This appears to be a mediocre piece of undergrad "research", which is to say, the theoretical material is entirely plagiarized without citation. Even the applied analysis (Beethoven's 9th, 2nd mvt. mm. 143-172) is directly lifted from another work.
The application of group-theoretical ideas to music theory (in particular, the rules underpinning voice leading) is quite interesting, however! People have been interested in the link between symmetry and musical beauty since the time of the Pythagoreans. Looking through the modern lens of group theory shows a delightful simplicity: if you look at the world of musical operations in this way, the ones that sound best are often small deviations away from maximum symmetry.
There are far better places to start, if you're interested, covering much of the same (plagiarized!) material:
I know you've been in the music business for a few hundred years, Josquin, but there's no need to be so harsh about this undergrad poster! From what I could figure, it's something like a final project for a class on Mathematics and Music at UW Eau Claire. The background material on transformations of sets of pitch classes is in the course textbook. (This doesn't take away from all the citations you helpfully provided, though.)
I'm trying to decide if I should go down this rabbit hole. The math looks easy enough, but I know nothing about music. Nothing. And I'd love to learn, of course, but I can see myself buying a keyboard and plinking out some of these transformations and that seems like a long time to go without a payoff.
I've been down this rabbit hole for a few years, except making procedural music with code rather than physically playing an instrument. No musical background at all (I started by googling "chord" etc).
My take is, the basics - the math behind chords and chord changes, summarized on the left page of the PDF - are embarrassingly simple; you can know everything there seems to be to know in a weekend or two. However for the application - using those basics to create or understand music - I'm still not entirely sure there's anything to know.
That is, one can find scads of sources online similar to the right-hand page of the PDF, which analyze the structure of some phrase or composition. But it all seems to pretty much boil down to giving names to things; there don't seem to be any generally applicable principles. The closest thing we have seems to be voice leading rules, but they're very much "doing X sounds good except when it doesn't" sorts of things.
No particular sources, but topic-wise what made things click for me was roman numeral notation and secondary chords. Or rather, implementing the math behind those things made me feel like I understood what was going on, and that's the nomenclature/data structure I stuck with. But the notation in the PDF is equivalent - it all boils down to having an array of numbers and then either reindexing the array or adding to the elements modulo 12; everything beyond that is just nomenclature.
The only thing difficult about it is finding sources that just explain the (extremely simple) math without tons of opaque terminology. E.g. here's the second sentence of wikipedia's definition for "secondary chord":
> Because tonic triads are either major or minor, you would not expect to find diminished chords (either the viio in major or the iio in minor) tonicized by a secondary dominant.
Blech. For years I've meant to write a bare-bones "music for programmers" article but haven't done it yet. There are other such articles around but I never saw one I was enthusiastic enough about to bookmark.
For dev setup, if you're a browser/JS person I recommend Tone.js, which is a very short path from zero to "here's an array of integers, play quarter notes for their midi values". I outgrew it at some point and now roll everything by hand, but I probably shouldn't have.
> The math looks easy enough, but I know nothing about music. Nothing. And I'd love to learn, of course, but I can see myself buying a keyboard and plinking out some of these transformations
read musictheory.net, get musescore, and then you can start experimenting with some of this. don't need a keyboard.
Was just reading on this topic while learning Arvo Pärt's Spiegel im Spiegel, and couldn't help interpret it as being heavy on critical-woo and light on functional analysis.
(see linked papers mystical treatment of a for loop.)
People talk about his Tintinnabuli style as being algorithmic and generated, but this particular paper wasn't persuasive, and reading the score while listening to it, I don't think it is. There are probably some symmetries encoded in his work, given how demanding they are on your attention, but the idea of discovering hidden meaning is itself superstitious.
However, the idea of reasoning about relationships and intent in music could have some applications to reasoning about code, and machine language and behaviour in particular. So I'm not entirely dismissive of the poster, just a bit wary of some of the baggage the ideas may have collected along the way.
The "complex" math in this poster disguises the simplicity of these chords and transitions.. If you're really interested in music theory (and why those transitions sound the way they do), "The Songwriting Secrets of The Beatles" is one of the best sources available today.
Sounded interesting, so I started reading it.. (jazz pianist here, I learnt basic harmony in large part from playing Beatles songs as a kid) Hmm yeah, it's entertainingly written. I'd never heard the term "Three Chord Trick" (i.e. songs using just I,IV,V.)
In "The power of I - the one-chord trick", Tomorrow Never Knows is described as "paradoxically structured around a single, simply embellished G chord, around which hypnotic modal melodies work their textural magic". Uh, no. It's in C, and there's distinctly a Bb/C chord in bars 5 and 6 of an 8 bar pattern - kind of like the IV chord in the first 8 bars of a blues. (Also, come to think of it, like the standard pattern in North Indian classical of not playing open tabla sounds in bars 5 & 6 of each 8 bars)
Then "The Chuck Berry rhythm", "a rock'n'roll guitar pattern popularised most famously by Chuck Berry"..turns out to be the boogie-woogie piano left hand pattern, "a two-note diad...a basic root-and-fifth perfect interval...From there, a brief extension from the fifth to the sixth degree (and sometimes on to b7) can be made"–he makes it sound like rock guitarists invented it. Anyway, Rosetta Tharpe invented most of what Chuck Berry "invented", decades before he did it, the rock-n-roll guitar and the dance moves. Check her out! (on youtube) She's amazing.
It looks like the author is applying Set Theory [1] and concepts from 12-Tone [2] to tonal music. It is an interesting exercise - though I would expect that this music essentially boils down to "Tonic -> Domainant" relationships. Schenker would be proud :)
The application of group-theoretical ideas to music theory (in particular, the rules underpinning voice leading) is quite interesting, however! People have been interested in the link between symmetry and musical beauty since the time of the Pythagoreans. Looking through the modern lens of group theory shows a delightful simplicity: if you look at the world of musical operations in this way, the ones that sound best are often small deviations away from maximum symmetry.
There are far better places to start, if you're interested, covering much of the same (plagiarized!) material:
https://www.math.drexel.edu/~dp399/musicmath/algebraicmusict...
https://sites.math.washington.edu/~morrow/336_09/papers/Ada....
https://alpof.wordpress.com/category/music/math-music/neo-ri...
http://www-personal.umd.umich.edu/~tmfiore/1/mathmusiccolloq...
https://www.maa.org/sites/default/files/pdf/upload_library/2...
Or, of course, follow the citation chains on Wikipedia:
https://en.wikipedia.org/wiki/Neo-Riemannian_theory https://en.wikipedia.org/wiki/Transformational_theory