simplifying ratios and fractions

http://mathforum.org/library/drmath/view/65801.html. I meant to link that earlier. Not too hard to read and should get you up to speed factoring integers by hand or with a basic calculator.

Do you know if a cheapish calculator can find the prime factors these days, or is it a job for a PC? Prime factorization, not that I know of, but even the cheapies will reduce fractions (as long as the calculator is designed to work with fractions.) Something like http://www.calculator.com/shop/item.php?m=electronics&s=casio&ASIN=B00004TVDR goes for under 20 bucks. http://h10025.www1.hp.com/ewfrf/wc/manualCategory?product=240101&lc=en&cc=us&dest_page=product&dlc=en& These calculators probably do the job using one of the algorithms discussed above.

Thanks Roger, I'll have a look. ac, I'm still learning this stuff myself but for example you can make a scale with 7:4, 7:5 and 7:6 in and the pitches share a particular quality (flinty, dark). and seem to fit together as a 'family'. I don't know why this is mathematically, but I'd imagine they share harmonics in common. Obviously a lot of numbers will have the prime factor 2 and not be especially related, but I think looking at the numbers helps get my subconscious learning rather than just getting the results off a spreadsheet.

gleuschk, where are you? We have a chance to get all of MeFi's math profs in one Answer. As escabeche emphasized, Euclid's algorithm is much faster than factoring numerator and denominator, even when the latter is already computer-approachable. Since this question has now been answered... Knuth's Art of Computer Programming Vol 1 has an analysis of how fast Euclid's algorithm is. The worst case turns out to be two successive Fibonacci numbers, taking log N steps where N is the larger number and the base of the logarithm is the golden ratio. The average case takes log N steps where the base of the logarithm is "Khinchin's constant", exp(pi^2/12 ln 2).

The TI line of calculators have GCD functions. And they will do just about any other math function you can think of, and program them to do even more. They aren't cheapish, but maybe you could pick up a used one on Ebay.

my knowledge of musical theory is limited, but i thought that the "flavour" depended on the reduced fraction, rather than primes. often the numbers in the reduced fraction are primes, but not always, and there's no argument i know of that says that the timbre of 13/12 is related to 1/2 any more than 13/11, even though 12 includes the prime factor 2. have i misunderstood? are you saying that the "flavour" of 13/12 is related to 1/2 (say) in a way that 13/11 isn't? just curious...

Thanks Wolfdog et al. That links not opening now, but I'll try it later. I'm not writing code or anything, I'm working with http://en.wikipedia.org/wiki/Just_intonation Most of the basic ratios are simple e.g. 3:2, but some calculations can end up more complicated. I'm not likely to get numbers much bigger than 1000 though. I've been looking for a way of doing this that let's me see the calculation rather than just giving the answer. It is handy to see the prime factors, as ratios based on different primes can have a different flavour.

http://ask.metafilter.com/mefi/29210#460304

A couple of points: everyone is correct to say that simplifying a fraction is easy by Euclid's algorithm, and that finding the prime factors of a large number is hard. But note that "large" here means hundreds of digits; a number like 504 (or 5000004) can be factored instantaneously by computer. No need to program this yourself; googling "factorization applet" will give you lots of sites that will factor numbers of this size for you. For instance, you can do it http://www.alpertron.com.ar/ECM.HTM