Modern approach to music theory
June 16, 2021 11:50 AM   Subscribe

Is there a curriculum that teaches music theory starting with something like serialism or the 12 tone scale, then goes back to classical Western music as a special case? I want to learn assuming that dividing an octave into 12 frequencies is the basis of Western music and then treats things like scales and keys as special cases.

I'm taking Coursera's intro to music theory course. I don't play any instruments, nor do I sing, nor do I make music. I do have a better-than-average musical appreciation education though, including college level listening / criticism courses on 20th century classical music and jazz. I also have a very strong math and computer science background and I keep thinking "but this is all simple algorithms: why does music theory have to be so complicated?"

The Coursera course teaches things the way I suspect is standard. "This is the major scale. It's composed of intervals of tone, tone, semitone, ...." But it doesn't explain why it's that way. It's Just So and you need to learn it. And so then you get to confusing things like "is this note called A-sharp or B-flat?" and I just throw my hands up in exasperation.

My impression is that normal music theory is taught from a sort of historical construction, taking the codified music of Bach's era as the basis of Western m usic and only later moving to the gradual use of dissonance, atonal scales, jazz harmony, etc. That's great for people who play music or come from a basic historical background of music.

But I think I'd understand it better from an inverted perspective. Start with fundamental physical realities, like frequency ratios and harmonics and the key role of dividing an octave up into 12 (nearly) equal parts. Then introduce things like picking 7 notes from the 12 as a scale or the names we give things or why the piano keys are colored the way they are. As historical artifacts or conveniences, not the basis of music theory that only later got complicated by the 20th century weirdos.

Does what I'm asking for make sense? Does anyone have a book or course like that?
posted by Nelson to Media & Arts (31 answers total) 21 users marked this as a favorite
 
I don't have a great resource off the top of my head for a course/video that "justifies" the basics of Western music theory starting from core principles, but I am 100% sure that such things exist (I've seen them). But I want to set up some background here that may help you as you look for one.

The division of an octave into 12 equal parts is historically a result of Western harmony, not a cause. The super short version is: a frequency ratio between two pitches of 2:1 (octave) sounds great, and a ratio of 3:2 ("perfect fifth", but don't worry about the reason for that now) sounds almost as good (consonant). If you apply two perfect fifths in a row to generate more nice pitches, you get 9:4, or 9:8 if you divide the frequency by two (bring it down an octave) to get it closer to the original. It is a fantastic coincidence that if you do this twelve times you almost get 1:1 back to where you started (you actually get ~1.014). Anyway, the upshot is that using exactly twelve evenly spaced pitch classes lets you use all the scales you were going to want to use anyway because they sounded good to your 17th century Western brain.

There are later approaches like serialism that use those 12 pitch classes as a basis for composition without reference to classical harmony and melody, but it would be kind of backwards to start there and then build up Western theory in reverse.

Anyway, what I expect will serve you best is some video that takes half an hour or so to cover the above paragraph in a lot more detail, which will give you enough background to go back to "tone, tone, semitone" and understand "why" it works, and then you can pick up the regular theory curriculum from there. I don't think you really want to do the whole thing backwards.

I (and probably dozens of others!) am happy to answer specific questions about the fundamentals here, too.
posted by dfan at 12:20 PM on June 16 [9 favorites]


This is not an easy suggestion, but you could do worse than to try reading Schoenberg's Theory of Harmony. Then again, you may be more interested in a straight-up Physics of Music course (why does it sound that way?) followed by a Music Cognition course (why does it sound that way to us?). A book I wouldn't be surprised to see in either course is Juan Roederer's Physics and Psychophysics of Music.
posted by All hands bury the dead at 12:28 PM on June 16 [3 favorites]


For a less Western-centric approach, try W.A. Mathieu's Harmonic Experience, or his Bridge of Waves.

To supplement dfan's excellent explanation, you can also search on "the overtone series."
posted by dum spiro spero at 12:32 PM on June 16 [2 favorites]


Yeah, no one really presents it that way because musicians (of any period, including modern or 20th Century) really thought of it that way. Musicians start by making sounds and then figuring out how they fit together via listening to them. And they start with the basis of music as they have learned it, then explore that further, react against it, etc. They don't start with either the math or physics of sound, as interesting as those are to study.

