L

Laegna Math

frequency calculator

Open lab

∞ · base-4 floating point · octave emulation

Calculate the frequency of infinite octaves

Analyze Laegna conceptions with classic math. Compose and decompose the Z, X and Y axes — each reflects one number from a different aspect, fourier-decomposed into separate dimensional factors.

base f₀ · hz
Hz
position p
p
p sweep · 1.0000
fractal depth
2
driver axis
f₀ presets

mouse: drag or scroll a field · shift ×10 · alt fine · alt+shift ultra-fine · ctrl ×100
keys: Z X Y axis · C compose/decompose · 1 2 3 depth · ← → p step (shift = whole octave) · ↑ ↓ f₀ octave · R reset

octave analyzer

base-4 · I O A E

Z · logarithmicX · linearY · exponential
I · innerO · outerA · apexE · edge

axis readout

Z

Yin · differential 1

p 1.0000 → 220.000 Hz

X

Dao · integral 0

p 1.0000 → 220.000 Hz

Y

Yang · integral 1

p 1.0000 → 220.000 Hz

f(X) at p 1.00 →220.000

derived quantities · axis X · f₀ · p

units · intermediates

frequencyf
220.0000Hz
frequencyf / 10³
0.2200kHz
angular frequencyω = 2πf
1382.3rad/s
periodT = 1/f
4.5455ms
periodT · 10⁶
4545.5µs
wavelength (air)λ = c/f, c = 343 m/s
1.5591m
wavenumberk = 2π/λ
4.0300rad/m
ratio to f₀r = f/f₀
1.0000×
binary octaveslog₂ r
0oct
base-4 octaveslog₄ r
0oct₄
interval from f₀1200 · log₂ r
0cents
level ratio20 · log₁₀ r
0dB
decadeslog₁₀ r
0dec
nearest note69 + 12·log₂(f/440)
A312-TET
base-4 positionaxis argument
1.0000p
Nyquist ratefₛ ≥ 2f (sampling theorem)
440.0000Hz

intermediate calculation per axis

axislawpf · Hzω · rad/sT · msλ · mcentsnotebase-4
Zf₀ · (1 + log₄ p)1.0000220.0001382.34.54551.55910.0A33.1300₄ · 4³
Xf₀ · p1.0000220.0001382.34.54551.55910.0A33.1300₄ · 4³
Yf₀ · 4^(p−1)1.0000220.0001382.34.54551.55910.0A33.1300₄ · 4³

Wavelength uses c = 343 m/s (dry air, 20 °C); cents are measured against f₀; notes are 12-TET with A4 = 440 Hz.

Three basic frequencies

Zf₀ · (1 + log₄ p)

Differential axis

220.000Hz

Fourier compound in base-4 at differential level 1 in infinity. Grows through logarithm while X stays linear.

Yin · 3.1300₄ · 4³

Xf₀ · p

Linear axis

220.000Hz

Zeroeth integral or differential, growing linearly. The Dao — the fixed reference every other axis reflects against.

Dao · 3.1300₄ · 4³

Yf₀ · 4^(p−1)

Integral axis

220.000Hz

A single part of whole-octave growth in X reads as fourier decomposition. Exponential, accelerated, growing to infinity.

Yang · 3.1300₄ · 4³

∞

Compose & decompose

Every axis reflects the same number from a different aspect, or fourier-decomposes into separate dimensional factors — one factor, one frequency change.

base-4 fractal division · axis X

I

220.000

O

275.000

A

330.000

E

385.000

Each basic frequency divides into four parts — the fractal cell of the whole-octave system.

Base-4 fractal readout

axis X · depth 2

cellpositionhz

Sources & further reading

Fourier analysis

The Z axis is a fourier compound: a whole-octave step in X read as a sum of harmonic parts.

classic source

Théorie analytique de la chaleur

Joseph Fourier · 1822

Original statement of decomposing any periodic quantity into sines and cosines.

modern reading

Gauss, normal curve & least squares

Decomposition of one octave into four parts distributes error; Gauss gives the classical law for it.

classic source

Theoria motus corporum coelestium

Carl Friedrich Gauss · 1809

Introduces least squares and the normal law of errors.

modern reading

Sampling, spectra & infinity

Base-4 floating point positions are sample points on an infinite octave line.

classic source

Communication in the Presence of Noise

Claude E. Shannon · 1949

The sampling theorem: fₛ ≥ 2f, the bound behind every Nyquist readout here.

modern reading