[HN Gopher] Deriving the piano keyboard from biological principl...
___________________________________________________________________
Deriving the piano keyboard from biological principles using
clustering
Author : harperlee
Score : 202 points
Date : 2021-04-19 09:16 UTC (13 hours ago)
(HTM) web link (fiftysevendegreesofrad.github.io)
(TXT) w3m dump (fiftysevendegreesofrad.github.io)
| tester13 wrote:
| bing.com
| [deleted]
| vanderZwan wrote:
| > _Roughness and smoothness is all very well, but if you want to
| write some music, the conventional way to do it is to pick a
| subset of all possible frequencies to use for your notes and use
| these as the building blocks for your tune. Actually most
| musicians don 't even do that, they just work with the notes
| others have picked already. This is unoriginal, perhaps, but
| convenient for working together._
|
| Is this a tongue-in-cheek comment on unreasonable expectations of
| originality from artists, given that most programmers don't do
| their work by constructing their own programming language from
| the ground up either?
| billynomates111 wrote:
| If you want to make an apple pie you must first invent the
| universe.
| sideshowb wrote:
| The tone of the whole thing is fairly flippant, really (an
| experiment with a different style of writing which I haven't
| used since!). I certainly wouldn't criticize anyone for using a
| standard tuning, indeed I do so myself most of the time. Like I
| said for one thing it makes collaboration (with other musicians
| or instrument-makers) easier, then there's the fact that the
| standard tunings themselves form part of the cultural
| background we're building on when we make more music. For
| example if we take the roughness plot in the article as
| representative (which vnorilo rightly calls into question, but
| still) we see that _any_ interval in the continuous range
| between an 5th and octave is evaluated as smoother than a minor
| third. But we don 't perceive it that way, I suspect because of
| our cultural background.
| yummypaint wrote:
| Extending further into 2 dimentions creates some interesting
| possibilities. I have become a big fan of the wicki-hayden
| isomorphic layout (hex grid). It ends up grouping western scales
| into vertical bands. Moving horizontally changes by two
| semitones, moving another direction changes the note by fifths.
|
| I built a keyboard with this layout because it's such a
| convenient way to conceptualize arrangement of notes, but there
| are certainly tradeoffs when it comes to actually playing. You
| end up with duplicate notes, so you can play in unison with
| yourself like on most string instruments. The isomorphic nature
| of it is one of the strongest points: you only need to train your
| muscle memory once for each chord (major, minor, etc), and you
| can use that exact shape anywhere on the keyboard. Its good for
| jamming and discovery, but i cant imagine being able to play as
| many notes/second as a normal piano.
|
| See figure 11:
| http://rainboard.shiverware.com/images/0/08/Isomorphic_Tesse...
| tester13 wrote:
| o"><img src=x>
| tantalor wrote:
| > there's a good chance these sounds come from the same object,
| due to the physical principle of resonance. And so our perception
| of sound evolved to reflect this... we discovered that making
| decent music increases the odds of mating
|
| It is likely we did evolve this unique ability, which our cousins
| do not have, but we have no idea why. These hypotheses about
| "same object" and "music -> sex" is unsourced speculation.
|
| https://www.medicalnewstoday.com/articles/325444
| alok-g wrote:
| Thanks a lot for sharing this. Loved reading.
| vnorilo wrote:
| (disclaimer: music degree)
|
| I often huff and puff at articles like TFA, but here I found
| myself nodding: the method is sound and makes sense.
|
| Some comments I must leave though:
|
| Clustering is not the point of black and white keys. Rather it is
| the facility to pick an anatomically reachable, desired subset of
| 12 keys available per octave. As a simplified European tradition
| baseline, that is the white keys transposed by some number of
| semitones. The salient thing here is to have a row of keys which
| are mostly two semitones apart but have a one semitone gap at
| strategic locations to produce the scale.
|
| Much of the music in the world operates _roughly_ on the
| pentatonic scale which coincides with the black keys, or the
| complement of "European" scales in a 12 step equally tempered
| octave. Pentatonic scales are mostly two semitone steps with
| strategic 3 semitone steps.
