[HN Gopher] Why 12 notes in Western music?
___________________________________________________________________
Why 12 notes in Western music?
Author : xchip
Score : 154 points
Date : 2022-08-29 14:51 UTC (8 hours ago)
(HTM) web link (github.com)
(TXT) w3m dump (github.com)
| mikewarot wrote:
| 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.
| dhosek wrote:
| 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 (1/2): A' (up an octave)
|
| Twelfth ( 1/3 ): E (up a fifth from the octave)
|
| Double Octave (1/4): A''
|
| 1/5 : C#2
|
| 1/6 : E
|
| 1/7: 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 1/3
| 1/3 1/4, 1/3 1/4 1/5 , etc.
| ajross wrote:
| 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.
| jbverschoor wrote:
| 12 notes, 12 hours, 12 months., even 12 monkeys.
| lioeters wrote:
| > The duodecimal system, which is the use of 12 as a division
| factor for many ancient and medieval weights and measures,
| including hours, probably originates from Mesopotamia.
|
| https://en.wikipedia.org/wiki/Duodecimal#Advocacy_and_%22doz...
| retrac wrote:
| 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 wrote:
| > 60 is the smallest composite number with three prime
| factors
|
| Er, that'd be 30 I think, but the main point stands.
| BitwiseFool wrote:
| 2 is such a nifty prime, we multiplied by it twice. It
| was just too perfect not to.
| 752963e64 wrote:
| kazinator wrote:
| 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/
|
| And:
| 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.
| aidenn0 wrote:
| You also get the western[1] chromatic scale if you go up by a
| fifth (which is pleasant sounding for many reasons) ad
| infinitum.
|
| C -> G -> D -> A -> E -> B -> F# -> C# -> G# -> D# -> A# -> E#
| -> B#(C)
|
| Of course the B# you end up with at the end is 531441/4096
| which is 1.3% higher frequency than 7 octaves above the
| starting C. If you want to generate flats as well, by traveling
| in the opposite direction, you end up with different notes for
| the flats. 12-TET is just the modern way of using a constant
| frequency ratio to divide the octave to match the 12 notes used
| by Pythagoras. The ancient greeks were unlikely to come up with
| it due to the reliance on irrational numbers.
|
| 1: Pythagoras is often credited with this scale, China also
| independently invented this scale and it's not clear which came
| first (https://en.wikipedia.org/wiki/Sh%C3%AD-%C3%A8r-l%C7%9C)
| kazinator wrote:
| That's just from modulo math. A fifth is 7 semitones, which
| is relatively prime to 12. Thus 7x (mod 12) hits all the
| elements of the modulo 12 congruence for x in 0..11. We cover
| all notes in the first twelve steps.
|
| But say we are not assuming a twelve note system in the first
| place; how do we get twelve notes?
|
| Going up a fifth and then down a fourth is very close to a
| tone. We can do that five times before we approximately hit
| an octave, yielding six notes. The fifths above those notes
| are six additional notes.
|
| We see that in your diagram:
|
| C -> G -> D -> A -> E -> B -> F# -> C# -> G# -> D# -> A# ->
| E# -> B#(C)
|
| in that we can interpret every other note as the whole tone
| scale:
|
| C -> D -> E -> F# -> G# -> A# -> B#(C)
|
| and their fifths:
|
| G -> A -> B -> C# -> D# -> E# -> Fx(G)
|
| Fifths fill the gaps in the whole tone scale to recover the
| other whole tone scale.
|
| Going back to the 12 tone math again, 2 and 12 have a common
| divisor, so steps of 2 modulo 12 cycle through 6 symbols.
| There are 6 others left out, reachable by some relatively
| prime step like 7 (perfect fifth).
| aidenn0 wrote:
| > That's just from modulo math. A fifth is 7 semitones,
| which is relatively prime to 12. Thus 7x (mod 12) hits all
| the elements of the modulo 12 congruence for x in 0..11. We
| cover all notes in the first twelve steps.
|
| > But say we are not assuming a twelve note system in the
| first place; how do we get twelve notes?
