15 ms·

Why 12 notes in Western music?

253 points by xchip 4y ago

MatthiasWandel 4y ago
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.
- sabotista 4y ago
  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 unwieldy for physical instruments (although some computer music does use 53-TET).
  - not2b 4y ago
    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. :-)
    - dhosek 4y ago
      It may not be true, but I still loved telling that story to my students.
- thirteenfingers 4y ago
  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 characters of the different keys to the best possible advantage. I can't speak to this personally, I keep my piano at equal temperament ;) (Big fan of your videos btw)
  - joegahona 4y ago
    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. Some of those keys sound really "out of tune" to modern ears -- I have a friend with perfect pitch who legit can't listen to them. Owen Jorgensen's "Tuning the Historical Temperaments by Ear" is good bedtime reading on this topic, if you have a couple hundred bucks burning a hole in your pocket. https://www.amazon.com/Tuning-historical-temperaments-ear-eighty-nine/dp/091861600X https://www.amazon.com/Tuning-historical-temperaments-ear-ei...
    - wizofaus 4y ago
      > 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.
      - joegahona 4y ago
        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_Temperament_on_Ch.html?id=O6b2tgAACAAJ https://books.google.com/books/about/The_Effects_of_Unequal_... - https://www.proquest.com/openview/b22142f819768ae82464aa2679f4ef68/1?pq-origsite=gscholar&cbl=18750&diss=y https://www.proquest.com/openview/b22142f819768ae82464aa2679...
- dahart 4y ago
  > it was Bach that pushed equal temperament (equal spacing). Not really. Equal temperament was being advocated both in China and Europe long before Bach was born. In Europe, it was the lute players that pushed for it, because it matters more for fretted instruments, where any temperament other than equal causes conflicts and inconsistencies on the neck. https://en.m.wikipedia.org/wiki/12_equal_temperament https://en.m.wikipedia.org/wiki/12_equal_temperament
locusofself 4y ago
I'm not a music theory expert, just a guy who has played guitar by ear for 28 years. Equal temperament is definitely a compromise .. I find myself constantly trying to "sweeten" the tuning of my guitar strings relative to the song/key I am playing. You can tune your guitar with the most accurate guitar tuner in the world (I have strobe tuners by Sonic Research and Peterson), and some things still just sound out of tune.
- criddell 4y ago
  There are guitars with frets that aren't straight lines that try to address some of the problems: https://guitargearfinder.com/faq/true-temperament-frets/ https://guitargearfinder.com/faq/true-temperament-frets/
  - locusofself 4y ago
    I've seen those! Pretty cool. I've also seen people put intermediate frets on certain notes.
  - khitchdee 4y ago
    FWIW, A sitar also has a 12 note scale, but the frets are moveable so you can tune by ear. The drawback is, you only play melodies on the top string and there are no chords. It's possible to design a guitar where the frets are spaced at harmonic intervals and not equal tempered, but then, you can only play in one key, the key of E and you can't capo or play barre chords. However, if you do play such a redesigned guitar in the key of E, it will sound sweeter because all your notes will be harmonic and you could play chords.
  - tshaddox 4y ago
    I'm a bit confused by that article. It says things like "the problem is that equal temperament isn’t perfect" and "because guitars are imperfect instruments, they can never be completely in-tune" and "if you tune your open strings perfectly to pitch, you may still notice that some chords sound slightly out-of-tune." That doesn't sound like a description of a problem with equal temperament. In fact, the whole point of equal temperament is that any given chord has precisely the same tuning in every key. Isn't the purpose of these staggered frets to tune the guitar to more accurately match equal temperament? In other words, a guitar string played on the 7th fret is supposed to be tuned exactly 7 12-TET semitones above that string played open, but with completely straight frets (in lines exactly perpendicular to the strings) it's difficult to get every fret on every string to be accurately tunes. I believe this is what's referred to as intonation. Many guitars have some form of manual adjustment, like a movable bridge saddle for each string, and it's common to use that to slightly adjust the length of each string so that e.g. the 12th fret is accurately in tune with the open string. These "true temperament" guitars seem to have done a similar thing but with small adjustments to each fret on each string. Presumably the fret arrangements are designed to match that exact guitar fretboard with some specific guitar strings? The article says "with a normal guitar, if you play an A Major chord, then play a D Major chord, those chords will be slightly out-of-tune with each other." Again, this doesn't sound like a description of a problem with equal temperament. From what I can tell, these "true temperament" frets would help specifically with playing the same chord/interval in different positions.
SeanLuke 4y ago
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.
- deleted 4y ago
  [deleted]
- ledauphin 4y ago
  i don't think this post is making any arguments about _accuracy_ - the point is that 12 is the _simplest_ (smallest) number that gets reasonably close. Simplicity is incredibly powerful.
- bryanlarsen 4y ago
  According to https://news.ycombinator.com/item?id=32641527 https://news.ycombinator.com/item?id=32641527, 12 is substantially more accurate than 31. 41 is slightly more accurate, and 53 is a lot more accurate.
- lynndotpy 4y ago
  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!)
- elihu 4y ago
  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_size 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/ https://kiteguitar.com/
TheOtherHobbes 4y ago
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 theory you can play with more precision on fretless instruments (including strings), but it's hard enough to get beginners to pitch 12-TET accurately. Electronic microtonal keyboards exist, but they require extra learning and the music you can make with them isn't compelling enough to justify the complexity. https://www.youtube.com/watch?v=9ZozXzKOf8o&t=94s https://www.youtube.com/watch?v=9ZozXzKOf8o&t=94s
- shams93 4y ago
  Thats a decent explanation but in other cultures like Turkish music you have 9 other notes in the space of one half step so losing those extra notes will make the music sound like faked Turkish music without those extra pieces.
- dmarchand90 4y ago
  There used to be a lot of tempers floating around. For example a lot bach's music is not really supposed to be played in modern equal temper https://www.ethanhein.com/wp/2020/what-does-the-well-tempered-clavier-sound-like-in-actual-well-temperament/#:~:text=The%20main%20tuning%20system%20in,fifths%2C%20the%20worse%20everything%20sounds https://www.ethanhein.com/wp/2020/what-does-the-well-tempere....
  - analog31 4y ago
    I consider the 12 tone scale to be a technology. The historical temperaments were compromise solutions to the problem of getting a useable scale within the skills and patience of the musician. A harpsichord had to be tuned before every performance, by the musician.
    - dmarchand90 4y ago
jbverschoor 4y ago
12 notes, 12 hours, 12 months., even 12 monkeys.
- lioeters 4y ago
