Projection: Difference between revisions

Line 259:

=== With respect to the generator tuning map===

A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref>Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map|1.000 1.585 2.232}}·{{ket|{{map|1 1 0}} {{map|0 0 0}} {{map|0 1/4 1}}}} = {{map|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map|log₂2 log₂3 log₂5}} ≈ {{map|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map|1.000 1.585 2.232}}·{{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}} = {{map|1.000 0.580}}, or in cents instead of octaves, that's {{~~map~~|1200.000 696.578}}.

A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref>Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map|1.000 1.585 2.232}}·{{ket|{{map|1 1 0}} {{map|0 0 0}} {{map|0 1/4 1}}}} = {{map|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map|log₂2 log₂3 log₂5}} ≈ {{map|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map|1.000 1.585 2.232}}·{{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}} = {{map|1.000 0.580}}, or in cents instead of octaves, that's {{rbra|1200.000 696.578}}.

Many popular regular temperament tuning schemes work by optimizing for the entries of <math>𝒈</math> directly, and many times it's not helpful or insightful to view the generators in non-integer vector form, which are reasons for <math>𝒈</math>'s popularity over <math>G</math>. Some practitioners may not even view tuning as an optimization problem and will simply choose values for <math>𝒈</math> on gut feeling. This is all to say that this idea of approximating and then re-embedding, AKA projecting, is not an inherently necessary feature of RTT; it is only one way to look at it which may be valuable to some musicians and theoreticians but completely bonkers-seeming and convoluted to others.

@@ Line 259: / Line 259: @@
 === With respect to the generator tuning map===
-A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref>Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map|1.000 1.585 2.232}}·{{ket|{{map|1 1 0}} {{map|0 0 0}} {{map|0 1/4 1}}}} = {{map|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map|log₂2 log₂3 log₂5}} ≈ {{map|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map|1.000 1.585 2.232}}·{{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}} = {{map|1.000 0.580}}, or in cents instead of octaves, that's {{map|1200.000 696.578}}.
+A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref>Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map|1.000 1.585 2.232}}·{{ket|{{map|1 1 0}} {{map|0 0 0}} {{map|0 1/4 1}}}} = {{map|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT: units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map|log₂2 log₂3 log₂5}} ≈ {{map|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map|1.000 1.585 2.232}}·{{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}} = {{map|1.000 0.580}}, or in cents instead of octaves, that's {{rbra|1200.000 696.578}}.
 Many popular regular temperament tuning schemes work by optimizing for the entries of <math>𝒈</math> directly, and many times it's not helpful or insightful to view the generators in non-integer vector form, which are reasons for <math>𝒈</math>'s popularity over <math>G</math>. Some practitioners may not even view tuning as an optimization problem and will simply choose values for <math>𝒈</math> on gut feeling. This is all to say that this idea of approximating and then re-embedding, AKA projecting, is not an inherently necessary feature of RTT; it is only one way to look at it which may be valuable to some musicians and theoreticians but completely bonkers-seeming and convoluted to others.