Intro to Mappings: Difference between revisions

Line 11:

<math>\left\{440\cdot 2^a\cdot 3^b\middle|a,b\in\mathbb Z\right\}</math>

Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab below it is -1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955... cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to 12a+19b.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<~~span style="vertical-align: super;"~~>12</~~span~~>/2<~~span style="vertical-align: super;"~~>19</~~span~~> (the Pythagorean comma) is mapped to 0, the same note as 1/1.

Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab below it is -1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955... cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to 12a+19b.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<sup>12</ups>/2<sup>19</sup> (the Pythagorean comma) is mapped to 0, the same note as 1/1.

==Contrast with rounding==

@@ Line 11: / Line 11: @@
 <math>\left\{440\cdot 2^a\cdot 3^b\middle|a,b\in\mathbb Z\right\}</math>
-Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab below it is -1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955... cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to 12a+19b.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<span style="vertical-align: super;">12</span>/2<span style="vertical-align: super;">19</span> (the Pythagorean comma) is mapped to 0, the same note as 1/1.
+Let's use integers to represent the 12edo notes, so that A440 is note 0, the Bb above that is 1, the Ab below it is -1, and so on. Then the mapping is simply expressed by saying that each factor of 2 counts for 12 steps, and each factor of 3 counts for 19 steps (because 3/1, or 1901.955... cents, is approximated as 1900 cents, or 19 steps of 12edo). (If you want a mathematical formula, that means that the above expression is mapped to 12a+19b.) So, for example, 1/1 is mapped to note 0, which is exactly A440; 2/1 is mapped to note 12, the A one octave higher; 3/2 is mapped to note 7 (the E above A440); and 3<sup>12</ups>/2<sup>19</sup> (the Pythagorean comma) is mapped to 0, the same note as 1/1.
 ==Contrast with rounding==