Harmony Explained: Progress Towards a Scientific Theory of Music (2012) | Hacker News Reader

Harmony Explained: Progress Towards a Scientific Theory of Music (2012)

arxiv.org

105 pointssurprisetalk114 comments

I read through a lot of this article and it does a good job explaining the ratios and math behind harmonics and how that makes chords, but ideas like lower ratios are more pleasant and minor chords feel like something is missing (from the article "I do not hear the Harmonic Series itself -- something is missing") is just wrong - if it was right, then the happiest music would be entirely octaves. Though it's built on harmonics and overtones which can be explained objectively, you cannot make a fully scientific theory of harmony since cultural expectations factor in just as much (and maybe even more) than the expectations around harmonics. Music is pleasurable when it mostly meets expectations but surprises you as well, IE it sits just outside of what we expect. It then becomes part of what you expect in the future and music evolves.

Ultimately minor sounds sad because it's been used that way for a long time and it's what we expect. I was listening to Trout Mask Replica and my wife said it sounded like her home town funeral music (I later heard some myself and yeah she was right) - Captain Beefheart doesn't sound sad to me, but it does to her. The funeral music uses major 5 note scale.

And on "But I've Been a Musician All My Life / Studied Music In College and I've Never Heard Any of This Before!" - ratios and how chords derive from that was absolutely part of my education at Berklee. This article is good for the objective parts, but really self-congratulating for the subjective/experiential part of it.

Right on. In my experience across different art fields, I’d say the only generalizable facet of ‘good’ art/music/etc is the “it sits just outside of what we expect”.

These days I liken it more to the idea of ‘surprise’ in information entropy which denotes an informative signal then to revealing any sort of objective truth.

You might like the book Emotion and Meaning in Music - it starts from some outdated psychological ideas (that emotions are only intensity and that we assign positive/negative valence post-hoc) but ties that to early information theory and makes a really compelling theory of music that's all about the interplay between expectation and surprise, across both the physical harmonic ratio stuff this paper talks about and the cultural associations too.

Fantastic! I think you've read me right in terms of interests. I'll check it out. Thank you!

While I agree that "good" art "sits just outside of what we expect", I must play devil's advocate here and say that the uncanny valley of CGI/AI/... also "sits just outside of what we expect".

So there must be more to it.

Eh, turning that uncanny valley to 11 can be an artform in itself: https://www.youtube.com/watch?v=TGIvO4eh190

By human? I don't doubt that.

Yes, there's certainly more to it. Just to jump on the uncanny valley vector, I'd argue that the artist having a coherent message they're trying to get across and the viewer having some affordance primed for receiving that message is critical.

I find the major/minor thing so fascinating. I remember it having the "happy"/"sad" vibes from my earliest memories and have a really hard time with the idea that it could be cultural. But if it's not cultural, then why? Also is the fact that major 9ths have a beach/island vibe cultural due to Brazilian jazz? :-)

Yeah, that's really getting to the heart of why the idea of a totally objective theory of music seems legit. Major/minor doesn't exist in a vacuum, there's a lot of other signifiers in the music that are highly correlated. It's not just the major triad, it's the energy of the beat, playing, singing, the bright instruments. If you were a 2 year old and just heard a major chord played on a piano, then a minor chord, then you probably wouldn't get "happy" or "sad" from it. Maybe by 3 or 4 you've already picked up the correlation between "happy music" and "major triad". You pick this stuff up really quick and it's part of your core memories/mental model, so it can feel inherent. It's like the fish that doesn't know what water is.

I've been working with some people building a music model from scratch and it really surprises me how much correlation there is across different features. It's been difficult applying that to general sound effects, because music is highly correlated in ways more general sound isn't.

> have a really hard time with the idea that it could be cultural

I played in a gamelan ensemble in college. The music was completely foreign. A piece that sounded like a funeral procession was played at weddings. I have a hard time believing the arrangement of music is not cultural - or, at least partially culturally bound.

Is "Take Five" sad-sounding even though it's written in a minor key? Maybe it sounds somewhat "cooler", not as bright as a major key piece of the same tempo and energy would be, but it doesn't sound sad.

I play piano. It's not just cultural, it is partly like that checkerboard lighting illusion. What is objectively one shade, can in a different context, seem like the opposite shade. Major and minor keys in classical music can sometimes exhibit this property.

Silly theory for why minor sounds sad: Say "ah", like you've just discovered something interesting. Now say "ah" like you've been told something sad. Notice the pitch difference? And notice how both pitches are about a major third and minor third above normal speaking pitch. Whether that's hard-wired or cultural, I couldn't say. But it would certainly be an interesting thing to research and compare across cultures and musical traditions.

Harry Partch has a book Genesis of a Music, which is half autobiography and half a theory that all melody derives from the micro pitch variations of speech - as in melody is a stretched out version of how we speak. I don't know if this idea has ever been pursued more academically but I find it very interesting. If it's true, I also wonder if there's feedback, like our vocal inflections are influenced by music melodies just as much as vice versa.

Metal music is mostly minor.

Does metal sound "sad"? Opinions differ. :)

> you cannot make a fully scientific theory of harmony since cultural expectations factor in just as much (and maybe even more) than the expectations around harmonics.

That's only true if you're expecting a "fully scientific theory of harmony" to mean "a theory that can precisely predict how popular/enjoyable any given piece of music will be." But I don't think that's a reasonable goal, since (as you say) individuals are complex and culture is even more complex.

It's not unlike, say, pain or discomfort, which we can also develop scientific theories for. We can discover how pain works physiologically and psychologically, but our theories are never going to fully explain why individuals or cultures might welcome and even seek out certain types of pain in certain contexts.

