Uniform map: Difference between revisions

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A '''uniform map''' is any near-just [[map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit <math>p</math>, {{map|log₂2 log₂3 log₂5 ... log₂<math>p</math>}}, by the same positive real number before rounding them to integers.

[[File:~~Near_linings_up_rare2~~.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.]]

[[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.

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 A '''uniform map''' is any near-just [[map]] found by ''uniformly'' multiplying every entry of the [[just intonation point]] (JIP) for some prime limit <math>p</math>, {{map|log₂2 log₂3 log₂5 ... log₂<math>p</math>}}, by the same positive real number before rounding them to integers.
-[[File:Near_linings_up_rare2.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.]]
+[[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.