Pythagoras Was Wrong: There Are No Universal Musical Harmonies, Study Finds (cam.ac.uk) 73
An anonymous reader shares a report: According to the Ancient Greek philosopher Pythagoras, 'consonance' -- a pleasant-sounding combination of notes -- is produced by special relationships between simple numbers such as 3 and 4. More recently, scholars have tried to find psychological explanations, but these 'integer ratios' are still credited with making a chord sound beautiful, and deviation from them is thought to make music 'dissonant,' unpleasant sounding.
But researchers from the University of Cambridge, Princeton and the Max Planck Institute for Empirical Aesthetics, have now discovered two key ways in which Pythagoras was wrong. Their study, published in Nature Communications, shows that in normal listening contexts, we do not actually prefer chords to be perfectly in these mathematical ratios. "We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us," said co-author, Dr Peter Harrison, from Cambridge's Faculty of Music and Director of its Centre for Music and Science.
The researchers also found that the role played by these mathematical relationships disappears when you consider certain musical instruments that are less familiar to Western musicians, audiences and scholars. These instruments tend to be bells, gongs, types of xylophones and other kinds of pitched percussion instruments. In particular, they studied the 'bonang,' an instrument from the Javanese gamelan built from a collection of small gongs.
But researchers from the University of Cambridge, Princeton and the Max Planck Institute for Empirical Aesthetics, have now discovered two key ways in which Pythagoras was wrong. Their study, published in Nature Communications, shows that in normal listening contexts, we do not actually prefer chords to be perfectly in these mathematical ratios. "We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us," said co-author, Dr Peter Harrison, from Cambridge's Faculty of Music and Director of its Centre for Music and Science.
The researchers also found that the role played by these mathematical relationships disappears when you consider certain musical instruments that are less familiar to Western musicians, audiences and scholars. These instruments tend to be bells, gongs, types of xylophones and other kinds of pitched percussion instruments. In particular, they studied the 'bonang,' an instrument from the Javanese gamelan built from a collection of small gongs.
Pythagoras was right (Score:5, Interesting)
"There is geometry in the humming of the strings, there is music in the spacing of the spheres." - Pythagoras
"Silence is better than unmeaning words." -Pythagoras
"There is no science that accounts for poor taste in music." - also Pythagoras, probably, when dealing with some pretentious critic.
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Maybe they though the comments are too soon. Give it 10,000 years at least.
Re:Pythagoras was right (Score:4, Funny)
I happen to agree. This study was run by people who don't understand the fundamental philosophy involved. Sure, you can "like" discordant music, but only after your soul has been poisoned with suffering. And that's not to say I don't like discordant music. I love Radiohead, for example, but I happen to know that's only because my soul has been poisoned with suffering already.
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The Blues is one note that doesn't belong in the scale, the blue note. A tiny change but everyone feels the sorrow because we know it isn't right. Pythagoras was.
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Q: How does a blues solo end well?
A: With a gunshot.
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Music - and what you like - is an intensely social event. If you're not inculcated into the cult of music, then you've no reason to apply social judgements like "nice" or beautiful" to the random sounds people throw together. You just nod, mutter platitudes, and go and do something interesting after be
Re:Pythagoras was right (Score:4, Insightful)
On the other hand musical taste is so relative to the individual one person's favorite band is another's agony to listen to which tells me there's a lot more to musical taste than math. Anyone who claims their own taste is definitive in regards to what's "good" and "bad" is just being ridiculous.
Yes, of course. (Score:4, Insightful)
We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us...
Of course. That's why live performances by people who can actually sing are generally way better than auto-tuned studio recordings. It's also why things like microtonality are played with in a lot of music.
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We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us...
Of course. That's why live performances by people who can actually sing..
Im glad you clarified that statement, since the chasm between auto-tuned entertainers and actual singers grows with every superficial contract.
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We prefer slight amounts of deviation. We like a little imperfection because this gives life to the sounds, and that is attractive to us...
Of course. That's why live performances by people who can actually sing are generally way better than auto-tuned studio recordings. It's also why things like microtonality are played with in a lot of music.