|
| My diagram showed an (approximate) 12 note cycle assuming
| only a 3:2 ratio for a fifth. There are lots of good
| reasons to use a fifth as the basic interval[1]. In no way
| does this assume a 12-note system.
|
| The 12 notes don't come from "filling in" between the 7
| notes of the diatonic major scale, they come from
| continuing the pattern until a near-cycle happens; is your
| argument that the 1.2% error in the cycle is arbitrary?
| it's less than 1/4 the next largest difference and slightly
| more than the rule of thumb for how much "anybody" can
| hear. The next time we get closer to a cycle is at 41, and
| we don't get closer by an order of magnitude until 53.
|
| 1: And in fact the fifth is used as a basis for many other
| scales both western and otherwise (Note that the first 5
| notes are the major Pentatonic scale and the first 7 are
| the major diatonic scale).
| plq wrote:
| The system of the Turkish Classical Music is a bit different:
|
| * https://www.sufi.gen.tr/nota-sistemi/en
|
| * http://www.turkishmusicportal.org/en/types-of-turkish-music
| j7ake wrote:
| 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 wrote:
| There are keyboards that will do this. I remember seeing this
| advertised back in the 90s.
| abetusk wrote:
| 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_...
|
| [1]
| https://github.com/abetusk/scratch/blob/release/src/music/be...
|
| [2] https://www.johndcook.com/blog/2010/10/20/best-rational-
| appr...
|
| [3] https://en.wikipedia.org/wiki/Farey_sequence
| buildsjets wrote:
| "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-mackenzi...
| mcbrit wrote:
| 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.
| nikeee wrote:
| These numbers are called "Highly Composite Numbers" [0].
| Basically, it is a series of numbers where each number has more
| factors than the number before it (and is the first number with
| that number of factors). As you hinted, they are especially
| useful if your number system does not have fractions or decimal
| places and you still want to divide things.
|
| You may recognize the beginning of the series: 1 2 4 6 12 24 36
| 48 60 120 180 240 360 720
|
| Numberphile [1] calls them "Anti-Primes".
|
| [0]: https://en.wikipedia.org/wiki/Highly_composite_number [1]:
| https://www.youtube.com/watch?v=2JM2oImb9Qg
| coldtea wrote:
| > _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.
| sleepdreamy wrote:
| Micro-Tonal bands such as King Gizzard play around/bend the rules
| on the whole 12 notes in music thing.
| joegahona wrote:
| Familiar with the band, but not examples of this. Can you
| recommend any tracks that demonstrate?
| inkcapmushroom wrote:
| 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.
| dr_dshiv wrote:
| I suspect it has something to do with resonance. Resonance occurs
| when frequencies match approximately. It isn't just in the
| fundamental or pitch frequency of two notes-- resonance can also
| occur via frequency matching in the shared overtones of two
| notes.
|
| Consonant notes tend to share a lot of overtones. I have heard
| that the pentatonic scale maximizes internote resonance. This
| seems relatively straightforward to test empirically.
|
| The first known scientific experiment (empirical test of
| mathematical model) was the attested case of the pythagoreans
| casting bronze chimes in the same rational proportions of lengths
| of a string. The experiment demonstrated that small integer
| ratios produce consonance.
|
| Here is the most recent and up-to-date theory of harmony in music
| (that I know):
| https://downloads.spj.sciencemag.org/research/2019/2369041.p...
| euroderf wrote:
| 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).
| locusofself wrote:
| 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 wrote:
| 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/
| khitchdee wrote:
| 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.
| locusofself wrote:
| I've seen those! Pretty cool. I've also seen people put
| intermediate frets on certain notes.
| tshaddox wrote:
| 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.
| xhevahir wrote:
| I mentioned this fretboard yesterday in another thread (
| https://news.ycombinator.com/item?id=32626218 ). One of the
| main proponents of the design talks about its advantages
| here: https://youtu.be/D8EjCTb88oA
|
| I'm not crazy about the sound, myself. It's kind of like a
| guitar sample played through a MIDI keyboard.
