The pythagorean theorem of musical mathematics

DENADE, AH. SCIENCE IN BOOKS THE PYTHAGOREAN THEOREM OF MUSICAL MATHEMATICS Cr He really wasn’t stringing his fellow Greeks along; he picked it out in intervals between philosophizing. l we listen to melody and harmony all over the world, and through history, we find so many things that turn up over and over that we are entitled to suspect the existence of physineed to know the musical answer beforehand as evidence that the numerical method of organizing music is a sort of peculiar accident. They are tempted to assert that all musical rules are a matter of custom and agreement, and that there is no absolute right or wrong way to go about choosing notes in music. My own belief is that music is very strongly affected by the manner in which things vibrate, and by the manner in which our ears work, It is certainly easy enough to show mechanical reasons why certain pitch intervals are special cal causes for them. The Greek philosopher Pythagoras was one of the earliest we know of who thought about this in an orderly way. He based his theory of music on the fact that some of the musical intervals (that is, differences in pitch) between notes in the Greek scales seemed to be related in a direct way to the sounds given out by a string when it is plucked with some simple fraction and distinct from their neighbors, in a of its length left free to vibrate. In more musical sense. modern language, we say that a halflength string sounds the important interA 28 OM 44 Ei on val of one octave in pitch above the fulllength string, and a string which is coni 8 28 EC strained to vibrate over only one third of 1004 $ : its length sounds higher by the musical interval of a “twelfth,” and so on, makBA CA DA EA FA GA, ing what musicians call a harmonic series of notes. | wo j 60 | soo fj Too A certain amount of number juggling dagradient frequencies can show that almost all the musically important intervals of modern western music (and of most other forms as well) I shall make use only of some mecan be more or less deduced from the chanical facts of life. I shall say nothing whole-number fractions of the lengths about all the complex things which go of a vibrating string. Now, it must be on between the mechanical ear and the confessed that in order to do this one brain. These also have a big effect on used to need not only a great deal of music, but would lead us out of our faith in the magic properties of numdepth and away from physics. bers, but also a good knowledge of the Let us start by asking what happens musical customs which were to be ex- when we play successively two simple plained. To the Greeks, and to many harmonic sounds of differing frequency. people who came later, the magic of If the first note is played quite loud, we pure number was so potent that they hear not only the frequency of that note, were quite willing to settle for the nu- but also integral multiples of that fremerical relations they found, without quency, 2f, 3f, 4f, and so on. The reever asking why they came out that sponse of the ear mechanism is responsiway. Some musicians with more logical ble for this behavior, and the higher correctness than physical knowledge multiples are relatively weak in comhave seized upon the Pythagoreans’ parison with the lower ones. If now the 74 second note played happens to be the same frequency as one of the family of sounds produced by the first note, we recognize it (unconsciously) as a repetition of something heard before, and so consider a note of this second frequency as being in a special relation to the first. Everyone who has fooled with a guitar, or played a violin, knows that the pitch rises as the vibrating length of a string is decreased, and many of you will have heard in school that the frequency of vibration of a string is inversely proportional to the length of the string. This means that cutting down the length to one half will double the frequency, reducing it to one-third will triple the frequency, and so on. We can now see very clearly why Pythagoras’ experiment with strings might be expected to correlate with his knowledge that there are certain especially interesting musical pitch relations. You may feel that this little demonstration is not enough to lay the scoffing ghosts of untrammeled artistic freedom, and you are right. But it is a step in the proper direction, and a bigger step than it may appear at first. I am being a little rude here toward people who make a hobby of not knowing enough about their subject of specialization. Composers who, for example, preach the extremist dogma of the “twelve-tone scale,” in which every note is on an even footing with every other one, are no better in their disregard of the physical world than are sculptors in wood who try to ignore effects of the grain on the texture, working, and strength carvings. of their —ARTHUR H. BENADE in “Horns, Strings and Harmony” Copyright © 1960 By Educational Services, Inc. (Doubleday Anchor Science Series). SR/December 2, 1961