Mapping: Difference between revisions

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<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 B♭ above that is 1, the A♭ below it is &~~minud~~;, 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 12''a'' + 19''b''.) 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 312/219 (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 B♭ above that is 1, the A♭ 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 {{nowrap|12''a'' + 19''b''}}.) 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 312/219 (the Pythagorean comma) is mapped to 0, the same note as 1/1.

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

=== Notation ===

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* A♯ = A440 + 4 octaves − 7 perfect fourths

In other words, every note can be represented as an ordered pair of integers (''x'', ''y'') where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.

In other words, every note can be represented as an ordered pair of integers {{nowrap|(''x'', ''y'')}} where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.

== Temperamental rank ==

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At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:

{{val| a b c }} – period

{{val| d e f }} – generator

The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.

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5/1 is simpler – we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:

5/1 is simpler—we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:

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This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves – it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.

This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves—it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.

If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):

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== Units ==

It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: units~~ analysis]].

It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].

== See also ==

* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT~~: mappings~~]], for a step-by-step textbook tutorial style introduction to this topic

* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] – for a step-by-step textbook tutorial style introduction to this topic

[[Category:Regular temperament theory]]

@@ Line 17: / Line 17: @@
 <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 B&#x266D; above that is 1, the A&#x266D; below it is &minud;, 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 12''a'' + 19''b''.) 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</sup>/2<sup>19</sup> (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 B&#x266D; above that is 1, the A&#x266D; below it is &minus;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 {{nowrap|12''a'' + 19''b''}}.) 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</sup>/2<sup>19</sup> (the Pythagorean comma) is mapped to 0, the same note as 1/1.
 === 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 B&#x266D;. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (B&#x266D;). 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 B&#x266D;. Mapping it to B&#x266D; would require one of the three Pythagorean 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 B&#x266D;. However, if you apply the mapping formula, you see that it is mapped to note 0 (A), not note 1 (B&#x266D;). 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 {{nowrap|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 B&#x266D;. Mapping it to B&#x266D; would require one of the three Pythagorean commas to be represented by 1 complete step (100 cents)! This illustrates the difference between regular mapping and rounding.
 === Notation ===
@@ Line 39: / Line 39: @@
 * A&#x266F; = A440 + 4 octaves &minus; 7 perfect fourths
-In other words, every note can be represented as an ordered pair of integers (''x'', ''y'') where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.
+In other words, every note can be represented as an ordered pair of integers {{nowrap|(''x'', ''y'')}} where ''x'' is the number of octaves from A440 (positive is up, negative is down), and ''y'' is the number of perfect fourths.
 == Temperamental rank ==
@@ Line 55: / Line 55: @@
 At first, we'll consider a 5-limit rank 2 example. A list of vals for such a temperament will take the following form:
-{{val| a b c }} – period
+{{val| a b c }} &ndash; period
-{{val| d e f }} – generator
+{{val| d e f }} &ndash; generator
 The top val is taken by convention to represent the generator chain which is the period, and the bottom one is taken to represent the one which is not.
@@ Line 73: / Line 73: @@
 {{val| 0 1 _ }}
-/1 is simpler – we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:
+/1 is simpler&mdash;we know that four meantone 3/2 generators gets us to 5/1. Since it lands us directly on 5/1, rather than something like 5/2 or 10/1, we don't need to shift by any octaves, and 4 generators and 0 periods is all we need:
 {{val| 1 1 0 }}
@@ Line 92: / Line 92: @@
 {{val| 0 -1 -4 }}
-This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves – it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.
+This is still rather intuitive: the 2/1 still maps to one period and no generators. The 3/1 is now reachable by two periods ''minus'' a generator, which is to say that it's just two octaves minus a perfect fourth. The 5/1 is a bit more complicated, but maps as four 4/3's ''down'', plus four octaves&mdash;it is left as an exercise to the reader to prove that in a meantone system this will actually yield 5/1.
 If you wanted your basis to be 2/1 and 3/1, you'd end up with the following list of vals (left as an exercise to the reader to derive):
@@ Line 105: / Line 105: @@
 == Units ==
-It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis]].
+It may be helpful to think of the units of each entry of a mapping as <math>\small 𝗴/𝗽</math>, read "generators per prime." Each mapping row corresponds to a different generator, and each mapping column corresponds to a different prime, and so the value of the mapping entry at the intersection of a given row and column tells how many of the corresponding generator are part of the temperament's approximation of the corresponding prime. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]].
 == See also ==
-* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT: mappings]], for a step-by-step textbook tutorial style introduction to this topic
+* [[Dave Keenan %26 Douglas Blumeyer%27s guide to RTT/Mappings]] &ndash; for a step-by-step textbook tutorial style introduction to this topic
 [[Category:Regular temperament theory]]