Constrained tuning: Difference between revisions
→Simple fast closed-form algorithm: cleanup: unify variable names with the other sections. Fix markup. Also important clarifications on the dimensionalities of these variables. |
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The problem is feasible if | The problem is feasible if | ||
# rank(''M'') ≤ rank(''V''), and | # rank(''M'') ≤ rank(''V''), and | ||
# The subgroups of ''M'' and | # The subgroups of ''M'' and nullspace(''V'') are {{w|linear independence|linearly independent}}. | ||
== Computation == | == Computation == | ||
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Notice we introduced the vector of lagrange multipliers ''Λ'', with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be discarded. | Notice we introduced the vector of lagrange multipliers ''Λ'', with length equal to the number of constraints. The lagrange multipliers have no concrete meaning for the resulting tuning, so they can be discarded. | ||
=== Simple fast closed-form algorithm === | ==== Simple fast closed-form algorithm ==== | ||
Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse. | Another way to compute the CTE and CWE tunings, and the CTWE tuning in general, is to use the pseudoinverse. | ||
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}} | }} | ||
=== Interpolating TE/CTE === | ==== Interpolating TE/CTE ==== | ||
We can also interpolate between the TE and CTE tunings, if we want. To do this, 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. Without comments, docstrings, etc, the calculation is only about five lines of python code: | We can also interpolate between the TE and CTE tunings, if we want. To do this, 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. Without comments, docstrings, etc, the calculation is only about five lines of python code: | ||
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J = 1200*np.log2(limit) | J = 1200*np.log2(limit) | ||
# Main calculation: get the generator map g such that g@M@W ≈ J@W. Use pinv | # Main calculation: get the generator map g such that g @ M @ W ≈ J @ W. Use pinv | ||
G = (J@W) @ np.linalg.pinv(M@W) | G = (J @ W) @ np.linalg.pinv(M @ W) | ||
T = G @ M | T = G @ M | ||
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</pre> | </pre> | ||
== | == Comparison of tunings == | ||
{{Todo|inline=1| rework |comment=More properly and concisely summarize each side's POV. }} | |||
=== Criticism of CTE === | === Criticism of CTE === | ||
People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle. | People have long noted, since the early days of the tuning list, that the CTE tuning, despite having very nice qualities on paper, can give surprisingly strange results.{{citation needed}} One good example is blackwood, where the 4:5:6 chord is tuned to 0–386–720 cents, so that the error is not even close to evenly divided between the 5/4, 6/5, and 3/2. The reasons for this are subtle. | ||
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So, one simple solution is to interpolate between the two, giving the '''Tenney–Weil–Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: {{nowrap|k {{=}} 0}} is the TE norm, {{nowrap|k {{=}} 1}} is the WE norm, and in between we have intermediate norms. This also gives a '''Constrained Tenney–Weil–Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE. | So, one simple solution is to interpolate between the two, giving the '''Tenney–Weil–Euclidean norm''': a weighted average of the TE and WE norms, with free weighting parameter k. This can be thought of as adjusting how much we care about the span: {{nowrap|k {{=}} 0}} is the TE norm, {{nowrap|k {{=}} 1}} is the WE norm, and in between we have intermediate norms. This also gives a '''Constrained Tenney–Weil–Euclidean''' or '''CTWE''' tuning as a result, which interpolates between CTE and CKE. | ||
=== | === Examples === | ||
These tunings can be very different from each other. | These tunings can be very different from each other. | ||