Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 509471130 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 509471662 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2014-05-17 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-05-17 01:26:52 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>509471662</tt>.<br> | ||
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<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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=Second definition of a Fokker block= | =Second definition of a Fokker block= | ||
Let us define a new set of vals by uk = P*vk - vk(2)*vn. To apply these vals to S[i], note first that floor((en*i+an)/P) = floor(i+an/P) = i, so that vn(S[i]) = i. Hence for k& | Let us define a new set of vals by uk = P*vk - vk(2)*vn. To apply these vals to S[i], note first that floor((en*i+an)/P) = floor(i+an/P) = i, so that vn(S[i]) = i. Hence un(S[i]) = P*vn - vn(2)*vn = 0, while for k<n, 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 vk and vn, which are both vals of the rank two temperament defined by the set of chromas 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 this rank two temperament; 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 vn∧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 the epimorph V. 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 arena of the Fokker block is defined equally well by the n-1 wedgies defining the n-1 distinct temperings as by the n-1 chromas introduced previously; these are dual points of view: if we take all but one of the n-1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely. | The val uk is a linear combination of vk and vn, which are both vals of the rank two temperament defined by the set of chromas 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 this rank two temperament; 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 vn∧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 the epimorph V. 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 arena of the Fokker block is defined equally well by the n-1 wedgies defining the n-1 distinct temperings as by the n-1 chromas introduced previously; these are dual points of view: if we take all but one of the n-1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely. | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Second definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:4 -->Second definition of a Fokker block</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Second definition of a Fokker block"></a><!-- ws:end:WikiTextHeadingRule:4 -->Second definition of a Fokker block</h1> | ||
Let us define a new set of vals by uk = P*vk - vk(2)*vn. To apply these vals to S[i], note first that floor((en*i+an)/P) = floor(i+an/P) = i, so that vn(S[i]) = i. Hence for k& | Let us define a new set of vals by uk = P*vk - vk(2)*vn. To apply these vals to S[i], note first that floor((en*i+an)/P) = floor(i+an/P) = i, so that vn(S[i]) = i. Hence un(S[i]) = P*vn - vn(2)*vn = 0, while for k&lt;n, 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 /> | <br /> | ||
The val uk is a linear combination of vk and vn, which are both vals of the rank two temperament defined by the set of chromas 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 this rank two temperament; 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 vn∧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 the epimorph V. 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. The arena of the Fokker block is defined equally well by the n-1 wedgies defining the n-1 distinct temperings as by the n-1 chromas introduced previously; these are dual points of view: if we take all but one of the n-1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely.<br /> | The val uk is a linear combination of vk and vn, which are both vals of the rank two temperament defined by the set of chromas 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 this rank two temperament; 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 vn∧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 the epimorph V. 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. The arena of the Fokker block is defined equally well by the n-1 wedgies defining the n-1 distinct temperings as by the n-1 chromas introduced previously; these are dual points of view: if we take all but one of the n-1 chromas, they define one of the wedgies, and if we take all but one of the wedgies, they define a chroma. The Fokker group basis is the dual basis of the chroma basis, and conversely.<br /> | ||