Defactoring algorithms: Difference between revisions

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{{Todo|inline=1| rework |text=Try explaining without wedgies outside historical notes.}}

This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring|defactor]] it.

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<math>

\left[ \begin{array} {ccc|cc}

0 & -11 & 4 & 13 & -6

0 & -11 & 4 & 13 & -6 \\

2 & 5 & 4 & -2 & 1 \\

\end{array} \right]

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\end{matrix} \right]</math>

The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 × 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with &~~inus~~;3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (−3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (−1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.

The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 × 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with −3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (−3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (−1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.

}}

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<blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group | ''Standardizing our wedge product'']</ref></blockquote>

Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[~~Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_EA_for_RTT~~#~~Multicomma_entries~~:~~_tempered_lattice_fractions_generated_by_prime_combinations~~|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.

Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: tempered lattice fractions generated by prime combinations|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.

Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.

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[[Category:Math]]

[[Category:Pages with proofs]]

[[Category:Algorithms]]

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+{{Todo|inline=1| rework |text=Try explaining without wedgies outside historical notes.}}
 This article discusses how to identify [[enfactoring]] in [[regular temperament]] [[mapping]]s and then [[defactoring|defactor]] it.
@@ Line 339: / Line 341: @@
 <math>
 \left[ \begin{array} {ccc|cc}
-& -11 & 4  &  13 & -6
+& -11 & 4  &  13 & -6 \\
 & 5 & 4  &  -2 & 1 \\
 \end{array} \right]
@@ Line 782: / Line 784: @@
 \end{matrix} \right]</math>
-The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 &#215; 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with &inus;3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (&minus;3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (&minus;1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.
+The pivots are 1 and 11, so that 11 tells us that we had a common factor of 11<ref group="note">In the doubly-enfactored case of {{rket|{{map|17 16 -4}} {{map|4 -4 1}}}}, i.e. with a common factor of {{nowrap|33 {{=}} 3 &#215; 11}}, the two pivots of the HNF are 3 and 11, putting each of them on display separately.</ref><ref group="note">It's interesting to observe that while the 11-enfactoring can be observed in the original matrix as a linear combination of 2 of the 1st row with &minus;3 of the 2nd row, i.e. 2{{map|6 5 -4}} + (&minus;3){{map|4 -4 1}} {{=}} {{map|0 22 -11}}, the linear combination of ''columns'', i.e. slicing the original {{rket|{{map|6 5 -4}} {{map|4 -4 1}}}} mapping the other direction like {{rbra|{{vector|6 4}} {{vector|5 -4}} {{vector|-4 1}}}}, that leads to the revelation of this 11 is completely different: (&minus;1){{vector|6 4}} + 2{{vector|5 -4}} + 1{{vector|-4 1}} {{=}} {{vector|0 11}}.</ref>. You could say that the HNF is useful for identifying common factors, but not for removing them. But if you leave them behind in the column-style HNF, the information that is retained in the unimodular matrix which is the other product of the Hermite decomposition, is enough to preserve everything important about the temperament, to get you back to where you started via an inverse and a trimming of extraneous rows.
 }}
@@ Line 792: / Line 794: @@
 <blockquote>''Since this is an invariant of the temperament, it would be a good thing to use to refer to it, but for the fact that it is opaque and does not immediately tell us how to define the temperament.''<ref group="note">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_1545.html#1545 Yahoo! Tuning Group | ''Standardizing our wedge product'']</ref></blockquote>
-Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_EA_for_RTT#Multicomma_entries:_tempered_lattice_fractions_generated_by_prime_combinations|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.
+Regarding any other advantages EA brought to the RTT table for beginners: they did not find any. The only minor advantage identified was how the largest-minors of the mapping which wedgies are a list of could also be read as a list of denominators of unit fractions of the tempered lattice which are capable of being generated by the associated combination of primes whose columns in the mapping were used in the calculation of the corresponding largest-minor (this idea is discussed in more detail [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT#Multicomma entries: tempered lattice fractions generated by prime combinations|here]]). Furthermore, several disadvantages of EA were identified, the main one being that it is more challenging to learn and use, involving higher level mathematical concepts than LA.
 Regarding the development of a canonical form for temperaments using only linear algebra, Dave and Douglas did manage to develop such a form, which is documented here: [[defactored Hermite form]]. It was Gene himself who first described this form (as the result of his "saturation" algorithm), so he either did not realize the full implications of his discovery, or it was simply not popularized and plugged in with the rest of the hive knowledge.
@@ Line 905: / Line 907: @@
 [[Category:Math]]
 [[Category:Pages with proofs]]
+[[Category:Algorithms]]