So these kinds of discussions typically center around harmonic considerations. What about the melodic ones?
Why 12 notes in Western music?
121–130 of 164 posts
Re: Why 12 notes in Western music?
#122Earlier quoted context omitted.
Somewhere there is a perfect universe where 12 fifths form an octave.
That would have to be a universe in which the fundamental theorem of arithemetic was false. Otherwise, the only way to cross an interval that is an integer number of octaves is to take steps that are also octaves.
Yes, that's correct.
Interesting tidbit: the ratio of a circle to its diameter isn't necessarily always Pi, but Pi will always be Pi as long as the foundation of mathematics holds.
Re: Why 12 notes in Western music?
#123Earlier quoted context omitted.
I don't remember where I learned this, but what you said is more accurate than what I said -- the concept was definitely known well before Beethoven, but my understanding is it wasn't the standard tuning on keyboard instruments until much later, and came to its fruition with all the atonal music of the early 20th century. I think composers even went so far as to assign emotions/moods to various keys based on each key…
More generally, I'd be curious to know how they'd practically tune a keyboard to 12TET before the electronic chromatic tuner got around. Start with Pythagorean fifths then compress ever-so slightly? How'd you keep them… equal?
Oo I can take this one! I'm not a registered piano technician, but I've tuned my piano (and helped a few friends) for some 15 years. Aurally (as opposed to electronically) tuning a piano is actually pretty straightforward. I'll stop short of saying it's easy, but once you learn the method, it's very sensible and just a matter of practice.
The general idea is to achieve consistent beat rates for a given interval -- major thirds being the most useful for its relatively high beat rate compared to other equally tempered intervals -- as you play it chromatically. By that I mean play A and C#, then Bb and D, then B and D#, etc. and if the beat rate hardly changes (but does consistently climb) then you've achieved equal temperament.
Say you've got A tuned to a reference (tuning fork). Then set the A above that so there's no beating, since octaves are always perfectly 2:1 regardless of temperament (until the extremes, when you need to stretch a bit, but I digress). Then tune the C# and then tune the F. Basically it's an augmented triad, or a stack of three major thirds (including from F back up to A). Get all of them to beat by about the same amount, but the higher ones just slightly faster than the lower ones. The fact that this feat utilizes two A's is the key to pulling it off. You frame out the octave and then fill in the augmented triad.
But now you need to do the next set: Bb, D, F#, Bb. How to get here without another external reference? Well, well, well. We have our ways. The perfect fourth between A and D is an option, but be careful not to make it actually a perfect integer (no beating), as that wouldn't be equal tempered; it should beat maybe about half as fast as the nearby major thirds, IIRC.
Re: Why 12 notes in Western music?
#124Article only looks at the 4th and 5th, but I think the more interesting observation is that if you pick any small number ratio between 1:1 and 2:1 (i.e. using numbers from 1..10 and lying within a single octave), they almost all have a reasonable 12-EDO approximation.
1:1 is the unison. Not terribly interesting.
2:1 is the octave, which is exact.
3:2 is the perfect fifth. About 2 cents of error.
4:3 is the perfect fourth. Also about 2 cents of error.
5:3 is the major sixth. About 15 cents or so of error.
5:4 is the major third. About 13 cents or so of error.
6:5 is the minor third. About 15 cents or so of error.
7:4 is the the first one that doesn't really have a 12-EDO equivalent, though the minor 7th is generally used, with about 31 cents of error.
7:6 similarly doesn't have an equivalent. It's about 44 cents flat of the 12-EDO minor third. In fact, most ratios with 7s are right out.
7:5 is in the ballpark of the 12-EDO tritone.
8:5 is the minor sixth. About 13 cents of error.
9:7 is a really sharp third, again no equivalent in 12-EDO.
9:8 is the major second. About 4 cents of error.
