
Harmony Explained: Progress Towards a Scientific Theory of Music (2014) - adamnemecek
https://arxiv.org/html/1202.4212v2
======
stfwn
> You can't start a science textbook like that.

'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](https://pastebin.com/ZYKJDFRN)

~~~
treehau5
> I wouldn't criticise cooks for not describing taste by the molecular
> contents of their ingredients.

Now _theres_ a good analogy.

------
theabacus
The author is correct in saying that music theory is not a "theory" the
scientific sense.

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.

~~~
_red
I think what many miss is that Music Theory is designed to be after the fact.
It seeks to explain - after a piece has already been written - why something
works harmonically / rhythmically / etc.

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...

~~~
rhizome
I think another aspect is that music theory is absolutely not required for
appreciation nor composition.

~~~
jancsika
It's just a framework. Like music history, ethnomusicology, music cognition,
music lessons, instrument building, etc., it will deepen one's appreciation
for music if you learn it.

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.

------
the_cat_kittles
hey guess what, western classical harmony that bach kind of codified, is but
_one_ harmonic system. its accidentally super arrogant to ignore the insanely
deep musical traditions that are not present in western classical music, since
they have different ideas about harmony. one of the great things about music
is that it has really opened my eyes to this kind of of cultural myopia. its
so obviously implicit in pretentious analyses like these, even if their
content is interesting and correct, that the writers believe that western
music is the _real_ music. its trained into alot of us that that is the case.
well, its not. its not superior, nor inferior. someone is going to read this
and think im being an sjw, but i assure you! i once implicitly thought this
way too, but i studied other music besides my first love, which was the music
of bach, and its true! the musical traditions of roma people, indian classical
music, american jazz, bluegrass, flamenco, romanian folk music, turkish music
(just whatever is coming to me in this moment) etc etc these things are so
deep, and have such rich traditions of insane dedication. so, dispose of this
idea that bach's music is the only real music.

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!

~~~
phlakaton
The author did attempt to address this, by complaining that you "aren't really
paying attention":

* 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. :-)

------
unholiness
The author starts out by complaining that current music theory is not
scientific. I completely agree. The field is astoundingly full of technical-
seeming, well-accepted textbooks containing a biological "explanation" of
harmony that is either totally unproven or trivially false.

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.

-

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](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.

~~~
TheOtherHobbes
We don't even perceive the Fourier transform. What we perceive is pre-
processed by complex "sonic object" recognition systems we barely understand.

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.

~~~
unholiness
True!

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.

------
laretluval
Does this work address the fact that people raised in non-Western musical
cultures don't experience pleasure/displeasure based on harmony?

[https://www.sciencemag.org/news/2016/07/your-culture-not-
you...](https://www.sciencemag.org/news/2016/07/your-culture-not-your-biology-
shapes-your-musical-taste)

~~~
whiddershins
Yes, well, sort of, and also perhaps you have a citation for your claim?

------
mtdewcmu
I came up with something of a theory of rhythm. I don't know anything about
music theory, so, for all I know, the theory is not new or a better theory
already exists. But it's simple, so I'll share the main idea.

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)
    

Since they occur together, they are perceived as being causally related. Now,
back to music. Within each repeating unit (usually a measure), the listener
always has a working theory of causation to explain the rhythm. The theory has
the form of a tree, with child nodes caused by the parent node.

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.
    

If rhythm2 is added to rhythm1 after rhythm1 has been playing already, rhythm2
is perceived as a distinct chain, which is caused by the main beat. Hence,
rhythm2 can be removed as a unit without violating causation.

~~~
jancsika
> [...] beat #3 is an echo of beat #2 [...]

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.

~~~
mtdewcmu
There's so much in your comment that I could respond to. I probably won't get
to everything.

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)
    

where the hardness was like 3-2-2-1 or 3-1-2-1. If I'm using terminology
correctly: I hit it hardest on the down beat and second hardest on up beats.
It just sounded right.

So the echo pattern would be like

    
    
      #1 Downbeat
      #2  (echo of #1)
      #3   (echo of #2)
      #4    (echo of #3)

(On the high hat)

OR

    
    
      #1 Downbeat
      #2    (Small echo of #1)
      #3  (Big echo of #1)
      #4    (Small echo of #3)

(On the high hat)

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.

~~~
lgas
I don't know anything about music, but it sounds like you said "Here's proof
that beats are perceived as echoes: I play them like echoes."

------
jimsmart
Dupe?
[https://news.ycombinator.com/item?id=12085844](https://news.ycombinator.com/item?id=12085844)

------
kaonashi
Music is language.

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.

~~~
rhizome
Music is _like_ a language, but unlike languages you don't need to be familiar
with it in order to understand it.

~~~
kaonashi
That depends entirely on the level of understanding.

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.

~~~
rhizome
_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.

------
TheOtherHobbes
For a more professional and thoughtful look at music and combinatorial math I
suggest:

[http://dmitri.mycpanel.princeton.edu/geometry-of-
music.html](http://dmitri.mycpanel.princeton.edu/geometry-of-music.html)

There's also

[https://www.amazon.co.uk/Topos-Music-Geometric-Concepts-
Perf...](https://www.amazon.co.uk/Topos-Music-Geometric-Concepts-
Performance/dp/3764357312)

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.

~~~
elihu
> 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.

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.

------
lyra_comms
Reads as a working lit review (huge collection of ideas the author has come
across) rather than a compact synthesis.

Discuss on Lyra: [https://hellolyra.com/c/382](https://hellolyra.com/c/382)

~~~
acjohnson55
What is Lyra?

~~~
mrkgnao
From the homepage:

> Lyra is a non-profit, non-invasive communication service which aims to
> enable deep and meaningful conversations.

Seems like a Slack/wiki hybrid.

------
slaymaker1907
I think a key challenge with music theory is that it is tackling some of the
same problems as in linguistics.

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.

------
acjohnson55
I don't have time to thoroughly read this before commenting, but upon scanning
it, I echo the disdain of many of the other commenters. The lack of humility
seems to reflect the lack of important prior work cited in the paper.

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](http://dmitri.tymoczko.com/geometry-of-music.html)

\- Bill Sethares's Tuning Timbre Spectrum Scale approach:
[http://sethares.engr.wisc.edu/ttss.html](http://sethares.engr.wisc.edu/ttss.html)

\- Andrew Duncan's Combinatorial Music Theory:
[http://andrewduncan.net/cmt/](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.

~~~
tprice7
In relation to Bill Sethare's work, I recommend listening to example (1) here
:
[http://sethares.engr.wisc.edu/html/soundexamples.html](http://sethares.engr.wisc.edu/html/soundexamples.html)
, which is an amazing demonstration of the relationship between overtone
beating and consonance / dissonance.

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.

------
jancsika
Does the author acknowledge the field of music cognition, and that it is a
scientific field that produces verifiable/falsifiable results?

------
xamuel
Authors who claim to understand music scientifically should be judged by the
music they produce. If an author really does have some deep understanding of
what makes music work, it should be fairly easy for the author to get a record
deal, get a bunch of #1 pop hits, etc. Until then, it's alchemy.

------
michaf
There is a newer version 2 from 2014:
[https://arxiv.org/html/1202.4212v2](https://arxiv.org/html/1202.4212v2)

~~~
sctb
Thanks! We've updated the link from
[https://arxiv.org/html/1202.4212v1](https://arxiv.org/html/1202.4212v1).