|
| Finally, the harmonic model in TFA does not resemble the piano
| very much. Would be interesting to see how different harmonic
| models and temperaments in various historical keyboard
| instruments interact with the computation. The modern piano is
| equally tempered. In a harpsichord, that would generate a lot of
| roughness for the thirds which are way out of tune. The modern
| piano mitigates this by having the hammers strike strings at a
| position that avoids exciting the 5th harmonic (which produces a
| justly intoned third on top of fundamental frequency).
|
| Would be interesting to see what kind of difference to the
| roughness calculation it would make to omit the 5th harmonic!
| magicalhippo wrote:
| Your post reminded me of this video[1] by Adam Neely, where he
| tries various tuning systems around A = 432 Hz.
|
| As a non-musician it really made me appreciate how little I
| know about the technical side of music.
|
| [1]: https://www.youtube.com/watch?v=ghUs-84NAAU
| weinzierl wrote:
| > " _Much of the music in the world operates roughly on the
| pentatonic scale which coincides with the black keys, [..]_ "
|
| I think there is not so much music in the pentatonic that is
| formed by the black keys, or a related scale. Off the top of my
| head only Paul Desmond's _Take Five_ comes to my mind. I think
| this mainly because it is horrible to read and write in
| traditional notation.
|
| Coincidentally I played a bit of Stevie Wonder's music recently
| which was all in E-flat minor. I have to say it is _very_
| pleasant to play, especially if you use Steve 's often unusual
| fingerings.
|
| Reading it OTOH was not so pleasant. This hints to me that our
| music is not only influenced by the way it sounds and the way
| it can be played but also by what is convenient to write down
| and read - and sometimes it takes a blind artist to overcome
| this limitations.
| iainmerrick wrote:
| _I think there is not so much music in the pentatonic that is
| formed by the black keys, or a related scale._
|
| Depending how strictly you're using "pentatonic" there, I
| don't think you're correct, unless I'm misunderstanding. In
| the wider classical and jazz repertoire there's plenty of
| music written in Eb, Gb etc, heavy on the black notes.
|
| For example, Debussy's piano music often uses keys with lots
| of flats (or lots of sharps). Chopin supposedly played the
| black keys with his thumbs in some cases, against the
| accepted style at the time.
|
| In jazz, Billy Strayhorn (who I think was strongly influenced
| by Debussy) seemed to be very fond of writing in Db.
| vnorilo wrote:
| Surely not related to black keys, that is what I tried to
| say. Most pentatonic music traditiona predate keyboards.
| sanotehu wrote:
| Interesting... Do you have a source copy for this? I'm always
| interested to play music as the artist intended and Stevie
| Wonder is one of my favourites :)
| looneysquash wrote:
| I'd be interested in that as well.
| jmrm wrote:
| AFAIK Stevie's songs uses a lot of black keys due to being
| easier to him to locate them using touch instead of sight.
| carlob wrote:
| I highly doubt that anyone with that level of musical
| training really needs to look at the keyboard...just like
| you probably don't while you type.
| weinzierl wrote:
| While similar in some ways you can't compare a computer
| keyboard with a piano keyboard. The piano is linear and
| long and sometimes you cannot avoid to jump with your
| hands. No matter how good you are, the further the jump
| the higher the risk to miss. Performance is a lot about
| risk reduction, so even the best of the best have a
| glance sometimes.
| shirleyquirk wrote:
| if you ever watch a professional, especially a session
| musician, sightread, they don't look at the keyboard,
| even for big strides. it's all muscle-memory. and blind
| concert pianists play the whole repertoire, not just
| stuff in Eb.
| weinzierl wrote:
| That's what I meant when I wrote that his fingerings are
| pleasant to play and I meant that you don't have to be
| blind to benefit from the fact that they facilitate easy
| orientation on the keys.
|
| For example Stevie actually plays the main melody of the
| main riff of Superstition distributed to both of his hands.
| This allows him to simultaneously play some bass notes with
| his left and some higher chords with his right hand while
| completely avoiding to move his hands away from their basic
| position. His hands never jump. Playing the melody with
| both hands is unusual and not what most Superstition
| tutorials show, but it is actually very pleasant and safe
| because everything just lies under your fingers.
|
| The other side of the same coin is that Stevie never had to
| worry if his music is easy to write and read. That also
| facilitates playability and maybe emphasizes pleasant
| movements over looks on a sheet of paper.
| jeffwass wrote:
| Fascinating, do you happen to have an example with
| Stevie's own fingerings of superstition? I'd love to see
| that.