I meant "fully scientific theory of harmony" in the same way your second paragraph does, not as predictor of popularity but predictor of what kind of effect it raises in the listener. A big part of how music effects the listener is sociology, which sure is scientific, but I tried to use scientific in the way the article uses the term, IE from objective first principles and not culturally specific/learned response.

It's worse than that though -- the things that you are born enjoying, are loud oceanic white-noises and a deep thumping backbeat with occasional muddied high-pitched vocals.

I had a similar dream as a freshman in physics, I had learned that the organ of Corti was a spiral-shaped transducer in a little special bone and I dreamed of snail shells and proving that harmonies formed interesting geometric patterns on the darned thing, and how maybe people who seemed chronically tuneless might be shown to have malformed ear bones -- until I finally got to see images of them in a bio textbook and realized that my dreams had to be crushed, this blobby thing, whatever else you say, is not working geometrically.

So a science of how music works physiologically and psychologically, is able to explain why rock and later certain forms of ambient and trance "felt good" to folks -- it took them right back into the womb: the distortion on the guitars felt like the swishing amniotic fluid, the four-on-the-floor sounded like a heartbeat, the indecipherable high-pitched vocals sounded like Mama's voice peeking through the noise. But beyond that, it's our experience with twanging strings and air blowing through wind instruments and the like, which informs what sounds familiar to our grown ears. And the proof is how robotic and alien pure sine waves sound: this is a part of your organ of Corti complaining, "hey, when these hairs are tickled usually those ones also tickle -- but they're just standing flat, what the hell is going on."

Lots of everyday things tickle the organ of Corti in patterns that resemble a major chord, so that sounds familiar to it. When you play a minor chord you deprive it of stimulation to some of those hairs but stimulate the ones a little more outward, and the organ says "there's something off, this thing is lower than I'd expect" and it's some sociology that you can do to say "when people are hurt they can communicate this most effectively to other pack animals through lower more mournful cries" and so there are semi-biological semi-cultural expectations that govern "lower = sadder" especially at low tempos. But you can do the same in reverse, play a sus4 chord, and it sounds weirdly brighter and you feedback through those semi-biological expectations in reverse, weird but with more energy than expected. But you can also just play minor scales uptempo and then they also change, the minor scale at a higher tempo has somehow more energy and less sadness than the major scale -- indeed rock music is full of minor pentatonic licks that nobody would ever describe as sad or painful or discomforting.

I think you could make a scientific theory of harmony (and perhaps music or even sound more generally) through the physiological lens of human perception. Straying a bit from music, but some sounds are fairly universally unpleasant, so there may be some objective basis for some sounds, and harmonies, being more “pleasurable” than others.

I don't buy that minor being "sad" is all nurture and no nature. No matter what tempo or rhythm you put on it or what instruments you use to play it, it may not sound sad, it may sound dramatic and powerful instead, but it never sounds upbeat and happy the way major does (or at least can).

Listening to "Fiddler on the Roof" and klezmer music unsettled to me the idea that major is happier than minor. I still think minor tends to sound "sad", but those wedding songs and the Russian dance in FotR are clearly upbeat and celebratory. (Literally upbeat; the Russian dance is more upward-beat, as it gets faster and faster!)

Yes, the klezmer “default” is minor and upbeat.

It’s my favorite example for the “minor != universally sad across cultures” thing since it’s not so “out there” for the Western ear that people would struggle to even hear minor/major/“music” in it at all.

I’m not seeing how that’s inconsistent with it being culturally derived?

Your brain is still just pattern matching “minor = sad” based on thousands of hours of examples from the western canon consumed throughout your life regardless of how you dress it up with rhythmic variations.

> but TO ME it never sounds upbeat and happy the way major does (or at least can)

FTFY. Notice when you conflate subjective experiences with objective ones. You have been molded by the culture you grew up with. Trying to disentangle yourself from it is like for a monolingual person to express thoughts without using a language.

> if it was right, then the happiest music would be entirely octaves

Not necessarily: if you read the essay, one assumption is that the brain is halving and doubling frequencies to fit them all into one frequency magnitude, an obvious optimization for reducing circuitry. If so, then an octave is too simple to fire the recognizer for a particular harmonic series; instead we need an input that places it relative to the other tones even after the halving/doubling.

Consider the fact that we name both middle C and high C as "C" because, even though they are different notes, they sound so similar that we literally deliberately confuse them notationally as they play the same role in harmony: one can be used to replace the other.

> Music is pleasurable when it mostly meets expectations but surprises you as well, IE it sits just outside of what we expect.

Ok, but where do the expectations come from? Your assertion that it is all culturally relative ("nurture") and none from the structure of the most efficient algorithm for recognizing the harmonic series ("nature") has no basis in fact; you just assert it without proof.

In contrast, there is strong evidence for the absolute nature of auditory processing. In the essay, there is a quotation from Daniel Levitin's book "This is your Brain on Music" where it discusses an experiment he witnessed in graduate school where music was played through wiring directly into the brain of an owl. The music was played into the brain with the root removed; when coming out of the brain of the owl, brain had restored the root (see the essay for details).

That means that virtual pitch (1) is created by the structure of the brain, not by a cultural expectation, and (2) further, that it is created by the brain of, not just a human, but of an owl! This is a very strong argument for the assertion that the processing of sound and the artifacts thereof are quite universal, not just across cultures, but across species.

> ratios and how chords derive from that was absolutely part of my education at Berklee

So at Berklee they teach recognizer algorithms and artifacts thereof? I did not know that Berklee is that strong in computer science. The explanation of the experience of the minor triad in "Harmony Explained" requires the concept of an inconsistency arising from a redundant recognizer algorithm.

In this case, one recognizer fires, that of the conjunction of three pairwise intervals of tones, whereas another recognizer does not fire, that if the triple of tones in order. This combination of firings is inconsistent because it is not possible for it to occur by playing any single normal harmonic series on, say, a flute. To generate it, one must play multiple notes and then use the effect of the recognizer of intervals pairing up tones from across the notes.

