[HN Gopher] We do not prefer chords to be perfectly in mathemati...
       ___________________________________________________________________
        
       We do not prefer chords to be perfectly in mathematical ratios:
       study
        
       Author : glitcher
       Score  : 65 points
       Date   : 2024-03-27 17:28 UTC (5 hours ago)
        
 (HTM) web link (www.cam.ac.uk)
 (TXT) w3m dump (www.cam.ac.uk)
        
       | jimbob45 wrote:
       | _These consonances cannot be replicated on a Western piano, for
       | instance, because they would fall between the cracks of the scale
       | traditionally used._
       | 
       | Sounds like the author doesn't realize you can tune a piano.
       | Actually, it doesn't sound like they know anything about music
       | because the octave is surely universally harmonic.
        
         | phailhaus wrote:
         | Of course you can tune any instrument to any scale. "Western
         | piano" implies a standard tuned piano.
        
           | karencarits wrote:
           | Perhaps, but baroque ensembles often play with weird tunings
           | where the intervals are different from the modern ones:
           | https://en.wikipedia.org/wiki/Musical_temperament
        
             | ben7799 wrote:
             | Octaves stay the same though. Other temperaments are not
             | "weird", they just make different rules about how perfectly
             | in tune different intervals are with each other.
        
           | ByQuyzzy wrote:
           | There's no such thing as an Eastern, Southern, or Northern
           | piano. The poor author was trying to say something but ended
           | up flubbing it.
        
             | phailhaus wrote:
             | I really don't think anyone is actually confused when he
             | says that a "Western piano" can't replicate certain
             | harmonies. We all know what he's saying here, because
             | pianos aren't known for having custom tunings. This kind of
             | pedantry is not useful.
        
         | holri wrote:
         | Octaves on a piano are tuned slightly wider and "out of tune"
         | because of the Inharmonicity of the strings.
        
           | itishappy wrote:
           | TIL!
           | 
           | https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_.
           | ..
        
         | beardyw wrote:
         | Yes, I agree. No mention of well tempered scales and the
         | contortions that were needed to achieve them. Even at the time
         | they knew it was a compromise. Yet today we (in the west) take
         | them for granted.
         | 
         | And not only octaves. Most instruments will yield various
         | harmonics and no civilisation will have been unaware of them.
         | It is no coincidence that for us a natural harmonic lies
         | between a major 3rd and a minor one.
        
       | leblancfg wrote:
       | Pythagoras from the grave would nicely like to ask if the
       | researchers could _also_ give those 4000 participants a two-
       | stringed instrument, and ask them to tune them in a way they find
       | pleasant to hear - and see if that spicy headline of theirs still
       | holds.
        
         | phailhaus wrote:
         | The 4000 participants who have been living in cultures
         | _steeped_ with Western musical harmonies? I don 't think it
         | would be surprising that they would pick tunings that reflect
         | the music that surrounds them every day.
        
           | leblancfg wrote:
           | GP here.                   s/those 4000/another extremely
           | diverse 4000/
           | 
           | would make the same point IMO.
        
           | fsckboy wrote:
           | > _have been living in cultures steeped with Western musical
           | harmonies? I don 't think it would be surprising that they
           | would pick tunings that reflect the music that surrounds them
           | every day_
           | 
           | you are making a massive assumption that we prefer the same
           | old same old. Some of us are always on the lookout for a bit
           | of strange
        
         | PeterCorless wrote:
         | He would give them a one-stringed instrument and tell them to
         | head down to the beach to try to find a tone that most closely
         | resonated with the sound of the waves crashing on the shore.
        
         | nicklecompte wrote:
         | This is irrelevant to the study, because the musical ear is
         | attuned to human voices and that's what instruments are trying
         | to _emulate._ A specific tuning system is a sacrifice that
         | human voices don't have to make (e.g. the barbershop 7th versus
         | a dominant 7th on a piano, or wolf intervals in Pythagorean
         | tuning).
         | 
         | Pythagorean tuning makes a ton of sense if you're trying to
         | tune a fretted stringed instrument (and haven't tried playing
         | anything with a key change), but it's an oversimplification of
         | how music works. The Pythagorean philosophical mistake is
         | treating the tuning system as a theory of music itself, when
         | it's really a simple framework for certain types of music.
        
