Math and Music: The Deeper Links (1982)
nytimes.comI’m working on this software for writing music that uses some cool math to make you more productive. I’m close to being done. Check it out http://ngrid.io
Also recommended: two-volume set "Musimathics" by Gareth Loy.[1]
The work's topics are, mostly, independent so I find it enjoyable to pick up and read a chapter from time to time. He tries to cover the math of everything musical from scales and composition to synthesis and signal processing to acoustics and physics.
Is there something in the books about the mathematics of chords and chord progressions?
IMO, the books are great but they don't go very deeply into the math of chords or relationships between chords, no. Certainly there is math coverage of intonation (JI, 12TET, etc.), which leads to "why" chords might sound the way they do as stacked intervals with more or less pure dissonance or consonance. But the "why" of progressions are a level above that yet again. So I'm sure there are better works on chords. From a guitar (but also keyboard) perspective a fascinating book is Werner Pohlert's "Basic Harmony" -- it's thick and analytical of chords and progressions. It's comprehensive though I suppose not particularly mathematical. And you probably know there is some great writing on Just Intonation which is rooted in the ratio math of "correct" intervals and chords. "The JI Primer" by David Doty is a nice short starting point on JI with references, with emphasis on the math of intervals, obviously, but it does have some coverage of stacking these into triads and beyond. If you find a pleasing rabbit hole for chord and progression math, let us know!
Thanks for a detailed response. I'll check the references out.
I am a lot less knowledgeable than you have imagined here. :-) Well, I know practically all the physics and electronics part, but have not found much that connects to music theory. I could figure the mathematical why's of scales and chords ("stacked intervals with more or less pure dissonance or consonance") by myself. But ever since have been struggling to find about which chords/progressions would fit which melody. Most musicians are doing this naturally, "by the ear" as they say. :-) And music theory books I have looked at so far (including the thick ones) do not talk about mathematics at all. :-( :-)
I hear you. My $0.25 ... I bet the expert authors of many music theory books would be capable of thinking in mathematical terms but (now I'm guessing) there is likely some undeserved "ew, yuck, math!" culture in the arts so rather than turn off their audience, they avoid talking about the quantitative underpinnings of why things sound the way they do.
Two more enjoyable books on the math and physics of music (though, again, probably not far enough up the tree of abstraction for chords):
"The Science of Musical Sound" by Pierce (lovely little book, not too deep though, quite coffee tableable)
"Fundamentals of Musical Acoustics" by Benade (old book, considered a classic, reads like a science text)
Good luck!
Thanks a lot! :-)
The last paragraph has something lovely to take home:
"Stravinsky, in discussing ''the art of combination which is composition'' quoted the mathematician Marston Morse: ''Mathematics are the result of mysterious powers which no one understands, and in which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.'' Morse, Stravinsky says, could as well have been talking about music. It is not only in the clarity of things, but in their beauty and mystery that the two arts join."
there is infinite complexity in both, i think this is part of the allure of music for me. i keep realizing the “true” nature of something only to later discover an even deeper truth. repeat forever
Very true. People who try to reduce music to math rarely succeed.
It looks like it's a simple problem, then you realise there are edge cases, then you realise the edge cases are where all the interesting detail is, then you realise your models are braindead and actually kind of useless, and then maybe you start again with a better model.
Repeat forever.
I also liked his quote that "the musician should find in mathematics a study 'as useful to him as the learning of another language is to a poet.'"