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Feb. 24th, 2004

11:47 pm

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In a manner of speaking, I have been researching links between Music and Math (which shall be capitalised out of politeness) for my Extended Essay. Edging away from the Fourier series out of hearstopping awe for the power of said mathematics, I stumbled across the ratio of frequencies. And G-d said: "And thou shall create the notes in a geometric sequence. And the frequency of each note over the frequency of the preceeding note shall be equal to a constant, which shall be 2^(1/12)". And it was done. "And the frequency 440 shall be termed A". And it was done. "And two notes with the frequency ratio 3/2 when divided shall be named perfect fifths". And it was done. A and E are perfect fifths. A is 440. E is 659.25511382573985947168352209311. 659.25511382573985947168352209311/440 does not equal 3/2. It is a bit less. The proper perfect fifth of 440 A is 660. Why are A and E still perfect fifths?

Comments:

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From:carbonic_acid
Date:February 25th, 2004 02:08 pm (UTC)
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The overtones diminish slightly over time.
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From:darth_phoenix
Date:February 25th, 2004 03:42 pm (UTC)
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Meaning that holding the note will diminish the 440n and 659.25n (where n is an integer not equal to 1) frequencies? That doesn't make sense... Explain please?
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From:carbonic_acid
Date:February 25th, 2004 04:06 pm (UTC)
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Here's a definition that may help:
\O"ver*tone`\, n. [A translation of G. oberton. See Over,Tone.] (Mus.) One of the harmonics faintly heard with and above a tone as it dies away, produced by some aliquot portion of the vibrating sting or column of air which yields the fundamental tone; one of the natural harmonic scale of tones, as the octave, twelfth, fifteenth, etc.; an aliquot or ``partial'' tone; a harmonic. See Harmonic, and Tone. --Tyndall.

Anyways, when you play the note itself, there is a momentary high then the low. So think of the high as the 660, and then it degrades to the actual note of 659.25. Make sense?
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From:darth_phoenix
Date:February 26th, 2004 01:00 am (UTC)
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That's what I meant by 440n and 659.25n (the harmonics are integral multiples of the original note). But 660 is not an integral multiple of either of them. From which note comes the 660?
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From:dearestapathy
Date:February 26th, 2004 07:33 am (UTC)

A slight digression on the topic of scales and keys

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An F# in D-major will be played, by most professional violinists, slightly higher than the "real" F#.* It is also evident that in g-minor, the B-flat will be flattened, albeit almost inaudibly,in order to change the note's colour; thereby adding effectiveness to the music. I'm not sure why I said all that but I believe it had something to do with what a real note is and isn't. Mathematically, music can be defined; however, sometimes these definitions do not fit or sound right and sometimes they do. The tempered scale says that E should be 659.25 (i am most likely wrong here as I'm simply speaking on hearsay) and who SAID that E was a fifth from A? Certainly, E is the fifth degree of both A-Major and a-minor, but, that does not necessarily mean it has to be a perfect fifth away. It is assumed that a fifth always has to be "perfect" (unless it's diminished or augmented) but, "perfect" is definable. What "sounds" right to us simply "sounds" that way because we have been trained to appreciate that sound. Some people always hear flat and others always hear sharp; with this in mind, the flat ones would have been very comfortable in one of Mozart's orchestras as he enjoyed a particularly low A.

I'm sorry for not making sense and it is most certainly due to my lack of sleep and food. I will try to be more coherent next time I post here and this was merely a digression and not actually an answer to any question.




*you can look up the numbers and plug them in if you like, you'll see that they concur**
**yes, numbers can concur, lol
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From:darth_phoenix
Date:February 26th, 2004 09:15 am (UTC)

Re: A slight digression on the topic of scales and keys

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I agree that musical perception and appreciation is subjective, but then are there ever "perfect" fifths? Perfect here being 3:2, not subjectively perfect. Why a 12 tone scale? Why not an 8 tone scale? Or a 3 tone scale? Of course that limits the possibilities, but then why not a... (ooh let me do the math...)... some ratio $r$ where 2==r^n, where n is the number of tones plus one, where r^(5/n) is a whole number.. I don't know if that exists, but I know n=9 comes close.
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From:dearestapathy
Date:February 26th, 2004 09:18 am (UTC)

Re: A slight digression on the topic of scales and keys

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A perfect fifth could of course exist. Using your example, why not simply raise the E by .75 hz?
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From:darth_phoenix
Date:February 26th, 2004 11:00 am (UTC)

Re: A slight digression on the topic of scales and keys

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Because then it wouldn't be an E... of course you were saying how the change the pitch somewhat to make it sound right, and of course yes it would make a perfect E, but it woulnd't then be a proper E. Actually, I did my math, and it turns out the scales with good fifths come from the equation 2^x=(3/2)^y, where x is the number of tones, and y is close to a whole number. The closer y is to a whole number, the better fifth you will have. Twelve tones is actually very good for this, but 41 or 53 is even better. They $y$th tone in the scale will therefore be the fifth.

x - y
1 - 0.58
2 - 1.17
3 - 1.754
4 - 2.34
..
11 - 6.43459
12 - 7.01955
13 - 7.60451
...
41 - 23.9835
...
53 - 31.003

I say we change the scale to a 41 tone =).
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From:dearestapathy
Date:February 27th, 2004 01:01 am (UTC)

Re: A slight digression on the topic of scales and keys

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Believe it or not, in the fifteen-seventeen hundreds they actually had these very discussions and sometime in 18something, they settled for a tempered 12 tone scale. Take a look at this site, that's where I found a lot of information for this discussion although I didn't bother actually relaying it.
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From:darth_phoenix
Date:February 27th, 2004 01:19 am (UTC)

Re: A slight digression on the topic of scales and keys

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Whoa! That's a lot of info.. thanks I'll have to read over that.
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From:darth_phoenix
Date:February 26th, 2004 11:00 am (UTC)

Re: A slight digression on the topic of scales and keys

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I was wrong... it's not r^(5/n) is a whole number, but r^(y/n), where y is the tone which is the fifth.
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From:carbonic_acid
Date:February 26th, 2004 11:28 am (UTC)
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Oh, ok. i was going to ask if maybe 3/2 was rounded or something.
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From:darth_phoenix
Date:February 26th, 2004 11:44 am (UTC)
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My mistake. But of course, the $y$th harmonic is rounded from $y$, as $y$ will never be an integer with an integral number of tones. Of course, this whole digression is based on the assumption that an octave consists of the the frequencies between $n$ and $2n$.... *shrug* but that's just convention.