Synthesizer.Basic.Wave
 Portability requires multi-parameter type classes Stability provisional Maintainer synthesizer@henning-thielemann.de
 Contents Definition and construction Operations on waves Examples unparameterized discretely parameterized continuously parameterized
Description

Basic waveforms

If you want to use parametrized waves with two parameters then zip your parameter signals and apply uncurry to the wave function.

Synopsis
newtype T t y = Cons {
 decons :: T t -> y
}
fromFunction :: (t -> y) -> T t y
raise :: C y => y -> T t y -> T t y
amplify :: C y => y -> T t y -> T t y
distort :: (y -> z) -> T t y -> T t z
overtone :: (C t, C n) => n -> T t y -> T t y
apply :: T t y -> T t -> y
phaseOffset :: C a => T a b -> a -> T a b
sine :: C a => T a a
cosine :: C a => T a a
helix :: C a => T a (T a)
fastSine2 :: (Ord a, C a) => T a a
fastSine4 :: (Ord a, C a) => T a a
saw :: C a => T a a
sawCos :: (C a, C a) => T a a
sawComplex :: (Power a, C a) => T a (T a)
square :: (Ord a, C a) => T a a
squareCos :: (C a, C a) => T a a
squareComplex :: (Power a, C a) => T a (T a)
triangle :: (Ord a, C a) => T a a
truncOddCosine :: C a => Int -> T a a
truncOddTriangle :: C a => Int -> T a a
truncCosine :: C a => a -> T a a
truncTriangle :: C a => a -> T a a
powerNormed :: (C a, C a) => a -> T a a
power01Normed :: (C a, C a) => a -> a -> a
powerSigned :: (C a, C a) => a -> a -> a
logitSaw :: C a => a -> T a a
logitSine :: C a => a -> T a a
sineSquare :: (C a, C a) => a -> T a a
piecewiseParabolaSaw :: (C a, Ord a) => a -> T a a
piecewiseSineSaw :: (C a, Ord a) => a -> T a a
sineSawSmooth :: C a => a -> T a a
sineSawSharp :: C a => a -> T a a
affineComb :: C a => a -> (a, a) -> a
sawPike :: (Ord a, C a) => a -> T a a
trianglePike :: (C a, C a) => a -> T a a
trianglePikeShift :: (C a, C a) => a -> a -> T a a
squarePike :: C a => a -> T a a
squarePikeShift :: C a => a -> a -> T a a
squareAsymmetric :: (Ord a, C a) => a -> T a a
squareBalanced :: (Ord a, C a) => a -> T a a
triangleAsymmetric :: (Ord a, C a) => a -> T a a
trapezoid :: (C a, C a) => a -> T a a
trapezoidAsymmetric :: (C a, C a) => a -> a -> T a a
trapezoidBalanced :: (C a, C a) => a -> a -> T a a
trapezoidSkew :: (Ord a, C a) => a -> T a a
data Harmonic a = Harmonic {
 harmonicPhase :: T a harmonicAmplitude :: a
}
harmonic :: T a -> a -> Harmonic a
composedHarmonics :: C a => [Harmonic a] -> T a a
Definition and construction
 newtype T t y Source
Constructors
Cons
 decons :: T t -> y
Instances
 Simple Flat t (T t) (T T) C Flat (T t) (T t) C amp => C amp (T t) (T amp (T t)) C a y => C a (T t y) C y => C (T t y)
 fromFunction :: (t -> y) -> T t y Source
Operations on waves
 raise :: C y => y -> T t y -> T t y Source
 amplify :: C y => y -> T t y -> T t y Source
 distort :: (y -> z) -> T t y -> T t z Source
 overtone :: (C t, C n) => n -> T t y -> T t y Source
 apply :: T t y -> T t -> y Source
 phaseOffset :: C a => T a b -> a -> T a b Source
Turn an unparametrized waveform into a parametrized one, where the parameter is a phase offset. This way you express a phase modulated oscillator using a shape modulated oscillator.
Examples
unparameterized
 sine :: C a => T a a Source
map a phase to value of a sine wave
 cosine :: C a => T a a Source
 helix :: C a => T a (T a) Source
 fastSine2 :: (Ord a, C a) => T a a Source
Approximation of sine by parabolas. Surprisingly not really faster than sine.
 fastSine4 :: (Ord a, C a) => T a a Source
Approximation of sine by fourth order polynomials.
 saw :: C a => T a a Source
saw tooth, it's a ramp down in order to have a positive coefficient for the first partial sine
 sawCos :: (C a, C a) => T a a Source
This wave has the same absolute Fourier coefficients as saw but the partial waves are shifted by 90 degree. That is, it is the Hilbert transform of the saw wave. The formula is derived from sawComplex.
 sawComplex :: (Power a, C a) => T a (T a) Source
`sawCos + i*saw`

This is an analytic function and thus it may be used for frequency shifting.

The formula can be derived from the power series of the logarithm function.

 square :: (Ord a, C a) => T a a Source
square
 squareCos :: (C a, C a) => T a a Source
This wave has the same absolute Fourier coefficients as square but the partial waves are shifted by 90 degree. That is, it is the Hilbert transform of the saw wave.
 squareComplex :: (Power a, C a) => T a (T a) Source
`squareCos + i*square`

This is an analytic function and thus it may be used for frequency shifting.