  > 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_%22dozenalism%22 https://en.wikipedia.org/wiki/Duodecimal#Advocacy_and_%22doz...
  - retrac 4y ago
    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.
    - mandmandam 4y ago
      > 60 is the smallest composite number with three prime factors Er, that'd be 30 I think, but the main point stands.
      - BitwiseFool 4y ago
        2 is such a nifty prime, we multiplied by it twice. It was just too perfect not to.
- marginalia_nu 4y ago
  The numbers 12 (2x2x3), 24 (2x2x2x3), 30 (2x3x5), 60 (2x2x3x5) and 360 (2x2x2x3x3x5) crop up in a lot of older counting systems because they're so conveniently divisible.
  - 4y ago
waffletower 4y ago
Bach explored another consideration, key changes within compositions, quite thoroughly: for example: https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier. You can also consider the group theoretic explanation as well (sorry paywall): https://www.jstor.org/stable/3679467 https://www.jstor.org/stable/3679467
- josephg 4y ago
  Modern electric pianos support changing from 12TET to other things. I wonder how it would sound to have an electric instrument retune itself throughout a performance as the key changes. Would it be worth the hassle?
abetusk 4y ago
This comes up every so often and in my mind, there is an answer and it has to do with how well the notes in the "temperament" combine to produce near-enough approximations to simple fractions. That is, take a temperament, combine each pairs of notes together. For each pairs of notes, find a close-enough fraction to it and give it a score depending on how many of these pairs produce simple fractions. The 12 note equal temperament produces one of the best scores, assuming some (perhaps arbitrary) constraints. There are some papers getting at this idea [0]. I even wrote a small program to try and do this [1]. Farey sequences are used for best rational approximation [2] [3]. I think this even gets at why some chords sound "sour/sad" while others sound "happy/full", because they have less or more constructive interference between the notes in the range of where we can hear. Obviously this has a lot to do with culture, so it's not as clear cut but at least this approach is better than just thinking it's completely arbitrary. [0] https://www.researchgate.net/publication/267806865_Measures_of_Consonances_in_a_Goodness-of-fit_Model_for_Equal-tempered_Scales https://www.researchgate.net/publication/267806865_Measures_... [1] https://github.com/abetusk/scratch/blob/release/src/music/best-scale-subdivision.py https://github.com/abetusk/scratch/blob/release/src/music/be... [2] https://www.johndcook.com/blog/2010/10/20/best-rational-approximation/ https://www.johndcook.com/blog/2010/10/20/best-rational-appr... [3] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence
j7ake 4y ago
With digital pianos, I imagine it is easy to switch to different tunings so that you can play each piece in a tuning that fits the key? Would be a major advantage over acoustic pianos.
- dhosek 4y ago
  There are keyboards that will do this. I remember seeing this advertised back in the 90s.
euroderf 4y ago
A long-winded way of saying that if you want to hit the 3:2 and 4:3 sweet spots "closely enough", dividing the octave into 12 logarithmically equidistant bins works very well, and better than any other number of bins less than 50 (or maybe 30).
- elihu 4y ago
  Actually, 41 has a better 4th and 5th, only being off by about half a cent, as opposed to being off by about 2 cents. It also has thirds and sixths that are quite a bit better (though still not great), and it has very good 7-limit intervals, which is something 12-EDO has nothing even remotely close to. 31 is generally decent all around, but it has worse 4ths and 5ths than 12-EDO. 53 EDO is even better than 41, having 4ths and 5ths that are off by about 7 hundredths of a cent. It also has much better 3rds and 6ths than 41, but the 7-limit intervals are slightly worse. https://en.wikipedia.org/wiki/31_equal_temperament#Interval_size https://en.wikipedia.org/wiki/31_equal_temperament#Interval_... https://en.wikipedia.org/wiki/41_equal_temperament#Interval_size https://en.wikipedia.org/wiki/41_equal_temperament#Interval_... https://en.wikipedia.org/wiki/53_equal_temperament#Interval_size https://en.wikipedia.org/wiki/53_equal_temperament#Interval_...
  - euroderf 4y ago
    I guess I could imagine a piano that is 53/12 times the size of a current piano, but I wouldn't want to have to play it :)
    - elihu 4y ago
      The H-Pi Tonal Plexus is 205-EDO, and reasonably playable as far as I know. I'm not sure what the biggest version is, but it's about 6 or 7 octaves.
acadapter 4y ago
The 12 tone scale provides good approximations of many aesthetically pleasing (mathematically simple) intervals.
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.
plq 4y ago
The system of the Turkish Classical Music is a bit different: * https://www.sufi.gen.tr/nota-sistemi/en https://www.sufi.gen.tr/nota-sistemi/en * http://www.turkishmusicportal.org/en/types-of-turkish-music http://www.turkishmusicportal.org/en/types-of-turkish-music
buildsjets 4y ago
"Flying Microtonal Banana" - because why not fret a whole bunch of guitars for 24-Tet and record an album? https://guitar.com/news/music-news/king-gizzard-stu-mackenzie-discusses-microtonal/ https://guitar.com/news/music-news/king-gizzard-stu-mackenzi...
- flobosg 4y ago
  Also Syzygys[1], a duo using a 43-tone scale[2]. [1]: https://www.youtube.com/watch?v=HkYWccNb0-4 https://www.youtube.com/watch?v=HkYWccNb0-4 [2]: https://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale https://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale
sleepdreamy 4y ago
Micro-Tonal bands such as King Gizzard play around/bend the rules on the whole 12 notes in music thing.
- joegahona 4y ago
  Familiar with the band, but not examples of this. Can you recommend any tracks that demonstrate?
  - inkcapmushroom 4y ago
    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.
captain_trips 4y ago
I usually think of it as someone hundreds of years ago playing with a tighly stretched string and noticing stuff: - Half a string's length is the same thing, just higher - The harmonic at half a string is the same thing, just higher - The harmonics at 3:2 and 4:3 are really loud and distinct - If you chop up the rest of the string at the same interval as the 3:2 and 4:3, some have strong harmonics that match tones on the string, some don't Then I figure with the Octave, 5th, 4th, and M3 as the strongest harmonics that matches other lengths (or octaves thereof), they went from there...
srcreigh 4y ago
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 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 https://www.youtube.com/watch?v=fJT5Q6HooyA
dr_dshiv 4y ago
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.pdf https://downloads.spj.sciencemag.org/research/2019/2369041.p...
thomasqbrady 4y ago
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…
- herghost 4y ago
  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.
  - pdonis 4y ago
    > 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).
- bryanlarsen 4y ago
  The why is because the 5th is the primary non-octave overtone of a vibrating string, and the 4th is an inverted 5th.
- diffeomorphism 4y ago
  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....
752963e64 4y ago
dhosek 4y ago
Some 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. ⸻ 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.
- ajross 4y ago