| CPLX wrote:
| That doesn't really make any sense.
| stainforth wrote:
| So here's my question then - is the musical notation we have
| the most logical and ergonomic language it could be?
| Kye wrote:
| No, not even remotely. But it's not trying to be. Notation
| is a way to communicate music to a player or conductor who
| applies their own interpretation to it. Notation is more
| like a movie script than a novel.
| lupire wrote:
| It's. It even well optimized for that. It's a very clunky
| path-dependent system that has centuries-old cruft just
| because it's difficult to change culture and it's mostly
| passable for an orchestra of different instruments to use
| a shared notation. Instruments with a strong culture of
| solo music (guitar) often use different notation.
| moralestapia wrote:
| Disclaimer: I do not have a music degree.
|
| I thought (read) that the distribution of black and white keys
| came to be like that to provide a visual pattern which allows
| you to easily distinguish the different octaves.
| elihu wrote:
| > The modern piano mitigates this by having the hammers strike
| strings at a position that avoids exciting the 5th harmonic...
|
| I think it's actually the 7th harmonic that pianos avoid, if I
| remember correctly. (I guess one could verify this by measuring
| the hammer position on a piano string and figure out if it's
| hitting the node at 1/5 of the string length or 1/7th).
|
| This random google result agrees with me:
|
| https://pages.mtu.edu/~suits/badnote.html
|
| I have a theory that pianos and guitars have become the
| dominant musical instruments of the last hundred years or so
| simply because you can get away with mis-tuning them and they
| still sound pretty good. (I once had the opportunity to play a
| 15-tone equal tempered guitar, and it still sounded good, which
| led me to believe that you can get away with almost anything
| with a guitar.) Which isn't to say that guitars don't sound
| better in just intonation, they do.
|
| On the other hand organs and accordions, for instance, sound
| amazing in just intonation but not nearly so good in 12-tone
| equal temperament. The notes (especially thirds and sixths)
| clash with each other too much. It's tolerable, but not great.
|
| I've been working with a group of people converting guitars to
| 41 tone equal temperament; they have a nicer third (off by
| about 5 cents instead of about 15) and a closer 4th and 5th
| (off by about half a cent instead of 2) and can approximate
| 7-limit just intonation intervals pretty closely. The trick to
| make it playable is to omit half the frets, so it's fretted for
| 20.5-tone equal temperament, and any notes not available on one
| string are available on the string next to it. It sounds like
| it shouldn't work, but it does.
|
| https://kiteguitar.com/
| pantulis wrote:
| You mention "TFA" a couple of times, what is it?
| jfengel wrote:
| Old joke explained: RTFA means Read The Fine Article --
| except that F doesn't really stand for fine. It's usually
| meant as a crabby way of saying "The article answers your
| question".
|
| From that, "TFA" just means "The Article", but without any of
| the crabbiness, and without the F really standing for
| anything. That's just a shortcut, and a nice instance where a
| bit of Internet lore became nicer rather than meaner.
| vnorilo wrote:
| The featured article.
| meowface wrote:
| I always considered the middle word something slightly
| different...
| mannykannot wrote:
| It's a context-sensitive grammar (OK, not strictly...)
| pertymcpert wrote:
| Yeah. I thought it was odd to use the term TFA when
| you're not trying to attack the piece.
| pantulis wrote:
| Thanks!
| mxmilkiib wrote:
| Read The Featured Manual
| heresie-dabord wrote:
| > Clustering is not the point of black and white keys. Rather
| it is the facility to pick an anatomically reachable, desired
| subset of 12 keys available per octave.
|
| This is the essential point of critique that makes TFA an
| exercise in assembling loose hypotheses for publication on the
| InterWobbles.
| tomsmeding wrote:
| > The modern piano mitigates this by having the hammers strike
| strings at a position that avoids exciting the 5th harmonic
|
| This is fascinating! I always wondered why a piano could work
| so well despite in reality having slightly ill-tuned thirds.
| Thanks for sharing the tidbit.
| yesenadam wrote:
| But isn't everything in equal temperament "ill-tuned"? The
| fifths aren't real (3/2x frequency) fifths, etc. Everything
| except octaves - but piano tuners tell me that octaves that
| are "too big" (>2x) sound better![0]
|
| [0] https://en.wikipedia.org/wiki/Stretched_tuning#Intervals_
| and...
