Mathematical theory of Fokker blocks - Revision history

https://en.xen.wiki/index.php?action=history&feed=atom&title=Mathematical_theory_of_Fokker_blocks Mathematical theory of Fokker blocks - Revision history 2026-06-29T13:51:50Z Revision history for this page on the wiki MediaWiki 1.43.6 https://en.xen.wiki/index.php?title=Mathematical_theory_of_Fokker_blocks&diff=205229&oldid=prev Fredg999: /* Definitions */ Fix heading levels 2025-07-28T04:49:57Z

<p><span class="autocomment">Definitions: </span> Fix heading levels</p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 04:49, 28 July 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l35">Line 35:</td> <td colspan="2" class="diff-lineno">Line 35:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The {{nowrap|''n'' − 1}} vals '''u'''<sub>1</sub>, '''u'''<sub>2</sub>, …, '''u'''<sub style="white-space: nowrap;">(''n'' − 1)</sub> defined in the previous section gave us {{nowrap|''n'' − 1}} inequalities {{nowrap| ''a''<sub>''k''</sub> − ''P'' < '''u'''<sub>''k''</sub> (''q'') }} ≤&nbsp;''a''<sub>''k''</sub>, which apply to any ''q'' in the Fokker block. If we restrict ''q'' to {{nowrap| 1 ≤ ''q'' < 2 }}, and regard it as representing a pitch class, then it is associated to a lattice point in an {{nowrap|(''n'' − 1)}}-dimensional vector space, and in that space the {{nowrap|''n'' − 1}} inequalities define the boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepiped around (in '''R'''<sup>2</sup>) in all ways which retain the same orientation and have the unison inside them, we obtain an arena.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The {{nowrap|''n'' − 1}} vals '''u'''<sub>1</sub>, '''u'''<sub>2</sub>, …, '''u'''<sub style="white-space: nowrap;">(''n'' − 1)</sub> defined in the previous section gave us {{nowrap|''n'' − 1}} inequalities {{nowrap| ''a''<sub>''k''</sub> − ''P'' < '''u'''<sub>''k''</sub> (''q'') }} ≤&nbsp;''a''<sub>''k''</sub>, which apply to any ''q'' in the Fokker block. If we restrict ''q'' to {{nowrap| 1 ≤ ''q'' < 2 }}, and regard it as representing a pitch class, then it is associated to a lattice point in an {{nowrap|(''n'' − 1)}}-dimensional vector space, and in that space the {{nowrap|''n'' − 1}} inequalities define the boundaries of a parallelepiped. The Fokker blocks can be defined as the pitch classes lying within such a parallelepiped. By moving the parallelepiped around (in '''R'''<sup>2</sup>) in all ways which retain the same orientation and have the unison inside them, we obtain an arena.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">=</del>=== Fourth definition of a Fokker block <del style="font-weight: bold; text-decoration: none;">=</del>===</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=== Fourth definition of a Fokker block ===</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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|step pattern product]] taken. This entails that every Fokker block leads to a step pattern product, and the process can be reversed, so that step pattern 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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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|step pattern product]] taken. This entails that every Fokker block leads to a step pattern product, and the process can be reversed, so that step pattern 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.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Fredg999 https://en.xen.wiki/index.php?title=Mathematical_theory_of_Fokker_blocks&diff=203849&oldid=prev FloraC: Style 2025-06-29T11:11:50Z

<p>Style</p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:11, 29 June 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4">Line 4:</td> <td colspan="2" class="diff-lineno">Line 4:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Preliminaries ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Preliminaries ==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[<del style="font-weight: bold; text-decoration: none;">Harmonic </del>limit|''p''-limit]] situation.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>While the idea generalizes easily to [[just intonation subgroups]] and tempered groups, for ease of exposition we will suppose that we are in a [[<ins style="font-weight: bold; text-decoration: none;">harmonic </ins>limit|''p''-limit]] situation.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Suppose n is equal to the number of primes up to and including p, and that we have {{nowrap|''n'' − 1}} commas. Call the n<del style="font-weight: bold; text-decoration: none;">-</del>1 commas ''c''<sub>1</sub>, ''c''<sub>2</sub>, … , ''c''<sub>''n-1<del style="font-weight: bold; text-decoration: none;">''</del></sub>. We can pick some uniformizing step ''c''<sub>''n''</sub> which allows us to find ''n'' vals '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, … , '''v'''<sub>''n''</sub> such that '''v'''<sub>i</sub> tempers out all ''c''<sub>''k''</sub> except ''c''<sub>''i''</sub>, which it maps to 1 step. If ''q'' is a ''p''-limit rational number, we may write it as</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Suppose <ins style="font-weight: bold; text-decoration: none;">''</ins>n<ins style="font-weight: bold; text-decoration: none;">'' </ins>is equal to the number of primes up to and including <ins style="font-weight: bold; text-decoration: none;">''</ins>p<ins style="font-weight: bold; text-decoration: none;">''</ins>, and that we have {{nowrap|<ins style="font-weight: bold; text-decoration: none;">(</ins>''n'' − 1<ins style="font-weight: bold; text-decoration: none;">)</ins>}} commas. Call the <ins style="font-weight: bold; text-decoration: none;">{{nowrap|(''</ins>n<ins style="font-weight: bold; text-decoration: none;">'' − </ins>1<ins style="font-weight: bold; text-decoration: none;">)}} </ins>commas ''c''<sub>1</sub>, ''c''<sub>2</sub>, …, ''c''<sub><ins style="font-weight: bold; text-decoration: none;">(</ins>''n<ins style="font-weight: bold; text-decoration: none;">'' </ins>- 1<ins style="font-weight: bold; text-decoration: none;">)</ins></sub>. We can pick some uniformizing step ''c''<sub>''n''</sub> which allows us to find ''n'' vals '''v'''<sub>1</sub>, '''v'''<sub>2</sub>, …, '''v'''<sub>''n''</sub> such that '''v'''<sub><ins style="font-weight: bold; text-decoration: none;">''</ins>i<ins style="font-weight: bold; text-decoration: none;">''</ins></sub> tempers out all ''c''<sub>''k''</sub> except ''c''<sub>''i''</sub>, which it maps to 1 step. If ''q'' is a ''p''-limit rational number, we may write it as</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>q = c_1^{\vec v_1(q)} c_2^{\vec v_2(q)} \cdots c_n^{\vec v_n(q)}.</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>q = c_1^{\vec v_1(q)} c_2^{\vec v_2(q)} \cdots c_n^{\vec v_n(q)}.</math>.</div></td></tr> </table>

FloraC https://en.xen.wiki/index.php?title=Mathematical_theory_of_Fokker_blocks&diff=203847&oldid=prev FloraC: Split 2025-06-29T11:05:30Z

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