Fokker block: Difference between revisions

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-01-27 21:30:30 UTC</tt>.<br>

: The original revision id was <tt>401882282</tt>.<br>

Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep&gt;, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get w2*e2+w3*e3+...+wp*ep where the w2, w3 ... wp are integers. We interpret this as the [[Vals and Tuning Space|val]] v = &lt;w2 w3 ... wp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if w2&lt;0 we reverse sign, and we have a val V which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us V = &lt;22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit.

Now choose an example step for the Fokker block, which is a p-limit interval c such that V(c) = 1; that is, if m is the monzo for c, then &lt;V|m&gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that V(c)=1, the determinant of this matrix will be ∓1. It is therefore a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]], that is, a square matrix with coefficients which are integers and with determinant ∓1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the [[http://en.wikipedia.org/wiki/Kronecker_delta|Kronecker delta]]. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1. If vi(2)=0 the val and corresponding comma play no role, and we discard them; otherwise, by inverting the comma ci when vi(2) is negative, we can normalize the vals so that vi(2) is always positive, which we will assume.

Let us set ei = vi(2), and also P = e1 = v1(2), and choose n non-negative integers a1, ...., an with 0 ≤ ak &lt; P. Let ti = log2(ci), so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by

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 collection of all Fokker blocks for any of the allowed values of the ak offsets is an //arena//; a Fokker arena is defined entirely by its set of commas.

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|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 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 commas introduced previously; these are dual points of view: if we take all but one of the n-1 commas, it defines one of the wedgies, and if we take all but one of the wedgies, they define a comma.

  Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep&amp;gt;, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get w2*e2+w3*e3+...+wp*ep where the w2, w3 ... wp are integers. We interpret this as the &lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&gt;val&lt;/a&gt; v = &amp;lt;w2 w3 ... wp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if w2&amp;lt;0 we reverse sign, and we have a val V which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 225/224, 100/99, 176/175 and 385/384, the above procedure gives us V = &amp;lt;22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit.&lt;br /&gt;

Now choose an example step for the Fokker block, which is a p-limit interval c such that V(c) = 1; that is, if m is the monzo for c, then &amp;lt;V|m&amp;gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 225/224, 100/99, 176/175 and 385/384, we could for instance choose 22/21, 25/24, 28/27, 33/32, 36/35, 45/44 or 49/48. Having selected a step, form the n by n matrix whose first row is the monzo for the step c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that V(c)=1, the determinant of this matrix will be ∓1. It is therefore a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow"&gt;unimodular matrix&lt;/a&gt;, that is, a square matrix with coefficients which are integers and with determinant ∓1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow"&gt;Kronecker delta&lt;/a&gt;. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1. If vi(2)=0 the val and corresponding comma play no role, and we discard them; otherwise, by inverting the comma ci when vi(2) is negative, we can normalize the vals so that vi(2) is always positive, which we will assume.&lt;br /&gt;

  Let us set ei = vi(2), and also P = e1 = v1(2), and choose n non-negative integers a1, ...., an with 0 ≤ ak &amp;lt; P. Let ti = log2(ci), so that e1*t1+e2*t2+...+en*tn=1. Now define a function on the integers by&lt;br /&gt;

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. The collection of all Fokker blocks for any of the allowed values of the ak offsets is an &lt;em&gt;arena&lt;/em&gt;; a Fokker arena is defined entirely by its set of commas.&lt;br /&gt;

  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;

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. 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 commas introduced previously; these are dual points of view: if we take all but one of the n-1 commas, it defines one of the wedgies, and if we take all but one of the wedgies, they define a comma.&lt;br /&gt;