Generalized Tenney dual norms and Tp tuning space: Difference between revisions
Wikispaces>genewardsmith **Imported revision 511015690 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 515964958 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014- | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-07-10 12:53:36 UTC</tt>.<br> | ||
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[[math]] | [[math]] | ||
for all f in **Tq<span style="font-size: 10px; vertical-align: super;">G</span>***. 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 maps 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. | for all f in **Tq<span style="font-size: 10px; vertical-align: super;">G</span>***. 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 maps 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. | ||
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--><script type="math/tex">||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | --><script type="math/tex">||f||_\mathbf{Tq*} = \text{sup}\left \{\frac{|f(\vec{v})|}{||\vec{v}||_\mathbf{Tp}}: \vec{v} \in \textbf{Lp}\right \}</script><!-- ws:end:WikiTextMathRule:0 --><br /> | ||
<br /> | <br /> | ||
for all f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>*. This normed space, for which the group of vals on <strong>G</strong> comprise the lattice of covectors with integer coefficients, is called <strong>Tq* Tuning Space</strong>. Other vectors in this space may be interpreted as tuning maps that send intervals in <strong>G</strong> to a certain number of cents (or other logarithmic units), although only tuning maps lying near the <strong>JIP</strong> will be of much musical relevance.<br /> | for all f in <strong>Tq<span style="font-size: 10px; vertical-align: super;">G</span></strong>*. This normed space, for which the group of vals on <strong>G</strong> comprise the lattice of covectors with integer coefficients, is called <strong>Tq* Tuning Space</strong>. Other vectors in this space may be interpreted as tuning maps that send intervals in <strong>G</strong> to a certain number of cents (or other logarithmic units), although only tuning maps lying near the <strong><a class="wiki_link" href="/JIP">JIP</a></strong> will be of much musical relevance.<br /> | ||
<br /> | <br /> | ||
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 &lt;7 11 16|) are rated less complex than vals which are further from the origin (such as &lt;171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map <strong>t</strong> and the <strong>JIP</strong> by looking at the quantity ||<strong>t</strong> - <strong>JIP</strong>||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for <strong>t - JIP</strong> over all intervals, and hence also gives us the maximum error for <strong>t</strong> over all intervals.<br /> | 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 &lt;7 11 16|) are rated less complex than vals which are further from the origin (such as &lt;171 271 397|). Additionally, if this metric is used on tuning maps, we can evaluate the average error for any tuning map <strong>t</strong> and the <strong>JIP</strong> by looking at the quantity ||<strong>t</strong> - <strong>JIP</strong>||. As per the definition of dual norm above, the Tq* norm of this vector gives us the maximum Tp-weighted mapping for <strong>t - JIP</strong> over all intervals, and hence also gives us the maximum error for <strong>t</strong> over all intervals.<br /> | ||