15 ms·

Harmony Explained: Progress Towards a Scientific Theory of Music

213 points by colund 10y ago

jkingsbery 10y ago
"You can't start a science textbook like that" ... actually, lots of math and science textbooks assume a certain level of knowledge, and start more-or-less that way. There also seems to be confusion about what Music Theory actually is - it's (usually!) not an attempt to axiomize music, rather it's an attempt to take a musical corpus and explain how pieces in it tend to work (which is why one sees different music theory text books for Jazz and Classical, for example).
- codeulike 10y ago
  I've seen a lot of articles/books confuse music notation with music theory.
- tomdell 10y ago
  Good point! This is an important distinction - to put it a little more simply, music theory tends to be more the study of the craft of music than the study of the science of music.
- squeaky-clean 10y ago
  > There also seems to be confusion about what Music Theory actually is - it's (usually!) not an attempt to axiomize music, rather it's an attempt to take a musical corpus and explain how pieces in it tend to wor I think they try to address this in 1.3 but it comes out more as a complaining and misunderstanding. People get way too hung up on the term "music theory". I have friends who are professional musicians who refuse to learn music theory. Not because they don't like learning it, or they think it stifles their creativity (which are fine reasons to not learn it, you don't have to). But they refuse because they "don't believe in it" or "my music doesn't fit the definition of music theory". Which is totally incorrect, because like you said, music theory is an attempt to explain existing pieces and how they tie together. Not describe some magical pre-existing rules created by our ears and the universe. It's like learning the grammatical rules behind a natural language. English isn't carved into a mountain somewhere, or programmed into our DNA, it's entirely made up by people, and you're free to break all the rules. But then you're unlikely to be understood. And if you are understood, we'll probably end up giving a name to whatever you did!
  - ilostmykeys 10y ago
    "or programmed into our DNA," English is not, but our perception (the automatic processing of cognitive input to build context) which is essential to language learning/use, is programmed into our DNA.
  - TheOtherHobbes 10y ago
    The problem is that in many books music theory means 18th century classical theory. This is fine as far as it goes, but music has moved on a lot since then and no one - not even film composers - write in that style any more. So you have the idea of a music theory - which is the set of elements that defines a style - and then you have one particular music theory, which isn't used any more, although some of the ideas are still creatively useful. Not many people understand that hip hop, dubstep, deep house, death metal, jazz, and shoegaze all have unique theories of their own. They're defined by elements - some of which are simple cliches, and some of which are more creative - that make the style recognisable. If you know the elements and how to put them together, you know the style. Simple as. Classical book theory gives you a small start towards understanding how this works, if you want to take it. But you have to do the rest yourself. A lot of producers don't bother. They learn one style intuitively, and that's all they know. Other producers want more. Neither has a monopoly on great music. Does axiomising music help? It may give you a few hints about harmony. But that's all. It's not even close to being a complete description of classical theory, and it completely fails to say anything useful about the theories of more modern pop styles. The math part is simply irrelevant to most musicians. Musicians use chords and lines as expressive tools, not as theorems. Knowing where frequency ratios come from is kind of interesting, but there's a lot more going on in music. Trying to mathify it won't get you closer to that.
    - auggierose 10y ago
      So I like your explanation a lot, it's actually the best explanation of music theory I've come across so far (if its true, that is :-)) Nevertheless, I think math can be very useful to describe the various music styles, elements that make up these styles, and the ways to compose these elements. That would be using math in a descriptive way. I like for example the 2 book series "Musimathics" (http://www.musimathics.com/ http://www.musimathics.com/). In the end, nobody knows how much of music can be understood by viewing it through the mathematical lens. I suspect quite a lot, if not all.
- virgil_disgr4ce 10y ago
  > "You can't start a science textbook like that" ... actually, lots of math and science textbooks assume a certain level of knowledge, and start more-or-less that way. Fine, but regardless, there don't seem to be any music theory books that start with the very beginnings (as this submission is trying to). I share the author's deep frustration with existing music theory books: they all assume some magical vocabulary as though you're just supposed to know what they mean already and how dare you question what 'diatonic' means! Fuck that shit. I teach a class on sound, of which music theory plays a part, and I start with raw sound and human perception: consonance and dissonance. This is the only way you can ever understand WHY there are music theories at all (and don't get me started on how most music theory textbooks don't even MENTION the history of tuning systems. equal temperament ONLY makes sense once you consider the historic alternatives, like Pythagorean tuning). So I absolutely applaud this effort. This will be a huge help to music students of all ages who aren't comfortable with just accepting a bunch of mystical (and exclusively western) 'wisdom' from on high.
  - gardano 10y ago
    In 13th century music, the major 3rd actually comes across as a dissonance, and also to mention that music during that period was taught as a species of mathematics. Music theory is as easy (and as difficult) as trig. Learn the language. Learn the axioms. Understand the assumptions, please. And also understand that those assumptions can change at any time, depending on the musical period you're looking at!
privong 10y ago
Note that the post currently links to version 1, which was uploaded in 2012 (https://arxiv.org/html/1202.4212v1 https://arxiv.org/html/1202.4212v1). There is an updated version from 2014 (https://arxiv.org/html/1202.4212v2 https://arxiv.org/html/1202.4212v2). For arXiv posts, unless a specific version is under discussion, it's probably best to link to the abstract page without the version tag: https://arxiv.org/abs/1202.4212 https://arxiv.org/abs/1202.4212 This lets folks see if there are multiple versions of the preprints and multiple formats to view those preprints.
- sctb 10y ago
  Thank you! We've updated the submission link.
dbranes 10y ago
There exists a related and impressive body of work towards understanding the geometric structure of elements of music theory, see e.g. https://www.google.com/url?sa=t&source=web&rct=j&url=http://dmitri.mycpanel.princeton.edu/geometry.pdf&ved=0ahUKEwjPn6mK1PDNAhVl0oMKHYySC_4QFgggMAE&usg=AFQjCNE222eyXk0bv3DtrJdG28jY9DX7lQ https://www.google.com/url?sa=t&source=web&rct=j&url=http://... and references therein.
- dangerlibrary 10y ago
  without the google referral link (pdf) http://dmitri.mycpanel.princeton.edu/geometry.pdf http://dmitri.mycpanel.princeton.edu/geometry.pdf
gardano 10y ago
Given that in the first few pages of this article, the author never mentioned temperament (tuning) systems, I had a heck of a time parsing the argument. Well-Tempered? Equal Tempered? Mean Tone? Pythagoric? I had a hard time even concentrating on the argument when the basic assumption regarding horizontal (melodic) vs vertical (harmonic) had some badly unstated assumptions. Should I have kept reading?
- squeaky-clean 10y ago
  > Should I have kept reading? If you know a bit of music theory, I wouldn't recommend it. I skimmed the whole thing and it basically just ruined my morning by pissing me off, haha.
  - thevibesman 10y ago
    Thanks for the warning! I guess I'll save it for after my meetings today so I'm more smiley ;) Hopefully I can still comment / reply by then because the discussion here is a bit interesting. EDIT: Okay, I'm actually more interested in reading this because of how much I disagree with the first sentence of the abstract: > Most music theory books are like medieval medical textbooks: they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases. I'd have to disagree with all three of these points. As others have pointed out, I get the sense the author mistook the fact that the subject is called "Music Theory" to mean that "Music Theory" == "The Theory of Music", when instead it is an umbrella topic of things like "species counterpoint" and "tonal harmony" which are both a study of how music was written in a particular style, not an overarching theory of how music works. > funny symbols glorified by Latin phrases I think the is a reason in school, students often study music theory and music history concurrently, is so they understand how and why these "funny symbols" evolved (i.e. to use music to help standardize the liturgy of the early Catholic church---western music traces its roots back to Gregorian Chant, those glorifying "Latin phrases").
  - gardano 10y ago
    Yeah. I'm definitely a western-music oriented person, but the assumptions baked into this article just made me see red.
dahart 10y ago
