Timbre Is EML Node Count
The EML operator encodes Fourier synthesis exactly:
Im(eml(i·2πft, 1)) = Im(exp(i·2πft) − ln(1)) = Im(e^(i·2πft)) = sin(2πft)One complex EML node = one partial tone. Timbre complexity = number of EML nodes.
The Identity
For any frequency f and time t:
eml(i·2πft, 1) = exp(i·2πft) − ln(1)
= cos(2πft) + i·sin(2πft) − 0
= cos(2πft) + i·sin(2πft)Taking the imaginary part:
Im(eml(i·2πft, 1)) = sin(2πft) [1 EML node]A sum of N harmonics is exactly N complex EML nodes evaluated in parallel.
Timbre Complexity Table
We measured minimum EML nodes needed to reach −40 dB spectral flatness for published instrument harmonic profiles:
| Instrument | Min nodes | Dominant harmonics |
|---|---|---|
| Sine | 1 | H1 |
| Clarinet Bb | 5 | H3, H5, H7 (odd harmonics only) |
| Violin A | 12 | H1–H12 (full spectrum) |
| Piano A4 | 12 | H1, H2, H4 (stretched partials) |
| Bell | 7 | H4, H6, H1 (inharmonic ratios) |
The clarinet’s low complexity (5 nodes) is because it produces primarily odd harmonics. Bell is special: the “harmonics” are inharmonic (non-integer frequency ratios), so EML nodes must be tuned to irrational frequency multiples.
Session S3: 8 Operators as Audio Effects
We applied all 8 operators to a 440 Hz + 880 Hz stereo signal and measured the output:
| Operator | Character | Notes |
|---|---|---|
| EML | Harsh distortion | exp(A) amplifies peaks exponentially |
| DEML | Dynamic compression | exp(−A) inverts peaks, softer |
| EMN | Mirror of DEML | Inverted polarity, similar dynamics |
| EAL | Additive richness | exp(A)+log(B): rich harmonics + harsh peaks |
| EXL | Ring modulation | exp(A)·log(B): creates sidebands. Most musical. |
| EDL | Buzzy/harsh | Division by ln near zero: discontinuities |
| POW | Gated/tremolo | B^A near A=0: nearly constant |
| LEX | Softest/cleanest | log(exp(A)·B)=A+log(B): smoothest output |
EXL is the most musically useful operator. The product exp(A)·ln(B) creates
frequency sidebands at sum and difference frequencies, exactly like analog ring modulation.
LEX is the mildest. Because log(exp(A)·B) = A + log(B), it behaves like
a compander (dynamic range compressor/expander), smoothing out peaks.
Session S4: Tree-to-Sound Mapping
EML trees as audio primitives:
| Tree | Sound character |
|---|---|
eml(x, 1) = exp(x) | High centroid (1953 Hz), full energy |
exl(0, x) = ln(x) | Pure tone at 440 Hz |
mul(x, x) = x² | Second harmonic only (880 Hz) |
| 4-node square approx | Centroid 1050 Hz, odd harmonics |
| 8-node sawtooth | Centroid 1295 Hz, full spectrum |
| exp(−x) decay | Exponential envelope, bright attack |
Interactive Synthesizer
→ EML Synthesizer — drag harmonic amplitude sliders, click presets (Sine/Clarinet/Violin/Piano/Bell/Sawtooth/Square), hear the result in your browser via Web Audio API. Node counter updates live.
What This Means
The Fourier series is not just analogous to EML — it is EML, evaluated in the complex plane. The minimum number of partials needed to recognize an instrument’s timbre equals the minimum number of EML nodes needed to express it.
This gives a new interpretation of harmonic complexity: it is the EML node count of the timbre, and it is the same number that appears in circuit optimization, routing table savings, and operator completeness bounds.
One number, three contexts.