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

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.

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.

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.

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.

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/

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.

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.

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.

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

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.

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...)

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.

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.

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.

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!

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).

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.

(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?

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