Tp tuning: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-24 14:54:44 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-26 14:01:37 UTC</tt>.<br>

: The original revision id was <tt>~~347655880~~</tt>.<br>

: The original revision id was <tt>348181842</tt>.<br>

: The revision comment was: <tt></tt><br>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>

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=Dual norm=

We can extend the Lp norm on monzos to a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]] on [[Monzos and Interval Space|interval space]], thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[http://en.wikipedia.org/wiki/Dual_norm|dual norm]]~~, which is the same as the PEps(T) previously defined~~. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then <cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).</pre></div>

We can extend the Lp norm on monzos to a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]] on [[Monzos and Interval Space|interval space]], thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[http://en.wikipedia.org/wiki/Dual_norm|dual norm]]. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then <cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).

=L2 tuning=

In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups. Starting from a [[gencom]] for the temperament, we may find the tuning by the following proceedure:

# Convert the gencom into monzo form.

# Convert the monzos to weighted coordinates.

# Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.

# Take the dot product of M with a vector consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; call that X.

# Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so this is easily done via Lagrange multipliers.

# The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.

# Find the point T on the subspace of G-tuning space spanned by the [[Gencom|gencom mapping]] closest to the JIP, which as usual is <1 1 1 ... 1|.

# Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.</pre></div>

<h4>Original HTML content:</h4>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tp tuning</title></head><body><a href="#Definition">Definition</a> | <a href="#Dual norm">Dual norm</a>

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Tp tuning</title></head><body><a href="#Definition">Definition</a> | <a href="#Dual norm">Dual norm</a> | <a href="#L2 tuning">L2 tuning</a>

<br />

<br />

<h1 id="toc0"><a name="Definition"></a>Definition</h1>

<strong>Lp tuning</strong> is a generalzation of <a class="wiki_link" href="/TOP%20tuning">TOP</a> and <a class="wiki_link" href="/Tenney-Euclidean%20tuning">TE</a> tuning. If p ≥ 1, define the Lp norm, which we may also call the Lp complexity, of any monzo in weighted coordinates b as <br />

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<br />

<h1 id="toc1"><a name="Dual norm"></a>Dual norm</h1>

We can extend the Lp norm on monzos to a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dual_norm" rel="nofollow">dual norm</a>~~, which is the same as the PEps(T) previously defined~~. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define <a class="wiki_link" href="/Smonzos%20and%20Svals">smonzos</a> for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).</body></html></pre></div>

We can extend the Lp norm on monzos to a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow">vector space norm</a> on <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">interval space</a>, thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dual_norm" rel="nofollow">dual norm</a>. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define <a class="wiki_link" href="/Smonzos%20and%20Svals">smonzos</a> for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).<br />

<br />

<h1 id="toc2"><a name="L2 tuning"></a>L2 tuning</h1>

In the special case where p = 2, the Lp norm becomes the L2 norm, which is the <a class="wiki_link" href="/Tenney-Euclidean%20metrics">Tenney-Euclidean</a> norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups. Starting from a <a class="wiki_link" href="/gencom">gencom</a> for the temperament, we may find the tuning by the following proceedure:<br />

