From ocanter@richp.com Thu Jan 8 23:14:37 2004 Received: from mv.richp.com (mv.richp.com [205.217.153.230]) by sanford.math.niu.edu (8.12.10/8.12.10) with ESMTP id i095Eb25015879 for ; Thu, 8 Jan 2004 23:14:37 -0600 (CST) Received: from localhost (ocanter@localhost) by mv.richp.com (8.12.6/8.12.6) with ESMTP id i095Elrf066114 for ; Thu, 8 Jan 2004 21:14:48 -0800 (PST) (envelope-from ocanter@richp.com) Date: Thu, 8 Jan 2004 21:14:47 -0800 (PST) From: Orville Canter X-X-Sender: ocanter@localhost To: rusin@math.niu.edu Subject: equal temperament Message-ID: <20040108185431.G65209-100000@localhost> MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Spam-Status: No, hits=-0.9 required=5.0 tests=BAYES_30 version=2.55 X-Spam-Level: X-Spam-Checker-Version: SpamAssassin 2.55 (1.174.2.19-2003-05-19-exp) Content-Length: 5597 Status: RO dear Dr. Rusin, you are probably aware that your web page "Mathematics and Music" (http://www.math.niu.edu/~rusin/papers/uses-math/music/) has developed a strong presence on the world wide web. for example, the search "google.com/search?q=music math" returns your page at the top. while your discussion of equal temperament presents some interesting arithmetic, it does not answer the question of why instrument makers divide the octave into twelve notes. what it answers is why we use equal temperament for tuning pianos and other instruments that do not allow to the performer to tune the notes as he plays.. the chromatic scale was not invented by multiplying a given note by 3/2 twelve times. the twelve notes of the chromatic scale were in use a long time before equal temperament caught on. originally, as I'm sure you're aware, keyboards were tuned according to "just temperament": http://www.phy.mtu.edu/~suits/scales.html at this time, and in fact at all times before this, very few musicians were troubled that an instrument with non-adjustable frequencies (it's really only the keyboard instruments that we're talking about here--other instruments, such as strings, do not use equal temperament unless they are playing with keyboards, marimbas, etc.) could not transpose intervals perfectly to distant keys. for one thing, transposition was very rare in the earliest music, and in most non-Western and "folk" music even today. for example, Indian music, which actually has a rich classical tradition, never modulates. most folk melodies do not modulate, and if they do, it is to a very closely related key, and only for a few bars. for another, the few composers, such as Gesualdo, who did, before Bach, write highly chromatic music, were content to use instruments such as the human voice that are capable of adjusting the temperament of intervals dynamically, and so achieve a more perfect chromatic scale than either just or equal temperament. this is, of course, what an ensemble of string players or vocalists does today. I think if you look at the Michigan Tech page above, you'll see how the twelve-tone chromatic scale was actually constructed. first, the octaves of C were tuned perfectly. then the fifth was tuned perfectly to the root. Then the fourth was tuned perfectly, then the major third to its harmonic value, and the minor third to make a major third with the fifth. then the major second to make a perfect fifth with the fifth. then the major sixth to make a perfect major third with the fourth (rather than a perfect fifth with the major second). I'm sure you get the idea, but note especially the tuning of the tritone--it was tuned to make it a perfect major third with D. in other words, they tuned it to make it sound good as a leading tone to G, when the keyboardist played in G. it would have sounded terrible in F#, but nobody minded very much, because most composers never modulated to the tritone anyway, and if they did, they would simply use strings or voice. so that, Dr. Rusic, is how the twelve-note chromatic scale was constructed, and that is why we still use twelve notes on keyboards today. it apparently had nothing to do with the desire of anybody to make all the intervals transposable to every key on a fixed-pitch instrument. an interesting question, then, is why, after developing the twelve note scale independently of any desire to make the intervals equal, did musicians suddenly discover that the twelve notes could be adapted quite nicely to equal temperament, enabling the keyboardist to modulate to twelve keys? the answer is that it was *not* adapted quite nicely at all. when musicians first encountered equal temperament, thought it sounded terrible. it does sound terrible, if you're used to only playing in C and G in just termperament. but over time, the desire of keyboardists to stray further and further from the tonic center prevailed over the purists' sense of perfect harmony. your arithmetic explains how we were able to adapt the *number* of tones we use to equal temperament, and you assume correctly that equal temperament became popular because keyboardists wanted the ability to modulate to twelve keys, like the other instruments, but it does not explain why we use twelve tones in the first place, nor does it answer the question I came to your page hoping to answer: why did we arrive at one of the few numbers that could be adapted to equal temperament? was it a happy coincidence? or was there something in the construction of the seven-note scale, and thus in the ratios of the fourth, fifth, and third themselves, that necessitated the eventual inclusion of twelve notes? your observations about five-note and seven-note scales seem irrelevant, because those scales do not use equal temperament. nobody but piano manufacturers, etc., use equal temperament, and they only use it because it approximates decently fourths, fifths, and thirds in every key. it is certainly not a requirement of making "good music". when you hear C and E, you do not perceive two notes four musical fifths away from each other; you perceive something very close to 5:4. in other words, it is E's place in the harmonic series on C that you hear, not its place in the circle of fifths. if you don't believe me, play a C for a trained singer and ask him to sing E. then compare that E to the E on your keyboard. the one he sang is the harmonic E, and it sounds better. the one on your keyboard is an approximation that is given by equal temperament. Peace, Orville "The sun is that which shines." --George Boole