Fokker block: Difference between revisions
Wikispaces>genewardsmith **Imported revision 497655076 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 497655448 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2014-03-23 01: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-03-23 01:29:51 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>497655448</tt>.<br> | ||
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Using the first definition of Fokker block, since the epimorph V may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding an plays no role and may be taken as 0, the block is entirely determined by the chroma basis, C = [c1, c2, ..., c_(n-1)] together with the offet values A = [a1, a2, ..., a_(n-1)]. Hence we may define a function Fokblock(C, A) from n-1 element listings of the chroma basis and corresponding offset values to a Fokker mode within the arena defined by C. If the list of wedgies [w1, w2, ..., w_(n-1)] is the dual Fokker group basis to the chroma basis C, then the period Pi of wi may as usual be found by taking the GCD of the first n-1 elements of wi. If S = Fokbloack(C, A) is a Fokker mode, the smallest value of ai giving S is always divisble by Pi, and fixing the other elements of A there are Pi successive values for ai which all give S. In terms of [[modal UDP notation]], the value of U for the MOS resulting from tempering S by Wi is ai/Pk, where ai is the smallest value giving S, and the value for D is V(2)/Pk - U - 1. Hence, the UDP notation for the MOS is U|D(Pk), with these values. | Using the first definition of Fokker block, since the epimorph V may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding an plays no role and may be taken as 0, the block is entirely determined by the chroma basis, C = [c1, c2, ..., c_(n-1)] together with the offet values A = [a1, a2, ..., a_(n-1)]. Hence we may define a function Fokblock(C, A) from n-1 element listings of the chroma basis and corresponding offset values to a Fokker mode within the arena defined by C. If the list of wedgies [w1, w2, ..., w_(n-1)] is the dual Fokker group basis to the chroma basis C, then the period Pi of wi may as usual be found by taking the GCD of the first n-1 elements of wi. If S = Fokbloack(C, A) is a Fokker mode, the smallest value of ai giving S is always divisble by Pi, and fixing the other elements of A there are Pi successive values for ai which all give S. In terms of [[modal UDP notation]], the value of U for the MOS resulting from tempering S by Wi is ai/Pk, where ai is the smallest value giving S, and the value for D is V(2)/Pk - U - 1. Hence, the UDP notation for the MOS is U|D(Pk), with these values. | ||
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 | 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 mode, 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 modes 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]], 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 | In terms of the rational intonation of the modes 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]], 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 = <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]] is Fokblock([64/63, 729/686, 5], [3, 4, 0]). We may also use this method on other subgroups. For instance, the11^3 modes Fokblock([225/224, 1728/1715, 7/5], [a1, a2, a3]) define an arena of the 2.5.7/3 subgroup.</pre></div> | 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]] is Fokblock([64/63, 729/686, 5], [3, 4, 0]). We may also use this method on other subgroups. For instance, the11^3 modes Fokblock([225/224, 1728/1715, 7/5], [a1, a2, a3]) define an arena of the 2.5.7/3 subgroup.</pre></div> | ||
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Using the first definition of Fokker block, since the epimorph V may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding an plays no role and may be taken as 0, the block is entirely determined by the chroma basis, C = [c1, c2, ..., c_(n-1)] together with the offet values A = [a1, a2, ..., a_(n-1)]. Hence we may define a function Fokblock(C, A) from n-1 element listings of the chroma basis and corresponding offset values to a Fokker mode within the arena defined by C. If the list of wedgies [w1, w2, ..., w_(n-1)] is the dual Fokker group basis to the chroma basis C, then the period Pi of wi may as usual be found by taking the GCD of the first n-1 elements of wi. If S = Fokbloack(C, A) is a Fokker mode, the smallest value of ai giving S is always divisble by Pi, and fixing the other elements of A there are Pi successive values for ai which all give S. In terms of <a class="wiki_link" href="/modal%20UDP%20notation">modal UDP notation</a>, the value of U for the MOS resulting from tempering S by Wi is ai/Pk, where ai is the smallest value giving S, and the value for D is V(2)/Pk - U - 1. Hence, the UDP notation for the MOS is U|D(Pk), with these values. <br /> | Using the first definition of Fokker block, since the epimorph V may be calculated from the chroma basis, the choice of uniformizer does not affect the resulting block, and the corresponding an plays no role and may be taken as 0, the block is entirely determined by the chroma basis, C = [c1, c2, ..., c_(n-1)] together with the offet values A = [a1, a2, ..., a_(n-1)]. Hence we may define a function Fokblock(C, A) from n-1 element listings of the chroma basis and corresponding offset values to a Fokker mode within the arena defined by C. If the list of wedgies [w1, w2, ..., w_(n-1)] is the dual Fokker group basis to the chroma basis C, then the period Pi of wi may as usual be found by taking the GCD of the first n-1 elements of wi. If S = Fokbloack(C, A) is a Fokker mode, the smallest value of ai giving S is always divisble by Pi, and fixing the other elements of A there are Pi successive values for ai which all give S. In terms of <a class="wiki_link" href="/modal%20UDP%20notation">modal UDP notation</a>, the value of U for the MOS resulting from tempering S by Wi is ai/Pk, where ai is the smallest value giving S, and the value for D is V(2)/Pk - U - 1. Hence, the UDP notation for the MOS is U|D(Pk), with these values. <br /> | ||
<br /> | <br /> | ||
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 | 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 mode, 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.<br /> | ||
<br /> | <br /> | ||
In terms of the rational intonation of the modes of a Fokker arena, this definition of &quot;chroma positive&quot; is the correct one if we want increasing &quot;up&quot; 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 <a class="wiki_link" href="/collapar">collapar</a>, 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 | In terms of the rational intonation of the modes of a Fokker arena, this definition of &quot;chroma positive&quot; is the correct one if we want increasing &quot;up&quot; 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 <a class="wiki_link" href="/collapar">collapar</a>, 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 = &lt;2 0 11 12 7|. Note that pajara∨245/242 = V, so it is up in pajara.<br /> | ||
<br /> | <br /> | ||
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 <a class="wiki_link" href="/nofives">nofives</a> is Fokblock([64/63, 729/686, 5], [3, 4, 0]). We may also use this method on other subgroups. For instance, the11^3 modes Fokblock([225/224, 1728/1715, 7/5], [a1, a2, a3]) define an arena of the 2.5.7/3 subgroup.</body></html></pre></div> | 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 <a class="wiki_link" href="/nofives">nofives</a> is Fokblock([64/63, 729/686, 5], [3, 4, 0]). We may also use this method on other subgroups. For instance, the11^3 modes Fokblock([225/224, 1728/1715, 7/5], [a1, a2, a3]) define an arena of the 2.5.7/3 subgroup.</body></html></pre></div> | ||