10:9 is also the major second, about 17 cents off in the other direction. (12-EDO makes no distinction between 9:8 and 10:9. That's actually fairly important, as it lets you get away with chord progressions that don't mathematically work out.)
It's really amazing to have so many decent approximations of ratios with only 12 notes. Different EDOs might have better or worse approximation of various musical intervals. It takes going all the way up to 41-EDO to find something that's better at basically everything -- it even has a more accurate 4th and 5th, which is the one thing that 12-EDO is amazingly good at.
Re: Why 12 notes in Western music?
#125A long-winded way of saying that if you want to hit the 3:2 and 4:3 sweet spots "closely enough", dividing the octave into 12 logarithmically equidistant bins works very well, and better than any other number of bins less than 50 (or maybe 30).
Actually, 41 has a better 4th and 5th, only being off by about half a cent, as opposed to being off by about 2 cents. It also has thirds and sixths that are quite a bit better (though still not great), and it has very good 7-limit intervals, which is something 12-EDO has nothing even remotely close to.
31 is generally decent all around, but it has worse 4ths and 5ths than 12-EDO.
53 EDO is even better than 41, having 4ths and 5ths that are off by about 7 hundredths of a cent. It also has much better 3rds and 6ths than 41, but the 7-limit intervals are slightly worse.
https://en.wikipedia.org/wiki/31_equal_temperament#Interval_...
https://en.wikipedia.org/wiki/41_equal_temperament#Interval_...
https://en.wikipedia.org/wiki/53_equal_temperament#Interval_...
Re: Why 12 notes in Western music?
#126Another possible explanation, which I'm surprised the author didn't go through is the "Circle of Fifths" which basically says: Since Fifths sound so great, why not just keep doing that? When we get to the next octave, then come back down. If we get to a place that's "pretty darn close" to another note, then stop. The Python explanation looks like: f = 440 for i in range(13): print(i,f) f = f * 3/2 if f > 880: f=f/2.0…
> Note that after exactly 12 steps, we're back at 446 which is "pretty close" to 440.
If you’re not careful you’ll summon the ~elders~ people who are convinced that frequency scale is wrong and that there’s a more ideal (to human ears) frequency step for the same 12 note scale. They might even be right, but goodness… prepare yourself for it to get weirder than finding out whether someone really believes it’s legal to be barefoot in all public settings.
Re: Why 12 notes in Western music?
#127Earlier quoted context omitted.
Somewhere there is a perfect universe where 12 fifths form an octave.
I've heard there is a secret chord, that prophets play to please their lords. But if you care for music less than algebra, just know it goes like this; the fourth, the fifth: the minor falls, the major lifts.
But, you don't really care for music; do ya?
Re: Why 12 notes in Western music?
#128Check out Musimathics volume 1 if you want an absurdly detailed explanation of why Western 12 TET is the way that it is.
http://www.musimathics.com/
Re: Why 12 notes in Western music?
#129Earlier quoted context omitted.
It's worth mentioning that stacking fifths this way creates something pretty close to an octave, but it's still noticeably different from an octave. The difference between an octave and 12 fifths is called "Pythagorean comma", it's about 23.46 cents and it'll be obvious to all humans who don't have a speech/hearing impediment, even if you were never musically trained. (It's believed humans are sensitive to small inte…
Somewhere there is a perfect universe where 12 fifths form an octave.
That would be a non-Euclidian (ie. curved) universe?
Maybe it is possible to construct a non-Euclidean universe for sound, by modifying properties of the propagation medium as a function of space or time?
Re: Why 12 notes in Western music?
#13012 notes, 12 hours, 12 months., even 12 monkeys.
The numbers 12 (2x2x3), 24 (2x2x2x3), 30 (2x3x5), 60 (2x2x3x5) and 360 (2x2x2x3x3x5) crop up in a lot of older counting systems because they're so conveniently divisible.
One twelfth the atomic weight of one carbon-12 atom is the atomic mass constant.
What does string theory sound like?