  I've never liked explaining the scale as a Pythagorean derivation. It's not really correct historically (Greek music didn't have anything approximating a full major scale) or mathematically (it doesn't understand the idea of a "third" interval the way tonal music does, so playing triads with pythagorean tuning sounds awful!). Here's my take: late medieval singers discovered The Major Chord. That's the combination of three (!) notes that is "most consonant" (mathematically: beats in the shortest period). This combines two notes a major fifth apart (ratio 3:2), with a third note that is 5:4 with the low note. You can write some code to prove this if you like. So now take that "best" chord with its three notes, and start moving it around. If you go up a fifth (i.e. by "the most consonent interval", that is the "closest best chord to your first best chord") you can play the same chord, adding two needed notes that weren't in the scale before. You can likewise go down a fifth to add two new notes. Then you compress these seven notes into a single octave, and you get... the major scale! It's just there. All you need is that one "best" three-note chord and an obvious metric for "nearest" (i.e. transpose by a fifth) and you have almost all of modern tonal music. Play the same tunes starting on different notes and you get "modalities", etc... You can transpose up and down to nearby keys and keep playing by "cheating" with your tunings to move a note half way up or down. And the practice of formalizing those transpositional cheats because what we now know as the equitempered scale. But they're still just cheats. And the fact that pow(2, 1.0/12) happens to work is, basically, just dumb luck.
kazinator 4y ago
When 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 alterations of these notes add five more, if you aren't picky about microtonal differences. If you have do-re-mi-fa-so-la-ti-do, there is a small step between "mi-fa" and "ti-do" which is about half of the longer step that is observed in the five remaining successive pairs. If you identify some half-step note between those other pairs like "do-re" or "so-la", you end up with five more notes, giving you twelve. That's all it is; if we back fill 7 notes with enough notes to have chromatic half steps, we get 12. Now, early practitioners of western music did know that that's not all there is to it: that a G# is not the same as an Ab. They tried using the in-between notes for transposing to other keys and found that the keys sounded different. They knew all about the mathematics behind it and the Pythagorean comma: that if you go around the circle of fifths 13 times, you don't end up at exactly the same note (modulo octave); there is a discrepancy. Various technical devices were devised, such as splitting the small keys of keyboard instruments, so that the G# key actually had a G# split and an Ab split. Various tunings were also used, like well temperament. Bach's Well-Tempered Clavier is basically a set of test cases for tuning. We settled on equal temperament because it distributes the error such that all the keys sound the same; when music is transposed to any key, the pitch relationships are preserved. Going back to the first concept; why wouldn't more than five additional tones be added to add color to a seven tone scale? It's because Western music traditionally hadn't been oriented toward recognizing microtonal differences, or at least into organizing them (where they exist) into a single system. In Indian music, there are 22 notes (shrutis). They are needed because there are numerous scales which have the same approximation on a western instrument. For instance, there are multiple scales that resemble "do-re-mi-fa-so-la-ti-do": the Pythagorean scale, but which use different microtones chosen from the 22 shrutis. Those scales all have different names; they are not just different tunings for obtaining different flavors of do-re-mi. But in Indian music, there is still a significance in 12 tones in an octave! "There are 12 universally identifiable notes ('Swaraprakar' in Sanskrit) in any Octave (Saptak). As we play them from one end on any string, the perception of each of these 12 changes 'only' at 22 points given by nature (See numbers in green in the slide below). The sounds produced at these 22 points are the '22 Shrutis' and the 3 types of distances in-between are called as 'Shrutyantara' (in Sanskrit) (See Legend below)" http://www.22shruti.com/ http://www.22shruti.com/ And: https://en.wikipedia.org/wiki/Shruti_(music)#Identification_of_a_shruti https://en.wikipedia.org/wiki/Shruti_(music)#Identification_... It seems there is no getting away from the situation of there being identifiable 7 note scales (Swaras), into which we can stuff five more notes to obtain some kind of twelve-note chromatic scale.
papandada 4y ago
3^12 is within 1.5% of 2^19
coldtea 4y ago
>You may have noticed that 24 is also includes the 5ths and 4ths, the problem is that having twice as many notes would require instruments with twice as many keys or buttons making them more expensive and complicated to play, also probably we wouldn't notice the difference between notes that are so close We actually would, and it's quite noticable. Several cultures use microtonals intervals with half-half-steps or similar. The main reason we stuck with 12, is complexity in making AND playing an instrument with so many notes - that, or the halved range, if we keep the number of notes on the instrument the same. But there are cultures (and instruments) which have more.
mikewarot 4y ago
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.
mcbrit 4y ago
Let's talk about splitting things up in useful ways. 12=2 * 2 * 3. Splitting something in half is useful; splitting it half again remains useful. Splitting in half a third time is arguably less useful than splitting it into a third. So 12 is the made-to-order number that lets you split it in half, twice, and in thirds, once. Which naturally leads to seconds and minutes, or 60: 60=2 * 2 * 3 * 5 Because dividing the whole into fifths is more useful than a second 3, or a third 2. So, there's your basic argument for why you would see a 12 or a 60 instead of a 10 or some other number. You have a whole that you want to divide into useful parts. I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis; we like hearing useful parts would be the somewhat surprising thing to talk about.
- lynndotpy 4y ago
  No: Having 12 notes is a neat but accidental outcome of the musical scale. On the derivation of the scale: Two pure tones go good together when they have "ratios" of frequency. So, 440Hz and 660Hz would interfere in a pleasing way. This is the same way that it's "nice" when tiles on a floor match your gait in a way you can follow a pattern, or when two blinkers sync up. So, it's nice when tones are fractions of one another like 3/2, 4/3, 5/3, etc. A ratio like pi/2 would sound weird, and very close frequencies (like 440Hz and 440.5Hz) would interfere to make a beat of 0.5Hz. (I'm sure we can all agree that a ratio of 4/3 is a much nicer fraction 440/441. In practice, this doesn't matter much, because we rarely use pure tones. This is why equal temperament scales don't sound abysmal.) A natural ratio is just "2/1". This division gives you your octave. One octave up from 440 is 880. One octave down from 440 is 220. This plays nicely into the second problem: Human perception is logarithmic in many things, tone included. For a musical scale to be perceived to have equal differences in pitch between notes, it needs to be roughly evenly spaced on the logarithmic scale. So, we need to select a set of frequencies in [440Hz, 880Hz) (where 440Hz is arbitrary) that (1) arenice fractions of one another, but are also (2) evenly spaced on