{-# LANGUAGE NoImplicitPrelude #-} {-# LANGUAGE MultiParamTypeClasses #-} {-# LANGUAGE FlexibleInstances #-} {- | Copyright : (c) Henning Thielemann 2006 License : GPL Maintainer : synthesizer@henning-thielemann.de Stability : provisional Portability : requires multi-parameter type classes Basic waveforms If you want to use parametrized waves with two parameters then zip your parameter signals and apply 'uncurry' to the wave function. -} module Synthesizer.Basic.Wave where import qualified Synthesizer.Basic.Phase as Phase import qualified Algebra.RealTranscendental as RealTrans import qualified Algebra.Transcendental as Trans import qualified Algebra.RealField as RealField import qualified Algebra.Algebraic as Algebraic import qualified Algebra.Module as Module import qualified Algebra.Field as Field import qualified Algebra.Real as Real import qualified Algebra.Ring as Ring import qualified Algebra.Additive as Additive import qualified Algebra.ToInteger as ToInteger import qualified MathObj.Polynomial as Poly import qualified Number.Complex as Complex import Data.Bool.HT (select, if', ) import NumericPrelude -- import qualified Prelude as P import PreludeBase {- * Definition and construction -} newtype T t y = Cons {decons :: Phase.T t -> y} {-# INLINE fromFunction #-} fromFunction :: (t -> y) -> (T t y) fromFunction wave = Cons (wave . Phase.toRepresentative) {- * Operations on waves -} {-# INLINE raise #-} raise :: (Additive.C y) => y -> T t y -> T t y raise y = distort (y+) {-# INLINE amplify #-} amplify :: (Ring.C y) => y -> T t y -> T t y amplify k = distort (k*) {-# INLINE distort #-} distort :: (y -> z) -> T t y -> T t z distort g (Cons f) = Cons (g . f) {-# INLINE overtone #-} overtone :: (RealField.C t, ToInteger.C n) => n -> T t y -> T t y overtone n (Cons f) = Cons (f . Phase.multiply n) {-# INLINE apply #-} apply :: T t y -> (Phase.T t -> y) apply = decons instance Additive.C y => Additive.C (T t y) where {-# INLINE zero #-} {-# INLINE (+) #-} {-# INLINE (-) #-} {-# INLINE negate #-} zero = Cons (const zero) (+) (Cons f) (Cons g) = Cons (\t -> f t + g t) (-) (Cons f) (Cons g) = Cons (\t -> f t - g t) negate = distort negate instance Module.C a y => Module.C a (T t y) where {-# INLINE (*>) #-} s *> w = distort (s*>) w {- | 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. -} {-# SPECULATE phaseOffset :: (T Double b) -> (Double -> T Double b) #-} {-# INLINE phaseOffset #-} phaseOffset :: (RealField.C a) => T a b -> (a -> T a b) phaseOffset (Cons wave) offset = Cons (wave . Phase.increment offset) {- * Examples -} {- ** unparameterized -} {- | map a phase to value of a sine wave -} {-# SPECULATE sine :: Double -> Double #-} {-# INLINE sine #-} sine :: Trans.C a => T a a sine = fromFunction $ \x -> sin (2*pi*x) {-# INLINE cosine #-} cosine :: Trans.C a => T a a cosine = fromFunction $ \x -> cos (2*pi*x) {-# INLINE helix #-} helix :: Trans.C a => T a (Complex.T a) helix = fromFunction $ \x -> Complex.cis (2*pi*x) {- | Approximation of sine by parabolas. Surprisingly not really faster than 'sine'. -} {-# INLINE fastSine2 #-} fastSine2 :: (Ord a, Ring.C a) => T a a fastSine2 = fromFunction $ \x -> if 2*x<1 then 1 - sqr (4*x-1) else sqr (4*x-3) - 1 {- | Approximation of sine by fourth order polynomials. -} {-# INLINE fastSine4 #-} fastSine4 :: (Ord a, Trans.C a) => T a a fastSine4 = fromFunction $ \x -> -- minimal least squares fit let pi2 = pi*pi pi3 = pi2*pi c = 3*((10080/pi2 - 1050) / pi3 + 1) -- 0.2248391014 {-# INLINE bow #-} bow y = let y2 = y*y in 1-y2*(1+c*(1-y2)) in if 2*x<1 then bow (4*x-1) else - bow (4*x-3) {- add a residue to fastSine2 and choose 'c' which minimizes the squared error in if 2*x<1 then let y = (4*x-1)^2 in 1-y-c*y*(1-y) else let y = (4*x-3)^2 in y-1+c*y*(1-y) -} {- GNUPlot.plotFuncs [] (GNUPlot.linearScale 1000 (0,1::Double)) [sine, fastSine2, fastSine4] -} {- | saw tooth, it's a ramp down in order to have a positive coefficient for the first partial sine -} {-# SPECULATE saw :: Double -> Double #-} {-# INLINE saw #-} saw :: Ring.C a => T a a saw = fromFunction $ \x -> 1-2*x {- | 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'. -} {-# INLINE sawCos #-} sawCos :: (Real.C a, Trans.C a) => T a a sawCos = fromFunction $ \x -> log (2 * sin (pi*x)) * (-2/pi) {- | @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. -} {-# INLINE sawComplex #-} sawComplex :: (Complex.Power a, RealTrans.C a) => T a (Complex.T a) sawComplex = fromFunction $ \x -> log (1 + Complex.cis (-pi*(1-2*x))) * (-2/pi) {- GNUPlot.plotFuncs [] (GNUPlot.linearScale 100 (0,1::Double)) [Complex.real . sawComplex, sawCos] GNUPlot.plotFuncs [] (GNUPlot.linearScale 100 (0,1::Double)) [sawCos, composedHarmonics (take 20 $ harmonic 0 0 : map (\n -> harmonic 0.25 ((2/pi) / fromInteger n)) [1..])] -} {- Matching implementation that do not match 'saw' exactly. sawCos :: (Real.C a, Trans.C a) => T a a sawCos = fromFunction $ \x -> log (2 * abs (cos (pi*x))) sawComplex :: (Complex.Power a, Trans.C a) => T a (Complex.T a) sawComplex = fromFunction $ \x -> log (1 + Complex.cis (2*pi*x)) -} {- | square -} {-# SPECULATE square :: Double -> Double #-} {-# INLINE square #-} square :: (Ord a, Ring.C a) => T a a square = fromFunction $ \x -> if 2*x<1 then 1 else -1 {- | 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. -} {-# INLINE squareCos #-} squareCos :: (RealField.C a, Trans.C a) => T a a squareCos = fromFunction $ \x -> log (abs (tan (pi*x))) * (-2/pi) -- sawCos x - sawCos (fraction (0.5-x)) {- | @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. -} {-# INLINE squareComplex #-} squareComplex :: (Complex.Power a, RealTrans.C a) => T a (Complex.T a) squareComplex = fromFunction $ \x -> {- these formulas are equivalent but wrong log (0 +: 2 * sine x) * (2/pi) log ((1 - Complex.cis (-2*pi*x)) * (1 + Complex.cis ( 2*pi*x))) * (2/pi) sawComplex x + sawComplex (0.5-x) -} {- The Fourier series is equal to the power series of 