That is, this is a new kind of auditory perception that can be created as an artifact of the auditory recognizer algorithms of the brain, but cannot be heard otherwise. This is harmony.

If this this was taught to you at Berklee, please tell me which course that was and what the textbook is so I can look it up in the course syllabus and table of contents.

> one assumption is that the brain is halving and doubling frequencies to fit them all into one frequency magnitude

You've got an awful lot of assumptions in there that I don't think are at all supported, and this is one. It's also contradicted by 3.5.2 in the article: if the brain were actually doing this, then you could, say, replace the 9th by the 2nd.

> Ok, but where do the expectations come from? Your assertion that it is all culturally relative ("nurture") and none from the structure of the most efficient algorithm for recognizing the harmonic series ("nature") has no basis in fact; you just assert it without proof.

I interpreted "mostly meets expectations but surprises you" as referring to things like chord progressions creating tension and resolving it, for which you don't need major triads. (and of which there's no discussion in the article).

> In contrast, there is strong evidence for the absolute nature of auditory processing

I'm perfectly happy to grant that an owl experienced the sensation of a pitch at a frequency that was not actually present, but there's something missing here that's also totally missing from the article: the role of the cochlea. The brain doesn't experience sounds, i.e. the pressure waves in the air, the brain experiences electrical signals transmitted to it that, tinnitus aside, come from excitement of the cochlea by those pressure waves. How do we know that there isn't some kind of mechanical resonance somewhere in the ear that fills in the missing root? And if we know there isn't in the human ear, how do we know the same for the owl?

> The explanation of the experience of the minor triad in "Harmony Explained" requires the concept of an inconsistency arising from a redundant recognizer algorithm.

Once upon a time the explanation of the propagation of light required the concept of a luminiferous ether.

I ask that you please read the comments before you make your own comments, just as you asked people to read the paper before commenting (I did!)

> Your assertion that it is all culturally relative ("nurture") and none from the structure of the most efficient algorithm for recognizing the harmonic series ("nature") has no basis in fact

I never said that. I said "Though it's built on harmonics and overtones which can be explained objectively, you cannot make a fully scientific theory of harmony since cultural expectations factor in just as much (and maybe even more) than the expectations around harmonics." To put it in your terms, "you cannot make a fully scientific theory since nurture factors in just as much (and maybe even more) than nature". The world is not black and white, just because I believe sociology to be critical to understanding the emotional effect of music on people, does not mean I'm claiming there is no objective, measurable component.

> If so, then an octave is too simple to fire the recognizer"

Ok so the happiest music is perfect fifths? Or is 3/2 ratio also too simple? So when does this recognizer get complicated enough to fire? 5/4? For someone claiming to have a scientific theory, you are making a lot of assumptions to make this theory work. Scientific theory is about testing hypothesis to reveal truth and not making hypothesis up to match what you already believe. You make an assumption that the brain wraps/aliases frequencies, create a theory of the recognizer, and then make that recognizer have a very convenient threshold that it makes a physical process suddenly the reasoning for "happy". To the "fully scientfic theory" above, I absolutely agree that ratios play a part, simple ratios are boring, and get more interesting as they get complex, then get too complex and it sounds dissonant. That part I think we, and most people, can agree is objectively true, and fits in the theory of expectation vs surprise at a microscale. I just don't agree this translates to "happy" or "sad". Please show me an experiment that indicates the kind of recognizer that you're talking about, or some kind of cross cultural link between ratios and emotional quality. I'd actually be very curious about that, as it's been a while since I've been reading papers about this.

> The music was played into the brain with the root removed; when coming out of the brain of the owl, brain had restored the root

Yes, I'm sure you know this, I'm just showing I know this too - this is called psychoacoustic bass, and happens as part of the feedback loop between the brain processing and the actual ear - the ear actually generates the root tone! This could also be a product of pattern matching though, as we're used to hearing related harmonics (say 200, 300hz) with a root (100hz), so if we just get the harmonics without a root (200, 300hz) the psychoacoustic part of your brain helps out by filling in the 100hz - this is an alternate theory to the aliasing/halving/doubling theory you put out, not saying either one is absolutely true, just that there are other explanations for the owl thing.

It also has nothing to do with the emotional effect of the music.

> So at Berklee they teach recognizer algorithms and artifacts thereof?

Again, I did not say that. I'm not talking about the speculative, non-experimentally backed parts of your theory, just the parts of your essay that are part of the common scientific consensus. I said "ratios and how chords derive from that was absolutely part of my education at Berklee". Nothing more.

>happiest music would be entirely octaves

While chords certainly do have a perception of emotion, it's resolution and chord progression that seals the deal whether it's perceived as joyful or sad or otherwise.

Yep, I have come to believe that as complex adaptive systems made of complex adaptive systems the part of us that experience music do so through being able to predict new information.

If there is no ability to predict it’s random and just noise.

If there is no uncertainty and it’s just purely repetitive then there’s also no new information and it loses the music (see what happens with overplayed pre-recorded songs).

The sweet spot is when we can to some degree predict what’s coming and thus understand it to be information while not being sure of what’s coming so we get the reward of encoding it.

I think this works for pretty much all genres and contexts. Even listening to techno in a dark warehouse while off ones face: the altered state makes most things seem novel no matter how repetitive and your mind doesn’t tune the information out.

But that’s IMO why octave harmonies largely don’t work: the mind perceives no information in the pure octave, and so little joy is gained from it.

