Harmony Explained: Progress Towards a Scientific Theory of Music (2014)
arxiv.org
arxiv.org
'Music theory books' make no claim to be scientific. Western music is a cultural construct and there are books that explain how to practice this particular field. For insiders, the three initial statements the author attempts to burn down make perfect sense. What it means to locate a (part of) a music piece in a 'harmonic system' makes just as much sense to musicians as the rest of us calling a lollypop 'sweet'. I wouldn't criticise cooks for not describing taste by the molecular contents of their ingredients.
While it's interesting to dissect music differently from how musicians may analyse it, completely rejecting local experts' understanding of a phenomenon runs counter to finding out more. Is a farmer a complete fail whale for knowing when to plant his crops based on when a certain type of flower first blooms, rather than the precise position of the earth relative to the sun?
If I would only read music theory by this author, I would still not have a clue how to play the piano. He/she wastes a lot of words on what could otherwise be described as the difference between Metis and Techne, or local knowledge and technical knowledge.
I pasted two long quotes about the distinction between Metis and Techne here: https://pastebin.com/ZYKJDFRN
Now theres a good analogy.
1) Much of this literature is indeed in inside books which also discuss the "cookbook" parts of music theory, and
2) It's shocking how much incorrect, unscientific nonsense there is in these texts, and
3) There's nothing close to a scientific consensus about why humans and animals nearly universally feel good when hearing harmonious sounds.
The author is attempting to answer that question, and I think his initial complaints are perfectly valid.
But as a musician/producer/composer I'd never want that anyway. Having studied many forms of music, I know that there is no "right" and so a proscriptive/predictive "theory" of music is not at all what we want. On the contrary, what is "right" in music comes from a long line of collective cultural reasoning. This is much more evident in the distinctions between the development of eastern and western music, each having their own "theory". Notably, eastern music has a lot lower emphasis on harmonic progression and tends to use different tuning, sometimes "perfect" tuning (something which western music must eschew to support flexible harmonic motion). This leads me to my thesis, that music theory is primarily descriptive in nature and there are many theories of music, driven by historical-cultural development.
Music "theory" should lie outside of the domain and concern of science. There are fields which address the human response and other scientific natures of this such as psychoacoustics, electrical engineering and signal processing, and mathematics. Music "theory" has in its view aesthetics which lie outside of the domain of science and is a tool for use in describing things which have been found pleasing aurally and culturally.
It has no prescriptive power, it was never intended to be a set of formulas that can then be used to generate "good music". Only to create a language to explain whats already there...
Personally, I think western music is stuck in a 12-tone-equal-temperament rut. It's not as if we've discovered all possible 12-TET music, but it's getting harder to find anything truly novel harmonically or melodically. (Timbre and rhythm are still good ways to innovate within 12-TET.)
Noting a neighboring comment, theory may not be necessary for composition, but some specific forms of composition may be prohibitively difficult without theory. I have musical colleague who composes multi part jazz harmonies for small to large ensembles. They are steeped in theory.
On the other hand, my spouse attended a seminar given by a fairly well known film composer. She asked him if he followed any particular school of theory, and he said no.
As far as composition-- I've had some encounters with talented musicians who get stuck putting together a song or electronic composition because "something just isn't sounding right." In those encounters, a process of elimination improved a chord/voicing that an understanding of fundamental music theory principles would have revealed without my help.
The problem is that without the fundamental understanding, it's nearly impossible to reverse-engineer such an improvement and deduce the general principle at work. Lacking that basic understanding makes one reliant on others just to figure out how to get the sounds out of one's head and into the air.
Said every carpenter, craftsman and artisan that was eventually replaced by machines. There's a place for both, but it seems silly to argue against the march of progress given where it's gotten us.
the same kind of weird cultural narcissism / racism / whatever it is happens in language, with "correct" usage of words. its the same damn thing.
and can i just add, for all the overly left brained harmonic analysers out there- it would be nice to see someone address the fact that harmony, even in the restricted western classical sense, i guess "diatonic", depends greatly on timbre and range as well. of course i didnt read this paper, and instead ranted somewhat incoherently about topics which might not even apply to the paper, but hey, i had a good time!
