Tenney–Euclidean temperament measures: Difference between revisions
Sintel's simple and logflat badnesses |
General formula for simple badness |
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== TE error == | == TE error == | ||
We can consider '''TE error''' to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in TE tuning. TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s. | We can consider '''TE error''' to be a weighted average of the error of each [[prime harmonic]]s in [[TE tuning]], that is, a weighted average of the [[error map]] in TE tuning. In this regard, TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s. | ||
By Graham Breed's definition<ref name="primerr"/>, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If ''T''<sub>''W''</sub> is the Tenney-weighted tuning map, then the TE error ''G'' can be found by | By Graham Breed's definition<ref name="primerr"/>, TE error may be accessed via [[Tenney–Euclidean tuning|TE tuning map]]. If ''T''<sub>''W''</sub> is the Tenney-weighted tuning map, then the TE error ''G'' can be found by | ||
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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. | ||
By Gene Ward Smith's definition, | By Gene Ward Smith's definition, TE error is derived from the relationship of TE simple badness and TE complexity. See the next section. We denote this definition of TE error ''Ψ''. | ||
From the ratio {{nowrap|(‖''J''<sub>''W''</sub> ∧ ''M''‖ / ‖''M''‖)<sup>2</sup>}} we obtain {{nowrap|{{sfrac|''C''(''n'', ''r'' + 1)|''n''⋅''C''(''n'', ''r'')}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank 1 with this for rank ''r'', the ''n'' cancels, and we get {{nowrap|{{sfrac|''n'' − 1|2}} · {{sfrac|''r'' + 1|''n'' − ''r''}} {{=}} {{sfrac|(''r'' + 1)(''n'' − 1)|2(''n'' − ''r'')}}}}. It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank-''r'' temperament then | From the ratio {{nowrap|(‖''J''<sub>''W''</sub> ∧ ''M''‖ / ‖''M''‖)<sup>2</sup>}} we obtain {{nowrap|{{sfrac|''C''(''n'', ''r'' + 1)|''n''⋅''C''(''n'', ''r'')}} {{=}} {{sfrac|''n'' − ''r''|''n''(''r'' + 1)}}}}. If we take the ratio of this for rank 1 with this for rank ''r'', the ''n'' cancels, and we get {{nowrap|{{sfrac|''n'' − 1|2}} · {{sfrac|''r'' + 1|''n'' − ''r''}} {{=}} {{sfrac|(''r'' + 1)(''n'' − 1)|2(''n'' − ''r'')}}}}. It follows that dividing TE error by the square root of this ratio gives a constant of proportionality such that if Ψ is the TE error of a rank-''r'' temperament then | ||
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== TE simple badness == | == TE simple badness == | ||
The '''TE simple badness''' of a temperament, which we may also call the '''relative error''' of a temperament, may be considered error relativized to the complexity of the temperament. It is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. | The '''TE simple badness''' of a temperament, which we may also call the '''relative error''' of a temperament, may be considered error relativized to the complexity of the temperament. It is error proportional to the complexity, or size, of the multival; in particular for a 1-val, it is (weighted) error compared to the size of a step. | ||
In general, if ''C'' is the complexity and ''E'' is the error of a temperament, then TE simple badness ''B'' is found by | |||
$$ B = C \cdot E $$ | |||
Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''J''<sub>''W''</sub> ∧ ''M''<sub>''W''</sub>‖<sub>RMS</sub>}}, where {{nowrap|''J''<sub>''W''</sub> {{=}} {{val| 1 1 … 1 }}}} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that {{nowrap|''a''<sub>''i''</sub> {{=}} ''J''<sub>''W''</sub>·('''v'''<sub>''w''</sub>)<sub>''i''</sub>/''n''}} is the mean value of the entries of ('''v'''<sub>''w''</sub>)<sub>''i''</sub>. Then note that {{nowrap|''J''<sub>''W''</sub> ∧ (('''v'''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>1</sub>''J''<sub>''W''</sub>) ∧ (('''v'''<sub>''w''</sub>)<sub>2</sub> − ''a''<sub>2</sub>''J''<sub>''W''</sub>) ∧ … ∧ (('''v'''<sub>''w''</sub>)<sub>''r''</sub> − ''a''<sub>''r''</sub>''J''<sub>''W''</sub>) {{=}} ''J''<sub>''W''</sub> ∧ ('''v'''<sub>''w''</sub>)<sub>1</sub> ∧ ('''v'''<sub>''w''</sub>)<sub>2</sub> ∧ … ∧ ('''v'''<sub>''w''</sub>)<sub>''r''</sub>}}, since wedge products with more than one term ''J''<sub>''W''</sub> are zero. The Gram matrix of the vectors ''J''<sub>''W''</sub> and {{nowrap|('''v'''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>''i''</sub>''J''<sub>''W''</sub>}} will have ''n'' as the {{nowrap|(1, 1)}} entry, and 0's in the rest of the first row and column. Hence we obtain: | Gene Ward Smith defines the simple badness of ''M'' as {{nowrap|‖''J''<sub>''W''</sub> ∧ ''M''<sub>''W''</sub>‖<sub>RMS</sub>}}, where {{nowrap|''J''<sub>''W''</sub> {{=}} {{val| 1 1 … 1 }}}} is the JIP in weighted coordinates. Once again, if we have a list of vectors we may use a Gramian to compute it. First we note that {{nowrap|''a''<sub>''i''</sub> {{=}} ''J''<sub>''W''</sub>·('''v'''<sub>''w''</sub>)<sub>''i''</sub>/''n''}} is the mean value of the entries of ('''v'''<sub>''w''</sub>)<sub>''i''</sub>. Then note that {{nowrap|''J''<sub>''W''</sub> ∧ (('''v'''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>1</sub>''J''<sub>''W''</sub>) ∧ (('''v'''<sub>''w''</sub>)<sub>2</sub> − ''a''<sub>2</sub>''J''<sub>''W''</sub>) ∧ … ∧ (('''v'''<sub>''w''</sub>)<sub>''r''</sub> − ''a''<sub>''r''</sub>''J''<sub>''W''</sub>) {{=}} ''J''<sub>''W''</sub> ∧ ('''v'''<sub>''w''</sub>)<sub>1</sub> ∧ ('''v'''<sub>''w''</sub>)<sub>2</sub> ∧ … ∧ ('''v'''<sub>''w''</sub>)<sub>''r''</sub>}}, since wedge products with more than one term ''J''<sub>''W''</sub> are zero. The Gram matrix of the vectors ''J''<sub>''W''</sub> and {{nowrap|('''v'''<sub>''w''</sub>)<sub>1</sub> − ''a''<sub>''i''</sub>''J''<sub>''W''</sub>}} will have ''n'' as the {{nowrap|(1, 1)}} entry, and 0's in the rest of the first row and column. Hence we obtain: | ||
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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 '' | 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 | ||
$$ L = B \cdot C^{r/(n - r)} $$ | $$ L = B \cdot C^{r/(n - r)} $$ | ||