the logarithmic scale. By nice mathematical luck, the 12-tone chromatic scale fulfills that! --- On the questionable qualities of "12": I don't think the nice divisibility of the number 12 matters here, in a scale where only 7 of the notes (with the most pleasing ratio) get the title of "major". Notation is written on a musical staff where the other 5 notes are folded away. Furthermore, if you swap out the base of 2 for a base of 3 and re-derive the scale, you get other scales. One is the Bohlen-Pierce scale, which has 13 notes. I derived a similar scale one time, and I remember finding a 7 or 11 note scale. I forget which exactly, but the point being that these are prime numbers. So, I'm wondering, how would 12 being divisible matter for music? I don't compose much, so I mean this question genuinely. --- On the main point: > I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis. I ask this in good faith: Did you read the linked article before commenting? If you didn't, then :\ If you did, then I'm curious: The scale derivation I described here is described in the article, with nice visualizations. If you did read the article, how do you now have this take? How could a 10-tone scale fit into this, and how would having a less nicely-divisible number matter?
analog31 4y ago
An alternative question is: Why not more? There are approximately rational scales with more than 12 notes. Something I wonder is how the complexity of music relates to its use. For instance, instruments for music that's primarily ceremonial, or used in centralized locations by trained experts, could adopt scales with more notes, or more difficult tunings. This includes 12TET, which was difficult for an untrained musician to replicate, and unlikely to stay in tune for an entire performance on some instruments. Simpler tunings might lend themselves to instruments that were homemade, used for folk music, carried by travelers, played at home, etc. In fact those two things could coexist within a single culture. There were pipe organs and folk fiddles in Europe during the same time period, after all. Once the 4 strings are tuned by means of an easily discerned interval, you can fill in with a tolerable scale by ear. "Carried by travelers" suggests an advantage for a tuning system that can be restored by a non-expert and used for music that spreads from town to town.
im3w1l 4y ago
So these kinds of discussions typically center around harmonic considerations. What about the melodic ones?
elihu 4y ago
Article only looks at the 4th and 5th, but I think the more interesting observation is that if you pick any small number ratio between 1:1 and 2:1 (i.e. using numbers from 1..10 and lying within a single octave), they almost all have a reasonable 12-EDO approximation. 1:1 is the unison. Not terribly interesting. 2:1 is the octave, which is exact. 3:2 is the perfect fifth. About 2 cents of error. 4:3 is the perfect fourth. Also about 2 cents of error. 5:3 is the major sixth. About 15 cents or so of error. 5:4 is the major third. About 13 cents or so of error. 6:5 is the minor third. About 15 cents or so of error. 7:4 is the the first one that doesn't really have a 12-EDO equivalent, though the minor 7th is generally used, with about 31 cents of error. 7:6 similarly doesn't have an equivalent. It's about 44 cents flat of the 12-EDO minor third. In fact, most ratios with 7s are right out. 7:5 is in the ballpark of the 12-EDO tritone. 8:5 is the minor sixth. About 13 cents of error. 9:7 is a really sharp third, again no equivalent in 12-EDO. 9:8 is the major second. About 4 cents of error. 10:9 is also the major second, about 17 cents off in the other direction. (12-EDO makes no distinction between 9:8 and 10:9. That's actually fairly important, as it lets you get away with chord progressions that don't mathematically work out.) It's really amazing to have so many decent approximations of ratios with only 12 notes. Different EDOs might have better or worse approximation of various musical intervals. It takes going all the way up to 41-EDO to find something that's better at basically everything -- it even has a more accurate 4th and 5th, which is the one thing that 12-EDO is amazingly good at.
dangan 4y ago
Check out Musimathics volume 1 if you want an absurdly detailed explanation of why Western 12 TET is the way that it is. http://www.musimathics.com/ http://www.musimathics.com/
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simonebrunozzi 4y ago
> pairs of notes whose frequencies are multiples of 4/3 or 3/2 sound very pleasing when played together. Yes, but why is that? Why did we evolve to like these particular combination of notes? Or, why do they remind us of some other natural sound?
- afiori 4y ago
  It it could also very well be that it is just unusual and not associated with danger, so it could be relaxing by lack of associations. In general anything more complex than "birds like blue feathers because it shows a strong immune system" is unlikely to have a simple satisfying evolutionary reason to exist that is less than 80% wrong.
zarzavat 4y ago
12 is a special number, it’s a highly composite number but it’s also a small number, this means that a large percentage of the numbers less than 12 divide it. This gives the group Z12 some interesting properties, there’s a wide range of element orders. Tritones come in pairs, major thirds come in 4 sets of 3 thirds, minor thirds come in 3 sets of diminished seventh chords, etc, major seconds form the two whole tone scales. Fourths (5) and fifths (7) don’t divide 12, so they generate the group, producing the cycle of fourths and fifths. If we look at the groups next door, Z11 and Z13, they are of prime order, so every element generates the group => there is a cycle of “fifths” for each interval. Beautifully symmetric, perhaps, but less structure to enjoy.
lachlan_gray 4y ago
Another nice thing about 12 notes is that 12 has lots of factors. It leads to a lot of cool symmetries which I think have to do with how we understand music. For example, since 12 is divisible by 3 and 4, diminished chords (4 notes separated by intervals of 3) and augmented chords (3 by 4) don’t have a definitive centre to our ear. As a result they both create different flavours of uncertainty and suspense. But if you break the symmetry, e.g. by moving any note in an augmented chord one to the left on a piano, the chord takes on a colour (in this case becomes a major chord). Other equal temperament scales would end up having similar patterns, but at the cost of complexity. having closer to 20 or 30 notes in an octave would increase the number of patterns to recognize, and probably make them less distinct to us without practice. It’s also interesting to me that we put 12 numbers on clocks as well.
- pyinstallwoes 4y ago
  And 12 months in a year. Around around frequencies abound. In ratio and interval, all things are sound. Rational reasoning abound, just don’t be too loud. A note here. A note there. A harmonic there and there. You say blue, I say G, and now we aught to see the cymatics inside you and me! Altogether now, ratio, ratios, frequency make everything between you and me. Hey now, hey now, just make music, it’s what we all need now.
dublin 4y ago
Thanks for this. It is perhaps the clearest explanation ever of the musical scale, and why it is the way it is. I've literally watched dozens of YouTube videos on this topic in the past week, and none of them are as clear and insightful as this was. You do have to wonder what an optical octave would look like, though - our visual range is, unfortunately, a bit short of an octave there... Maybe the same "tone" of a kind of extreme magenta?

