Piano Tuning Math
By Robbie Rhodes
The logarithmic arithmetic developed by Napier and Briggs in the 17th century was soon applied to tuning the musical scale: the result was the "Equal Temperment Tuning", which was adopted immediately by composers who enjoyed lots of key-changes. True, the new tuning algorithm didn't sound quite as sweet with simple "diatonic" music, and so the Traditionalists continued to compose and perform using older methods of "Just Tuning". Others simply observed: "With the new tuning, now the Klavier sounds equally _bad_ in all keys!" Well, it's not a panacea, it's a compromise, with a sound mathematical description.
(Since I enjoy playing with lots of key- and chord-changes myself, I tune my piano with the logarithmic Equal Temperment, like practically everyone else. Perhaps someday I'll get to play a fancy synthesizer, which dynamically adjusts the temperment for the "sweetest sound" for whichever chord (or dischord) I play at the moment!)
Here's the mathematics in simple form, and you don't have to worry about logarithms if you have an engineering calculator (or equivalent in a computer "assistant").
The basis is simply that the frequency of each adjacent note on the piano keyboard differs by a constant _ratio_. Since the "octave" of the keyboard has 12 different notes, and since the frequency doubles in each octave, the ration is the twelfth root of two, which is 1.059463094.
So, if I define my standard pitch as 220 Hz for the A below middle C, then when I multiply 220 by 1.056... _twelve_ times, I get the next octave higher, which is A = 440 Hz.
220.00 233.08 246.94 261.63 .... 369.9994 391.995 415.30 440.000
The piano tuner also geometrically sub-divides the ratio between adjacent tones into 100 parts, called "Cents". 1 Cent is the twelve-hundredth root of two, which is 1.000577790. Not much of a change, is it! But when I multiply 220.00 Hz by this tiny ratio one hundred times, I get the correct frequency of the next note, A# 233.08 Hz.
Suppose, for example, the piano tuner reports that my "new" old piano is 20 Cents flat compared to concert pitch. Is this a lot? Well, since 100 Cents flat would amount to one note (key), the piano is 1/5-note flat. 5 times 12 is 60, so I find the sixtyth root of two, which is 1.0116, and then I divide 440 Hz by 1.0116 to discover that "A" on my piano is really at 434.95 Hz. That's 5 Hz (cycles per second) difference from concert pitch, and I can easily hear the 5 Hz beat frequency when the tuner strikes his A440 tuning fork and the piano key together.
So, I instruct the tuner to "wind it up to concert pitch" and then I leave him alone for an hour or so. The result -- a properly tuned piano -- always pleases me!
-- Robbie Rhodes
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(Message sent Tue 5 Nov 1996, 03:58:42 GMT, from time zone GMT-0800.) |
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