But based on what you are saying, you might enjoy:

- Something like a more modern approach to music theory. That is to say, starting with a basis of modern jazz, pop, rock etc usage of melody, scale, chord, harmony, rhythm, etc instead of the more usual way of using Bach or the "common practice" as the starting point and then arriving at modern usage as the end product of a few centuries of development. The Berklee College of Music is one that takes more of that approach and you could look at some of their materials.

- You might like to learn Set Theory, which is super-math-based and kind of cool, and used as an interesting tool by some theorists. (But not really how much of anyone actually writes music).

- You might enjoy learning about the music theory of ancient Greece. They started with a lot of the interests and concerns you are talking about. I don't know if you can find a really comprehensive step-by-step learning course for this stuff, though--but there are plenty of web sites to get you started and then books for more details.
posted by flug at 12:40 PM on June 16 [2 favorites]


I would imagine that if you really want to marry math and music theory, you would want to look into overtone theory (or, more or less as dfan writes, plus, on preview, dum spiro spero). Overtones have the pleasant characteristic of being a physical reality, so they don't need history, chronology, or Bach (nor Schönberg) to become understandable once properly explained.

The equal division of the octave in 12 steps is, on the other hand, a construction or an agreement, to make certain things in music work (at the expense of other things), so: a cultural thing that, while it can be expressed mathematically, does not represent a physical core concept without which music isn't possible. To understand what such a compromise does to music, one does need some measure of historical or chronological awareness.
Music theory has engaged mathematicians for many centuries, so you're in good hands. Outstanding musicians were mathematicians and vice versa. Not many would have expected music theory not to be complicated in one way or another, however.
posted by Namlit at 12:42 PM on June 16 [2 favorites]


(By the way, since you have a math background, be aware that "set theory" in music has nothing to do with mathematical set theory. It's really a subfield of group theory.)
posted by dfan at 12:46 PM on June 16 [1 favorite]


I used to take singing lessons from Aaron Wolf, who I suspect may have some relevant stuff on his website. In particular, the "Ear Training and Theory" section on his Software Recommendations and More page starts out,
For learning the standard stuff that most people think of as "music theory" (but which is not a theory, i.e. an explanation, but is mostly just learning culturally-biased jargon and notation), there are many resources. The vast majority are based on the assumption that the piano and the Western major scale are the basis of music. While familiarity with that classical system is useful (especially in communicating with others who have that background), it is a flawed approach to understanding music overall.
Keep scrolling from there and you'll find a book recommendation. If you can't find what you want on his site, Aaron lists his contact email and I bet he'd be happy to hear from you.
posted by aws17576 at 1:49 PM on June 16 [2 favorites]


Seconding Harmonic Experience. It doesn't start with the 12-tone chromatic scale. But it does start with math — specifically, with the stuff about low prime numbers and their ratios that dfan is talking about. And it turns out if you start there, you can build up to a good explanation of music that uses that scale (Western Classical and Romantic stuff, jazz, chromaticism) and music that doesn't (old modal harmony, Indian classical music, blue notes and quarter tones and all kinds of note-bendy stuff).
posted by nebulawindphone at 1:52 PM on June 16 [4 favorites]


The first thing I would do in your shoes is find an entry-level physics textbook and learn about the physics of sound: standing waves, frequencies, wavelengths, harmonics, intervals, consonance, dissonance (beats), etc. As you read, try to answer this question: "Which intervals sound best together, and why?" This will be your guide to why music developed the way it did.

An octave is the most consonant interval. The next one is the perfect fifth. If you start on a given note and then add a bunch of perfect fifths in a row, you eventually get to a note that is exactly 7 octaves above the original note. This is very useful for harmonizing, and it's called the Circle of Fifths. See if you can find some resources (that focus more about the physics of sound rather than chord progressions and key signatures) and read about how it was developed.

How do you subdivide the octave? This was (is) a whole debate, and there have been many solutions proposed. Read about just intonation vs. well temperament vs. equal temperament and taming the wolf interval. Equal temperament is what we have now, 12 evenly spaced tones in an octave. People have some strong opinions about how it "ruins" harmonies, but it's also what makes harmonies more possible on a keyed instrument such as a piano that doesn't have microadjustments (unlike, say, a violin, where you can put your finger anywhere on the string). Again, see if you can find some resources about historically how Western music settled on equal temperament and the compromises that went into it.