| tshaddox wrote:
| Yeah, that video is linked from the article I was
| responding to. The video certainly makes sense: he's
| demonstrating that intervals sound the same in every
| position and specifically that octaves are very in tune
| across huge distances on the fretboard.
|
| From the small clips I've heard, it sounds great to me,
| especially when the guitar is playing alone. Back when I
| played a lot of guitar I was often pretty bothered by
| intonation issues, even with decent gear that was setup
| well and didn't seem to bother other (much better)
| guitarists. But of course guitars often play together
| with 12-TET instruments that are likely to be much more
| accurately tuned (like a piano or organ), and it makes
| sense that the every-so-slightly out-of-tune nature of
| guitars has become part of what sounds distinctively
| _guitar-like_ , especially the specific tuning you're
| likely to hear a lot with common guitar chord voicing in
| many styles of music.
| InCityDreams wrote:
| Apart from ensuring my intonation was spot on, and my frets
| nicely shaped, scalloping my frets allowed me a lot of what may
| be the 'sweetness' you are after - a slightly harder pressure,
| without finger gymnastics, allows minor pitch corrections (but
| you can't go flatter). Regular guitars just seem dead to me. I
| also shaved parts of my neck - significantly - along different
| parts of it ie it's not 'smooth', it's shaped only for me, and
| how i want to play in the different registers.
| SeanLuke wrote:
| 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.
| ledauphin wrote:
| 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.
| elihu wrote:
| For the intervals they look at in the article, the perfect 4th
| and 5th, 31-EDO is worse -- about 5 cents of error, versus
| about 2. What 31-EDO has is a major third that's almost dead-
| on, and a minor third that's a lot closer.
|
| https://en.wikipedia.org/wiki/31_equal_temperament#Interval_...
|
| 41-EDO though has a perfect 4th and 5th that are closer than
| 12-EDO, being off by about half a cent rather than about 2
| cents. In fact, 41-EDO is better at every commonly-used
| interval than 12-EDO, plus it adds a lot of very good 7-limit
| intervals too (i.e. ratios with sevens in them like 7:4, which
| is way off in 12-EDO).
|
| By a weird set of mathematical coincidences, 41-EDO is actually
| quite playable on guitar with the right layout. The trick is to
| omit half the frets and tune the strings so that each string
| has the notes that the strings above and below it lack. Tuning
| by major 3rds, you get a whole lot of useful notes clustered
| where they're easy to play. There's a handful of us (in
| Portland mostly) trying to promote this idea:
| https://kiteguitar.com/
| [deleted]
| bryanlarsen wrote:
| According to 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 wrote:
| 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!)
| thomasqbrady wrote:
| 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...
| bryanlarsen wrote:
| The why is because the 5th is the primary non-octave overtone
| of a vibrating string, and the 4th is an inverted 5th.
| diffeomorphism wrote:
| 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....
| thomasqbrady wrote:
| I guess I wasn't clear. I'm not saying the 4th and 5th notes
| don't have a special sound to anyone. I'm saying that 1) just
| exactly how important their resonance is to you is influenced
| by your culture. It's not that atonal musicians didn't notice
| the resonance. They weren't drawn to it as much. 2) The logic
| given was "if you want your scale to include the 4th and 5th,
| then 12 notes is inevitable." The "if you want your scale to
| include the 4th and 5th" is the a built in assumption that
| you want something like the western scale, so it doesn't seem
| that impressive to me that they then arrive at the 12-tone
| scale.
| herghost wrote:
| 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 wrote:
| _> 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).
| mcphage wrote:
| That's not really circular, though. It does start from the
| assumption that 4ths and 5ths sound pleasant, but uses that to
| build possible scales, some of which are more compatible with
| 4ths and 5ths than others.
| jollybean wrote:
| No, 4ths and 5ths are resonant. They sound better to a lot of
| people. They are completely distinguishable from arbitrary
| intervals right near there.
|
| Most of the notes on the Western scale fall into this
| orientation, i.e. there is reasoning for it. It's not just
| arbitrary.
| papandada wrote:
| 3^12 is within 1.5% of 2^19
| analog31 wrote:
| 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.
| MatthiasWandel wrote:
| 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.
