seam.math.lib

SEAM Math library. Its official prefix is sma.

General-purpose mathematical helpers used across SEAM instruments: angular and frequency quantities, squared trigonometry, angle and coordinate conversions, the golden ratio, combinatorics, and acoustic distance/delay utilities. Only elements not already provided by the standard Faust maths.lib are kept here.

References

Constants and Angular Frequency

Angular constants and helpers that turn a frequency in Hz into the angular quantities used by oscillators and filter design.

(sma.)twoPI

The constant 2*pi (a full turn in radians). Convenience for angular-rate math, where a cycle spans 2*pi radians.

Usage

sma.twoPI : _

Test

sma = library("seam.math.lib");
twoPI_test = sma.twoPI;

(sma.)omega

Normalised angular frequency of fc in radians per sample: omega = fc*2*pi/SR. This is the per-sample phase increment of a sinusoid at fc Hz.

Usage

omega(fc) : _

Where:

  • fc: frequency in Hz

Test

sma = library("seam.math.lib");
omega_test = sma.omega(1000);

(sma.)prewarp

Bilinear-transform frequency pre-warping: tan(pi*fc/SR/2). Maps a target cutoff fc (Hz) to the warped value the bilinear transform expects, so an analog prototype lands on the intended digital frequency. Used by the seam.filters designs.

Usage

prewarp(fc) : _

Where:

  • fc: cutoff frequency in Hz

Test

sma = library("seam.math.lib");
prewarp_test = sma.prewarp(1000);

Trigonometry

Squared trigonometric helpers (e.g. for equal-power laws and identities).

(sma.)cosq

Squared cosine: cos(x)^2.

Usage

cosq(x) : _

Where:

  • x: angle in radians

Test

sma = library("seam.math.lib");
cosq_test = sma.cosq(ma.PI);

(sma.)sinq

Squared sine: sin(x)^2.

Usage

sinq(x) : _

Where:

  • x: angle in radians

Test

sma = library("seam.math.lib");
sinq_test = sma.sinq(3/2*ma.PI);

Angle Conversion

(sma.)d2r

Degrees to radians: d*pi/180.

Usage

d2r(d) : _

Where:

  • d: angle in degrees

Test

sma = library("seam.math.lib");
d2r_test = sma.d2r(90);

(sma.)r2d

Radians to degrees: r*180/pi.

Usage

r2d(r) : _

Where:

  • r: angle in radians

Test

sma = library("seam.math.lib");
r2d_test = sma.r2d(ma.PI);

Coordinates

Conversions between spherical (azimuth, elevation, distance) and Cartesian (x, y, z) coordinates. Angles are in radians; elevation is measured from the horizontal plane (e=0 on the plane, e=pi/2 straight up). aed2xyz and xyz2aed are exact inverses.

(sma.)aed2xyz

Azimuth / elevation / distance to Cartesian x, y, z: x = d*cos(a)*cos(e), y = d*sin(a)*cos(e), z = d*sin(e).

Usage

aed2xyz(a,e,d) : _,_,_

Where:

  • a: azimuth in radians
  • e: elevation in radians (from the horizontal plane)
  • d: distance (radius)

Test

sma = library("seam.math.lib");
aed2xyz_test = sma.aed2xyz(0,0,1);

(sma.)xyz2aed

Cartesian x, y, z to azimuth / elevation / distance: a = atan2(y,x), e = atan2(z, sqrt(x^2+y^2)), d = sqrt(x^2+y^2+z^2). The exact inverse of aed2xyz (elevation measured from the horizontal plane).

Usage

xyz2aed(x,y,z) : _,_,_

Where:

  • x, y, z: Cartesian coordinates

Test

sma = library("seam.math.lib");
xyz2aed_test = sma.xyz2aed(1,1,1) : sma.r2d, sma.r2d, _;

Golden Ratio

(sma.)phi

Scales x by the golden ratio phi = (1+sqrt(5))/2 ~= 1.618.

Usage

phi(x) : _

Where:

  • x: value to scale

Test

sma = library("seam.math.lib");
phi_test = sma.phi(1);

(sma.)rphi

Scales x by the reciprocal golden ratio 1/phi = (sqrt(5)-1)/2 ~= 0.618.

Usage

rphi(x) : _

Where:

  • x: value to scale

Test

sma = library("seam.math.lib");
rphi_test = sma.rphi(1);

(sma.)srphi

Reciprocal-golden progression: applies rphi to x a total of i times, i.e. the i-th term of the geometric sequence with ratio 1/phi.

Usage

srphi(i,x) : _

Where:

  • i: number of iterations (compile-time integer)
  • x: starting value

Test

sma = library("seam.math.lib");
srphi_test = par(i,16,sma.srphi(i,ma.SR/2));

Combinatorics

(sma.)factorial

Factorial n! = n*(n-1)*...*1 (with 0! = 1), by compile-time recursion.

Usage

factorial(n)

Where:

  • n: non-negative integer (compile-time constant)

Test

sma = library("seam.math.lib");
factorial_test = sma.factorial(5);

(sma.)permutation

Number of ordered arrangements of k items taken from n: P(n,k) = n!/(n-k)!. Requires n >= k >= 0.

Usage

permutation(n,k)

Where:

  • n: size of the set (compile-time integer)
  • k: number of items chosen (compile-time integer, <= n)

Test

sma = library("seam.math.lib");
permutation_test = sma.permutation(5,2);

Acoustics

Speed of sound and distance-to-delay helpers for physical spatialisation. SEAM distinguishes an exterior (open-air) and an interior (room) speed of sound.

(sma.)esos

Exterior speed of sound: 344 m/s (open air, ~20 degrees C).

Usage

sma.esos : _

Test

sma = library("seam.math.lib");
esos_test = sma.esos;

(sma.)isos

Interior speed of sound: 331.4 m/s (~0 degrees C reference).

Usage

sma.isos : _

Test

sma = library("seam.math.lib");
isos_test = sma.isos;

(sma.)emt2samp

Metres to whole samples using the exterior speed of sound: int(mt*SR/esos).

Usage

emt2samp(mt)

Where:

  • mt: distance in metres

Test

sma = library("seam.math.lib");
emt2samp_test = sma.emt2samp(3.4);

(sma.)imt2samp

Metres to whole samples using the interior speed of sound: int(mt*SR/isos).

Usage

imt2samp(mt)

Where:

  • mt: distance in metres

Test

sma = library("seam.math.lib");
imt2samp_test = sma.imt2samp(3.4);

(sma.)imdelay

Pure integer-sample delay for a distance mt in metres, using the interior speed of sound. The one stateful function in this library. Delay line is 1<<15 samples (~0.68 s at 48 kHz, ~226 m). Used by the SEAM-LTM DDELAY plugin (per channel).

Usage

_ : imdelay(mt) : _

Where:

  • mt: distance in metres

Test

no = library("noises.lib");
sma = library("seam.math.lib");
imdelay_test = no.noise : sma.imdelay(3.4);