           | BobaFloutist wrote:
           | > the musical ear is attuned to human voices and that's what
           | instruments are trying to emulate.
           | 
           | That's quite the claim, do you have anything to support it
           | beyond your intuition?
        
         | omnicognate wrote:
         | They compared people's subjective judgements of consonances
         | tuned to perfect ratios to ones slightly off from those ratios
         | and found they preferred the latter.
         | 
         | A) Most people wouldn't be able to tune an instrument like what
         | you describe accurately enough to contradict the result.
         | Practically all the intervals would be slightly off, and
         | 
         | B) The result really does contradict Pythagoras, who postulated
         | that musical beauty arose from the mathematical perfection of
         | those ratios. Contradicting Pythagoras in this is, of course,
         | hardly new. More interesting is the idea that there may be
         | entirely new forms of naturally appealing harmony out there to
         | explore.
        
           | anigbrowl wrote:
           | I disagree with B. People like slightly detuned oscillators
           | because it makes the sound thicker and bigger. But they like
           | slight detuning _around_ the mathematically perfect ratio.
           | The imperfection _emphasizes_ the ratio in the brain 's
           | auditory processing.
           | 
           | This is why many subtractive synthesizers have a detune
           | parameter, to slightly offset oscillators that are nominally
           | tuned to the same frequency. Chorus, phasers, and flangers
           | are similarly used to add texture and color to sounds by
           | using slight frequency or phase delays to create interference
           | signals that make the sound 'move' and make it sound larger
           | and more distinct.
        
             | dr_dshiv wrote:
             | It's like the "humanize" button on drum machines that will
             | make the rhythms slightly off. We still like the small
             | integer relations of the drum beats but when they are a bit
             | off, they sound warmer and more human.
        
         | buildsjets wrote:
         | That sounds more like an experiment that Von Neumann and
         | Stanislaw Ulam would have come up with than the product of an
         | ancient Greek philosopher. But he probably would have been
         | pleased by it.
        
       | odyssey7 wrote:
       | Cambridge researchers have discovered that music is enjoyed by
       | actual flesh-and-blood humans, refuting Pythagoras?
       | 
       | Clearly a revival of the quadrivium is needed.
        
       | re wrote:
       | Some previous discussion:
       | https://news.ycombinator.com/item?id=39532586 (28 days ago, 6
       | comments)
       | 
       | The other article posted also has a audiovisual version of some
       | of the study findings which helps clarify the results:
       | https://phys.org/news/2024-02-pythagoras-wrong-universal-mus...
        
       | FLT8 wrote:
       | > "We prefer slight amounts of deviation. We like a little
       | imperfection because this gives life to the sounds, and that is
       | attractive to us,"
       | 
       | If you've ever tried building an additive synth, this probably
       | seems obvious. If you play two tones with exact ratios together,
       | the brain has a tendency to interpret them as a single tone with
       | a different timbre. By making the ratios slightly off (don't have
       | to be by much at all), or by making the ratios vary a little over
       | time (eg. Subtle vibrato), the brain is able to pick out the
       | "consonance" as a chord, and it sounds far more interesting.
        
         | TheOtherHobbes wrote:
         | It's almost as if Patterson has no clue about the history of
         | tunings, in the West or elsewhere.
         | 
         | Perfect harmonic tunings sound incredibly exotic and out of
         | tune to Western ears - not least because we're so used to
         | slightly-imperfect equal temperament. We have decades of
         | exposure to it, so it sounds normal. A _perfect_ perfect fifth
         | sounds very strange, even a little disturbing, and thirds are
         | almost dissonant.
         | 
         | I don't think anyone who has even the first idea about this has
         | ever suggested that harmonic tunings are a goal, an ideal, or
         | even a practical possibility. (See also, Pythagorean Comma.)
        
           | mrob wrote:
           | Just intonation tuning of chords (but not melodies) is the
           | ideal in barbershop music. Barbershop music also uses
           | harmonic sevenths, which are not found in standard 12EDO
           | equal temperament. I've never heard of anybody calling
           | barbershop "strange" or "disturbing".
           | 
           | I don't think anybody is likely to call a perfect fifth or
           | third those things either. You can easily check using the
           | "Generate, Tone..." feature in Audacity. I recommend using a
           | sawtooth wave, which contains prominent high harmonics that
           | make it easy to hear subtle difference in tuning. The worst
           | criticism I'd make of perfect just intonation is that it can
           | sound boring, for the reason the GP post suggests.
        
