It's the same reason 12 was used as a number base so often, it divides an doubling of frequency (misnamed an octave) evenly into 1,2,3,4,6 and 12 parts (on a logarithmic scale), which then have pleasant overtones.
Why 12 notes in Western music?
91–100 of 164 posts
Re: Why 12 notes in Western music?
#92Earlier quoted context omitted.
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.
2 is such a nifty prime, we multiplied by it twice. It was just too perfect not to.
Re: Why 12 notes in Western music?
#93When 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 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 approximately hit an octave, yielding six notes. The fifths above those notes are six additional notes.
We see that in your diagram:
C -> G -> D -> A -> E -> B -> F# -> C# -> G# -> D# -> A# -> E# -> B#(C)
in that we can interpret every other note as the whole tone scale:
C -> D -> E -> F# -> G# -> A# -> B#(C)
and their fifths:
G -> A -> B -> C# -> D# -> E# -> Fx(G)
Fifths fill the gaps in the whole tone scale to recover the other whole tone scale.
Going back to the 12 tone math again, 2 and 12 have a common divisor, so steps of 2 modulo 12 cycle through 6 symbols. There are 6 others left out, reachable by some relatively prime step like 7 (perfect fifth).
Re: Why 12 notes in Western music?
#94Earlier quoted context omitted.
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.
Viewing sound waves that don't synchronize with each other as being better matched than sound waves that do synchronize is less plausible than violating the fundamental theorem of arithmetic. There's no element of coincidence in whether two frequencies harmonize.
That hypothetical species wouldn't recognize two notes an octave apart as being similar, so there would be no reason to imagine a circle of fifths in the first place.
Re: Why 12 notes in Western music?
#95Another 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…
Fifths don't sound so great after a while. This tuning leads to the dissonant "wolf interval" - https://en.wikipedia.org/wiki/Wolf_interval - so it was largely replaced by the well temperament - https://en.wikipedia.org/wiki/Well_temperament - used by Bach in The Well-Tempered Klavier.
Re: Why 12 notes in Western music?
#96We'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.
For the intervals they look at in the article, the perfect 4th and 5th, 31-EDO is worse -- about 5 cents of error, versus about 2. What 31-EDO has is a major third that's almost dead-on, and a minor third that's a lot closer.
https://en.wikipedia.org/wiki/31_equal_temperament#Interval_...
41-EDO though has a perfect 4th and 5th that are closer than 12-EDO, being off by about half a cent rather than about 2 cents. In fact, 41-EDO is better at every commonly-used interval than 12-EDO, plus it adds a lot of very good 7-limit intervals too (i.e. ratios with sevens in them like 7:4, which is way off in 12-EDO).
By a weird set of mathematical coincidences, 41-EDO is actually quite playable on guitar with the right layout. The trick is to omit half the frets and tune the strings so that each string has the notes that the strings above and below it lack. Tuning by major 3rds, you get a whole lot of useful notes clustered where they're easy to play. There's a handful of us (in Portland mostly) trying to promote this idea: https://kiteguitar.com/
Re: Why 12 notes in Western music?
#97Earlier quoted context omitted.
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.
Viewing sound waves that don't synchronize with each other as being better matched than sound waves that do synchronize is less plausible than violating the fundamental theorem of arithmetic. There's no element of coincidence in whether two frequencies harmonize. That hypothetical species wouldn't recognize two notes an octave apart as being similar, so there would be no reason to imagine a circle of fifths in the fi…
>Viewing sound waves that don't synchronize with each other as being better matched than sound waves that do synchronize is less plausible than violating the fundamental theorem of arithmetic
Actually it's perfectly plausible. People couldn't imagine others enjoying hearing a tritone -- and nobody in 1800 would imagine we'd enjoy listening to punk, hip hop, or Death Metal, and yet, millions do. We can surely consider a race that doesn't require intervals to absolutely synchronize.
>That hypothetical species wouldn't recognize two notes an octave apart as being similar, so there would be no reason to imagine a circle of fifths in the first place.
Note how I said that this imagined alien race would consider the "divisions in 12-tet as perfect". Note how those do include a perfect octave, that we already recognize as such. The alien race wouldn't change that, they'd just need to also consider perfect the slightly off ratios in 12-tet.
Re: Why 12 notes in Western music?
#98Another 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…
are the post and your comment not mathematically equivalent statements?
No, the article is essentially pointing out that the 12th root of two to the seventh power is really close to 1.5, whereas the comment you're replying to is saying that if you raise 1.5 to the 12th power, you get really close to a power of 2.
The former is more interesting when it comes to how music works psychoacoustically: the interval of a perfect fifth is fundamental to almost all music. Whereas the "circle of fifths" is more of a convenience that makes it easier to think about keys. Few songs would ever traverse the whole circle and come back to where it started, and if you stick with strict just intonation there is no circle of fifths anyways. (Maybe you could better call it a "spiral of fifths" or something.)
Re: Why 12 notes in Western music?
#99Earlier quoted context omitted.
Viewing sound waves that don't synchronize with each other as being better matched than sound waves that do synchronize is less plausible than violating the fundamental theorem of arithmetic. There's no element of coincidence in whether two frequencies harmonize. That hypothetical species wouldn't recognize two notes an octave apart as being similar, so there would be no reason to imagine a circle of fifths in the fi…
> Viewing sound waves that don't synchronize with each other as being better matched than sound waves that do synchronize is less plausible than violating the fundamental theorem of arithmetic Actually it's perfectly plausible. People couldn't imagine others enjoying hearing a tritone -- and nobody in 1800 would imagine we'd enjoy listening to punk, hip hop, or Death Metal, and yet, millions do. We can surely conside…
> Actually it's perfectly plausible.
No, for this to be plausible, you would need to have some theory of why the two notes matched with each other. There is no such theory; they have been chosen to be as unmatched as possible.
> nobody in 1800 would imagine we'd enjoy listening to punk, hip hop, or Death Metal
This is false.
>> That hypothetical species wouldn't recognize two notes an octave apart as being similar
> Note how I said that this imagined alien race would consider the "divisions in 12-tet as perfect". Note how those do include a perfect octave, that we already recognize as such. The alien race wouldn't change that
Sure. In that case, we can also imagine an alien race that perceives all and only the light that fails to reach its eyes.
Then again, perhaps being able to form a sentence describing something doesn't guarantee that the situation described is possible.
Re: Why 12 notes in Western music?
#10012 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…
41-EDO is surprisingly playable on guitar, if you omit half the frets. The trick is to tune the strings so that each string only has half the notes, but the notes that aren't there are available on neighboring strings. It seems like it shouldn't work, but it does.
https://kiteguitar.com/theory/fretboard-charts-downmajor-tun...