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Attention, music and math lovers. I found this fascinating site while trying to find something else. He starts off with how fractions are used to denote intervals in music; for ex., 5/12 = 5/6 = 5/3 = 10/3 = 20/3. The last section is one I thought you might be particularly interested in, Musick: "4. What Do Pure Intervals Sound Like?"
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Carpal Tunnel
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Running perty fast and loose with them equalses there, Jackie. Gonna post the site's url to back them up?
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Joined: Sep 2001
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Carpal Tunnel
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I don't know what pure intervals are, but I would guess that octaves, perfect 4ths and perfect 5ths might be good contenders for the title.
Will be interesting to read here whether 'tis so...
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Carpal Tunnel
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About to check it out, but it looks like http://home.earthlink.net/~kgann/tuning.htmlis the one. Entry #2 on a Google search.
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"What Do Pure Intervals Sound Like?"Am I right in thinking it's impossible to get a "pure" interval? I recall (possibly incorrectly) that all tunings - including the well-known Well-Tempered - deviate from "purity" to favour particular keys. In fact that definitely strikes a chord. 
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Carpal Tunnel
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you can have pure intervals, just not a whole string of them(no pun intended. yeah, right!) stacked on top of each other. you can tune all the octaves(say "C"), and the 5ths to them, but if you try to tune the 5ths of the 5ths, bingo! trouble. the math just doesn't add up. the 5ths of the 5ths would be the seconds and ninths of the octaves, and the frequencies wouldn't quite add up. you are left with a hole. or too much, I forget.  Well-Tempered tuning alters all of the tunings so that all the intervals work pretty well; all your major thirds are fairly major like, etc. it's a huge and deep topic, and if you get a chance to hear a group play with Just Intonation, it's really quite interesting. it can be hard to accept to our compromised ears, however. this of course, really only deals with music from Western Civilization, as other cultures use different tuning systems, with different numbers of notes between the "octaves".
formerly known as etaoin...
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Carpal Tunnel
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Uh oh. Don't get Faldage started on Pythagorus.
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Don't get Faldage started on PythagorusYeah, we don't want the summation of squares. 
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It's certainly not impossible to get a pure interval. THe fifth you get out of a well tuned piano won't be one. It'll be close but.
The fifth of the fifth isn't where the problem is noticed. The fifth is not half an octave so we shouldn't expect two fifths to make an octave. If you tune a perfect fifth (it's a wee tiny bit sharp of your piano fifth) from a C and call it a G, then tune a fifth from that to a D and so on, you'll get the famous circle of fifths:
C-G-D-A-E-B-F#-C#-G#-Eb-Bb-F-C
but the C you end up with won't be the same C you started with. If you read the page wofa posted you'll see that the pure interval fifth is 702 cents (Don't panic folks, one cent is one one-hundredth of a half step. That's the half step on your well tuned piano.) Add up the seven half steps in the fifth and you get only 700 cents. This is two cents flat of the pure interval fifth. Add up all those two centses in the circle of fifths and you'll see that the C you get when you're done with piling all these pure interval fifths on top of each other is 24 cents sharp of the C you started out with. Oh, oops. This is called the Pythagorean comma and it's not a problem as long as you're only playing in one key with no accidentals, but as soon as you want to modulate or have an instrument that will play in any key (such as a piano) you're in big trouble. It took us a couple of hundred years to get this all settled out. We got a lot of compromises along the way that left us with the idea that different keys have different characters. They used to even as recently as Mozart's time. If they still do, it is merely because composers think they do and write music accordingly.
There. Din't even sum no squares.
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Carpal Tunnel
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I read this three times and for the first time in many years I am at a loss as to how to translate it. My job is to take complicated stuff and reduce it to simple stuff, but in the beginning I have to have English language. Whatever this stuff is it AIN'T English.
To me a fifth of a fifth is nothing more than a damned strong drink, about 6 oz. A cent is a hundredth of a dollar, not of a half step.
A comma is used for punctuation, Pythagorean has to do with right triangles and other stuff like that, and accidental is how Sasha (well never mind about that.)
TEd
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