The formula can be derived from the power series of the area tangens function.

 triangle :: (Ord a, C a) => T a a Source
triangle
discretely parameterized
 truncOddCosine :: C a => Int -> T a a Source
A truncated cosine. This has rich overtones.
 truncOddTriangle :: C a => Int -> T a a Source
For parameter zero this is saw.
continuously parameterized
 truncCosine :: C a => a -> T a a Source

A truncated cosine plus a ramp that guarantees a bump of high 2 at the boundaries.

It is truncCosine (2 * fromIntegral n + 0.5) == truncOddCosine (2*n)

 truncTriangle :: C a => a -> T a a Source
 powerNormed :: (C a, C a) => a -> T a a Source

Power function.

Roughly the map x p -> x**p but retains the sign of x and normalizes the mapping over [-1,1] to L2 norm of 1.

 power01Normed :: (C a, C a) => a -> a -> a Source
auxiliary
 powerSigned :: (C a, C a) => a -> a -> a Source
auxiliary
 logitSaw :: C a => a -> T a a Source
Tangens hyperbolicus allows interpolation between some kind of saw tooth and square wave. In principle it is not necessary because you can distort a saw tooth oscillation by map tanh.
 logitSine :: C a => a -> T a a Source
Tangens hyperbolicus of a sine allows interpolation between some kind of sine and square wave. In principle it is not necessary because you can distort a square oscillation by map tanh.
 sineSquare Source
 :: (C a, C a) => a 0 for sine, 1 for square -> T a a Interpolation between sine and square.
 piecewiseParabolaSaw Source
 :: (C a, Ord a) => a 0 for fastSine2, 1 for saw -> T a a Interpolation between fastSine2 and saw. We just shrink the parabola towards the borders and insert a linear curve such that its slope matches the one of the parabola.
 piecewiseSineSaw Source
 :: (C a, Ord a) => a 0 for sine, 1 for saw -> T a a Interpolation between sine and saw. We just shrink the sine towards the borders and insert a linear curve such that its slope matches the one of the sine.
 sineSawSmooth Source
 :: C a => a 0 for sine, 1 for saw -> T a a Interpolation between sine and saw with smooth intermediate shapes but no perfect saw.
 sineSawSharp Source
 :: C a => a 0 for sine, 1 for saw -> T a a Interpolation between sine and saw with perfect saw, but sharp intermediate shapes.
 affineComb :: C a => a -> (a, a) -> a Source
 sawPike Source
 :: (Ord a, C a) => a pike width ranging from 0 to 1, 1 yields saw -> T a a saw with space
 trianglePike Source
 :: (C a, C a) => a pike width ranging from 0 to 1, 1 yields triangle -> T a a triangle with space
 trianglePikeShift Source
 :: (C a, C a) => a pike width ranging from 0 to 1 -> a shift ranges from -1 to 1; 0 yields trianglePike -> T a a triangle with space and shift
 squarePike Source
 :: C a => a pike width ranging from 0 to 1, 1 yields square -> T a a square with space, can also be generated by mixing square waves with different phases
 squarePikeShift Source
 :: C a => a pike width ranging from 0 to 1 -> a shift ranges from -1 to 1; 0 yields squarePike -> T a a square with space and shift
 squareAsymmetric Source
 :: (Ord a, C a) => a value between -1 and 1 controlling the ratio of high and low time: -1 turns the high time to zero, 1 makes the low time zero, 0 yields square -> T a a square with different times for high and low
 squareBalanced :: (Ord a, C a) => a -> T a a Source
Like squareAsymmetric but with zero average. It could be simulated by adding two saw oscillations with 180 degree phase difference and opposite sign.
 triangleAsymmetric Source
 :: (Ord a, C a) => a asymmetry parameter ranging from -1 to 1: For 0 you obtain the usual triangle. For -1 you obtain a falling saw tooth starting with its maximum. For 1 you obtain a rising saw tooth starting with a zero. -> T a a triangle
 trapezoid Source
 :: (C a, C a) => a width of the plateau ranging from 0 to 1: 0 yields triangle, 1 yields square -> T a a Mixing trapezoid and trianglePike you can get back a triangle wave form
 trapezoidAsymmetric Source
 :: (C a, C a) => a sum of the plateau widths ranging from 0 to 1: 0 yields triangleAsymmetric, 1 yields squareAsymmetric -> a asymmetry of the plateau widths ranging from -1 to 1 -> T a a Trapezoid with distinct high and low time. That is the high and low trapezoids are symmetric itself, but the whole waveform is not symmetric.
 trapezoidBalanced :: (C a, C a) => a -> a -> T a a Source
trapezoid with distinct high and low time and zero direct current offset
 trapezoidSkew Source
 :: (Ord a, C a) => a width of the ramp, that is 1 yields a downwards saw ramp and 0 yields a square wave. -> T a a parametrized trapezoid that can range from a saw ramp to a square waveform.
 data Harmonic a Source
This is similar to Polar coordinates, but the range of the phase is from 0 to 1, 0 to 2*pi.
Constructors
Harmonic
 harmonicPhase :: T a harmonicAmplitude :: a
 harmonic :: T a -> a -> Harmonic a Source
 composedHarmonics :: C a => [Harmonic a] -> T a a Source

Specify the wave by its harmonics.

The function is implemented quite efficiently by applying the Horner scheme to a polynomial with complex coefficients (the harmonic parameters) using a complex exponential as argument.