Now, how do you combine tones and semitones in sequence to make scales? The pitches of the modern scale (what you might think of as C major, for example) again come from the Circle of Fifths. Wikipedia says: "the seven natural pitch classes that form the C major scale can be obtained from a stack of perfect fifths starting from F: F—C—G—D—A—E—B." (Why do you start from F? That I'm not sure about.) Read about the different types of diatonic scales. The C major scale is C—D—E—F—G—A—B—C. Now what happens if you start on a different note, like if you still play only the white keys on the keyboard but start at D or E instead of C? Now you have different modes — Mixolydian, Dorian, etc. What we call "major" is the same as Ionian mode. What we call "minor" is the same as Aeolian mode. Some of these are a lot more popular than others.

I don't have specific resources to recommend, but I think you'll get a lot out of these search terms, especially equal temperament and the Circle of Fifths. Also, you may find it enlightening to read about how other cultures have subdivided the octave and created their own scales and tunings. Equal temperament is by no means the only way to solve this problem!
posted by danceswithlight at 1:58 PM on June 16 [2 favorites]


Response by poster: Thanks for the responses! The answer to my basic question ("is there a curruciulum for this") seems to be "no", or more accurately "there is no introductory curriculum". Not surprised by that answer.

But I'm getting a lot of references and ideas for things I can read as a supplement to the traditional approach I'm studying now and it's very helpful. For instance I've gotten a lot of mileage from reading Wikipedia on overtones and equal temperament vs just intonation. Feel free to keep it coming!
posted by Nelson at 2:20 PM on June 16 [1 favorite]


If you just want to muck about with some different microtonal scales take a look at Leimma & Apotome. Liemma is a tool for setting intervals of a scale and setting which intervals are primary or secondary. There are a wide variety of presets ranging from Gamelan to Persian to Modern Western experimental tunings. Apotome is a rule based generative sequencer that uses the scales created in Liemma. They are both web based, but pretty powerful tools.
posted by clockwork at 2:21 PM on June 16 [1 favorite]


Edit: almost exactly 7 octaves above the original note. Thanks to dfan for the details!
posted by danceswithlight at 2:22 PM on June 16


My formal music education included taking music theory and music history courses concurrently, and that helped a lot with the questions you seem to be dealing with.
posted by The Underpants Monster at 3:10 PM on June 16 [2 favorites]


Here's a different way to think about music. I don't know why nobody spelled it out to me over years of study, but here it is. Music is a sequence of sounds that people find pleasing. To be pleasing, it has to have elements of the familiar and predictable, alternating with departures into the jarring and unstable.

Church hymns, children's songs, and much of Western music up to Mozart are nothing more than alternations of the tonic (home) and dominant (away from home), mostly in the major and minor scales. To really crack the code, to really get the layout of the land, you need to always be aware of the tonic and the dominant.

Further developments are elaborations of the dominant, taking it to stranger and stranger territory. Beethoven played all sorts of tricks with the dominant - inversions, omitting the root , extending the chord up into 9ths and 13ths, substituting in the diminished seventh, overlaying modal scales, etc. He added new colors to the palette, but he always came home to the tonic. The sonata form just codifies a particular itinerary, a Grand tour if you will.

There are others here with more jazz theory, but it can also be seen as further elaboration of the dominant, going even further afield, spending all their time in it, but still, always coming home to the tonic.

Stasis and kinesis, theme and variation, and of course a beautiful sound - there is a lifetime of enjoyment and study.
posted by dum spiro spero at 3:34 PM on June 16


There are analogies to language that work pretty well here.

Yes, you can study linguistics, and it will give you some useful tools that apply across languages. And some of that's a result of underlying physical realities about how sound and mouths and ears and brains work. But in the end linguists study actual human languages--and those are human creations that have evolved over time in weird ways and are full of irregularities, and you can't avoid learning those quirks, or some relevant culture and history, along the way.

There are some physical and mathematical realities that have an important influence on the development of music, but they'll never explain everything.

The 12-tone scale is a weird compromise that takes advantage of some numerical coincidences to allow you to play and talk about certain kinds of music. It's really important--but it's not perfect or universal, either.