* He notes that octaves are common across cultures (per Levitin, "This is Your Brain On Music")
* He asserts that 12 tones represent a practical limit to harmony; more than 12 tones is more complex than the brain can handle in its "note-expectation engine".
* He thinks other cultures could be working within his theory, but simply not taking advantage of _all_ of the universal harmonic features that Western music offers. He cites pentatonic scales as an example.
* He is baffled by the Nasca Indians of Peru, who apparently form their pitches by linear steps in frequency rather than logarithmic (I won't even attempt to use the word "harmony" in this sense), but he asserts that 1) their tuning system was probably derived because it was mechanically easier to produce a set of pipes to that tuning system, 2) they're a weird case because they were isolated.
* To finish that reasoning off, he suggests there's a "musical sophistication" of a culture which develops from linear scales (easy to carve) on the simple end to equal temperament on the sophisticated side (because arbitrary key changes without wolf intervals is the ne plus ultra, right?).
So, yeah: I'd say there's some serious cultural narcissism going on. It would be OK if he confined his analysis to Western harmony and simply tried to explain what was going on there, but by trying to derive universal musical principles from barbershop quartets, his project was pretty much doomed from the get-go. :-)
Nature is the aspects of harmony that independent societies have all agreed upon, likely due to an unknown biological reason harmony sounds pleasant. Nurture is the connotations society and culture put on top of that.
The author is only attempting to explain the nature part of things.
You wouldn't fault the theory of gravity for not explaining cell biology, so let's not fault this guy for ignoring culture.
But then he precedes to dive into a mountain of unsubstantiated conjecture, saying:
1. Our brains are just computers
2. Evolution motivates less computation
3. Finding the fundamental frequency of a sound is worthwhile
4. Doing that is computationally easier if all other frequencies are in harmony
5. That is what has lead to harmony giving us pleasure
(The rest is essentially fluff around this core argument)
It's a nice theory, but there's nothing to substantively back up any of these points. He comes the closest on point #4, but only by making strong assumptions about the flavor of #1. The whole thrust of his argument is just totally tangential to addressing his initial complaints about the lack of science and rigor within the subject of biological harmony.
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One thing he does get right is avoiding most common mistake that textbooks and laymen alike think about hearing: Our ears do not perceive the waveform of sound directly (for the most part[1]). Rather, we perceive sympathy vibrations at different parts our cochlea, rather directly sending nerve signals according to the Fourier transform of the original waveform. This can be measured (scientifically)!
This Vi Hart video does a remarkably good job at giving a layman's tour of our some of the phenomena in our ears: https://www.youtube.com/watch?v=i_0DXxNeaQ0
[1] If this were totally true, we wouldn't be able to perceive phase at all! But it is the case that these sympathy vibrations are perceived with much more sensitivity than our perception of phase, and are linked more closely to peoples' idea of harmony.
Obvious example: you don't hear formants and spectral distributions when someone is speaking. You hear words, which are parsed into sentences, which are processed into meaning.
Along the way a lot of redundant elements - background noises, breath sounds, ums and ahs, and even distortion from electronic transmission systems - are all eliminated.
Music is similar but even more complicated, because while there are stylistic conventions, there doesn't seem to any single consistent grammar.
There's a superficial listening level which parses a mix of sounds into the sounds of different instruments, a "theory" level which listens for horizontal and vertical groupings and structures, an "emotional" level which reacts to implied emotion and mood, and sometimes a "metaphorical" level which listens poetically and associatively.
It's not a simple process.
I won't risk getting out of my depth. My comment was simply saying there are unique nerve signals from unique points across the cochlea which uniquely convey the information "this frequency is currently being heard at this volume". To say that that information is what we perceive... that's not an argument I'd make. But it is a step along the process.
Everyone knows we "see" in rgb, but few people seem to know we "hear" in frequencies more than we "hear" in waveforms. That's my point.
Of course there is.
1. All of physics is computable, and dualistic theories of mind are far more problematic than monistic ones. Also, computational minds are consistent with all of our scientific knowledge, and require no further ontological commitments; not true of other theories of mind.
