Posts tagged with affine transformation

Dmitri Tymoczko — author of The Geometry of Music

  • how to make visual representations of music
  • (in paintings, video games, sculpture)
  • 5 constraints on a composition that are necessary (but not sufficient) for it to sound good
  • global statistical properties of songs
  • why 20th century classical music had little audience
  • a random painting is much less offensive to the eye than random notes are to the ear
  • "I came up with these 5 principles using my brain, which is a kind of crude statistical device”
  • the piano is essentially a line
  • [NB: linear ⊃ monotonic ⊃ totally ordered]
  • violin/voice musicians know that notes ⊂ continuous space, but the piano does us a favour by constraining us to a subset of those notes
  • line mod 13 = circle
  • (equivalence classes of octaves — A1=A2=A9 and E4=E7=E12 etc.)
  • directed segments, unordered tuples
  • musical translation = mathematical transposition, musical inversion = mathematical rotation
  • The fact that most people don’t have most perfect pitch (things sound the same in different keys) may be so that we can understand that, despite pitch differences in male/female adults’ speech and children’s speech, they are saying the same words.
  • "It’s as if we couldn’t tell the difference between red and blue, but we were highly sensitive to the-difference-between-red-and-orange and the-difference-between-blue-and-green.
  • [Also: this.]
  • Minor vs major is the other isometry of the circle (besides rotation): reflection.
  • "Harmonic progression is like zone defence"
  • Minute 26: Awesome. Watch how to move around in 2-chord space — seen on a circle and on Tymoczko’s grid

Hint: it’s not 50 degrees Fahrenheit.

100 ℉ = 311 K, half of which is 105.5 K = −180℉


Yup — half of 100℉ is −180℉.

The difference between the Kelvin scale ℝ⁺ and the Fahrenheit scale is like the difference between a linear scale and an affine scale.

You were taught in 6th form that y = mx + b is a “linear” equation, but it’s technically affine. The +b makes a huge difference when the mapping is iterated (like a Mandelbrot fractal) or even when it’s not, like in the temperature example above.

(The difference between affine and linear is more important in higher dimensions where y = Mx means M is a matrix and y & x vectors.)

Abstract algebraists conceive of affine algebra and manifolds like projective geometry — “relaxing the assumption” of the existence of an origin.

(Technically Fahrenheit does have a bottom just like Celsius does. But I think estadounidenses conceive of Fahrenheit being “just out there” while they conceive of Celsius being anchored by its Kelvin sea-floor. This conceptual difference is what makes Fahrenheit : Celsius :: affine : linear.)

It’s completely surprising and rad that mere linear equations can describe so many relevant, real things (examples in another post). Affine equations — that barely noticeable +b — do even more, without reaching into nonlinear chaos or anything trendy sounding like that.