'atanh'. -} atanh (Complex.cis (2*pi*x)) * (4/pi) {- GNUPlot.plotFuncs [] (GNUPlot.linearScale 100 (0,1::Double)) [squareCos, composedHarmonics (take 20 $ zipWith (\b n -> harmonic 0.25 (if b then (4/pi) / fromInteger n else 0)) (cycle [False,True]) [0..])] -} {- | triangle -} {-# SPECULATE triangle :: Double -> Double #-} {-# INLINE triangle #-} triangle :: (Ord a, Ring.C a) => T a a triangle = fromFunction $ \x -> let x4 = 4*x in select (2-x4) [(x4<1, x4), (x4>3, x4-4)] {- int(arctan(x)/x,x); - polylog(2, x*I)*1/2*I + polylog(2, x*(-I))*1/2*I series(int(arctan(x)/x,x),x,10); x - 1/9*x^3 + 1/25*x^5 - 1/49*x^7 + 1/81*x^9 + O(x^11) int(arctan(I*x)/(I*x),x); int(arctanh(x)/(x),x); 1/2*polylog(2, x) - 1/2*polylog(2, -x) int(1/x*arctanh(x), x) polylog(2,x) = dilog(1-x); -- dilog is implemented in GSL for complex arguments polylog(2,x) = hypergeom([1,1,1],[2,2],x) * x; series(int(arctan(I*x)/(I*x),x),x,10); x + 1/9*x^3 + 1/25*x^5 + 1/49*x^7 + 1/81*x^9 + O(x^11) -} {- ** discretely parameterized -} {- | A truncated cosine. This has rich overtones. -} truncOddCosine :: Trans.C a => Int -> T a a truncOddCosine k = let f = pi * fromIntegral (2*k+1) in fromFunction $ \ x -> cos (f*x) {- | For parameter zero this is 'saw'. -} truncOddTriangle :: (RealField.C a) => Int -> T a a truncOddTriangle k = let s = fromIntegral (2*k+1) in fromFunction $ \ x -> let (n,frac) = splitFraction (s*x) in if even (n::Int) then 1-2*frac else 2*frac-1 {- ** continuously parameterized -} {- | 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)@ -} truncCosine :: Trans.C a => a -> T a a truncCosine k = let f = 2 * pi * k s = 2 * (sin (f*0.5) - 1) in fromFunction $ \ x0 -> let x = x0-0.5 in - sin (f*x) + s*x {- GNUPlot.plotFuncs [] (GNUPlot.linearScale 1000 (0,1::Double)) (map truncCosine [0.5,0.7..2.5]) -} truncTriangle :: (RealField.C a) => a -> T a a truncTriangle k = let tr x = let (n,frac) = splitFraction (2*k*x+0.5) in if even (n::Int) then 1-2*frac else 2*frac-1 s = 2 * (1 + tr 0.5) in fromFunction $ \ x0 -> let x = x0-0.5 in tr x - s*x {- GNUPlot.plotFuncs [] (GNUPlot.linearScale 1000 (0,1::Double)) (map truncTriangle [0,0.25..2.5]) -} {- | 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. -} {-# INLINE powerNormed #-} powerNormed :: (Real.C a, Trans.C a) => a -> T a a powerNormed p = fromFunction $ \x -> power01Normed p (2*x-1) -- | auxiliary {-# INLINE power01Normed #-} power01Normed :: (Real.C a, Trans.C a) => a -> a -> a power01Normed p x = (p+0.5) * powerSigned p x -- | auxiliary {-# INLINE powerSigned #-} powerSigned :: (Real.C a, Trans.C a) => a -> a -> a powerSigned p x = signum x * abs x ** p {- | 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@. -} logitSaw :: (Trans.C a) => a -> T a a logitSaw c = distort tanh $ amplify c saw {- | 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@. -} logitSine :: (Trans.C a) => a -> T a a logitSine c = distort tanh $ amplify c sine {- | Interpolation between 'sine' and 