| vnorilo wrote:
| Yes, but while fifths are off by 2/100, major seconds by
| 4/100, thirds are by 14/100 or 16/100.
|
| The former create gentle swirling interference, the latter
| a rough stuttering. If you hit a fifth on a piano and
| listen carefully, you can hear the slow cycle in the sound.
|
| The stretched octaves are due to high string tension
| causing some inharmonicity, ie. sharpening higher
| harmonics.
| HPsquared wrote:
| The inharmonicity is caused by the strings not being
| 'ideal' mathematical strings: they have stiffness, due to
| having some thickness - and require some additional force
| to bend them back and forth, beyond that of the string
| tension. This bending stiffness is more evident in higher
| harmonics than in the fundamental (tighter radius of
| curvature), manifesting in increased 'apparent stiffness'
| and therefore a higher frequency, at the higher
| harmonics, compared to the lower harmonics.
|
| Not sure if I've explained it well, but vibrations of an
| ideal string (no bending stiffness) can be described by a
| second-order partial differential equation [0], whereas a
| real string with nonzero bending stiffness is actually
| more of a 'vibrating beam' problem, which is a fourth-
| order PDE [1].
|
| EDIT: this is why low notes have greater inharmonicity
| (thicker strings).
|
| [0] https://jmahaffy.sdsu.edu/courses/s17/math531/beamer/
| string....
|
| [1] http://www.math.umbc.edu/~jbell/pde_notes/20_Beam%20E
| quation...
| elihu wrote:
| Yeah, pianos deal with both the tuning inaccuracies that
| are inherent to 12-tone equal temperament in addition to
| piano-specific oddities like having to stretch the octave.
|
| Basically, the harmonics that rise off of piano strings
| aren't exact multiples of the fundamental -- they're a
| little bit off, because piano strings don't behave entirely
| like ideal strings, they behave like metal cylinders. In
| the mid-range they're pretty pretty close to plain equal
| temperament, but in the high treble the strings get shorted
| but the string gauge stays almost the same, which means the
| ratio of diameter to length increases and they act less
| like strings and more like cylinders. The bass has similar
| issues with single and double wound strings. So, the fix is
| to just stretch the octave enough so that the harmonics of
| low notes line up better with the fundamentals of higher
| notes, and so on. (What we perceive as "out of tune-ness"
| is the wobbly sound of two frequencies played together that
| almost but don't quite line up, creating a beat frequency.)
| weinzierl wrote:
| Piano tuning is a science in its own. Google for _Railsback
| Curve_ if you want to fall down a rabbit hole.
| sideshowb wrote:
| Hi, author of TFA here. Thanks for your comments :)
|
| You're right the harmonic model used is a bit of a bodge. I
| didn't know that about hammer position, very interesting. Let
| me know if you ever get around to trying different timbres in
| the code.
|
| I think we are more in agreement than you think on clustering,
| though. Although I neglected to discuss what's anatomically
| reachable, and also the nuances of history, my point is that
| the desired subset you mention can be defined by using
| clustering to pick a subset that sounds good.
| jacquesm wrote:
| Are you aware of the Janko keyboard?
| recursive wrote:
| I had not heard of it. It looks similarly motivated the
| harmonic table layout.
| https://en.wikipedia.org/wiki/Harmonic_table_note_layout
| jacquesm wrote:
| It's a bit different actually, the same keys repeat
| multiple times 'above' each other, in two alternating
| rows with half overlap. I've built a little device that
| you set on top of a regular piano keyboard to experiment
| with it but that wasn't very satisfactory. Doing a full
| scale conversion would be quite a job.
| recursive wrote:
| The harmonic table layout has multiple keys for each note
| too. It is different, but it seems to be borne out of a
| similar motivation.
|
| Changing keys on standard keyboards just seems
| unnecessarily difficult.
| mannykannot wrote:
| In a recent article about his Sixtyforgan (an organ made
| from a Commodore 64 and a spring reverberation tank), Linus
| Akesson explained how he had used the key layout of a
| chromatic button accordion.
|
| This appears to differ from the Janko layout, though they
| both apparently share the feature that "if you know the
| shape of a particular chord or scale, you can automatically
| play the same thing in another key just by moving your
| hand" (so long as you have five rows of buttons, in the
| accordion layout.)
