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Will VogtmanFeb 13, 2009 6:19 PM GMT
I downloaded some data and figured out that for the nth tone in a 12 tone scale,
f(n) = 2^(n/12)*f_o,
where f_o is the considered root.
Since the interals are chosen as ratios of 2 harmonic numbers, why couldn't I choose a rational number so close to any given interval that there is no discernable difference between the Just system and the 12TET system?
Now I do understand the prime number limit system as--
Neither number in the ratio chosen for the given interval can have a prime factor larger than the given limit.
So, why not choose a prime number large enough to approximate 12TET?
Just Limit (very large prime limit) = 12TET?
At one time, complicated fractions would be a problem. But now, decimal approximations are easy to find and use with calculators, spreadsheets, digital tuners, etc.
Now, I know that seems crazy. But, what is the "MAGIC" behind choosing a limit?
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Sorry . . . math geek.
Sorry, Will, this is primarily an English language forum for harmonica players. We do our best to respond to all alien languages here, but we don't pretend to be prepared to serve the needs of all nine of the known universes. Perhaps you could take this up with Howard Levy? ;)
(BTW, string theory presently suggests 11 dimensions.)
Well, the Magic is about chording. 12ET really is the notes perfectly tuned, the truest form of the note. Just had some or most notes flatten so when played together they blend together easier as equal temperament sounds a little harsher.
One thing that you need to remember is that the only instrument that gives off more harmonic overtones than the reeds of a harmonicas a piano, but the piano has a very powerful fundamental overtone, wheras on the harmonica, it's considerably weaker. Tho they both give off many more overtones than any other instruments, on piano, many of those overtones are even numbered, which to the human ear, sounds pleasing, much like the way tube amp distortion does, and harmonica reeds give off tons of odd numbered overtones, which especially if tuned to 12TET, will tend to sound harsh to the human ear, much like the way distortion from a solid state amp does. Every tuning employed involves some sort of a comprimise, like it or not.
Chords on an accordion or a melodica, both of which are tuned to 12TET, the chords tend to sound harsh and beat like hell for the very same reason as does a harmonica. Until the 1700's, most musical instruments were tuned to some sort of just intonation and on the site http://www.justintonation.net, every one of those historical temperaments are all listed there.
If you were to give a root note for an acapella group to sing and no instrumental accompaniment, once the root note was set, if you were to check all of the harmony parts being sung with a strobe tuner, you'd have an 80% chance that when closely monitored, the harmony parts would be sung consciously or unconsciously in some form of just intontion for everything to be in full, exact harmony.
When instrument makers decided to go with ET, they knew for a fact that chords would often end up sounding rough, with some instruments less, and some more, but it allowed all instruments to be played together in any key, so they knew they had to comprimise somehow and live with it.