I love the dissection of harmony from a physics/computation point of view; I'm going to read this in more detail. But the wrapping is quite off-putting to me; the author seems to misunderstand both music theory and science, and rather deeply. I don't see a scientific theory here at all. To be called science, it needs to have a hypothesis and be falsifiable, and after that get tested experimentally and proved - I don't see that, nor even a stab at a metric that is verifiable. What I see here is note relations described using science terminology instead of music terminology. That's not science, that's just someone using the terminology they're comfortable with rather than learn the accepted paradigm. "The more factored a theory and the more emergent the observed phenomena from the theory, the more satisfying the theory." - Feynman. It's ironic that this better explains existing music theory than the one presented here, even though Feynman wasn't talking about music theory. The author accuses music theory of being pseudo-science, when music theory is not and never was attempting to be science, and yet the author proposes a theory of music claiming to be science that in fact isn't science. Unfortunately, this 'theory' here only seems to contend with the first two of the 32 chapters in my copy of "Harmony and Voice Leading", and this mainly talks about what makes notes sound good, not what makes music sound good. Also unfortunately, the author doesn't seem to be aware of what's going on in modern music theory, which is exploring things like how to compose music out of the beat frequencies that exist as harmonic dissonances between two or more notes. A lot of the physics of harmonics is currently already being incorporated into music theory.
- Bromskloss 10y ago
  > to compose music out of the beat frequencies that exist as harmonic dissonances between two or more notes Do you have any links regarding this?
  - dahart 10y ago
    Yeah, I was thinking of "Spectral Music": https://en.wikipedia.org/wiki/Spectral_music https://en.wikipedia.org/wiki/Spectral_music Tristan Murail is a well known composer of spectral music. http://www.tristanmurail.com/en/outils.html http://www.tristanmurail.com/en/outils.html https://en.wikipedia.org/wiki/Tristan_Murail https://en.wikipedia.org/wiki/Tristan_Murail Also worth exploring in terms of modern music theory are microtonal music and atonal music, both of those are playing with incorporating the physics of notes, overtones and harmonic relationships. https://en.wikipedia.org/wiki/Microtonal_music https://en.wikipedia.org/wiki/Microtonal_music https://en.wikipedia.org/wiki/Atonality https://en.wikipedia.org/wiki/Atonality Apologies for only bringing Wikipedia links, but they are pretty good launching points for learning more.
squeaky-clean 10y ago
I don't like the tone of this book, it gets too hung up on the literal word "theory" in music theory, and also badly explains a lot of concepts in music theory while insulting them. For example, the circle of fifths doesn't explain at all why the circle is cool or useful for chord transitions. It's not a coincidence at all. By rotating around the circle, your key changes by only one note at a time. It's also just waaaay to objective about subjective things. Other cultures use of limited or different musical scales or temperaments is because they are missing out on real harmony? A major triad sounds best because a factor of 3 and 5 are the "most interesting" intervals? Section 4.3 is where I'm just quitting. > "Play C and F# on a piano; it sounds awful" [...] " because someone has even written a piece of music based entirely in the most un-harmonic of intervals, the Augmented Fourth, and gotten away with it." [...] "There are people who can abuse themselves to the point of re-calibrating their expectations to all kinds of strange inputs, including thinking that getting beaten with whips is fun or that McDonald's tastes good. That doesn't mean that those inputs are natural or good or beautiful or true. " So I don't really like dissonant music, I've only tricked myself into thinking it's good? The Ben Franklin quote after it just sounds like someone comparing pop music to anything else. Just because it's Ben Franklin doesn't make it definitive. It starts by insulting the assumptions that the diatonic scale and triads are a given in conventional music theory. But then goes on to make even greater assumptions as to why some harmonies sound "better" than others. Please prove it, or it's just as meaningless as the work you're insulting.
- mazelife 10y ago
  Yeah, his discussion of the tritone was where any remaining sympathy or interest in his argument evaporated for me. His discussion of it is completely simplistic and not grounded in any way in the complex historical context of that interval. Listen to the opening of Debussy's "Prelude to the Afternoon of a Faun" (https://www.youtube.com/watch?v=EvnRC7tSX50 https://www.youtube.com/watch?v=EvnRC7tSX50). Tritones everywhere: flute plays a melody that outlines a tritone (C# to G). In come the oboes and horns playing a tritone which is part of the A# 7th chord the harp outlines. Then a Bb dominant seventh, with horns and strings playing another tritone. Tritones abound until the music finally settles into the tonic nine bars later, and yet who would argue with the idea that this is one of the most beautiful openings in music? It's ironic that he likens works based on the tritone to McDonalds (!?). When I hear someone savage a work of music they've never heard, and describe the sound of an augmented fourth as "awful" and call any piece based around the tritone as "[not] natural or good or beautiful or true" I would suggest that person's musical taste is limited and simplistic, sort of the musical equivalent of the McDonald's hamburger.
apalmer 10y ago
Ehhh.. fundamentally I think this argument is flawed. It assumes harmony as the underlying driver of music. Majority of music across history is not harmonic oriented. Really only in western music is harmony the driving force behind music. now western music is extremely popular but that is really driven by historical coincidence that europe colonized the world, rather than the appreciation for the music itself. I do think the points are well thought out, and seem reasonable... but ignores way too much empirical reality to be taken very serious as a 'scientific' explanation for music... also its more or less the standard explanation for western musical theory, not sure what is novel here... but i like it.
- toomim 10y ago
  Well, there's harmony, melody, rhythm, and words. Melody derives from harmony. I agree that this doesn't cover rhythm or words.
nirajshr 10y ago
This article helped me grasp how the major/minor scales are constructed by using the idea of consonance and dissonance. It is sort of like a derivation of a mathematical formula. So many other books just give you the scales as given. For that alone, it was an insightful read.
tobr 10y ago
The writer starts out complaining that the distinction between "chord" and "scale" is arbitrary. He then proceeds to derive the major chord and major scale using a bunch of arbitrary decisions. For example, why does he stop adding notes to the major chord when he has three of them? And look at what he does when he starts constructing the major scale from the major chord: > Well, we like the Major Triad, so let's make another one, but starting with a different note as the fundamental. To preserve as much theme with the previous triad, let's start with the "closest" notes to the C that we have in our first triad: The first note other than C that we hit was 3/2 times the Root, also called the Perfect Fifth; therefore let's build a triad using 3/2 times C4 = G4 as the fundamental. And then > Ok, that was so much fun let's go in the other direction as well. Why not just continue with the harmonic series, rather than construct a new chord from the fifth? Why continue with a fifth below the root, and not continue in the established direction? Because these changes would lead to a completely different answer. He knows which answer he wants to end up with, so he picks a path that will lead there. This is numerology.
- baddox 10y ago
  > For example, why does he stop adding notes to the major chord when he has three of them? This particular complaint isn't very valid. The author explicitly writes that he could keep going, but is going to stop at three for reasons that will be explained later. He then explains later that the next note (the 7:4 interval) doesn't work well on equal tempered instruments because it ends up sounding like a dominant/minor seventh chord instead of a harmonic/barbershop seventh chord.
beat 10y ago
"Music theory" tends to be incredibly narrowminded and Eurocentric. Harmonic theory? Outside of Europe and European-derived forms, harmony is very rare. There are melodic traditions that do not use harmony, yet are just as traditional and sophisticated (Arabic, Hindustani, and Carnatic music all come to mind), and there are purely rhythmic traditions (various African, Arabic, central Asian, etc). And in the modern world, there is ambient, music built out of tone rather than rhythm/harmony/melody. Having studied and played several different traditions to varying degrees, one thing I've learned is that music is equally sophisticated everywhere. There's this idea that classical and jazz are more sophisticated than "folk" forms. It's BS.
- mrob 10y ago
  "Sophisticated" has no objective meaning, but not all music is equally complex. I mostly listen to techno. This genre has high timbral complexity, average rhythmic complexity, and almost no melodic or harmonic complexity. Compared to, eg. Beethoven's 9th, it's obviously artistically inferior. Despite this I find it more enjoyable to listen to.
  - lucasnemeth 10y ago
    But if we see this in a more broad perspective, there is electronic music (not necessarily techno) that is as complex as Beethoven 9th. Specially if more avant-guarde electronic music is included, not just EDM.
- aridiculous 10y ago
  Perhaps I'm being a tad too pedantic in this followup, but can you elaborate further? It's difficult for me to square how every style of music is equally sophisticated. I'm taking sophisticated to roughly mean 'complex'. I'm not promoting classical music as superior to folk, but can we not reasonably say that a Bach fugue is more complex than Simon and Garfunkel? Using two Eurocentric examples to control for bias.
  - the_cat_kittles 10y ago
GFK_of_xmaspast 10y ago
It's a discredit to the arxiv that this paper is still up, and I have to assume it's because there are just many fewer 'music theory nuts' out there than 'quantum physics nuts.'