'square'. -} {-# INLINE sineSquare #-} sineSquare :: (Real.C a, Trans.C a) => a {- ^ 0 for 'sine', 1 for 'square' -} -> T a a sineSquare c = distort (powerSigned (1-c)) sine {- | 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. -} {-# INLINE piecewiseParabolaSaw #-} piecewiseParabolaSaw :: (Algebraic.C a, Ord a) => a {- ^ 0 for 'fastSine2', 1 for 'saw' -} -> T a a piecewiseParabolaSaw c = let xb = (1 - sqrt c) / 2 y x = 1 - ((4*x - (1-c))/(1-c))^2 in fromFunction $ \ x -> select ((2*x - 1)/(2*xb - 1) * y xb) [(x < xb, y x), (x > 1-xb, - y (1-x))] {- equ0 c x = let y = 1 - ((4*x - (3+c))/(1-c))^2 secant = y/(x-1/2) tangent = - 8 * (4*x - (3+c))/(1-c)^2 in (tangent, secant) equ1 c x = let secant = (1 - ((4*x - (3+c))/(1-c))^2)/(x-1/2) tangent = - 8 * (4*x - (3+c))/(1-c)^2 in (tangent, secant) equ2 c x = (1, ((4*x - (3+c))/(1-c))^2 - 8 * (x-1/2) * (4*x - (3+c))/(1-c)^2) equ3 c x = ((1-c)^2, (4*x - (3+c) - 4 * (2*x-1)) * (4*x - (3+c))) equ4 c x = (4*x - (1-c)) * (4*x - (3+c)) + (1-c)^2 equ5 c x = (4*x - 2) ^ 2 - (1+c)^2 + (1-c)^2 equ6 c x = (4*x - 2) ^ 2 - 4*c -} {- | 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. -} {-# INLINE piecewiseSineSaw #-} piecewiseSineSaw :: (Trans.C a, Ord a) => a {- ^ 0 for 'sine', 1 for 'saw' -} -> T a a piecewiseSineSaw c = let {- This simple fix point iteration converges very slow for small 'c', maybe we should use a Newton iteration. -} iter z = iterate (\zi -> pi + atan (zi - pi / (1-c))) z !! 10 xb = (1-c)/(2*pi) * iter 0 -- iter (xInit * (2*pi) / (1-c)) -- xb = (1 - sqrt c) / 2 -- y x = sine (x/(1-c)) y x = sin (2*pi*x/(1-c)) in fromFunction $ \ x -> select ((2*x - 1)/(2*xb - 1) * y xb) [(x < xb, y x), (x > 1-xb, - y (1-x))] {- equ0 c x = let secant = 2 * sin (2*pi*x/(1-c)) / (2*x - 1) tangent = 2*pi/(1-c) * cos (2*pi*x/(1-c)) in (tangent, secant) iter0 c x = -- secant / tangent -- (x - 1/2) = tan (2*pi*x/(1-c)) * (1-c) / (2*pi) tan (2*pi*x/(1-c)) * (1-c) / (2*pi) + 1/2 iter1 c x = (1-c)/(2*pi) * (pi + atan ((x - 1/2) * (2*pi) / (1-c))) iter2 c x = let iter z = iterate (\zi -> pi + atan (zi - pi / (1-c))) z !! 10 in (1-c)/(2*pi) * iter (x * (2*pi) / (1-c)) -} {- | Interpolation between 'sine' and 'saw' with smooth intermediate shapes but no perfect saw. -} {-# INLINE sineSawSmooth #-} sineSawSmooth :: (Trans.C a) => a {- ^ 0 for 'sine', 1 for 'saw' -} -> T a a sineSawSmooth c = distort (\x -> sin (affineComb c (pi * x, asin x * 2))) saw {- | Interpolation between 'sine' and 'saw' with perfect saw, but sharp intermediate shapes. -} {-# INLINE sineSawSharp #-} sineSawSharp :: (Trans.C a) => a {- ^ 0 for 'sine', 1 for 'saw' -} -> T a a sineSawSharp c = distort (\x -> sin (affineComb c (pi * x, asin x))) saw affineComb :: Ring.C a => a -> (a,a) -> a affineComb phase (x0,x1) = (1-phase)*x0 + phase*x1 {- {- | Smooth saw generated by a quintic polynomial function. Unfortunately if 'c' approaches the right border, the function will overshoot the 'y' range (-1,1). -} quinticSaw :: Field.C