Tenney–Euclidean temperament measures: Difference between revisions
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Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the third definition, as do parts of his [https://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the third definition, as do parts of his [https://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | ||
It is also possible to normalize the metrics to allow us to meaningfully compare temperaments across subgroups and even ranks. [[Sintel]]'s scheme in 2023 | It is also possible to normalize the metrics to allow us to meaningfully compare temperaments across subgroups and even ranks. [[Sintel]]'s scheme in 2023 is the first attempt at this goal<ref name="sintel">Sintel. [https://github.com/Sin-tel/temper/blob/c0d5c36e3c189f64860f4aea288ff3ff3bc34982/lib_temper/temper.py "Collection of functions for dealing with regular temperaments"], Temperament Calculator.</ref>. | ||
== TE complexity == | == TE complexity == | ||
TE complexity is the average hypervolume of the parallelepipeds formed by any ''n'' linearly independent generators which form and saturate that lattice. Since this is exactly the same thing as the magnitude of the multivector formed by those vectors, TE complexity is the exact same thing as the average of the coefficients of the [[wedgie]] for that temperament. | |||
Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the prime basis {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }}, | Let us define the val weighting matrix ''W'' to be the {{w|diagonal matrix}} with values 1, 1/log<sub>2</sub>3, 1/log<sub>2</sub>5 … 1/log<sub>2</sub>''p'' along the diagonal. For the prime basis {{nowrap|''Q'' {{=}} {{val| 2 3 5 … ''p'' }} }}, | ||
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If ''V'' is the mapping matrix of a temperament, then ''V<sub>W</sub>'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals (''v''<sub>''w''</sub>)<sub>''i''</sub>. | If ''V'' is the mapping matrix of a temperament, then ''V<sub>W</sub>'' {{=}} ''VW'' is the mapping matrix in the weighted space, its rows being the weighted vals (''v''<sub>''w''</sub>)<sub>''i''</sub>. | ||
Given a wedgie ''M'', that is a canonically reduced ''r''-val correspondng to a temperament of rank ''r'', the norm ‖''M''‖ is a measure of the complexity of ''M''; that is, how many notes in some sort of weighted average it takes to get to intervals. For 1-vals, for instance, it is approximately equal to the number of scale steps it takes to reach an octave. | |||
Our first complexity measure of a temperament is given by the ''L''<sup>2</sup> norm of the Tenney-weighted wedgie ''M''<sub>''W''</sub>, which can in turn be obtained from the Tenney-weighted mapping matrix ''V''<sub>''W''</sub>. This complexity can be easily computed either from the wedgie or from the mapping matrix, using the {{w|Gramian matrix|Gramian}}: | Our first complexity measure of a temperament is given by the ''L''<sup>2</sup> norm of the Tenney-weighted wedgie ''M''<sub>''W''</sub>, which can in turn be obtained from the Tenney-weighted mapping matrix ''V''<sub>''W''</sub>. This complexity can be easily computed either from the wedgie or from the mapping matrix, using the {{w|Gramian matrix|Gramian}}: | ||
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: '''Note''': that is the definition used by Graham Breed's temperament finder. | : '''Note''': that is the definition used by Graham Breed's temperament finder. | ||
Gene Ward Smith defines the TE error as the ratio ‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows: | Gene Ward Smith defines the TE error as the ratio {{nowrap|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}}, derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. From {{nowrap|‖''M''<sub>''W''</sub> ∧ ''J''<sub>''W''</sub>‖/‖''M''<sub>''W''</sub>‖}} we can extract a coefficient {{nowrap| sqrt(''C''(''n'', ''r'' + 1)/''C''(''n'', ''r'')) {{=}} sqrt((''n'' − ''r'')/(''r'' + 1)) }}, which relates ''Ψ'' with ''E'' as follows: | ||
$$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$ | $$ \Psi = \sqrt{\frac{r + 1}{n - r}} E $$ | ||
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''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap| ''G'' {{=}} sin ''θ'' }} octaves, where ''θ'' is the angle between ''J''<sub>''W''</sub> and the val in question. | ''G'' and ''ψ'' error both have the advantage that higher-rank temperament error corresponds directly to rank-1 error, but the RMS normalization has the further advantage that in the rank-1 case, {{nowrap| ''G'' {{=}} sin ''θ'' }} octaves, where ''θ'' is the angle between ''J''<sub>''W''</sub> and the val in question. | ||
Sintel defines the TE error as the ratio {{nowrap|''G'' {{=}} ‖''M''<sub>''U''</sub> ∧ ''J''<sub>''U''</sub>‖/‖''M''<sub>''U''</sub>‖}}, using ''U''-weighted norm (see the next section), and it results to the same value of Graham's definition. | |||
== TE simple badness == | == TE simple badness == | ||
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== TE logflat badness == | == TE logflat badness == | ||
Some consider the simple badness to be a sort of badness which favors complex temperaments. The '''logflat badness''' is developed to address that. If we define ''B'' to be the simple badness (relative error) of a temperament, and ''C'' to be the complexity, then the logflat badness ''L'' is defined by the formula | Some consider the simple badness to be a sort of badness which favors complex temperaments. The '''logflat badness''' (called ''Dirichlet coefficients'' in Sintel's scheme), is developed to address that. If we define ''B'' to be the simple badness (relative error) of a temperament, and ''C'' to be the complexity, then the logflat badness ''L'' is defined by the formula | ||
$$ L = B \cdot C^{r/(n - r)} $$ | $$ L = B \cdot C^{r/(n - r)} $$ | ||
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$$ L = \norm{ M_W \wedge J_W } \norm{M_W}^{r/(n - r)} $$ | $$ L = \norm{ M_W \wedge J_W } \norm{M_W}^{r/(n - r)} $$ | ||
In Sintel's | In Sintel's derivation, | ||
$$ L = \norm{ M_U \wedge J_U } \norm{M_U}^{r/(n - r)} / \norm{J_U} $$ | $$ L = \norm{ M_U \wedge J_U } \norm{M_U}^{r/(n - r)} / \norm{J_U} $$ | ||
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| 5.400 | | 5.400 | ||
| 2.763 | | 2.763 | ||
| | | 1.244×10<sup>−2</sup> | ||
|- | |- | ||
| Septimal magic | | Septimal magic | ||
| 7.195 | | 7.195 | ||
| 2.149 | | 2.149 | ||
| | | 1.288×10<sup>−2</sup> | ||
|} | |} | ||
{| class="wikitable center-all left-1" | {| class="wikitable center-all left-1" | ||
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| 2.631 | | 2.631 | ||
| 6.441×10<sup>−3</sup> | | 6.441×10<sup>−3</sup> | ||
|} | |||
{| class="wikitable center-all left-1" | |||
|+ style="font-size: 105%;" | Sintel's norm | |||
|- | |||
! Temperament | |||
! Complexity | |||
! Error (¢) | |||
! Simple badness | |||
|- | |||
| Septimal meantone | |||
| 17.357 | |||
| 1.382 | |||
| 1.999×10<sup>−2</sup> | |||
|- | |||
| Septimal magic | |||
| 23.126 | |||
| 1.074 | |||
| 2.070×10<sup>−2</sup> | |||
|} | |} | ||