Intro to Mappings: Difference between revisions

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==Contrast with rounding==

Now, consider the pitch 336/257. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 336/257is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 336/2<~~span style="vertical-align: super;"~~>57 </~~span~~>must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.

Now, consider the pitch 336/257. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 336/257is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 336/257 must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.

==Notation==

@@ Line 15: / Line 15: @@
 ==Contrast with rounding==
-Now, consider the pitch 3<sup>36</sup>/2<sup>57</sup>. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 3<sup>36</sup>/2<sup>57</sup>is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 3<sup>36</sup>/2<span style="vertical-align: super;">57 </span>must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.
+Now, consider the pitch 3<sup>36</sup>/2<sup>57</sup>. In JI, this pitch is 70.38... cents above A440, so the closest 12edo note to it is Bb. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (Bb). Why is this? The pitch 3<sup>36</sup>/2<sup>57</sup>is three Pythagorean commas above A. If each Pythagorean comma is represented by 0 steps, then since 0+0+0=0 the pitch 3<sup>36</sup>/2<sup>57 </sup>must be represented by A, even though in JI it's closer to Bb. Mapping it to Bb would require one of the three Pythaogrean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.
 ==Notation==