I studied harmony and music analysis many years ago as a piano performance major in school. One thing I will say is that harmony really is not an abstract set of symbols on a page representing chords, but rather is a taxonomy of common specific sequences of simultaneously sounding pitches, abstracting away instruments or larger structural considerations and providing a dictionary that defines the most common patterns. Here's an example: Perhaps you're familiar with the idea of major and minor, but you might be unfamiliar with a Picardy third, where a composition is in a minor ("sad") key throughout but ends in that same pitch but on a major ("happy") chord. J.S. Bach used Picardy thirds regularly to end his compositions, but the Picardy third became rarer after Bach's death. Still, it's a beautiful and striking harmonic progression that continues to appear now and then. And now you know EXACTLY why that 1967 song by the Turtles, "Happy Together", has that really nice and memorable ending; that's a modern pop example of a Picardy third. Again, harmony is not abstract; it's the names musicians have given for specific sequential combinations, many of which have been named centuries ago. It's not about nicer or worse; sounds are sounds, but some are combined more often than others and those combined most frequently have well-known names that can be used to explain concisely the "harmonic" aspect of a piece of music.

I think the word "taxonomy" is a really good way to look at music theory. Really, the study is about giving names and notation to things in music so we can write and talk about it at all.

There's an interesting idea though that there could be a deeper why specific sounds and their combinations are the things we create names for, but that's an intersection of cognition/physics/culture that goes beyond elementary music theory.

where would I learn a taxonomy of such motifs? I trained for classical piano as a kid, and I'm slowly trying to learn jazz piano as I ease back into it! I'd love to learn more about these types of harmonic ideas and motifs as I get back to playing

Can't go wrong with Alfred's Essentials of Music Theory

There are much better overviews of this from Sethares and Tymoczko.

Consonance is somewhat frequency-dependent. For more, Google Critical Band and ERB.

The happy/sad idea is pretty much a high-school oversimplification. It's true for beginners, not at all true beyond that.\

It's quite rare to find music with straight major and/or minor triads with no other colours.

to add, all of this doesn’t account for music outside of that which is a direct descendant of 15th and 16th century contrapuntal music, aka the sort defined by exactly the medieval texts it pokes fun at. any fusions are outside scope, let alone entire non european cultures of music. incredibly ethnocentric and blinkered premise.

Are there any readings would you recommend to read if I play classical piano, and especially for Romantic and post-Romantic music ranging from Chopin/Liszt to Debussy/Ravel?

If you want to go way deep, check out Tymoczko's first book: https://dmitri.mycpanel.princeton.edu/geometry-of-music.html

It's a challenging read, but extremely thorough and enlightening. Tymoczko wrote a follow up that extends and reformulates his theories, but I haven't gotten through it yet.

I also jumped straight to the bibliography to look for those two. Not seeing any reference to their work is disappointing and makes me a little pessimistic that digesting the entire piece will be worthwhile, as a scientific piece. He references Helmholtz, but his theories have been surpassed for more than a century.

But hopefully I'll get around to reading this at some point. It's one of my favorite topics. I wrote up my own attempt to explain the Western music system here: https://fakestoryofmusic.com

I've also been working on something smaller that focuses on major/minor tonality, specifically. But it's further from being in a place where I want to share it.

A good overview on the topic.

My general issue with things like these is the simple fact, that as a lifelong musician I find the "less worse" harmonic combinations (to quote the abstract) often very boring. Aside from great composers like Bach et al I find the harmonic relationship of the notes played one of the least interesting aspects of the music, and expression, timbre, rhythm, phrasing much more important and noteworthy. A set of great musicians can play a single note for minutes and get it to sound amazing, this has always fascinated me more.

Music theory IMO has historically a too strong focus on the harmony (or the lack thereof) between notes. Not that it isn't important, but with some texts you get the feeling it wouldn't matter to the authors whether the piece was heard expressively played on a grand piano or cut together from equal leveled sine waves.

In my experience harmonies are greatly influenced by the relative levels of the notes and the overtone content within each note, to a degree you can make a supposed "harmonic" chord sound bad and a supposed disharmonic one sound good.

Do you have any examples of "a single note for minutes and get it to sound amazing"? I'm finding this difficult to imagine. A lead voice doing this while some accompaniment is changing chords underneath makes sense to me, but then you're back to harmonic relationships.

Half joking, but Berlin techno :)

You can have a whole set developing motifs, then multiple minutes of a single note being played in the middle, and somehow that sounds awesome (if you’re into it)

A long hold of one note doesn't mean there isn't harmonic progression. It just means the progression is slow in that part of the song. It's often done to create a massive contrast that occurs when the progression finally...progresses. In electronic music, this often comes as the drop or some similar kind of break.

No, in electronic music harmonic progressions are done by manipulating timbre.

Try it yourself - play one note on repeat and start twirling the cutoff knob.

Not quite one note, but "Open Eye Signal" (2023 remaster) by Jon Hopkins has wide spectrum background and heavy rhythms, but once the song gets going the main instrument moves between fifth -> seventh-flat -> sixth -> fifth on repeat, and that's the melody for the song. And yet, to me, it's captivating.

Thanks for the suggestion, that’s a really neat discovery! I will try to integrate that in my own music

I'm finding this difficult to imagine

Maybe go on Youtube and listen to music that you might not be familiar with?

Cage, Schaffer, Stockhousen, Boulez are some composers in the tradition of European classical music.

If you go outside that tradition, there’s even more. Endlessly more.

I can only think of examples which as you describe have accompaniment moving underneath. Neil Young's solo in Cinnamon Girl is one example. Another is Bob Dylan's Like a Rolling Stone in which he sings one note as the chords run up C Dm Em F and only breaks on the G.

Adam Neely has an interesting video about one-note muisc, with some good (and bad) examples: https://www.youtube.com/watch?v=eSuK_5zW2iM

Chariots of Fire is pretty dope.

