diff --git a/src/Synthesizer/ApplicativeUtility.hs b/src/Synthesizer/ApplicativeUtility.hs
--- a/src/Synthesizer/ApplicativeUtility.hs
+++ b/src/Synthesizer/ApplicativeUtility.hs
@@ -76,7 +76,7 @@
 
 {-# INLINE ($#) #-}
 ($#) :: (Functor f) => f (a -> b) -> a -> f b
-($#) f x = fmap ($x) f
+($#) f x = fmap ($ x) f
 -- ($#) f x = f $: pure x
 
 
diff --git a/src/Synthesizer/Basic/Wave.hs b/src/Synthesizer/Basic/Wave.hs
--- a/src/Synthesizer/Basic/Wave.hs
+++ b/src/Synthesizer/Basic/Wave.hs
@@ -42,6 +42,7 @@
    saw,
    sawCos,
    sawComplex,
+   superSaw,
    square,
    squareCos,
    squareComplex,
@@ -424,6 +425,51 @@
    T a (Complex.T a)
 sawComplex = fromFunction $ \x -> log (1 + Complex.cis (2*pi*x))
 -}
+
+
+{-
+sum [0..n-1] $ \k -> fraction (x+k*d)
+sum [0..n-1] $ \k -> fraction (x+k*(floor d + fraction d))
+sum [0..n-1] $ \k -> fraction (x+k*floor d + k*fraction d)
+sum [0..n-1] $ \k -> fraction (x + k*fraction d)
+sum [0..n-1] $ \k -> fraction (fraction x + k*fraction d)
+X = fraction x (0<=X<1)
+D = fraction d (0<=D<1)
+sum [0..n-1] $ \k -> X+k*D - floor (X+k*D)
+n*X + n*(n-1)/2*D - sum [0..n-1] (\k -> floor (X+k*D))
+
+n=1: sum = 0
+n=2: sum = floor(X+D)
+n=3: sum = floor(X+D)(<2) + floor(X+2*D)(<3)
+n=4: sum = floor(X+D) + floor(X+2*D) + floor(X+3*D)
+
+N=n-1
+special case: N*D<1   => X+N*D<2
+
+X+(k-1)*D < 1
+k-1 < (1-X)/D
+k = ceiling((1-X)/D)
+
+sum: max 0 (n - ceiling((1-X)/D))
+   = n - min n (ceiling((1-X)/D))
+
+https://math.stackexchange.com/questions/116689/counting-integral-lattice-points-in-a-triangle-that-may-not-have-integer-coordin
+-}
+{- |
+@superSaw n d@ requires @1 <= n@ and @n*d <= 1@.
+-}
+superSaw, _superSaw :: (RealRing.C a, Field.C a) => Int -> a -> T a a
+superSaw n =
+   let nf = fromIntegral n
+   in \d ->
+      fmap (\y -> 1-2*y) $
+      fromFunction $ \x ->
+         (x + (nf-1)/2*d - 1) + min 1 (ceiling((1-x)/d) / nf)
+
+_superSaw n d =
+   fmap (\y -> 1 - 2/fromIntegral n * y) $
+   fromFunction $ \x ->
+      sum $ map fraction $ take n $ iterate (d+) x
 
 
 {- | square -}
diff --git a/src/Synthesizer/CausalIO/Gate.hs b/src/Synthesizer/CausalIO/Gate.hs
--- a/src/Synthesizer/CausalIO/Gate.hs
+++ b/src/Synthesizer/CausalIO/Gate.hs
@@ -67,7 +67,7 @@
 
 instance Mn.Monoid (Chunk a) where
    mempty = error "Gate.mempty cannot be defined"
-   mappend = error "Gate.mappend cannot be defined"
+   mappend = (Sg.<>)
 
 instance CutG.Transform (Chunk a) where
    take n (Chunk dur mrel) =
diff --git a/src/Synthesizer/Frame/Stereo.hs b/src/Synthesizer/Frame/Stereo.hs
--- a/src/Synthesizer/Frame/Stereo.hs
+++ b/src/Synthesizer/Frame/Stereo.hs
@@ -1,5 +1,5 @@
 module Synthesizer.Frame.Stereo (
-   T, left, right, cons, map,
+   T, left, right, cons, map, swap,
    arrowFromMono, arrowFromMonoControlled, arrowFromChannels,
    Stereo.Channel(Left, Right), Stereo.select,
    Stereo.interleave,
diff --git a/src/Synthesizer/Interpolation/Custom.hs b/src/Synthesizer/Interpolation/Custom.hs
--- a/src/Synthesizer/Interpolation/Custom.hs
+++ b/src/Synthesizer/Interpolation/Custom.hs
@@ -91,7 +91,7 @@
       combineMany
          "Interpolation.element: list of functions empty"
          "Interpolation.element: list of samples empty" $
-            Sig.map ($t) $ Sig.fromList $ reverse ps
+            Sig.map ($ t) $ Sig.fromList $ reverse ps
 
