Fokker block: Difference between revisions

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==== 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: {{nowrap|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 mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap|'''a''' {{=}} −{{val| 10 16 23 28 34 }} and {{nowrap|'''b''' {{=}} {{val| 12 19 28 34 42 }}}}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap|'''c''' {{=}} −{{val| 3 5 7 9 10 }}}} and {{nowrap|'''d''' {{=}} {{val| 19 30 44 53 66 }}}}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap|'''e''' {{=}} {{val| 9 14 21 25 31 }}}} and {{nowrap|f {{=}} −{{val| 13 21 30 37 45 }}}}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap|'''g''' {{=}} {{val| 7 11 16 20 24 }}}} and {{nowrap|'''h''' {{=}} −{{val| 15 24 35 42 52 }}}}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word]]s, 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.

Starting from our example 22 note per octave scale, we can produce a list of 22 steps: {{nowrap|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 mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap|'''a''' {{=}} −{{val| 10 16 23 28 34 }}}} and {{nowrap|'''b''' {{=}} {{val| 12 19 28 34 42 }}}}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap|'''c''' {{=}} −{{val| 3 5 7 9 10 }}}} and {{nowrap|'''d''' {{=}} {{val| 19 30 44 53 66 }}}}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap|'''e''' {{=}} {{val| 9 14 21 25 31 }}}} and {{nowrap|'''f''' {{=}} −{{val| 13 21 30 37 45 }}}}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap|'''g''' {{=}} {{val| 7 11 16 20 24 }}}} and {{nowrap|'''h''' {{=}} −{{val| 15 24 35 42 52 }}}}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word]]s, 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. For example, if we take 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 {{multival|rank=3| 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 {{nowrap|'''w''' {{=}} {{multival| 1 -3 5 -1 -7 5 -5 20 8 -20 }}|'''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }}|'''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }}|and '''z''' {{=}} {{multival| -6 0 -3 -3 14 12 16 -7 -7 2 }}}}. If we set {{nowrap|Orw[''i''] {{=}} orwell ∨ steps[''i'']}} and {{nowrap|Por[''i''] {{=}} porcupine ∨ steps[''i'']}}, then {{nowrap|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.

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 ==== 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: {{nowrap|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 mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap|'''a''' {{=}} &minus;{{val| 10 16 23 28 34 }} and {{nowrap|'''b''' {{=}} {{val| 12 19 28 34 42 }}}}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap|'''c''' {{=}} &minus;{{val| 3 5 7 9 10 }}}} and {{nowrap|'''d''' {{=}} {{val| 19 30 44 53 66 }}}}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap|'''e''' {{=}} {{val| 9 14 21 25 31 }}}} and {{nowrap|f {{=}} &minus;{{val| 13 21 30 37 45 }}}}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap|'''g''' {{=}} {{val| 7 11 16 20 24 }}}} and {{nowrap|'''h''' {{=}} &minus;{{val| 15 24 35 42 52 }}}}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word]]s, 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.
+Starting from our example 22 note per octave scale, we can produce a list of 22 steps: {{nowrap|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 mosses, each of which has two kinds of steps, expressed as vals. If {{nowrap|'''a''' {{=}} &minus;{{val| 10 16 23 28 34 }}}} and {{nowrap|'''b''' {{=}} {{val| 12 19 28 34 42 }}}}, then pajara applied to the steps gives '''abababaabababababaabab'''. If {{nowrap|'''c''' {{=}} &minus;{{val| 3 5 7 9 10 }}}} and {{nowrap|'''d''' {{=}} {{val| 19 30 44 53 66 }}}}, then magic gives '''cccdccccccdccccccdcccc'''. If {{nowrap|'''e''' {{=}} {{val| 9 14 21 25 31 }}}} and {{nowrap|'''f''' {{=}} &minus;{{val| 13 21 30 37 45 }}}}, then orwell gives '''efeefefeefefeefefeefef'''. Finally, if {{nowrap|'''g''' {{=}} {{val| 7 11 16 20 24 }}}} and {{nowrap|'''h''' {{=}} &minus;{{val| 15 24 35 42 52 }}}}, then porcupine gives '''ghggghgghgghgghgghgghg'''. By taking [[product word]]s, 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. For example, if we take 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 {{multival|rank=3| 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 {{nowrap|'''w''' {{=}} {{multival| 1 -3 5 -1 -7 5 -5 20 8 -20 }}|'''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }}|'''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }}|and '''z''' {{=}} {{multival| -6 0 -3 -3 14 12 16 -7 -7 2 }}}}. If we set {{nowrap|Orw[''i''] {{=}} orwell ∨ steps[''i'']}} and {{nowrap|Por[''i''] {{=}} porcupine ∨ steps[''i'']}}, then {{nowrap|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.