Fokker block: Difference between revisions

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-03-26 03:09:15 UTC</tt>.<br>

: The original revision id was <tt>314535908</tt>.<br>

Note that because any ak can be chosen satisfying 0 ≤ ak &lt; P, each Fokker block comes associated not only with various modes, but also [[Dome|domes]], which are variations on the same basic theme.  

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.

The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P &lt; uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &lt; 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.

Consider the periodic scale S[i] with quasiperiod P = 22 whose values for i from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that v = &lt;22 35 51 62 76| sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {&lt;&lt;1 9 -2 -6 12 -6 -13 -30 -45 -10||,  &lt;&lt;2 -4 -4 -12 -11 -12 -26 2 -14 -20||, &lt;&lt;6 10 10 8 2 -1 -8 -5 -16 -12||, &lt;&lt;2 -4 -4 10 -11 -12 9 2 37 42||}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22, which are pajara, magic = pajara+hedgehog-suprapyth-pajarous, orwell = pajara+hedgehog-suprapyth, porcupine = suprapyth+pajarous; hence, S is a Fokker block.

Note that because any ak can be chosen satisfying 0 ≤ ak &amp;lt; P, each Fokker block comes associated not only with various modes, but also &lt;a class="wiki_link" href="/Dome"&gt;domes&lt;/a&gt;, which are variations on the same basic theme. &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;

The n-1 vals u2, u3, ..., un defined in the previous section gave us n-1 inequalities ak - P &amp;lt; uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q &amp;lt; 2, and regard it as representing a pitch class, then it is associated to a lattice point in an n-1 dimensional vector space, and in that space the n-1 inequalities define the boundries of a parallepiped. The Fokker blocks can be defined as the pitch classes lying within such a paralellepiped.&lt;br /&gt;

Consider the periodic scale S[i] with quasiperiod P = 22 whose values for i from 0 to 22 are 1, 33/32, 16/15, 11/10, 9/8, 75/64, 6/5, 5/4, 165/128, 33/25, 11/8, 45/32, 35/24, 3/2, 99/64, 8/5, 33/20, 12/7, 7/4, 231/128, 15/8, 77/40, 2. By solving for the val, or simply testing to see if the patent val works, we quickly find that v = &amp;lt;22 35 51 62 76| sorts the scale in ascending order. A basis for the commas of this val is {50/49, 55/54, 64/63, 99/98}, and by taking three element subsets we find a basis for the wedgies to be {&amp;lt;&amp;lt;1 9 -2 -6 12 -6 -13 -30 -45 -10||,  &amp;lt;&amp;lt;2 -4 -4 -12 -11 -12 -26 2 -14 -20||, &amp;lt;&amp;lt;6 10 10 8 2 -1 -8 -5 -16 -12||, &amp;lt;&amp;lt;2 -4 -4 10 -11 -12 9 2 37 42||}, which is to say, {suprapyth, pajara, hedgehog, pajarous}. Taking Z-linear (integer coefficient) combinations, we quickly find that there are four and only four wedgies which give a Graham complexity for the scale less than 22, which are pajara, magic = pajara+hedgehog-suprapyth-pajarous, orwell = pajara+hedgehog-suprapyth, porcupine = suprapyth+pajarous; hence, S is a Fokker block.&lt;br /&gt;