A lot of modern metal is mostly one note played rhythmically. And it is can sound awesome, however timbre is obviously part of this awesomeness.

I'm pretty sure you've heard plenty of ambient music and just never thought about it. Timbre and rhythm have been the focus of modern music for at least 40 years.

A professor taught us genre is defined by how you resolve dissonance...

And dissonance has cultural and physiological components...

And other traditions focus on rhythm over harmony...

That said this looks like a good introduction.

That's a great line from the professor. Really great.

Also, other traditions focus on melody over harmony (e.g. Carnatic and Hindustani), although they have to consider harmony as a by-product of polyphonic music (i.e. more than 1 instrument).

> Also, other traditions focus on melody over harmony

Like Western electronic music written after 1960, for example. :)

(And electronic music is, like, basically all we have today.)

This is really not true.

Firstly, "western electronic music" covers an enormous range, from still, wall-of-sound ambient to four-on-the-floor EDM to music that sounds like any other but is generated using electronic/digital instruments.

Secondly, I could give you tons of examples of western electronic music that is almost entirely about harmony and not melody. The first one that springs to mind is any of Steve Roach's "immersion" series, or even more so "Mystic Chords and Sacred Spaces". But basically anything that might get played on SomaFM's "Drone Zone" channel is in the same category.

So, I suspect you have a much more specific idea of what you actually meant by "western electronic music", and you should likely lay it out more precisely.

"Western electronic music" is anything that uses an instrument with a filter cutoff knob.

Filter knobs and polyphony don't play together well.

That's why they gravitate towards "leads" or "pads" over a separate bass or percussion track. Unlike normal Western polyphony you want the two to NOT form chords or counterpoint.

This just isn't true. Try listening to any deep ambient, and you'll be hearing (a) instruments with filter cutoffs in use (b) chords (c) no percussion (d) no lead. Again, Roach's "Mystic Chords and Sacred Spaces" is as good a place to start as any.

What is true in ambient is that you frequently do not want the third in the chords, because it makes the chords "too obvious". A lot of drone music involves a variety of chords with the third omitted.

IMO "chords" is the wrong word to use here, because chords are a fingering technique for polyphonic instruments. (So, guitar, piano, accordion, and ... that's it?)

What I mean is voice leading and counterpoint - once you have a filter sweep as an instrument you now need to deliberately avoid all that when composing.

People act as if these Western music theory concepts are grand universals, but they're really just useful conventions for the particular music instruments common at the time. They're not even universal for all of Western music; when the instruments evolved so did the conventions and the "theory".

"Chords" is definitely the wrong word to use if you mean "voice leading and counterpoint" or more generally, western tonal music theory. And yes, that theory is highly localized in both time and space.

Have you read Sethares' Tuning Timbre Spectrum Scale book or the shorter paper? It gets right in on the topic of the interrelationship between timbre and dissonance. As you say, the timbres of the instruments are very important to how dissonance is or isn't experienced by the listener.

Also a lifelong musician here, and I think you're completely wrong about harmony. Western music (assuming that's what we're talking about here) IS directly derived from harmonic progression. It is truly the core of the music. The underlying chords are the obvious harmonic progression. Melody is just an expression of underlying harmonic structure. Rhythm is meant to emphasize the changes and movement of harmonic structure. Timbre itself is meaningless without a note to carry it, and a note is meaningless without the harmony it implies.

All of the many parts of western music serve the harmonic progression. This is because human emotion attaches itself most strongly to harmonic progression, and music is the art of manipulating human emotion through sound.

>A set of great musicians can play a single note for minutes and get it to sound amazing, this has always fascinated me more.

Someone already asked, but give me an example.

Not a one-note solo, but Miles Davis's solo on So What is really minimalistic.

>Timbre itself is meaningless without a note to carry it

I suppose that is literally true but I think of timbre (especially after reading Sethares who I mentioned earlier) as telling me about the notes that are brought along for the ride but aren't shown on the sheet music. It's well known that a distorted powerchord introduces a major 3rd (and not the 12TET major third) as well as a phantom or missing bass note and other higher notes which may or may not be perceptible in a given piece. Quite a bit different from what two flutes playing a root and fifth will give you.

That makes some sense, but if you look at a spectrum analysis of any signal that isn't a sine wave (and specially for highly clipped signals like you mention here), they consist of many tens or hundreds of even and odd harmonics, which is what you're referring to here when you say a root/fifth chord under distortion produces a major third phantom note. In truth, it creates many, many phantom notes, and pretty much any instrument, even a flute, does as well. Yes, that's the timbre - the relationship of the harmonics that result from a waveform. That's the very idea behind additive synthesis, where the harmonic nodes are individually created from a series of sine waves. So I guess it could be said that a single non-sine note would produce a harmony (multiple notes with a harmonic relationship), but this is not the same thing as a harmonic progression, where that relationship would change.

I would actually invert that and say that melody and rhythm are the most fundamental aspects of music. A harmonic progression with no sense of melody or rhythm is quite boring.

Harmony is derived from composing music with multiple melodic parts. This is most obvious with counterpoint, but even pop/rock music can be seen this way. The essence of pop/rock is a melody and bass line as the two primary parts, and the rest of the instrumentation often plays harmonies that imply inner voices.

I personally consider this to be a better intro, albeit in video form:

https://www.youtube.com/watch?v=tCsl6ZcY9ag

This is a more recent paper (2019) that I think is better constructed and more aware of the in-field work than TFA:

https://pmc.ncbi.nlm.nih.gov/articles/PMC7006947/

Yes, not having had the time to read TFA or the paper you shared yet, it immediately stands out to me that the bibliography of the latter paper is far more comprehensive.

The logic that something sounds worse simply by combining dissonant notes is questionable, because music constantly combines elements—notes, chords, even different rhythms—without that necessarily implying displeasure for the listener. Convincing.

