TOP tuning: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2015-05-15 16:36:32 UTC</tt>.<br>

: The original revision id was <tt>551220834</tt>.<br>

If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes in the simple form of prime tunings which can be expressed by way of [[fractional monzos]]; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. The same thing can happen in higher ranks: 7-limit starling has the 3 and 7 eigenvalue tuning, and 11 and 13 limit thrush the 3 and 11 eighenvalue tuning, etc.

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 νₚ is the valuation val from prime p, meaning all coefficients bu the one for p are zero and the p coefficient is 1, then this "moving" can be accomplished by adding νₚ, 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.

If we want a pure-octaves tuning, we may divide the TIPTOP tuning by the tuning of 2, giving what may be called the Pure Octaves TIPTOP tuning, or POTT. The POTT tuning is sometimes in the simple form of prime tunings which can be expressed by way of &lt;a class="wiki_link" href="/fractional%20monzos"&gt;fractional monzos&lt;/a&gt;; for 5- and 7-limit meantone and 11-limit meanpop, we get the 1/4-comma tuning which is also the eigenmonzo 5 minimax tuning. For 7- and 11-limit pajara, we also get the eigenmonzo 5 tuning, with pure 5/4s, and for 13-limit POTT, we get the eigenmonzo 13 tuning. For 5-, 7-, 11-, and 13-limit myna, the POTT tuning is pure 3s. And so forth, for many other examples. The same thing can happen in higher ranks: 7-limit starling has the 3 and 7 eigenvalue tuning, and 11 and 13 limit thrush the 3 and 11 eighenvalue tuning, etc.&lt;br /&gt;

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 νₚ is the valuation val from prime p, meaning all coefficients bu the one for p are zero and the p coefficient is 1, then this &amp;quot;moving&amp;quot; can be accomplished by adding νₚ, 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 &lt;em&gt;TOP comma&lt;/em&gt;.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.&lt;br /&gt;