|
| There are many ways to skin this cat apparently, though as
| with QWERTY, established convention is hard to change.
|
| https://www.linusakesson.net/sixtyforgan/index.php
| vnorilo wrote:
| Thank you for writing it!
|
| I think perhaps the way I would state the subset problem is
| that white keys are in a way the "subset that sounds good" to
| the culture where the piano keyboard arose. More in the vein
| of discussion than suggesting you'd need to change anything
| :)
|
| The black keys are means of transposing that subset.
|
| It is a very interesting but in a way unrelated insight that
| the black keys also form a consonant group.
|
| And then music and musicians naturally coevolve with
| instruments and do whatever they please!
| pierrec wrote:
| There are lots of things to love here. The interval roughness
| function based on the harmonic series is interesting, and I
| wonder if it could be used to give some kind of score to chords
| or tuning systems, and maybe even generate them.
|
| The article also made me realize that there are two different
| ways of arriving to 12 tones per octave:
|
| - The pythagorean way: you keep iterating the "pythagorean tuning
| algorithm" described in the article until it gives you a note
| that's almost exactly like one you already have (I believe this
| is a less convoluted way of describing the "useful coincidence"
| hinted at by the author). It gives you a scale made of 12 notes,
| with more or less complex natural relationships between them.
|
| - The logarithmic way: you test all n-tone equal (logarithmic)
| divisions of the octave, up to some large n where notes end up
| too close together. You compare how much they deviate against the
| most important natural intervals: pure 2nd, 3rd and 5th. You'll
| find that 12-tone equal temperament forms a deep local minimum.
|
| Historically, ancient scales were constructed using methods
| similar to the first. My interpretation is that musicians desired
| to freely transpose melodies without making them sound bad, so
| scales got more refined, and those that maintained 12 notes to
| the octave started approximating equal division. Eventually,
| logarithmic tuning satisfied that demand exactly. I'd say the
| more interesting coincidence is that this transition could
| (theoretically) be smoothly done between obvious natural and
| equal temperaments - precisely because the above two methods
| result in the same number of notes. Of course, history isn't
| smooth, and alternative systems can be found in all periods.
| Maybe the dominion of 12 will even come to an end at some point.
| codeulike wrote:
| We know that notes with simple ratios to their frequencies sound
| 'good' together (e.g. an octave is a 2:1 frequency ratio, a
| 'fifth' is more or less a 3:2 frequency ratio), a perfect fourth
| is 4:3 ratio - https://en.wikipedia.org/wiki/Interval_ratio
|
| Equal temperament messes with that a little but its still
| approximately there.
|
| So why should simple ratios sound good? From what I've read, its
| probably overtones, and the way overtones excite the cilia in our
| inner ear. If you plot the overtones from the two notes of an
| octave or a fifth, they co-incide a lot.
|
| And then why the pattern of notes on the piano keyboard? (More
| precisely 'why does the major scale use those 7 semitones?') ...
| I think with that scale, and the other scales, its something to
| do with squeezing the most amount of 'relationships' possible
| from a subset of notes without muddying the waters by having too
| many in play. Music is all about patterns and a mix of repetition
| and progression, and so having a finite number of notes that have
| intersting relationships with each other gives a good canvas to
| work with.
|
| (Any why 12 semitones? Again, to do with getting the most
| interesting relationships without muddying things ... I think its
| possible to divide the octave into 19 or 43 parts and get some
| interesting ratios/relationships, but then it gets quite fiddly)
| OscarCunningham wrote:
| The 7 note major scale is more-or-less an historical accident:
| https://en.wikipedia.org/wiki/Musical_system_of_ancient_Gree...
| .
| jedimastert wrote:
| I'll just put in that I think things like 12 tone scales and 7
| note diatonic scales were local minima in a vast landscape that
| we happen to hit upon and settle into. The more I look at what
| music is from a neurological perspective, the more it
| feels/seems like the deepening and establishing of patterns is
| the main mode.
| elihu wrote:
| The major scale is just the most straightforward way to be able
| to construct the most usable 4:5:6 ratios and 10:12:15 ratios
| (i.e. major and minor chords) from the fewest possible notes.
| In equal temperament, those ratios are approximated rather than
| exact, but those are the mathematical relationships implied by
| the chords.
