Polynomial (x^2 + 1)
p0-d2-w0-c0- root note
- C3
- tempo
- 74 bpm
- voices
- 2
- waveform
- sine
Polynomial. No transcendental work, no chain. Pure ground tone in a sine.
Sound of Math
Each formula gets a note. The note's pitch, timbre, and ornament come from the formula's Pfaffian cost profile, not from any hand-tuned mapping. Two formulas in the same cost class sound the same. The Stefan-Boltzmann law and a polynomial both ring as a pure C3 sine. A formula that uses Bessel J0 or erf gets a high bell ping because it has stepped outside the elementary family.
Mapping
- pfaffian_r → pitch (log). Higher chain order sits higher on the keyboard.
- predicted_depth → tempo. Deeper formulas play faster.
- eml_depth → chord voices. One voice for trivial depth; up to four for stacked transcendentals.
- witness count → timbre. Sine, triangle, sawtooth, square in increasing brightness.
- is_pfaffian_not_eml → bell ping. The teaching moment: the formula has left the elementary family.
exp(x)
p1-d1-w1-c0- root note
- E3
- tempo
- 67 bpm
- voices
- 1
- waveform
- triangle
One chain element. The triangle's single warm overtone matches one transcendental layer.
sin(x)
p2-d3-w2-c1- root note
- G3
- tempo
- 81 bpm
- voices
- 2
- waveform
- sawtooth
Two chain elements. The brighter sawtooth captures the second transcendental layer.
cos²(x)
p2-d4-w2-c1- root note
- G3
- tempo
- 88 bpm
- voices
- 3
- waveform
- sawtooth
Same chain as sin, one extra structural layer from the square. Three voices instead of two.
Gaussian e^(-x^2)
p1-d3-w1-c0- root note
- E3
- tempo
- 81 bpm
- voices
- 2
- waveform
- triangle
One chain element wrapped in structure. Triangle stays warm; tempo lifts because depth is 3.
sigmoid 1/(1+e^-x)
p1-d1-w1-c-1- root note
- E3
- tempo
- 74 bpm
- voices
- 1
- waveform
- triangle
Sigmoid is one chain element under F-family fusion. Same warm tone as exp because the substrate hears the fused operator.
Stefan-Boltzmann (T⁴)
p0-d1-w0-c0- root note
- C3
- tempo
- 67 bpm
- voices
- 1
- waveform
- sine
A famously deep physical law. The sound shows it as a polynomial: pure C3 sine, the same tone as x² + 1.
Planck spectral radiance
p1-d3-w1-c0- root note
- E3
- tempo
- 81 bpm
- voices
- 2
- waveform
- triangle
Single exp inside a polynomial wrapper. Same cost class as a Gaussian; the sound is the same shape too.
GELU (uses erf)
p3-d5-w3-c0- root note
- A3
- tempo
- 102 bpm
- voices
- 3
- waveform
- square
- ornament
- bell ping (Pfaffian-not-EML)
GELU contains erf, which lives outside the elementary family. The bell ping marks the boundary; the square chord sits high.
Bessel J₀
p3-d1-w3-c0- root note
- A3
- tempo
- 81 bpm
- voices
- 1
- waveform
- square
- ornament
- bell ping (Pfaffian-not-EML)
Bessel functions are Pfaffian but not elementary. Same outlier ping as GELU; tempo is calmer because EML depth is just 1.