Studio guys, and even home artists, know the value in futzing with tuning slightly between takes to give a natural "movement" to the sounds. Like when you hum against a blow-dryer to get some oscillation? It's also sometimes accomplished by tossing in some stereo chorus or flanger. That doesn't negate what Pythagoras said. It just means that we like some "life" to the sound.
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Depends on the musical act. I've gone to shows of bands who have multiple albums I regularly enjoy only to discover their live sets are god awful while there are other bands I will only listen to live (although the later is mostly because they feature strong live acts).
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You mean this guy [youtube.com]?
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Hahaha. I wasnt thinking of this specifically but you illustrated what I was getting at perfectly.
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Of course. That's why live performances by people who can actually sing are generally way better than auto-tuned studio recordings. It's also why things like microtonality are played with in a lot of music.
Jimi Hendrix was the first person I found to do this. Listen to his guitar playing, how he subtly messes with the timings but is still musically perfect. It adds a whole new dimension to the music.
I am sure others have done it for a very long time previously, I am just not aware of them.
No Way (Score:3)
This undermines my core understanding of aesthetics that I learned from watching Donald Duck in Mathmagic Land. I just can't accept it.
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Donald Duck in Mathmagic Land was overproduced but taught me how to hustle Pool
Well that sucked. (Score:2, Funny)
When starting to read title, I thought the whole BS about hypotenuse was disproved and I could sue my middle school. Instead, this article is some crap about harmonics nobody gives a shit about. What a waste.
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When starting to read title, I thought the whole BS about hypotenuse was disproved and I could sue my middle school. Instead, this article is some crap about harmonics nobody gives a shit about. What a waste.
I thought it was the stuff about fava beans holding people's souls. Turns out that must be true as well!
Preferences (Score:2)
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"people don't remain enamored with the same sounds for hundreds of years"
Define sounds. Musical styles may change but the musical scales remain the same. Eg the western piano - now synth - keyboard and notes haven't changed in 300 years.
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Define sounds. Musical styles may change but the musical scales remain the same. Eg the western piano - now synth - keyboard and notes haven't changed in 300 years.
From the article:
The researchers found that the bonang’s consonances mapped neatly onto the particular musical scale used in the Indonesian culture from which it comes. These consonances cannot be replicated on a Western piano, for instance, because they would fall between the cracks of the scale traditionally used.
This is what I was referring to when I was talking about novel sounds, precisely ones that do not fit into the notes found on a western piano.
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Thats why I said "musical scales", not "scale". It would be fairly easy to make up your own scale out of the infinite possibilities but the fact only certain ones persist shows there's something specific about them wrt frequency spacing that works for the human brain.
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It's obvious that the authors weren't keen on getting someone who could say, bend the strings on a guitar, like a blues or jazz player [youtube.com]. Or a singer who can emulate that. [youtube.com]
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Correct. It's not what they were researching and would have been a useless complication.
Percussive instruments? (Score:5, Insightful)
The point of percussive instruments is that they make a quick noise. If they have a main frequency rather than a white noise spectrum then great, but their job isn't to provide the harmony so using them as a counter example is silly.
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Depends on the instrument.
Gongs, chimes, bells, tubular bells, celesta, and even the humble triangle have long sustains.
There's a whole bunch called keyboard percussion which are there to provide clearly pitched tones.
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A piano is technically both a percussion & string instrument. When you press a key, it's basically hitting a little hammer on a string.
Re:Percussive instruments? (Score:4, Informative)
The point of percussive instruments is that they make a quick noise. If they have a main frequency rather than a white noise spectrum then great, but their job isn't to provide the harmony so using them as a counter example is silly.
Xylophone, marimba, glockenspiel, hammered dulcimer, handbells/chimes, etc. can all used for melody or harmony.
What makes the argument silly is that western music hasn't used anything approaching exact frequency ratios since the advent of equal temperament in the 1700s, because nobody wants to have separate instruments for playing in different keys, nor retuning something the size of a piano or harpsichord or harp every time you change keys. So instead, all of the notes are just slightly out of tune, so that everything sounds equally close to being in tune regardless of what key you're playing in. We've been doing that for three centuries, and it doesn't seem to bother anyone.
