From: Aguanut24@aol.com Date: Wed, 21 May 1997 22:55:27 -0400 (EDT) To: rusin@math.niu.edu Subject: Student needs fast answers PLEASE!!! Mr. Rusin, I have a speech in math on Friday involving frequencies of notes in music. I got your letter written to bert@netcom.com off the internet reguarding this way in which math is related to music. I read it thoroughly and found it very fascinating and a lot of the information useful. However, I did not understand how (log 3/log 2) is useful as a power of two in the circle of fifths. The scale of music is based on a standard of the note A above middle C. A is at a frequency of 440 Hertz or cycles per second. (An octave above that is 880 Hertz. Octaves are twice the frequency.) I could not derive a ratio using the form of log 3/log 2 that when calculating out the circle of fifths was more accurate than the ratio of 3:2. In another letter I found, the ratio of 2^(7/12):1 was used. When calculating the circle all the way through, I ended up with A at exactly 440 Hertz. In trying to figure out how log 3/log 2 plays a part in this, I figured out that .58333...= 7/12. Adding 1 to 7/12 is 1.58333... which is close to log 3/log 2 (1.58496). When calculating the ratio using this knowledge, I found that 2^[(log 3/log 2)+1]=1.5 or 3/2 and ended up right where I started. I would greatly apprieciate an explanation of how you used log 3/log 2 in tempering the circle of fifths ratio to end up with 440 Hertz for A. If you could; please include how you determined using logs would get you an accurate answer. Your help is greatly apprieciated. Thankyou very much. Sincerely, --Kristi Larson ============================================================================== Date: Thu, 22 May 1997 00:24:44 -0500 (CDT) From: Dave Rusin To: Aguanut24@aol.com Subject: Re: Student needs fast answers PLEASE!!! >that is 880 Hertz. Octaves are twice the frequency.) I could not derive a >ratio using the form of log 3/log 2 that when calculating out the circle of >fifths was more accurate than the ratio of 3:2. In another letter I found, >the ratio of 2^(7/12):1 was used. When calculating the circle all the way >through, I ended up with A at exactly 440 Hertz. In trying to figure out how >log 3/log 2 plays a part in this, I figured out that .58333...= 7/12. > Adding 1 to 7/12 is 1.58333... which is close to log 3/log 2 (1.58496). > When calculating the ratio using this knowledge, I found that 2^[(log 3/log >2)+1]=1.5 or 3/2 and ended up right where I started. I would greatly Try this: tune the first A to 440Hz, as you suggested. What should be the frequency of the E above this note? Your ear will tell you that a freuency ratio of 3:2 is very natural, so you try to tune the E to a frequency of 660Hz. Of course the next E down an octave lower is then tuned to 330 Hz. Then you tune the B just above your initial A; it's exactly a fifth above that low E, so you give it a frequency of (3/2)*(330) =495 Hz. Continue in this way to get all 12 tones in the scale. When you finally tune the D just below the initial A, you will find that you are selecting a frequency of (3^11/2^18)*440Hz. As a final check, you then compare this D to the initial A. Since they're a fifth apart, the frequencies should be in a 3:2 ratio, no? But they're not: the ratio is 1 : 3^11/2^18 , which works out to about 1.48 : 1, that is, they're a little too close; the original A now seems a little flat. Well you could keep doing this forever -- sharpen that A a little more than 440Hz, then of course you have to sharpen the E again, etc. But it will never work. As long as you keep insisting that the frequencies be in a strict 3:2 ratio, you'll never get the circle to completely close around. (The problem is simply that no power of three will ever be even, must less a power of two!). The alternative is to fudge _all_ of the intervals equally. What's the best ratio of frequencies to use if we want all fifths to have their frequencies in a common ratio? Let's see, a fifth consists of seven halfsteps on the keyboard, so if the common ratio between successive halfsteps is R, then the ratio between the