Fokker block: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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Suppose we have n-1 commas, which we will assume are greater than 1, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep>, 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 = <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<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 = <22 35 51 62 76|, and we will be looking at a 22-note scale in the 11-limit. We may call the val V the epimorph val, and the n-1 commas, which form a basis for the kernel of V, the chroma basis.

Now choose a uniformizing step for the Fokker block, by which is meant a p-limit interval c such that V(c) = 1; that is, if m is the monzo for c, then <V|m>=1. Precisely which interval with this property we choose doesn't actually matter, so if our chromas 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 chromas. 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 "cn", and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(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.

Now choose a uniformizing step for the Fokker block, by which is meant a p-limit interval c such that V(c) = 1; that is, if m is the monzo for c, then <V|m>=1. Precisely which interval with this property we choose doesn't actually matter, so if our chromas 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 last row is the monzo for the step c, and whose other rows are the monzos of the n-1 chromas. 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 "cn", and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(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.

These unimodular matricies define a [[http://en.wikipedia.org/wiki/Change_of_basis|change of basis]] for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as

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=Second definition of a Fokker block=

Let is define a new set of vals by uk = P*vk - vk(2)*vn. 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.

Let is 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>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 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 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~~.

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.

=Third definition of a Fokker block=

The n-1 vals u2~~, u3~~, ..., un defined in the previous section gave us n-1 inequalities ak - P < uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q < 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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.

The n-1 vals u1, u2, ..., u_(n-1) defined in the previous section gave us n-1 inequalities ak - P < uk(q) ≤ ak, which apply to any q in the Fokker block. If we restrict q to 1 ≤ q < 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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.

=Fourth definition of a Fokker block=

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=Determining if a scale is a Fokker block=

The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ~~∓Wk~~ for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i < P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.

The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i < P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.

Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.

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=Example=

==Using a Fokker group basis==

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 = <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 {<<1 9 -2 -6 12 -6 -13 -30 -45 -10||, <<2 -4 -4 -12 -11 -12 -26 2 -14 -20||, <<6 10 10 8 2 -1 -8 -5 -16 -12||, <<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, in the pajara-magic-orwell-porcupine arena.

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 = <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 {<<1 9 -2 -6 12 -6 -13 -30 -45 -10||, <<2 -4 -4 -12 -11 -12 -26 2 -14 -20||, <<6 10 10 8 2 -1 -8 -5 -16 -12||, <<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, and porcupine = suprapyth+pajarous; hence, S is a Fokker block, in the pajara-magic-orwell-porcupine arena.

If Q(a,b,c,d) is the ∑(T[i] - μ)^2 quadratic form on a*suprapyth+b*pajara+c*hedgehog+d*pajarous, then explicitly we have Q = 2205.5*a^2 + 880*b^2 + 2904*c^2 + 1254*d^2 + 264*a*b + 2992*a*c - 2574*a*d - 1848*b*c - 440*b*d - 880*c*d. From this we can find Q(pajara) = 880, Q(magic) = 885.5, Q(orwell) = 885.5, and Q(porcupine) = 885.5, with the Graham complexity of S being 21 in magic, orwell and porcupine, and 20 in pajara. If we look at the extrema of a, b, c, and d separately after setting Q = 900, we find they are all less than 2 in absolute value, so we need look no farther than the 27 Z-linear combinations of suprapyth, pajara, hedgehog and pajarous with coefficients less than 2 in absolute value. Had the block not been Fokker, we could have used the analysis of extrema to show it was not.

==Generator range and the first definition of a Fokker block==

From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. By forming the 5x5 matrix whose first row is v1, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 36/35, 385/384, 175/176, 100/99 and 224/225, inverting and transposing, we obtain [<22 35 51 62 76|, <12 19 28 34 42|, <3 5 7 9 10|, <9 14 21 25 31|, <7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that

From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13.

S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)

We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the [[Tenney-Euclidean Tuning#The Frobenius projection map|Frobenius projection map]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k

By forming the 5x5 matrix whose last row is V, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 385/384, 175/176, 100/99, 224/225 and 36/35, inverting and transposing, we obtain <12 19 28 34 42|, -<3 5 7 9 10|, <9 14 21 25 31|, -<7 11 16 20 24|], [<22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that

S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((-3*i+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)

is the periodic scale with which we began this analysis.