And we've used tritones in music for centuries, and apart from the early church calling it the devil's interval, the roughly 45:32 ratio is as far from a small-integer ratio as you can plausibly get in modern western music, but most people wouldn't call it dissonant. I mean, listen to "Maria" from West Side Story, and tell me that it's dissonant.
We're used to what we're used to, and we find it familiar and comfortable. For closed voicing chords, that's about all I'd agree is probably universally true.
That said, when you get into ratios greater than 2:1, you do start to get something interesting because of the harmonic series, and lower notes reinforcing the upper notes. And that creates a musical richness that can't really exist with other ratios, but that's talking about an octave, an octave plus a fifth, two octaves, two octaves and a third, two octaves and a fifth, two octaves and a horribly flat minor 7th, three octaves, three octaves and a major second, major third, flat tritone, fifth, etc. But that's open voicing, and the only reason the intervals matter in closed voicing is because there are usually notes down below whose harmonics overlap with the chord as a result.
Yeah (Score:5, Insightful)
Anyone with a passing curiosity in musical theory learns already knows that consonant ratios aren't perfect in every key for any of the several tunings and that non-western, non-12-tone music also exists. And that other harmonies sound interesting, musical, or pleasant depending on the context and instruments.
Maybe the paper is interesting because it puts these questions to survey with a few thousand participants, but the Cambridge article on it is youtube-tier clickbait dogshit. "Pythagoras was wrong", "Challenges centuries of western music theory," as if Haydn was the only game in town and nothing new has been learned since. Come the fuck on.
Summary doesn't make much sense (Score:5, Informative)
Right off the bat, there is a major misconception in the summary. Musical intervals with simple integer ratios (like 3:2, the interval of a fifth) are associated with a feeling of "consonance". But this feeling of "consonance" has nothing to do with the feeling that a musical interval is "pleasant" or "beautiful" (as the summary suggests), and conversely, a feeling of "dissonance" has nothing to do with the feeling that a musical interval is "unpleasant" or "ugly". Really, this is basic Intro-To-Music-Theory stuff.
(It goes without saying that if you wrote a musical composition that contained nothing but "consonant" intervals, the resulting composition would probably be perceived by most listeners as bland and boring, and rather fatiguing to listen to-- not "pleasant" or "beautiful" at all).
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You can do plenty with just a pentatonic scale, so much so that it is an entire oevre of composition and has been for centuries. Some might call it "plinky plonky" but others appreciate a "purity" not present in a different scale containing more tones.
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Sure, but even the pentatonic scale has some moderately dissonant intervals in it-- for example the major second is "dissonant" (compared with the octave, the fifth, the major thirds and the major sixths). You're right that it's not a scale that produces a lot of dissonance (it's impossible to get the tritone interval). And maybe that's why songs in a *pure* pentatonic scale tend to sound kind of bland or plinky-plonky (case in point, the opening of "Oh Susanna"). Not to be confused with songs in a blues
That's not what it says (Score:5, Insightful)
The paper itself is much more interesting than this "summary."
Most music systems make extensive use of simple harmonic ratios, or close approximations. The results of their simple pleasantness vs. harmony seem to pretty strongly support the idea that simple ratios are "more pleasant."
The study itself is interested in competing psychoacoustical theories, one of which predicts that relationship is unaffected by timbre, and the other that predicts timbre has an effect. They found that sounds with more high harmonics could differ more from the simple harmonic ratios and still be pleasant, but roll off of those harmonics did not matter much, showing that timbre matters, but not in exactly the same way either model predicts.
Also, yeah, westerners are and were totally unfamiliar with things like bells. It's not like there's a massive western cultural institution that's wielded insane power for a couple thousand years that likes to install them in all their buildings or anything.