             | ben_w wrote:
             | My suspicion is that learning music theory (for any
             | specific cultural norm) is a bit like learning about font
             | kerning: https://xkcd.com/1015/
             | 
             | Many of the auditory things that trained musicians describe
             | as creepy or weird, seem unremarkable to me.
        
             | xotesos wrote:
             | I would say that is because there is nothing to compare
             | barber shop music too.
             | 
             | I am a big fan of Terry Riley's - Harp of New Albion
             | exactly because its five-limit just intonation piano sounds
             | so incredibly weird to my ears.
        
               | dekhn wrote:
               | See also La Monte Young- Forever Bad Blues Band. I
               | listened to it and it sounded "different" but not
               | particularly weird (I was already listening to Phish, The
               | Dead, and indian music, so I think I was prepared for
               | blues in just intonation).
        
           | hax0ron3 wrote:
           | I don't think everyone is equally sensitive to this. Just
           | intonation sounds a bit different to me from equal
           | temperament, but it doesn't sound disturbing. Likewise,
           | although I have heard a lot of Indian classical music that is
           | not in equal temperament, so I am pretty acclimated to it,
           | when I hear Indian classical music played on equal
           | temperament instruments it does not sound that bad. Some
           | subtleties are definitely lost, and I prefer the non equal
           | temperament tunings for that kind of music, but I do not find
           | it disturbing.
        
         | gnramires wrote:
         | Related, a friend likes to say instruments that produce too
         | pure sounds (too much like a sine wave) sound somewhat boring,
         | because of their lack of timbre characteristics. I think this
         | is a somewhat general property; although in general art is more
         | of a language. I think a pure sine can sound right in the right
         | context (e.g. perhaps how we remember fondly, and still enjoy,
         | chiptunes and retro music), and it can have a particular
         | meaning within that context that makes sense.
        
           | WorldMaker wrote:
           | It is indeed a common complaint of many of the chiptune
           | genres that they can sound boring, especially for audiences
           | too old or young to have as much nostalgia for the hardware
           | whose constraints set the genre constraints. The basic NES
           | "instruments" were simple wave generators that were about as
           | pure as waveforms get, plus one simple white noise generator.
           | 
           | I've met critics that even think that "pure" chiptune sub-
           | genres such as the ones that stay within the bounds of
           | physical NES hardware "can't" survive long-term. I think they
           | are wrong both because I've heard some extraordinary things
           | done within those constraints in my lifetime and for some of
           | the reasons that "minimalism" will always be a viable
           | subgenre of "Classical" (no matter how boring some people
           | find that, too). But I understand where that idea comes from
           | of how such a unique set of hardware constraints gave us some
           | of the "purest" waveforms possible and why such pure
           | waveforms are generally an anomaly in music.
        
             | immibis wrote:
             | Ah yes, [simple and
             | boring](https://www.youtube.com/watch?v=1lScTUyEsA4). You
             | get 3 voices or whatever's on your chip, but you can also
             | re-adjust them up to hundreds of times a second any way you
             | like. Of course if you just leave them fixed for long times
             | they will sound a bit boring.
        
         | vr46 wrote:
         | This is intriguing and clearly not new as you say - my strobe
         | tuner offers "sweetened" tunings for guitar. Going to have to
         | experiment with this variation now...
        
         | pje wrote:
         | "Imperfect" tunings are pretty common even in western musical
         | instruments.
         | 
         | Fender Rhodes and Wurlitzer electric pianos, for example, which
         | are staples of pop and jazz, usually use what's called Stretch
         | Tuning, wherein notes an octave apart are _just slightly_
         | higher than 2:1 frequency ratios.
         | 
         | https://en.wikipedia.org/wiki/Stretched_tuning
        
           | denotational wrote:
           | A "normal" piano with strings uses stretch tuning too.
        
         | StableAlkyne wrote:
         | Even playing a single note, having your oscillators a couple
         | cents out of tune with each other contributes to a warm sound!
         | 
         | Old polyphonic synths like the venerable Oberheim OB-Xa would
         | have their voices go slightly out of tune as the instrument
         | warmes up. It's subtle, but once you hear it you start to
         | notice the color it adds
        
       | pfgallagher wrote:
       | What makes a song sound "good" is another question - there seem
       | to be countless factors. Most people can easily tell you if they
       | subjectively like or dislike a song, but typically have a hard
       | time explaining exactly why.
        
