Fokker block: Difference between revisions
m Undo (If mos doesn't depend on rtt; neither does fokker block) Tag: Undo |
Cmloegcmluin (talk | contribs) use available templates, correct typos, use correct angle brackets, and correct use of the similar-looking MASCULINE ORDINAL INDICATOR to the actual degree symbol |
||
| Line 9: | Line 9: | ||
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 {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get ''w''<sub>2</sub>''e''<sub>2</sub> + ''w''<sub>3</sub>''e''<sub>3</sub> + … + ''w''<sub>''p''</sub>''e''<sub>''p''</sub> where the ''w''<sub>2</sub>, ''w''<sub>3</sub> … ''w''<sub>''p''</sub> are integers. We interpret this as the [[Vals and Tuning Space|val]] v = {{val| w<sub>2</sub> w<sub>3</sub> … w<sub>''p''</sub> }}. 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 ''w''<sub>2</sub> < 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 = {{val| 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. | 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 {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … ''e''<sub>''p''</sub> }}, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get ''w''<sub>2</sub>''e''<sub>2</sub> + ''w''<sub>3</sub>''e''<sub>3</sub> + … + ''w''<sub>''p''</sub>''e''<sub>''p''</sub> where the ''w''<sub>2</sub>, ''w''<sub>3</sub> … ''w''<sub>''p''</sub> are integers. We interpret this as the [[Vals and Tuning Space|val]] v = {{val| w<sub>2</sub> w<sub>3</sub> … w<sub>''p''</sub> }}. 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 ''w''<sub>2</sub> < 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 = {{val| 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 | 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 [[Wikipedia: 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'' "''c''<sub>''n''</sub>", and label the chromas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>(''n'' - 1)</sub>; and if we consider the columns of the inverse matrix to be vals and call them v<sub>1</sub>, v<sub>2</sub>, … , v<sub>''n''</sub>, then by the definition of the inverse of a matrix, v<sub>''i''</sub> (c<sub>''j''</sub>) = δ (''i'', ''j''), where δ (''i'', ''j'') is the [[Wikipedia: Kronecker delta|Kronecker delta]]. Stated another way, v<sub>i</sub> (''c''<sub>j</sub>) is 0 unless ''i'' equals ''j'', in which case v<sub>''i''</sub> (''c''<sub>''i''</sub>) = 1. | ||
These unimodular matricies define a [[Wikipedia: 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 ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>''n''</sub> with integer exponents. To determine the exponents, we use v<sub>1</sub>, v<sub>2</sub>, … , v<sub>''n''</sub>, so that if ''q'' is a ''p''-limit rational number, we may write it as | These unimodular matricies define a [[Wikipedia: 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 ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>''n''</sub> with integer exponents. To determine the exponents, we use v<sub>1</sub>, v<sub>2</sub>, … , v<sub>''n''</sub>, so that if ''q'' is a ''p''-limit rational number, we may write it as | ||
| Line 54: | Line 54: | ||
A Fokker block as we have so far defined it is an epimorphic periodic scale S with period P repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = n-1 different rank-two wedgies {Wk} such that S has Graham complexity less than P for each Wk. If we unpack that definition we can extend it in several distinct ways. | A Fokker block as we have so far defined it is an epimorphic periodic scale S with period P repeating at the octave, with values in p-limit rational intonation, such that there exist pi(p)-1 = n-1 different rank-two wedgies {Wk} such that S has Graham complexity less than P for each Wk. If we unpack that definition we can extend it in several distinct ways. | ||