253 points by xchip 4y ago

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.

sabotista 4y ago
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 unwieldy for physical instruments (although some computer music does use 53-TET).
- not2b 4y ago
  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. :-)
  - dhosek 4y ago
    It may not be true, but I still loved telling that story to my students.
thirteenfingers 4y ago
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 characters of the different keys to the best possible advantage. I can't speak to this personally, I keep my piano at equal temperament ;) (Big fan of your videos btw)
- joegahona 4y ago
  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. Some of those keys sound really "out of tune" to modern ears -- I have a friend with perfect pitch who legit can't listen to them. Owen Jorgensen's "Tuning the Historical Temperaments by Ear" is good bedtime reading on this topic, if you have a couple hundred bucks burning a hole in your pocket. https://www.amazon.com/Tuning-historical-temperaments-ear-eighty-nine/dp/091861600X https://www.amazon.com/Tuning-historical-temperaments-ear-ei...
  - wizofaus 4y ago
    > 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.
    - joegahona 4y ago
      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_Temperament_on_Ch.html?id=O6b2tgAACAAJ https://books.google.com/books/about/The_Effects_of_Unequal_... - https://www.proquest.com/openview/b22142f819768ae82464aa2679f4ef68/1?pq-origsite=gscholar&cbl=18750&diss=y https://www.proquest.com/openview/b22142f819768ae82464aa2679...
dahart 4y ago
> it was Bach that pushed equal temperament (equal spacing). Not really. Equal temperament was being advocated both in China and Europe long before Bach was born. In Europe, it was the lute players that pushed for it, because it matters more for fretted instruments, where any temperament other than equal causes conflicts and inconsistencies on the neck. https://en.m.wikipedia.org/wiki/12_equal_temperament https://en.m.wikipedia.org/wiki/12_equal_temperament

locusofself 4y ago

I'm not a music theory expert, just a guy who has played guitar by ear for 28 years. Equal temperament is definitely a compromise .. I find myself constantly trying to "sweeten" the tuning of my guitar strings relative to the song/key I am playing. You can tune your guitar with the most accurate guitar tuner in the world (I have strobe tuners by Sonic Research and Peterson), and some things still just sound out of tune.

criddell 4y ago
There are guitars with frets that aren't straight lines that try to address some of the problems: https://guitargearfinder.com/faq/true-temperament-frets/ https://guitargearfinder.com/faq/true-temperament-frets/
- locusofself 4y ago
  I've seen those! Pretty cool. I've also seen people put intermediate frets on certain notes.
- khitchdee 4y ago
  FWIW, A sitar also has a 12 note scale, but the frets are moveable so you can tune by ear. The drawback is, you only play melodies on the top string and there are no chords. It's possible to design a guitar where the frets are spaced at harmonic intervals and not equal tempered, but then, you can only play in one key, the key of E and you can't capo or play barre chords. However, if you do play such a redesigned guitar in the key of E, it will sound sweeter because all your notes will be harmonic and you could play chords.
- tshaddox 4y ago
  I'm a bit confused by that article. It says things like "the problem is that equal temperament isn’t perfect" and "because guitars are imperfect instruments, they can never be completely in-tune" and "if you tune your open strings perfectly to pitch, you may still notice that some chords sound slightly out-of-tune." That doesn't sound like a description of a problem with equal temperament. In fact, the whole point of equal temperament is that any given chord has precisely the same tuning in every key. Isn't the purpose of these staggered frets to tune the guitar to more accurately match equal temperament? In other words, a guitar string played on the 7th fret is supposed to be tuned exactly 7 12-TET semitones above that string played open, but with completely straight frets (in lines exactly perpendicular to the strings) it's difficult to get every fret on every string to be accurately tunes. I believe this is what's referred to as intonation. Many guitars have some form of manual adjustment, like a movable bridge saddle for each string, and it's common to use that to slightly adjust the length of each string so that e.g. the 12th fret is accurately in tune with the open string. These "true temperament" guitars seem to have done a similar thing but with small adjustments to each fret on each string. Presumably the fret arrangements are designed to match that exact guitar fretboard with some specific guitar strings? The article says "with a normal guitar, if you play an A Major chord, then play a D Major chord, those chords will be slightly out-of-tune with each other." Again, this doesn't sound like a description of a problem with equal temperament. From what I can tell, these "true temperament" frets would help specifically with playing the same chord/interval in different positions.

SeanLuke 4y ago

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.

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ledauphin 4y ago
i don't think this post is making any arguments about _accuracy_ - the point is that 12 is the _simplest_ (smallest) number that gets reasonably close. Simplicity is incredibly powerful.
bryanlarsen 4y ago
According to https://news.ycombinator.com/item?id=32641527 https://news.ycombinator.com/item?id=32641527, 12 is substantially more accurate than 31. 41 is slightly more accurate, and 53 is a lot more accurate.
lynndotpy 4y ago
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!)
elihu 4y ago
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_size 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/ https://kiteguitar.com/

TheOtherHobbes 4y ago

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 theory you can play with more precision on fretless instruments (including strings), but it's hard enough to get beginners to pitch 12-TET accurately. Electronic microtonal keyboards exist, but they require extra learning and the music you can make with them isn't compelling enough to justify the complexity. https://www.youtube.com/watch?v=9ZozXzKOf8o&t=94s https://www.youtube.com/watch?v=9ZozXzKOf8o&t=94s

shams93 4y ago
Thats a decent explanation but in other cultures like Turkish music you have 9 other notes in the space of one half step so losing those extra notes will make the music sound like faked Turkish music without those extra pieces.
dmarchand90 4y ago
There used to be a lot of tempers floating around. For example a lot bach's music is not really supposed to be played in modern equal temper https://www.ethanhein.com/wp/2020/what-does-the-well-tempered-clavier-sound-like-in-actual-well-temperament/#:~:text=The%20main%20tuning%20system%20in,fifths%2C%20the%20worse%20everything%20sounds https://www.ethanhein.com/wp/2020/what-does-the-well-tempere....
- analog31 4y ago
  I consider the 12 tone scale to be a technology. The historical temperaments were compromise solutions to the problem of getting a useable scale within the skills and patience of the musician. A harpsichord had to be tuned before every performance, by the musician.
  - dmarchand90 4y ago

jbverschoor 4y ago

12 notes, 12 hours, 12 months., even 12 monkeys.