- gardano 10y ago
  I'd love to read a paper that modernises pythagorean music theory with modern mathematics. However, medieval music theory would balk at modern piano tuning -- much less 2nd Viennese School compositions -- so at first blush, I can't see how that could be done.
  - mrob 10y ago
    The Pythagorean theory (simple ratios are consonant) is a consequence of the design of most Western musical instruments. They're typically built around either a string or a column of air, which is constrained at both ends. The overtones are unavoidably integer multiples of the fundamental because of this constraint. If they were not then at least one end of the string would be oscillating, which is not possible because it is clamped in place. Perception of consonance of a sound can be approximated by decomposing it into sine waves and calculating the sum of the values on the Plomp and Levelt curve (which is empirically measured and a consequence of human biology) for the intervals of all pairs of sine waves, weighted by amplitude. For instruments based around the harmonic series this results in standard Western harmony. There is nothing mystical about it, and the human brain does not have any ratio detector hardware. It's purely an artefact of a simpler underlying rule. Other instruments, eg. tuned percussion, do not have this constraint. If you clamp a bar at one end and strike it you can get all kinds of inharmonic overtones. This means cultures with music based around these types of instruments (eg. Indonesian classical music) need completely different harmonic theory. See http://sethares.engr.wisc.edu/paperspdf/consonance.pdf http://sethares.engr.wisc.edu/paperspdf/consonance.pdf This generalizes all existing theories of musical harmony, and explains things like stretched tuning in pianos (a consequence of non-ideal strings).
- drauh 10y ago
  I don't think it's a discredit. arXiv is a preprint database, which means nothing is refereed. It's under "Computer Science", but that's just the author's own classification. I don't think there's a way for the admins of arXiv to remove a paper. Once a paper is in the arXiv, it's there permanently: even the authors themselves can only update.[0] E.g. this bit of quackery "Electromagnetic Signals from Bacterial DNA"[1] [0] https://arxiv.org/help/withdraw https://arxiv.org/help/withdraw [1] https://arxiv.org/abs/1104.3113v1 https://arxiv.org/abs/1104.3113v1
bluetwo 10y ago
As a life-long lover of music and programmer who decided in his 30's to try learning an instrument, I share his frustration with traditional music theory instruction and desire to explain using more physics-based models. I'm curious if his spent any time developing tools that might leverage his theories. I ended up developing a midi-toolset for composition based on my attempts at understanding the space.
- drauh 10y ago
  Using physics-based models will make it even more frustrating and complicated. You'd have to take into account "temperament" or tuning, which someone else has already mentioned. Then, the relationship between frequencies is dependent on what intervals one decides should sound good.[0] Music theory is an abstraction: it tries to make a transient activity (musical performance) amenable to being described on paper. Maybe an appropriate analogy is mathematics to physics. You can't learn to program a computer by learning only abstract grammar. [0] https://en.wikipedia.org/wiki/Equal_temperament https://en.wikipedia.org/wiki/Equal_temperament
sampo 10y ago
The harmony of of popular music and jazz is based on the diatonic of major scale. Each of the twelve scales is a frame forming the harmonic system. Translation: Western music is based on the diatonic scale. Diatonic harmony moves in two directions: Horizontal and Vertical. Translation: Music consists of notes played one after each other, and at the same time as each other. By combining these two movements... we derive the scale-tone seventh chords in the key of C Translation: We can build the scale-tone 7-chords (root + third + fifth + seventh played at the same time) on top of any note in the scale. Wanting to sound unnecessary grandiose can be a problem with textbooks in almost any field.
mrob 10y ago
William Sethares has already done some excellent work towards formalizing consonance and dissonance. See: http://sethares.engr.wisc.edu/consemi.html http://sethares.engr.wisc.edu/consemi.html This new paper looks worse than Sethares' as it only considers timbres based around the harmonic series, ignoring important sounds such as tuned percussion, and ignoring the possibilities for genuine harmonic novelty with synthesisers. It cites Terhardt (1974) as "modern" while completely ignoring more recent work.
hkailahi 10y ago
For anyone interested the intersection of rhythm and harmony: https://www.youtube.com/watch?v=_gCJHNBEdoc https://www.youtube.com/watch?v=_gCJHNBEdoc
holri 10y ago
"...phenomena of any machine that must make sense of sound, such as the human brain" A machine is a man made artifact. A brain is a living artifact, we do not understand it and it can not be built by us.
- kefka 10y ago
  I don't know about that. Sex is pretty simple. Certainly simpler than making an engine :)
mazelife 10y ago
Other commenters here have done a good job picking out specific problems with the author's arguments. It's always frustrating when someone who is capable and knowledgeable about one subject decides to try their hand in a different field without having bothered to do any real research or study in it. And it's doubly so when they proceed to then dismiss, literally, the entire field as "superstitions," and "unscientific" even as everything they've written betrays that they haven't bothered to familiarize themselves with basically any of the work that's been done in music theory or psychoacoustics in the last 100 years. I usually love it when articles about the intersection of music and computation show up on HN, but in this case it's really unfortunate. At least it's sparking some good discussion, I guess.
- gardano 10y ago
  You know -- you're right. Music is incredibly complex, and I'm so glad to have a conversation about it. Perhaps I've been too strenuous in my objections to the article, but at the same time, music to me is an intersection between theory and practice; science and art, and I just hate to see it simplified on one side or on the other side. I look forward to the day where theory and practice can be brought together. It will take someone smarter than me to do such a thing.
- toomim 10y ago
  It's telling that the top-ranked comment is an ad-hominem attack on the author and his style of writing, without critiquing the content at all. Your comment is stuck in the lowest 3 levels of Paul Graham's Hierarchy of Disagreement: DH1: Ad-hominem DH2: Responding to Tone http://paulgraham.com/disagree.html http://paulgraham.com/disagree.html https://commons.wikimedia.org/wiki/File:Graham%27s_Hierarchy_of_Disagreement.svg https://commons.wikimedia.org/wiki/File:Graham%27s_Hierarchy... You haven't come close to addressing, or even acknowledging the central point of this paper. You can't even reach level DH3 without acknowledging the point. DH3 is where "we finally get responses to what was said, rather than how or by whom." (Paul Graham) > It's always frustrating when someone who is capable and knowledgeable about one subject decides to try their hand in a different field without having bothered to do any real research or study in it. It looks to me that the reason you're frustrated and offended is that he critiques your field. You're clearly flustered; using black-and-white language like "always" that is factually incorrect, and saying "real research" as if his research didn't happen in reality -- but I assure you it's real. This is not a dream. He actually did just critique your field. Calling it unreal is a tactic of denial. This is groundbreaking work, and the establishment can't have it. You're no better than the Catholic Church.
nabla9 10y ago
Here is much better theory: Music And Measure Theory – A connection between a classical puzzle about rational numbers and what makes music harmonious. https://www.youtube.com/watch?v=cyW5z-M2yzw https://www.youtube.com/watch?v=cyW5z-M2yzw
AnthonyNagid 10y ago
I have a theory that papers seeking a true break through in music theory should be exploring Christopher Alexander's work 'The Nature of Order'. We need music thoerists / scientists / programmers to be fleshing out Alexander's thoeries in relation to music. Shouldn't the scientific spirit take us where we haven't looked yet? To illustrate my point with a musical example I'd like to refer to Keith Jarrett's recording 'Hymns / Spheres'. This is a musician who has become as intimate as a human can with classical and jazz. In a C. Alexandrian fashion, Jarrett takes all of these elements and applies them on an ancient cathedral organ in Germany and trancends both genres and his usual way of playing. The lowest hanging fruit in terms of understanding what's going on here more completely is not in music theory or acoustics but in Christopher Alexander's 'Nature of Order'. https://www.youtube.com/watch?v=lXymPInuMkM https://www.youtube.com/watch?v=lXymPInuMkM https://en.wikipedia.org/wiki/The_Nature_of_Order https://en.wikipedia.org/wiki/The_Nature_of_Order https://en.wikipedia.org/wiki/Keith_Jarrett https://en.wikipedia.org/wiki/Keith_Jarrett
multinglets 10y ago
Pulses and waves compounded in whole number ratios form periodic and stable, i.e. consonant, waveforms. But oh yeah some people just play all random frequencies or like a dog. A dog can be music. It's such a huge mystery, music.
denfromufa 10y ago
You should perhaps watch this incredible video from PyCon 2016 given by Lars Bark about music, road trips, software development, and fractals! https://www.youtube.com/watch?v=bSfe5M_zG2s https://www.youtube.com/watch?v=bSfe5M_zG2s
pcsanwald 10y ago
I've mentioned this before, but for me, the definitive work on this vast subject is "The Harmonic Experience" by W.A. Mathieu. He really gets into low prime ratios, why they sound good to us, and also why the building blocks of modern western harmony are approximations of those ratios, rather than the real thing. A fascinating read for anyone interested in the subject.