       | ben7799 wrote:
       | I've always been taught Music theory exists to explain why we
       | like things. Not to dictate what we like, it's like some grand
       | unifying theory.
       | 
       | Whenever we find something new people like Music theory has to
       | adapt.
       | 
       | I think it would be impossible to prove any unifying theory like
       | this unless you can actually test all possible things someone
       | could hear and western scales & such obviously don't encompass
       | all that.
        
         | dboreham wrote:
         | True but in my experience few music people know this.
         | 
         | Music theory isn't a theory. It's just a scheme for naming
         | patterns that turn out to sound good/interesting.
        
       | henjodottech wrote:
       | In other news; Golden ratio? More like bronze
        
       | uwagar wrote:
       | so a pitch and its octaves, or a pitch and its perfect fifth
       | above or below sounding harmonious is not universal?
        
         | 082349872349872 wrote:
         | Anecdote whose source I've forgotten: a missionary family to
         | some Pacific islands had brought a shortwave set and tried to
         | get the locals interested in their favourite classical and
         | opera. The locals enjoyed the radio all right, but much
         | preferred US Navy stations, because they usually played both
         | kinds of music: country and western.
         | 
         | EDIT: something that might be consonant with the article would
         | be frequent hammer-on and pull-off, as well as voice-leading in
         | the slide parts?
        
       | mrob wrote:
       | Pythagoras' ideas about tuning only make sense with instruments
       | that have harmonic or almost harmonic timbres, which is true for
       | most instruments. However, tuned percussion is typically
       | inharmonic. It's long been known that inharmonic instruments
       | sound most consonant in chords using non-integer frequency
       | ratios.
       | 
       | William Sethares developed a model of consonance and dissonance
       | that accounts for this:
       | 
       | https://sethares.engr.wisc.edu/paperspdf/consonance.pdf
       | 
       | Informal explanation:
       | 
       | https://sethares.engr.wisc.edu/consemi.html
        
         | omnicognate wrote:
         | The paper references Sethares (and Plomp & Levelt). This is a
         | large empirical study building on their theoretical work. The
         | paper itself is worth a read:
         | https://www.nature.com/articles/s41467-024-45812-z
        
       | riffic wrote:
       | This reminds me, have we had a good 432 Hz vs 440 Hz discussion
       | lately?
        
         | thuuuomas wrote:
         | I feel like 432hz vs 440hz is merely the Just Intonation /
         | Equal Temperment debate reiterated by people who don't know any
         | better. 432hz advocates make a lot of appeals (to "natural
         | frequencies" that sound "more human") which are quite similar
         | to arguments made by proponents of Just Intonation during the
         | Industrial Revolution.
        
       | dsign wrote:
       | I tend to see music as an idiosyncratic language. I know from
       | experience I can't hear harmonies that well, and that I had to
       | find out about chords in Wikipedia and was completely deaf to
       | them until I trained myself. Meaning and emotions that _I_ can
       | hear in music are most definitely culturally learned, even if I
       | won 't fight anybody swearing a major C is so much uplifting than
       | a minor one and that that is an essential part of Creation--to
       | each their own.
       | 
       | With that said, just as languages tend to have culturally
       | accepted "proper form" --grammar and spelling--so does music.
       | Music theory is about what spelling and grammar the locals accept
       | and like in their music, and is worth studying for that reason
       | alone.
       | 
       | Now, I wonder what would Pythagoras think of our current musical
       | genres...
        
       | blitzar wrote:
       | I hope this result calls into question all the other work done by
       | Pythagoras.
        
       | andoando wrote:
       | So Im guessing non western music still still satisfies other
       | mathematical relationships? I dont think it's correct to say the
       | mathematical explanation is incorrect or even limited.
       | 
       | Even considering deviations/slight variations from ratios:
       | 
       | 1. You have to consider that these are deviations from the
       | mathematical ratios. 2. The variations themselves can follow
       | ratios. After all, not all deviations or variations are equally
       | interesting.
       | 
       | For example if you were to play 1xxx2xxx3xxx4
       | 
       | you can vary the next verse 1xx2xxxx3xxx4 and the next
       | 1xxxx2xx3xxx4
       | 
       | Here the second note is oscillating one before the "ideal" ratio
       | and one after.
        
