Defactoring algorithms: Difference between revisions
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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> | <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 [[ | 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. | 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. | ||
{{Databox|Failed defactoring methods| | |||
When in development on an ideal defactoring method—the effort which culminated in column Hermite defactoring—Dave and Douglas experimented on other methods, which are imperfect (don't work all the time, are very slow, or too complicated). | When in development on an ideal defactoring method—the effort which culminated in column Hermite defactoring—Dave and Douglas experimented on other methods, which are imperfect (don't work all the time, are very slow, or too complicated). | ||
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==== Addabilization defactoring ==== | ==== Addabilization defactoring ==== | ||
This defactoring technique is used specifically in the process of preparing matrices for [[temperament addition]]; it defactors while managing to change only a single row of the original matrix, a necessary constraint of that problem. But this method is not computationally efficient or easier to understand, so unless you have this specific need, it is not your best option. Details can be found in [[Temperament addition #3. Addabiliziation defactoring]]. | This defactoring technique is used specifically in the process of preparing matrices for [[temperament addition]]; it defactors while managing to change only a single row of the original matrix, a necessary constraint of that problem. But this method is not computationally efficient or easier to understand, so unless you have this specific need, it is not your best option. Details can be found in [[Temperament addition #3. Addabiliziation defactoring]]. | ||
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== Finding the greatest factor == | == Finding the greatest factor == | ||