Uniform map: Difference between revisions

Line 3:

[[File:Uniform maps.png|thumb|right|500px|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]

What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ~~GPV~~ sequence]]s given for many [[regular temperament]]s across the wiki.

What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.

For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1·{{map|log₂2 log₂3 log₂5}} = {{map|17.1 27.103 39.705}} which rounds to {{map|17 27 40}}. This is one of the many uniform maps for 17-EDO.

@@ Line 3: / Line 3: @@
 [[File:Uniform maps.png|thumb|right|500px|This visualizes EDO maps. We could call it the uniform map continuum. Generator size decreases to the right, so the EDO number goes up. Each colored cell is the closest approximation of steps to a prime for that generator size. We can see uniform maps as any set of step counts for primes found as perfectly vertical lines drawn through such a diagram. And integer uniform maps would be any such vertical line that is also drawn straight through a number which appears in the row for n-ED2.]]
-What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal GPV sequence]]s given for many [[regular temperament]]s across the wiki.
+What this means is that somewhere along the continuum of all possible [[equal-step tuning]] [[generator]] sizes, we can find one whose individually closest approximations for each of the primes is given by this map. In this sense, uniform maps are those that give reasonable tunings (further discussion of this idea may be found [[Patent_val#Generalized_patent_val|here]]). The map with the overall best tuning accuracy for an EDO will always be a uniform map. In many [[regular temperament theory]] contexts, uniform maps are the only maps used, such as in the famous [[projective tuning space]] diagrams from [[Paul Erlich]]'s seminal A Middle Path paper, or the [[optimal ET sequence]]s given for many [[regular temperament]]s across the wiki.
 For an example, choosing 17.1 as our uniform multiplier, we find the map 17.1·{{map|log₂2 log₂3 log₂5}} = {{map|17.1 27.103 39.705}} which rounds to {{map|17 27 40}}. This is one of the many uniform maps for 17-EDO.