lioeters 4y ago
> 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_%22dozenalism%22 https://en.wikipedia.org/wiki/Duodecimal#Advocacy_and_%22doz...
- retrac 4y ago
  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.
  - mandmandam 4y ago
    > 60 is the smallest composite number with three prime factors Er, that'd be 30 I think, but the main point stands.
    - BitwiseFool 4y ago
      2 is such a nifty prime, we multiplied by it twice. It was just too perfect not to.
marginalia_nu 4y ago
The numbers 12 (2x2x3), 24 (2x2x2x3), 30 (2x3x5), 60 (2x2x3x5) and 360 (2x2x2x3x3x5) crop up in a lot of older counting systems because they're so conveniently divisible.
- 4y ago

waffletower 4y ago

Bach explored another consideration, key changes within compositions, quite thoroughly: for example: https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier https://en.wikipedia.org/wiki/The_Well-Tempered_Clavier. You can also consider the group theoretic explanation as well (sorry paywall): https://www.jstor.org/stable/3679467 https://www.jstor.org/stable/3679467

josephg 4y ago
Modern electric pianos support changing from 12TET to other things. I wonder how it would sound to have an electric instrument retune itself throughout a performance as the key changes. Would it be worth the hassle?

abetusk 4y ago

This comes up every so often and in my mind, there is an answer and it has to do with how well the notes in the "temperament" combine to produce near-enough approximations to simple fractions. That is, take a temperament, combine each pairs of notes together. For each pairs of notes, find a close-enough fraction to it and give it a score depending on how many of these pairs produce simple fractions. The 12 note equal temperament produces one of the best scores, assuming some (perhaps arbitrary) constraints. There are some papers getting at this idea [0]. I even wrote a small program to try and do this [1]. Farey sequences are used for best rational approximation [2] [3]. I think this even gets at why some chords sound "sour/sad" while others sound "happy/full", because they have less or more constructive interference between the notes in the range of where we can hear. Obviously this has a lot to do with culture, so it's not as clear cut but at least this approach is better than just thinking it's completely arbitrary. [0] https://www.researchgate.net/publication/267806865_Measures_of_Consonances_in_a_Goodness-of-fit_Model_for_Equal-tempered_Scales https://www.researchgate.net/publication/267806865_Measures_... [1] https://github.com/abetusk/scratch/blob/release/src/music/best-scale-subdivision.py https://github.com/abetusk/scratch/blob/release/src/music/be... [2] https://www.johndcook.com/blog/2010/10/20/best-rational-approximation/ https://www.johndcook.com/blog/2010/10/20/best-rational-appr... [3] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence

j7ake 4y ago

With digital pianos, I imagine it is easy to switch to different tunings so that you can play each piece in a tuning that fits the key? Would be a major advantage over acoustic pianos.

dhosek 4y ago
There are keyboards that will do this. I remember seeing this advertised back in the 90s.

euroderf 4y ago

A long-winded way of saying that if you want to hit the 3:2 and 4:3 sweet spots "closely enough", dividing the octave into 12 logarithmically equidistant bins works very well, and better than any other number of bins less than 50 (or maybe 30).

elihu 4y ago
Actually, 41 has a better 4th and 5th, only being off by about half a cent, as opposed to being off by about 2 cents. It also has thirds and sixths that are quite a bit better (though still not great), and it has very good 7-limit intervals, which is something 12-EDO has nothing even remotely close to. 31 is generally decent all around, but it has worse 4ths and 5ths than 12-EDO. 53 EDO is even better than 41, having 4ths and 5ths that are off by about 7 hundredths of a cent. It also has much better 3rds and 6ths than 41, but the 7-limit intervals are slightly worse. https://en.wikipedia.org/wiki/31_equal_temperament#Interval_size https://en.wikipedia.org/wiki/31_equal_temperament#Interval_... https://en.wikipedia.org/wiki/41_equal_temperament#Interval_size https://en.wikipedia.org/wiki/41_equal_temperament#Interval_... https://en.wikipedia.org/wiki/53_equal_temperament#Interval_size https://en.wikipedia.org/wiki/53_equal_temperament#Interval_...
- euroderf 4y ago
  I guess I could imagine a piano that is 53/12 times the size of a current piano, but I wouldn't want to have to play it :)
  - elihu 4y ago
    The H-Pi Tonal Plexus is 205-EDO, and reasonably playable as far as I know. I'm not sure what the biggest version is, but it's about 6 or 7 octaves.

acadapter 4y ago

The 12 tone scale provides good approximations of many aesthetically pleasing (mathematically simple) intervals.

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.

plq 4y ago

The system of the Turkish Classical Music is a bit different: * https://www.sufi.gen.tr/nota-sistemi/en https://www.sufi.gen.tr/nota-sistemi/en * http://www.turkishmusicportal.org/en/types-of-turkish-music http://www.turkishmusicportal.org/en/types-of-turkish-music

buildsjets 4y ago

"Flying Microtonal Banana" - because why not fret a whole bunch of guitars for 24-Tet and record an album? https://guitar.com/news/music-news/king-gizzard-stu-mackenzie-discusses-microtonal/ https://guitar.com/news/music-news/king-gizzard-stu-mackenzi...

flobosg 4y ago
Also Syzygys[1], a duo using a 43-tone scale[2]. [1]: https://www.youtube.com/watch?v=HkYWccNb0-4 https://www.youtube.com/watch?v=HkYWccNb0-4 [2]: https://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale https://en.wikipedia.org/wiki/Harry_Partch%27s_43-tone_scale

sleepdreamy 4y ago

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

joegahona 4y ago
Familiar with the band, but not examples of this. Can you recommend any tracks that demonstrate?
- inkcapmushroom 4y ago
  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.

captain_trips 4y ago

I usually think of it as someone hundreds of years ago playing with a tighly stretched string and noticing stuff: - Half a string's length is the same thing, just higher - The harmonic at half a string is the same thing, just higher - The harmonics at 3:2 and 4:3 are really loud and distinct - If you chop up the rest of the string at the same interval as the 3:2 and 4:3, some have strong harmonics that match tones on the string, some don't Then I figure with the Octave, 5th, 4th, and M3 as the strongest harmonics that matches other lengths (or octaves thereof), they went from there...

srcreigh 4y ago

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

dr_dshiv 4y ago

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.pdf https://downloads.spj.sciencemag.org/research/2019/2369041.p...

thomasqbrady 4y ago

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…

herghost 4y ago
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.
- pdonis 4y ago
  > 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).
bryanlarsen 4y ago
The why is because the 5th is the primary non-octave overtone of a vibrating string, and the 4th is an inverted 5th.
diffeomorphism 4y ago
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....

752963e64 4y ago

dhosek 4y ago

Some 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. ⸻ 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.

ajross 4y ago
I've never liked explaining the scale as a Pythagorean derivation. It's not really correct historically (Greek music didn't have anything approximating a full major scale) or mathematically (it doesn't understand the idea of a "third" interval the way tonal music does, so playing triads with pythagorean tuning sounds awful!). Here's my take: late medieval singers discovered The Major Chord. That's the combination of three (!) notes that is "most consonant" (mathematically: beats in the shortest period). This combines two notes a major fifth apart (ratio 3:2), with a third note that is 5:4 with the low note. You can write some code to prove this if you like. So now take that "best" chord with its three notes, and start moving it around. If you go up a fifth (i.e. by "the most consonent interval", that is the "closest best chord to your first best chord") you can play the same chord, adding two needed notes that weren't in the scale before. You can likewise go down a fifth to add two new notes. Then you compress these seven notes into a single octave, and you get... the major scale! It's just there. All you need is that one "best" three-note chord and an obvious metric for "nearest" (i.e. transpose by a fifth) and you have almost all of modern tonal music. Play the same tunes starting on different notes and you get "modalities", etc... You can transpose up and down to nearby keys and keep playing by "cheating" with your tunings to move a note half way up or down. And the practice of formalizing those transpositional cheats because what we now know as the equitempered scale. But they're still just cheats. And the fact that pow(2, 1.0/12) happens to work is, basically, just dumb luck.

kazinator 4y ago

When 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 alterations of these notes add five more, if you aren't picky about microtonal differences. If you have do-re-mi-fa-so-la-ti-do, there is a small step between "mi-fa" and "ti-do" which is about half of the longer step that is observed in the five remaining successive pairs. If you identify some half-step note between those other pairs like "do-re" or "so-la", you end up with five more notes, giving you twelve. That's all it is; if we back fill 7 notes with enough notes to have chromatic half steps, we get 12. Now, early practitioners of western music did know that that's not all there is to it: that a G# is not the same as an Ab. They tried using the in-between notes for transposing to other keys and found that the keys sounded different. They knew all about the mathematics behind it and the Pythagorean comma: that if you go around the circle of fifths 13 times, you don't end up at exactly the same note (modulo octave); there is a discrepancy. Various technical devices were devised, such as splitting the small keys of keyboard instruments, so that the G# key actually had a G# split and an Ab split. Various tunings were also used, like well temperament. Bach's Well-Tempered Clavier is basically a set of test cases for tuning. We settled on equal temperament because it distributes the error such that all the keys sound the same; when music is transposed to any key, the pitch relationships are preserved. Going back to the first concept; why wouldn't more than five additional tones be added to add color to a seven tone scale? It's because Western music traditionally hadn't been oriented toward recognizing microtonal differences, or at least into organizing them (where they exist) into a single system. In Indian music, there are 22 notes (shrutis). They are needed because there are numerous scales which have the same approximation on a western instrument. For instance, there are multiple scales that resemble "do-re-mi-fa-so-la-ti-do": the Pythagorean scale, but which use different microtones chosen from the 22 shrutis. Those scales all have different names; they are not just different tunings for obtaining different flavors of do-re-mi. But in Indian music, there is still a significance in 12 tones in an octave! "There are 12 universally identifiable notes ('Swaraprakar' in Sanskrit) in any Octave (Saptak). As we play them from one end on any string, the perception of each of these 12 changes 'only' at 22 points given by nature (See numbers in green in the slide below). The sounds produced at these 22 points are the '22 Shrutis' and the 3 types of distances in-between are called as 'Shrutyantara' (in Sanskrit) (See Legend below)" http://www.22shruti.com/ http://www.22shruti.com/ And: https://en.wikipedia.org/wiki/Shruti_(music)#Identification_of_a_shruti https://en.wikipedia.org/wiki/Shruti_(music)#Identification_... It seems there is no getting away from the situation of there being identifiable 7 note scales (Swaras), into which we can stuff five more notes to obtain some kind of twelve-note chromatic scale.

papandada 4y ago

3^12 is within 1.5% of 2^19

coldtea 4y ago

>You may have noticed that 24 is also includes the 5ths and 4ths, the problem is that having twice as many notes would require instruments with twice as many keys or buttons making them more expensive and complicated to play, also probably we wouldn't notice the difference between notes that are so close We actually would, and it's quite noticable. Several cultures use microtonals intervals with half-half-steps or similar. The main reason we stuck with 12, is complexity in making AND playing an instrument with so many notes - that, or the halved range, if we keep the number of notes on the instrument the same. But there are cultures (and instruments) which have more.

mikewarot 4y ago

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.

mcbrit 4y ago

Let's talk about splitting things up in useful ways. 12=2 * 2 * 3. Splitting something in half is useful; splitting it half again remains useful. Splitting in half a third time is arguably less useful than splitting it into a third. So 12 is the made-to-order number that lets you split it in half, twice, and in thirds, once. Which naturally leads to seconds and minutes, or 60: 60=2 * 2 * 3 * 5 Because dividing the whole into fifths is more useful than a second 3, or a third 2. So, there's your basic argument for why you would see a 12 or a 60 instead of a 10 or some other number. You have a whole that you want to divide into useful parts. I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis; we like hearing useful parts would be the somewhat surprising thing to talk about.

lynndotpy 4y ago
No: Having 12 notes is a neat but accidental outcome of the musical scale. On the derivation of the scale: Two pure tones go good together when they have "ratios" of frequency. So, 440Hz and 660Hz would interfere in a pleasing way. This is the same way that it's "nice" when tiles on a floor match your gait in a way you can follow a pattern, or when two blinkers sync up. So, it's nice when tones are fractions of one another like 3/2, 4/3, 5/3, etc. A ratio like pi/2 would sound weird, and very close frequencies (like 440Hz and 440.5Hz) would interfere to make a beat of 0.5Hz. (I'm sure we can all agree that a ratio of 4/3 is a much nicer fraction 440/441. In practice, this doesn't matter much, because we rarely use pure tones. This is why equal temperament scales don't sound abysmal.) A natural ratio is just "2/1". This division gives you your octave. One octave up from 440 is 880. One octave down from 440 is 220. This plays nicely into the second problem: Human perception is logarithmic in many things, tone included. For a musical scale to be perceived to have equal differences in pitch between notes, it needs to be roughly evenly spaced on the logarithmic scale. So, we need to select a set of frequencies in [440Hz, 880Hz) (where 440Hz is arbitrary) that (1) arenice fractions of one another, but are also (2) evenly spaced on the logarithmic scale. By nice mathematical luck, the 12-tone chromatic scale fulfills that! --- On the questionable qualities of "12": I don't think the nice divisibility of the number 12 matters here, in a scale where only 7 of the notes (with the most pleasing ratio) get the title of "major". Notation is written on a musical staff where the other 5 notes are folded away. Furthermore, if you swap out the base of 2 for a base of 3 and re-derive the scale, you get other scales. One is the Bohlen-Pierce scale, which has 13 notes. I derived a similar scale one time, and I remember finding a 7 or 11 note scale. I forget which exactly, but the point being that these are prime numbers. So, I'm wondering, how would 12 being divisible matter for music? I don't compose much, so I mean this question genuinely. --- On the main point: > I'm not sure that the linked article, or the current top comment (Circle of Fifths) meaningfully extends beyond this "useful parts" hypothesis. I ask this in good faith: Did you read the linked article before commenting? If you didn't, then :\ If you did, then I'm curious: The scale derivation I described here is described in the article, with nice visualizations. If you did read the article, how do you now have this take? How could a 10-tone scale fit into this, and how would having a less nicely-divisible number matter?

