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Why 12 notes in Western music?

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Re: Why 12 notes in Western music?

#41

Post by wizofaus »

joegahona wrote: post #37
thirteenfingers wrote: post #6

Earlier quoted context omitted.

IIRC the system Bach was pushing wasn't actually equal temperament, but "well temperament" which was some sort of compromise between equal temperament and having pure fifths everywhere. The result was that all twelve keys sounded acceptable, but some keys had purer fifths or thirds than others. Some musicians/scholars say that Bach composed the different preludes and fugues specifically to use the resulting different…

This is my understanding too. I have a Kurzweil K2500 keyboard from around 1999 that has a bunch of the alternate tunings, including three from that era. The Bach-era tunings weren't what we know of as "equal temperament." Truly equal temperament didn't come around until well after Beethoven was dead. I've always interpreted the "Well-Tempered" in Bach's title to mean that he was brining out the strength of each key.…

> 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.

Re: Why 12 notes in Western music?

#42

Post by lynndotpy »

SeanLuke wrote: post #4

We've seen this before: and it's likely wrong. He ended his experiment too soon at 24 divisions, but even a little googling should have told him to go to 31, which is more accurate than 12. The 12-note scale long predates the notion of just or equal temperament.

Even better, if we abandon the idea of an octave base of 2, we can get other scales. What divisions lie between, say, powers of 3 or 5?

(You can find these naively by brute force!)

Re: Why 12 notes in Western music?

#43

Post by srcreigh »

There was 12 notes in Western music before 12 equal divisions.

This Mozart piece played in mean-tone temperament (historically accurate) has better tensions and resolutions than the equal tempered version.

https://www.youtube.com/watch?v=lzsEdK48CDY&t=700s

(The chipper ending to this piece is believed to be not written by Mozart.. the song was incomplete when he died.)

I even prefer Chopin in unequal temperament, but I'm not as confident about whether Chopin used 12 equal divisions.

https://www.youtube.com/watch?v=fJT5Q6HooyA

Re: Why 12 notes in Western music?

#44

Post by TakeBlaster16 »

TheOtherHobbes wrote: post #7

12 notes tuned in equal temperament is a workable compromise between musical expressiveness, harmonic ratio accuracy, readability, and finger precision. It's also an established standard, which is a huge deal because it means you have access to a huge established repertoire. A 31-TET acoustic piano would be huge, extremely complicated, and probably unplayable. Smaller instruments mostly just aren't practical. In theo…

I think 19-TET is the most viable alternative to 12-TET. It fits in a standard piano form factor by adding a few black keys, and uses the same note names everyone is used to. https://commons.wikimedia.org/wiki/File:19_equal_temperament...

And it sounds really unique! https://www.youtube.com/watch?v=bJfTu1Y2H44

Re: Why 12 notes in Western music?

#45

Post by not2b »

sabotista wrote: post #5
MatthiasWandel wrote: post #2

There was 12 notes before there was even division. It was Bach that pushed equal temprament (equal spacing). Before that, the ratios were actual ratios (perfect 4ths and 5ths), though you couldn't just transpose music and expect to sound good.

Pythagoras is believed to have come up with the just intonation (exact rational) figures. At the time, irrational numbers were distrusted and despised so, as you noted, the perfect fifth really was exactly 3:2. But it’s likely that a 12-tone system won out because lg(3/2) is so close to 7/12, even if this was never a conscious decision. 19, 31, and 53 are also credible candidates per continued fraction expansion, but…

Pythagoras and his followers at first thought that irrational numbers didn't even exist, though the story that they drowned a guy for proving by contradiction that sqrt(2) is irrational is probably not right. Rather, strings with length ratios made of small integers, like 2/3 or 3/4, sound good (harmonize) when played together. So the started with the ratios, because that's what made sense. Not to use ratios was considered, well, irrational. :-)

Re: Why 12 notes in Western music?

#46

Post by gnulinux »

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 intervals like this because it's required to process spoken human language). Traditionally, it's considered anything more than a synctonic comma (i.e. 21.51 cents) will feel different to even untrained ears. (but of course, this is just the theory, in reality there is some small variance between humans, background, culture etc).

https://en.wikipedia.org/wiki/Pythagorean_comma

This is pretty significant to mention, because even though 12 fifths are "very close" to an octave [1], they're far apart enough that no one will feel an octave. In music, near misses like this are very significant since they cause the feeling of harmonic "dissonance". Since 12 fifths is a very dissonant interval (since it's so close to an octave but still noticeably out-of-tune) Western music developed techniques (such as well-temperament, equal temperament etc) to make sure this "error" is blend in. We achieve this by changing other notes ("tempering") ever so slightly so that critical intervals like fifths (or in other cases thirds etc) are stable. Other cultures, such as classical Indian music, have their own way dealing with Pythagorean comma! Since music is a universal phenomena found in all cultures, but it doesn't manifest the same way in all cultures (e.g. not all cultures give the same kind of emphasis to pitch or harmony Western music gives) various cultures developed their own different and interesting ways to work around this "error".

[1] To be precise, we're referring to the difference between 12 fifths and 7 octaves. Since an octave is so consonant, sounds N octave(s) apart feel "equal" albeit with different timbre.

Re: Why 12 notes in Western music?

#47

Post by dr_dshiv »

I suspect it has something to do with resonance. Resonance occurs when frequencies match approximately. It isn’t just in the fundamental or pitch frequency of two notes— resonance can also occur via frequency matching in the shared overtones of two notes.

Consonant notes tend to share a lot of overtones. I have heard that the pentatonic scale maximizes internote resonance. This seems relatively straightforward to test empirically.

The first known scientific experiment (empirical test of mathematical model) was the attested case of the pythagoreans casting bronze chimes in the same rational proportions of lengths of a string. The experiment demonstrated that small integer ratios produce consonance.

Here is the most recent and up-to-date theory of harmony in music (that I know): https://downloads.spj.sciencemag.org/research/2019/2369041.p...

Re: Why 12 notes in Western music?

#48

Post by mandmandam »

retrac wrote: post #39
lioeters wrote: post #13

Earlier quoted context omitted.

> The duodecimal system, which is the use of 12 as a division factor for many ancient and medieval weights and measures, including hours, probably originates from Mesopotamia. https://en.wikipedia.org/wiki/Duodecimal#Advocacy_and_%22doz...

Yep. 60 seconds, 60 minutes, 12 hours. To the Sumerian mind that was apparently as nice and round as 100 seconds, 100 minutes, 20 hours. 60 is the smallest composite number with three prime factors, and divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30. Decimal only divides by 2 and 5. Makes arithmetic by hand a lot easier. Duodecimal has a similar advantage.

> 60 is the smallest composite number with three prime factors

Er, that'd be 30 I think, but the main point stands.

Re: Why 12 notes in Western music?

#49

Post by thomasqbrady »

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 4ths and 5ths aren’t so special…

Re: Why 12 notes in Western music?

#50

Post by herghost »

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…

Thank you! After reading the article I was left unsatisfied, but couldn't put my finger on it. I was somewhere around thinking that we hadn't yet established why 4th and 5th intervals were particularly special and so I couldn't see why the conclusion worked.

You nailed it.

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