diff --git a/masque/library.py b/masque/library.py
index efc3aad..8dc63a4 100644
--- a/masque/library.py
+++ b/masque/library.py
@@ -466,9 +466,11 @@ class ILibraryView(Mapping[str, 'Pattern'], metaclass=ABCMeta):
             memo = {}
 
         if transform is None or transform is True:
-            transform = numpy.zeros(4)
+            transform = numpy.array([0, 0, 0, 0, 1], dtype=float)
         elif transform is not False:
             transform = numpy.asarray(transform, dtype=float)
+            if transform.size == 4:
+                transform = numpy.append(transform, 1.0)
 
         original_pattern = pattern
 
diff --git a/masque/ref.py b/masque/ref.py
index 8a72167..6423394 100644
--- a/masque/ref.py
+++ b/masque/ref.py
@@ -191,10 +191,11 @@ class Ref(
         xys = self.offset[None, :]
         if self.repetition is not None:
             xys = xys + self.repetition.displacements
-        transforms = numpy.empty((xys.shape[0], 4))
+        transforms = numpy.empty((xys.shape[0], 5))
         transforms[:, :2] = xys
         transforms[:, 2] = self.rotation
         transforms[:, 3] = self.mirrored
+        transforms[:, 4] = self.scale
         return transforms
 
     def get_bounds_single(
diff --git a/masque/utils/transform.py b/masque/utils/transform.py
index 62e0f1e..ed0453b 100644
--- a/masque/utils/transform.py
+++ b/masque/utils/transform.py
@@ -87,33 +87,41 @@ def apply_transforms(
     Apply a set of transforms (`outer`) to a second set (`inner`).
     This is used to find the "absolute" transform for nested `Ref`s.
 
-    The two transforms should be of shape Ox4 and Ix4.
-    Rows should be of the form `(x_offset, y_offset, rotation_ccw_rad, mirror_across_x)`.
-    The output will be of the form (O*I)x4 (if `tensor=False`) or OxIx4 (`tensor=True`).
+    The two transforms should be of shape Ox5 and Ix5.
+    Rows should be of the form `(x_offset, y_offset, rotation_ccw_rad, mirror_across_x, scale)`.
+    The output will be of the form (O*I)x5 (if `tensor=False`) or OxIx5 (`tensor=True`).
 
     Args:
-        outer: Transforms for the container refs. Shape Ox4.
-        inner: Transforms for the contained refs. Shape Ix4.
-        tensor: If `True`, an OxIx4 array is returned, with `result[oo, ii, :]` corresponding
+        outer: Transforms for the container refs. Shape Ox5.
+        inner: Transforms for the contained refs. Shape Ix5.
+        tensor: If `True`, an OxIx5 array is returned, with `result[oo, ii, :]` corresponding
             to the `oo`th `outer` transform applied to the `ii`th inner transform.
-            If `False` (default), this is concatenated into `(O*I)x4` to allow simple
+            If `False` (default), this is concatenated into `(O*I)x5` to allow simple
             chaining into additional `apply_transforms()` calls.
 
     Returns:
-        OxIx4 or (O*I)x4 array. Final dimension is
-            `(total_x, total_y, total_rotation_ccw_rad, net_mirrored_x)`.
+        OxIx5 or (O*I)x5 array. Final dimension is
+            `(total_x, total_y, total_rotation_ccw_rad, net_mirrored_x, total_scale)`.
     """
     outer = numpy.atleast_2d(outer).astype(float, copy=False)
     inner = numpy.atleast_2d(inner).astype(float, copy=False)
 
+    if outer.shape[1] == 4:
+        outer = numpy.pad(outer, ((0, 0), (0, 1)), constant_values=1.0)
+    if inner.shape[1] == 4:
+        inner = numpy.pad(inner, ((0, 0), (0, 1)), constant_values=1.0)
+
     # If mirrored, flip y's
     xy_mir = numpy.tile(inner[:, :2], (outer.shape[0], 1, 1))   # dims are outer, inner, xyrm
     xy_mir[outer[:, 3].astype(bool), :, 1] *= -1
 
+    # Apply outer scale to inner offset
+    xy_mir *= outer[:, None, 4, None]
+
     rot_mats = [rotation_matrix_2d(angle) for angle in outer[:, 2]]
     xy = numpy.einsum('ort,oit->oir', rot_mats, xy_mir)
 
-    tot = numpy.empty((outer.shape[0], inner.shape[0], 4))
+    tot = numpy.empty((outer.shape[0], inner.shape[0], 5))
     tot[:, :, :2] = outer[:, None, :2] + xy
 
     # If mirrored, flip inner rotation
@@ -122,6 +130,7 @@ def apply_transforms(
 
     tot[:, :, 2] = rotations % (2 * pi)
     tot[:, :, 3] = (outer[:, None, 3] + inner[None, :, 3]) % 2             # net mirrored
+    tot[:, :, 4] = outer[:, None, 4] * inner[None, :, 4]                   # net scale
 
     if tensor:
         return tot