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.
> We show to be incomplete the theory of the heretofore agreed-upon authority on this subject, 19th-century Physicist Hermann Helmholtz [Helmholtz1863]: he says notes are in "concord" because the sound playing them together is "less worse" than that of some other notes. But note that, in this theory, more notes can only penalize, some merely less than others, and so the most harmonious sound should be a single note by itself(!) and harmony would not exist as a phenomenon of music at all.
A bunch of the material available in this paper is sufficiently common knowledge to appear e.g. on Wikipedia: https://en.wikipedia.org/wiki/Equal_temperament .
The most major novelty here appears to be an attempt to hand-wave some general aesthetic principles for what humans like, and apply these to the brain via some sort of fuzzy "computational approach" (which never, y'know, broaches the point of an actual computer program). A lot of the overarching explanation can be ignored to the benefit of the theory in many cases: for example, it is likely more helpful to a neurobiological understanding of harmony to steal his insight that the minor chord probably sounds "good but off" because it has the right intervals in the wrong order, than to couch it in the author's full explanation with computational assumptions etc.
It's kind of fun to think that we've just discovered some local optimum for enjoyable music and that going in a completely different direction will lead to something both unrecognizable and mind-blowingly awesome. I'm not even remotely convinced that's the case, but it's fun to dream.
The frequency ratio of two or more pitches
> Scale and mode
The set of frequency ratios out of which a given piece of music can be constructed
> The 12-note Western chromatic scale
The modern western chromatic scale is formed out frequency ratios which are powers of the twelfth root of 2, or ~1.059. This is called the "12 note" scale, because 1.059^12 == 2, and our brains are predisposed to find pitches who are multiples of two apart similar. This is because pitch classification in humans (along with many other human senses) measures inbound signal in exponential, rather than linear, terms, and 2:1 is also the simplest ratio by which two pitches can have constructive interference.
As far as why the 12-note scale is the Western scale, there's a fair amount of debate, but most are in agreement that it likely comes from the fact that 12 is a low number with a fair number of integer factors (1, 2, 3, 4, 6). Integer factors translates very directly into constructive wave interference (3:2 has less destructive interference than 101:100). So, one can easily construct many constructive interference ratios out of the (2^^(1/12)) atomic element (e.g. the "major chord", the most common set of three pitches in western music, is (1.059^7):(1.059^4):1 ~= 6:5:4). Each exponent of the twelfth root of two is very close to, respectively: 16/15, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 16/9, 15/8
So! Now the only remaining question is just a question of the antropological evidence: do all other cultures also use the twelfth root of two as their simplest harmonic difference? And the answer is very clearly no:
* Indian music uses the twenty second root of two -- the "shruti" https://en.wikipedia.org/wiki/Shruti_(music)
* Indonesian music has many different scales, some based on the 9th root of 2 ( https://en.wikipedia.org/wiki/Pelog ), others based on the 5th root of 2 ( https://en.wikipedia.org/wiki/Slendro which is often paired with a scale based on the 17th root of 2)
* Arabic music uses the twentyfourth root of two ( https://en.wikipedia.org/wiki/Arabic_maqam#Notation although to be fair, this is just the square root of the western twelfth root of two, but this conceit is largely used for notation, while actual performance uses adjustments smaller than the twenty-fourth root of two)
* And plenty of other cultures besides ( https://en.wikipedia.org/wiki/Microtonal_music#Microtonal, https://danielpaulschnee.wordpress.com/2015/05/11/microtonal... etc )
Does this constitute "A large portion of the world (possibly a majority)"? I'd probably say not a majority, but certainly more than 15%, likely at least 25%.
His reference to the major scale as being alternately called "Diatonic" is very confused, as that mode is called "Ionian."
Also, for example, note the naivete of viewing a C Major scale as fundamentally outlined by a piano, while ignoring the interesting tidbit that the notes are named starting with A.
If the piano fundamentally outlines anything, it is the A minor (aeolian) scale, saying it outlines a C Major is no more accurate than saying it outlines a B Locrian.