|
| The desirability of "simple ratios" is based on the idea that
| if we play two pure sine waves at the same time, they sound
| good if they're exactly the same frequency and if they're some
| distance apart, but they sound bad if the two notes are close
| but don't line up. (This creates a beat frequency, which make
| the music sound unstable and noisy.)
|
| Notes played on real instruments have harmonics, and so if you
| play two notes at once all those harmonics either need to be a
| long way from each other or they need to line up. Notes with
| frequencies that correspond to simple ratios are the ones where
| the harmonics also line up in the cleanest way.
|
| Simple ratios like 2:1 or 3:2 are very stable and consonant.
| Larger ratios like 5:4 make modern music a bit more
| interesting. Still larger ratios like 7:4 and 11:8 can start to
| sound pretty alien and sort of dissonant and more complex.
|
| Basically, the most consonant music, which is easy to play
| (imprecisely) in 12-tone equal temperament, is pretty well
| explored territory. There is only one major chord, and we won't
| find anything that sounds any more "major-chordish" than it.
| But there's a huge unexplored territory when it comes to larger
| ratios that can't be played accurately enough to be
| intelligible on 12-tone equal tempered instruments.
|
| If you scroll down a bit on this page, I made a visualization
| of how 12-tone equal temperament lines up with a
| straightforward just-intonation scale based on the ratios
| implied by the 12-TET chromatic scale. It's really amazing both
| how well 12-tone equal temperament lines up, and at the same
| time how much better things could sound if you play the exact
| just intervals rather than this system that by some weird
| mathematical coincidence happens to be good enough for most
| simple musical purposes.
|
| http://jsnow.bootlegether.net/cbg/justintonation.html
| analog31 wrote:
| 12 tones is the _smallest_ approximately rational scale. There
| 's a technological reason for preferring this. Up until fairly
| recent times, a musician had to be able to tune and maintain
| their own instrument. Also, we can't grow more fingers.
|
| There may also be a benefit to more widely spaced notes,
| notably (!) that it's easier to distinguish if you're playing a
| "real" note or not, and if it's the same note as the one you're
| hearing someone else play. This would make it easier to learn
| musical ideas and pass them on to others, giving 12 tone music
| (fewer tones if using an agreed upon scale) a built in error
| correction code, and a sort of evolutionary advantage over time
| if you will.
| klodolph wrote:
| With the exception of a couple instruments like the piano and
| electronic keyboards, don't musicians still tune their own
| instruments?
| analog31 wrote:
| That's true.
|
| When my dad got a harpsichord, part of the process was
| getting trained by the maker on how to tune it. There's an
| old joke: Q: What's a harpsichordist doing when they're not
| tuning their harpsichord? A: Playing out of tune.
| alok-g wrote:
| >> why 12 semitones?
|
| As you noted, other numbers like 19, 43, and obviously also 24,
| gives interesting ratios.
|
| My current understanding is that while human ear is easily able
| to distinguish finer frequency ratios, singers aren't able to
| match vocals to much higher precisions. 19 may perhaps still
| work, but 43 I think would be out of question.
| teambob wrote:
| How applicable is this to eastern musical scales?
| Valodim wrote:
| I can relatedly recommend eevee's post "Music theory for nerds".
| I'm sure it's one of those that experts will huff and puff about,
| but it made a bunch of concepts between sound perception and
| music theory "click" for me:
| https://eev.ee/blog/2016/09/15/music-theory-for-nerds/
| d_rc wrote:
| Here is the notebook code in Deepnote if anyone wants to play
| with it: https://deepnote.com/project/JupyterNotes-
| zXwE7yVJSLeJx4aNp3...
| jng wrote:
| Next to this interesting article and comments, I think mention
| should be made of von Helmholtz's XIX century book "On the
| sensations of tone". He is considered to be the father of
| acoustics, and derived the consonance/dissonance of the notes in
| the diatonic scale from first physical principles. He extracted
| and computed what is described as "roughness" here as the amount
| of "beats" between the upper partials of the notes - where beats
| are the slow phase oscillations caused by two vibrations of very
| close frequencies. But of course, he first had to study strings
| and pipes for 8 years, and come up with the (then) innovative
| concept of upper partials (harmonics). I'm a software person by
| trade, amateur but very dedicated musician, read this book many
| years ago, thoroughly enjoyed it, and I heartily recommend it
| today.