         | anon291 wrote:
         | The non-western music I'm familiar with oftentimes still
         | emphasizes things like I-V consonance.
        
       | igorbark wrote:
       | so, yes, pythagoras did not create a dead simple mathematical
       | model that captures the entire complexity of human musical
       | experience several thousand years ago. BUT i think the ongoing
       | study of consonance/dissonance is a very interesting area of the
       | intersection of math and music
       | 
       | some key words/links to get you started:
       | 
       | - "local consonance"
       | 
       | - "consonance/dissonance curves"
       | 
       | - a seminal paper:
       | https://sethares.engr.wisc.edu/paperspdf/consonance.pdf
       | 
       | - a more recent re-implementation with a cool video at the end:
       | https://www.sebastianjiroschlecht.com/post/ondissonance/
       | 
       | the basic idea being, different timbres lend themselves
       | differently to different tuning systems. so we can parameterize
       | our models of tuning systems based on timbre
       | 
       | an important thing to keep in mind: consonant/dissonant doesn't
       | mean "good/bad" or "pleasant/unpleasant". they're the output
       | values of a mathematical model which we have a complex intuitive
       | relationship with. other ways of thinking about it might be
       | "simple/complex", "resolved/unresolved", "release/tension", but
       | all are inaccurate in their own way
       | 
       | some areas i'd love to see progress in: - the work i've seen
       | focuses on computational models, i.e. take a simple mathematical
       | model of timbre, and directly compute the consonance/dissonance
       | curve from it. but real instruments' timbre varies across many
       | dimensions, some prominent ones being pitch, time, and dynamics.
       | can we instead burn some CPU cycles and generate curves from a
       | waveform? - what does this look like for triads? tetrads? ...? -
       | put this in the browser! would make it so much easier to play
       | with and present the ideas to less technical audiences - how can
       | we use this to generate new instruments? can a synth
       | automatically adjust its tuning system based on its parameters?
       | can we start from a set of desired consonant/dissonant intervals
       | and generate an instrument with a matching curve?
        
         | andoando wrote:
         | To simplify what youre saying - I think its just a matter of
         | opposites/differences. Music is a pattern of differences. You
         | can take any two things that are different and create a
         | pattern, a rhythm and humans will find it interesting to listen
         | to and make sense of it. This moreover has a 1:1 relationship
         | with a visual representation. That is, every piece of music has
         | an equivalent visual representation.
        
       | reilly3000 wrote:
       | Musica Universalis is all about overfitting ratios and
       | frequencies to satisfy an aesthetic. I want to believe, but the
       | data isn't there. I dive deep into this in college music theory
       | and acoustics but came up wanting.
        
         | giraffe_lady wrote:
         | Yeah this is one of those things that so so _looks_ like it
         | should work out a certain way. As soon as you start learning
         | you _want_ it to work out that way. Some number of people in
         | every generation have _tried_ to make it work out, or become
         | convinced that it does.
         | 
         | It would just be so aesthetically satisfying. One of the most
         | perfect congruences there could be, it just makes so much
         | sense, would imply so much about the universe and our place in
         | it. Musical and maybe human maturity is finally accepting that
         | it isn't there.
        
           | reilly3000 wrote:
           | Yep, agreed. Once you get there you can really appreciate
           | what music is and does all on its own.
        
         | andoando wrote:
         | Can you explain more? What do you mean the data isnt there?
        
