The Mathematics of Music: Constructive or Descriptive?

Interesting stuff here! Felt the itch to throw in some more… I guess the birds really sing out of tune, if “tune” means “inside the equal tempered 12 note chromatic scale”. But I guess the birds are just non conformists…

I was just reading the first chapter of the aforementioned “Harmony Explained”, feeling right at home with the reference to richard feynman and the first principles approach. On the other hand, I think the rant about music theory being too mystic is fruitless, as many music theory books don’t claim to be scientific.

I can only recommend “Harmonic Experience” by “W. A. Mathieu” if you want to get into the foundation of harmony (and also some rhythm).

FYI, I’ve been writing some more or less interactive blog posts, for example on Pure Intervals or an algorithmic approach to Chord Scales (and many many more).

I know about Scala :> You can download about 5000 SCL files.

Being an engineer of rather dubious musical talent, I thought I found my safe space in category theory and procedural generation of music … until I ran out of mathematical ability too :>

A few books and people I have come across;

  • “Topos of Music”. Guerino Mazzola.
  • Ervin Wilson. Web. Documentary. App.
  • “Microtonality and the Tuning Systems of Erv Wilson”. Terumi Narushima.
  • Categories of Instruments. Wikipedia.
  • “On Concepts and Classifications of Musical Instruments”. Margaret Kartomi.
  • Microtonal Resources including TUN file collection.
  • “An Improviser’s OS v.2”. Wayne Krantz.

Some of my initial googling around that lead me down this path was;

Something I got from the “Topos of Music” was these figures illustrating the differing perspectives of the creative process.

This made me think documentation is part of the creative process. I started to look into VexFlow and Lilypond. I think music can be more than just an aural experience that should be extended into the production and documentation processes. Perhaps something more user friendly to the creative process than PureData, SonicPi, SuperCollider, etc.

There is always someone better than you and always someone worse. (I won’t explain the mathematics of that here.) Enjoy wallowing in the mire of mediocre then shine bright in the spotlight of occasionally excellence. It is better than burning bright all the time worrying about failing at any moment. Sorry, what were we talking about?

Part of being an engineer involves never letting not knowing what you are doing to stop you from doing it :>

Indeed… I had better put some trousers on and go and fool the normal people that it is difficult.

Sorry, definition, I mean ‘classic blues’ as Robert Johnson and Furry Lewis, etc.

Thanx, I misspelled Scalia for Scala. I got the RP version.

That sounds like what I call Consonance. These numbers evaluate the harmoniousness of the scale.

I argue for the descriptive view. Music existed before refined mathematical descriptions of that music existed.

Finally had time to look at Ian Ring’s ‘A Study of Scales.’

The Major Scale in different mathematical notations:

Decimal 2741
Binary 101010110101
Intervals [2,2,1,2,2,2,1]
Pitch Class Set {0,2,4,5,7,9,11}

The modal relationships between scales define the Grabdeha Groups in the Carnatic Melakarta system.

The 2048 scales are simply 2^11. Elliot’s ‘Group Actions…’ starts with the eleventh row of Pascal’s triangle, and goes on to enumerate the complete permutations of 12 tones, musical or not. Elliot focuses on the Hindustani system of 32 scales, although he also describes the Carnatic system of 72 scales.

Forte vs Rahn. Both systems are technically mathematically correct, but they are unwieldy, unintuitive, and calculating with them is like doing logarithms with Roman Numerals. No musician is ever going to start a scale on zero–ONE has been beaten into us from childhood. ONE, two. ONE, two, three. ONE, two, THREE, four. ONE, two, three, FOUR, five. &c. But I say nothing about rhythm, only scales.

I prefer to use the Intervals, any of the other systems can be derived from them. Also, the Roman Numeral system can be expanded by adding altered tones, as for instance, Vb6 or ii#7, which can be named Melodic Myxolydian and Harmonic Dorian, respectively. This is purely descriptive, not calculable, well maybe.

My intention is merely to introduce the Melakarta system to Western musicians of all levels of skills, from the self taught to the Juliard graduates.

The question remains: which math is most useful for musical software? Probably the Binary. I will, however, try adding scales in Binary. I’ve been working on a group addition (or rather averaging) of scales using Intervals.

But I have convinced myself that the Intervals are more easily understood by humans. It takes a ‘Rain Man’ to think in Binary; Decimal and Pitch Class are also too far abstracted from the actual physical scales.