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Why 12 notes in Western music?

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Re: Why 12 notes in Western music?

#81

Post by ajuc »

jacquesm wrote: post #75
gnulinux wrote: post #71

Earlier quoted context omitted.

This may or may not be true. In fact, it seems unlikely to me your claim is true. In nature, sounds produce harmonics i.e. when two objects collide they usually create waves of frequency f, 2f, 3f, 4f... in various (usually exponentially decreasing) weights. It's very rare to find pure sounds (i.e. only f frequency) in nature. The interval between f and 2f is an octave apart (1:2 ratio); the interval between 2f and 3…

There is also the shape of the individual waveforms to take into account a piano has a more or less sinusoidal wave and a violin is more of sawtooth (due to the stickslip of the bow moving across the string(s)).

You're saying the same thing - the combined sawtooth wave is just the sum of all the sinus harmonics.

Re: Why 12 notes in Western music?

#82

Post by gnulinux »

jacquesm wrote: post #75
gnulinux wrote: post #71

Earlier quoted context omitted.

This may or may not be true. In fact, it seems unlikely to me your claim is true. In nature, sounds produce harmonics i.e. when two objects collide they usually create waves of frequency f, 2f, 3f, 4f... in various (usually exponentially decreasing) weights. It's very rare to find pure sounds (i.e. only f frequency) in nature. The interval between f and 2f is an octave apart (1:2 ratio); the interval between 2f and 3…

There is also the shape of the individual waveforms to take into account a piano has a more or less sinusoidal wave and a violin is more of sawtooth (due to the stickslip of the bow moving across the string(s)).

Is it not true that the shape of the waveform (sinusoidal, saw-like etc) is created by the relative weights of each harmonic? E.g. if you take any random sound wave, Fourier-transform it, you'll find the weight of each harmonic. Or are you saying there is a separate quality to sound waves that can cause their shape to be different even if each harmonic has the same relative weight with respect to the fundamental?

Re: Why 12 notes in Western music?

#83

Post by thaumasiotes »

jacquesm wrote: post #61
gnulinux wrote: post #46

Earlier 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 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.

Re: Why 12 notes in Western music?

#84

Post by coldtea »

>You may have noticed that 24 is also includes the 5ths and 4ths, the problem is that having twice as many notes would require instruments with twice as many keys or buttons making them more expensive and complicated to play, also probably we wouldn't notice the difference between notes that are so close

We actually would, and it's quite noticable. Several cultures use microtonals intervals with half-half-steps or similar.

The main reason we stuck with 12, is complexity in making AND playing an instrument with so many notes - that, or the halved range, if we keep the number of notes on the instrument the same.

But there are cultures (and instruments) which have more.

Re: Why 12 notes in Western music?

#85

Post by joegahona »

wizofaus wrote: post #72
joegahona wrote: post #55

Earlier 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…

I still feel that way about certain keys, even playing on an exactly equal tempered keyboard. I don't think the degree to which certain intervals might vary between keys is necessarily the important factor.

Those descriptors ("austere," etc.) have always struck me as subjective -- I'm not one to tell people what mood they're getting from certain keys. But a root major chord will have a much different feel in, e.g., C#-major on an 18th-century tuning than in equal temperament.

I have this CD [1] in a box somewhere but can't find it on Youtube. It's a few Beethoven sonatas in the temperament he would've used. Just sounded out-of-tune to me in certain parts (especially during the Waldstein), but I don't have perfect pitch. The booklet that came with that CD is really helpful in understanding all this, and I think that might be where I learned about that Owen Jorgensen tome.

There's no shortage of similar experiments on Youtube. This one [2] has a wild one in just intonation, but I doubt that temperament was still used when Mozart was composing.

[1] https://www.amazon.com/Beethoven-Temperaments-Historical-Tun...

[2] https://www.youtube.com/watch?v=lzsEdK48CDY

Re: Why 12 notes in Western music?