Agreed that "Harmonic Experience" is something you might enjoy. The author's kinda chatty for my tastes, but the approach of developing everything starting from simple ratios is neat. I seem to recall it does eventually assume some basic music theory, though. Definitely stick with the music theory class!
posted by bfields at 3:34 PM on June 16 [5 favorites]


You might enjoy this online course: https://www.wondrium.com/how-music-and-mathematics-relate
posted by danceswithlight at 3:59 PM on June 16


Bill Sethares is I think exactly your jam, right on the nose. He's an EE and music theorist and you might want his $140 book but you can get a long way on what he's put online and his published papers.

His analysis of scales for inharmonic instruments like gamelan metallophones is one of those things that are mind-bendingly obvious in hindsight.
posted by away for regrooving at 4:06 PM on June 16 [1 favorite]


Or if I have slightly misread your jam, John Chalmers may be on its nose instead. His book Divisions of the Tetrachord is online, about the math of different tunings for the harmonic series. You may appreciate that Chapter 1 is about contemporary experimental music before he dives back to Pythagoras.
posted by away for regrooving at 4:12 PM on June 16 [1 favorite]


You can explore Miller S. Puckette's books, lectures, and software. He is the guy who wrote Max/MSP and pd.
posted by niicholas at 4:26 PM on June 16


Music theory is far more a cultural than a mathematical topic in contemporary scholarship. The reason it’s taught the way it is is that it is taught as a developmental sequence analogous to the development of western art music. It isn’t at all true that “music before Mozart was just tonic and dominant,” for example, and to grasp why you’d have to understand the transition from “modal” to “tonal” composition in the context of the Renaissance and parallel movements in architecture and art history (perspective in visual art is almost exactly analogous, right down to being connected to a shift in patronage from church to landed aristocracy). You’d have to grasp equal temperament and uniform tuning as signs of an emerging era of a science and transnational capitalism. All the organs being tuned the same is like all the computers running the same software now.

The founding text of modern ethnomusicology is Alexander Ellis’ 1885 “On the musical scales of various nations,” which advanced the first scholarly case that the world’s many musical scales are culturally explicable, not universal or stages of a universal evolution toward any particular scale system.

The very idea of “music theory” as an abstract discipline that can be separated from “musicianship” at all is essentially a mid-19th century German thing, and equally cultural for that. How German 19th century romanticism and scientism came to shape modern music curricula all over the world is another story.

You may be interested in recent books by Mariusz Kozak and Benjamin Steege, both of whom take a historical view of how “music theory” came to exist and have the strangely acultural character it has had through most of the last century, and whence it derives its largely pseudoscientific pretensions to mathematical abstraction and mentalism. You may also be intrigued by the connection of this history to white supremacy, as argued in Philip Ewell’s bombshell recent paper “Music Theory and the White Racial Frame” in MTO, easily readable online as its blown up the entire conversation in the last year.
posted by spitbull at 5:46 PM on June 16 [10 favorites]


You might enjoy joining the Music Theory Reddit. They often have discussions similar to the one you are having here (another that is pretty on the nose for y oru question), and sometimes good resources and alternative approaches are mentioned.

Here is a discussion about where to start studying music theory with various resources mentioned (and another).

The Reddit Music Theory FAQ is also very useful and might address many of the questions you have from a practical perspective, and provide some alternative viewpoints and resources to consider. There is also a good list of resources for people wanting to learn music theory.

Also you could pose this question there--the responses would definitely be worthwhile.
posted by flug at 5:53 PM on June 16 [1 favorite]


By the way, my credentials include having taken both a year of modal counterpoint and a seminar in set theory from David Lewin. I’m an ethnomusicologist who is very critical of a great deal of music theory as pseudoscience, especially on its border with cognitive science and neuroscience these days. It is not that musical *sound* cannot be mathematically modeled. Of course it can. It’s a physical entity in the material world, subject to the laws of physics and the constraints of biology. But there is no such thing as “Music” in the abstract, detachable from culture, history, or human agency. Musicking, in Small’s famous term, is intentional communicative behavior, and listening is as much a part of it as making sounds. We listen from culturally and historically situated positions, just as some of us compose from such positions. It is no accident that the core western music curriculum is centered almost exclusively on the musical languages of European white men who enjoyed elite patronage in their times, and that the dimensions of musical sound we foreground as central to musical meaning in “music theory” curricula (battles over which I’ve fought for much of my career) are derived from that narrow canon.

I heartily second looking sideways at acoustics and psychoacoustics on the one hand, and linguistics on the other. Both areas of study profoundly shaped me as a music scholar and a musician. Indeed, I became a linguistic anthropologist because the aspects of musical sound in cultural context that interested me most — how and why people sing like they do — was just barely centered at all in the music departments of my grad school years.
posted by spitbull at 6:04 PM on June 16 [8 favorites]


The other thing that freed my mind so much was deeply studying non-western music theoretical systems, for me raga and tala in Hindustani music, but almost any other tradition will have the desired mind-opening effects.
posted by spitbull at 6:09 PM on June 16


And so then you get to confusing things like "is this note called A-sharp or B-flat?"

The logic here is that, in Western music, there are twelve notes that are expressed using only seven letters (A, B, C, D, E, F and G), each diatonic scale uses exactly seven notes, and, crucially, each letter must be represented in the scale for any given key. So the decision to describe a note as either A-sharp or B-flat depends on whether there is already another note in the scale that is an A or a B.

For example, in the F major scale, the notes are F, G, A, Bb, C, D and E. We use Bb and not A# because there already is a note for A and it gets confusing if there are two. If the scale was written as F, G, A, A#, C, D and E, then every time an A note is written, the composer would have to indicate whether it is an A or an A#, and if the song had a chord that used an A and the A# that is only a half-tone above, it could not be expressed clearly in the written form. But if we write the notes as A and Bb, the problem is eliminated. The scale has seven notes and each of the seven letters are included.

This also explains why you might sometimes see E# and Fb, which can be even more confusing at first glance. But the logic is simple: The key of F#/Gb requires that six of the seven notes be accidentals, and so must be written as:

F# major scale = F#, G#, A#, B, C#, D#, E#
Gb major scale = Gb, Ab, Bb, Cb, Db, Eb, F

So if you see an E# or an Fb, there is a good chance you are looking at a piece of music in this key signature.
posted by obscure simpsons reference at 10:04 PM on June 16 [1 favorite]


I want to correct myself. Globally, the abstraction of “music theory” from “music practice” is found in many cultures, not just the modern west.
Ethnomusicologist Hugo Zemp argued that all cultures have music theory, convincingly. (How else do you know how to build a flute that plays in tune?) But its elaboration into an abstract discipline appears in societies with division of labor, cities, literacy, and professional classes of artists (and scholars) employed by patrons — thus for example China, India, Japan, and the Islamic world (the source of modern western music theory) all have long traditions of formal music theory, often as a mathematical discipline, often completely detached from contemporary musical practice.
posted by spitbull at 1:54 AM on June 17 [1 favorite]


By the way imagine a “music theory” curriculum that centers rhythm or timbre or dance or texture instead of pitch and consonance/dissonance and functional harmony.

Imagine a scale that isn’t an equal division of the octave. Now imagine that each octave in that scale has a different intervallic structure. Now imagine that each village has its own version, and tunes elaborate fixed-pitch metal gong orchestras accordingly. Welcome to Java. And now you know why it’s been an ethnomusicological obsession at least since Jaap Kunst wrote the book that gave the discipline its name in 1955. Imagine thinking of melody in terms of tetrachordal divisions of the octave that permit modulations between modes (Arabic music). Imagine a concept of melody and drone that has no use for theorizing vertical sonorities as “chords” in functional relationships of suspension and resolution. Welcome to India and Pakistan. Imagine a music in which repetition of rhythmic cycles and tempo modulations carry most of the musical meaning? Welcome to African diasporic music, arguably the single most influential music “theoretical” system in our modern world (“riff based music” as some call it). Imagine a music in which blocks of sonorous texture provide the main structural elements, and welcome to the ancient form of Japanese Gagaku. Or in which the whole point of music is to induce physical ecstasy through repetition of physical movement, and you are talking about many forms of Sufi music. Or in which musical sound is not evaluated for its “aesthetic” properties at all, but for its functional efficacy in ritual, and you have many of the world’s Indigenous traditions of making what westerners call — with false confidence in its obvious universality — “Music” with a capital “M” (as I teach this to students, a distinction I have found very useful to port over from linguistics where we clearly understand that all “languages” are examples of the same underlying “Language”).