| dahart wrote:
| > 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
| sabotista wrote:
| 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 wrote:
| 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 wrote:
| It may not be true, but I still loved telling that story to
| my students.
| thirteenfingers wrote:
| 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 wrote:
| 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-
| ei...
| wizofaus wrote:
| > 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 wrote:
| 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_Une
| qual_...
|
| - https://www.proquest.com/openview/b22142f819768ae82464a
| a2679...
| wizofaus wrote:
| I still feel that way about certain keys, even playing on
| an exactly equal tempered keyboard. I don't think the
| degree to which certain intervals might vary between keys
| is necessarily the important factor.
| joegahona wrote:
| Those descriptors ("austere," etc.) have always struck me
| as subjective -- I'm not one to tell people what mood
| they're getting from certain keys. But a root major chord
| will have a much different feel in, e.g., C#-major on an
| 18th-century tuning than in equal temperament.
|
| I have this CD [1] in a box somewhere but can't find it
| on Youtube. It's a few Beethoven sonatas in the
| temperament he would've used. Just sounded out-of-tune to
| me in certain parts (especially during the Waldstein),
| but I don't have perfect pitch. The booklet that came
| with that CD is really helpful in understanding all this,
| and I think that might be where I learned about that Owen
| Jorgensen tome.
|
| There's no shortage of similar experiments on Youtube.
| This one [2] has a wild one in just intonation, but I
| doubt that temperament was still used when Mozart was
| composing.
|
| [1] https://www.amazon.com/Beethoven-Temperaments-
| Historical-Tun...
|
| [2] https://www.youtube.com/watch?v=lzsEdK48CDY
| wizofaus wrote:
| Thanks for that link, I don't know if it demonstrates
| "just intonation" though? But the 1/4 Comma Meantone
| tuning just sounds horrible the moment a diminished chord
| comes into the picture.
| tripa wrote:
| More generally, I'd be curious to know how they'd
| practically tune a keyboard to 12TET before the
| electronic chromatic tuner got around. Start with
| Pythagorean fifths then compress ever-so slightly? How'd
| you keep them... equal?
| captain_trips wrote:
| 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 wrote:
| There was 12 notes in Western music before 12 equal divisions.
|
| This Mozart piece played in mean-tone temperament (historically
| accurate) has better tensions and resolutions than the equal
| tempered version.
|
| https://www.youtube.com/watch?v=lzsEdK48CDY&t=700s
|
| (The chipper ending to this piece is believed to be not written
| by Mozart.. the song was incomplete when he died.)
|
| I even prefer Chopin in unequal temperament, but I'm not as
| confident about whether Chopin used 12 equal divisions.
|
| https://www.youtube.com/watch?v=fJT5Q6HooyA
| TheOtherHobbes wrote:
| 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
| gnulinux wrote:
| Not a musician, but as a hobbyist composer and music theory
| enthusiast, I think anything more than 24-TET is overkill in
| terms of daily practice. I don't think there is sufficient
| expression to adding anything more than quartertone to justify
| making your theory and practice so much more complicated.
| 24-TET is convenient because all your 12-TET theory works
| exactly the same, except now you have quartertone, in addition
| to semitone. This gives really interesting intervals, although
| not all of them will be usable in practice. In terms of
| instrument practice, 19-TET is a good middle ground since it
| maps unambiguously to 12-TET while introducing useful and
| expressive intervals such as septimal minor third.
| TakeBlaster16 wrote:
| I think 19-TET is the most viable alternative to 12-TET. It
| fits in a standard piano form factor by adding a few black
| keys, and uses the same note names everyone is used to.
| https://commons.wikimedia.org/wiki/File:19_equal_temperament...
|
| And it sounds really unique!
| https://www.youtube.com/watch?v=bJfTu1Y2H44
| diydsp wrote:
| 22-TET is a sweet spot in the curve of "Average Error
| Distance From Harmonic Interval" http://www.gweep.net/%7Eshif
| ty/portfolio/musicratios/index.h...
| shams93 wrote:
| 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 wrote:
| 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-tempere....
| analog31 wrote:
| 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.