Explicitly, S is a [http://en.wikipedia.org/wiki/Quasiperiodic_function quasiperiodic function] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + P] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[P]) = V(2) = P, so that V = | Explicitly, S is a [http://en.wikipedia.org/wiki/Quasiperiodic_function quasiperiodic function] from the integers to the p-limit rational numbers, such that S[0] = 1 and S[i + P] = 2S[i], for which there is a val V such that V(S[i]) = i. This entails that V(S[P]) = V(2) = P, so that V = {{val|P ... }}, with P a positive integer; in other words, V is a P-edo val. For each of the n-1 wedgies Wk, we can form an abstract temperament periodic scale, meaning a periodic scale taking values in an [[Abstract_regular_temperament|abstract regular temperament]], by Tk[i] = Wk∨S[i]. The values Tk[i] are p-limit vals, and since Tk[P] = Wk∨S[i] = Wk∨2, Tk[P](2) = 0, and so Tk[i + P](2) = (Tk[i] + Tk[P])(2) = Tk[i](2). Hence Tk[i](2) takes on P or fewer values, with a ≤ Tk[i](2) ≤ b. The Graham complexity G(Wk) of S with respect to Wk is b-a, and if S is a Fokker block, for each Wk, G(Wk) < P. | ||
One way to generalize this is to allow the [[Just_intonation_subgroups|group of the scale]] to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). | One way to generalize this is to allow the [[Just_intonation_subgroups|group of the scale]] to be something other than the full p-limit group, adjusting the basis for vals, monzos and wedgies to correspond with a basis for this subgroup. We may also replace the interval of equivalence 2 with any rational number E which is not a power, so that S[i + P] = E S[i] and replacing Tk[i](2) with Tk[i](E). | ||
| Line 64: | Line 64: | ||
==== Using a Fokker group basis ==== | ==== 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 = | 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 = {{val|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 {{{wedgie|1 9 -2 -6 12 -6 -13 -30 -45 -10}}, {{wedgie|2 -4 -4 -12 -11 -12 -26 2 -14 -20}}, {{wedgie|6 10 10 8 2 -1 -8 -5 -16 -12}}, {{wedgie|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 | 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 | ||
| Line 76: | Line 76: | ||
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. | 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. | ||
We can pass from a Fokker group basis to a chroma basis in various ways. One way 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 has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain | We can pass from a Fokker group basis to a chroma basis in various ways. One way 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 has the property that each chroma except c_k is an eigenvector with eigenvalue 1. Hence, the matrix product of the Q_i with i≠k has a single eigenvalue of 1, corresponding to c_k, which allows us to find c_k. From the Fokker group basis [pajara, magic, orwell, porccupine] we may find in this way the dual chroma basis [385/384, 176/175, 100/99, 225/224]. Taking the monzo matrix for 385/384, 175/176, 100/99, 225/224 and 36/35, inverting and transposing, we obtain [ {{val|12 19 28 34 42}, -{{val|3 5 7 9 10}, {{val|9 14 21 25 31}}, -{{val|7 11 16 20 24}}, {{val|22 35 51 62 76}} ]. From this and the previously obtained generator ranges, we find that | ||
<math>S[i] = (36/35)^i (385/384)^{\lfloor(12i+14)/22\rfloor} (175/176)^{\lfloor(-3i+9)/22\rfloor} (100/99)^{\lfloor(9i+4)/22\rfloor} (224/225)^{\lfloor(-7i+13)/22\rfloor}</math> | <math>S[i] = (36/35)^i (385/384)^{\lfloor(12i+14)/22\rfloor} (175/176)^{\lfloor(-3i+9)/22\rfloor} (100/99)^{\lfloor(9i+4)/22\rfloor} (224/225)^{\lfloor(-7i+13)/22\rfloor}</math> | ||
| Line 84: | Line 84: | ||
==== Product words and the fourth definition of a Fokker block ==== | ==== Product words and the fourth definition of a Fokker block ==== | ||
Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = - | Starting from our example 22 note per octave scale, we can produce a list of 22 steps: steps[i] = 33/32, 512/495, 33/32, 45/44, 25/24, 128/125, 25/24, 33/32, 128/125, 25/24, 45/44, 28/27, 36/35, 33/32, 512/495, 33/32, 80/77, 49/48, 33/32, 80/77, 77/75, 80/77. We can apply the four wedgies for pajara, magic, orwell and porcupine to these steps to obtain four abstract temperament MOS, each of which has two kinds of steps, expressed as vals. If a = -{{val|10 16 23 28 34}} and b = {{val|12 19 28 34 42}}, then pajara applied to the steps gives abababaabababababaabab. If c = -{{val|3 5 7 9 10}} and d = {{val|19 30 44 53 66}}, then magic gives cccdccccccdccccccdcccc. If e = {{val|9 14 21 25 31}} and f = -{{val|13 21 30 37 45}}, then orwell gives efeefefeefefeefefeefef. Finally, if g = {{val|7 11 16 20 24}} and h = -{{val|15 24 35 42 52}}, then porcupine gives ghggghgghgghgghgghgghg. By taking product words, we get not only the Fokker block itself, but also the various temperings in the associated temperaments. Here "product" means product in a quite literal sense, since these can be construed as wedge product words. | ||
As noted above, pajara, magic, orwell and porcupine correspond to the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[The_dual|dual]] we obtain the wedgie for zeus, which is | As noted above, pajara, magic, orwell and porcupine correspond to the commas 385/384, 176/175, 100/99 and 225/224. If we take for example 385/384 and 176/175, we get zeus temperament. Wedging the monzos for these two commas and taking [[The_dual|dual]] we obtain the wedgie for zeus, which is ⟨⟨⟨2 -3 1 -1 -1 2 11 3 -10 4]]]. Taking the interior product of this with the steps of our scale gives wxwwyzywzywxwwxwyzwyzy, where w = {{wedgie|1 -3 5 -1 -7 5 -5 20 8 -20}}, x = {{wedgie|-3 5 -9 1 15 -6 12 -35 -15 34}}, y = {{wedgie|4 2 -1 3 -6 -13 -9 -8 0 12}}, and z = {{wedgie|-6 0 -3 -3 14 12 16 -7 -7 2}}. If we set Orw[i] = orwell∨steps[i] and Por[i] = porcupine∨steps[i], then Zeus[i] = Orw[i]∧Por[i], which exhibits the scale tempered in zeus as a product word of the orwell MOS with the porcupine MOS. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka. | ||
==== The tempered scales of a Fokker block ==== | ==== The tempered scales of a Fokker block ==== | ||
| Line 100: | Line 100: | ||
Let S be the abstract scale defined by, for scale steps from 1 to 22: | Let S be the abstract scale defined by, for scale steps from 1 to 22: | ||
[ | [ | ||
⟨⟨⟨1 -2 0 1 1 -2 9 5 -10 -4]]], ⟨⟨⟨0 1 1 -1 -1 2 -4 -4 4 4]]], ⟨⟨⟨1 -1 1 0 0 0 5 1 -6 0]]], | |||
⟨⟨⟨0 0 0 2 2 -4 3 3 -6 -8]]], ⟨⟨⟨0 -2 -2 1 1 -2 6 6 -4 -4]]], ⟨⟨⟨0 1 1 1 1 -2 -1 -1 -2 -4]]], | |||
⟨⟨⟨0 -1 -1 0 0 0 2 2 0 0]]], ⟨⟨⟨1 -3 -1 1 1 -2 11 7 -10 -4]]], ⟨⟨⟨1 0 2 1 1 -2 4 0 -8 -4]]], | |||
⟨⟨⟨1 -2 0 0 0 0 7 3 -6 0]]], ⟨⟨⟨0 -1 -1 2 2 -4 5 5 -6 -8]]], ⟨⟨⟨1 -1 1 -1 -1 2 3 -1 -2 4]]], | |||
⟨⟨⟨0 0 0 1 1 -2 1 1 -2 -4]]], ⟨⟨⟨1 -2 0 2 2 -4 10 6 -12 -8]]], ⟨⟨⟨0 1 1 0 0 0 -3 -3 2 0]]], | |||
⟨⟨⟨1 -1 1 1 1 -2 6 2 -8 -4]]], ⟨⟨⟨-1 0 -2 1 1 -2 -2 2 4 -4]]], ⟨⟨⟨1 0 2 0 0 0 2 -2 -4 0]]], | |||
⟨⟨⟨2 -2 2 1 1 -2 11 3 -14 -4]]], ⟨⟨⟨0 -1 -1 1 1 -2 3 3 -2 -4]]], ⟨⟨⟨2 -1 3 0 0 0 7 -1 -10 0]]], | |||
⟨⟨⟨0 0 0 0 0 0 -1 -1 2 0]]] | |||
] | |||
This represents an abstract scale defined in terms of 11-limit trivals derived from taking interior products of an unknown scale with an unknown 11-limit rank four temperament. Working with it directly is more difficult than dealing with the [[Transversal|transversal]] we may obtain by [[The_wedgie#Truncation of wedgies|truncation]]. If we truncate each scale step to the 7-limit, we obtain a list of 7-limit trivals. Each of these is [[The_dual|dual]] to a monzo, which we may express in terms of a 7-limit rational number, leading to the following scale, from 1 to 22: 525/512, 16/15, 35/32, 9/8, 