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

## What’s half of 100 degrees Fahrenheit?

Hint: it’s not 50 degrees Fahrenheit.

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

$\large \dpi{200} \bg_white \begin{matrix} 100 \, ^{\circ} \rm{F} & \longrightarrow & 311 \, \rm{K} \\ \\ && \downarrow \\ \\ -180 \, ^{\circ} \rm{F} & \longleftarrow & 155 \, ^1\!\!/\!_2 \, \rm{K} \end{matrix}$

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.