TOP tuning: Difference between revisions

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== TOP commas and TOP extensions ==

Suppose ''T'' is a TOP tuned temperament with ''i'' intrinsic primes, ''e'' extrinsic primes, and a sharp semigroup of rank ''k'' + 1. Then the dimensionality of ''T'' is ''n'' = ''e'' + ''i''; the corank (rank of the comma group) is ''i'' - ''k'' and so the rank of the temperament is ''n'' - (''i'' - ''k'') = ''e'' + ''k''. If we move a prime from intrinsic to extrinsic, the rank is therefore increased by 1 and the corank decreased by 1, leaving the dimensionality the same. If ''ν''''p'' is the valuation val from prime ''p'', meaning all coefficients but the one for ''p'' are zero and the ''p'' coefficient is 1, then this "moving" can be accomplished by adding ''ν''''p'', for some prime ''p'' which is intrinsic but not a prime or inverse prime of the sharp semigroup, as the bottom row of the val list (mapping matrix) ~~for T, or equivalently wedging it with the wedgie~~ for T. This process can continue until all intrinsic primes except those for the sharp semigroup are moved to extrinsic primes. In this case, ''i'' = ''k'' + 1 so the corank is ''i'' - ''k'' = (''k'' + 1) - ''k'' = 1, and there is only one comma, defined as usual as a rational number number greater than one which is not a square, cube or other power, generating the kernel. Since either this comma or its inverse is a product in the sharp semigroup, its absolute proportional error is equal to APE (''T''). The result is that for any regular temperament, there is a unique comma of the temperament such that the absolute proportional error in any TOP tuning is equal to the maximal absolute proportional error for the temperament. This comma we may call the ''TOP comma''. The TOP comma in a sense encapsulates the error of the temperament. Any TOP tuning of the temperament, including TIPTOP, is also a TOP tuning of the codimension one temperament defined by the TOP comma.

Suppose ''T'' is a TOP tuned temperament with ''i'' intrinsic primes, ''e'' extrinsic primes, and a sharp semigroup of rank ''k'' + 1. Then the dimensionality of ''T'' is ''n'' = ''e'' + ''i''; the corank (rank of the comma group) is ''i'' - ''k'' and so the rank of the temperament is ''n'' - (''i'' - ''k'') = ''e'' + ''k''. If we move a prime from intrinsic to extrinsic, the rank is therefore increased by 1 and the corank decreased by 1, leaving the dimensionality the same. If ''ν''''p'' is the valuation val from prime ''p'', meaning all coefficients but the one for ''p'' are zero and the ''p'' coefficient is 1, then this "moving" can be accomplished by adding ''ν''''p'', for some prime ''p'' which is intrinsic but not a prime or inverse prime of the sharp semigroup, as the bottom row of the val list (mapping matrix) for T. This process can continue until all intrinsic primes except those for the sharp semigroup are moved to extrinsic primes. In this case, ''i'' = ''k'' + 1 so the corank is ''i'' - ''k'' = (''k'' + 1) - ''k'' = 1, and there is only one comma, defined as usual as a rational number number greater than one which is not a square, cube or other power, generating the kernel. Since either this comma or its inverse is a product in the sharp semigroup, its absolute proportional error is equal to APE (''T''). The result is that for any regular temperament, there is a unique comma of the temperament such that the absolute proportional error in any TOP tuning is equal to the maximal absolute proportional error for the temperament. This comma we may call the ''TOP comma''. The TOP comma in a sense encapsulates the error of the temperament. Any TOP tuning of the temperament, including TIPTOP, is also a TOP tuning of the codimension one temperament defined by the TOP comma.

For example, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, {{monzo| 0 -11 15 0 -5 }}; in the 13 limit, {{monzo| 0 0 46 0 -19 -11 }}.

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 == TOP commas and TOP extensions ==
-Suppose ''T'' is a TOP tuned temperament with ''i'' intrinsic primes, ''e'' extrinsic primes, and a sharp semigroup of rank ''k'' + 1. Then the dimensionality of ''T'' is ''n'' = ''e'' + ''i''; the corank (rank of the comma group) is ''i'' - ''k'' and so the rank of the temperament is ''n'' - (''i'' - ''k'') = ''e'' + ''k''. If we move a prime from intrinsic to extrinsic, the rank is therefore increased by 1 and the corank decreased by 1, leaving the dimensionality the same. If ''ν''<sub>''p''</sub> is the valuation val from prime ''p'', meaning all coefficients but the one for ''p'' are zero and the ''p'' coefficient is 1, then this "moving" can be accomplished by adding ''ν''<sub>''p''</sub>, for some prime ''p'' which is intrinsic but not a prime or inverse prime of the sharp semigroup, as the bottom row of the val list (mapping matrix) for T, or equivalently wedging it with the wedgie for T. This process can continue until all intrinsic primes except those for the sharp semigroup are moved to extrinsic primes. In this case, ''i'' = ''k'' + 1 so the corank is ''i'' - ''k'' = (''k'' + 1) - ''k'' = 1, and there is only one comma, defined as usual as a rational number number greater than one which is not a square, cube or other power, generating the kernel. Since either this comma or its inverse is a product in the sharp semigroup, its absolute proportional error is equal to APE (''T''). The result is that for any regular temperament, there is a unique comma of the temperament such that the absolute proportional error in any TOP tuning is equal to the maximal absolute proportional error for the temperament. This comma we may call the ''TOP comma''. The TOP comma in a sense encapsulates the error of the temperament. Any TOP tuning of the temperament, including TIPTOP, is also a TOP tuning of the codimension one temperament defined by the TOP comma.
+Suppose ''T'' is a TOP tuned temperament with ''i'' intrinsic primes, ''e'' extrinsic primes, and a sharp semigroup of rank ''k'' + 1. Then the dimensionality of ''T'' is ''n'' = ''e'' + ''i''; the corank (rank of the comma group) is ''i'' - ''k'' and so the rank of the temperament is ''n'' - (''i'' - ''k'') = ''e'' + ''k''. If we move a prime from intrinsic to extrinsic, the rank is therefore increased by 1 and the corank decreased by 1, leaving the dimensionality the same. If ''ν''<sub>''p''</sub> is the valuation val from prime ''p'', meaning all coefficients but the one for ''p'' are zero and the ''p'' coefficient is 1, then this "moving" can be accomplished by adding ''ν''<sub>''p''</sub>, for some prime ''p'' which is intrinsic but not a prime or inverse prime of the sharp semigroup, as the bottom row of the val list (mapping matrix) for T. This process can continue until all intrinsic primes except those for the sharp semigroup are moved to extrinsic primes. In this case, ''i'' = ''k'' + 1 so the corank is ''i'' - ''k'' = (''k'' + 1) - ''k'' = 1, and there is only one comma, defined as usual as a rational number number greater than one which is not a square, cube or other power, generating the kernel. Since either this comma or its inverse is a product in the sharp semigroup, its absolute proportional error is equal to APE (''T''). The result is that for any regular temperament, there is a unique comma of the temperament such that the absolute proportional error in any TOP tuning is equal to the maximal absolute proportional error for the temperament. This comma we may call the ''TOP comma''. The TOP comma in a sense encapsulates the error of the temperament. Any TOP tuning of the temperament, including TIPTOP, is also a TOP tuning of the codimension one temperament defined by the TOP comma.
 For example, in the 5 and 7 limits, the TOP comma for magic temperament is 3125/3072; in the 11-limit, {{monzo| 0 -11 15 0 -5 }}; in the 13 limit, {{monzo| 0 0 46 0 -19 -11 }}.