Meanwhile, there's interesting stuff in there nonetheless.
An interesting phenomenon is that C Ionian harmony can be 'flipped' to A Aeolian by mirroring all the notes around the central D. The C chord becomes A minor, G7 becomes Dm6, F6 becomes Em7. The melody can switch as well; the contour is reversed yet the tune retains its essence.
Do you have any links regarding this?
Tristan Murail is a well known composer of spectral music.
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.
Apologies for only bringing Wikipedia links, but they are pretty good launching points for learning more.
Oh. It has not been my understanding that spectral music is about beat frequencies.
Beside the point: My experience of spectral music has been that it's a neat idea that seems worthy of trying out. Then, when you do it – test the hypothesis, so to say – it turns out that nope, didn't work, sounds awful.
I agree that it's not producing pleasant music to the masses. It's more of an academic exercise, but it is influencing modern music theory and bridging classical music theory with a more recent understanding of harmonics.
I've got a piece in C#m at 87 bpm . I'll play a piano instrument which is being fed into a distortion (the distortion makes the beat frequency much more apparent) and slam an Ab1 and A1 at the same time. That's 55.00hz - 51.91hz , which gives a 3.09 hertz beating. Or a period of 323.6ms . So you have two notes close together, which sounds very dissonant, they're bass notes which makes it muddy, and it's so distorted, you can barely tell what the actual notes are. But it swells up and down on the tempo like a rhythmic device.
 Anyone who looked up a tempo -> ms chart is going to say "WTF, that's not in time with 87 bpm." Which is correct, but because pianos are stretched tuned (and so is the virtual instrument I use), the numbers I provided aren't exactly true. I never bothered to measure the exact frequencies, but I kept adjusting things until everything sounded right by ear.
Let me explain it for you:
Hypothesis / Theory: The keys on the keyboard are laid out in their particular queer way because of X.
Evidence: (1) the Major Scale, (2) the Standard Chord Dictionary, and (2) the difference between the Major and Minor Triads.
Falsifiability: Do any of the evidence in 1, 2, and 3 fit the theory?
Fitness test: How simple is the theory, as compared to competing theories? How many assumptions (aka "free parameters") are there?
The author explains how competing theories have more assumptions and free parameters that his theory does not require.
This is clearly science. It's just written in a different style than you're accustomed to. Please read and critique the content of the article. If you so firmly believe that it isn't science, then you should be able to easily refute it using science, rather than just superficially (superstitiously) claiming "That's not science! You don't understand!"
Prove it! Critique the content, man!
* the Major Scale,
* the Standard Chord Dictionary, and
* the difference in feeling between the Major and Minor Triads.
• Standard Chord Dictionary. What does that even mean? Please map that onto gregorian chant or even Monteverdi Vespers of 1610.
• As I've indicated elsewhere, a Major 3rd in 13th century France is actually a discord, not a resolution.
If the author had indicated that he was only referring to 19th century European classical music, I'd have been cool with it. But the author isn't even referring to the European musical tradition in general.
I'm not understanding what you're explaining. Feel free to ignore me, or explain it more, but I still don't see any actual science in what you're talking about. I see technical terminology, but not scientific method.
The layout of keys on the keyboard is not a natural phenomenon that needs a scientific theory to explain, that was a (somewhat arbitrary) choice that has been canonized. The layout was decided in past history for known reasons that mix music theory, engineering and human factors, it wasn't something that science ever had a direct say in.
If this paper is science, what precisely does it explain that we didn't already know? Music theory isn't a science, and nobody ever claimed it was. Music theory is an art.
What, exactly, does this sciency sounding theory here demonstrate? I acknowledge that it's a useful framework for understanding intervals for physicists and computer scientists, but I don't see it actually showing why the major scale is major, and how to use that fact to compose music, do you? Does it somehow prove why a minor chord following a major chord sounds good to most people? Does it scientifically prove why a minor chord sounds sad and major happy? Does it prove why Turkish music sounds good to Turkish people and not to Canadians? I don't see anywhere in this document that anything about harmony is proposed and then proved via scientific method. I'm not sure it's possible, or a particularly useful goal either.
