Generalized Tenney dual norms and Tp tuning space: Difference between revisions
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== Dual norms == | == Dual norms == | ||
Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity: | Given any [[Generalized_Tenney_Norms_and_Tp_Interval_Space|Tp norm]] on an interval space '''Tp<sup>G</sup>''' associated with a group '''G''', we can define a corresponding '''dual Tq* norm''' on the dual space '''Tq<sup>G</sup>'''* which satisfies the following identity: | ||
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Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16|) are rated less complex than vals which are further from the origin (such as <171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity ||'''t''' - '''JIP'''||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals. | Note that this norm enables us to define something like a complexity metric on vals, where vals that are closer to the origin (such as <7 11 16|) are rated less complex than vals which are further from the origin (such as <171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map '''t''' and the '''JIP''' by looking at the quantity ||'''t''' - '''JIP'''||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for '''t - JIP''' over all intervals, and hence also gives us the maximum error for '''t''' over all intervals. | ||
==Prime | == Prime power interval groups == | ||
In the simplest case where '''G''' has as its chosen basis only primes and prime powers, || · ||'''<sub>Tp</sub>''' is given by | In the simplest case where '''G''' has as its chosen basis only primes and prime powers, || · ||'''<sub>Tp</sub>''' is given by | ||
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[[Category:Math]] | [[Category:Math]] | ||
[[Category:Tuning space]] | [[Category:Tuning space]] | ||
[[Category:Tenney]] | [[Category:Temperament complexity measures]] | ||
[[Category:Tenney-weighted measures]] | |||
{{Todo| cleanup }} | {{Todo| cleanup }} | ||