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Suppose we have n-1 commas, which we will assume are greater than 1, 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 <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> 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. We may call the val V the epimorph val, and the n-1 commas, which form a basis for the kernel of V, the chroma basis.<br />

<br />

Now choose a uniformizing step for the Fokker block, by which is meant 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 chromas 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 chromas. Because we have chosen c so that V(c)=1, the determinant of this matrix will be ±1. It is therefore a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>, 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 &quot;cn&quot;, and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(i,j) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow">Kronecker delta</a>. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.<br />

Now choose a uniformizing step for the Fokker block, by which is meant 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 chromas 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 last row is the monzo for the step c, and whose other rows are the monzos of the n-1 chromas. Because we have chosen c so that V(c)=1, the determinant of this matrix will be ±1. It is therefore a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>, 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 &quot;cn&quot;, and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(i,j) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow">Kronecker delta</a>. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.<br />

<br />

These unimodular matricies define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Change_of_basis" rel="nofollow">change of basis</a> for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as<br />

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<br />

<h1 id="toc2"><a name="Second definition of a Fokker block"></a>Second definition of a Fokker block</h1>

Let is define a new set of vals by uk = P*vk - vk(2)*vn. 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 />

Let is 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&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 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 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. 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~~.<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 />

<br />

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

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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.<br />

The n-1 vals u1, u2, ..., u_(n-1) 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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.<br />

<br />

<h1 id="toc4"><a name="Fourth definition of a Fokker block"></a>Fourth definition of a Fokker block</h1>

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<br />

<h1 id="toc5"><a name="Determining if a scale is a Fokker block"></a>Determining if a scale is a Fokker block</h1>

The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. <a class="wiki_link" href="/Scala">Scala</a> does this as a part of its &quot;Show data&quot; suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ~~∓Wk~~ for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a> to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &lt; P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.<br />

The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. <a class="wiki_link" href="/Scala">Scala</a> does this as a part of its &quot;Show data&quot; suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a <a class="wiki_link" href="/Graham%20complexity">Graham complexity</a> to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &lt; P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.<br />

<br />

Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.<br />

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<h1 id="toc7"><a name="Example"></a>Example</h1>

<h2 id="toc8"><a name="Example-Using a Fokker group basis"></a>Using a Fokker group basis</h2>

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, in the pajara-magic-orwell-porcupine arena.<br />

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, and porcupine = suprapyth+pajarous; hence, S is a Fokker block, in the pajara-magic-orwell-porcupine arena.<br />

<br />

If Q(a,b,c,d) is the ∑(T[i] - μ)^2 quadratic form on a*suprapyth+b*pajara+c*hedgehog+d*pajarous, then explicitly we have Q = 2205.5*a^2 + 880*b^2 + 2904*c^2 + 1254*d^2 + 264*a*b + 2992*a*c - 2574*a*d - 1848*b*c - 440*b*d - 880*c*d. From this we can find Q(pajara) = 880, Q(magic) = 885.5, Q(orwell) = 885.5, and Q(porcupine) = 885.5, with the Graham complexity of S being 21 in magic, orwell and porcupine, and 20 in pajara. If we look at the extrema of a, b, c, and d separately after setting Q = 900, we find they are all less than 2 in absolute value, so we need look no farther than the 27 Z-linear combinations of suprapyth, pajara, hedgehog and pajarous with coefficients less than 2 in absolute value. Had the block not been Fokker, we could have used the analysis of extrema to show it was not.<br />

<br />

<h2 id="toc9"><a name="Example-Generator range and the first definition of a Fokker block"></a>Generator range and the first definition of a Fokker block</h2>

From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. By forming the 5x5 matrix whose ~~first~~ row is v1, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for ~~36/35,~~ 385/384, 175/176, 100/99 ~~and~~ 224/225, inverting and transposing, we obtain ~~[&lt;22 35 51 62 76|,~~ &lt;12 19 28 34 42|, &lt;3 5 7 9 10|, &lt;9 14 21 25 31|, &lt;7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that<br />

From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. <br />

<br />

We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The Frobenius projection map">Frobenius projection map</a> P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k <br />