2. That evolution favours less computation, all else being equal, follows trivially from natural selection. Using more computation than is needed requires more food, which would not compete well against mutations that are more efficient.
3. Finding a fundamental frequency is adaptive for identification of predators, prey and allies.
4. Well defined frequency separation does make the problem of separating frequencies easier. It's practically tautological.
5. This is the only thing requiring additional empirical support.
1, for instance, is not philosophical. He's literally carrying the metaphor down to additions and multiplications of waveform samples. There's no evidence that this is mechanistically similar to how our brain thinks, nor is there evidence that "easy" problems of addition are easy problems of our mind.
2 is an argument of strong evolutionary force... strong enough to overcome the fact that my emotional reactions to harmony will change my behavior, despite them not being a good signal of how to survive
And the rest is pure conjecture. I'm not saying it's a bad hypothesis. It's just not anything beyond hypothesis.
https://www.sciencemag.org/news/2016/07/your-culture-not-you...
I think that rhythms are perceived like echos. So, in 4/4 time with beats on quarter notes, beat #2 is an echo of beat #1 (the main beat), beat #3 is an echo of beat #2, and beat #4 is an echo of beat #3. Then, the main beat sounds again and the whole thing repeats.
Now, if that makes sense, you can forget about echos. What matters is that there is a chain of causation. The listener perceives that Beat #1 causes beat #2, which causes beat #3, and so on. The listener unconsciously creats a theory of causation that explains the rhythm.
Now set aside music for a second. Imagine that you hear a sequence of sounds:
1. "Oops!"
2. (Crash!)
3. (Baby crying)
When rhythms mutate over time, the change will sound small if the change only affects a leaf. The change will sound bigger if the change affects a complete branch. And, the change will sound jarring if it violates the theory of causation, i.e. by removing the cause without removing the effect.
Multiple rhythms can be superposed over each other, so the theory of causation can be a tree with multiple branches. For ex.:
Main beat -> rhythm1
AND ALSO
Main beat -> rhythm2.
This doesn't match the hierarchy of beats present in most music written in 4/4, where beats #1 and #3 are strong beats-- with beat #1 getting a greater emphasis than beat #3-- and beats #2 and #4 being weak beats. Much of the music of the classical period supports this fundamental pattern-- in 4/4 cadential formulae are generally articulated on beat #1 or beat #3, and rarely on beat #2 or #4.
One could make the argument that beat #3 is an echo of #1. But then beat #2 couldn't also be an echo of beat #1. (And there isn't a similar hierarchy of weak beats with #2 and #4.)
Furthermore, this hierarchy usually extends out exponentially in larger durations. So in a Mozart string quartet you'll hear how rhythmic material of measures 1 and 2 is often repeated (or slightly varied) in measures 3 and 4, with measures 2 and 4 being the "weak" measures. A lot of repertoire from the classical period extends at least two levels further so that the first four measures are a group balanced by another group of four. That 8 measure phrase then has a matching 8 measure phrase, although once we get that far out the rhythms aren't usually strictly symmetrical or imitative anymore.
I haven't read the literature but I'd imagine music cognitive research can shed light on just how far out humans can track those rhythmic groups. On the one hand, I know it has to be smaller than, say, an entire section of a sonata (I've never perceived a development section as the "weak" rhythmic group, for example). But techno music makes me speculate that at least in simplistic musical structures these groups can extend quite far.
But here is a "proof" that beats are perceived as if they were echos, using a rock beat. A (natural) echo is softer than the original, because the sound is traveling a greater distance to get to your ear. So, an artificial echo should also be softer. I used to play drums and I remember instinctively hitting the hi hat like (on quarter notes):
(DAH-Dah)-(Dah-dah)
So the echo pattern would be like
#1 Downbeat
#2 (echo of #1)
#3 (echo of #2)
#4 (echo of #3)OR
#1 Downbeat
#2 (Small echo of #1)
#3 (Big echo of #1)
#4 (Small echo of #3)The way to test a theory of causation is to remove beats and see if it sounds right to your ear. Pay attention to how the transition sounds. Does the transition sound natural? This is kind of hard to do with rock rhythms. The theory becomes more interesting as the rhythms get more complex. When I was thinking about this, I was imagining house music or other electronic dance music.