"Worse" is definitely a subjective appreciation influenced by culture and personal taste.

That's why it's questionable, because it's subjective. It would be factual to find the keys to determine what sounds "worse" across the majority of subjective opinions, independent of culture and personal taste.

Yeah. If anyone doubts this go listen to some gamelan music and see if they can make any sense of it (harmonically speaking).

https://www.youtube.com/watch?v=nGCSrC8RN6c

In the 00s, I sang in the Holy Name Cathedral choir and we did a lot of 20th/21st century choral works with harmonies that strayed very far from the classical tonal vocabulary to the point where when I was doing my own composition work, I realized that I didn’t hear dissonance any more.

I don't find this convincing at all. The actual interesting attempt at scientific theory of harmony in this could be distilled to a couple of lines, yet the author rambles endlessly in ways that don't add to the piece at all. I already knew his additions from traditional music theory already, so I can't really see what the purpose of this is.

If this essay can be distilled into a couple of lines, please do that.

I only skimmed the article. There's a lot of good stuff (debunking a lot of classical harmony) but they miss some important things too. There's nothing about repetition. Repetition legitimizes. Our brain is really good at pattern recognition, and as soon as it can predict something you're more than half way there to make it sound good. There's plenty of chromaticism that works because the brain "gets" the sense of direction.

I was wondering whether this might have an explanation for why I don't really like major music, preferring minor / modal / non-functional harmony... it does:

> Throughout this derivation of different chords, you will note that a gradual progression or degradation from high-theme/low-complexity (Major Triad, Harmonic Series chords) to low-theme/high-complexity (Minor and Ambiguous chords). This progression is suggested by our measure of interestingness from Section 2.4 "Interestingness: Just Enough Complexity". That is, as we progress, more and more of the theme of the Harmonic Series is lost and more and more complexity is introduced. Notice that this progression seems to mirror that of musical sophistication as well: musically untrained listeners like Major chords while more musically trained listeners are more tolerant to loss of theme and more interested in complexity. (I met a signal processing engineer who had played piano for something like 18 years and who simply did not like Major chords at all.) Other fields seem to progress similarly: white wine is preferred by new wine drinkers, whereas more "complex" red wines are an acquired taste.

Consider that you're not actually hearing major and minor intervals, you're hearing approximations of them due to 12TET.

Depends on the instrument. Temperament is mostly applicable to keyboard and some electronic instruments.

Awfully dismissive of non western tonal harmonic systems for a universal theory of music? The whole idea that musical traditions that use less harmonic structures just haven't unlocked all the structures available is kind of absurd - those musical traditions have been developing for the same amount of time - and there's a pretty huge blind spot for other traditions. The whole microtones just being ornamentations - no that's absolutely not the case. Trying to ignore/hand wave away all the counterexamples and force them into your system is how you get epicycles. If you're gonna do a universal theory of music it would be a good idea to engage deeply with more than one system. As a universal theory of harmonica maybe that's fine but it's sloppy and motivated reasoning

More directly the stuff about ranking different cultures on a scale of musical sophistication that ends in western harmonic music and dismissing obvious counterexamples as cultures that just haven't gotten there yet or just use other feature as ornamentation or just happened to not stumble on things because they were at the lowest level of musical sophistication is almost comically ethnocentric. Its an easy hammer to reach for when your counterarguments are not European (though in this case you're having to ignore an awful lot of European music and the huge influence of Greek music on the middle East) but yeesh

I think the most fascinating thing about music (from a mathematical/scientific perspective) is the fact that the principal harmonics of equal temperament are so close to the mathematical ideals. Fourths and fifths are so close trained musicians typically cannot hear the difference. Major thirds are farther from the ideal but still are less than 1% off. That can be heard by the trained ear, but in the real world there are chorus effects (multiple voices/instruments at nominal but slightly varied pitch), vibrato, reverb etc. such that in most cases a just major 3rd sounds fine.

Without just temperament, we would never have had "The Well Tempered Clavier", Bobby Daren's "Mac The knife", and basically all of western music from the baroque period on.

I have to highly recommend the book 'Fundamentals of Musical Acoustics' for mathematical/physical explanations of sound and music. Bells having multiple resonant modes, tuning kettle drums, how string materials and plucking methods changes harmonic profiles, why having lots of high frequency harmonics sounds bright (and electric guitars, pianos optimise for this), why sounds without high frequency harmonics sound hollow (nylon strings), virtual pitch, acoustics of rooms, etc.

My impression is that most sounds have their effect on us due to psychological association with familiar sources of those sounds, and that melody and harmony are fundamentally more subjective - they depend on looser associations with motion, mood, the human voice, the changes and developments of sounds in our environment.

Are you talking about "Fundamentals of Musical Acoustics" by Arthur H. Benade? I'm interested in buying the book, but I'm not sure which one you mean. Do you perhaps even have the ISBN?

That's the one. I read the 2nd edition on kindle. There is only one short chapter on harmony but discussion of tone, intervals, etc is comprehensive.

Alright, thanks a lot :)

I stopped automatically judging things 10 years ago and all of a sudden could enjoy listening to everything. Music theory based on pleasure without accounting for this self-manipulation we're brought up to do isn't solid at all. And expectation making is another thing I stopped doing. I now make music that tends to defy expectations and tonality.

Here's a statistical mechanics paper that I don't quite understand yet, but seems more aligned with the physics of harmony.

https://www.science.org/doi/10.1126/sciadv.aav8490

Harmony isn't just the chords made, though. It's about the progression between chords,too.

I agree that there's a lot to be desired in the pedagogy of music theory, but this is pretty bad. There are so many little things wrong with it that if I started trying to address them all I'd be here forever.