- soundwave106 10y ago
  Seconded. The one law I think you can say about harmony and melody, is that there seems to be a universal preference towards low frequency ratio combinations. EG: in all of world music, it seems octaves (2/1) fifths (3/2), fourths (4/3), and thirds (5/4) occur quite often and are the fundamental driving force of so many styles. They are seen as more "pure sounding" to the ear. Regarding the linked paper, the low ratio concept has been known for a while, and linking it to the harmonic series is fine. But there's more to music than simple pure ratios. Even many simple pop songs often offers transitions between "purity" and "impurity" (the simple dominant seven to tonic resolutions is enough to show this.) This is where the linked paper fell short... way short. It seemed to imply in 4.3 that there is no room for dissonance (or, to describe it without using Western terms, stepping away from the pure ratios, perhaps to resolve to purity... or perhaps not...) whatsoever in music, which is very laughable to anyone who's actually composed. 4.2.1 also almost seemed to apply a certain sense of Western cultural superiority... even though I would argue that Western culture actually is a case against his ideas. (As in... equal temperament decreases the "purity" by detuning some of the ratios, particularly the thirds. But a greater freedom is allowed for key modulation, and I agree with Mathieu that the ambiguity 12TET opens up is a feature in itself. Let's not go into the large amount of dissonance in some art such as 20th century classical, jazz, even some pop particularly in the alternative realm). I'm currently in the middle of Mathieu's book. The main disadvantage of his style is that it is a bit "flowery" which may not appeal to those who like reading their math more exactly. But one big advantage of W.A. Mathieu is that he seems well versed in three very different backgrounds (Indian classical music, jazz and Western classical music, although definitely heavier on the first two). For those familiar with neo-Riemannian theory, I consider his book a good tie-off with that side of music theory, not completely the same, but tangentially related. Unlike some neo-Riemannian books, because of Mathieu's background, I believe it avoids both some of the cultural mistakes that other music theory books seem to make and also avoids the mistake of trying to be the "one theory to rule them all".
costcopizza 10y ago
Does anyone else not care to know what makes music sound good and enjoy the mystery and emotional complexity of it?
- theseatoms 10y ago
  Even if we knew exactly what physics made music sound good, that doesn't take any of the mystery out of physics itself.
jng 10y ago
I urge anyone interested in this topic to read Helmholtz's "On the sensations of tone", a fascinating 19th-century tour-de-force trying to understand music sensation as coming from physical principles, deriving the standard western intervals and consonance/dissonance from first principles (beats between upper partials), and for which he is considered the father of acoustics. He spent 8 years researching in order to produce it, and is to me a great example of a 19th century hacker.
simbalion 10y ago
I've read a number of music theory books and guides, and not one of them contained: "unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases". In fact all were based on mathematics and at least some science.
toomim 10y ago
Wonderful paper. And wow, the author has really managed to piss a lot of people off! People get really upset when you present work that points out flaws in their own thinking, using a different style than they've come to accept. It comes across as an attack from a different tribe, and they get all tribal at you. You can tell because they attack the style as much as (or more than) the content. This paper provides one of the first theories at the foundations of music! How exciting is this!
wry_discontent 10y ago
Maybe I'll finally be able to understand music.
metaxy2 10y ago
The best work happening in this area is by a Princeton professor named Dmitri Tymoczko [1]. He's discovered some pretty fundamental results that tie together a lot of the previous theories of harmony based on geometry. He was the first music theorist ever published in Science [2] and he wrote an amazing book called A Geometry of Music [3] (technical but readable, a must for anyone trying to really understand why music sounds good). He also teaches a 2-semester intro level music class at Princeton that is infused with his ideas, and makes the lecture notes available online [4]. Those notes are so good they're better than any intro music textbook I know of, especially for us geeks who like to hear the scientific explanation for everything. [1] http://dmitri.mycpanel.princeton.edu/bio.html http://dmitri.mycpanel.princeton.edu/bio.html [2] http://dmitri.tymoczko.com/sciencearticle.html http://dmitri.tymoczko.com/sciencearticle.html [3] https://www.amazon.com/Geometry-Music-Counterpoint- https://www.amazon.com/Geometry-Music-Counterpoint- Extended-Practice/dp/0195336674 [4] http://dmitri.tymoczko.com/teaching.html http://dmitri.tymoczko.com/teaching.html
- dfan 10y ago
  Another enthusiastic thumbs-up for A Geometry of Music. I wouldn't take all his conclusions as far as he does, but it's all very thought-provoking and he's obviously a really sharp guy. (Some prior music theory knowledge is probably necessary, if only to understand the context of his arguments.) His music's pretty good too [1]. [1] http://dmitri.tymoczko.com/scores.html http://dmitri.tymoczko.com/scores.html
- j_s 10y ago
  Broken link; here's the Kindle edition: http://amzn.com/dp/B005254IXS http://amzn.com/dp/B005254IXS
- acjohnson55 10y ago
  I read that several years ago, when it first came out, and it blew my mind. I also highly recommend Tuning, Timbre, Spectrum, Scale by William Sethares [1]. It's a different approach to deriving our familiar harmonic structures, with a greater emphasis on consonance within a given timbre than voice leading. [1] https://www.amazon.com/Tuning-Timbre-Spectrum-William-Sethares/dp/1852337974 https://www.amazon.com/Tuning-Timbre-Spectrum-William-Sethar...
bigjcr 10y ago
The questions the author asks in the first chapter make me wonder whether I want to keep reading or not; his comments to those questions look full of arrogance or just sarcasm. You don't start learning music theory in a jazz theory book, so you wouldn't be asking those questions if you started learning music from a music theory book. Or at least you wouldn't have that kind of comments. Or are they just intended jokes? Seems interesting tho, I might keep reading, I would like to see another approach to harmony.
dmritard96 10y ago
"music: * the Major Scale, * the Standard Chord Dictionary, and * the difference in feeling between the Major and Minor Triads. " Doesn't this already carry a ton of biases in that these are largely constructs of Western music? Certain pieces are based on physics, an octave is a doubling of frequency and the way a major chord fits together to some degree is a constructive interference. I studied music performance and compsci and its amazing to me how much magic and mysticism the music schools believe in. It was also pretty eye opening to see how much they believed that western music was the center of the human sound intersection world. It wasn't until taking a world music class that you realize how myopic even these researchers can be.
- dmritard96 10y ago
  Also very relevant: http://www.theatlantic.com/science/archive/2016/07/music-to-our-western-ears/491081/ http://www.theatlantic.com/science/archive/2016/07/music-to-...
pjdorrell 10y ago
I take this opportunity to re-advertise my own alternative approach to developing a scientific theory of music: http://whatismusic.info/ http://whatismusic.info/.
kingkawn 10y ago
Anything to say about music is better said by making it.
sunnyps 10y ago
Not exactly on topic but I wanted to ask: I have never learnt how to play a musical instrument (sadly) but I would like to dabble in computer generated music and music theory. What's the best way to get started with this?
- ZenoArrow 10y ago
  If you want to dabble in computer generated music, I'd suggest Sonic Pi is a good place to start, especially if you're comfortable with coding. http://sonic-pi.net/ http://sonic-pi.net/ You may also be interested in Syntorial, which is a synthesizer design course, teaching you how to design synthesizer sounds: http://www.syntorial.com/ http://www.syntorial.com/ Music theory is a huge field. I'd say the circle of fifths is a good starting point into exploring music theory. https://en.m.wikipedia.org/wiki/Circle_of_fifths https://en.m.wikipedia.org/wiki/Circle_of_fifths https://m.youtube.com/watch?v=d1aJ6HixSe0 https://m.youtube.com/watch?v=d1aJ6HixSe0
- dahart 10y ago
  I've played with SuperCollider a bit and enjoyed it. It's an audio programming language, at the low level of construct and compose your own waveforms. Higher level, less technical, many of my pro audio friends have recommended Ableton. Learn a musical instrument! It's never too late, you'll enjoy it, and it will definitely complement dabbling in computer generated music.
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dminor14 10y ago
There is nothing fundamental about the major scale or triad, they didn't even exist 300 years ago, they are an invention of western culture. The author is not aware of previous work of Hindemith, who derived them from purely acoustic phenomena, or the advanced mathematical/musical theories of Mazzola. In other words, like most scientists who delve into art, they look at it as a trivial pursuit and are not diligent.
- baddox 10y ago
  I don't see why the alleged fact that the major triad didn't exist 300 years ago implies that there is nothing fundamental about them. Even if the biological reasons for their sound perception are as claimed in this article, people would still have to discover that and start using it to create music.

