Generalized Tenney dual norms and Tp tuning space: Difference between revisions
Cmloegcmluin (talk | contribs) add link to tuning map |
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<math>||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math> | <math>||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</math> | ||
for all f in '''Tq<sup>G</sup>'''*. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq* Tuning Space'''. Other vectors in this space may be interpreted as tuning | for all f in '''Tq<sup>G</sup>'''*. This normed space, for which the group of vals on '''G''' comprise the lattice of covectors with integer coefficients, is called '''Tq* Tuning Space'''. Other vectors in this space may be interpreted as [[tuning map]]s that send intervals in '''G''' to a certain number of cents (or other logarithmic units), although only tuning maps lying near the '''[[JIP]]''' will be of much musical relevance. | ||
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. | ||