Projection: Difference between revisions

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=== Mapping non-uniqueness===

To be clear, a mapping matrix does ''not'' uniquely represent approximation information. Multiple mappings can be found that describe the same temperament, in the sense that the same set of [[commas]] ~~are~~ [[Tempering_out|~~vanished~~]]. This non-uniqueness is the reason why a [[canonical form]] for mappings was developed, which can be understood as a function which takes any equivalent mapping and converts it to the same exact mapping.

To be clear, a mapping matrix does ''not'' uniquely represent approximation information. Multiple mappings can be found that describe the same temperament, in the sense that the same set of [[commas]] [[Tempering_out|vanish]]. This non-uniqueness is the reason why a [[canonical form]] for mappings was developed, which can be understood as a function which takes any equivalent mapping and converts it to the same exact mapping.

What distinguishes these equivalent mappings from each other is the sizes of the generators they use.

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which technically ~~vanishes~~ the [[meantone comma]] <math>\frac{81}{80}</math> ~~— the~~ main requirement of being a meantone temperament — but it has a period of about 76 ¢ and generator of about 41 ¢, which is pretty strange indeed (we're not being specific about tuning here, just giving the ballpark sizes).

which technically makes the [[meantone comma]] <math>\frac{81}{80}</math> vanish — the main requirement of being a meantone temperament — but it has a period of about 76 ¢ and generator of about 41 ¢, which is pretty strange indeed (we're not being specific about tuning here, just giving the ballpark sizes).

What we can say is that generator form information differentiates these forms of the meantone mapping.<ref>There is a way to represent approximation information without generator form information, which therefore means a data structure which inherently uniquely represents temperaments, and that is a [[multimap]].</ref>

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A good way to understand the idempotence of projections is geometrically. This also agrees with our natural language intuitions about projections, such as projecting shadows of objects onto a wall. A projection is any transformation that reduces the dimensionality. What we've done is taken information in <math>d</math>-dimensional space and projected it down by one dimension, into <math>(d-1)</math>-dimensional space. For example we might take some 3D object and project it down to its silhouette. If we take the projection we just made and try to project it again, nothing is left to change.

But how to connect this to RTT? Well, when we vanish ~~commas~~, we reduce dimensionality. For example, 5-limit JI is three-dimensional, or ''rank-3'', because it is represented by a mapping matrix with three rows:

But how to connect this to RTT? Well, when we make commas vanish, we reduce dimensionality. For example, 5-limit JI is three-dimensional, or ''rank-3'', because it is represented by a mapping matrix with three rows:

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On the left, we've highlighted the diagonalized scaling factors with green if they are for unchanged-intervals, and with red if they are for ~~vanished~~ commas. Then on the right, we've colored the entire corresponding vector columns, and placed a vertical line between the two green columns corresponding to the two green scaling factors of 1 and the one red column corresponding to the one red scaling factor of 0. And so we can see that the green-colored part of <math>\mathrm{V}</math> is <math>\textrm{U}</math>, and the red-colored part of <math>V</math> is <math>\textrm{C}</math>.

On the left, we've highlighted the diagonalized scaling factors with green if they are for unchanged-intervals, and with red if they are for vanishing commas. Then on the right, we've colored the entire corresponding vector columns, and placed a vertical line between the two green columns corresponding to the two green scaling factors of 1 and the one red column corresponding to the one red scaling factor of 0. And so we can see that the green-colored part of <math>\mathrm{V}</math> is <math>\textrm{U}</math>, and the red-colored part of <math>V</math> is <math>\textrm{C}</math>.

==Generator information types==

@@ Line 283: / Line 283: @@
 === Mapping non-uniqueness===
-To be clear, a mapping matrix does ''not'' uniquely represent approximation information. Multiple mappings can be found that describe the same temperament, in the sense that the same set of [[commas]] are [[Tempering_out|vanished]]. This non-uniqueness is the reason why a [[canonical form]] for mappings was developed, which can be understood as a function which takes any equivalent mapping and converts it to the same exact mapping.
+To be clear, a mapping matrix does ''not'' uniquely represent approximation information. Multiple mappings can be found that describe the same temperament, in the sense that the same set of [[commas]] [[Tempering_out|vanish]]. This non-uniqueness is the reason why a [[canonical form]] for mappings was developed, which can be understood as a function which takes any equivalent mapping and converts it to the same exact mapping.
 What distinguishes these equivalent mappings from each other is the sizes of the generators they use.
@@ Line 331: / Line 331: @@
-which technically vanishes the [[meantone comma]] <math>\frac{81}{80}</math> — the main requirement of being a meantone temperament — but it has a period of about 76 ¢ and generator of about 41 ¢, which is pretty strange indeed (we're not being specific about tuning here, just giving the ballpark sizes).
+which technically makes the [[meantone comma]] <math>\frac{81}{80}</math> vanish — the main requirement of being a meantone temperament — but it has a period of about 76 ¢ and generator of about 41 ¢, which is pretty strange indeed (we're not being specific about tuning here, just giving the ballpark sizes).
 What we can say is that generator form information differentiates these forms of the meantone mapping.<ref>There is a way to represent approximation information without generator form information, which therefore means a data structure which inherently uniquely represents temperaments, and that is a [[multimap]].</ref>
@@ Line 908: / Line 908: @@
 A good way to understand the idempotence of projections is geometrically. This also agrees with our natural language intuitions about projections, such as projecting shadows of objects onto a wall. A projection is any transformation that reduces the dimensionality. What we've done is taken information in <math>d</math>-dimensional space and projected it down by one dimension, into <math>(d-1)</math>-dimensional space. For example we might take some 3D object and project it down to its silhouette. If we take the projection we just made and try to project it again, nothing is left to change.
-But how to connect this to RTT? Well, when we vanish commas, we reduce dimensionality. For example, 5-limit JI is three-dimensional, or ''rank-3'', because it is represented by a mapping matrix with three rows:
+But how to connect this to RTT? Well, when we make commas vanish, we reduce dimensionality. For example, 5-limit JI is three-dimensional, or ''rank-3'', because it is represented by a mapping matrix with three rows:
@@ Line 1,119: / Line 1,119: @@
-On the left, we've highlighted the diagonalized scaling factors with green if they are for unchanged-intervals, and with red if they are for vanished commas. Then on the right, we've colored the entire corresponding vector columns, and placed a vertical line between the two green columns corresponding to the two green scaling factors of 1 and the one red column corresponding to the one red scaling factor of 0. And so we can see that the green-colored part of <math>\mathrm{V}</math> is <math>\textrm{U}</math>, and the red-colored part of <math>V</math> is <math>\textrm{C}</math>.
+On the left, we've highlighted the diagonalized scaling factors with green if they are for unchanged-intervals, and with red if they are for vanishing commas. Then on the right, we've colored the entire corresponding vector columns, and placed a vertical line between the two green columns corresponding to the two green scaling factors of 1 and the one red column corresponding to the one red scaling factor of 0. And so we can see that the green-colored part of <math>\mathrm{V}</math> is <math>\textrm{U}</math>, and the red-colored part of <math>V</math> is <math>\textrm{C}</math>.
 ==Generator information types==