Fascinating paper! Somehow, this escaped my attention when I was researching this topic. I was interested in the question of how the heck we evolved such a crazy system of 7 white notes + 5 black notes (i.e. the diatonic (and pentatonic) scale overlaid on the 12-tone equal temperament), as well as the origin of the Western concepts of major and minor scales and chords.
Long story short, it seems the answer is that our scales and chords derive from a combination of what makes for interesting voice leading and what notes sound good together in a given timbre. The choice of approximating ideal chords and scales using 12 equally spaced notes derives from a instrumental playability concerns and the desire to be able to freely transpose and to modulate between distant keys.
It's a deep topic, but if anyone's interested, I recommend two books:
- Timbre, Tuning, Spectrum, Scale by William Sethares
- A Geometry of Music by Dmitri Tymoczko
Worth noting that this use of the word "combinatorial" in this article is not the same "combinatorial" as used by Milton Babbitt describing a row set characteristic in twelve tone music composition. For further reading on that, https://en.wikipedia.org/wiki/Combinatoriality is as good a start as any.
Tangential, but the quote saying that musicians don't think about counting is wrong. For example, Chuck Berry said "My biggest influence was my mathematics teacher. Music is so much mathematics that it’s pathetic."
Musicians do a lot of counting, just that they stop at four. On the subject of which, music would make a lot more sense if the normal count went 'oh and one and two and three and'.
I've been playing with music and mathematics for a while, and I had a thought that someone far more mathematical and musical than myself could correlate tried and true mathematical constructs to music other than defining patterns via Markov processes, neural or evolutionary algorithms, or simply generating random melodies. Pulling these patterns into 12 point space is really interesting.
I was just turned on to SYZYGYS's music based on Harry Partch's 43-microtone scale. It is hard to stop listening to it today, but maybe because it's novel for me.
Mathematics is all about patterns and relationships, so music is an aural expression of mathematics as has been said by so many others, yet I never tire of seeing (and hearing) examples of this.
Thanks! Will have a read through.
I often wonder if we could develop software to make chord/harmony discovery easier when arranging a melody. At the moment it's down to a combination of knowledge of music theory, intuition and trial and error.
I'm talking about more advanced harmony constructions like Jacob Colliers work (check him out if you haven't heard)-
Harmony (and it's relation to melody) is a time based problem, you can't just join random chords together that match the current melody note. LSTM works for time series learning but I am not convinced it "gets" concepts of music theory, so maybe some kind of hybrid of engineered music theory features + LSTM with good training material could work
Long story short, it seems the answer is that our scales and chords derive from a combination of what makes for interesting voice leading and what notes sound good together in a given timbre. The choice of approximating ideal chords and scales using 12 equally spaced notes derives from a instrumental playability concerns and the desire to be able to freely transpose and to modulate between distant keys.
It's a deep topic, but if anyone's interested, I recommend two books:
- Timbre, Tuning, Spectrum, Scale by William Sethares - A Geometry of Music by Dmitri Tymoczko