Anybody interested in this stuff would be better off with something like Tuning Timbre Spectrum Scale (https://sethares.engr.wisc.edu/ttss.html) or Music a Mathematical Offering (https://homepages.abdn.ac.uk/d.j.benson/pages/html/maths-mus...)

Responses that say "If I started listing criticisms I would never finish" usually have none. Name 3.

I gave a couple in my other reply, the most damning (imo) was that you completely ignore the cochlea.

My biggest overall critique is that you start with the assumption that everything is an algorithm, that the brain works like a computer, and the world can be derived from that.

"To the intuition of anyone who has seen hardware designed it seems very likely that the brain is halving/doubling frequencies by many different powers of two in parallel and then running all of the results through the frequency recognizer at once. If any one matches, the harmonic has been found. If this were so, then tones (and notes) that differ from each other by a factor of two would sound very much alike. "

I don't see why I should care about a hardware designer's opinion on the matter any more than a clockmaker's or a dog trainer's.

"1.5.1 Timbre: Systematic Distortions from the Ideal Harmonic Series" I don't believe you ever mention how you want the intensities in your "ideal harmonic series" to be distribute, so why does changing the relative intensities to get timbre matter? If I have an ideal harmonic series with peaks at 220Hz, 440Hz, 880Hz, etc, what should its intensities be? Timbre also has to take into account the full ADSR envelope.

"That is, two notes (series-es of overtones) made by the same (kind of) instrument will be distorted from the ideal Harmonic Series in the same (or similar) way. This must be the case in order for an instrument or instrument kind to have a uniform, recognizable timbre"

If you get a guitar in standard tuning and play, say, the 5th fret on the 6th string it sounds different than the open 5th string, despite them both being the same 'A'.

"But The Nasca People Of Peru Use A Linear, Not A Logarithmic, Scale!" "Who knows what is going on with the linear scales of the Nasca. I suspect the following: Linear-scale flutes are easy to make ... This culture was small and isolated and so no one ever noticed harmony"

I checked the cited article (Joerg Haeberli (1979). Twelve Nasca Panpipes: A Study. Ethnomusicology, 23(1), 57–74. doi:10.2307/851338 ). Among the things I learned: (1) At least some of the pipes were ceramic (picture here: https://artmuseum.princeton.edu/art/collections/objects/1374...) (and the Brooklyn Museum claims that "hundreds" have been found: https://www.brooklynmuseum.org/objects/51700), this is not consistent with a culture that is just figuring things out (2) "Intervals close to the fifth, major third, minor third, and major second occur".

You know those HN posts where someone who barely knows how to program decides to tell you why language X is terribly designed? This is that.

Anyone with a decent understanding of music and acoustics can tell it's silly from the title and abstract alone.

Does Daniel Levitin, author of "This is your brain on music", count as someone with "a decent understanding of music and acoustics" ? He thinks that Helmholtz explained harmony (he told me personally). However, as the essay points out, (1) The American Acoustical Society collection has an experiment stretching frequencies from 2.0 to 2.1 that they conclude shows that Helmholtz was wrong, and (2) if Helmholtz was right, there would be not harmony at all, as the essay points out. So even someone like Levitin is wrong.

Harmony has always been a principle that refers to more than music, let alone chords. Here is a recent review: https://www.sciencedirect.com/science/article/pii/S240587262...

It also refers to the “first” scientific experiment: an empirical test of the generalization of a mathematical model of harmony — from ratios of string lengths to ratios of chime thickness (as recorded by Aristotle’s student Aristoxenus)

The central insight (consonance involves both physical properties of sound and active perceptual pattern recognition) is interesting, but as a musician I see the following issue: the harmonic series can motivate certain local sonorities, but it cannot generate the operative grammar of common-practice Western harmony; the paper recognizes the tuning problem, yet treats it as a regrettable engineering compromise, when it is actually constitutive of the harmonic system it claims to explain.

I don't think I want a scientific theory of music. That would take the fun out of it. I like music that matches my tastes while still managing to surprise me. What else do I need to know about music?

And what about the discovery that happens when you make your own music? If this was replaced by the perfect scientific theory and a computer program cranking out perfect songs, our incentive for being creative would be greatly reduced.

This is like saying we should not study physics because if we did then all art could be automatically generated.

Are you saying demystifying music is as important as understanding physics?

No worries. This one is based on so many untested assumptions that you can easily ignore it.

The slight of hand in standard Western European music theory is that stacking 5ths gives you Lydian and its tritone as the foundational seven pitch scale. Major/Ionian is the fourth mode of this scale.

Not that there’s anything wrong with that…except when it comes to science as a basis for an argument from authority.

Like all music, people like what they like.

For that matter, the first 5 stacked fifths gives you the standard pentatonic scale. Extending by a fifth on either end and you get the most common accidentals in Western music (if your first seven give C major you’ll get F# and B♭). I think (but I’m not sure) that some of the non-Western microtonal scales extend the tonality by not ignoring the Pythagorean comma when you’ve done the 12th fifth, but by continuing to stack fifths for more harmonies.

From C the first six stacked fifths give C Lydian, not C major. (You get an F#, the 5th of B as the seventh pitch, not F the 5th of Bb).

You don’t get the perfect fourth until you go all the way around the circle of fifths.

You cannot stack perfect fifths and get a major scale.

I was being sloppy and should have said diatonic scale.

Sloppiness is why music theory is not scientific.

You can’t get to a scientific rationalization of the primacy of the major scale without it.

The primacy of the major scale happened many centuries after the diatonic scale came to be predominant in Western music. The primary development that led to major tonality was the acceptance of thirds as stable consonances. As that happened, the Ionian mode (previously little used) began to stand out because it happens to have a tritone (B and F) surrounding its tonic major third (C and E). This is the core feature of the authentic cadence which gives us the tension <-> resolution progression that characterizes tonal music. Tonal music practice and theory develop in all sorts of interesting ways, spilling out from that basis.