Scientists rediscover temperament? (Score:5, Informative)
In 1722, JS Bach published "Das wohltemperirte Clavier (The Well-Tempered Clavier)", a demonstration in all 24 keys of his proposed method of compromising the the purity of fifths (made by evenly subdividing frequencies) in order to align them at each octave. If you just subdivide the frequencies as Pythagoras noted, every octave would be slightly higher than its successor. Bach's method distributed small adjustments throughout the octave so everything comes out exactly even.
Not everyone was a fan, of course, but his method was eventually adopted to the tuning system we have today.
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Julliard has a nice write-up on Bach's temperament: https://journal.juilliard.edu/... [juilliard.edu]
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> but his method was eventually adopted to the tuning system we have today.
The tuning system that is common today. However vocal harmonies and unfretted string instruments enable other scale temperaments and the difference in the sound of the harmonies is clear when you hear them back to back. I'm not a keyboardist, but I assume any modern synth can be tweaked for closer fifths for a chosen key. I play guitar and pulling a string a little bit to improve the sound of a dog not interval (one where the inte
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> Not everyone was a fan, of course, but his method was eventually adopted to the tuning system we have today.
Bach's proposed technique was to demonstrate musically how (approximate) equal temperament can make certain musical activities easier, such as transposing a tune (or phrase) to different pitches (frequencies) without losing what we call a "consistent melody".
Since Bach liked to reuse the same (sub?) melody in different pitches and speed in a given composition, he needed a tuning conducive to that
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Addendum: tuning is rarely close to perfect on analog instruments such that the article's point may not really matter much to Bach's time. Even if the best tuner tuned a harpsichord, a slight change in temperature or humidity will throw it off from perfection.
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This is why pianos are tuned after they're rolled out onto the stage. Even now, they don't stay in tune when mechanically jostled too much. However, most harpsichords have one string per note (some more modern ones have two) rather than the two or three of a piano (except in the bass), and not as wide a compass, and it is practical to completely tune one in fifteen or twenty minutes if it's not ridiculously far out to start with. The string tension is also far lower than that of a piano, which reduces labor
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Back then they didn't have HVAC, so the temperature roamed more than we are used to even indoors.
> have two instruments, one of which is being tuned for the next set while the other is being used for performance.
How would the tuning keep quiet enough to not leak through the show? These days they have insulating cones that are pretty good at that, or even elevators to take it to a different level.
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Roll it off stage. Harpsichords aren't as prone to drift from just being rolled across a flat surface as pianos are.
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Thank you for saying approximately equal. The well temperaments in use were reasonably close to 12-ET, in the sense that you could play a well-tempered instrument and a 12-ET fretted instrument, and they'd be within 10 cents or so of each other even in the worst cases. This meant no key was so bad as to be unusable, but they still did have characteristic colors and grinds to them.
I would recommend you look up Stanley Lehman and his paper on what Bach's intended tuning was, based on the squiggles on the cove
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Sorry, BRADLEY Lehman! Brain fart.
And to think that when I pointed out a typo in his paper (the specific frequency in Hz of a note had a wrong digit -- but this did not carry over into any other calculations), he said I was the first person to notice in the seven years the paper had been online to that point. Sorry, man!
Choir / unison (Score:2)
Also the second argument about percussion is very weak, as pitch is a secondary (at least) characteristic of percussion.
This is just clickbait bullshit "Pythagoras was wrong". What's next: newsflash that we don't prefer perfectly symmetrical or perfect faces?
People aren't used to Pythagorean tuning (Score:2)
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Pythagorean tuning has been out of style for so long it sounds wildly off-key to most listeners today. We've been using mostly Equal Temperament tuning since the 1800s.
Fair point, but music hasn’t become less subjective since the 1800s. The invention of Autotune has probably warped the concept of music far more,
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Disagree, the move to ET tuning enabled all kinds of crazy modulations on instruments without dynamic tuning capabilities. This opened up a huge door for new musical ideas. All autotune did was make people who didn't practice singing and do 20 takes sound more passable.