<br />

By forming the 5x5 matrix whose last row is V, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 385/384, 175/176, 100/99, 224/225 and 36/35, inverting and transposing, we obtain &lt;12 19 28 34 42|, -&lt;3 5 7 9 10|, &lt;9 14 21 25 31|, -&lt;7 11 16 20 24|], [&lt;22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that<br />

<br />

S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)<br />

S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((-3*i+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)<br />

<br />

is the periodic scale with which we began this analysis.<br />

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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>2014-03-20 11:40:12 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-20 13:52:38 UTC</tt>.<br>
-: The original revision id was <tt>497131986</tt>.<br>
+: The original revision id was <tt>497165174</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 13: / Line 13: @@
 Suppose we have n-1 commas, which we will assume are greater than 1, 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. We may call the val V the epimorph val, and the n-1 commas, which form a basis for the kernel of V, the chroma basis.
-Now choose a uniformizing step for the Fokker block, by which is meant 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 chromas 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 chromas. 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 "cn", and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(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.
+Now choose a uniformizing step for the Fokker block, by which is meant 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 chromas 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 last row is the monzo for the step c, and whose other rows are the monzos of the n-1 chromas. 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 "cn", and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(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.
 These unimodular matricies define a [[http://en.wikipedia.org/wiki/Change_of_basis|change of basis]] for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as
@@ Line 33: / Line 33: @@
 =Second definition of a Fokker block=
-Let is define a new set of vals by uk = P*vk - vk(2)*vn. 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.
+Let is 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&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 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 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.
+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.
 =Third definition of a Fokker block=
-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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.
+The n-1 vals u1, u2, ..., u_(n-1) 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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.
 =Fourth definition of a Fokker block=
@@ Line 44: / Line 44: @@
 =Determining if a scale is a Fokker block=
-The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ∓Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &lt; P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.
+The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. [[Scala]] does this as a part of its "Show data" suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a [[Graham complexity]] to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &lt; P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.
 Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.
@@ Line 59: / Line 59: @@
 =Example=
 ==Using a Fokker group basis==
-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, in the pajara-magic-orwell-porcupine arena.
+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, and porcupine = suprapyth+pajarous; hence, S is a Fokker block, in the pajara-magic-orwell-porcupine arena.
 If Q(a,b,c,d) is the ∑(T[i] - μ)^2 quadratic form on a*suprapyth+b*pajara+c*hedgehog+d*pajarous, then explicitly we have Q = 2205.5*a^2 + 880*b^2 + 2904*c^2 + 1254*d^2 + 264*a*b + 2992*a*c - 2574*a*d - 1848*b*c - 440*b*d - 880*c*d. From this we can find Q(pajara) = 880, Q(magic) = 885.5, Q(orwell) = 885.5, and Q(porcupine) = 885.5, with the Graham complexity of S being 21 in magic, orwell and porcupine, and 20 in pajara. If we look at the extrema of a, b, c, and d separately after setting Q = 900, we find they are all less than 2 in absolute value, so we need look no farther than the 27 Z-linear combinations of suprapyth, pajara, hedgehog and pajarous with coefficients less than 2 in absolute value. Had the block not been Fokker, we could have used the analysis of extrema to show it was not.
 ==Generator range and the first definition of a Fokker block==
-From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. By forming the 5x5 matrix whose first row is v1, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 36/35, 385/384, 175/176, 100/99 and 224/225, inverting and transposing, we obtain [&lt;22 35 51 62 76|, &lt;12 19 28 34 42|, &lt;3 5 7 9 10|, &lt;9 14 21 25 31|, &lt;7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that
+From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13.
-S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)
+We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the [[Tenney-Euclidean Tuning#The Frobenius projection map|Frobenius projection map]] P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k