213 points by colund 10y ago

"You can't start a science textbook like that" ... actually, lots of math and science textbooks assume a certain level of knowledge, and start more-or-less that way. There also seems to be confusion about what Music Theory actually is - it's (usually!) not an attempt to axiomize music, rather it's an attempt to take a musical corpus and explain how pieces in it tend to work (which is why one sees different music theory text books for Jazz and Classical, for example).

codeulike 10y ago
I've seen a lot of articles/books confuse music notation with music theory.
tomdell 10y ago
Good point! This is an important distinction - to put it a little more simply, music theory tends to be more the study of the craft of music than the study of the science of music.
squeaky-clean 10y ago
> There also seems to be confusion about what Music Theory actually is - it's (usually!) not an attempt to axiomize music, rather it's an attempt to take a musical corpus and explain how pieces in it tend to wor I think they try to address this in 1.3 but it comes out more as a complaining and misunderstanding. People get way too hung up on the term "music theory". I have friends who are professional musicians who refuse to learn music theory. Not because they don't like learning it, or they think it stifles their creativity (which are fine reasons to not learn it, you don't have to). But they refuse because they "don't believe in it" or "my music doesn't fit the definition of music theory". Which is totally incorrect, because like you said, music theory is an attempt to explain existing pieces and how they tie together. Not describe some magical pre-existing rules created by our ears and the universe. It's like learning the grammatical rules behind a natural language. English isn't carved into a mountain somewhere, or programmed into our DNA, it's entirely made up by people, and you're free to break all the rules. But then you're unlikely to be understood. And if you are understood, we'll probably end up giving a name to whatever you did!
- ilostmykeys 10y ago
  "or programmed into our DNA," English is not, but our perception (the automatic processing of cognitive input to build context) which is essential to language learning/use, is programmed into our DNA.
- TheOtherHobbes 10y ago
  The problem is that in many books music theory means 18th century classical theory. This is fine as far as it goes, but music has moved on a lot since then and no one - not even film composers - write in that style any more. So you have the idea of a music theory - which is the set of elements that defines a style - and then you have one particular music theory, which isn't used any more, although some of the ideas are still creatively useful. Not many people understand that hip hop, dubstep, deep house, death metal, jazz, and shoegaze all have unique theories of their own. They're defined by elements - some of which are simple cliches, and some of which are more creative - that make the style recognisable. If you know the elements and how to put them together, you know the style. Simple as. Classical book theory gives you a small start towards understanding how this works, if you want to take it. But you have to do the rest yourself. A lot of producers don't bother. They learn one style intuitively, and that's all they know. Other producers want more. Neither has a monopoly on great music. Does axiomising music help? It may give you a few hints about harmony. But that's all. It's not even close to being a complete description of classical theory, and it completely fails to say anything useful about the theories of more modern pop styles. The math part is simply irrelevant to most musicians. Musicians use chords and lines as expressive tools, not as theorems. Knowing where frequency ratios come from is kind of interesting, but there's a lot more going on in music. Trying to mathify it won't get you closer to that.
  - auggierose 10y ago
    So I like your explanation a lot, it's actually the best explanation of music theory I've come across so far (if its true, that is :-)) Nevertheless, I think math can be very useful to describe the various music styles, elements that make up these styles, and the ways to compose these elements. That would be using math in a descriptive way. I like for example the 2 book series "Musimathics" (http://www.musimathics.com/ http://www.musimathics.com/). In the end, nobody knows how much of music can be understood by viewing it through the mathematical lens. I suspect quite a lot, if not all.
virgil_disgr4ce 10y ago
> "You can't start a science textbook like that" ... actually, lots of math and science textbooks assume a certain level of knowledge, and start more-or-less that way. Fine, but regardless, there don't seem to be any music theory books that start with the very beginnings (as this submission is trying to). I share the author's deep frustration with existing music theory books: they all assume some magical vocabulary as though you're just supposed to know what they mean already and how dare you question what 'diatonic' means! Fuck that shit. I teach a class on sound, of which music theory plays a part, and I start with raw sound and human perception: consonance and dissonance. This is the only way you can ever understand WHY there are music theories at all (and don't get me started on how most music theory textbooks don't even MENTION the history of tuning systems. equal temperament ONLY makes sense once you consider the historic alternatives, like Pythagorean tuning). So I absolutely applaud this effort. This will be a huge help to music students of all ages who aren't comfortable with just accepting a bunch of mystical (and exclusively western) 'wisdom' from on high.
- gardano 10y ago
  In 13th century music, the major 3rd actually comes across as a dissonance, and also to mention that music during that period was taught as a species of mathematics. Music theory is as easy (and as difficult) as trig. Learn the language. Learn the axioms. Understand the assumptions, please. And also understand that those assumptions can change at any time, depending on the musical period you're looking at!

privong 10y ago

Note that the post currently links to version 1, which was uploaded in 2012 (https://arxiv.org/html/1202.4212v1 https://arxiv.org/html/1202.4212v1). There is an updated version from 2014 (https://arxiv.org/html/1202.4212v2 https://arxiv.org/html/1202.4212v2). For arXiv posts, unless a specific version is under discussion, it's probably best to link to the abstract page without the version tag: https://arxiv.org/abs/1202.4212 https://arxiv.org/abs/1202.4212 This lets folks see if there are multiple versions of the preprints and multiple formats to view those preprints.

sctb 10y ago
Thank you! We've updated the submission link.

dbranes 10y ago

There exists a related and impressive body of work towards understanding the geometric structure of elements of music theory, see e.g. https://www.google.com/url?sa=t&source=web&rct=j&url=http://dmitri.mycpanel.princeton.edu/geometry.pdf&ved=0ahUKEwjPn6mK1PDNAhVl0oMKHYySC_4QFgggMAE&usg=AFQjCNE222eyXk0bv3DtrJdG28jY9DX7lQ https://www.google.com/url?sa=t&source=web&rct=j&url=http://... and references therein.

dangerlibrary 10y ago
without the google referral link (pdf) http://dmitri.mycpanel.princeton.edu/geometry.pdf http://dmitri.mycpanel.princeton.edu/geometry.pdf

gardano 10y ago

Given that in the first few pages of this article, the author never mentioned temperament (tuning) systems, I had a heck of a time parsing the argument. Well-Tempered? Equal Tempered? Mean Tone? Pythagoric? I had a hard time even concentrating on the argument when the basic assumption regarding horizontal (melodic) vs vertical (harmonic) had some badly unstated assumptions. Should I have kept reading?

squeaky-clean 10y ago
> Should I have kept reading? If you know a bit of music theory, I wouldn't recommend it. I skimmed the whole thing and it basically just ruined my morning by pissing me off, haha.
- thevibesman 10y ago
  Thanks for the warning! I guess I'll save it for after my meetings today so I'm more smiley ;) Hopefully I can still comment / reply by then because the discussion here is a bit interesting. EDIT: Okay, I'm actually more interested in reading this because of how much I disagree with the first sentence of the abstract: > Most music theory books are like medieval medical textbooks: they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases. I'd have to disagree with all three of these points. As others have pointed out, I get the sense the author mistook the fact that the subject is called "Music Theory" to mean that "Music Theory" == "The Theory of Music", when instead it is an umbrella topic of things like "species counterpoint" and "tonal harmony" which are both a study of how music was written in a particular style, not an overarching theory of how music works. > funny symbols glorified by Latin phrases I think the is a reason in school, students often study music theory and music history concurrently, is so they understand how and why these "funny symbols" evolved (i.e. to use music to help standardize the liturgy of the early Catholic church---western music traces its roots back to Gregorian Chant, those glorifying "Latin phrases").
- gardano 10y ago
  Yeah. I'm definitely a western-music oriented person, but the assumptions baked into this article just made me see red.

dahart 10y ago

I love the dissection of harmony from a physics/computation point of view; I'm going to read this in more detail. But the wrapping is quite off-putting to me; the author seems to misunderstand both music theory and science, and rather deeply. I don't see a scientific theory here at all. To be called science, it needs to have a hypothesis and be falsifiable, and after that get tested experimentally and proved - I don't see that, nor even a stab at a metric that is verifiable. What I see here is note relations described using science terminology instead of music terminology. That's not science, that's just someone using the terminology they're comfortable with rather than learn the accepted paradigm. "The more factored a theory and the more emergent the observed phenomena from the theory, the more satisfying the theory." - Feynman. It's ironic that this better explains existing music theory than the one presented here, even though Feynman wasn't talking about music theory. The author accuses music theory of being pseudo-science, when music theory is not and never was attempting to be science, and yet the author proposes a theory of music claiming to be science that in fact isn't science. Unfortunately, this 'theory' here only seems to contend with the first two of the 32 chapters in my copy of "Harmony and Voice Leading", and this mainly talks about what makes notes sound good, not what makes music sound good. Also unfortunately, the author doesn't seem to be aware of what's going on in modern music theory, which is exploring things like how to compose music out of the beat frequencies that exist as harmonic dissonances between two or more notes. A lot of the physics of harmonics is currently already being incorporated into music theory.