This document only covers a very tiny fraction of music theory, so it simply can't be a good replacement for music theory until it's much, much bigger.
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 . 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.
I also highly recommend Tuning, Timbre, Spectrum, Scale by William Sethares . It's a different approach to deriving our familiar harmonic structures, with a greater emphasis on consonance within a given timbre than voice leading.
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.
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.
Now there is an ad-hominem attack.
It was my comment you accused, and so I'm completely biased, but I did critique the content. I critiqued the stated premise of the argument, which is in fact part of the content. I did not critique the tone above, but I will now.
"Diatonic harmony moves in two directions:
Horizontal and Vertical."
Really?! They both look pretty diagonal to me.
Oh, but it's Diatonic! That sounds Latin so I
guess these people are smart.
That argument accuses music theorists of using Latin to try to sound smart, when in fact the music theory terms originated in Latin. Western music theory predates almost the entirety of physics, which is why we have terms in music theory that sound old, and part of the reason the author of this paper doesn't feel comfortable with them.
This one line is evidence that the author does not understand music theory. He is using abrasive and confrontational language to attempt to discredit music theory, when he's actually provably demonstrating his ignorance.
FWIW, I gave you up-votes for both of your comments for engaging, but I disagree with you and think you're also demonstrating a misunderstanding of science.
This whole paper would be so much better if it just said "Hey, let's look at harmonic relationships through the looking glass of physics." This paper has value, I actually like what it's presenting. But trying to disprove music theory, or suggest that an art can be explained by science, well, that's just peeing into the wind.
In fact, this paper references Paul Hindemith in the 1940s, so I suspect that the notion of linking the harmonic series to music has been around a very long time.
I think the real problem is the very dismissive topic in section 4. That's what completely turned me off this paper.
Dissonance (in Western music terms) has a huge role in music, and even that 20th century purely dissonant music has a place. (Take music like Penderecki's early stuff eg: https://www.youtube.com/watch?v=Dp3BlFZWJNA . It sounds "awful" in a sense, but that's really the point, and certainly has its place -- The Shining wouldn't have been the same if Kubrick had used Thomas Tallis instead. :) ).
Beyond fully atonal music, a lot of Western art music relies on "dissonances" of some sort to advance melodic development (even in standard pop there's a little of it sometimes). I don't know many pieces based solely on the tritone. A lot of Western composers in the past used the tritone in key places because it was rather nasty sounding, and it was condemned as "the Devil's interval" culturally. Thus it could easily represent "evil" or "sinister" in your tone poems and operas.
Voice leading is a huge core principle behind the circle of fifths or the reverse circle of fourths; from the paper (in bashing the circle as a "red herring") it sounds like he doesn't understand the core behind why the circle exists at all, or has chosen to ignore the principle of voice leading for some reason. The circle of fifths is more melody than harmony, I personally felt puzzled why he felt it would be a criticism for a harmony paper.
I also found the dismissal of other cultural scales (which indeed generally are less, with pentatonic probably the most common), er, tone deaf. The best way I think to explore the scales of a large portion of music cultures (there are exceptions) is to think more on Pythagorean terms, not on chromatic terms. Saying that "their culture has simply never made use of the rest of the available parameter space" is looking at it the wrong way.
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.
Listen to the opening of Debussy's "Prelude to the Afternoon of a Faun" (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.
If you wouldn't mind me sharing an example which isn't nearly as beautiful :P . "43% Burnt" by The Dillinger Escape Plan (https://www.youtube.com/watch?v=Zv8b6RPbnAc). It starts with an incredibly dissonant and jarring intro. Semitone intervals on distorted guitars to an almost unpredictable rhythm (there's a pattern, but it's not apparent in the intro). And the song doesn't really get any gentler from there. The last bridge before the outro is an aggressive halftone-wholetone diminished riff on the guitars and non-stop 16th note blast on the drums. Stop... And then... It comes back to that disgusting intro which loops until the track fades out. But now, it feels familiar, like home. And with each loop it sounds better. What started off sounding awful becomes comfortable in a different context.