+By forming the 5x5 matrix whose last row is V, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 385/384, 175/176, 100/99, 224/225 and 36/35, inverting and transposing, we obtain &lt;12 19 28 34 42|, -&lt;3 5 7 9 10|, &lt;9 14 21 25 31|, -&lt;7 11 16 20 24|], [&lt;22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that
+S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((-3*i+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)
 is the periodic scale with which we began this analysis.
@@ Line 109: / Line 113: @@
   Suppose we have n-1 commas, which we will assume are greater than 1, 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. We may call the val V the epimorph val, and the n-1 commas, which form a basis for the kernel of V, the chroma basis.&lt;br /&gt;
 &lt;br /&gt;
-Now choose a uniformizing step for the Fokker block, by which is meant 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 chromas 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 chromas. 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 &amp;quot;cn&amp;quot;, and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(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.&lt;br /&gt;
+Now choose a uniformizing step for the Fokker block, by which is meant 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 chromas 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 last row is the monzo for the step c, and whose other rows are the monzos of the n-1 chromas. 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 &amp;quot;cn&amp;quot;, and label the chromas c1, c2, ... c_(n-1); 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) = δ(i,j), where δ(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.&lt;br /&gt;
 &lt;br /&gt;
 These unimodular matricies define a &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Change_of_basis" rel="nofollow"&gt;change of basis&lt;/a&gt; for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as&lt;br /&gt;
@@ Line 129: / Line 133: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Second definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Second definition of a Fokker block&lt;/h1&gt;
-  Let is define a new set of vals by uk = P*vk - vk(2)*vn. 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;
+  Let is 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&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 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 &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal val list&lt;/a&gt;, 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 &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;
+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 &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal val list&lt;/a&gt;, 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 &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 the epimorph V. 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 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.&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;
-  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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.&lt;br /&gt;
+  The n-1 vals u1, u2, ..., u_(n-1) 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 boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepipeds around in all ways which retain the same orientation and have the unison inside them, we obtain an arena.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Fourth definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Fourth definition of a Fokker block&lt;/h1&gt;
@@ Line 140: / Line 144: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Determining if a scale is a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Determining if a scale is a Fokker block&lt;/h1&gt;
-  The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt; does this as a part of its &amp;quot;Show data&amp;quot; suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ∓Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a &lt;a class="wiki_link" href="/Graham%20complexity"&gt;Graham complexity&lt;/a&gt; to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &amp;lt; P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.&lt;br /&gt;
+  The second definition of Fokker block can be used to determine if a given periodic JI scale is a Fokker block. The first step is to find if it is epimorphic; this can be done by starting with a val V with indeterminate coefficients, and finding if the linear equations V(S[i]) = i have a solution. &lt;a class="wiki_link" href="/Scala"&gt;Scala&lt;/a&gt; does this as a part of its &amp;quot;Show data&amp;quot; suite of scale analytics. Now we take note of the fact that if r is the rank of the group generated by the scale (which is therefore the minimal JI system it is defined in) the Fokker group of bivals associated to V is a free abelian group of rank r-1. We will assume we are working in a full p-limit group, but nothing essential is changed in Fokker block theory in the case of subgroups. The free group, defined by addition of bivals, has a basis consisting of ±Wk for some set of wedgies, and we may assume the sign is positive and the basis is a basis of wedgies. Using this basis, we may either find a basis of r-1 wedgies each of which gives a &lt;a class="wiki_link" href="/Graham%20complexity"&gt;Graham complexity&lt;/a&gt; to the scale reduced to the octave; that is, to S = {S[i]| 0 ≤ i &amp;lt; P} which is less than P, in which case the scale is a Fokker block, or determine no such basis exists, in which case it is not Fokker.&lt;br /&gt;