       | srcreigh wrote:
       | It cannot be overstated that the actual study data without
       | exception points to the opposite conclusion, that pythagorean
       | intervals (mathematical ratios) are in fact what we prefer.
       | 
       | Study link: https://www.nature.com/articles/s41467-024-45812-z
       | 
       | Start with figure 3. Non-musicians generally favour pythagorian
       | style harmonic ratios, while trained musicians tend to prefer
       | equal temperament.
       | 
       | Each of the following intervals in the data confirm this:
       | 
       | - The pythagorian major 3rd ratio 5/4 is slightly flat compared
       | to 5 semitones. Non-musicians nailed it, musicians prefer the
       | sharp ET version.
       | 
       | - The range from 10-11 semitones is rated relatively high by non-
       | musicians. This range includes the 7th harmonic, 7/4
       | 
       | - Even the 11th harmonic, taking place between 6(tritone) and
       | 7(perfect fifth) is rated highly by non-musicians relatively to
       | musicians.
       | 
       | - The "9" major 2nd octave, 9/4 JI, aka 14 semitones, is tuned
       | more perfectly by non musicians (9/4) compared to musicians
       | 
       | Later the authors experiment with a artificial gong style
       | instrument (fig 5). However its harmonics are remarkably close to
       | several standard pythagorean ratios. You see 1.52f0 (slightly
       | sharp perfect 5th 3/2); 3.46f0 (slightly flat harmonic 7th 7/4);
       | 3.92f0 (slightly flat double octave 4/1). These harmonics are so
       | close to a guitar or piano or human voice, it can't really even
       | be considered inharmonic at all.
       | 
       | With this instrument, you get, yet again, more pythagorean
       | intervals:
       | 
       | - sharp major 2nd, 9/8 ratio (makes sense; started with a sharp
       | 3/2 and 9/8 is just (3/2)^2 modulo octaves).
       | 
       | - People _still_ correctly tune the major 3rd 5 /4 ratio
       | (slightly less than 5 semitones) even without a 5th harmonic.
       | bonkers
       | 
       | - Sharp 5th, which makes sense as the harmonic is 1.52f0 instead
       | of 1.5f0
       | 
       | - Heavily emphasized harmonic 7th interval between 9 and 10
       | semitones
       | 
       | Later, the authors experiment with various mis-tunings of the
       | pythagorean intervals (Fig 8). The most highly rated intervals
       | are slightly sharp or flat from JI. This is evidence _for_ JI
       | intervals being pleasing--slightly sharp or flat creates a
       | pleasing rhythm of Just harmony. It 's like just harmony but
       | better.
       | 
       | In conclusion, essentially every plot in the study points towards
       | pythagorean, mathematically tuned harmony being more pleasing.
       | Even in surprising ways. I could not find a single data point
       | indicating otherwise, except for indoctrination of musicians
       | towards ET leading to things like preferring sharp major 3rds and
       | not recognizing the harmonic 7th interval.
        
         | pclmulqdq wrote:
         | It's very possible that the averages over the population we see
         | in the paper reflect a gradient of musical experience rather
         | than an abstract notion of "pleasantness." Many trained
         | musicians I know with a great ear have a very hard time with
         | non-ET fifths and thirds, even knowing that ET is a grand
         | compromise (all tunings are). We must learn through exposure -
         | and admonitions from teachers to have better intonation - that
         | ET is correct and everything else is wrong.
         | 
         | I also have to say that the idea of making "pleasant" intervals
         | is a bit reductive to begin with: unequal temperament tunings
         | in modern use are often chosen to give a piece a particular
         | "color" regardless of how "pleasant" the intervals are in a
         | vacuum.
        
           | srcreigh wrote:
           | Agreed. Western musicians have a sort of
           | indoctrination/familiarity towards ET, leaving the untrained
           | folks better equipped to recognize Just intoned harmony.
           | 
           | Also agreed re: pleasant intervals. My favourite example
           | (which I have posted before) is this Mozart piece in 6th
           | comma meantone. The back and forth between dissonance and
           | pure harmonic resolution with great 3rds tuning is
           | extraordinary. Anyone, give it one listen, you will be
           | convinced.
           | 
           | https://youtu.be/lzsEdK48CDY?si=rtBE-M0cCRQsDHNx&t=700
           | 
           | (Note the last, cheerful section was added after mozart's
           | death.)
        
             | pclmulqdq wrote:
             | I don't personally like the tuning on that - it feels a
             | little too rough. However, many classical and even
             | romantic-period composers had pianos tuned in Werckmeister,
             | Kirnberger, or Vallotti temperaments, which had toned-down
             | versions of similar dissonances you hear here (similar to
             | the second rendition of the Mozart in that video).
        
         | anigbrowl wrote:
         | Quite, and this has been known for years - any good
         | ethnomusicology textbook addresses it chapter 1. Equal
         | temperament and 'western harmony' are not all that.
        
       | HPsquared wrote:
       | Maybe it has something to do with phase. If they are ever so
       | slightly out of tune, the phase is kind of "smeared" out, whereas
       | with "exactly in tune" the locked phase can affect the timbre.
        
