Constrained tuning: Difference between revisions

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== Simple Fast Algorithm ==

A much simpler way to compute the CTE tuning is just to note that it's what we get if we modify the TE tuning so that the weighting of the 2's coefficient is very large. As the weighting goes to infinity, we get the CTE tuning. Thus, we can set it to some sufficiently large number, so that we get whatever numerical precision we want, and compute the result in closed-form using the pseudoinverse. The ~~result~~ is only ~~a few~~ lines of ~~numpy~~ code:

A much simpler way to compute the CTE tuning is just to note that it's what we get if we modify the TE tuning so that the weighting of the 2's coefficient is very large. As the weighting goes to infinity, we get the CTE tuning. Thus, we can set it to some sufficiently large number, so that we get whatever numerical precision we want, and compute the result in closed-form using the pseudoinverse. The calculation is only about five lines of python code:

import numpy as np

def modified_TE(limit, M, H=1):

def modified_TE(limit, M, H=1000000):

"""

Computes the ~~modified~~ TE/CTE tuning of a *full-limit* temperament given

Computes the interpolated TE/CTE tuning of a *full-limit* temperament given

a limit and mapping matrix ~~H. To compute the TE/CTE tuning of a subgroup~~

a limit and mapping matrix M. For H = 0 we have TE, and as H -> inf we have

~~temperament, first get the tuning of the full-limit temperament with the~~

CTE. To compute CTE to arbitrary precision, make H sufficiently large.

~~same kernel, and then restrict to the subgroup in question~~.

For subgroup temperaments, first compute the tuning of the full-limit

~~This function interpolates between the regular TE and the CTE: for~~ H = 0 we

temperament with same kernel, then multiply by the subgroup basis matrix.

have ~~the~~ TE, and as H -> ~~infinity~~ we have ~~the~~ CTE.

To compute ~~the~~ CTE to ~~any~~ precision, ~~set~~ H ~~to some~~ sufficiently large ~~value~~.

"""

# Compute adjusted TE weighting matrix and JIP

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return G, T

# %% Compute the CTE of ~~Sensi~~ temperament

# %% Compute the CTE of septimal meantone temperament

H = ~~1_000_000~~

H = 1000000

limit = np.array([2, 3, 5, 7])

M = np.array( # mapping matrix for ~~Sensi~~

M = np.array( # mapping matrix for meantone

[[1, -1, -1, -2],

[[1, 0, -4, -13],

[0, 7, ~~9, 13~~]]

[0, 1, 4, 10]]

)

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print("Generator map: " + str(G))

print("Tuning map: " + str(T))

The output to the above is:

Generator map: [ 1200.0000 1896.9521]

Tuning map: [ 1200.0000 1896.9521 2787.8086 3369.5214]

== CTE tuning vs POTE tuning CWE tuning vs CTWE tuning ==

@@ Line 185: / Line 185: @@
 == Simple Fast Algorithm ==
-A much simpler way to compute the CTE tuning is just to note that it's what we get if we modify the TE tuning so that the weighting of the 2's coefficient is very large. As the weighting goes to infinity, we get the CTE tuning. Thus, we can set it to some sufficiently large number, so that we get whatever numerical precision we want, and compute the result in closed-form using the pseudoinverse. The result is only a few lines of numpy code:
+A much simpler way to compute the CTE tuning is just to note that it's what we get if we modify the TE tuning so that the weighting of the 2's coefficient is very large. As the weighting goes to infinity, we get the CTE tuning. Thus, we can set it to some sufficiently large number, so that we get whatever numerical precision we want, and compute the result in closed-form using the pseudoinverse. The calculation is only about five lines of python code:
   import numpy as np
-  def modified_TE(limit, M, H=1):
+  def modified_TE(limit, M, H=1000000):
       """
-      Computes the modified TE/CTE tuning of a *full-limit* temperament given
+      Computes the interpolated TE/CTE tuning of a *full-limit* temperament given
-      a limit and mapping matrix H. To compute the TE/CTE tuning of a subgroup
+      a limit and mapping matrix M. For H = 0 we have TE, and as H -> inf we have
-     temperament, first get the tuning of the full-limit temperament with the
+     CTE. To compute CTE to arbitrary precision, make H sufficiently large.
-     same kernel, and then restrict to the subgroup in question.
+     For subgroup temperaments, first compute the tuning of the full-limit
-     This function interpolates between the regular TE and the CTE: for H = 0 we
+     temperament with same kernel, then multiply by the subgroup basis matrix.
-     have the TE, and as H -> infinity we have the CTE.
-     To compute the CTE to any precision, set H to some sufficiently large value.
       """
       # Compute adjusted TE weighting matrix and JIP
@@ Line 212: / Line 209: @@
       return G, T
-  # %% Compute the CTE of Sensi temperament
+  # %% Compute the CTE of septimal meantone temperament
-  H = 1_000_000
+  H = 1000000
   limit = np.array([2, 3, 5, 7])
-  M = np.array( # mapping matrix for Sensi
+  M = np.array( # mapping matrix for meantone
-      [[1, -1, -1, -2],
+      [[1, 0, -4, -13],
-       [0,  7,  9, 13]]
+       [0, 1,  4,  10]]
   )
@@ Line 224: / Line 221: @@
   print("Generator map: " + str(G))
   print("Tuning map: " + str(T))
+The output to the above is:
+ Generator map: [ 1200.0000   1896.9521]
+ Tuning map: [ 1200.0000   1896.9521   2787.8086   3369.5214]
 == CTE tuning vs POTE tuning CWE tuning vs CTWE tuning ==