Uniform map: Difference between revisions

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== Integer uniform map ==

A uniform map whose multiplier is an integer is called an '''integer uniform map'''~~, or [[simple map]]~~. For example, we could use the integer 17 itself directly. So 17·{{map|log₂2 log₂3 log₂5}} = {{map|17 26.944 39.473}}, which also rounds to {{map|17 27 39}}.

A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly. So 17·{{map|log₂2 log₂3 log₂5}} = {{map|17 26.944 39.473}}, which also rounds to {{map|17 27 39}}. Every EDO has one integer uniform map.

== Simple map ==

Another name for an [[integer uniform map]] is a '''simple map'''. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:

* In contexts pertaining to tuning accuracy, "simple map" works well. This is probably the more common context.

* In contexts pertaining to other uniform maps, "integer uniform map" works well.

So the simple map is not necessarily the ''best'' map for its EDO in terms of overall tuning accuracy, but it is the ''simplest'' map to calculate. The classic example of a simple map which is not the best map is 17p in the 5-limit, 17·{{map|log₂2 log₂3 log₂5}} = {{map|17 26.944 39.473}} which rounds to {{map|17 27 39}}. The approximation of prime 5 is really bad here; it's about exactly halfway between 39 and 40 steps, but slightly below, which is why it rounds down. But it turns out that if we round up instead, using 40 steps to approximate prime 5, then the absolute errors in the primes remain about the same. However, the error in 5/3 is much less, because the error in 5 and the error in 3 are now in the same direction, canceling each other out, and so overall {{map|17 27 40}} has less error than {{map|17 27 39}}.

== Vs. related terminology ==

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A uniform map is the same thing as a [[generalized patent val]], or GPV.

An integer uniform map is the same thing as a [[patent val]].

An integer uniform map or simple map is the same thing as a [[patent val]].

As for the difference between "map" and "val", there is none, at least in most [[RTT]] cases; this is discussed here: [[Map]].

@@ Line 15: / Line 15: @@
 == Integer uniform map ==
-A uniform map whose multiplier is an integer is called an '''integer uniform map''', or [[simple map]]. For example, we could use the integer 17 itself directly.  So 17·{{map|log₂2 log₂3 log₂5}} = {{map|17 26.944 39.473}}, which also rounds to {{map|17 27 39}}.
+A uniform map whose multiplier is an integer is called an '''integer uniform map'''. For example, we could use the integer 17 itself directly.  So 17·{{map|log₂2 log₂3 log₂5}} = {{map|17 26.944 39.473}}, which also rounds to {{map|17 27 39}}. Every EDO has one integer uniform map.
+== Simple map ==
+Another name for an [[integer uniform map]] is a '''simple map'''. The two different terms provide two different ways of presenting the same object, which can be helpful in different contexts:
+* In contexts pertaining to tuning accuracy, "simple map" works well. This is probably the more common context.
+* In contexts pertaining to other uniform maps, "integer uniform map" works well.
+So the simple map is not necessarily the ''best'' map for its EDO in terms of overall tuning accuracy, but it is the ''simplest'' map to calculate. The classic example of a simple map which is not the best map is 17p in the 5-limit, 17·{{map|log₂2 log₂3 log₂5}} = {{map|17 26.944 39.473}} which rounds to {{map|17 27 39}}. The approximation of prime 5 is really bad here; it's about exactly halfway between 39 and 40 steps, but slightly below, which is why it rounds down. But it turns out that if we round up instead, using 40 steps to approximate prime 5, then the absolute errors in the primes remain about the same. However, the error in 5/3 is much less, because the error in 5 and the error in 3 are now in the same direction, canceling each other out, and so overall {{map|17 27 40}} has less error than {{map|17 27 39}}.
 == Vs. related terminology ==
@@ Line 21: / Line 29: @@
 A uniform map is the same thing as a [[generalized patent val]], or GPV.
-An integer uniform map is the same thing as a [[patent val]].
+An integer uniform map or simple map is the same thing as a [[patent val]].
 As for the difference between "map" and "val", there is none, at least in most [[RTT]] cases; this is discussed here: [[Map]].