| WalterSear wrote:
| It's still twelve notes.
| elihu wrote:
| 41-EDO is surprisingly playable on guitar, if you omit half the
| frets. The trick is to tune the strings so that each string
| only has half the notes, but the notes that aren't there are
| available on neighboring strings. It seems like it shouldn't
| work, but it does.
|
| https://kiteguitar.com/
|
| https://kiteguitar.com/theory/fretboard-charts-downmajor-tun...
| acadapter wrote:
| The 12 tone scale provides good approximations of many
| aesthetically pleasing (mathematically simple) intervals.
| waffletower wrote:
| Bach explored another consideration, key changes within
| compositions, quite thoroughly: for example:
| 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
| josephg wrote:
| 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?
| smlacy wrote:
| 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 wrote:
| are the post and your comment not mathematically equivalent
| statements?
| elihu wrote:
| 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.)
| smlacy wrote:
| 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 wrote:
| 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.
| layer8 wrote:
| In other words, 1.5^12 = 129.746... [?] 2^7.
| gnulinux wrote:
| It's worth mentioning that stacking fifths this way creates
| something pretty close to an octave, but it's still noticeably
| different from an octave. The difference between an octave and
| 12 fifths is called "Pythagorean comma", it's about 23.46 cents
| and it'll be obvious to all humans who don't have a
| speech/hearing impediment, even if you were never musically
| trained. (It's believed humans are sensitive to small intervals
| like this because it's required to process spoken human
| language). Traditionally, it's considered anything more than a
| synctonic comma (i.e. 21.51 cents) will feel different to even
| untrained ears. (but of course, this is just the theory, in
| reality there is some small variance between humans,
| background, culture etc).
|
| https://en.wikipedia.org/wiki/Pythagorean_comma
|
| This is pretty significant to mention, because even though 12
| fifths are "very close" to an octave [1], they're far apart
| enough that no one will feel an octave. In music, near misses
| like this are very significant since they cause the feeling of
| harmonic "dissonance". Since 12 fifths is a _very_ dissonant
| interval (since it 's so close to an octave but still
| noticeably out-of-tune) Western music developed techniques
| (such as well-temperament, equal temperament etc) to make sure
| this "error" is blend in. We achieve this by changing other
| notes ("tempering") ever so slightly so that critical intervals
| like fifths (or in other cases thirds etc) are stable. Other
| cultures, such as classical Indian music, have their own way
| dealing with Pythagorean comma! Since music is a universal
| phenomena found in all cultures, but it doesn't manifest the
| same way in all cultures (e.g. not all cultures give the same
| kind of emphasis to pitch or harmony Western music gives)
| various cultures developed their own different and interesting
| ways to work around this "error".
|
| [1] To be precise, we're referring to the difference between 12
| fifths and 7 octaves. Since an octave is so consonant, sounds N
| octave(s) apart feel "equal" albeit with different timbre.
| jacquesm wrote:
| Somewhere there is a perfect universe where 12 fifths form an
| octave.
| [deleted]
| CobrastanJorji wrote:
| 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.
| gnulinux wrote:
| This may or may not be true. In fact, it seems unlikely
| to me your claim is true.
|
| In nature, sounds produce harmonics i.e. when two objects
| collide they usually create waves of frequency f, 2f, 3f,
| 4f... in various (usually exponentially decreasing)
| weights. It's very rare to find pure sounds (i.e. only f
| frequency) in nature. The interval between f and 2f is an
| octave apart (1:2 ratio); the interval between 2f and 3f
| is a perfect fifth (2:3 ratio). So, when you actually
| hear a sound, you actually hear an octave and a fifth
| too, and how dominant this octave and fifth changes the
| "timbre" of the sound. This way, you know the source of
| the sound independent of the frequency. For example, both
| a violin and a piano can produce the note A4 at 440Hz,
| but anyone can easily determine if it's a piano or
| violin. The reason is, when a piano produces A4, it
| sounds not only just 440Hz but also 880Hz and 1320Hz
| etc... too and the relative volume of 880Hz and 1320Hz
| will be different than that of violin. Your brain
| automatically interprets these volume weights as "timbre"
| and the fundamental frequency 440Hz as "pitch".
