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Why 12 notes in Western music?

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Re: Why 12 notes in Western music?

#101

Post by inopinatus »

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.

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.

Re: Why 12 notes in Western music?

#102

Post by abstrakraft »

gnulinux wrote: post #82
jacquesm wrote: post #75

Earlier quoted context omitted.

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?

Yes, this quantity is the relative phase of the harmonics, although the human ear is generally considered to be insensitive to phase.

Re: Why 12 notes in Western music?

#103

Post by aidenn0 »

kazinator wrote: post #93
aidenn0 wrote: post #88

Earlier quoted context omitted.

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 wit…

That's just from modulo math. A fifth is 7 semitones, which is relatively prime to 12. Thus 7x (mod 12) hits all the elements of the modulo 12 congruence for x in 0..11. We cover all notes in the first twelve steps. But say we are not assuming a twelve note system in the first place; how do we get twelve notes? Going up a fifth and then down a fourth is very close to a tone. We can do that five times before we approx…

> That's just from modulo math. A fifth is 7 semitones, which is relatively prime to 12. Thus 7x (mod 12) hits all the elements of the modulo 12 congruence for x in 0..11. We cover all notes in the first twelve steps.

> But say we are not assuming a twelve note system in the first place; how do we get twelve notes?

My diagram showed an (approximate) 12 note cycle assuming only a 3:2 ratio for a fifth. There are lots of good reasons to use a fifth as the basic interval[1]. In no way does this assume a 12-note system.

The 12 notes don't come from "filling in" between the 7 notes of the diatonic major scale, they come from continuing the pattern until a near-cycle happens; is your argument that the 1.2% error in the cycle is arbitrary? it's less than 1/4 the next largest difference and slightly more than the rule of thumb for how much "anybody" can hear. The next time we get closer to a cycle is at 41, and we don't get closer by an order of magnitude until 53.

1: And in fact the fifth is used as a basis for many other scales both western and otherwise (Note that the first 5 notes are the major Pentatonic scale and the first 7 are the major diatonic scale).

Re: Why 12 notes in Western music?

#104

Post by rrrrrrrrrrrryan »

tshaddox wrote: post #89
rrrrrrrrrrrryan wrote: post #80

Earlier quoted context omitted.

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 equiv…

I don't think you need to imagine "creatures" - there are humans on earth whose culture's music doesn't have the concept of octaves, let alone fifths. All the real action in their music is in its rhythmic complexity.

But my comment was in reply to this statement specifically:

> Somewhere there is a perfect universe where 12 fifths form an octave.

For this to be true, I think you'd indeed need a new universe where 3^12 = 2^x, where X is a whole integer.

Re: Why 12 notes in Western music?

#105

Post by mcbrit »

Let's talk about splitting things up in useful ways.

12=2 * 2 * 3.

Splitting something in half is useful; splitting it half again remains useful. Splitting in half a third time is arguably less useful than splitting it into a third. So 12 is the made-to-order number that lets you split it in half, twice, and in thirds, once.

Which naturally leads to seconds and minutes, or 60:

60=2 * 2 * 3 * 5

Because dividing the whole into fifths is more useful than a second 3, or a third 2.

So, there's your basic argument for why you would see a 12 or a 60 instead of a 10 or some other number. You have a whole that you want to divide into useful parts.

I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis; we like hearing useful parts would be the somewhat surprising thing to talk about.

Re: Why 12 notes in Western music?

#106

Post by tshaddox »

rrrrrrrrrrrryan wrote: post #104
tshaddox wrote: post #89

Earlier quoted context omitted.

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 equiv…

I don't think you need to imagine "creatures" - there are humans on earth whose culture's music doesn't have the concept of octaves, let alone fifths. All the real action in their music is in its rhythmic complexity. But my comment was in reply to this statement specifically: > Somewhere there is a perfect universe where 12 fifths form an octave. For this to be true, I think you'd indeed need a new universe where 3^1…

I'm not talking about explicit notions of octaves and octave equivalence, although those do exist in very many musical traditions and appear to be extremely widespread in musical traditions where tones have names (if not ubiquitous—I'm not aware of any exceptions). I was referring to the claims that octave equivalence is in some sense hard-wired in the human brain, or sometimes claimed to be all or many mammalian brains. I'm not qualified to evaluate these claims or even whether how well-accepted they are among experts, but such claims do seem to pop up all over the place when discussing music perception.

Re: Why 12 notes in Western music?

#107

Post by thomasqbrady »

diffeomorphism wrote: post #57
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…

No, not at all. > 4th and 5th being intervals ON a Western scale. That is where you went wrong. They are not. They naturally arise as small integer frequency multiples/fractions on any string instrument (wave lengths 1, 1/2, 1/3,...). They are hence quite obvious/loud (and humans recognize patterns as pleasant for whatever reason). Once you have 4ths and 5ths you repeat to get the Western system (handwaving away that…

I guess I wasn’t clear. I’m not saying the 4th and 5th notes don’t have a special sound to anyone. I’m saying that 1) just exactly how important their resonance is to you is influenced by your culture. It’s not that atonal musicians didn’t notice the resonance. They weren’t drawn to it as much. 2) The logic given was “if you want your scale to include the 4th and 5th, then 12 notes is inevitable.” The “if you want your scale to include the 4th and 5th” is the a built in assumption that you want something like the western scale, so it doesn’t seem that impressive to me that they then arrive at the 12-tone scale.

Re: Why 12 notes in Western music?

#108

Post by nikeee »

mcbrit wrote: post #105

Let's talk about splitting things up in useful ways. 12=2 * 2 * 3. Splitting something in half is useful; splitting it half again remains useful. Splitting in half a third time is arguably less useful than splitting it into a third. So 12 is the made-to-order number that lets you split it in half, twice, and in thirds, once. Which naturally leads to seconds and minutes, or 60: 60=2 * 2 * 3 * 5 Because dividing the wh…

These numbers are called "Highly Composite Numbers" [0]. Basically, it is a series of numbers where each number has more factors than the number before it (and is the first number with that number of factors). As you hinted, they are especially useful if your number system does not have fractions or decimal places and you still want to divide things.

You may recognize the beginning of the series: 1 2 4 6 12 24 36 48 60 120 180 240 360 720

Numberphile [1] calls them "Anti-Primes".

[0]: https://en.wikipedia.org/wiki/Highly_composite_number [1]: https://www.youtube.com/watch?v=2JM2oImb9Qg

Re: Why 12 notes in Western music?

#109

Post by wizofaus »

joegahona wrote: post #85
wizofaus wrote: post #72

Earlier quoted context omitted.

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…

Thanks for that link, I don't know if it demonstrates "just intonation" though? But the 1/4 Comma Meantone tuning just sounds horrible the moment a diminished chord comes into the picture.

Re: Why 12 notes in Western music?

#110

Post by analog31 »

An alternative question is: Why not more? There are approximately rational scales with more than 12 notes. Something I wonder is how the complexity of music relates to its use. For instance, instruments for music that's primarily ceremonial, or used in centralized locations by trained experts, could adopt scales with more notes, or more difficult tunings. This includes 12TET, which was difficult for an untrained musician to replicate, and unlikely to stay in tune for an entire performance on some instruments.

Simpler tunings might lend themselves to instruments that were homemade, used for folk music, carried by travelers, played at home, etc. In fact those two things could coexist within a single culture. There were pipe organs and folk fiddles in Europe during the same time period, after all. Once the 4 strings are tuned by means of an easily discerned interval, you can fill in with a tolerable scale by ear.

"Carried by travelers" suggests an advantage for a tuning system that can be restored by a non-expert and used for music that spreads from town to town.

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