It's funny, but I've noticed that as my musical tastes have broadened over the years, the line between what sounds "beautiful" and "ugly" has become considerably more blurry. I can still recognize particular harmonies as consonant or dissonant, but that mapping in my brain where consonant -> beautiful and dissonant -> ugly has started to really break down.
One of the first times I noticed this was happening was probably 15 years ago, listening to Salomé by Richard Strauss. The last scene has this incredible dissonance, a polychord that's an A7 with an F# stacked on top: two chords that absolutely do not belong together in any diatonic piece of music. When I first heard that I thought it sounded awful. Now it has a kind of strange beauty to my ears and is one of those musical moments that always sends a shiver up my spine.
 Give a listen here if you like: https://youtu.be/Op1VoQXXARs?t=431 I started the clip at 7:15, the chord comes in around 8:53 but that whole section is shot through which these ugly/beautiful sonorities.
I think the line between beautiful and ugly has gotten blurrier for me too. Artistically, I don't actually think they're opposites anymore. One needs the other to be fully realized, otherwise it just sounds too plastic.
I first got into really dissonant music around high school, I think mostly just in an attempt to be "edgy" and different. There's a lot of music that just tries too hard to sound heavy (I can't listen to like 99% of grindcore anymore. It just sounds like trying to piss of my parents). But I started finding things that used dissonance artistically and not just for shock, and I love it. Now the music that hits hardest and sounds "heaviest" to me aren't the songs that just shove abrasive sounds in your face as quickly and loudly as possible. But the ones with lots of contrast and empty space, and slowly building tension before slamming into something. You know what they say, if everything is loud, then nothing is loud.
I'm listening to Strauss now because of your examples. I think I might have to go on a classical binge for the next week or so just because of this conversation!
In my opinion, King Crimson's Red  is among the most beautiful music ever made. It's a terrible beauty in this case, but that shouldn't stop us from listening and feeling awe. Sure we have our own moods, and want different feelings to go with them. But we can't have an all-saccharine diet.
(And nice pick with Dillinger Escape Plan. I assume you're familiar with Devin Townsend too?)
 Live recording of the title track: https://youtu.be/KpJaJt3NWsM?list=RDGFQbQGCnOR0
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.
It's not rare, but it has different forms than western music. Maybe the most common form is pedal, which can be heard in this Tuvan form: https://www.youtube.com/watch?v=1rmo3fKeveo
Almost any time there are musical instruments accompanying singing, you'll get some form of harmony, like with this Japanese shamisen music: https://youtu.be/_k0wyAIkPhM?t=916
Native Australians have harmony that I would call pedal, too: https://www.youtube.com/watch?v=V1pDPuetPdg
Nope, they're harmony, just not western harmony (as you said). You don't need a full tonic scale to get harmony.
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.
Take a Bach Fugue, where the notes and harmonies are very important, and 'complex' with lots of rules, patterns, intertwining melodies. The specific instruments used doesn't always matter, most pieces could interchangeably be played on piano or harpsichord or organ and sound as beautiful.
Then let's look at something like Scary Monsters by Skrillex, where the entire drop is pretty much just a G with a small melody coming in every 4 measures. There's not much melodic complexity going on there. But the sound production, synthesis and processing is insane. After that song came out everyone was trying to figure out how to make a talking bass that sounded like that. And even years later, barely anyone has figured it out, I've only heard really famous synthesists like Rob Swire get a close sound.
Is a Skrillex drop as complex as a Bach Fugue? If you're just comparing the arrangement, or the melodies, not at all. But the Skrillex song isn't about the melody as much as the texture and dynamics. Bach didn't specify precisely what harmonics should exist in the instruments playing his music. Skrillex probably could not write a Fugue to the standards of Bach's best. But I bet if you sat Bach down in front of a laptop with an FM synthesizer, he couldn't create a "sick drop" either.
