Fokker block: 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>2012-03-09 16:49:21 UTC</tt>.<br>

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-09 17:57:21 UTC</tt>.<br>

: The original revision id was <tt>~~309544642~~</tt>.<br>

: The original revision id was <tt>309555482</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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Let is define a new set of vals by uk = P*vk - vk(2)*v1. To apply these vals to S[i], note first that floor((e1*i+a1)/P) = floor(i+a1/P) = i, so that v1(S[i]) = i. Hence for k>1, uk(S[i]) = P*vk(S[i]) - vk(2)*i. Since x-1 < floor(x) ≤ x, we have (ek*i + ak)/P-1 < floor((ek*i + ak)/P) ≤ (ek*i + ak)/P, so that ek*i + ak - P < P*vk(S[i]) ≤ ek*i + ak. Since ek = vk(2), this gives us ak - P < uk(S[i]) ≤ ak. This means that for each of the vals uk, the scale is mapped to a set of P integers.

The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a [[Normal lists|normal val list]], or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a [[The wedgie|~~wedge~~]] Wk, then the [[Interior product|interior products]] Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of v1. It follows that the abstract [[periodic scale]] Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.

The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a [[Normal lists|normal val list]], or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a [[The wedgie|wedgie]] Wk, then the [[Interior product|interior products]] Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of v1. It follows that the abstract [[periodic scale]] Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.

=Third definition of a Fokker block=

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Let is define a new set of vals by uk = P*vk - vk(2)*v1. To apply these vals to S[i], note first that floor((e1*i+a1)/P) = floor(i+a1/P) = i, so that v1(S[i]) = i. Hence for k&gt;1, uk(S[i]) = P*vk(S[i]) - vk(2)*i. Since x-1 &lt; floor(x) ≤ x, we have (ek*i + ak)/P-1 &lt; floor((ek*i + ak)/P) ≤ (ek*i + ak)/P, so that ek*i + ak - P &lt; P*vk(S[i]) ≤ ek*i + ak. Since ek = vk(2), this gives us ak - P &lt; uk(S[i]) ≤ ak. This means that for each of the vals uk, the scale is mapped to a set of P integers.<br />

<br />

The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a <a class="wiki_link" href="/Normal%20lists">normal val list</a>, or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a <a class="wiki_link" href="/The%20wedgie">~~wedge~~</a> Wk, then the <a class="wiki_link" href="/Interior%20product">interior products</a> Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the <a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals">Fokker group</a> of v1. It follows that the abstract <a class="wiki_link" href="/periodic%20scale">periodic scale</a> Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.<br />

The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a <a class="wiki_link" href="/Normal%20lists">normal val list</a>, or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a <a class="wiki_link" href="/The%20wedgie">wedgie</a> Wk, then the <a class="wiki_link" href="/Interior%20product">interior products</a> Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the <a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals">Fokker group</a> of v1. It follows that the abstract <a class="wiki_link" href="/periodic%20scale">periodic scale</a> Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.<br />

<br />

<h1 id="toc3"><a name="Third definition of a Fokker block"></a>Third definition of a Fokker block</h1>

@@ 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>2012-03-09 16:49:21 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-09 17:57:21 UTC</tt>.<br>
-: The original revision id was <tt>309544642</tt>.<br>
+: The original revision id was <tt>309555482</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 36: / Line 36: @@
 Let is define a new set of vals by uk = P*vk - vk(2)*v1. To apply these vals to S[i], note first that floor((e1*i+a1)/P) = floor(i+a1/P) = i, so that v1(S[i]) = i. Hence for k&gt;1, uk(S[i]) = P*vk(S[i]) - vk(2)*i. Since x-1 &lt; floor(x) ≤ x, we have (ek*i + ak)/P-1 &lt; floor((ek*i + ak)/P) ≤ (ek*i + ak)/P, so that ek*i + ak - P &lt; P*vk(S[i]) ≤ ek*i + ak. Since ek = vk(2), this gives us ak - P &lt; uk(S[i]) ≤ ak. This means that for each of the vals uk, the scale is mapped to a set of P integers.
-The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a [[Normal lists|normal val list]], or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a [[The wedgie|wedge]] Wk, then the [[Interior product|interior products]] Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of v1. It follows that the abstract [[periodic scale]] Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.
+The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a [[Normal lists|normal val list]], or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a [[The wedgie|wedgie]] Wk, then the [[Interior product|interior products]] Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the [[Minkowski reduced bases for Fokker groups of certain vals|Fokker group]] of v1. It follows that the abstract [[periodic scale]] Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.
 =Third definition of a Fokker block=
@@ Line 104: / Line 104: @@
 Let is define a new set of vals by uk = P*vk - vk(2)*v1. To apply these vals to S[i], note first that floor((e1*i+a1)/P) = floor(i+a1/P) = i, so that v1(S[i]) = i. Hence for k&amp;gt;1, uk(S[i]) = P*vk(S[i]) - vk(2)*i. Since x-1 &amp;lt; floor(x) ≤ x, we have (ek*i + ak)/P-1 &amp;lt; floor((ek*i + ak)/P) ≤ (ek*i + ak)/P, so that ek*i + ak - P &amp;lt; P*vk(S[i]) ≤ ek*i + ak. Since ek = vk(2), this gives us ak - P &amp;lt; uk(S[i]) ≤ ak. This means that for each of the vals uk, the scale is mapped to a set of P integers.&lt;br /&gt;
 &lt;br /&gt;
-The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal val list&lt;/a&gt;, or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a &lt;a class="wiki_link" href="/The%20wedgie"&gt;wedge&lt;/a&gt; Wk, then the &lt;a class="wiki_link" href="/Interior%20product"&gt;interior products&lt;/a&gt; Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the &lt;a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals"&gt;Fokker group&lt;/a&gt; of v1. It follows that the abstract &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.&lt;br /&gt;
+The val uk is a linear combination of v1 and vk, which are both vals of the set of commas {c2, c3, ... cn} minus {ck}. Since uk(2)=0, uk is a multiple of the generator step val of a &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal val list&lt;/a&gt;, or map, for the rank two temperament tempering out {c2, c3, ... ,cn} minus {ck}; in fact it is ∓mGk, where Gk is the generator step val and m is the number of periods to the octave. If we take the wedge product v1∧Gk and reduce it to a &lt;a class="wiki_link" href="/The%20wedgie"&gt;wedgie&lt;/a&gt; Wk, then the &lt;a class="wiki_link" href="/Interior%20product"&gt;interior products&lt;/a&gt; Wk∨S[i] for i from 1 to P are P distinct vals wi, each of which have wi(2) in a range of P successive values. The Wk are a basis for the &lt;a class="wiki_link" href="/Minkowski%20reduced%20bases%20for%20Fokker%20groups%20of%20certain%20vals"&gt;Fokker group&lt;/a&gt; of v1. It follows that the abstract &lt;a class="wiki_link" href="/periodic%20scale"&gt;periodic scale&lt;/a&gt; Wk∨S represents a MOS of the temperament defined by Wk. The Fokker block can be tempered in n-1 distinct rank two temperament ways to n-1 distinct MOS, and this provides another definition of a Fokker block: a periodic JI scale is Fokker if and only if from the rank n JI group it generates it can be tempered in n-1 ways to n-1 distinct MOS.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Third definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Third definition of a Fokker block&lt;/h1&gt;