Wedgie: Difference between revisions

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== Derivation from mapping matrices ==

Below is a script that finds the wedgie from a mapping matrix in [https://www.python.org/ Python], using [https://scipy.org/ Scipy].

import itertools

import numpy as np

from scipy import linalg

def wedgie (breeds):

combinations = itertools.combinations (range (breeds.shape[1]), breeds.shape[0])

wedgie = np.array ([linalg.det (breeds[:, entry]) for entry in combinations])

if wedgie[0] < 0:

wedgie *= -1

# convert to integer type if possible

wedgie_rd = np.rint (wedgie)

if np.allclose (wedgie, wedgie_rd, rtol = 0, atol = 1e-6):

wedgie = wedgie_rd.astype (int)

return wedgie

</syntaxhighlight>

== Derivation from edo joins ==

[[File:Wedgediagram .png|thumb|504x504px|Wedge product diagram]]

Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val| 5 8 12 }} and {{val| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival| -1 -4 -4 }}~~, which is the wedgie for 5 & 7, a.k.a. meantone~~. Note that ~~if you take a~~ wedgie ~~and~~ flip all the signs, it ~~still results in the same wedgie, so~~ {{multival| -1 -4 -4 }} is the ~~same as {{multival| 1 4 4 }} from before~~.

Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val| 5 8 12 }} and {{val| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival| -1 -4 -4 }}. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain {{multival| 1 4 4 }}, which is the wedgie for 5 & 7, a.k.a. meantone.

More than two vals can be combined into a higher-rank wedgie by an analogous method.

@@ Line 70: / Line 70: @@
 == Derivation from mapping matrices ==
 {{Todo|inline=1| expand }}
+Below is a script that finds the wedgie from a mapping matrix in [https://www.python.org/ Python], using [https://scipy.org/ Scipy].
+<syntaxhighlight lang="python">
+import itertools
+import numpy as np
+from scipy import linalg
+def wedgie (breeds):
+    combinations = itertools.combinations (range (breeds.shape[1]), breeds.shape[0])
+    wedgie = np.array ([linalg.det (breeds[:, entry]) for entry in combinations])
+    if wedgie[0] < 0:
+        wedgie *= -1
+    # convert to integer type if possible
+    wedgie_rd = np.rint (wedgie)
+    if np.allclose (wedgie, wedgie_rd, rtol = 0, atol = 1e-6):
+        wedgie = wedgie_rd.astype (int)
+    return wedgie
+</syntaxhighlight>
 == Derivation from edo joins ==
-{{See also| Linear algebra formalism #Wedge product }}
+{{See also|Wedge product }}
+[[File:Wedgediagram .png|thumb|504x504px|Wedge product diagram]]
-Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val| 5 8 12 }} and {{val| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival| -1 -4 -4 }}, which is the wedgie for 5 & 7, a.k.a. meantone. Note that if you take a wedgie and flip all the signs, it still results in the same wedgie, so {{multival| -1 -4 -4 }} is the same as {{multival| 1 4 4 }} from before.
+Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val| 5 8 12 }} and {{val| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival| -1 -4 -4 }}. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain {{multival| 1 4 4 }}, which is the wedgie for 5 & 7, a.k.a. meantone.
 More than two vals can be combined into a higher-rank wedgie by an analogous method.