I don't know much classical music, so I haven't really even thought about it.
That isn't true! Beat #1 can have an infinite number of echos.
N.B.: When I was initially thinking about this I was thinking of subsequent beats as "echos," and I mentioned the echo concept here in the hope that it would be something concrete and instantly relatable. But the echo concept only goes so far, because the "echos" don't have to sound like the original (orig: crash! echo: whistle), and the echos can be removed and replaced over time (say, a change every 4 bars).
The concept of an "echo" is that
1. It comes after the "cause"
2. It's less accentuated
For example, when listening to a rhythm, you construct rules like:
"Boom" comes before "whoosh"
I think it's actually a bit of a trick. It causes the song to sound faster. The reason is because the snare or whatever on #3 is perceived as being incongruously loud, too loud to be an echo. Which gives the impression that it's a second main beat, splitting the measure into two halves and making it sound like 2/4.
Having said that, it might also make sense to consider it an echo of the bass drum. You could remove the snare and leave the bass drum and it would sound ok. If you tried to remove the bass drum and leave snare on #3, it really wouldn't sound right.
I agree that that's a weak part of the theory and it could use more thought.
>>The pattern of dah-DUM is pretty common, and the dah would typically be a quarter or eight note before the DUM on the 1.
I thought about that case, and I considered whether to create the concept of a pre-echo.
Another edge case is when beat #3 is accentuated, like in almost any rock song:
BASS-tick-SNARE-tick
The echo pattern is what the listener expects to hear. When a beat is louder than it's "supposed" to be, it makes that beat sound louder and brings it out. I think the snare sounds more powerful as a consequence of being unexpectedly loud.
Wrt pre-echos, I think you could consider them a deliberate deviation from the echo pattern to achieve a particular sound. The difference between theory and actual rhythms is what we call music, right?
The pattern can change, say, every 4 bars. Then you are manipulating the chain.
Playing drums is probably the ideal background to be able to follow along with what I wrote.
The idea is kind of hard to explain to begin with.
Music 'theory' is somewhat akin to linguistics (or language subjects). It can help you dissect and understand how things work in an abstract sense, but when it comes to actually talking (or playing music), you're operating in a different mindset.
The people who rely on music theory are non-experts in the same way that the people who rely on language class to speak are non-native speakers.
This is because music has evolved as a mutated copy of part of the human language acquisition instinct. The mood change caused by music is a mutated copy of the mood change that accompanies initial language acquisition, and it serves a different purpose.
The mood change that accompanies initial language acquisition helps to focus the infant's mind on language as a separate important thing that has its own (somewhat arbitrary) rules that need to be understood separately from everything else.
The mood change that accompanies music serves to intensify emotions that accompany daydreaming - this helps us to "better" daydream, in a manner that does not cause us to confuse daydreamed emotions with real emotions (because we are always consciously aware that the mood change is directly caused by listening to music).
(More details at http://whatismusic.info/.)
Recognizing the harmony underneath a line isn't something a casual listener is going to do, but is essential for a musician who is trying to communicate with other musicians in real-time.
My point is that none of that is essential to recognizing it as music and getting meaning from it.
http://dmitri.mycpanel.princeton.edu/geometry-of-music.html
There's also
https://www.amazon.co.uk/Topos-Music-Geometric-Concepts-Perf...
Although IMO it's mostly blather - unlike Tymoczko, who's worth reading.
There are no complete scientific theories of music and there never will be, because music is a huge and semi-random collection of cultural experiments and stylistic conventions built only very loosely on perceptual psychology.
Academic theory books define an alphabet of terms, but the rest is learned by practice and experiment, oral teaching, and by listening, copying, and varying.
You could say much the same thing about architecture. In that domain, I find Christopher Alexander's book A Pattern Language to be a very good reference for describing many concrete reasons why certain buildings feel better than others that seem to be pretty universal. (And we could reasonably expect to be able to test that if we were in doubt.)