<br />

<ol><li>Convert the gencom into monzo form.</li><li>Convert the monzos to weighted coordinates.</li><li>Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.</li><li>Take the dot product of M with a vector consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; call that X.</li><li>Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so this is easily done via Lagrange multipliers.</li><li>The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.</li><li>Find the point T on the subspace of G-tuning space spanned by the <a class="wiki_link" href="/Gencom">gencom mapping</a> closest to the JIP, which as usual is &lt;1 1 1 ... 1|.</li><li>Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.</li></ol></body></html></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-24 14:54:44 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-26 14:01:37 UTC</tt>.<br>
-: The original revision id was <tt>347655880</tt>.<br>
+: The original revision id was <tt>348181842</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 22: / Line 22: @@
 =Dual norm=
-We can extend the Lp norm on monzos to a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]] on [[Monzos and Interval Space|interval space]], thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[http://en.wikipedia.org/wiki/Dual_norm|dual norm]], which is the same as the PEps(T) previously defined. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).</pre></div>
+We can extend the Lp norm on monzos to a [[http://en.wikipedia.org/wiki/Normed_vector_space|vector space norm]] on [[Monzos and Interval Space|interval space]], thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a [[http://en.wikipedia.org/wiki/Dual_norm|dual norm]]. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define [[Smonzos and Svals|smonzos]] for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).
+=L2 tuning=
+In the special case where p = 2, the Lp norm becomes the L2 norm, which is the [[Tenney-Euclidean metrics|Tenney-Euclidean]] norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups. Starting from a [[gencom]] for the temperament, we may find the tuning by the following proceedure:
+# Convert the gencom into monzo form.
+# Convert the monzos to weighted coordinates.
+# Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.
+# Take the dot product of M with a vector consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; call that X.
+# Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so this is easily done via Lagrange multipliers.
+# The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.
+# Find the point T on the subspace of G-tuning space spanned by the [[Gencom|gencom mapping]] closest to the JIP, which as usual is &lt;1 1 1 ... 1|.
+# Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.</pre></div>
 <h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tp tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:6:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;!-- ws:start:WikiTextTocRule:8: --&gt; | &lt;a href="#Dual norm"&gt;Dual norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;
+<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Tp tuning&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Definition"&gt;Definition&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt; | &lt;a href="#Dual norm"&gt;Dual norm&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#L2 tuning"&gt;L2 tuning&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;
-&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;br /&gt;
+&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Definition"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Definition&lt;/h1&gt;
 &lt;strong&gt;Lp tuning&lt;/strong&gt; is a generalzation of &lt;a class="wiki_link" href="/TOP%20tuning"&gt;TOP&lt;/a&gt; and &lt;a class="wiki_link" href="/Tenney-Euclidean%20tuning"&gt;TE&lt;/a&gt; tuning. If p ≥ 1, define the Lp norm, which we may also call the Lp complexity, of any monzo in weighted coordinates b as &lt;br /&gt;
@@ Line 42: / Line 54: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Dual norm"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Dual norm&lt;/h1&gt;
-We can extend the Lp norm on monzos to a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;vector space norm&lt;/a&gt; on &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dual_norm" rel="nofollow"&gt;dual norm&lt;/a&gt;, which is the same as the PEps(T) previously defined. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define &lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;smonzos&lt;/a&gt; for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &amp;lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).&lt;/body&gt;&lt;/html&gt;</pre></div>
+We can extend the Lp norm on monzos to a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Normed_vector_space" rel="nofollow"&gt;vector space norm&lt;/a&gt; on &lt;a class="wiki_link" href="/Monzos%20and%20Interval%20Space"&gt;interval space&lt;/a&gt;, thereby defining the real normed interval space Lp. This space has a normed subspace generated by monzos belonging to the just intonation group G, which in the case where G is a full p-limit will be the whole of Lp but otherwise might not be; this we call G-interval space. The dual space to G-interval space is G-tuning space, and on this we may define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Dual_norm" rel="nofollow"&gt;dual norm&lt;/a&gt;. If r1, r2, ... rn are a set of generators for G, which in particular could be a normal list and so define &lt;a class="wiki_link" href="/Smonzos%20and%20Svals"&gt;smonzos&lt;/a&gt; for G, then corresponding generators for the dual space can in particular be the sval generators. On this standard basis for G-tuning space we can express the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the normal G generator list, then &amp;lt;cents(r1) cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning space, and the nearest point to it under the G-sval norm on the subspace of tunings of some abstract G-temperament S, meaning svals in the null space of its commas, is precisely the Lp tuning Lp(S).&lt;br /&gt;
+&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="L2 tuning"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;L2 tuning&lt;/h1&gt;
+In the special case where p = 2, the Lp norm becomes the L2 norm, which is the &lt;a class="wiki_link" href="/Tenney-Euclidean%20metrics"&gt;Tenney-Euclidean&lt;/a&gt; norm, or TE complexity. Associated to this norm is L2 tuning, or TE tuning, extended to arbitrary JI groups. Starting from a &lt;a class="wiki_link" href="/gencom"&gt;gencom&lt;/a&gt; for the temperament, we may find the tuning by the following proceedure:&lt;br /&gt;
+&lt;br /&gt;
+&lt;ol&gt;&lt;li&gt;Convert the gencom into monzo form.&lt;/li&gt;&lt;li&gt;Convert the monzos to weighted coordinates.&lt;/li&gt;&lt;li&gt;Multiply each monzo by an indeterminate, creating a parametrized n-dimensional weighted monzo, where n is the number of primes in the prime limit; call that M.&lt;/li&gt;&lt;li&gt;Take the dot product of M with a vector consisting of n new indeterminates x1, x2, ..., xn, and take the dot product of that with M; call that X.&lt;/li&gt;&lt;li&gt;Maximize X^2 subject to the constraint that the M.M, the dot product of M with itself, is equal to 1. This is maximinzing a quadric subject to a quadratic constraint, and so this is easily done via Lagrange multipliers.&lt;/li&gt;&lt;li&gt;The result of the maximization process is a positive definite quadratic form Q(x1, x2, ..., xn) in n indeterminates x1, x2, ..., xn.&lt;/li&gt;&lt;li&gt;Find the point T on the subspace of G-tuning space spanned by the &lt;a class="wiki_link" href="/Gencom"&gt;gencom mapping&lt;/a&gt; closest to the JIP, which as usual is &amp;lt;1 1 1 ... 1|.&lt;/li&gt;&lt;li&gt;Since the primes may or may not be in the group G, the mapping of primes in T may or may not make sense by itself; however, now unweight T and apply it to the generators of the gencom, and obtain the TE tuning of those generators, which defines the TE tuning for the temperament defined by the gencom.&lt;/li&gt;&lt;/ol&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>