It gets entirely too subjective at this point. A lot of people will disagree with me that the production and synthesis techniques should be considered part of music theory or composition. But I think they're concepts that you couldn't replicate certain music without. In my opinion the palindromic structure of a Boards of Canada album is on par with something like Bach's Crab Canon. They just get there in totally different ways.
A somewhat poor, though adequate, performance of a Bach fugue with an instrument that doesn't have great tone will still convey a great deal of Bach's meaning.
A performance by average vocalists of a Simon and Garfunkle tune, backed by some average lounge band will just be a laughable farce.
It can't be saved by the musical content, because that is there in insufficient quantity.
Pop music is all in the nuance and tone that doesn't get notated in sheet music.
The result is that performing a convincing cover of some pop music requires a very high musical standard. Whereas a five-year-old kid can play some passable Bach on the piano that we can enjoy as Bach, just by hitting most of the right notes from the sheet music at mostly the right time.
Sure, you can get pleasure from that, but at the highest level classical music is very much about interpretation. People may think that classical music is about following the notes exactly, but it's not quite true, the way the player interprets a piece is what brings a piece to life (or not), and these subtle touches are not directly transcribed in the sheet music.
As for forgotten interpreters, before the days of recording I'd agree with you, but now we have recordings of greats from the past that's no longer the case. For example, in the case of classical guitar, people still admire the playing of players that are no longer with us, such as Segovia, and will still play their recordings. Whilst it's good to support new players too, a great performance is a great performance, it doesn't matter how long ago it was performed if it's still possible to listen to it.
Music isn't just a series of clever notes, strung together. Sure, it's that too, but there's so much more to it than that.
I agree with what you're saying here.
Then why do people pack concert halls to listen to Joshua Bell instead of listening to their kid cousin playing the violin?
> The result is that performing a convincing cover of some pop music requires a very high musical standard.
The same is absolutely true of classical music. Just because you can recognize what a Bach piece is by hearing the notes doesn't mean it's a good or convincing performance.
Precisely because the raw content of Bach or Brahms, though monumentally important, isn't enough to pack halls.
He was at a raga concert, and admitted to feeling totally lost. As the piece progressed, about 30 minutes into the performance, everybody in the audience sighed a breath of satisfaction, and nodded and smiled to each other when the cycles had reached conclusion.
At that moment he realised that there was so much more to music than he realised, and that the audience was much more capable of concentration, too.
I'd counter that Sidney Bechet or Muddy Waters 12-bar blues can be sophisticated without being complex
If all you're doing is counting notes, Bach is more complex. If you bring expressiveness into it, the Carter family trounces Bach. And if you want to hear Bach ruined, just plug the notation directly into a midi sequencer with no human involved in the performance.
Music is both intellectual and emotional. Measuring only the intellectual aspect is a disservice to music. The emotional and cultural roles of music matter. This is often lost in "music theory".
I remember a long time ago, watching Peter Gabriel perform at a charity concert for human rights. A quarter million people in the audience, and he got them to do a sing-along of the chorus to "Biko", a song about the death of South African activist Steve Biko, who was beaten/tortured to death in police custody. Peter Gabriel was sitting on the edge of the stage, swinging his legs, providing minimal guidance. A quarter million people raised their voices together. How is that less sophisticated than a Bach fugue?
You raised the example of Simon and Garfunkel. "The Sound of Silence" will last in human society as long as Bach does. It will still be heard centuries from now. That's sophistication.
Often it's forgotten that articulation, ornamentation, and improvisation are all supposed to be assumed in baroque performance practice. And that in many cases (nowadays), all three of the above are tossed aside (thankfully, less so nowadays -- with performers skilled in performance practice).
Remember: Bach was much better known as a performer/improviser rather than as a composer in his own days.
Combining the complexity of Bach's compositions with the complexity of his (or someone else's) actual performance just adds additional layers of the type of sophistication that you're speaking about.
Classical music is in general much more complex and sophisticated than pop music. This is undeniably true by any rational yardstick. Whether its better or not is completely a different question.