It's not a complete theory of architecture and can't be, but partial theories are still tremendously useful and more likely to produce good results than blindly following stylistic rules without knowing what purpose they serve.
I've been hoping for awhile that someone will write a good music theory book in the same format.
Discuss on Lyra: https://hellolyra.com/c/382
We did not take up rhythm in depth in this article, however I think it is quite likely that rhythm can be explained in a manner similar to harmony.
- Wile E. Coyote, Super Genius
Look, it's true that music theorists aren't scientists and the most innovative musicians are often terrible at explaining the underlying structures or musical strange attractors that shape what they do. I'm a terrible musician and I've been asking and groping towards a scientific theory of music for years. But at every music tech trade show, there are a couple of people like this guy showing off their automatic music composition system. They're very very proud of the theoretical abstractions they've been able to identify and automate, and they should be, it's impressive work. But the music is always terrible, boring stuff that doesn't evoke any emotion. And I say that as someone who listens to tons of techno and 'difficult' electronic music from artists like Autechre.
Theory gives you a framework on which you can build your composition, where that's musical, paint or whatever. There is lots of bad art produced without such a framework that typically has a short half-life, and lots of bad art that merely decorates the framework and is thus pleasing but formulaic, also having a short half-life. Enduring art can develop and even comment upon the framework, but it's not primarily about the framework. The choice of subject is one of the most difficult and hard to teach or communicate aspects of art, and the problem with theoretical exegeses like this is that they attempt to eliminate the subjective altogether and thus kill the very thing that attracts people to art in the first place, the ability to project themselves into it.
> Lyra is a non-profit, non-invasive communication service which aims to enable deep and meaningful conversations.
Seems like a Slack/wiki hybrid.
You have a somewhat formal system that follows some sort of methodology. However, while there is a system, it seems nearly impossible to actually describe the axioms of this system formally.
With language or music, how do you determine some bases from which all other concepts are composed? I suspect that there is probably no such bases since new concepts seem to be able to be spontaneously created from human experience and often have circular relations.
Most glaringly, the author assaults Helmholtz without acknowledging the fact that people who study this topic in academia are way past that. It's like if I were to tell you that I've got an amazing new theory of physics and justify it by referencing Newtonian mechanics -- every academic knew it to be incomplete a century ago. Pointing that out is not a novel idea.
If you're interested in understanding how music works on a psychological level, "music theory" is the wrong thing to search for. The fields you're interested in are psychoacoustics and music perception.
Music theory is the study of compositional practice. It's a field that collects a whole bunch of descriptive theories on how composers in different periods and genres tend to work. You can use it to help yourself compose pieces in a given paradigm.
Psychoacoustics and music perception are the study of how we actually understand sound and music. The theories therein are what gave us things like lossy compression (e.g. "exactly what raw sound data can we discard and still preserve a perceptually identical recording?"). This is an incredible achievement in that it only works if we have a thorough understanding of how the brain processes sound and music. Psychoacoustics has little to say about what makes something qualitatively musical, but that's where study of music perception picks up.
In the overlap between these two fields, there are a bunch of people working on unified theories of harmonic/Western/tonal music. The author fails to reference the most interesting work done in this area. The ones I've found most convincing are:
- Dmitri Tymoczko's Geometry of Music: http://dmitri.tymoczko.com/geometry-of-music.html
- Bill Sethares's Tuning Timbre Spectrum Scale approach: http://sethares.engr.wisc.edu/ttss.html
- Andrew Duncan's Combinatorial Music Theory: http://andrewduncan.net/cmt/
Some of this work is pretty old at this point. Granted, it's surprisingly difficult to do a full literature review in this area. I know from experience, because I studied this topic in grad school. Someone could probably earn a Ph.D. by properly synthesizing all of these idea into a music theory that combines the analytical power of Tymoczko's approach with the first-principle work of Setheres.
Even if the statement "lack of beating between overtones sounds good" doesn't hold true as a universal rule (and I'm sure this has occurred to many people prior to Wilkerson), you can really see what Helmholtz was getting at here.