|
| Consequently, in order for your brain to be able to
| process the timbre of a sound it needs to find octaves
| and fifths between each fundamental note it hears. This
| means there might be something universal about octave and
| fifth (and other decreasingly consonant intervals such as
| major third etc...). Maybe we "understand" music because
| our brain is hard-wired to search for octaves and fifths
| in all sounds, in order to analyze timbre and in order to
| process spoken language. If this hypothesis is true,
| maybe an alien species could have octave/fifth/major
| third based music too! (if they have music at all, of
| course)
| jacquesm wrote:
| There is also the shape of the individual waveforms to
| take into account a piano has a more or less sinusoidal
| wave and a violin is more of sawtooth (due to the
| stickslip of the bow moving across the string(s)).
| ajuc wrote:
| You're saying the same thing - the combined sawtooth wave
| is just the sum of all the sinus harmonics.
| gnulinux wrote:
| Is it not true that the shape of the waveform
| (sinusoidal, saw-like etc) is created by the relative
| weights of each harmonic? E.g. if you take any random
| sound wave, Fourier-transform it, you'll find the weight
| of each harmonic. Or are you saying there is a separate
| quality to sound waves that can cause their shape to be
| different even if each harmonic has the same relative
| weight with respect to the fundamental?
| abstrakraft wrote:
| Yes, this quantity is the relative phase of the
| harmonics, although the human ear is generally considered
| to be insensitive to phase.
| rrrrrrrrrrrryan wrote:
| Eh, I think you'd need a new universe, as it's a pretty
| basic principle of math: 3^12 ~= 2^19
|
| You can take two long strings of equal length (A & B),
| and pluck them, and they'll make the same sound. Then you
| can take scissors cut string A in half, and it'll sound
| different. (This is an octave.)
|
| Then you can cut string B into thirds, and it too will
| sound different.
|
| If you pluck both of your new strings at the same time,
| you'll find they sound quite nice together (the
| difference between these two is called a "fifth").
|
| And after 12 rounds of cutting string B into thirds, and
| 19 rounds of cutting string A in half, you'll happen to
| have a string from each group that are _almost_ identical
| in length and pitch.
|
| But it won't line up exactly! They'll be about 1.4%
| different in length, which roughly works out to a
| quarter-semitone difference in pitch (i.e. 1/4th the
| distance from one piano key to the next).
| tshaddox wrote:
| But who says the creatures need to perceive sound in such
| a way that the harmonic series has sensory significance?
| To be honest I've never seen a compelling evolutionary
| explanation for why "hearing the harmonic series"
| developed in the first place. It obviously seems useful
| to be able to perceive sounds generated by (roughly)
| harmonic oscillators, since those occur naturally for
| various reasons, but why octave equivalence?
| rrrrrrrrrrrryan wrote:
| I don't think you need to imagine "creatures" - there are
| humans on earth whose culture's music doesn't have the
| concept of octaves, let alone fifths. All the real action
| in their music is in its rhythmic complexity.
|
| But my comment was in reply to this statement
| specifically:
|
| > Somewhere there is a perfect universe where 12 fifths
| form an octave.
|
| For this to be true, I think you'd indeed need a new
| universe where 3^12 = 2^x, where X is a whole integer.
| tshaddox wrote:
| I'm not talking about explicit notions of octaves and
| octave equivalence, although those do exist in very many
| musical traditions and appear to be extremely widespread
| in musical traditions where tones have names (if not
| ubiquitous--I'm not aware of any exceptions). I was
| referring to the claims that octave equivalence is in
| some sense hard-wired in the human brain, or sometimes
| claimed to be all or many mammalian brains. I'm not
| qualified to evaluate these claims or even whether how
| well-accepted they are among experts, but such claims do
| seem to pop up all over the place when discussing music
| perception.
| inopinatus wrote:
| I've heard there is a secret chord, that prophets play to
| please their lords.
|
| But if you care for music less than algebra, just know it
| goes like this; the fourth, the fifth: the minor falls, the
| major lifts.
