Temperament addition: Difference between revisions

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====Introduction====

The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that ~~form~~ a ~~basis for preimage intervals (this basis is typically called "the~~ [[~~generator~~]]s"). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> vectors corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).

The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that are a [[generators preimage transversal]]). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> vectors corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).

So consider this first example of such a diagram:

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 ====Introduction====
-The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that form a basis for preimage intervals (this basis is typically called "the [[generator]]s"). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> vectors corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).
+The diagrams used for this explanation were inspired in part by [[Kite Giedraitis|Kite]]'s [[gencom]]s, and specifically how in his "twin squares" matrices — which have dimensions <math>d×d</math> — one can imagine shifting a bar up and down to change the boundary between vectors that form a basis for the commas and those that are a [[generators preimage transversal]]). The count of the former is the nullity <math>n</math>, and the count of the latter is the rank <math>r</math>, and the shifting of the boundary bar between them with the total <math>d</math> vectors corresponds to the insight of the rank-nullity theorem, which states that <math>r + n=d</math>. And so this diagram's square grid has just the right amount of room to portray both the mapping and the comma basis for a given temperament (with the comma basis's vectors rotated 90 degrees to appear as rows, to match up with the rows of the mapping).
 So consider this first example of such a diagram: