Author: Tony Pay ★2017
Date: 2007-01-13 08:39
Pat wrote:
>>Does anyone know a website of formula that can churn out the oscillation value for each pitch?>>
You only need a pocket calculator. It's worthwhile UNDERSTANDING how it works, too, because then you'll understand string harmonics as well.
If an open string on a string instrument vibrates with a frequency f say, then if you stop the string half way up its length the resultant half-length string vibrates with a frequency 2f, an octave above. If you stop the string 2/3 of the way up its length, the resultant one-third-length string vibrates with a frequency 3f, an octave and a fifth above. If you stop the string 3/4 of the way up its length, the resultant one-quarter-length string vibrates with a frequency 4f, two octaves above. And so on, up what's called the harmonic series.
That means that given a note frequency (or a string length that produces that note) you can calculate the frequency corresponding to a fifth above (or the string length that would produce a note a fifth above). Because, in the example above, the two frequencies that correspond to the interval of a fifth are 2f and 3f, and so are in a RATIO of 3 to 2 (or equivalently, 1.5 to 1).
Therefore if the first note is A=440, the second note will be E=(440*1.5), that is, E=660.
Equivalently, if you have a string length that produces A=440, the string length that produces the E a fifth above is obtained by DIVIDING the original string length by 1.5 -- or multiplying by 2/3.
The principle is that given any frequency, say A=440, you can calculate the frequency of any note other than A (in the particular case I described, E) by finding the number that corresponds to the interval between A and that note (in our case, 1.5), and then multiplying 440 by that number. So 1.5 is the number for a fifth, and every interval has its own magic number.
Now that's how to calculate a 'pure' fifth. But in practice, we tune to equal temperament, which doesn't quite follow the harmonic series.
Fortunately, it's very easy to understand equal temperament from our point of view. Equal temperament makes all semitones the same, so once we've found what magic number corresponds to the interval of a semitone, then, starting at any note, we can work out the frequency of any other note by multiplying by the magic number several times. How many times? Well, however many times it takes to get to the other note by semitone steps!
So, what's the magic number? We want multiplying by it twelve times to get to an octave above, and THAT frequency is twice the first frequency. So we want a number which when multiplied by itself twelve times will give 2 -- and mathematicians call that the twelfth root of 2. The way to write that on a keyboard is 2^(1/12).
A pocket calculator will quickly give you this number -- or if I type 2^(1/12) into Google's search box and press return, Google tells me it's 1.05946309 -- and so we can work out the magic number corresponding to an EQUAL TEMPERED fifth by multiplying 1.05946309 by itself 7 times (there are 7 semitones in a fifth). If I type (2^(1/12))^7, Google says that's 1.49830708, which is AMAZINGLY close to 1.5, the magic number for a perfect fifth. This incredible coincidence is what makes western music possible!
Anyway, that's how Mark Charette was able to say what frequency A had to have if C=256. He just multiplied 256 by the magic number 9 times. Try it! (Google says 430.538965;-)
Tony
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The Clarinet BBoard
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