| thaumasiotes wrote:
| That would have to be a universe in which the fundamental
| theorem of arithemetic was false. Otherwise, the only way
| to cross an interval that is an integer number of octaves
| is to take steps that are also octaves.
| coldtea wrote:
| What the parent proposed doesn't require different math.
|
| Just a species that hears out current slightly offset
| divisions in 12-tet as perfect, as opposed to only
| hearing integral ratios as perfect.
| thaumasiotes wrote:
| Viewing sound waves that don't synchronize with each
| other as being better matched than sound waves that do
| synchronize is less plausible than violating the
| fundamental theorem of arithmetic. There's no element of
| coincidence in whether two frequencies harmonize.
|
| That hypothetical species wouldn't recognize two notes an
| octave apart as being similar, so there would be no
| reason to imagine a circle of fifths in the first place.
| coldtea wrote:
| > _Viewing sound waves that don 't synchronize with each
| other as being better matched than sound waves that do
| synchronize is less plausible than violating the
| fundamental theorem of arithmetic_
|
| Actually it's perfectly plausible. People couldn't
| imagine others enjoying hearing a tritone -- and nobody
| in 1800 would imagine we'd enjoy listening to punk, hip
| hop, or Death Metal, and yet, millions do. We can surely
| consider a race that doesn't require intervals to
| absolutely synchronize.
|
| > _That hypothetical species wouldn 't recognize two
| notes an octave apart as being similar, so there would be
| no reason to imagine a circle of fifths in the first
| place._
|
| Note how I said that this imagined alien race would
| consider the "divisions in 12-tet as perfect". Note how
| those _do_ include a perfect octave, that we already
| recognize as such. The alien race wouldn 't change that,
| they'd just need to also consider perfect the slightly
| off ratios in 12-tet.
| thaumasiotes wrote:
| > Actually it's perfectly plausible.
|
| No, for this to be plausible, you would need to have some
| theory of why the two notes matched with each other.
| There is no such theory; they have been chosen to be as
| unmatched as possible.
|
| > nobody in 1800 would imagine we'd enjoy listening to
| punk, hip hop, or Death Metal
|
| This is false.
|
| >> That hypothetical species wouldn't recognize two notes
| an octave apart as being similar
|
| > Note how I said that this imagined alien race would
| consider the "divisions in 12-tet as perfect". Note how
| those _do_ include a perfect octave, that we already
| recognize as such. The alien race wouldn 't change that
|
| Sure. In that case, we can also imagine an alien race
| that perceives all and only the light that fails to reach
| its eyes.
|
| Then again, perhaps being able to form a sentence
| describing something doesn't guarantee that the situation
| described is possible.
| posterboy wrote:
| I'm not comfortable with this first refering to a natural
| human tendency and then a western harmony, which is pretty
| much accquired, as if it were a natural consequence.
|
| A simpler way to go about this is using the chromatic scale,
| drawing multiples of C0 upto C8 so that C7 to C8 spans an
| octave, and then fixing F according to a table of equal
| temperament.
| coldtea wrote:
| > _I 'm not comfortable with this first refering to a
| natural human tendency and then a western harmony, which is
| pretty much accquired, as if it were a natural
| consequence._
|
| Well, western harmony is based on a set of natural human
| tendencies formalized.
|
| There are other ethnic music practices, also based on
| natural human tendencies.
|
| The parts that are acquired are built on top. But most/all
| music practices (western or otherwise) start with natural
| human tendencies, as their foundations.
| frereubu wrote:
| Fifths don't sound so great after a while. This tuning leads to
| the dissonant "wolf interval" -
| https://en.wikipedia.org/wiki/Wolf_interval - so it was largely
| replaced by the well temperament -
| https://en.wikipedia.org/wiki/Well_temperament - used by Bach
| in _The Well-Tempered Klavier_.
| jefftk wrote:
| 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%.
| tetris11 wrote:
| (For those wanting to hear these frequencies:)
|
| aplay -d 2 -r $freq
| xchip wrote:
| 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
| AndrewUnmuted wrote:
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