Search Loci: Convergence:
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See the article whose title is the above quotation by Barry Cipra, SIAM News, July 1994, 1, 12-13.
A Plague of Ratios
Rather a lot, it turns out. If you have two strings made of the same material, with the same thickness and tension, but different lengths, they will make different musical pitches when you pluck them (like a guitar) or stroke them with a bow (like a violin). The relationship between the two pitches determines whether they blend or not; whether they make a consonance or a dissonance. That relationship in turn is determined by the ratio of the two strings' lengths. If the ratio is 1:1, the two strings make the same pitch: they are in unison. If it is 2:1, they are an octave apart (e.g. from one C to the next C above it). If it is 3:2, they are a perfect fifth apart (e.g. from C to the G above it). If it is 4:3, they are a perfect fourth apart (e.g. from C to the F above it). Finally, if the ratio is 9:8, the two notes are a tone apart (e.g. from C to the D above it).
When mathematical music theory was done in ancient Greece, those were all the ratios and musical intervals that were used. You might notice certain relationships between them. For example, a fifth plus a fourth is an octave, since 3:2 'plus' 4:3 is 2:1. And the difference between a fifth and a fourth is a tone: 3:2 'minus' 4:3 is 9:8.
But what if we want to divide up the octave into small steps to make a scale? How many times, for instance, will a tone go into an octave?
Now you see the connection between music and ratio theory. To divide the octave into tones is the same as 'dividing' 2:1 by 9:8. (That is, how many times must we 'add' 9:8 to itself to get 2:1?) By trial and error we can see that it goes more than five times and less than six. It doesn't go exactly. If we want to know how many fourths or fifths there are in an octave, or how many tones in a fourth or fifth, they don't go exactly either, and we find ourselves needing again and again to be able to perform 'divisions' with ratios.
One solution is an approximation: divide the octave into six, or twelve, exactly equal portions, and use these to make approximations for the smaller intervals. This is what is normally done nowadays: the twelve notes in an octave are exactly equally spaced, a tuning that is called equal temperament. The twelfths of an octave are called semitones. A tone is taken to be two semitones, a fourth five, and a fifth seven. These are good approximations, but not very good ones, and in Mercator's time it hadn't been completely accepted that this was the best thing to do.
Before returning to Mercator and seeing how he went about finding better approximations and exact answers to these musical questions, you might like to experiment with the interactive applet. It shows a musical string and a modern keyboard, and you can see how lengths on the string correspond with positions on the keyboard. You can change either the string position or the keyboard position by dragging its slider, or by typing an exact value into the box.
For example, you can see that half of the string gives a note an octave - twelve semitones - above the whole string. What happens if you take two-thirds of the string, an interval of a fifth above the whole string? It is about seven semitones above the bottom note on the keyboard, but it is not a whole number of semitones. Experiment with other ratios, and see how they come out on the keyboard: or see what ratios correspond to particular whole numbers of semitones.