analog31 4y ago

An alternative question is: Why not more? There are approximately rational scales with more than 12 notes. Something I wonder is how the complexity of music relates to its use. For instance, instruments for music that's primarily ceremonial, or used in centralized locations by trained experts, could adopt scales with more notes, or more difficult tunings. This includes 12TET, which was difficult for an untrained musician to replicate, and unlikely to stay in tune for an entire performance on some instruments. Simpler tunings might lend themselves to instruments that were homemade, used for folk music, carried by travelers, played at home, etc. In fact those two things could coexist within a single culture. There were pipe organs and folk fiddles in Europe during the same time period, after all. Once the 4 strings are tuned by means of an easily discerned interval, you can fill in with a tolerable scale by ear. "Carried by travelers" suggests an advantage for a tuning system that can be restored by a non-expert and used for music that spreads from town to town.

im3w1l 4y ago

So these kinds of discussions typically center around harmonic considerations. What about the melodic ones?

elihu 4y ago

Article only looks at the 4th and 5th, but I think the more interesting observation is that if you pick any small number ratio between 1:1 and 2:1 (i.e. using numbers from 1..10 and lying within a single octave), they almost all have a reasonable 12-EDO approximation. 1:1 is the unison. Not terribly interesting. 2:1 is the octave, which is exact. 3:2 is the perfect fifth. About 2 cents of error. 4:3 is the perfect fourth. Also about 2 cents of error. 5:3 is the major sixth. About 15 cents or so of error. 5:4 is the major third. About 13 cents or so of error. 6:5 is the minor third. About 15 cents or so of error. 7:4 is the the first one that doesn't really have a 12-EDO equivalent, though the minor 7th is generally used, with about 31 cents of error. 7:6 similarly doesn't have an equivalent. It's about 44 cents flat of the 12-EDO minor third. In fact, most ratios with 7s are right out. 7:5 is in the ballpark of the 12-EDO tritone. 8:5 is the minor sixth. About 13 cents of error. 9:7 is a really sharp third, again no equivalent in 12-EDO. 9:8 is the major second. About 4 cents of error. 10:9 is also the major second, about 17 cents off in the other direction. (12-EDO makes no distinction between 9:8 and 10:9. That's actually fairly important, as it lets you get away with chord progressions that don't mathematically work out.) It's really amazing to have so many decent approximations of ratios with only 12 notes. Different EDOs might have better or worse approximation of various musical intervals. It takes going all the way up to 41-EDO to find something that's better at basically everything -- it even has a more accurate 4th and 5th, which is the one thing that 12-EDO is amazingly good at.

dangan 4y ago

Check out Musimathics volume 1 if you want an absurdly detailed explanation of why Western 12 TET is the way that it is. http://www.musimathics.com/ http://www.musimathics.com/

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simonebrunozzi 4y ago

> pairs of notes whose frequencies are multiples of 4/3 or 3/2 sound very pleasing when played together. Yes, but why is that? Why did we evolve to like these particular combination of notes? Or, why do they remind us of some other natural sound?

afiori 4y ago
It it could also very well be that it is just unusual and not associated with danger, so it could be relaxing by lack of associations. In general anything more complex than "birds like blue feathers because it shows a strong immune system" is unlikely to have a simple satisfying evolutionary reason to exist that is less than 80% wrong.

zarzavat 4y ago

12 is a special number, it’s a highly composite number but it’s also a small number, this means that a large percentage of the numbers less than 12 divide it. This gives the group Z12 some interesting properties, there’s a wide range of element orders. Tritones come in pairs, major thirds come in 4 sets of 3 thirds, minor thirds come in 3 sets of diminished seventh chords, etc, major seconds form the two whole tone scales. Fourths (5) and fifths (7) don’t divide 12, so they generate the group, producing the cycle of fourths and fifths. If we look at the groups next door, Z11 and Z13, they are of prime order, so every element generates the group => there is a cycle of “fifths” for each interval. Beautifully symmetric, perhaps, but less structure to enjoy.

lachlan_gray 4y ago

Another nice thing about 12 notes is that 12 has lots of factors. It leads to a lot of cool symmetries which I think have to do with how we understand music. For example, since 12 is divisible by 3 and 4, diminished chords (4 notes separated by intervals of 3) and augmented chords (3 by 4) don’t have a definitive centre to our ear. As a result they both create different flavours of uncertainty and suspense. But if you break the symmetry, e.g. by moving any note in an augmented chord one to the left on a piano, the chord takes on a colour (in this case becomes a major chord). Other equal temperament scales would end up having similar patterns, but at the cost of complexity. having closer to 20 or 30 notes in an octave would increase the number of patterns to recognize, and probably make them less distinct to us without practice. It’s also interesting to me that we put 12 numbers on clocks as well.

pyinstallwoes 4y ago
And 12 months in a year. Around around frequencies abound. In ratio and interval, all things are sound. Rational reasoning abound, just don’t be too loud. A note here. A note there. A harmonic there and there. You say blue, I say G, and now we aught to see the cymatics inside you and me! Altogether now, ratio, ratios, frequency make everything between you and me. Hey now, hey now, just make music, it’s what we all need now.

dublin 4y ago

Thanks for this. It is perhaps the clearest explanation ever of the musical scale, and why it is the way it is. I've literally watched dozens of YouTube videos on this topic in the past week, and none of them are as clear and insightful as this was. You do have to wonder what an optical octave would look like, though - our visual range is, unfortunately, a bit short of an octave there... Maybe the same "tone" of a kind of extreme magenta?