Bromskloss 10y ago
> to compose music out of the beat frequencies that exist as harmonic dissonances between two or more notes Do you have any links regarding this?
- dahart 10y ago
  Yeah, I was thinking of "Spectral Music": https://en.wikipedia.org/wiki/Spectral_music https://en.wikipedia.org/wiki/Spectral_music Tristan Murail is a well known composer of spectral music. http://www.tristanmurail.com/en/outils.html http://www.tristanmurail.com/en/outils.html https://en.wikipedia.org/wiki/Tristan_Murail https://en.wikipedia.org/wiki/Tristan_Murail Also worth exploring in terms of modern music theory are microtonal music and atonal music, both of those are playing with incorporating the physics of notes, overtones and harmonic relationships. https://en.wikipedia.org/wiki/Microtonal_music https://en.wikipedia.org/wiki/Microtonal_music https://en.wikipedia.org/wiki/Atonality https://en.wikipedia.org/wiki/Atonality Apologies for only bringing Wikipedia links, but they are pretty good launching points for learning more.

squeaky-clean 10y ago

I don't like the tone of this book, it gets too hung up on the literal word "theory" in music theory, and also badly explains a lot of concepts in music theory while insulting them. For example, the circle of fifths doesn't explain at all why the circle is cool or useful for chord transitions. It's not a coincidence at all. By rotating around the circle, your key changes by only one note at a time. It's also just waaaay to objective about subjective things. Other cultures use of limited or different musical scales or temperaments is because they are missing out on real harmony? A major triad sounds best because a factor of 3 and 5 are the "most interesting" intervals? Section 4.3 is where I'm just quitting. > "Play C and F# on a piano; it sounds awful" [...] " because someone has even written a piece of music based entirely in the most un-harmonic of intervals, the Augmented Fourth, and gotten away with it." [...] "There are people who can abuse themselves to the point of re-calibrating their expectations to all kinds of strange inputs, including thinking that getting beaten with whips is fun or that McDonald's tastes good. That doesn't mean that those inputs are natural or good or beautiful or true. " So I don't really like dissonant music, I've only tricked myself into thinking it's good? The Ben Franklin quote after it just sounds like someone comparing pop music to anything else. Just because it's Ben Franklin doesn't make it definitive. It starts by insulting the assumptions that the diatonic scale and triads are a given in conventional music theory. But then goes on to make even greater assumptions as to why some harmonies sound "better" than others. Please prove it, or it's just as meaningless as the work you're insulting.

mazelife 10y ago
Yeah, his discussion of the tritone was where any remaining sympathy or interest in his argument evaporated for me. His discussion of it is completely simplistic and not grounded in any way in the complex historical context of that interval. Listen to the opening of Debussy's "Prelude to the Afternoon of a Faun" (https://www.youtube.com/watch?v=EvnRC7tSX50 https://www.youtube.com/watch?v=EvnRC7tSX50). Tritones everywhere: flute plays a melody that outlines a tritone (C# to G). In come the oboes and horns playing a tritone which is part of the A# 7th chord the harp outlines. Then a Bb dominant seventh, with horns and strings playing another tritone. Tritones abound until the music finally settles into the tonic nine bars later, and yet who would argue with the idea that this is one of the most beautiful openings in music? It's ironic that he likens works based on the tritone to McDonalds (!?). When I hear someone savage a work of music they've never heard, and describe the sound of an augmented fourth as "awful" and call any piece based around the tritone as "[not] natural or good or beautiful or true" I would suggest that person's musical taste is limited and simplistic, sort of the musical equivalent of the McDonald's hamburger.

apalmer 10y ago

Ehhh.. fundamentally I think this argument is flawed. It assumes harmony as the underlying driver of music. Majority of music across history is not harmonic oriented. Really only in western music is harmony the driving force behind music. now western music is extremely popular but that is really driven by historical coincidence that europe colonized the world, rather than the appreciation for the music itself. I do think the points are well thought out, and seem reasonable... but ignores way too much empirical reality to be taken very serious as a 'scientific' explanation for music... also its more or less the standard explanation for western musical theory, not sure what is novel here... but i like it.

toomim 10y ago
Well, there's harmony, melody, rhythm, and words. Melody derives from harmony. I agree that this doesn't cover rhythm or words.

nirajshr 10y ago

This article helped me grasp how the major/minor scales are constructed by using the idea of consonance and dissonance. It is sort of like a derivation of a mathematical formula. So many other books just give you the scales as given. For that alone, it was an insightful read.

tobr 10y ago

The writer starts out complaining that the distinction between "chord" and "scale" is arbitrary. He then proceeds to derive the major chord and major scale using a bunch of arbitrary decisions. For example, why does he stop adding notes to the major chord when he has three of them? And look at what he does when he starts constructing the major scale from the major chord: > Well, we like the Major Triad, so let's make another one, but starting with a different note as the fundamental. To preserve as much theme with the previous triad, let's start with the "closest" notes to the C that we have in our first triad: The first note other than C that we hit was 3/2 times the Root, also called the Perfect Fifth; therefore let's build a triad using 3/2 times C4 = G4 as the fundamental. And then > Ok, that was so much fun let's go in the other direction as well. Why not just continue with the harmonic series, rather than construct a new chord from the fifth? Why continue with a fifth below the root, and not continue in the established direction? Because these changes would lead to a completely different answer. He knows which answer he wants to end up with, so he picks a path that will lead there. This is numerology.

baddox 10y ago
> For example, why does he stop adding notes to the major chord when he has three of them? This particular complaint isn't very valid. The author explicitly writes that he could keep going, but is going to stop at three for reasons that will be explained later. He then explains later that the next note (the 7:4 interval) doesn't work well on equal tempered instruments because it ends up sounding like a dominant/minor seventh chord instead of a harmonic/barbershop seventh chord.

beat 10y ago

"Music theory" tends to be incredibly narrowminded and Eurocentric. Harmonic theory? Outside of Europe and European-derived forms, harmony is very rare. There are melodic traditions that do not use harmony, yet are just as traditional and sophisticated (Arabic, Hindustani, and Carnatic music all come to mind), and there are purely rhythmic traditions (various African, Arabic, central Asian, etc). And in the modern world, there is ambient, music built out of tone rather than rhythm/harmony/melody. Having studied and played several different traditions to varying degrees, one thing I've learned is that music is equally sophisticated everywhere. There's this idea that classical and jazz are more sophisticated than "folk" forms. It's BS.

mrob 10y ago
"Sophisticated" has no objective meaning, but not all music is equally complex. I mostly listen to techno. This genre has high timbral complexity, average rhythmic complexity, and almost no melodic or harmonic complexity. Compared to, eg. Beethoven's 9th, it's obviously artistically inferior. Despite this I find it more enjoyable to listen to.
- lucasnemeth 10y ago
  But if we see this in a more broad perspective, there is electronic music (not necessarily techno) that is as complex as Beethoven 9th. Specially if more avant-guarde electronic music is included, not just EDM.
aridiculous 10y ago
Perhaps I'm being a tad too pedantic in this followup, but can you elaborate further? It's difficult for me to square how every style of music is equally sophisticated. I'm taking sophisticated to roughly mean 'complex'. I'm not promoting classical music as superior to folk, but can we not reasonably say that a Bach fugue is more complex than Simon and Garfunkel? Using two Eurocentric examples to control for bias.
- the_cat_kittles 10y ago

GFK_of_xmaspast 10y ago

It's a discredit to the arxiv that this paper is still up, and I have to assume it's because there are just many fewer 'music theory nuts' out there than 'quantum physics nuts.'

