Tenney–Euclidean temperament measures: Difference between revisions
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There have been several minor variations in the definition of TE temperament measures, which differ from each other only in their choice of multiplicative scaling factor. Each of these variations will be discussed below. | There have been several minor variations in the definition of TE temperament measures, which differ from each other only in their choice of multiplicative scaling factor. Each of these variations will be discussed below. | ||
== Note on scaling factors == | == Note on scaling factors == | ||
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where {{!}}''A''{{!}} denotes the determinant of ''A'', and ''A''{{t}} denotes the transpose of ''A''. | where {{!}}''A''{{!}} denotes the determinant of ''A'', and ''A''{{t}} denotes the transpose of ''A''. | ||
We denote the RMS norm as ‖''M''‖<sub>RMS</sub>. In Graham Breed's paper, an RMS norm is proposed as | We denote the RMS norm as ‖''M''‖<sub>RMS</sub>. In Graham Breed's paper<ref name="primerr">[http://x31eq.com/temper/primerr.pdf ''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''</ref>, an RMS norm is proposed as | ||
$$ \norm{M}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M}_2}{\sqrt {n^r}} $$ | $$ \norm{M}_\text{RMS} = \sqrt {\abs{\frac {V_W V_W^\mathsf{T}}{n}}} = \frac {\norm{M}_2}{\sqrt {n^r}} $$ | ||
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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. TE error may be expressed in any logarithmic [[interval size unit]]s such as [[cent]]s or [[octave]]s. | ||
By Graham Breed's definition, 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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| 6.441×10<sup>−3</sup> : 6.218×10<sup>−3</sup> | | 6.441×10<sup>−3</sup> : 6.218×10<sup>−3</sup> | ||
|} | |} | ||
<references /> | |||
== See also == | |||
* [[Cangwu badness]] – a derived badness measure with a free parameter that enables one to specify a tradeoff between complexity and error | |||
== Notes == | |||
<references/> | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||