       | graycat wrote:
       | Come ON guys, help me out here: Be more clear about your
       | terminology!!!
       | 
       | Gee, I'm plenty good with the math and applications of Fourier
       | theory. Sure, some physics course solved the differential
       | equation of vibrating strings. Early on my career did well with
       | Fourier theory, the FFT (fast Fourier transform), and Navy ocean
       | data. And with violin, I got a start in the music school of
       | Indiana University and eventually made it through several pages
       | of the Bach E-major preludio, the Bach Chaconne, and a
       | transcription, as I recall, up an octave and a fifth, of that
       | famous Bach solo cello piece. Been known to take the positive
       | whole number powers of 2^(1/12). Dissonance? Sure, I really like
       | the S. Barber _Adagio_ for strings!!! R. Wagner 's chords. The
       | Franz Schmidt Intermezzo to his Notre Dame. The Sibelius violin
       | concerto.
       | 
       | But, stilllll, a lot of your terminology I can't follow.
       | 
       | For an opinion: Mostly music is art, that is communication,
       | interpretation of human experience, emotion. Given the roles of,
       | say, vibrato, Fourier theory and 2^(1/12) do not directly explain
       | all of the art.
       | 
       | For a _perfect fifth_ , sure, as is standard, when take the
       | violin out of its case, first tune the four strings G, D, A, E in
       | perfect fifths and do this by listening for beats in the
       | overtones. Then with the strings so tuned, often when the music
       | calls for one of those 4 notes, just go ahead and play it on the
       | _open_ (look, Ma, no use of fingers on the left hand) string.
       | 
       | In particular, for
       | 
       | > perfectly in mathematical ratios
       | 
       | in real music from before Bach to after Barber, I doubt that
       | "perfectly" happens very often or is even attempted. E.g., with
       | violin, can't play much music if play only on the open strings,
       | and as soon as put a finger of the left hand on a string, with
       | vibrato or even without, there won't be much "perfectly
       | mathematical".
        
       | Nav_Panel wrote:
       | I really like this site for info on various tunings, including
       | Just Intonation, where the intervals ARE perfect mathematical
       | ratios: https://www.kylegann.com/microtonality.html
       | 
       | One tidbit I always think about:
       | 
       | > _I 've had interesting experiences playing just-intonation
       | music for non-music-major students. Sometimes they will identify
       | an equal-tempered chord as "happy, upbeat," and the same chord in
       | just intonation as "sad, gloomy." Of course, this is the first
       | time they've ever heard anything but equal temperament, and
       | they're far more familiar with the first sound than the second.
       | But I think they correctly hit on the point that equal
       | temperament chords do have a kind of active buzz to them, a level
       | of harmonic excitement and intensity. By contrast, just-
       | intonation chords are much calmer, more passive; you literally
       | have to slow down to listen to them. (As Terry Riley says,
       | Western music is fast because it's not in tune.) It makes sense
       | that American teenagers would identify tranquil, purely consonant
       | harmony as moody and depressing. Listening from the other side,
       | I've learned to hear equal temperament music as a kind of aural
       | caffeine, overly busy and nervous-making. If you're used to
       | getting that kind of buzz from music, you feel the lack of it as
       | a deprivation when it's not there. But do we need it? Most
       | cultures use music for meditation, and ours may be the only
       | culture that doesn't. With our tuning, we can't._
       | 
       | > _My teacher, Ben Johnston, was convinced that our tuning is
       | responsible for much of our cultural psychology, the fact that we
       | are so geared toward progress and action and violence and so
       | little attuned to introspection, contentment, and acquiesence.
       | Equal temperament could be described as the musical equivalent to
       | eating a lot of red meat and processed sugars and watching
       | violent action films. The music doesn 't turn your attention
       | inward, it makes you want to go out and work off your nervous
       | energy on something._
        
       | mingus88 wrote:
       | I haven't read the study but the article gives the impression
       | that researchers can get published for things as simple as
       | discovering gamelan orchestras?
       | 
       | I don't want to sound cynical but ethnomusicology is a field unto
       | itself and the idea that western tonal harmonies are a very small
       | piece of the pie is something we've talked about for at least a
       | hundred years
       | 
       | Even the concept of just intonation for a western scale is well
       | established
        
       | dr_dshiv wrote:
       | I spent a very long time trying to make sense of the broad
       | concept of harmony. Here's what I came up with:
       | https://www.sciencedirect.com/science/article/pii/S240587262...
       | 
       | You'll find some juicy details on Pythagoras (his Olympic days,
       | for instance) as well as an overview of harmony in neuroscience,
       | computer science, aesthetic theories, etc.
        
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