Tenney–Euclidean metrics: Difference between revisions

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To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo|1 0 0 … 0 }} representing 2 to the matrix B. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by A.

~~== TE logflat badness ==~~

Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank ''r'' of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if ''p'' is the largest such prime, then the dimension ''n'' is π(''p''), the number of primes to ''p''. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then '''logflat badness''' is defined by the formula

~~<math>\displaystyle~~

~~S(A)C(A)^{r/(n-r)}</math>~~

If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.

== Examples ==

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 To define the '''octave equivalent Tenney-Euclidean seminorm''', or '''OETES''', we simply add a column {{monzo|1 0 0 … 0 }} representing 2 to the matrix B. An alternative procedure is to find the [[Normal lists #Normal val list|normal val list]], and remove the first val from the list, corresponding to the octave or some fraction thereof, and proceed as in the previous section on temperamental complexity. This seminorm is a measure of the octave-equivalent complexity of a given ''p''-limit rational interval in terms of the ''p''-limit regular temperament given by A.
-== TE logflat badness ==
-Given a matrix A whose rows are linearly independent vals defining a regular temperament, then the rank ''r'' of the temperament is the number of rows, which equals the number of linearly independent vals. The dimension of the temperament is the number of primes it covers; if ''p'' is the largest such prime, then the dimension ''n'' is π(''p''), the number of primes to ''p''. If we define S(A) to be the simple badness (relative error) of A, and C(A) to be the complexity of A, then '''logflat badness''' is defined by the formula
-<math>\displaystyle
-S(A)C(A)^{r/(n-r)}</math>
-If we set a cutoff margin for logflat badness, there are still infinite numbers of new temperaments appearing as complexity goes up, at a lower rate which is approximately logarithmic in terms of complexity.
 == Examples ==