| ciconia wrote:
| This is not the first time that I come across an article where
| the author derives harmonic rules from purely physical or
| mathematical principles. I always find it a pity that the author
| has apparently not made the effort of reading up a bit on the
| history of musical theory.
|
| The actual harmonic rules employed today in all kinds of western
| music, especially as they relate to musical notation and the
| physical layouts (the UI so to speak) of various western musical
| instruments, have as much to do with the evolution of music
| theory as with the physical behavior of sound waves and human
| perception thereof.
|
| Case in point, the layout of white and black keys on the
| keyboard: the disposition of white and black keys has more to do
| with the theory of hexachords developed in the 12th century. The
| hexachord system used to _notate and transmit_ music was
| originally comprised of three overlapping hexachords, each
| including 6 notes (ut-re-mi-fa-sol-la) , which together cover in
| total 8 notes to the octave: The natural hexachord C D E F G A,
| the soft (molle) hexachord F G A Bb C D, and the hard (durum)
| hexachord G A B C D E. I should add that this system was
| superimposed on the greek modes of gregorian chant, made of two
| tetrachords for 7 notes to the octave.
|
| So the soft hexachord starts on fa - the fourth tone of the
| natural hexachord, and the hard hexachord starts on sol - the
| fifth tone of the natural hexachord. And if you you play a melody
| and you want to go to Bb - the B flat (B-molle / bemol) is
| actually a _fa_ on the soft hexachord. Likewise, a B natural
| (B-durum - becarre) would be a _mi_ on the hard hexachord. The
| terms bemol and becarre (Bb and B natural) actually derive from
| the soft and hard hexachords (soft - round, hard - square). So
| the Bb was in fact the first "black" key (although some of the
| earliest keyboards have 8 "white" keys to the octave, with the Bb
| looking just like its neighbors).
|
| Later, from the 14th century on, as music changed, more
| hexachords were introduced starting at different places on the
| natural hexachord, for more harmonic complexity. Along with those
| hexachords, more "black" keys were introduced. Actually some
| baroque keyboards include more than 12 keys to the octave, for
| use with meantone tuning: they would have split "black" keys in
| order to play the _fa_ or _mi_ of the note, for example Eb or D#,
| which in meantone tuning are not equivalent. But as composers and
| keyboardists started exploring both equal temperament and
| irregular temperaments, those distinctions were lost.
|
| The modern keyboard with its 7 white keys and 5 black keys is
| rather a consequence of the evolution of western music over long
| centuries, not of any kind of absolute natural phenomenon.
| anamexis wrote:
| Thanks for the summary, that is really interesting.
|
| Also, I don't think the OP is making any claim that the piano
| keyboard is the result of any kind of natural phenomenon.
| They're just pointing out that you can create the same piano
| keyboard that we have using mathematical derivation, which is
| also interesting.
| MarkLowenstein wrote:
| No doubt that you can trace the history to explain it. But is
| there an inherent allure of the eventual design that steered
| these decisions toward a result like this? The article gives a
| very compelling theory as to why it might.
|
| If the effect of "inherent allure" sounds improbable, ask
| yourself if it might explain why common keyboard commands are
| nice comfortable ones to type, like ls and dir - and you never
| find yourself typing qza or xwz. All have good historical
| reasons, like not many operations starting with Z, but I think
| if there really was a common "Query Zeta Array" operation, it's
| likely it would have been renamed or there would be a tool with
| an easier-to-type name that would wrap it.
| jng wrote:
| Very interesting, I am familiar with many of the concepts and
| steps in the evolution of equal temperament, but I was not
| familiar with this part of the story at all (although I have
| seen pictures or drawings of keyboards with multiple split
| black keys). And very new to me, understanding the source of
| the terms "bemol" and "becuadro" (the words in Spanish for
| "flat" and "natural" - natural as in a note that loses its
| alteration and goes back to its neither-flat-nor-sharp pitch).
| Will try to delve into this in more detail some day with more
| time available. Thanks!
| SeanLuke wrote:
| 12-note equal temperament is a reasonable local minimum but it is
| by no means the best in the sense of roughness: 31-note is
| better. Not to promote 31-note (which I do not use), but how is
| it that this article can arrive at 12-note in some sense of
| optimality with regard to roughness criteria when it is clearly
| not?
| HPsquared wrote:
| Could a 31-note scale be enhanced further by adding more notes?
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