But this, too, is just a subjective take from an amateur instrumentalist who would never have seen any benefit from autotune anyway
dumbassery (Score:5, Insightful)
Pythagoras is not wrong about consonance, we know that from simple observation. Minor variations or imperfections in pitch are a separate matter, Pythagoras after all didn't argue that vibrato was dissonant. And timbre, harmonic content, is yet another matter, instruments do not produce pure tones. The sheer hubris and ignorance here is amazing. Perfect for a /. article targeting the uneducated. SuperKendall approved.
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Maybe, but not really. The quality that differentiates the sound of different instruments is called timbre. It's made up of a bunch of things, but a very important one is the pattern of harmonics the instrument produces. A pure tone would just have the fundamental. Pure tones sound weird and synthetic, because you rarely encounter them.
The attack, decay and sustain of notes also contribute to timbre, but probably wouldn't be enough to tell most instruments apart without the harmonics.
BTW, the actual paper i
Blues musicians have known this for years (Score:2)
...and they don't need exotic instruments like the bonang
Jazz musicians have also figured this out
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Bonang players figured out out years ago ama didn't need exotic instruments like guitars
off topic (Score:1)
A432 > A440
Next you'll be telling us (Score:2)
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Funny that EWF tunes tend to modulate a lot, and unusual modulations like a major third at that. These would sound a whole lot jankier in Pythagorean, I'm sure.
Pythagoras was wrong in one very important way: he didn't accept five as a prime number basis for intervals, only two and three. Had he made that next step, he'd have been much more in line with the common practice to follow.
Whereas (Score:2)
Atonal music has never been popular in any culture or in any context, this study is clickbait nonsense.
Further, the notion Western musicians aren't familiar with bells, gongs and xylophones is the most asinine thing I've heard in months, and that's saying something considering the modern news cycle.
Music is math. They are two dialects of the only universal language, and their relationship is as precise as the universe they describe.
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> Atonal music has never been popular in any culture or in any context, this study is clickbait nonsense.
It didn't say "atonal", it just said "imperfect tuning". We westerners prefer our tuning "slightly imperfect" from the ideal, which is not the same as atonal. And there are atonal or semi-atonal genres that have decent followings. I'd say it's more a personal preference than a cultural preference, although there is some overlap.
> Further, the notion Western musicians aren't familiar with bells, gon
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The problem with mapping music precisely to math is that (3/2)^12 is 129.746... instead of being 128. And (4/3)^12 is 31.569... instead of being 32. And (5/4)^12 is 14.552... instead of being 16. As a result, none of the most consonant intervals (3/2, 4/3, 5/4) can be used to construct a scale that allows for equally consonant intervals in every key.
So, instead, Western music has decided that 2^(7/12), which equals 1.498..., is close enough to 3/2 to make a scale out of it. This allows for all keys to
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The worst "miss" actually is the harmonic seventh, at 31 cents sharp. It's significant enough that I make a point of including properly tuned harmonics for both the fifth and seventh harmonics of my guitar VSTs. I also tune the third and sixth harmonics up the appropriate 1.6 cents, but in my experience nobody would ever notice if I didn't.
Distortion, a key to rock-and-roll (Score:2)
This is akin to "distortion", something modern rock is built around. I was once trying to get my Midi sequencer to produce a mid-range "power chord" of sorts for my finale. It sounded thin no matter which instrument I used. I even tried doubling
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You didn't need a fancy gadget even then. Just record to tape with levels well into the red, and the saturation of the tape will give you a very nice soft clipping effect that rounds everything out.
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I didn't have good audio mixing equipment back then. I didn't have enough tracks for that and vocals, and when I merged them via re-recording, got a lot of tape hiss, so I avoided too many layers of merging. I suppose with enough fiddling maybe I could have learned a way around those problems. It was a matter of where I spend my fiddle time.
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Agreed, I just thought the money spent on your distortion pedal might have gone toward recording equipment which you could then abuse to produce your distortion effect (and still have a recording device).
Pythagoras (Score:2)
same thing (Score:2)
So Pythagoras said people universally like some musical harmonies and those researches found people universally like slight variation of those said harmonies. Not basically the same thing?
...and misunderstandings abound! (Score:2)
Yoko Ono (Score:2)