Saturation, torsion, and contorsion: Difference between revisions

Line 4:

A temperament matrix is '''saturated''' when it represents a temperament without any redundancies due to a common factor. A [[mapping]] is saturated when no common factor is found in its rows (i.e. generator maps). A [[comma basis]], the dual of a mapping, is saturated when no common factor is found in its columns (i.e. comma vectors).

To be more specific, a mapping is saturated if no [[Wikipedia: Coprime integers|coprime]]<ref>If the multiples used on the linear combinations themselves have a GCD > 1, the resulting row will always have a GCD >  1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{ket| {{map| 1 0 -4 }}, {{map| 0 1 4 }} }}; we can find the linear combination of rows {{nowrap|2 × {{map| 1 0 -4 }} + 4 × {{map| 0 1 4 }} {{=}} {{map| 2 4 8 }}}}, which has a GCD of 2, but that is clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.</ref> integer linear combination of its rows can produce another row whose entries have a common factor other than 1. For example, {{ket| {{map| 3 0 -1 }} {{map| 0 3 5 }} }} is ''not'' saturated, because {{nowrap|{{map| 3 0 -1 }} − {{map| 0 3 5 }} {{=}} {{map| 3 -3 6 }}}}, which has a common factor of 3. A mapping which consists of a single row with a common factor, such as {{ket| {{map| 24 38 56 }} }} with a visible common factor of 2, is also not saturated.

To be more specific, a mapping is saturated if no [[Wikipedia: Coprime integers|coprime]]<ref>If the multiples used on the linear combinations themselves have a GCD > 1, the resulting row will always have a GCD > 1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{ket| {{map| 1 0 -4 }}, {{map| 0 1 4 }} }}; we can find the linear combination of rows {{nowrap|2 × {{map| 1 0 -4 }} + 4 × {{map| 0 1 4 }} {{=}} {{map| 2 4 8 }}}}, which has a GCD of 2, but that is clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.</ref> integer linear combination of its rows can produce another row whose entries have a common factor other than 1. For example, {{ket| {{map| 3 0 -1 }} {{map| 0 3 5 }} }} is ''not'' saturated, because {{nowrap|{{map| 3 0 -1 }} − {{map| 0 3 5 }} {{=}} {{map| 3 -3 6 }}}}, which has a common factor of 3. A mapping which consists of a single row with a common factor, such as {{ket| {{map| 24 38 56 }} }} with a visible common factor of 2, is also not saturated.

The greatest factor possible to create as a GCD of a coprime linear combination of rows in this way is called the mapping's '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>, and so an alternative definition of saturation would be that the mapping must have a greatest factor of 1.

@@ Line 4: / Line 4: @@
 A temperament matrix is '''saturated''' when it represents a temperament without any redundancies due to a common factor. A [[mapping]] is saturated when no common factor is found in its rows (i.e. generator maps). A [[comma basis]], the dual of a mapping, is saturated when no common factor is found in its columns (i.e. comma vectors).
-To be more specific, a mapping is saturated if no [[Wikipedia: Coprime integers|coprime]]<ref>If the multiples used on the linear combinations themselves have a GCD &gt;&nbsp;1, the resulting row will always have a GCD &gt;&nbsp; 1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{ket| {{map| 1 0 -4 }}, {{map| 0 1 4 }} }}; we can find the linear combination of rows {{nowrap|2 × {{map| 1 0 -4 }} + 4 × {{map| 0 1 4 }} {{=}} {{map| 2 4 8 }}}}, which has a GCD of 2, but that is clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.</ref> integer linear combination of its rows can produce another row whose entries have a common factor other than 1. For example, {{ket| {{map| 3 0 -1 }} {{map| 0 3 5 }} }} is ''not'' saturated, because {{nowrap|{{map| 3 0 -1 }} − {{map| 0 3 5 }} {{=}} {{map| 3 -3 6 }}}}, which has a common factor of 3. A mapping which consists of a single row with a common factor, such as {{ket| {{map| 24 38 56 }} }} with a visible common factor of 2, is also not saturated.
+To be more specific, a mapping is saturated if no [[Wikipedia: Coprime integers|coprime]]<ref>If the multiples used on the linear combinations themselves have a GCD &gt;&nbsp;1, the resulting row will always have a GCD &gt;&nbsp;1, and such a linear combination therefore can not be used to demonstrate unsaturation. For example, consider the matrix {{ket| {{map| 1 0 -4 }}, {{map| 0 1 4 }} }}; we can find the linear combination of rows {{nowrap|2 × {{map| 1 0 -4 }} + 4 × {{map| 0 1 4 }} {{=}} {{map| 2 4 8 }}}}, which has a GCD of 2, but that is clearly a result of the fact that we used 2× and 4× of the original rows, and 2 and 4 have a GCD of 2. So this linear combination does not prove that the matrix is unsaturated. It is still possible that another linear combination might prove it, but this one does not.</ref> integer linear combination of its rows can produce another row whose entries have a common factor other than 1. For example, {{ket| {{map| 3 0 -1 }} {{map| 0 3 5 }} }} is ''not'' saturated, because {{nowrap|{{map| 3 0 -1 }} − {{map| 0 3 5 }} {{=}} {{map| 3 -3 6 }}}}, which has a common factor of 3. A mapping which consists of a single row with a common factor, such as {{ket| {{map| 24 38 56 }} }} with a visible common factor of 2, is also not saturated.
 The greatest factor possible to create as a GCD of a coprime linear combination of rows in this way is called the mapping's '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>, and so an alternative definition of saturation would be that the mapping must have a greatest factor of 1.