octave, you double the Hz. So a pitch that’s an octave higher than 440 Hz will have a frequency of 880 Hz. And I also want to be able to go up a fifth from either of these notes (the note we call E). To raise pitch by a fifth, you multiply the Hz by 3/2. So the E’s will be at 660 Hz and 1320 Hz. So here are the notes we have so far for our instrument, listed by frequency:440, 660, 880, 1320But we aren’t done yet because we don’t have the notes that are a fifth above E (that would be B). Remember, we want to be able to go up a fifth from *any* note that our instrument can play. So we have to add in (3/2)*660 and (3/2)*1320:440, 660, 880, 990, 1320, 1980Notice now that we’ve added in 990, and half of that is greater than 440 - so we’ll add in the B that’s an octave below 990 as well, giving us:440, 495, 660, 880, 990, 1320, 1980And then we have to add in F#, which is a fifth above B. By now you might be thinking, is this ever going to stop? The answer is…kind of. If you do this 11 times (counting the time we added E as the first time), you end up with these numbers (rounding to the nearest integer) between 440 and 880 (i.e., between concert A and the next A):440, 470, 495, 529, 557, 595, 626, 660, 705, 743, 793, 835, 880Now what happens if we try to multiply the most recent addition (595) by 3/2 as we have been doing: that gives us 892. And we don’t have that one yet, so to be completely accurate we have to add it in. But we have 880, and that’s pretty darn close. So we just say meh, good enough, and stop there, leaving us with our scale. All we’ve got to do is name the frequencies above with letters:A, A#, B, C, C#, D, D#, E, F, F#, G, G#, AThat’s why the scale has 12 unique notes - that’s as many as you need to make sure each note has its (almost) fifth included in the scale. But it doesn’t have exactly its fifth, as I’ve shown above. It’s just pretty close. The Harmonie-Hammerflügel tries to improve matters by adding in more notes - that is, I don’t have to play A at 880 Hz when what I really want is a fifth above D at 595 Hz (and therefore 892 Hz). I’ve got a different key for 892 Hz, so I play that instead.So the Harmonie-Hammerflügel is more precise. A purist might even say it’s better. But it’s a pain to keep in tune and it’s much more difficult to play. So it died. It’s now sitting in a museum and has to appeal to Beethoven’s having played it once for us to give it any credit or attention at all. But it was one of humankind’s leaps towards perfection, and we should honor it, even if pragmatism slapped it to the pavement.Quality content, like quality clothing, ages well. This article first appeared on the No Man blog in August 2016. -- source link