a => a {- ^ position of the right minimum -} -> a -> a quinticSaw c x = let (s,t) = ToneMod.solveSLE2 ((c^2-1, 3*c^2-1), (c^4-1, 5*c^4-1)) (-1/c,0) r = - s - t x2 = x^2 in x * (r + x2 * (s + x2*t)) {- r*x + s* x^3 + t* x^5 0 = r + s + t -1 = r*c + s* c^3 + t* c^5 0 = r + s*3*c^2 + t*5*c^4 -1/c = r + s* c^2 + t* c^4 -1/c = s*(c^2-1) + t*(c^4-1) 0 = s*(3*c^2-1) + t*(5*c^4-1) -} -} {- | saw with space -} {-# SPECULATE sawPike :: Double -> Double -> Double #-} {-# INLINE sawPike #-} sawPike :: (Ord a, Field.C a) => a {- ^ pike width ranging from 0 to 1, 1 yields 'saw' -} -> T a a sawPike r = fromFunction $ \x -> if x<r then 1-2/r*x else 0 {- | triangle with space -} {-# SPECULATE trianglePike :: Double -> Double -> Double #-} {-# INLINE trianglePike #-} trianglePike :: (Real.C a, Field.C a) => a {- ^ pike width ranging from 0 to 1, 1 yields 'triangle' -} -> T a a trianglePike r = fromFunction $ \x -> if x < 1/2 then max 0 (1 - abs (4*x-1) / r) else min 0 (abs (4*x-3) / r - 1) {- | triangle with space and shift -} {-# SPECULATE trianglePikeShift :: Double -> Double -> Double -> Double #-} {-# INLINE trianglePikeShift #-} trianglePikeShift :: (Real.C a, Field.C a) => a {- ^ pike width ranging from 0 to 1 -} -> a {- ^ shift ranges from -1 to 1; 0 yields 'trianglePike' -} -> T a a trianglePikeShift r s = fromFunction $ \x -> if x < 1/2 then max 0 (1 - abs (4*x-1+s*(r-1)) / r) else min 0 (abs (4*x-3+s*(1-r)) / r - 1) {- | square with space, can also be generated by mixing square waves with different phases -} {-# SPECULATE squarePike :: Double -> Double -> Double #-} {-# INLINE squarePike #-} squarePike :: (Real.C a) => a {- ^ pike width ranging from 0 to 1, 1 yields 'square' -} -> T a a squarePike r = fromFunction $ \x -> if 2*x < 1 then if abs(4*x-1)<r then 1 else 0 else if abs(4*x-3)<r then -1 else 0 {- | square with space and shift -} {-# SPECULATE squarePikeShift :: Double -> Double -> Double -> Double #-} {-# INLINE squarePikeShift #-} squarePikeShift :: (Real.C a) => a {- ^ pike width ranging from 0 to 1 -} -> a {- ^ shift ranges from -1 to 1; 0 yields 'squarePike' -} -> T a a squarePikeShift r s = fromFunction $ \x -> if 2*x < 1 then if abs(4*x-1+s*(r-1))<r then 1 else 0 else if abs(4*x-3+s*(1-r))<r then -1 else 0 {- | square with different times for high and low -} {-# SPECULATE squareAsymmetric :: Double -> Double -> Double #-} {-# INLINE squareAsymmetric #-} squareAsymmetric :: (Ord a, Ring.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 squareAsymmetric r = fromFunction $ \x -> if 2*x < r+1 then 1 else -1 {- | Like 'squareAsymmetric' but with zero average. It could be simulated by adding two saw oscillations with 180 degree phase difference and opposite sign. -} {-# SPECULATE squareBalanced :: Double -> Double -> Double #-} {-# INLINE squareBalanced #-} squareBalanced :: (Ord a, Ring.C a) => a -> T a a squareBalanced r = raise (-r) $ squareAsymmetric r {- | triangle -} {-# SPECULATE sawPike :: Double -> Double -> Double #-} {-# INLINE triangleAsymmetric #-} triangleAsymmetric :: (Ord a, Field.