8 ms·

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

by smlacy 4y ago

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 0 440 1 660.0 2 495.0 3 742.5 4 556.875 5 835.3125 6 626.484375 7 469.86328125 8 704.794921875 9 528.59619140625 10 792.894287109375 11 594.6707153320312 12 446.00303649902344 Note that after exactly 12 steps, we're back at 446 which is "pretty close" to 440. So, we take this set of notes, sort them, and just jigger it a little bit to get the 12 notes we know today.

ledauphin 4y ago
are the post and your comment not mathematically equivalent statements?
- smlacy 4y ago
  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.
  - dwringer 4y ago
    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.
    - posterboy 4y ago
      I think it's integral to the theorie seeing the fifth as 3 times, the octave as 4 times and the prime as 2 times an arbitrarily low root key. Trivially, the 2^n multiples form octave intervals but between C12 and C24 there's your twelve tones on a logarithmic scale. Naturally, it doesn't transpose in integer intervals if stepping down.
- elihu 4y ago
  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.)
gnulinux 4y ago
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 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.
- jacquesm 4y ago
  Somewhere there is a perfect universe where 12 fifths form an octave.
  - CobrastanJorji 4y ago
    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.
tetris11 4y ago
(For those wanting to hear these frequencies:) aplay -d 2 -r $freq
xchip 4y ago
Eh eh as other people say, you have discovered the Pythagorean scale, that is known to drift away slowly from the correct frequencies, that's is why for a long time people didn't use chords that overlapped octaves, because they sounded weird and they called those evil chords
jefftk 4y ago
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%.
AndrewUnmuted 4y ago
layer8 4y ago
In other words, 1.5^12 = 129.746… ≈ 2^7.
frereubu 4y ago
Fifths don't sound so great after a while. This tuning leads to the dissonant "wolf interval" - https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Well_temperament - used by Bach in The Well-Tempered Klavier.
eyelidlessness 4y ago
> Note that after exactly 12 steps, we're back at 446 which is "pretty close" to 440. If you’re not careful you’ll summon the ~elders~ people who are convinced that frequency scale is wrong and that there’s a more ideal (to human ears) frequency step for the same 12 note scale. They might even be right, but goodness… prepare yourself for it to get weirder than finding out whether someone really believes it’s legal to be barefoot in all public settings.
drBonkers 4y ago
I love math explanations in Python, thank you.
anon291 4y ago
This is the reason. You can keep going but it's just pointless. Things sound good enough at this point

by smlacy 4y ago

ledauphin 4y ago

are the post and your comment not mathematically equivalent statements?

smlacy 4y ago
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.
- dwringer 4y ago
  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.
  - posterboy 4y ago
    I think it's integral to the theorie seeing the fifth as 3 times, the octave as 4 times and the prime as 2 times an arbitrarily low root key. Trivially, the 2^n multiples form octave intervals but between C12 and C24 there's your twelve tones on a logarithmic scale. Naturally, it doesn't transpose in integer intervals if stepping down.
elihu 4y ago
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.)

gnulinux 4y ago

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 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.

jacquesm 4y ago
Somewhere there is a perfect universe where 12 fifths form an octave.
- CobrastanJorji 4y ago
  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.

tetris11 4y ago

(For those wanting to hear these frequencies:) aplay -d 2 -r $freq

xchip 4y ago

Eh eh as other people say, you have discovered the Pythagorean scale, that is known to drift away slowly from the correct frequencies, that's is why for a long time people didn't use chords that overlapped octaves, because they sounded weird and they called those evil chords

jefftk 4y ago

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%.

AndrewUnmuted 4y ago

layer8 4y ago

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

frereubu 4y ago

Fifths don't sound so great after a while. This tuning leads to the dissonant "wolf interval" - https://en.wikipedia.org/wiki/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 https://en.wikipedia.org/wiki/Well_temperament - used by Bach in The Well-Tempered Klavier.

eyelidlessness 4y ago

> Note that after exactly 12 steps, we're back at 446 which is "pretty close" to 440. If you’re not careful you’ll summon the ~elders~ people who are convinced that frequency scale is wrong and that there’s a more ideal (to human ears) frequency step for the same 12 note scale. They might even be right, but goodness… prepare yourself for it to get weirder than finding out whether someone really believes it’s legal to be barefoot in all public settings.

drBonkers 4y ago

I love math explanations in Python, thank you.

anon291 4y ago

This is the reason. You can keep going but it's just pointless. Things sound good enough at this point