Interior product: Difference between revisions

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The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = <<<1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit [[Marvel_family#Marvel|marvel temperament]]. Then Marv∨45/44 = <<4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = <<-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = <<11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = <<-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = <<5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = <<6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = <<-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = <<6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨2200/2187 = <<-1 8 14 -23 15 25 -33 10, -81 -113||, garibaldi, Marv∨9801/9800 = <<-12 2 -20 6 31 2 51 -52 7 86||, wizard.

The interior product is also useful in finding the temperament ~~map~~ given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the ~~map~~. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [<0 0 -1 -2 3|, <0 1 0 2 -1|, <0 2 -2 0 4|, <0 -3 1 -4 0|, <-1 0 0 5 -12|, <-2 0 -5 0 -9|, <3 0 12 9 0|, <2 5 0 0 19|, <-1 -12 0 -19 0|, <4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|], the normal val list for 11-limit marvel. In practice this method nearly always suffices.

The interior product is also useful in finding the temperament mapping given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the mapping. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [<0 0 -1 -2 3|, <0 1 0 2 -1|, <0 2 -2 0 4|, <0 -3 1 -4 0|, <-1 0 0 5 -12|, <-2 0 -5 0 -9|, <3 0 12 9 0|, <2 5 0 0 19|, <-1 -12 0 -19 0|, <4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [<1 0 0 -5 12|, <0 1 0 2 -1|, <0 0 1 2 -3|], the normal val list for 11-limit marvel. In practice this method nearly always suffices.

[[Category:Math]]

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 The interior product can also be used to add a comma to a p-limit temperament of rank r, producing a rank r-1 temperament which supports it. For instance, Marv = &lt;&lt;&lt;1 2 -3 -2 1 -4 -5 12 9 -19||| is the wedgie for 11-limit [[Marvel_family#Marvel|marvel temperament]]. Then Marv∨45/44 = &lt;&lt;4 -3 2 5 -14 -8 -6 13 22 7||, 11-limit negri, Marv∨64/63 = &lt;&lt;-2 4 4 -10 11 12 -9 -2 -37 -42||, pajarous, Marv∨245/242 = &lt;&lt;11 -6 10 7 -35 -15 -27 40 37 -15||, septimin, Marv∨99/98 = &lt;&lt;-7 3 -8 -2 21 7 21 -27 -15 22||, orwell, Marv∨100/99 = &lt;&lt;5 1 12 -8 -10 5 -30 25 -22 -64||, magic, Marv∨243/242 = &lt;&lt;6 -7 -2 15 -25 -20 3 15 59 49||, miracle, Marv∨3136/3125 = &lt;&lt;-1 -4 -10 13 -4 -13 24 -12 44 71||, meanpop, Marv∨6250/6237 = &lt;&lt;6 5 22 -21 -6 18 -54 37 -66 -135||, catakleismic, Marv∨2200/2187 = &lt;&lt;-1 8 14 -23 15 25 -33 10, -81 -113||, garibaldi, Marv∨9801/9800 = &lt;&lt;-12 2 -20 6 31 2 51 -52 7 86||, wizard.
-The interior product is also useful in finding the temperament map given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the map. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [&lt;0 0 -1 -2 3|, &lt;0 1 0 2 -1|, &lt;0 2 -2 0 4|, &lt;0 -3 1 -4 0|, &lt;-1 0 0 5 -12|, &lt;-2 0 -5 0 -9|, &lt;3 0 12 9 0|, &lt;2 5 0 0 19|, &lt;-1 -12 0 -19 0|, &lt;4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [&lt;1 0 0 -5 12|, &lt;0 1 0 2 -1|, &lt;0 0 1 2 -3|], the normal val list for 11-limit marvel. In practice this method nearly always suffices.
+The interior product is also useful in finding the temperament mapping given the wedgie. Given a rank r p-limit wedgie, we can find a collection of vals belonging to it by taking the interior product with every set of r-1 primes less than or equal to p, and reducing this to the mapping. For instance, for Marv we take [Marv∨2∨3, Marv∨2∨5, ..., Marv∨7∨11], which gives [&lt;0 0 -1 -2 3|, &lt;0 1 0 2 -1|, &lt;0 2 -2 0 4|, &lt;0 -3 1 -4 0|, &lt;-1 0 0 5 -12|, &lt;-2 0 -5 0 -9|, &lt;3 0 12 9 0|, &lt;2 5 0 0 19|, &lt;-1 -12 0 -19 0|, &lt;4 -9 19 0 0|]. Hermite reducing this to a normal val list results in [&lt;1 0 0 -5 12|, &lt;0 1 0 2 -1|, &lt;0 0 1 2 -3|], the normal val list for 11-limit marvel. In practice this method nearly always suffices.
 [[Category:Math]]