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A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford Studies in Music Theory)

Dmitri Tymoczko · 4 HN comments
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Amazon Summary
In this groundbreaking work, author Dmitri Tymoczko describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock. Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. In the process he sheds new light on an age-old question: what makes music sound good? A Geometry of Music provides an accessible introduction to Tymoczko's revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing relationships among familiar chords and scales, giving readers the tools to translate between the musical and visual realms and revealing surprising degrees of structure in otherwise hard-to-understand pieces. Tymoczko uses this theoretical foundation to retell the history of Western music from the eleventh century to the present day. Arguing that traditional histories focus too narrowly on the "common practice" period from 1680-1850, he proposes instead that Western music comprises an extended common practice stretching from the late middle ages to the present. He discusses a host of familiar pieces by a wide range of composers, from Bach to the Beatles, Mozart to Miles Davis, and many in between. A Geometry of Music is accessible to a range of readers, from undergraduate music majors to scientists and mathematicians with an interest in music. Defining its terms along the way, it presupposes no special mathematical background and only a basic familiarity with Western music theory. The book also contains exercises designed to reinforce and extend readers' understanding, along with a series of appendices tha
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You really went to town on that!

After studying a basic amount of music theory, I have come to the conclusion that there is no such thing as canonical analysis. Analysis, in most theories, is a way of rationalizing what's become common practice in a given style.

One thing I learned when trying to research the origins of our music system is that our chord progressions appear to be underpinned by rules for multi-voice counterpoint. But in pop music, voicing isn't particularly important, it's just background for the bass and melody lines. So pure chord progressions tend to harken back to harmonic motion that's familiar from the more formal genres. A book I often recommend called A Geometry of Music [1] goes way down the rabbit hole, if you want to obsess over this to the extent that I did.

So, modal chord progressions (especially Locrian) really only tend make sense to the extent that they analogize major and minor chord progressions, from a functional tonality perspective. I don't think this is precise, but the way I think of it is that you sell the listener on alterations of major or minor scales, and they buy it after a few repetitions, and other signals of what the cadences are :)


Sep 19, 2016 · acjohnson55 on Music theory for nerds
This is an interesting article, which touches on a lot of interesting musical topics. However, it's wrong on so many levels that I would never recommend it to someone who wants to understand music theory. I'll try to summarize.

I spent a long time researching where Western music "comes from". One important thing to understand is that the key feature of Western music is harmony. This is not the case for most other musical traditions. Harmony comes from a layering of independent melodic lines (called voices), such that they evolve in ways that create and resolve tensions. This is called counterpoint.

The ways in which these tensions resolve are called cadences, and classical music was all them. Musical form arises around how a piece of music is organized to have an arc (and arcs of arcs) that leads to a cadence. It's kind of like how there are only a few basic narrative structures in storytelling, yet infinite variations in types of stories.

The tension between the sounds in concurrent melodic voices emerges from the structure of the sounds. Harmonic sounds, like ones produced from wind and string instruments, sound good (consonant) when their fundamental frequencies are simple ratios. This is largely because of how the human auditory system works.

Our scales and chords come from picking collections of notes that approximate simple ratios. We deal with approximate ratios using nth-roots for two reasons. First, simple ratios don't compose well -- stack them (i.e. multiply them) and you get ratios that aren't so simple. nth-roots do stack nicely. Second, it lets us merge nearby flat and sharp notes. This wouldn't be possible if we used real ratios. That merger, known as enharmonicity, lets us do all sorts of cool compositional tricks that are generally considered more worthwhile than the slight improvement in sound quality we'd experience by using the real ratios. There are various ways to pick just which notes merge, but by far the most popular approach is to use 12 notes.

So, this is where we got our notes and scale from. This happened around the year 1800, give or take a couple decades.

The history of Western music since that point in time largely revolves around coming up with new ways to play with those 12 notes, in many ways erasing the musical features that caused the 12 notes to emerge in the first place. This is why the system of note names, with the flats and sharps, feels rather arbitrary today.

Unfortunately, there are not a lot of great resource that tell this full story. The two books I found most enlightening in piecing together the "real story" are:

- Tuning, Timbre, Spectrum, Scale: - A Geometry of Music:

> This is not the case for most other musical traditions.

Do you have examples of musical traditions that don't do this? Both the Indian and Chinese musical systems do something similar.


I'm not really sure if this post is "wrong". It gets the chronology mixed up a bit -- talking of twelfth roots of two before talking about ratios, but I thought that was just done to help explain things by reverse-engineering them. I don't think it claims that the notes "came from" roots of two.

I'm not an expert, but Indian classical music isn't known for having harmony in the same sense [1]. Modern Indian music is influenced by Western music. I know even less about Chinese music, which isn't mentioned in that article. But in all the sources I've read, the concept of deeply developed chordal harmony is pretty unique to the Western music tradition.


Right, it doesn't involve playing notes together the same way it's done in Western music.

I guess I misinterpreted what you were saying -- the Indian notes are still derived by looking at notes which sound harmonious when played together, roughly the same way Western notes are derived. They are not usually played together on the same instrument like you would with a piano or a guitar. Harmony between vocal parts is not uncommon as is harmony with background instruments like the tanpura. But yes, Indian music doesn't have harmony in the same sense as Western music. But the origins of the notes are very similar.

(I'm talking about classical Indian music, btw)

Yep! Many traditions got their notes the same way Western music did. The Tuning, Timbre, Spectrum, Scale book does a great job illuminating this.
Ooh, that sounds like a fun book to read!

I spent a lot of time studying this myself because I was fascinated with the fact that these 12 notes in particular seemed ubiquitous in many music systems, and that there wasn't any immediately obvious reason as to why these notes sounded pleasing. Ultimately learned some music theory in the process (I've studied Indian and some Western music, so a lot of it was stuff I already knew, but not in concrete form), but never in a very structured fashion -- more along the lines of asking people and reading snippets.

Put this book on my list, sounds like a structured way to learn something I've been super interested in for years.

Are you thinking of A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice? I learned about that book here on HN from another participant.

I don't think that's the one I'm recalling, though that title is interesting.

I'll have to ask him. I'll probably speak with him this weekend.

P.S. I hope the OP will take a look at tokenadult's title. It might be a good choice.

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