#86

Post by coldtea »

posterboy wrote: post #76
gnulinux wrote: post #46

Earlier 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…

I'm not comfortable with this first refering to a natural human tendency and then a western harmony, which is pretty much accquired, as if it were a natural consequence. A simpler way to go about this is using the chromatic scale, drawing multiples of C0 upto C8 so that C7 to C8 spans an octave, and then fixing F according to a table of equal temperament.

>I'm not comfortable with this first refering to a natural human tendency and then a western harmony, which is pretty much accquired, as if it were a natural consequence.

Well, western harmony is based on a set of natural human tendencies formalized.

There are other ethnic music practices, also based on natural human tendencies.

The parts that are acquired are built on top. But most/all music practices (western or otherwise) start with natural human tendencies, as their foundations.

Re: Why 12 notes in Western music?

#87

Post by coldtea »

thaumasiotes wrote: post #83
jacquesm wrote: post #61

Earlier 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.

What the parent proposed doesn't require different math.

Just a species that hears out current slightly offset divisions in 12-tet as perfect, as opposed to only hearing integral ratios as perfect.

Re: Why 12 notes in Western music?

#88

Post by aidenn0 »

kazinator wrote: post #62

When we divide the octave into various equal steps using equal temperament, we find that there is a local maximum at 12, which yields a good approximations for important intervals. But the "why" cannot be explained just using arithmetic. There is a history behind it. Twelve note instruments didn't begin with equal temperament. There are twelve notes in western music because the diatonic scale has 7 notes, and alterat…

You also get the western[1] chromatic scale if you go up by a fifth (which is pleasant sounding for many reasons) ad infinitum.

C -> G -> D -> A -> E -> B -> F# -> C# -> G# -> D# -> A# -> E# -> B#(C)

Of course the B# you end up with at the end is 531441/4096 which is 1.3% higher frequency than 7 octaves above the starting C. If you want to generate flats as well, by traveling in the opposite direction, you end up with different notes for the flats. 12-TET is just the modern way of using a constant frequency ratio to divide the octave to match the 12 notes used by Pythagoras. The ancient greeks were unlikely to come up with it due to the reliance on irrational numbers.

1: Pythagoras is often credited with this scale, China also independently invented this scale and it's not clear which came first (https://en.wikipedia.org/wiki/Sh%C3%AD-%C3%A8r-l%C7%9C)

Re: Why 12 notes in Western music?

#89

Post by tshaddox »

rrrrrrrrrrrryan wrote: post #80
CobrastanJorji wrote: post #65

Earlier quoted context omitted.

You don't need a new universe. You just need a species that hears sounds a little differently. We won't like listening to their music, but it'll be really great for them.

Eh, I think you'd need a new universe, as it's a pretty basic principle of math: 3^12 ~= 2^19 You can take two long strings of equal length (A & B), and pluck them, and they'll make the same sound. Then you can take scissors cut string A in half, and it'll sound different. (This is an octave.) Then you can cut string B into thirds, and it too will sound different. If you pluck both of your new strings at the same tim…

But who says the creatures need to perceive sound in such a way that the harmonic series has sensory significance? To be honest I’ve never seen a compelling evolutionary explanation for why “hearing the harmonic series” developed in the first place. It obviously seems useful to be able to perceive sounds generated by (roughly) harmonic oscillators, since those occur naturally for various reasons, but why octave equivalence?

Re: Why 12 notes in Western music?

#90

Post by mcphage »

thomasqbrady wrote: post #49

This piece is a good example of circular reasoning, isn’t it? The question “Why are there 12 notes in Western scales?” Is answered first by presuming that 4ths and 5ths sound pleasant (to whom? a Westerner?), the “4th” and “5th” being intervals ON a Western scale, which the author then reverse-engineers back to the 12-note scale which they assumed from the start. There are other scales you could start from, in which…

That's not really circular, though. It does start from the assumption that 4ths and 5ths sound pleasant, but uses that to build possible scales, some of which are more compatible with 4ths and 5ths than others.
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