Harmony Explained: Progress Towards a Scientific Theory of Music (2012)
arxiv.orgI 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You might like the book Emotion and Meaning in Music - it starts from some outdated psychological ideas (that emotions are only intensity and that we assign positive/negative valence post-hoc) but ties that to early information theory and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.
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.
> 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 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.
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
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.
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.
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:
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.
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.
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.
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...)
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.
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?
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)
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.
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.
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
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!
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.
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.
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).
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)
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
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.
Adam Neely has an interesting video about one-note muisc, with some good (and bad) examples: https://www.youtube.com/watch?v=eSuK_5zW2iM
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.
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.
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.
>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.
Not a one-note solo, but Miles Davis's solo on So What is really minimalistic.
It's all about big booties
The cuties with cooties
Corporate duties and African rubies
It's the doobies and boobies
That see K's like Louie
Shake out the loogies
And K-shape the Louis
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.