Join the discussion
Write your take first — we'll ask for email only when you're ready to publish.
- Hacker News
- 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?by schroffl
- 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.
by danbmil99 - 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 reasoningby vikramkr
- 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 yeeshby vikramkr
- 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.by breezybottom
- 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.by toolslive
- 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.by dswilkerson
- 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.by okzgn
- "Worse" is definitely a subjective appreciation influenced by culture and personal taste.by pier25
- 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:
- 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.by acjohnson55
- 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.
by atoav - 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.by antonyt
- 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.by klaff
- 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.
by IAmGraydon - 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.
by diydsp - 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.by pitchlatte
- 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?by calf
- 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.
by acjohnson55 - 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.by simonreiff
- 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 playingby bollu
- 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.
by duped - 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.
by klik99 - > 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.
by dswilkerson - >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.
by telesilla - 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.by nkrisc