Simple map: Difference between revisions

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#~~REDIRECT~~ [[~~Uniform~~ map~~#Simple~~ map]]

A '''simple map''' is a specific type of [[map]] used for [[EDO]]s. Every EDO has one simple map. The simple map for <math>n</math>-EDO is found by multiplying the [[just intonation point]] (JIP) {{map|log₂2 log₂3 log₂5 ...}} by <math>n</math>, then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19-EDO is 19·{{map|log₂2 log₂3 log₂5 log₂7}} = {{map|19 30.115 44.117 53.340}} which rounds to {{map|19 30 44 53}}.

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}}. We can see that 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 rounded down. It turns out that if we shrink our steps slightly so that it takes 40 steps to approximate 5 instead, then the approximations of primes 2 and 3 remain pretty good, and so overall {{map|17 27 40}} has less error than {{map|17 27 39}}. For more information on this issue, see: [[Douglas_Blumeyer's_RTT_How-To#A_multitude_of_maps]].

== Vs. integer uniform maps ==

Every simple map is also a [[integer uniform map]], and vice versa. They are identical objects, but the two different terms provide two different ways of presenting about the same object, which can be helpful in different contexts.

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

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

[[Category:Temperament]]

[[Category:Regular temperament theory]]

[[Category:Theory]]

[[Category:Terms]]

[[Category:Math]]

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-#REDIRECT [[Uniform map#Simple map]]
+A '''simple map''' is a specific type of [[map]] used for [[EDO]]s. Every EDO has one simple map. The simple map for <math>n</math>-EDO is found by multiplying the [[just intonation point]] (JIP) {{map|log₂2 log₂3 log₂5 ...}} by <math>n</math>, then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19-EDO is 19·{{map|log₂2 log₂3 log₂5 log₂7}} = {{map|19 30.115 44.117 53.340}} which rounds to {{map|19 30 44 53}}.
+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}}. We can see that 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 rounded down. It turns out that if we shrink our steps slightly so that it takes 40 steps to approximate 5 instead, then the approximations of primes 2 and 3 remain pretty good, and so overall {{map|17 27 40}} has less error than {{map|17 27 39}}. For more information on this issue, see: [[Douglas_Blumeyer's_RTT_How-To#A_multitude_of_maps]].
+== Vs. integer uniform maps ==
+Every simple map is also a [[integer uniform map]], and vice versa. They are identical objects, but the two different terms provide two different ways of presenting about the same object, which can be helpful in different contexts.
+* In contexts pertaining to tuning accuracy, "simple map" works great. This is probably the more common context.
+* In contexts pertaining to other uniform maps, "integer uniform map" works great.
+[[Category:Temperament]]
+[[Category:Regular temperament theory]]
+[[Category:Theory]]
+[[Category:Terms]]
+[[Category:Math]]