gardano 10y ago
I'd love to read a paper that modernises pythagorean music theory with modern mathematics. However, medieval music theory would balk at modern piano tuning -- much less 2nd Viennese School compositions -- so at first blush, I can't see how that could be done.
- mrob 10y ago
  The Pythagorean theory (simple ratios are consonant) is a consequence of the design of most Western musical instruments. They're typically built around either a string or a column of air, which is constrained at both ends. The overtones are unavoidably integer multiples of the fundamental because of this constraint. If they were not then at least one end of the string would be oscillating, which is not possible because it is clamped in place. Perception of consonance of a sound can be approximated by decomposing it into sine waves and calculating the sum of the values on the Plomp and Levelt curve (which is empirically measured and a consequence of human biology) for the intervals of all pairs of sine waves, weighted by amplitude. For instruments based around the harmonic series this results in standard Western harmony. There is nothing mystical about it, and the human brain does not have any ratio detector hardware. It's purely an artefact of a simpler underlying rule. Other instruments, eg. tuned percussion, do not have this constraint. If you clamp a bar at one end and strike it you can get all kinds of inharmonic overtones. This means cultures with music based around these types of instruments (eg. Indonesian classical music) need completely different harmonic theory. See http://sethares.engr.wisc.edu/paperspdf/consonance.pdf http://sethares.engr.wisc.edu/paperspdf/consonance.pdf This generalizes all existing theories of musical harmony, and explains things like stretched tuning in pianos (a consequence of non-ideal strings).
drauh 10y ago
I don't think it's a discredit. arXiv is a preprint database, which means nothing is refereed. It's under "Computer Science", but that's just the author's own classification. I don't think there's a way for the admins of arXiv to remove a paper. Once a paper is in the arXiv, it's there permanently: even the authors themselves can only update.[0] E.g. this bit of quackery "Electromagnetic Signals from Bacterial DNA"[1] [0] https://arxiv.org/help/withdraw https://arxiv.org/help/withdraw [1] https://arxiv.org/abs/1104.3113v1 https://arxiv.org/abs/1104.3113v1

bluetwo 10y ago

As a life-long lover of music and programmer who decided in his 30's to try learning an instrument, I share his frustration with traditional music theory instruction and desire to explain using more physics-based models. I'm curious if his spent any time developing tools that might leverage his theories. I ended up developing a midi-toolset for composition based on my attempts at understanding the space.

drauh 10y ago
Using physics-based models will make it even more frustrating and complicated. You'd have to take into account "temperament" or tuning, which someone else has already mentioned. Then, the relationship between frequencies is dependent on what intervals one decides should sound good.[0] Music theory is an abstraction: it tries to make a transient activity (musical performance) amenable to being described on paper. Maybe an appropriate analogy is mathematics to physics. You can't learn to program a computer by learning only abstract grammar. [0] https://en.wikipedia.org/wiki/Equal_temperament https://en.wikipedia.org/wiki/Equal_temperament

sampo 10y ago

The harmony of of popular music and jazz is based on the diatonic of major scale. Each of the twelve scales is a frame forming the harmonic system. Translation: Western music is based on the diatonic scale. Diatonic harmony moves in two directions: Horizontal and Vertical. Translation: Music consists of notes played one after each other, and at the same time as each other. By combining these two movements... we derive the scale-tone seventh chords in the key of C Translation: We can build the scale-tone 7-chords (root + third + fifth + seventh played at the same time) on top of any note in the scale. Wanting to sound unnecessary grandiose can be a problem with textbooks in almost any field.

mrob 10y ago

William Sethares has already done some excellent work towards formalizing consonance and dissonance. See: http://sethares.engr.wisc.edu/consemi.html http://sethares.engr.wisc.edu/consemi.html This new paper looks worse than Sethares' as it only considers timbres based around the harmonic series, ignoring important sounds such as tuned percussion, and ignoring the possibilities for genuine harmonic novelty with synthesisers. It cites Terhardt (1974) as "modern" while completely ignoring more recent work.

hkailahi 10y ago

For anyone interested the intersection of rhythm and harmony: https://www.youtube.com/watch?v=_gCJHNBEdoc https://www.youtube.com/watch?v=_gCJHNBEdoc

holri 10y ago

"...phenomena of any machine that must make sense of sound, such as the human brain" A machine is a man made artifact. A brain is a living artifact, we do not understand it and it can not be built by us.

kefka 10y ago
I don't know about that. Sex is pretty simple. Certainly simpler than making an engine :)

mazelife 10y ago

Other commenters here have done a good job picking out specific problems with the author's arguments. It's always frustrating when someone who is capable and knowledgeable about one subject decides to try their hand in a different field without having bothered to do any real research or study in it. And it's doubly so when they proceed to then dismiss, literally, the entire field as "superstitions," and "unscientific" even as everything they've written betrays that they haven't bothered to familiarize themselves with basically any of the work that's been done in music theory or psychoacoustics in the last 100 years. I usually love it when articles about the intersection of music and computation show up on HN, but in this case it's really unfortunate. At least it's sparking some good discussion, I guess.

gardano 10y ago
You know -- you're right. Music is incredibly complex, and I'm so glad to have a conversation about it. Perhaps I've been too strenuous in my objections to the article, but at the same time, music to me is an intersection between theory and practice; science and art, and I just hate to see it simplified on one side or on the other side. I look forward to the day where theory and practice can be brought together. It will take someone smarter than me to do such a thing.
toomim 10y ago
It's telling that the top-ranked comment is an ad-hominem attack on the author and his style of writing, without critiquing the content at all. Your comment is stuck in the lowest 3 levels of Paul Graham's Hierarchy of Disagreement: DH1: Ad-hominem DH2: Responding to Tone http://paulgraham.com/disagree.html http://paulgraham.com/disagree.html https://commons.wikimedia.org/wiki/File:Graham%27s_Hierarchy_of_Disagreement.svg https://commons.wikimedia.org/wiki/File:Graham%27s_Hierarchy... You haven't come close to addressing, or even acknowledging the central point of this paper. You can't even reach level DH3 without acknowledging the point. DH3 is where "we finally get responses to what was said, rather than how or by whom." (Paul Graham) > It's always frustrating when someone who is capable and knowledgeable about one subject decides to try their hand in a different field without having bothered to do any real research or study in it. It looks to me that the reason you're frustrated and offended is that he critiques your field. You're clearly flustered; using black-and-white language like "always" that is factually incorrect, and saying "real research" as if his research didn't happen in reality -- but I assure you it's real. This is not a dream. He actually did just critique your field. Calling it unreal is a tactic of denial. This is groundbreaking work, and the establishment can't have it. You're no better than the Catholic Church.

nabla9 10y ago

Here is much better theory: Music And Measure Theory – A connection between a classical puzzle about rational numbers and what makes music harmonious. https://www.youtube.com/watch?v=cyW5z-M2yzw https://www.youtube.com/watch?v=cyW5z-M2yzw

AnthonyNagid 10y ago

I have a theory that papers seeking a true break through in music theory should be exploring Christopher Alexander's work 'The Nature of Order'. We need music thoerists / scientists / programmers to be fleshing out Alexander's thoeries in relation to music. Shouldn't the scientific spirit take us where we haven't looked yet? To illustrate my point with a musical example I'd like to refer to Keith Jarrett's recording 'Hymns / Spheres'. This is a musician who has become as intimate as a human can with classical and jazz. In a C. Alexandrian fashion, Jarrett takes all of these elements and applies them on an ancient cathedral organ in Germany and trancends both genres and his usual way of playing. The lowest hanging fruit in terms of understanding what's going on here more completely is not in music theory or acoustics but in Christopher Alexander's 'Nature of Order'. https://www.youtube.com/watch?v=lXymPInuMkM https://www.youtube.com/watch?v=lXymPInuMkM https://en.wikipedia.org/wiki/The_Nature_of_Order https://en.wikipedia.org/wiki/The_Nature_of_Order https://en.wikipedia.org/wiki/Keith_Jarrett https://en.wikipedia.org/wiki/Keith_Jarrett

multinglets 10y ago

Pulses and waves compounded in whole number ratios form periodic and stable, i.e. consonant, waveforms. But oh yeah some people just play all random frequencies or like a dog. A dog can be music. It's such a huge mystery, music.

denfromufa 10y ago

You should perhaps watch this incredible video from PyCon 2016 given by Lars Bark about music, road trips, software development, and fractals! https://www.youtube.com/watch?v=bSfe5M_zG2s https://www.youtube.com/watch?v=bSfe5M_zG2s

pcsanwald 10y ago

I've mentioned this before, but for me, the definitive work on this vast subject is "The Harmonic Experience" by W.A. Mathieu. He really gets into low prime ratios, why they sound good to us, and also why the building blocks of modern western harmony are approximations of those ratios, rather than the real thing. A fascinating read for anyone interested in the subject.

