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Why 12 notes in Western music?

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Re: Why 12 notes in Western music?

#71

Post by gnulinux »

CobrastanJorji wrote: post #65
jacquesm wrote: post #61

Earlier quoted context omitted.

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

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.

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 3f is a perfect fifth (2:3 ratio). So, when you actually hear a sound, you actually hear an octave and a fifth too, and how dominant this octave and fifth changes the "timbre" of the sound. This way, you know the source of the sound independent of the frequency. For example, both a violin and a piano can produce the note A4 at 440Hz, but anyone can easily determine if it's a piano or violin. The reason is, when a piano produces A4, it sounds not only just 440Hz but also 880Hz and 1320Hz etc... too and the relative volume of 880Hz and 1320Hz will be different than that of violin. Your brain automatically interprets these volume weights as "timbre" and the fundamental frequency 440Hz as "pitch".

Consequently, in order for your brain to be able to process the timbre of a sound it needs to find octaves and fifths between each fundamental note it hears. This means there might be something universal about octave and fifth (and other decreasingly consonant intervals such as major third etc...). Maybe we "understand" music because our brain is hard-wired to search for octaves and fifths in all sounds, in order to analyze timbre and in order to process spoken language. If this hypothesis is true, maybe an alien species could have octave/fifth/major third based music too! (if they have music at all, of course)

Re: Why 12 notes in Western music?

#72

Post by wizofaus »

joegahona wrote: post #55
wizofaus wrote: post #41

Earlier quoted context omitted.

> Truly equal temperament didn't come around until well after Beethoven was dead Source for that? The concept and practice certainly existed well before Beethoven's time but it's less clear at which point it became the norm. Even the wikipedia article on 12 TET has "citation needed" for the claim that it happened in the early 19th century.

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.

Re: Why 12 notes in Western music?

#73

Post by jefftk »

smlacy wrote: post #27

Another 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…

You're both describing two similar consequences of the same mathematical fact: 2^(7/12) is close to 3/2:

* The reason that in their 12-note graph the red line very nearly overlaps with the seventh green line is that (2^(1/12))^7 is very close to 3/2.

* The reason that twelve fifths nearly make an octave -- (3/2)^12 is ~2^7 -- is that if you use 2^(7/12) to approximate 3/2 then it's (2^(7/12))^12 which is exactly 2^7.

Since you're applying the approximation twelve times instead of once, that also explains why we've gone from being off by 0.11% to 1.4%.

Re: Why 12 notes in Western music?

#74

Post by AndrewUnmuted »

smlacy wrote: post #27

Another 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…

No post body was provided.

Re: Why 12 notes in Western music?

#75

Post by jacquesm »

gnulinux wrote: post #71
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.

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)).

Re: Why 12 notes in Western music?

#76

Post by posterboy »

gnulinux wrote: post #46
smlacy wrote: post #27

Another 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…

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.

Re: Why 12 notes in Western music?

#77

Post by layer8 »

smlacy wrote: post #27

Another 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…

In other words, 1.5^12 = 129.746… ≈ 2^7.

Re: Why 12 notes in Western music?

#78

Post by tripa »

joegahona wrote: post #55
wizofaus wrote: post #41

Earlier quoted context omitted.

> Truly equal temperament didn't come around until well after Beethoven was dead Source for that? The concept and practice certainly existed well before Beethoven's time but it's less clear at which point it became the norm. Even the wikipedia article on 12 TET has "citation needed" for the claim that it happened in the early 19th century.

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?

Re: Why 12 notes in Western music?

#79

Post by inkcapmushroom »

joegahona wrote: post #38
sleepdreamy wrote: post #33

Micro-Tonal bands such as King Gizzard play around/bend the rules on the whole 12 notes in music thing.

Familiar with the band, but not examples of this. Can you recommend any tracks that demonstrate?

Their albums Flying Microtonal Banana, KG, and LW are all microtonal, so anything from those albums. Rattlesnake, Billabong Valley, Intrasport, the Hungry Wolf of Fate, etc.

Re: Why 12 notes in Western music?

#80

Post by rrrrrrrrrrrryan »

CobrastanJorji wrote: post #65
jacquesm wrote: post #61

Earlier quoted context omitted.

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

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 time, you'll find they sound quite nice together (the difference between these two is called a "fifth").

And after 12 rounds of cutting string B into thirds, and 19 rounds of cutting string A in half, you'll happen to have a string from each group that are almost identical in length and pitch.

But it won't line up exactly! They'll be about 1.4% different in length, which roughly works out to a quarter-semitone difference in pitch (i.e. 1/4th the distance from one piano key to the next).

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