- 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
mod 13= circle
- (equivalence classes of octaves —
- 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
Posts tagged with affine transformation
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 grade 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.
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.