Simple map

A simple map is a specific type of map used for EDOs. Every EDO has one simple map. The simple map for [math]\displaystyle{ n }[/math]-EDO is found by taking the just intonation point (JIP) for some prime limit [math]\displaystyle{ p }[/math], ⟨log₂2 log₂3 log₂5 ... log₂[math]\displaystyle{ p }[/math]], multiplying it by [math]\displaystyle{ n }[/math], then individually rounding entries to the nearest integer. For example, the simple map for 7-limit 19-EDO is 19·⟨log₂2 log₂3 log₂5 log₂7] = ⟨19 30.115 44.117 53.340] which rounds to ⟨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·⟨log₂2 log₂3 log₂5] = ⟨17 26.944 39.473] which rounds to ⟨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 ⟨17 27 40] has less error than ⟨17 27 39].

Vs. integer uniform maps

Another name for a simple map is an integer uniform 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.