Fokker block: Difference between revisions

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==== Fourth definition of a Fokker block ====

The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a ~~MOS pattern~~ product, and the process can be reversed, so that ~~products~~ of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.

The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a product word, and the process can be reversed, so that product words of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.

=== Determining if a scale is a Fokker block ===

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is the periodic scale with which we began this analysis.

==== ~~Scale pattern products~~ 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: {{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 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''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 of the orwell mos with the porcupine mos. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.

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 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''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.

==== The tempered scales of a Fokker block ====

@@ Line 54: / Line 54: @@
 ==== Fourth definition of a Fokker block ====
-The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a MOS pattern product, and the process can be reversed, so that products of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.
+The {{nowrap|''n'' − 1}} (tempered) abstract mos scales discussed with the second definition of Fokker blocks can be put into some chosen order, and the [[product word]] taken. This entails that every Fokker block leads to a product word, and the process can be reversed, so that product words of {{nowrap|''n'' − 1}} abstract mos scales (''not'' ignoring mode) lead to Fokker blocks. Given the {{nowrap|''n'' − 1}} vals obtained by taking the interior product with some interval ''q'', ''q'' can be recovered either by wedging the vals together and taking the [[the dual|dual]], or by taking the determinant of the {{nowrap| ''n'' × ''n'' }} matrix of vals whose first row consists of indeterminates, as in the [[#Preliminaries]] section.
 === Determining if a scale is a Fokker block ===
@@ Line 89: / Line 89: @@
 is the periodic scale with which we began this analysis.
-==== Scale pattern products 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: {{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 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''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 of the orwell mos with the porcupine mos. This procedure is easily turned into a formal proof which generalizes a result of Marek Zabka.
+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 }} }}, {{nowrap| '''x''' {{=}} {{multival| -3 5 -9 1 15 -6 12 -35 -15 34 }} }}, {{nowrap| '''y''' {{=}} {{multival| 4 2 -1 3 -6 -13 -9 -8 0 12 }} }}, and {{nowrap| '''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.
 ==== The tempered scales of a Fokker block ====