There has been some interest expressed in having tutorial(s) on the math involved in tuning. So, here we go... Imagine this scene: you're at a party. A lovely young woman arrives, and somehow you find yourself in conversation with her, and somehow it comes around to tuning, and she asks, "But isn't equal temperament the perfect way to tune? Why would you want to change it?". Are you ready? Could grab a calculator and dazzle her with math? Or are you left stammering? Dont'cha HATE it when that happens? Well, now you can be math literate, because Uncle John has prepared... BASIC TUNING MATH: converting frequency ratios to cents and back A conceptual introduction to logarithms The piano provides the perfect model, as well as an illustration of why we have to do these transformations. On the one hand there is the row of strings: with each octave higher, the frequency doubles, from A440 to 880 to 1760 hertz (cycles/second) as we go up to higher A's; 220, 110, and 55 hertz as we go down to lower A's. On the other hand, the keys move in linear progression, twelve semitones per octave. Logs are a mathematical way of relating doubling to linear motion. Exponents (powers) go the other way, relating linear motion to doubling. Whenever the whole process gets confusing, think of the piano. But, even if you're lost or indifferent so far, you can still do The math First you need a calculator with logs and exponents. A key might be labeled "log" or "ln" (for log natural). The other key we want is something like "X^Y", where Y is probably a superscript to X, indicating that the key will raise X to the Y power. Don't have a calculator with those keys? Get down to the store and buy one! They're cheap. Yes, you did vow most solemnly you'd never own anything of the sort, but holding out this long is enough to have made your point. Now buy (or borrow) one. Let's walk through some basic calculations. You gotta get this to work before we can go on. Confess your lack of knowledge to someone who might know, if you're stuck. Raise 2.0 to the 3.0 power; get 8.0. I wish I could help you with more specific instructions, but calculators are all different. Next exercise: divide the log of 1.5 by the log of 2. It'll go one of two ways: If your calculator uses natural logs: log(1.5) = 0.4055 log(2.0) = 0.6931 log(1.5) / log(2.0) = 0.5850 If your calculator uses base 10 logs: log(1.5) = 0.1761 log(2.0) = 0.3010 log(1.5) / log(2.0) = 0.5850 Note that the final answer comes out the same, so it doesn't matter; we're actually calculating "log to base 2" by either path. By the way, 0.5850 happens to be the fraction of an octave representing a perfect fifth (frequency ratio 1.5). Cents review: a cent is a measure defined to be 1/100 of a 12 tone equal temperament semitone. So, a cent is 1/1200 of an octave. Now you're ready! To convert frequency ratio to cents interval: Delta_cents = 1200 x log(freq_ratio) / log(2) OK, walk thru with an example: a major third, a 5/4 ratio, divides out to 1.25 frequency ratio; take the log of that; take the log of 2 and divide; you should see 0.3219 (if your calculator displays intermediate results); multiply by 1200 and get 386.31 cents (compare with 400 cents for a major third in 12-tET). To convert cents interval to frequency ratio: Freq_ratio = 2 ^ (delta_cents / 1200) That's "two to the power". Again, do an example: a 12-tET minor third, 3 semitones or 300 cents, divided by 1200 gives 0.25 (octaves). Two (2.0) raised to that power gives 1.1892; compare to 1.2000 for a perfect 6/5 minor third ratio. You are SO cool. Put a mortar board on your head! Note that everything we've talked about relates TWO notes. A single note in space, such as A440, may be a starting point for forming intervals, but there's little we can do with it by itself.

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