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>

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 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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 <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 & 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 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.