Tenney–Euclidean temperament measures: Difference between revisions
m "L2" seemed to be unclear |
primerr.pdf better mentioned at the top |
||
| Line 1: | Line 1: | ||
The '''Tenney-Euclidean temperament measures''' (or '''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness. | The '''Tenney-Euclidean temperament measures''' (or '''TE temperament measures''') consist of TE complexity, TE error, and TE simple badness. | ||
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. Nonetheless, the following relationship always holds: | 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. Nonetheless, the following relationship always holds: | ||
| Line 7: | Line 5: | ||
<math>\displaystyle | <math>\displaystyle | ||
\text{TE simple badness} = \text{TE complexity} \times \text{TE error} </math> | \text{TE simple badness} = \text{TE complexity} \times \text{TE error} </math> | ||
TE temperament measures have been extensively studied by [[Graham Breed]] (see [http://x31eq.com/temper/primerr.pdf|''Prime Based Error and Complexity Measures''], often referred to as ''primerr.pdf''), who also proposed [[Cangwu badness]], an important derived measure, which adds a free parameter to TE simple badness that enables one to specify a tradeoff between complexity and error. | |||
== Introduction == | == Introduction == | ||
| Line 26: | Line 26: | ||
# Any of the above and also dividing by the norm of the JIP | # Any of the above and also dividing by the norm of the JIP | ||
Graham Breed's original definitions from his | Graham Breed's original definitions from his ''primerr.pdf'' paper tend to use the second definition, as do parts of his [http://x31eq.com/temper/ temperament finder], although other scaling and normalization methods are sometimes used as well. | ||
Note that the above is mainly for comparing temperaments within the same subgroup; when making intra-subgroup comparisons, this can be more complicated. | Note that the above is mainly for comparing temperaments within the same subgroup; when making intra-subgroup comparisons, this can be more complicated. | ||
== TE Complexity == | == TE Complexity == | ||
Given a [[Wedgies_and_Multivals|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. | Given a [[Wedgies_and_Multivals|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. We may call it '''Tenney-Euclidean complexity''', or '''TE complexity''' since it can be defined in terms of the [[Tenney-Euclidean_metrics|Tenney-Euclidean norm]]. | ||
Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [ | Below shows various definitions of TE complexity. All of them can be easily computed either from the multivector or from the mapping matrix, using the [[wikipedia:Gramian_matrix|Gramian]]. | ||
Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The ''L''<sup>2</sup> norm is one of the standard measures, | Let's denote a weighted mapping matrix, whose rows are the weighted vals ''v<sub>i</sub>'', as V. The ''L''<sup>2</sup> norm is one of the standard measures, | ||
| Line 47: | Line 47: | ||
||M||_\text{RMS} = \sqrt {\operatorname{det} (\frac {VV^\mathsf{T}}{n})} = \frac {||M||_2}{\sqrt {n^r}}</math> | ||M||_\text{RMS} = \sqrt {\operatorname{det} (\frac {VV^\mathsf{T}}{n})} = \frac {||M||_2}{\sqrt {n^r}}</math> | ||
where ''n'' is the number of primes up to the prime limit ''p'', and ''r'' is the rank of the temperament, which equals the number of vals wedged together to compute the wedgie. Note: this is the definition used by | where ''n'' is the number of primes up to the prime limit ''p'', and ''r'' is the rank of the temperament, which equals the number of vals wedged together to compute the wedgie. Note: this is the definition used by Graham Breed's temperament finder. | ||
[[Gene Ward Smith]] has recognized that TE complexity can be interpreted as the RMS norm of the wedgie. That defines another RMS norm, | [[Gene Ward Smith]] has recognized that TE complexity can be interpreted as the RMS norm of the wedgie. That defines another RMS norm, | ||