saf@floyd.UUCP (07/13/83)
An article by ...mcnc!ebs (eddie) mentioned that if computers had existed 400 years ago, we might have a 16 note scale instead of a 12 note scale. This is probably not correct, the rational for 12 is rooted in physics, not "convenient mathematics". Here's a little of the theory for those interested. The ear seems to like to hear tones which have at least some of their harmonics at the same frequency. Thus, the octave, the "cleanest" musical interval, represents an exact doubling of frequency. The harmonics of two notes which are an octave apart line up perfectly (except that non-ideal strings have overtones which are not exactly the same as harmonics due to stiffness in the strings but that's another story). If you look at "nice" fractions like 1/2, 2/3, 3/4, 4/5 you will see similar patterns in the harmonics. This was the basis for the equally tempered 12 note scale. If you force the octave to be perfect, you have a 2:1 ratio. If you now geometrically divide that up EVENLY, you will use a multiplier like 2^(i/n) where i is the "distance" between any 2 notes and n is the total number of subdivisions in an octave. Let's pick n = 12 and see what we get. We can let C represent the 0'th note, C# be the 1'st, D be the 2'nd, etc. You now find, for example that the fifth note (F) is 2^(5/12) or 1.33+ times the frequency of the zeroth note. Well, that is essentially a 3/4 ratio. Similarly, 2^(7/12) (G) yields a ratio of 2/3 (to a very close approximation). Also, 2^(4/12) (E) gives 4/5 as a ratio. So, letting n = 12 gets us the kind of relationships the ear likes to hear. Now let's try it with n = 16. Obviously, 2:1 works by definition. How about 2/3? The closest we can come is 2^(9/16) which is about 1.5% low - with n = 12 we were only off by .11%, about an order of magnitude closer. How does 3/4 fare? 2^(7/16) is also about 1.5% low while 2^(5/12) is within .11% again. For 4/5, 2^(5/16) is .6% high while 2^(4/12) is .8% low so here n = 16 is actually better, but not by much. So n = 16 isn't very good. Those 1.5% errors are going to sound pretty discordant with very noticeable beating. I haven't tried other values of n but there is one modern composer who lets n = 53 and gets almost perfect ratios. Note that in this "microtonal" system, there are some really bizzare sharp and flat notations to handle all 53 subdivisions. Perhaps others will find "better" values for n. By now someone is asking, "Why not just use the exact ratios and forget finding a value for n?" Good question. If you use the exact ratios, you are presented with a mathematical impossibility. Take 2:1 through 7 "octave" intervals starting at C and you get back to C with a total ratio of 128:1. Take 3:2 through 12 "fifth" intervals starting at C and you get back to the same C (you have to write out C C# D D# E F F# G G# A A# B C C# D D# E... to see why you must get the same note. Another way is to look at a piano keyboard). Anyway, the ratio now is 129.75:1. So we really can't use exactly 2:1 and 3:2 because we have a contradiction. The thing that seems to work is to get as close to the "simple" ratios as possible, while still useing an equally tempered system (where the notes are always the same geometrical distance apart). Enough for now. If anyone is interested in delving further into the physics of music I can supply some references... Steve Falco BTL Whipany, NJ (201) 386-4865