The western chromatic scale is cultural. Not natural. One of thousands of possible options humans have developed in their cultures into a “theory” to explain musical practice. Same for equal temperament, absolute pitch, and functional harmony. None of these are “natural” expressions of a “universal” faculty for music in our species.

In fact, as a linguist (and pro musician) I maintain that the analogy of “Music” to “Language” is ultimately false and spurious. We know Language is a modular, discretely evolved function of the human mind/brain, in Chomsky’s famous term. We have NO SUCH CONFIDENCE aboutMusic.” And many decades of efforts tl prove the analogy holds have been unconvincing to me. Indeed, I flatly assert that what they mean is not “Music,” but “singing and dancing” as things all humans do, and I simply believe singing and dancing are dimensions of Language as such, not requiring or even wanting a separate conceptual abstraction (“Music”) to explain. Rather, that concept has been used to obscure the common human behavior of singing and dancing. “Absolute Music” is a distinctly ideological and cultural abstraction of social elites in literate and stratified societies.
posted by spitbull at 2:24 AM on June 17 [4 favorites]


Sorry to pepper this thread, obviously my heart is in it.

Wanna blow your mind? Read the great German sociologist/historian Max Weber’s “The Rational and Social Foundations of Music,” which when written in 1921 clearly linked the emergence of western tonal practice, equal temperament, and instrument design to the rise of capitalism. In 1921.

Few music scholars read Weber. All social scientists do. It took western musicology 70 years to realize the same thing.
posted by spitbull at 2:34 AM on June 17 [2 favorites]


(Sorry, Weber wrote RSFM in 1914, he died in 1920, it was posthumously published in 1921……/scholarlyparticulars )
posted by spitbull at 2:56 AM on June 17


Nthing Harmonic Experience. And do the singing exercises in it which give you a primordial encounter with the pure intervals. You shouldn't need more musicianship training for this than what Mathieu gives you. His aim in the book is to show how the meaning of equal temperament scales and harmony is derivable from just intonation a.k.a. pure intonation. I think he is pretty convincing by-the-by in arguing that the division of the octave into 12 pitches is neither the first division nor an inevitable result; though he does think it is a super useful one for composers.

It's a book that should dovetail nicely with any works on the physics of sound and acoustics you might choose to look at.

If you want to read some very thorny theory by a guy who did think the twelve tones were inevitable and that free twelve-tone music superior to everything that came before it in a melancholy truth-telling way, read Theodor Adorno's Philosophy of Modern Music. (There's no going back to tonal music, he relentlessly insisted while sipping tea at the Hotel Abyss.)
posted by bertran at 4:03 AM on June 17


Probably the reason you can't find the curriculum you want is because western music didn't start that way. It started with instruments that favored some intervals over others and keys that had different moods. For example, the flutes of Bach's and Mozart's days had much less keywork than a modern flute, and some sharps and flats had to be played using obscure fingerings resulting in problems with tone color and pitch. It wasn't until about 1850 that there was a flute that could be said to play in all keys equally well.

On the other hand, I suspect most any music primer is going to get to the circle of fifths pretty early. From what I've seen, that's the basis of modern music theory and mostly assumes the equal interval scale.

The book I have is The Musician's Guide To Theory And Analysis. By Clendinning and Marvin. Its tedious in its detail, too fat and too heavy in the way of modern textbooks.
posted by SemiSalt at 5:29 AM on June 17 [2 favorites]


I'd suggest "The Musician's Guide to Perception and Cognition" for the part from physics of acoustics to physiology to cognitive aspects of music. It is biased toward European music.

...normal music theory is taught from a sort of historical construction, taking the codified music of Bach's era as the basis of Western music...

You are right here. Bach's works are explorations of the raw psychoacoustic material you are thinking of within a frame of historical conventions. Western music accumulates new conventions mined from this raw psychoacoustic material as it gone along.

...starting with something like serialism or the 12 tone scale, then goes back to classical Western music as a special case...

As you learn about the practices of serialism and 12 tone music you will find they are not some return to a root source of music.
posted by bdc34 at 8:04 AM on June 17 [1 favorite]


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