The transitional period where this happened is roughly in the 1600s.

Check out the 3Blue1Brown video on this topic as well! It goes more in depth on music theory than any music-community one I've seen. (Which tend to be presented as "Here are the rules"; Grant, like this article, goes into the "why"; a step below. With great visuals, of course!

Please provide a link.

Oops! It was Minute Physics: https://www.youtube.com/watch?v=tCsl6ZcY9ag

Nice. But, is there a less “conversationally” written piece on the subject? In any event, this stuck with me: > Most music theory books are like medieval medical textbooks: they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases

Calling this a scientific theory seems premature. It is mostly a speculative model built on untested assumptions about cognition, with no empirical validation of its central claims.

i tend to think of this stuff in terms of top down information theoretic approaches to composition and bottom up psychophysics of auditory perception. there is some cool work out there where physicists attempt to compute information content of western classical music and then there is all the fun stuff around the biophysics of the cochlea, nonlinear frequency discrimination and frequency selectivity of the ear.

This seems to be the 9th time that my paper on harmony has been on Hacker News. I wonder if it has been on HN more than any other paper on arxiv: https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...

I have one request: if you are going to comment on the paper, please actually read it first. Here is a little test to check that you read it: did you notice the points below?

(1) This is a paper in the field of Computer Science. Having studied music all of your life, even at Berklee, does not qualify you in Computer Science, so be open to learning what Computer Science is. In science we have a practice of making the simplest model we can from which emerge the observed measurements. Hence the remark by Feynman quoted near the top of the paper on how to know when you are right:

"When you get it right, it is obvious that it is right -- at least if you have any experience -- because usually what happens is that more comes out than goes in. Your guess is, in fact, that something is very simple. If you cannot see immediately that it is wrong, and it is simpler than it was before, then it is right. The inexperienced, and crackpots, and people like that, make guesses that are simple, but you can immediately see that they are wrong, so that does not count. Others, the inexperienced students, make guesses that are very complicated and it sort of looks as if it is all right, but I know it is not true because the truth always turns out to be simpler than you thought."

Many of you seem to like very complex explanations of harmony. Note that Feynman is saying that is not an indication of a theory being right.

(2) This problem has not really been solved before. Even someone like Daniel Levitin was convinced (the last time I spoke to him) that Helmholtz had the correct theory of harmony. However, as I point out in the paper:

* The American Acoustical Society has a collection of experiments, one of which they say (not just me), proves Helmholtz's theory is wrong.

* If Helmholtz is right, playing multiple notes always sounds either a little worse or a lot worse, but never better, and so we would never play two notes at the same time at all; harmony would not exist and there would be nothing to explain.

(3) There is strong evidence for the absolute nature of auditory processing that is not only cross-cultural, but cross-species. In the paper, there is a quotation from Daniel Levitin's book "This is your Brain on Music", where he discusses an experiment he witnessed in graduate school where music was played through wiring directly into the brain of an owl. The music was played into the brain with the root removed; when coming out of the brain of the owl, brain had restored the root (see the paper for details).

(4) The explanation of the experience of the minor triad in "Harmony Explained" requires the concept of an inconsistency arising from a redundant recognizer algorithm. Being a musician does not make you an expert in recognizer algorithms and their emergent artifacts, so read what that is before you comment.

The theory of the minor goes like this: two redundant recognizers exist in the brain. When hearing a minor triad:

* one recognizer fires, that of the conjunction of three pairwise intervals of tones,

* whereas another recognizer does not fire, that of the triple of tones in order.

This combination of firings is inconsistent because it is not possible for it to occur by playing any single normal harmonic series on, say, a flute. To generate it, one must play multiple notes and then use the effect of the recognizer of intervals pairing up tones from across the notes. That is, this is a new kind of auditory perception that can be created as an artifact of the auditory recognizer algorithms of the brain, but cannot be heard otherwise.

If you object to my assertion that minor triad is well-described as "I recognize it, but something is wrong", try reading the explanation in the paper of the same phenomenon in visual perception called cubism. Say what you will about the minor triad, when you look at Picasso's "Head of Woman" (provided in the paper), does it not disturb you? Would not not describe it as "I recognize it, but something is wrong"? Does your objection to the analysis of the minor still stand in the context of visual art?

It's a good quote from Feynman. I think you'll find if you measure your paper against that honestly, you'll see where it continues to fall short, as far less falls out than goes in. All the counterexamples find themselves being dismissed by vague speculation, often not supported by further research, and that impulse towards dismissing things that don't nearly fit the theory means the theory is restricted from finding a path towards generalizing and capturing deeper truths - you end up with epicycles instead of universal equations

I don't think a redundant recognizer algorithm is required to explain minor.

Centuries before we had major and minor tonality, we had the diatonic scale.

Through the Renaissance, thirds began to be accepted as stable consonances, and from that, you get tertian chords (triads and seventh chords from stacked thirds, and their inversions). In the diatonic scale, you only have 3 types of tertian triad: 3 major, 3 minor, and 1 diminished. The diminished triad was not accepted as a stable consonance.

In another post, I summarize the history of the development of major tonality out of the convention of the authentic cadence: https://news.ycombinator.com/item?id=49095594

I have come to believe that minor tonality is quite simply a result of "what if we used a minor chord as the tonic, instead?" Note that the minor chord already existed and had been in use for quite some time. The conventions of minor tonality (the tonal use of Aeolian and Dorian modes, minor key signature, the raised seventh degree, relative and paralellel key conventions, etc), however, developed as a contrast to major tonality.

I'm not discounting your neurological explanation, I haven't read your whole paper yet, but just pointing towards historical contingency as another way to see it.

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