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 triangleAsymmetric r = fromFunction $ \x -> select ((2-4*x)/(1-r)) [(4*x < 1+r, 4/(1+r)*x), (4*x > 3-r, 4/(1+r)*(x-1))] {- | Mixing 'trapezoid' and 'trianglePike' you can get back a triangle wave form -} {-# SPECULATE trapezoid :: Double -> Double -> Double #-} {-# INLINE trapezoid #-} trapezoid :: (Real.C a, Field.C a) => a {- ^ width of the plateau ranging from 0 to 1: 0 yields 'triangle', 1 yields 'square' -} -> T a a trapezoid w = fromFunction $ \x -> if x < 1/2 then min 1 ((1 - abs (4*x-1)) / (1-w)) else max (-1) ((abs (4*x-3) - 1) / (1-w)) {- | Trapezoid with distinct high and low time. That is the high and low trapezoids are symmetric itself, but the whole waveform is not symmetric. -} {-# SPECULATE trapezoidAsymmetric :: Double -> Double -> Double -> Double #-} {-# INLINE trapezoidAsymmetric #-} trapezoidAsymmetric :: (Real.C a, Field.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 trapezoidAsymmetric w r = fromFunction $ \x -> let c0 = 1+w*r c1 = 1-w*r in if 2*x < c0 then min 1 ((c0 - abs (4*x-c0)) / (1-w)) else max (-1) ((abs (4*(1-x)-c1) - c1) / (1-w)) {- let c = w*r+1 in if 2*x < c then min 1 ((1 - abs (4*x/c-1))*c/(1-w)) else max (-1) ((abs (4*(1-x)/(2-c)-1) - 1)*(2-c)/(1-w)) -} {- let c = (w*r+1)/2 in if x < c then min 1 ((1 - abs (2*x/c-1))*2*c/(1-w)) else max (-1) ((abs (2*(1-x)/(1-c)-1) - 1)*2*(1-c)/(1-w)) -} {- | trapezoid with distinct high and low time and zero direct current offset -} {-# SPECULATE trapezoidBalanced :: Double -> Double -> Double -> Double #-} {-# INLINE trapezoidBalanced #-} trapezoidBalanced :: (Real.C a, Field.C a) => a -> a -> T a a trapezoidBalanced w r = raise (-w*r) $ trapezoidAsymmetric w r -- could also be generated by amplifying and clipping a saw ramp {- | parametrized trapezoid that can range from a saw ramp to a square waveform. -} trapezoidSkew :: (Ord a, Field.C a) => a {- ^ width of the ramp, that is 1 yields a downwards saw ramp and 0 yields a square wave. -} -> T a a trapezoidSkew w = fromFunction $ \t -> if' (2*t<=1-w) 1 $ if' (2*t>=1+w) (-1) $ (1-2*t)/w {- | This is similar to Polar coordinates, but the range of the phase is from @0@ to @1@, @0@ to @2*pi@. -} data Harmonic a = Harmonic {harmonicPhase :: Phase.T a, harmonicAmplitude :: a} {-# INLINE harmonic #-} harmonic :: Phase.T a -> a -> Harmonic a harmonic = Harmonic {- | 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. -} {-# INLINE composedHarmonics #-} composedHarmonics :: Trans.C a => [Harmonic a] -> T a a composedHarmonics hs = let p = Poly.fromCoeffs $ map (\h -> Complex.fromPolar (harmonicAmplitude h) (2*pi * Phase.toRepresentative (harmonicPhase h))) hs in distort (Complex.imag . Poly.evaluate p) helix {- GNUPlot.plotFunc [] (GNUPlot.linearScale 1000 (0,1::Double)) (composedHarmonics [harmonic 0 0, harmonic 0 0, harmonic 0 0, harmonic 0.25 1]) -}