soundwave106 10y ago
Seconded. The one law I think you can say about harmony and melody, is that there seems to be a universal preference towards low frequency ratio combinations. EG: in all of world music, it seems octaves (2/1) fifths (3/2), fourths (4/3), and thirds (5/4) occur quite often and are the fundamental driving force of so many styles. They are seen as more "pure sounding" to the ear. Regarding the linked paper, the low ratio concept has been known for a while, and linking it to the harmonic series is fine. But there's more to music than simple pure ratios. Even many simple pop songs often offers transitions between "purity" and "impurity" (the simple dominant seven to tonic resolutions is enough to show this.) This is where the linked paper fell short... way short. It seemed to imply in 4.3 that there is no room for dissonance (or, to describe it without using Western terms, stepping away from the pure ratios, perhaps to resolve to purity... or perhaps not...) whatsoever in music, which is very laughable to anyone who's actually composed. 4.2.1 also almost seemed to apply a certain sense of Western cultural superiority... even though I would argue that Western culture actually is a case against his ideas. (As in... equal temperament decreases the "purity" by detuning some of the ratios, particularly the thirds. But a greater freedom is allowed for key modulation, and I agree with Mathieu that the ambiguity 12TET opens up is a feature in itself. Let's not go into the large amount of dissonance in some art such as 20th century classical, jazz, even some pop particularly in the alternative realm). I'm currently in the middle of Mathieu's book. The main disadvantage of his style is that it is a bit "flowery" which may not appeal to those who like reading their math more exactly. But one big advantage of W.A. Mathieu is that he seems well versed in three very different backgrounds (Indian classical music, jazz and Western classical music, although definitely heavier on the first two). For those familiar with neo-Riemannian theory, I consider his book a good tie-off with that side of music theory, not completely the same, but tangentially related. Unlike some neo-Riemannian books, because of Mathieu's background, I believe it avoids both some of the cultural mistakes that other music theory books seem to make and also avoids the mistake of trying to be the "one theory to rule them all".

costcopizza 10y ago

Does anyone else not care to know what makes music sound good and enjoy the mystery and emotional complexity of it?

theseatoms 10y ago
Even if we knew exactly what physics made music sound good, that doesn't take any of the mystery out of physics itself.

jng 10y ago

I urge anyone interested in this topic to read Helmholtz's "On the sensations of tone", a fascinating 19th-century tour-de-force trying to understand music sensation as coming from physical principles, deriving the standard western intervals and consonance/dissonance from first principles (beats between upper partials), and for which he is considered the father of acoustics. He spent 8 years researching in order to produce it, and is to me a great example of a 19th century hacker.

simbalion 10y ago

I've read a number of music theory books and guides, and not one of them contained: "unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases". In fact all were based on mathematics and at least some science.

toomim 10y ago

Wonderful paper. And wow, the author has really managed to piss a lot of people off! People get really upset when you present work that points out flaws in their own thinking, using a different style than they've come to accept. It comes across as an attack from a different tribe, and they get all tribal at you. You can tell because they attack the style as much as (or more than) the content. This paper provides one of the first theories at the foundations of music! How exciting is this!

wry_discontent 10y ago

Maybe I'll finally be able to understand music.

metaxy2 10y ago

The best work happening in this area is by a Princeton professor named Dmitri Tymoczko [1]. He's discovered some pretty fundamental results that tie together a lot of the previous theories of harmony based on geometry. He was the first music theorist ever published in Science [2] and he wrote an amazing book called A Geometry of Music [3] (technical but readable, a must for anyone trying to really understand why music sounds good). He also teaches a 2-semester intro level music class at Princeton that is infused with his ideas, and makes the lecture notes available online [4]. Those notes are so good they're better than any intro music textbook I know of, especially for us geeks who like to hear the scientific explanation for everything. [1] http://dmitri.mycpanel.princeton.edu/bio.html http://dmitri.mycpanel.princeton.edu/bio.html [2] http://dmitri.tymoczko.com/sciencearticle.html http://dmitri.tymoczko.com/sciencearticle.html [3] https://www.amazon.com/Geometry-Music-Counterpoint- https://www.amazon.com/Geometry-Music-Counterpoint- Extended-Practice/dp/0195336674 [4] http://dmitri.tymoczko.com/teaching.html http://dmitri.tymoczko.com/teaching.html

dfan 10y ago
Another enthusiastic thumbs-up for A Geometry of Music. I wouldn't take all his conclusions as far as he does, but it's all very thought-provoking and he's obviously a really sharp guy. (Some prior music theory knowledge is probably necessary, if only to understand the context of his arguments.) His music's pretty good too [1]. [1] http://dmitri.tymoczko.com/scores.html http://dmitri.tymoczko.com/scores.html
j_s 10y ago
Broken link; here's the Kindle edition: http://amzn.com/dp/B005254IXS http://amzn.com/dp/B005254IXS
acjohnson55 10y ago
I read that several years ago, when it first came out, and it blew my mind. I also highly recommend Tuning, Timbre, Spectrum, Scale by William Sethares [1]. It's a different approach to deriving our familiar harmonic structures, with a greater emphasis on consonance within a given timbre than voice leading. [1] https://www.amazon.com/Tuning-Timbre-Spectrum-William-Sethares/dp/1852337974 https://www.amazon.com/Tuning-Timbre-Spectrum-William-Sethar...

bigjcr 10y ago

The questions the author asks in the first chapter make me wonder whether I want to keep reading or not; his comments to those questions look full of arrogance or just sarcasm. You don't start learning music theory in a jazz theory book, so you wouldn't be asking those questions if you started learning music from a music theory book. Or at least you wouldn't have that kind of comments. Or are they just intended jokes? Seems interesting tho, I might keep reading, I would like to see another approach to harmony.

dmritard96 10y ago

"music: * the Major Scale, * the Standard Chord Dictionary, and * the difference in feeling between the Major and Minor Triads. " Doesn't this already carry a ton of biases in that these are largely constructs of Western music? Certain pieces are based on physics, an octave is a doubling of frequency and the way a major chord fits together to some degree is a constructive interference. I studied music performance and compsci and its amazing to me how much magic and mysticism the music schools believe in. It was also pretty eye opening to see how much they believed that western music was the center of the human sound intersection world. It wasn't until taking a world music class that you realize how myopic even these researchers can be.

dmritard96 10y ago
Also very relevant: http://www.theatlantic.com/science/archive/2016/07/music-to-our-western-ears/491081/ http://www.theatlantic.com/science/archive/2016/07/music-to-...

pjdorrell 10y ago

I take this opportunity to re-advertise my own alternative approach to developing a scientific theory of music: http://whatismusic.info/ http://whatismusic.info/.

kingkawn 10y ago

Anything to say about music is better said by making it.

sunnyps 10y ago

Not exactly on topic but I wanted to ask: I have never learnt how to play a musical instrument (sadly) but I would like to dabble in computer generated music and music theory. What's the best way to get started with this?

ZenoArrow 10y ago
If you want to dabble in computer generated music, I'd suggest Sonic Pi is a good place to start, especially if you're comfortable with coding. http://sonic-pi.net/ http://sonic-pi.net/ You may also be interested in Syntorial, which is a synthesizer design course, teaching you how to design synthesizer sounds: http://www.syntorial.com/ http://www.syntorial.com/ Music theory is a huge field. I'd say the circle of fifths is a good starting point into exploring music theory. https://en.m.wikipedia.org/wiki/Circle_of_fifths https://en.m.wikipedia.org/wiki/Circle_of_fifths https://m.youtube.com/watch?v=d1aJ6HixSe0 https://m.youtube.com/watch?v=d1aJ6HixSe0
dahart 10y ago
I've played with SuperCollider a bit and enjoyed it. It's an audio programming language, at the low level of construct and compose your own waveforms. Higher level, less technical, many of my pro audio friends have recommended Ableton. Learn a musical instrument! It's never too late, you'll enjoy it, and it will definitely complement dabbling in computer generated music.

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dminor14 10y ago

There is nothing fundamental about the major scale or triad, they didn't even exist 300 years ago, they are an invention of western culture. The author is not aware of previous work of Hindemith, who derived them from purely acoustic phenomena, or the advanced mathematical/musical theories of Mazzola. In other words, like most scientists who delve into art, they look at it as a trivial pursuit and are not diligent.

baddox 10y ago
I don't see why the alleged fact that the major triad didn't exist 300 years ago implies that there is nothing fundamental about them. Even if the biological reasons for their sound perception are as claimed in this article, people would still have to discover that and start using it to create music.