 {-# INLINE piecewiseConstant #-}
 piecewiseConstant :: (Interpol.C t y) => T t y
diff --git a/src/Synthesizer/Interpolation/Module.hs b/src/Synthesizer/Interpolation/Module.hs
--- a/src/Synthesizer/Interpolation/Module.hs
+++ b/src/Synthesizer/Interpolation/Module.hs
@@ -72,7 +72,7 @@
    Int -> [t -> t] -> T t y
 piecewise center ps =
    cons (length ps) (center-1)
-      (\t -> Sig.linearComb (Sig.fromList (map ($t) (reverse ps))))
+      (\t -> Sig.linearComb (Sig.fromList (map ($ t) (reverse ps))))
 
 {-# INLINE piecewiseConstant #-}
 piecewiseConstant :: (Module.C t y) => T t y
diff --git a/src/Synthesizer/State/Signal.hs b/src/Synthesizer/State/Signal.hs
--- a/src/Synthesizer/State/Signal.hs
+++ b/src/Synthesizer/State/Signal.hs
@@ -944,7 +944,7 @@
 
 instance Monoid (T y) where
    mempty = empty
-   mappend = append
+   mappend = (<>)
 
 
 catMaybes :: T (Maybe a) -> T a
diff --git a/synthesizer-core.cabal b/synthesizer-core.cabal
--- a/synthesizer-core.cabal
+++ b/synthesizer-core.cabal
@@ -1,5 +1,5 @@
 Name:           synthesizer-core
-Version:        0.8.2.1
+Version:        0.8.3
 License:        GPL
 License-File:   LICENSE
 Author:         Henning Thielemann <haskell@henning-thielemann.de>
@@ -38,7 +38,7 @@
 
 
 Source-Repository this
-  Tag:         0.8.2.1
+  Tag:         0.8.3
   Type:        darcs
   Location:    http://code.haskell.org/synthesizer/core/
 
@@ -48,9 +48,9 @@
 
 Library
   Build-Depends:
-    sample-frame-np >=0.0.4 && <0.1,
+    sample-frame-np >=0.0.5 && <0.1,
     sox >=0.1 && <0.3,
-    transformers >=0.2 && <0.6,
+    transformers >=0.2 && <0.7,
     non-empty >=0.2 && <0.4,
     semigroups >=0.1 && <1.0,
     event-list >=0.1 && <0.2,
@@ -60,7 +60,7 @@
     numeric-quest >=0.1 && <0.3,
     utility-ht >=0.0.14 && <0.1,
     filepath >=1.1 && <1.5,
-    bytestring >=0.9 && <0.11,
+    bytestring >=0.9 && <0.12,
     binary >=0.1 && <1,
     deepseq >=1.1 && <1.5,
     storablevector >=0.2.5 && <0.3,
diff --git a/test/Test/Sound/Synthesizer/Plain/Interpolation.hs b/test/Test/Sound/Synthesizer/Plain/Interpolation.hs
--- a/test/Test/Sound/Synthesizer/Plain/Interpolation.hs
+++ b/test/Test/Sound/Synthesizer/Plain/Interpolation.hs
@@ -104,7 +104,7 @@
    (Interpol.C a v, Module.C a v, Eq v) =>
    a -> v -> [v] -> Bool
 constant t x0 xs =
-   equalList $ map ($(x0:xs)) $ map ($t) $
+   equalList $ map ($ (x0:xs)) $ map ($ t) $
       Interpolation.func ExampleCustom.constant :
       Interpolation.func ExampleCustom.piecewiseConstant :
       Interpolation.func ExampleModule.constant :
@@ -115,7 +115,7 @@
    (Interpol.C a v, Module.C a v, Eq v) =>
    a -> v -> v -> [v] -> Bool
 linear t x0 x1 xs =
-   equalList $ map ($(x0:x1:xs)) $ map ($t) $
+   equalList $ map ($ (x0:x1:xs)) $ map ($ t) $
       Interpolation.func ExampleCustom.linear :
       Interpolation.func ExampleCustom.piecewiseLinear :
       Interpolation.func ExampleModule.linear :
@@ -126,7 +126,7 @@
    (Interpol.C a v, VectorSpace.C a v, Eq v) =>
    a -> v -> v -> v -> v -> [v] -> Bool
 cubic t x0 x1 x2 x3 xs =
-   equalList $ map ($(x0:x1:x2:x3:xs)) $ map ($t) $
+   equalList $ map ($ (x0:x1:x2:x3:xs)) $ map ($ t) $
       Interpolation.func ExampleCustom.cubic :
       Interpolation.func ExampleCustom.piecewiseCubic :
       Interpolation.func ExampleModule.cubic :