Fokker block: Difference between revisions
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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 | ||
<math>q = c_1^{v_1(q)} c_2^{v_2(q)} \cdots c_n^{v_n(q)}.</math> | |||
=== Definitions === | === Definitions === | ||
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Let us set ''e''<sub>''i''</sub> = v<sub>''i''</sub> (2), and also ''P'' = ''e''<sub>''n''</sub> = v<sub>''n''</sub> (2), and choose ''n'' non-negative integers ''a''<sub>1</sub>, … , ''a''<sub>''n''</sub> with 0 ≤ ''a''<sub>''k''</sub> < ''P''. Here the choice of ''a''<sub>''n''</sub> doesn't matter and we can take it to be 0. Let ''t''<sub>''i''</sub> = log<sub>2</sub> (''c''<sub>''i''</sub>), so that ''e''<sub>1</sub>''t''<sub>1</sub> + ''e''<sub>2</sub>''t''<sub>2</sub> + … + ''e''<sub>''n''</sub>''t''<sub>''n''</sub> = 1. Now define a function on the integers by | Let us set ''e''<sub>''i''</sub> = v<sub>''i''</sub> (2), and also ''P'' = ''e''<sub>''n''</sub> = v<sub>''n''</sub> (2), and choose ''n'' non-negative integers ''a''<sub>1</sub>, … , ''a''<sub>''n''</sub> with 0 ≤ ''a''<sub>''k''</sub> < ''P''. Here the choice of ''a''<sub>''n''</sub> doesn't matter and we can take it to be 0. Let ''t''<sub>''i''</sub> = log<sub>2</sub> (''c''<sub>''i''</sub>), so that ''e''<sub>1</sub>''t''<sub>1</sub> + ''e''<sub>2</sub>''t''<sub>2</sub> + … + ''e''<sub>''n''</sub>''t''<sub>''n''</sub> = 1. Now define a function on the integers by | ||
S[ | <math>S[i] = \bigg\lfloor \dfrac{e_1 i + a_1}{P} \bigg\rfloor t_1 + \cdots + \bigg\lfloor \dfrac{e_n i + a_n}{P} \bigg\rfloor t_n.</math> | ||
Here | Here ⌊''x''⌋ is the [[Wikipedia: Floor and ceiling functions|floor function]], the [[Wikipedia: Quasiperiodic function|quasiperiodic function]] returning the largest integer less than or equal to ''x''. When ''i'' = 0, since ''a''<sub>''k''</sub> < P each term is 0 and so S[0] = 0. Since for integer ''j'', ⌊''x'' + ''j''⌋ = ⌊''x''⌋ + ''j'', we have | ||
S[ | <math>S[i+P] = S[i] + e_1 t_1 + e_2 t_2 + … + e_n t_n = S[i] + 1.</math> | ||
Hence S satisfies the conditions for being a [[periodic scale]], and since our unit of measurement is the octave, i.e. we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block. | Hence S satisfies the conditions for being a [[periodic scale]], and since our unit of measurement is the octave, i.e. we are using log base two to define intervals, the repetition interval 1 represents an octave. This gives us our first definition of Fokker block. | ||
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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. | 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 | 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 | ||
<math>Q = 2205.5 a^2 + 880 b^2 + 2904 c^2 + 1254 d^2 + 264ab + 2992 ac - 2574ad - 1848bc - 440bd - 880cd. </math> | |||
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 ==== | ==== Generator range and the first definition of a Fokker block ==== | ||
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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 <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 | 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 <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 | <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> | ||
is the periodic scale with which we began this analysis. | is the periodic scale with which we began this analysis. | ||