The sqr-root(-12)* works out to the ratio of each tone within the octave of the equal-tempered scale, the one in current use on the piano, and in theory the one everyone uses on post-Bach tuning systems (==temperaments).
For millennia people have been screwing around with the ineffable meaning of other tuning systems based on divvying up of rational numbers, which unfortunately doesn’t get you 11 nice notes to play around with if you want to keep the 2:1 “octave” ratio (the etymology of “octave” is itself is laden with historical baggage irrelevant at this point).
If only 3:2 were commensurate with 2:1. Sorry Pythagoras.
*In case the symbol doesn’t display right in the hed.
I can’t figure out what this post is about. Maths? Numerology? Or music?
Well in any case you mean the twelfth root of 2, not the square root of minus 12 (which would be imaginary). The tones of modern western music are 12 equally spaced tones within a given octave. Also, it’s not a pre-Bach post-Bach thing. Our modern equal temp thing didn’t really come about until the early 20th century when we could measure tone frequencies accurately enough to implement equal temperament. Bach’s “Well Tempered Clavier” was actually meant to promote a different system of tuning. Here’s something I found on the subject.
I think you’re somewhat confused. The square root of -12 is I times the square root of 12 (j times square root of 12 for you EEs), a purely imaginary number, and not good as the ratio of realtones.
I think you’re referring to Equal Temperament, where the relevant ratio is the twelfth root of 2, or 2[sup]1/12[/sup]
You know what? Lemons and lemonade, and all’s fair in love, war, and numerological silliness when music is concerned. I’ve asked mods to change to original-intent (heh) hed. But anyone who can come with anything neat to say on stuff you can do with positive, negative, 2, 12, and square roots will I assure you be looked on as a discoverer of something profoundly meaningful in music, by some.
Many temperaments have been used in music. Perhaps the oldest is the Pythagorean which gives perfect fourths and fifths, but does not allow you to transpose keys and keep the same ratios between the notes. The modern equal temperament ought to allow arbitrary transposition (and to my ear does), although some claim otherwise. Every major key ought to sound like every other major key and similarly for minor. I suspect that people who feel otherwise have been trained by the kinds of music heard in those keys. But the equal tempered scale does not allow for perfect fourths and fifths. What you want is the second overtone of one note to be exactly the third overtone of the note a fifth (or is it a fourth–I am too lazy to work this out) below it. That is what the Pythagorean scale does. I am sure there will be a Wiki article explaining all this. And it is the even tempered scale that uses the 12th root of 2 between half tones. So all ratios except the octave are irrational and you cannot have perfect fourths and fifths.
This is a key point that is often ignored. Pythagorean temperament (improved after Pythagoras by Archytas and others) was “perfectly” harmonic in a way Stevin’s equal temperament is not. (31-tone instruments have been built to allow greater harmony than the 12-tone scale.) The 31st root of 2 is 1.0226. The 10th, 13th and 18th powers of that number are respectively approx. 5/4, 4/3, 3/2. With one exception these are closer approximations than those of Stevin’s scale. (Stevin’s fifth is more perfect; the 31-tone version is good enough.)
A question I’ve asked in this forum before, without a clear answer, is: How often are instruments in concerts tuned for Pythagorean harmony, rather than equal-temperament?
I like this page about why the twelve-tone scale happens to be particularly suitable for equal temperament. It discusses other candidates including the 31-tone scale:
[ul][li]The 31-tone equal-tempered scale has all seven basic intervals to a good approximation, some with better accuracy than the twelve-tone scale, but the most important fifth (3/2) interval is less accurate than in the twelve-tone scale (218/31=1.495).[/li][li]The 41-tone equal-tempered scale is the first with a better fifth (3/2) interval than the twelve-tone scale (224/41=1.5004).[/li][*]The 53-tone equal-tempered scale has all seven basic intervals with a better accuracy than the twelve-tone scale (the fifth is 231/53=1.49994).[/ul]
There’s also the 19-tone equal-tempered scale, that that page seems to dismiss a little:
That’s not really true. The perfect fourth and fifth are not more nearly pure, but over all the basic consonant intervals he’s considering, the maximum error of the 19-tone scale is half that of the maximum error for the 12-tone scale.