ends of a fifth will be R^7. On the other hand, there are twelve half-steps in an octave, so R^12 will have to be exactly 2 if we want to keep octaves in a precise 2:1 ratio. So we have R = 2^(1/12) = 1.0595, and then the fifths have a frequency ratio of R^7 = 2^(7/12) = 1.4983. Do you see what happened? Rather than have 11 fifths in the circle be perfect and the last one noticeably short, we simply shortened each of them just a little. Now, about the log3/log2: It's a basic fact of logarithms that 2^(log3/log2) is exactly 3, so the fact that 2^(19/12) = 2.9966 is darned close to 3 is just a reflection of the fact that log3/log2 is pretty close to 19/12. (It happens to be pretty close to 1.5, too, but that has nothing to do with the 3:2 ratio discussed so far). The point is really that log3/log2 is the magic, perfect number to use. We want a perfect fifth to consist of tone whose frequency ratio is exactly 2^(log3/log2) : 2. We settle for something close to that -- namely 2^(7/12) : 1 -- because we like having octaves of 12 tones and fifths of 7 halfsteps. But that's just a cultural thing. The Chinese liked to break the octave up into just 5 tones, three consecutive ones being a pretty good fifth (that is, 2^(3/5) = 1.516 is pretty close to 1.5). Maybe on Mars they break the octave up into 19 tones, and they think 11 consecutive tones makes a pretty good fifth (that is, 2^(11/19) = 1.494 is really close to 1.5). The Chinese can tune their instruments faster: a "circle of fifths" has only 5 fifths in it, for them. The Martians take longer: After 5 fifths, they've come back pretty close to their initial A440, after 12 fifths they've come back even closer, but they're not done until they've gone through 19 fifths to complete the circle. All these methods -- 5, 12, or 19 tones per octave -- are more or less arbitrary, and each requires that all the "fifths" be off a little from the perfect 3:2 ratio that we want. It's just that you'll never get exactly a 3:2 ratio if you want a "circle of fifths" that closes up after a finite number of fifths, because 3/2 isn't 2^(p/q) for any whole numbers p and q. Restated using logarithms, this last statement says: log3/log2 isn't a rational number p/q. I hope I've made it clear how log3/log2 enters the picture, and why you must make some arbitrary decision about the number of tones per octave, and must then accept that all the "fifths" in your system will be a little untrue. There is some mathematics in here beyond this: it turns out that 5, 12, and 19 are _not_ really "arbitrary" choices. The binary basis of computers notwithstanding, it makes _no_ sense to divide an octave into 8 or 16 parts, say. How big would a fifth be with a 16-note scale? Well, if a fifth consists of n out of these 16 tones, the ratio of the frequencies would be 2^(n/16). If you used n=9, this would make your "fifths" have frequency ratios of 2^(9/16) = 1.477 -- noticeably too flat to be called a "fifth". The next interval would be even worse: an interval of n=10 of these tones has a frequency ratio of 2^(10/16)= 1.542 -- worse! So your 16-note scale wouldn't let you play anything nearly as close to a true fifth (3:2 ratio) as we can get in the 12-note scale (remember our "fifth" is a 2^(7/12)=1.498 : 1 ratio). It turns out there are some numbers of notes-per-scale which are better than others in this way: they allow a closer approximation to a perfect fifth than any smaller number of notes-per-scale. The first few such numbers are 1, 2, 3, 5, 7, 12, 17, 22, ... Of these, some will give a remarkably good approximation to a perfect fifth -- that is, the error between a fifth on the scale and a perfect fifth is small even compared to what you might expect given the size of the set of tones in an octave. The first few of these is 1, 2, 5, 12, 41, ... You can get an amazingly good "fifth" with a 665-note scale ( 2^(389/665) differs from 1.5 by less than 10^(-7). That's far and away better than you can get with, say, a 666-note scale.) But I'm not ready to divide every half-step on the piano into 55 or 56 smaller parts! Good luck on your presentation. dave