> “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.
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.)
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.
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).
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.
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.
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.
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.
"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.
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
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...
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:
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
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.
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.
>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
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.
First, I don't have a dog in the fight between parent and GP. Secondly, you're balking at a foundational commonplace in Western music.
Musical perception shares pathways with speech and it is believed this is also why certain timbres seem more expressive or interesting to us than others (that's over my head, but I first encountered that claim in the teaching company lecture series on mathematics and music). Let's consider that the first music must have been vocal or percussive, and that music making is innate to humans. Let's also consider that our hearing has to be able to finely distinguish sounds in the frequency range our voices can produce. This is why we don't make musical instruments only cats and dogs can hear at the very least -- our musical instruments have significant overlap with not only the sounds we can hear, but especially those we can produce vocally.
In terms of Western music, earliest polyphony was vocal. Contrapuntal technique likewise arose from vocal music (and this yielded our common practice harmonic tradition). I think every counterpoint manual since the renaissance (at least the ones I've read from Zarlino to Fux) begins with a discussion of the idiophone to establish the concepts of consonances and disonances based on proportions and then abruptly gets on with the business of counterpoint, where the ideal of the voice is mentioned whenever instrumental music is brought up. Additionally, there are special cases in counterpoint that arise from (or are explained as having arisen from) the sort of embellishments vocalist were fond of making (e.g. nota cambiata for one example). Not to beat a dead horse, but I even recall reading a fugue manual from the 1700s discussing an instrumental fugue and pointing out that the first note of the subject is longer, being a mannerism from vocal fugue writing (singers felt more secure if they could sit on the note a little at the beginning of an entrance).
Every single manual I've read on instrumental performance has stressed achieving a vocal quality. This is really deeply baked into Western music.
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.
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.
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.
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.
When slightly detuned, it's so close to the perfect ratio that it still satisfies our preference for perfect ratios, and it also fills more of the frequency spectrum (which is another thing we like).
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.
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.
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.
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.
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
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...
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.
> "these preferences for slight deviations disappeared upon elimination of the upper harmonics, presumably because this eliminates the slow beating effect"
It seems like the title is oversimplifying results for the sake of clickbait sensationalism.
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.
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.
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.
that's not how phase works, though you're thinking reasonable thoughts.
if you have a bicycle with two wheels, where one wheel is slightly bigger than the other, and you mark the wheels with a white spot, as the bicycle rolls, say the white spots start out in the same place (wrt a clock face). As the bicycle rolls, the white spots will "go out of phase" with each other, and with every roll, even more out of phase, till they are so out of phase that they start to slowly come a little bit more in phase with each other with every roll, till they match up again... except they are not out of phase, they are just rolling at two different frequencies, and that mix of frequencies is the "timbre" that you hear.
if you have two signals at the same frequency which are out of phase, and you measure at any point, you will just measure a signal at that frequency whose amplitude is the sum of the two signals at that point. You won't be able to tell there is any phasing of two signals taking place. If your two ears hear the same frequency, but with a phase difference, that will be interpreted in part as what direction the sound comes from; it's not definitive, it's just evidence that gets combined with any other senses you have of where the sound should be coming from.
there's a lot more to say about it but no room here for a treatise.
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
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?
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.
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.
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".
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?
EDIT: Hawai'ian slack key (listening, not dancing music) has some fancier influences (which I think they may have picked up from arabic music by way of the portuguese) but even there you can tell how much the locals pulled from {church hymns,country}: https://www.youtube.com/watch?v=tpfT8AzAl0c
Beyoncé is rumoured to have covered Dolly on her most recent album; everyone else seems to be hoping it's Jolene (which would be the commercially appropriate choice), but I wish it were Coat of Many Colors: https://www.youtube.com/watch?v=HYAdKXzGtcY
* I tried to listen to Silk Smitha once but guess I lack the background to appreciate her songs...
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.
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.
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.
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.
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.
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.
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.
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.
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).
I have a pet theory that one of the reasons the V7 to I resolution is so pleasing and so fundamental to Western harmony is that the ET b7 is the sharpest from the harmonic 7th, and the ear wants the V dominant’s b7 to resolve down to the 3rd of the I chord.
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.
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.
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.