Defactoring: Difference between revisions

Line 174:

DC form is arguably even more important for comma-bases than it is for mappings, because enfactored mappings at least have clear musical meaning, while enfactored comma-bases are little but a wellspring of confusion. In other words, {{map|24 38 56}} may not be a true temperament, but it still represents a temperoid and an EDO. However, {{

== generator manipulation ==

A canonical mapping form is an important standard to have as community for uniquely identifying temperaments, but it is not the only mapping form one should ever need.

If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches. For example, the canonical form of 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 fifth, respectively. But any pitch system constructed using an octave and a fifth could also have been constructed using an octave and a fourth, because the fourth is the [[octave complement]] of the fifth; specifically, any pitch you reached previously with a fifth could be instead reached by going up an octave and down a fourth. So in situations where you're approaching 5-limit meantone as a pitch system constructed by an octave and a fourth, you 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, you might prefer only to use generators that are approximations of primes, so you'd like meantone's mapping in the form where the generators are an octave and a tritave, which works for a similar reason: namely, that anything you could have reached with a fifth you 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}}}}. Now clearly all three of these mappings look related, and they are, but the exact relationships between them may not be immediately apparent. The purpose of this section is to break down which elementary row operations to use on an RTT mapping in order to achieve the generator sizes you're looking for.

###

@@ Line 174: / Line 174: @@
 DC form is arguably even more important for comma-bases than it is for mappings, because enfactored mappings at least have clear musical meaning, while enfactored comma-bases are little but a wellspring of confusion. In other words, {{map|24 38 56}} may not be a true temperament, but it still represents a temperoid and an EDO. However, {{
+== generator manipulation ==
+A canonical mapping form is an important standard to have as community for uniquely identifying temperaments, but it is not the only mapping form one should ever need.
+If two mappings are equivalent, i.e. they have the same canonical form and therefore represent the same temperament, then their corresponding generators are equivalent too. That doesn't mean their generators are the same sizes; it only means that in combination with each other, their generators reach the same set of pitches. For example, the canonical form of 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 fifth, respectively. But any pitch system constructed using an octave and a fifth could also have been constructed using an octave and a fourth, because the fourth is the [[octave complement]] of the fifth; specifically, any pitch you reached previously with a fifth could be instead reached by going up an octave and down a fourth. So in situations where you're approaching 5-limit meantone as a pitch system constructed by an octave and a fourth, you 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, you might prefer only to use generators that are approximations of primes, so you'd like meantone's mapping in the form where the generators are an octave and a tritave, which works for a similar reason: namely, that anything you could have reached with a fifth you 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}}}}. Now clearly all three of these mappings look related, and they are, but the exact relationships between them may not be immediately apparent. The purpose of this section is to break down which elementary row operations to use on an RTT mapping in order to achieve the generator sizes you're looking for.
+###