75/64, 6/5, 5/4, 2625/2048, 21/16, 175/128, 45/32, 35/24, 3/2, 1575/1024, 8/5, 105/64, 12/7, 7/4, 3675/2048, 15/8, 245/128, 2. This we may now test for Fokker properties in the usual way. | This represents an abstract scale defined in terms of 11-limit trivals derived from taking interior products of an unknown scale with an unknown 11-limit rank four temperament. Working with it directly is more difficult than dealing with the [[Transversal|transversal]] we may obtain by [[The_wedgie#Truncation of wedgies|truncation]]. If we truncate each scale step to the 7-limit, we obtain a list of 7-limit trivals. Each of these is [[The_dual|dual]] to a monzo, which we may express in terms of a 7-limit rational number, leading to the following scale, from 1 to 22: 525/512, 16/15, 35/32, 9/8, 75/64, 6/5, 5/4, 2625/2048, 21/16, 175/128, 45/32, 35/24, 3/2, 1575/1024, 8/5, 105/64, 12/7, 7/4, 3675/2048, 15/8, 245/128, 2. This we may now test for Fokker properties in the usual way. | ||
The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val | The first order of business is to determine if the scale is epimorphic, which it is, with 22 patent val {{val|22 35 51 62}}. Using a basis for the Fokker group, for instance the one listed [[Minkowski_reduced_bases_for_Fokker_groups_of_certain_vals|here]], pajara-magic-porcupine, we find that pajara, porcupine and orwell all temper it to a MOS, so that the scale is a Fokker block. This is enough to prove the original scale is an abstract Fokker block; however, we might want a result in terms of the original 11-limit problem. By solving for the condition that the interior product with each scale step is zero, we find that 176/175 is the unique comma tempered out by the rank-four temperament which tempered to the abstract scale. Adding 176/175 to the commas of pajara, porcupine and orwell leads to the 11-limit versions of each of these. Taking the interior product of the dual scale of bimonzos with each of these 11-limit wedgies leads to the conclusion that each of these temper the abstract scale to a MOS. | ||
=== Scale properties of Fokker blocks === | === Scale properties of Fokker blocks === | ||
| Line 124: | Line 126: | ||
By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank r Fokker block, meaning one which generates a group of rank r, has r-1 abstract MOS scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the r-1 abstract MOS, that means each interval class in the scale has at most 2^(r-1) possible values; in other words, it has maximum variety less than or equal to 2^(r-1). | By definition, a Fokker block is weakly epimorphic, which implies it is constant structure. Since the pitch classes are all of those contained in some parallelepiped, it is convex. A rank r Fokker block, meaning one which generates a group of rank r, has r-1 abstract MOS scales which can take at most two values for any interval class, by Myhill's property. Since the scale itself can be reconstituted from the r-1 abstract MOS, that means each interval class in the scale has at most 2^(r-1) possible values; in other words, it has maximum variety less than or equal to 2^(r-1). | ||
The reconstitution can be obtained as follows: for every note of S[i] except S[0], S[i] will be either the rational number obtained by finding the monzo of the wedge products of the r-1 abstract MOS vals for i, taking the dual, and dividing by i^(r-1), or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[i] by (v1∧v2∧...∧v_(r-1)) | The reconstitution can be obtained as follows: for every note of S[i] except S[0], S[i] will be either the rational number obtained by finding the monzo of the wedge products of the r-1 abstract MOS vals for i, taking the dual, and dividing by i^(r-1), or else the inverse of this number. Hence we may choose an ordering of the correct parity, and find the value associated to S[i] by (v1∧v2∧...∧v_(r-1))°/i^(r-1). | ||