(of a person, ideas, tastes, manners, etc.) altered by education, experience, etc., so as to be worldly-wise; not naive: a sophisticated young socialite;
the sophisticated eye of an experienced journalist.
pleasing or satisfactory to the tastes of sophisticates, or people who are educated, cultured, and worldly-wise:
complex or intricate, as a system, process, piece of machinery, or the like:
a sophisticated electronic control system.
of, for, or reflecting educated taste, knowledgeable use, etc.:
Many Americans are drinking more sophisticated wines now.
By any of these 5 distinct definitions classical music is more sophisticated than pop music. I understand that you do not like the connotations, but you have to use a word to mean what it actually means...
I think more to the point, if you have to think about what you're playing while you're playing it, you're hosed. It's like the old question of how does a centipede walk. If the centipede thinks about walking, it falls. So music, even very complex music, tends to be built from small, easy to grok bits of technique, small enough that a player can simply memorize them, the way we memorize how to walk, or how to conjugate verbs.
European classical music is rhythmically crude and primitive.
I spent a couple of years studying the frame drumming of the central Asian steppes with a good teacher. The concept of "time signature", a basic European idea, goes right out the window. If you're doing something best described as "17/16 with the occasional 13/8 counter-rhythm", you start to understand how crude European rhythm is. Yet it's actually easy to play! That's because it's not a "bar", it's a seqeuence of phrases that may be two, three, or four subdivisions long. Memorize the sequence and it's pretty straightforward.
Likewise, equal temperament severely limits European melodies. Arabic and Indian music are far more melodically complex.
But if you measure every other culture by the values of your own, they all look worse.
I can't disagree with the overall idea that traditional European classical music is pretty simple rhythmically, but the statement above is just flatly false.
For instance, I have here a copy of the first book of the Wohltemperierte Klavier, Bach's big collection of preludes and fugues. Prelude no 3 is in 3/8. Prelude no 4 is in 6/4. Prelude no 8 is in 3/2. Prelude no 9 is in 12/8. Prelude no 11 too. Prelude no 13 is in 12/16.
It's still all 2s and 3s in various combinations (though there's a little bit of 5/8 in Handel) and, again, I'm not disputing that this stuff is rhythmically much simpler than your central-Asian frame drumming. But it's also not all 3/4 and 4/4, nor is it just a steady stream of quavers.
It's kind of like arguing a Buick is better than a Chevy to someone driving a Tesla.
But what you actually said was, in so many words, "every Bach piece, and every similar piece from other composers, is written in 3/4 or 4/4 time". And that simply isn't true.
From time to time I listen to an Indian radio station. The melodies are often strange; yet the pitches are so accurate and nuanced, and a sour note is scarcely to be heard.
you can bet that if alot of people have been playing some kind of music for many generations, with lots of enthusiasm, then it has as much depth and sophistication as any other music with equivalent tradition. the problem might just be you dont know what to listen for.
> 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.
> 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.
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.
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).
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!
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.
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/). 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.
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.
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.
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!
A fascinating read for anyone interested in the subject.
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".
Still, for me, it continues to be a life-changing read which fundamentally altered the way I think about and hear music, and I've worked as a professional musician for more than 20 years.
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.
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.
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.
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?
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.
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").
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.
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'.
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.
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.
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
Seems interesting tho, I might keep reading, I would like to see another approach to harmony.
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.
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?
You may also be interested in Syntorial, which is a synthesizer design course, teaching you how to design synthesizer sounds:
Music theory is a huge field. I'd say the circle of fifths is a good starting point into exploring music theory.
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.
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.
For arXiv posts, unless a specific version is under discussion, it's probably best to link to the abstract page without the version tag:
This lets folks see if there are multiple versions of the preprints and multiple formats to view those preprints.
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.
E.g. this bit of quackery "Electromagnetic Signals from Bacterial DNA"
* https://arxiv.org/abs/gr-qc/0605023 (dated 2006)
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.
This generalizes all existing theories of musical harmony, and explains things like stretched tuning in pianos (a consequence of non-ideal strings).
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.
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!