Defactoring: Difference between revisions

Line 185:

For example, the canonical form of [[Meantone_family#Meantone_.2812.2619.2C_2.3.5.29|5-limit meantone]] is {{vector|{{map|1 1 0}} {{map|0 1 4}}}}, and form has generators with sizes of approximately an [[octave]] and a [[perfect fifth]], respectively. But any pitch system constructed using an octave and a perfect fifth could also have been constructed using an octave and a [[perfect fourth]], because the perfect fourth is the [[octave complement]] of the perfect fifth. Specifically, any pitch we reached previously with a perfect fifth could be instead reached by going up an octave and down a perfect fourth. So in situations where we're approaching 5-limit meantone as a pitch system constructed by an octave and a perfect fourth, we might prefer to have the mapping in that form, which looks like {{vector|{{map|1 2 4}} {{map|0 -1 -4}}}}.

As a further example, we might prefer only to use generators that are approximations of primes, so we'd like meantone's mapping in the form where the generators are an octave and a [[tritave]] (3/1). This works for a similar reason: anything we could have reached with a perfect fifth we could also reach by moving up a tritave and down an octave, and that form looks like {{vector|{{map|1 0 -4}} {{map|0 1 4}}}}.

As a further example, we might prefer only to use generators that are approximations of primes, so we'd like meantone's mapping in the form where the generators are an octave and a perfect twelfth, or [[tritave]] (3/1). This works for a similar reason: anything we could have reached with a perfect fifth we could also reach by moving up a tritave and down an octave, and that form looks like {{vector|{{map|1 0 -4}} {{map|0 1 4}}}}.

{| class="wikitable"

~~### insert a table~~

|+example meantone mapping forms

![⟨octave] ⟨fifth]⟩

|{{vector|{{map|1 1 0}} {{map|0 1 4}}}}

|-

![⟨octave] ⟨fourth]⟩

|{{vector|{{map|1 2 4}} {{map|0 -1 -4}}}}

|-

![⟨octave] ⟨tritave]⟩

|{{vector|{{map|1 0 -4}} {{map|0 1 4}}}}

|}

Now clearly all three of these mapping forms look related, and they are indeed, but the exact relationships between them may not be immediately apparent, or how those relationships correspond to the relationships between their generator sizes. The purpose of this section is to demonstrate tricks for transforming from one matrix form to another so that we can make the generators the sizes we want, and along the way we'll look at how the tricks work in order to explain these relationships.

@@ Line 185: / Line 185: @@
 For example, the canonical form of [[Meantone_family#Meantone_.2812.2619.2C_2.3.5.29|5-limit meantone]] is {{vector|{{map|1 1 0}} {{map|0 1 4}}}}, and form has generators with sizes of approximately an [[octave]] and a [[perfect fifth]], respectively. But any pitch system constructed using an octave and a perfect fifth could also have been constructed using an octave and a [[perfect fourth]], because the perfect fourth is the [[octave complement]] of the perfect fifth. Specifically, any pitch we reached previously with a perfect fifth could be instead reached by going up an octave and down a perfect fourth. So in situations where we're approaching 5-limit meantone as a pitch system constructed by an octave and a perfect fourth, we might prefer to have the mapping in that form, which looks like {{vector|{{map|1 2 4}} {{map|0 -1 -4}}}}.
-As a further example, we might prefer only to use generators that are approximations of primes, so we'd like meantone's mapping in the form where the generators are an octave and a [[tritave]] (3/1). This works for a similar reason: anything we could have reached with a perfect fifth we could also reach by moving up a tritave and down an octave, and that form looks like {{vector|{{map|1 0 -4}} {{map|0 1 4}}}}.
+As a further example, we might prefer only to use generators that are approximations of primes, so we'd like meantone's mapping in the form where the generators are an octave and a perfect twelfth, or [[tritave]] (3/1). This works for a similar reason: anything we could have reached with a perfect fifth we could also reach by moving up a tritave and down an octave, and that form looks like {{vector|{{map|1 0 -4}} {{map|0 1 4}}}}.
+{| class="wikitable"
-### insert a table
+|+example meantone mapping forms
+![⟨octave] ⟨fifth]⟩
+|{{vector|{{map|1 1 0}} {{map|0 1 4}}}}
+|-
+![⟨octave] ⟨fourth]⟩
+|{{vector|{{map|1 2 4}} {{map|0 -1 -4}}}}
+|-
+![⟨octave] ⟨tritave]⟩
+|{{vector|{{map|1 0 -4}} {{map|0 1 4}}}}
+|}
 Now clearly all three of these mapping forms look related, and they are indeed, but the exact relationships between them may not be immediately apparent, or how those relationships correspond to the relationships between their generator sizes. The purpose of this section is to demonstrate tricks for transforming from one matrix form to another so that we can make the generators the sizes we want, and along the way we'll look at how the tricks work in order to explain these relationships.