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Why 12 notes in Western music?
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Re: Why 12 notes in Western music?
#5212 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…
Not a musician, but as a hobbyist composer and music theory enthusiast, I think anything more than 24-TET is overkill in terms of daily practice. I don't think there is sufficient expression to adding anything more than quartertone to justify making your theory and practice so much more complicated. 24-TET is convenient because all your 12-TET theory works exactly the same, except now you have quartertone, in addition to semitone. This gives really interesting intervals, although not all of them will be usable in practice. In terms of instrument practice, 19-TET is a good middle ground since it maps unambiguously to 12-TET while introducing useful and expressive intervals such as septimal minor third.
Re: Why 12 notes in Western music?
#53Earlier quoted context omitted.
are the post and your comment not mathematically equivalent statements?
Its the use of the musical concept of Fifths that's the key here: We just derive the notes from what sounds good, not what makes sense mathematically? I'm just using Python to mirror the author's analysis -- you can derive "12 notes" pretty much just by using your ear and listening to the Fifths.
A perfect fifth is just 150% (3/2) the frequency of the root, just as an octave is 200% (2/1). "What sounds good" is in a certain sense based on the harmonic series, and in that sense it is equivalent to what makes sense mathematically.
A problem is that if you stack perfect fifths to get 12 notes then they won't sound in tune with each other across different keys. It's this issue which forms the crux of the linked post.
Re: Why 12 notes in Western music?
#54This 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…
The why is because the 5th is the primary non-octave overtone of a vibrating string, and the 4th is an inverted 5th.
Re: Why 12 notes in Western music?
#55Earlier quoted context omitted.
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.
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's sound. E.g. "E-flat major is austere, D-minor is sad," etc.
Might be worth reading:
- https://books.google.com/books/about/The_Effects_of_Unequal_...
- https://www.proquest.com/openview/b22142f819768ae82464aa2679...
Re: Why 12 notes in Western music?
#56Another 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…
(For those wanting to hear these frequencies:)
aplay -d 2 -r $freq
Re: Why 12 notes in Western music?
#57This 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 this does not actually close, but "rounds" to make it fit into twelve. That is a whole other subject).
This is a pretty good example of inductive reasoning. We want a system that for any note also includes its first few harmonics, show that this implies....
Re: Why 12 notes in Western music?
#58This 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, 4ths and 5ths are resonant. They sound better to a lot of people. They are completely distinguishable from arbitrary intervals right near there.
Most of the notes on the Western scale fall into this orientation, i.e. there is reasoning for it. It's not just arbitrary.
Re: Why 12 notes in Western music?
#59Some comments on tones. Pythagorean tuning is based on repeated ³⁄₂ increases in frequency with occasional halving to stay in the octave, so we have, e.g.,
A = 1
E = 3/2
B = 9/8 (here we halved to get back into our 1–2 range)
F# = 27/16
C# = 81/64 (another halving)
etc.
Another approach is to use harmonic overtones. When a string (or a column of air) vibrates, it vibrates not just at its fundamental, but in a series of integer divisions of the string.¹
Fundamental: A
Octave (½): A' (up an octave)
Twelfth (⅓): E (up a fifth from the octave)
Double Octave (¼): A''
⅕: C#²
⅙: E
⅟₇: G (but a bit flat from most tunings).
We invert wavelength to get frequencies and halve to get into the 1–2 range and our E matches up at 3/2 and C# comes out as 5/4 which is pretty close to the 81/64 of Pythagorean
I would also note that 24-tone music does occur with some moderate frequency in avant-garde music where half-flats and half-sharps have their own notation, although these notes are not easily accessible from many standard instruments, but the sound of a quarter-tone difference in pitch is definitely distinct. Many non-Western musics apply various micro-tonalities, such as Indonesian scales which are closest to a subset of a 9-tone equal temperament.
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1. In some cases, e.g., overtones of a cylindrical pipe vs conical pipe, or open at both ends vs open at one end, you won’t get all of these tones, so a flute, which is cylindrical and open at both ends can hit the fundamental and the octave, while a clarinet, which is cylindrical but closed at one ends hits the fundamental and then the third partial (the twelfth) but not the octave.
2. The place where you hear notes produced to these pitches most commonly is in bugle calls: Taps, for example, would be ⅓ ⅓ ¼, ⅓ ¼ ⅕, etc.
Re: Why 12 notes in Western music?
#60This 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.
> I was somewhere around thinking that we hadn't yet established why 4th and 5th intervals were particularly special
Other responses to the GP have explained that. The article itself also mentions the reason (small integer ratios).