 &lt;br /&gt;
 Graham complexity for S with respect to a wedgie W defines a complexity measure for the wedgies which makes the wedgies which determine if the scale S is a Fokker block precisely those of lowest complexity. However, for some purposes a quadratically (L2) defined complexity measure with similar properties is of use. We can define such a complexity measure for wedgies W by setting T[i] = (W∨S[i])(2), and then taking the sum ∑(T[i] - μ)^2 for i from 0 to P-1, where μ is the mean (∑T[i])/P. This can be analyzed in terms of the associated positive definite bilinear form on the linear combinations of basis elements giving W, and it is clear that past a certain range which can be determined the quadratic complexity measure will continue to increase, and that if needed one can in this way prove that a block is not Fokker. Like Graham complexity, this gives a slightly lower value to a MOS with more than one period to the octave. WE can make them exactly the same by modifying things slightly so that T[i] is (W∨S[i])(2) in the first period of the octave, (W∨S[i])(2) + 1 for the second period, and so forth. This makes all MOS to result in P contiguous values, so that the resulting quadratic form returns P(P^2-1)/12 in all cases when the wedgie results in a MOS of P notes per octave, and more otherwise.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc7"&gt;&lt;a name="Example"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Example&lt;/h1&gt;
   &lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc8"&gt;&lt;a name="Example-Using a Fokker group basis"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Using a Fokker group basis&lt;/h2&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, in the pajara-magic-orwell-porcupine arena.&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, and porcupine = suprapyth+pajarous; hence, S is a Fokker block, in the pajara-magic-orwell-porcupine arena.&lt;br /&gt;
 &lt;br /&gt;
 If Q(a,b,c,d) is the ∑(T[i] - μ)^2 quadratic form on a*suprapyth+b*pajara+c*hedgehog+d*pajarous, then explicitly we have Q = 2205.5*a^2 + 880*b^2 + 2904*c^2 + 1254*d^2 + 264*a*b + 2992*a*c - 2574*a*d - 1848*b*c - 440*b*d - 880*c*d. From this we can find Q(pajara) = 880, Q(magic) = 885.5, Q(orwell) = 885.5, and Q(porcupine) = 885.5, with the Graham complexity of S being 21 in magic, orwell and porcupine, and 20 in pajara. If we look at the extrema of a, b, c, and d separately after setting Q = 900, we find they are all less than 2 in absolute value, so we need look no farther than the 27 Z-linear combinations of suprapyth, pajara, hedgehog and pajarous with coefficients less than 2 in absolute value. Had the block not been Fokker, we could have used the analysis of extrema to show it was not.&lt;br /&gt;
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 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Example-Generator range and the first definition of a Fokker block"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Generator range and the first definition of a Fokker block&lt;/h2&gt;
-  From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. By forming the 5x5 matrix whose first row is v1, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 36/35, 385/384, 175/176, 100/99 and 224/225, inverting and transposing, we obtain [&amp;lt;22 35 51 62 76|, &amp;lt;12 19 28 34 42|, &amp;lt;3 5 7 9 10|, &amp;lt;9 14 21 25 31|, &amp;lt;7 11 16 20 24|]. From this and the previously obtained generator ranges, we find that&lt;br /&gt;
+  From the values for T[i] for each of the four temperaments, we find that the generator range for pajara is -7 to 3, since we obtain the even numbers from -14 to 6. The others are magic from -9 to 12, orwell from -4 to 17 and porcupine from -8 to 13. &lt;br /&gt;
+&lt;br /&gt;
+We can pass from a Fokker group basis to a chroma basis in various ways. One begins by finding the &lt;a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The Frobenius projection map"&gt;Frobenius projection map&lt;/a&gt; P_k corresponding to each temperament wedgie W_k, and from that the dual projection map Q_k. Q_k &lt;br /&gt;
+&lt;br /&gt;
+By forming the 5x5 matrix whose last row is V, the patent val for 22 equal, and whose other rows are pajara∨2, magic∨2, orwell∨2 and porcupine∨2, inverting, transposing, and multiplying by 22, we obtain a matrix whose rows are the monzos for 2, 385/384, 176/175, 100/99 and 224/225 respectively. Taking the monzo matrix for 385/384, 175/176, 100/99, 224/225 and 36/35, inverting and transposing, we obtain &amp;lt;12 19 28 34 42|, -&amp;lt;3 5 7 9 10|, &amp;lt;9 14 21 25 31|, -&amp;lt;7 11 16 20 24|], [&amp;lt;22 35 51 62 76|, . From this and the previously obtained generator ranges, we find that&lt;br /&gt;
 &lt;br /&gt;
-S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((3*i+12)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((7*i+8)/22)&lt;br /&gt;
+S[i] = (36/35)^i * (385/384)^floor((12*i+14)/22) * (175/176)^floor((-3*i+9)/22) * (100/99)^floor((9*i+4)/22) * (224/225)^floor((-7*i+13)/22)&lt;br /&gt;
 &lt;br /&gt;
 is the periodic scale with which we began this analysis.&lt;br /&gt;