=== The Fokblock function and modal UDP notation === | === The Fokblock function and modal UDP notation === | ||
| Line 132: | Line 134: | ||
Returning to our pajmagorpor22 example, we have that pajmagorpor22 = Fokblock([385/384, 176/175, 100/99, 225/224], [14, 9, 4, 13]). It is also equal to Fokblock([385/384, 176/175, 100/99, 225/224], [15, 9, 4, 13]), reflecting the fact that pajara has a period of half on octave, ie that P1 = 2. Hence the pajara MOS mode is 7|3(2) in UDP notation. Finding the others by the fact that for them Pk=1 and ak=U, we have that the block, in product word form, is (pajara 7|3(2))*(magic 9|12)*(orwell 4|17)*(porcupine 13|8). We can easily reverse this process, finding the chroma basis from the Fokker group basis, and the offset ai from the corresponding U and Pi as Pi*U, and so display S in terms of Fokblock. | Returning to our pajmagorpor22 example, we have that pajmagorpor22 = Fokblock([385/384, 176/175, 100/99, 225/224], [14, 9, 4, 13]). It is also equal to Fokblock([385/384, 176/175, 100/99, 225/224], [15, 9, 4, 13]), reflecting the fact that pajara has a period of half on octave, ie that P1 = 2. Hence the pajara MOS mode is 7|3(2) in UDP notation. Finding the others by the fact that for them Pk=1 and ak=U, we have that the block, in product word form, is (pajara 7|3(2))*(magic 9|12)*(orwell 4|17)*(porcupine 13|8). We can easily reverse this process, finding the chroma basis from the Fokker group basis, and the offset ai from the corresponding U and Pi as Pi*U, and so display S in terms of Fokblock. | ||
In terms of the rational intonation of the blocks of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values U to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the U and D found by considering the MOS deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar|collapar]], a 12-note 11-limit scale which tempers to a MOS in six different ways--pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for Fokblock([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8]) the chroma dual to pajaric, which is 245/242, is in the down direction considered as a MOS, since pajaric∨245/242 = -V, where V is the epimorph, wheras 3, which can be taken as the generator, is in the up direction since pajaric∨3 = | In terms of the rational intonation of the blocks of a Fokker arena, this definition of "chroma positive" is the correct one if we want increasing "up" values U to correspond with increasingly sharp intervals. However, in borderline cases it need not correspond to the U and D found by considering the MOS deriving by tempering by an element of the Fokker group basis taken separately. For example, consider the superwakalix [[collapar|collapar]], a 12-note 11-limit scale which tempers to a MOS in six different ways--pajaric, injera, august, diminished, demolished, and hemidim. The scale belongs to eight different arenas, in five of which pajaric is one of the Fokker group basis wedgies. In four of these, the chroma corresponding to pajaric goes in the up direction; however for Fokblock([245/242, 126/121, 50/49, 45/44], [8, 2, 3, 8]) the chroma dual to pajaric, which is 245/242, is in the down direction considered as a MOS, since pajaric∨245/242 = -V, where V is the epimorph, wheras 3, which can be taken as the generator, is in the up direction since pajaric∨3 = {{val|2 0 11 12 7}}. Note that pajara∨245/242 = V, so it is up in pajara. | ||
If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance [[nofives|nofives]] is Fokblock([64/63, 729/686, 5], [3, 4, 0]). | If we want to compute Fokker blocks in subgroups resulting from excluding one or more odd primes, we can do so by adding the primes to the list of chromas. For instance [[nofives|nofives]] is Fokblock([64/63, 729/686, 5], [3, 4, 0]). | ||