Fokker chord: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
__FORCETOC__
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
A '''Fokker chord''' is a chord which is also a strong [[Fokker block]]. Such chords belong to an [[arena]] of related chords, which includes its various inversions, but other chords besides. Often a Fokker chord is a [[wakalix]], so that it belongs to more than one arena.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2014-04-03 10:30:14 UTC</tt>.<br>
: The original revision id was <tt>500311060</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">[[toc|flat]]


By a //Fokker chord// is meant a chord which is also a strongly epimorphic Fokker block. Such chords belong to an [[Fokker blocks#First definition of a Fokker block|arena]] of related chords, which includes its various inversions, but other chords besides. Often a Fokker chord is a wakalix, so that it belongs to more than one arena.
== 5-limit triads ==
The major and minor 5-limit triads are wakalixes; in terms of the [[Mathematical theory of Fokker blocks#The fb function and modal UDP notation|Fokblock function]] the major triad in root position can be denoted by Fokblock([16/15, 10/9], [0, 1]), Fokblock([[25/24, 10/9], [2, 0]) or Fokblock([25/24, 16/15], [1, 0]) and the minor triad Fokblock([16/15, 10/9], [1, 0]), Fokblock([25/24, 10/9], [1, 0]), or Fokblock([25/24, 16/15], [0, 0]). Each of these arenas contain the three inversions of both the major and minor triads, plus the inversions of another triad. In the case of the [16/15, 10/9] arena, that's the qunital triad, in its quintal (1-9/8-3/2) and quartal (1-4/3-16/9) forms. [25/24, 10/9] adds the three inversions of the diminished triad, 1-6/5-5/3, and [25/24, 16/15] adds the three inversions of the augmented triad, 1-5/4-8/5.


=5-limit triads=
== 7-limit tetrads ==
The major and minor 5-limit triads are wakalixes; in terms of the [[Fokker blocks#The Fokblock function and modal UDP notation|Fokblock function]] the major triad in root position can be denoted by Fokblock([16/15, 10/9], [0, 1]), Fokblock([[25/24, 10/9], [2, 0]) or Fokblock([25/24, 16/15], [1, 0]) and the minor triad Fokblock([16/15, 10/9], [1, 0]), Fokblock([25/24, 10/9], [1, 0]), or Fokblock([25/24, 16/15], [0, 0]). Each of these arenas contain the three inversions of both the major and minor triads, plus the inversions of another triad. In the case of the [16/15, 10/9] arena, that's the qunital triad, in its quintal (1-9/8-3/2) and quartal (1-4/3-16/9) forms. [25/24, 10/9] adds the three inversions of the diminished triad, 1-6/5-5/3, and [25/24, 16/15] adds the three inversions of the augmented triad, 1-5/4-8/5.
{{Todo|rework|inline=1|text=Replace wedgies}}


=7-limit tetrads=
The major (otonal) and minor (utonal) 7-limit tetrads are Fokker blocks in eight different ways. In the table below, the chromas defining each of these arenas are listed in the first columns, and the offsets giving the otonal and utonal tetrads in the second. Hence, from the first row, we can find that the otonal tetrad is Fokblock([21/20, 15/14, 35/32], [2, 2, 2] ) and the utonal tetrad is Fokblock([21/20, 15/14, 35/32], [3, 3, 0]). The dual Fokker group basis is given in the last column in abbreviated form. Each number corresponds to a patent val, and that val, wedged with the patent val for four, gives the Fokker group element. Hence in the first row, 1, 3, 2 is shorthand for 1&amp;4, 3&amp;4, 2&amp;4, and these denote the wedgies &lt;&lt;2 -1 1 -6 -4 5||, &lt;&lt;2 1 -1 -3 -7 -5||, &lt;&lt;0 2 2 3 3 -1||.
The major (otonal) and minor (utonal) 7-limit tetrads are Fokker blocks in eight different ways. In the table below, the chromas defining each of these arenas are listed in the first columns, and the offsets giving the otonal and utonal tetrads in the second. Hence, from the first row, we can find that the otonal tetrad is Fokblock([21/20, 15/14, 35/32], [2, 2, 2] ) and the utonal tetrad is Fokblock([21/20, 15/14, 35/32], [3, 3, 0]). The dual Fokker group basis is given in the last column in abbreviated form. Each number corresponds to a patent val, and that val, wedged with the patent val for four, gives the Fokker group element. Hence in the first row, 1, 3, 2 is shorthand for 1&amp;4, 3&amp;4, 2&amp;4, and these denote the wedgies &lt;&lt;2 -1 1 -6 -4 5||, &lt;&lt;2 1 -1 -3 -7 -5||, &lt;&lt;0 2 2 3 3 -1||.


|| Chroma basis || Major offsets || Minor offsets || Dual basis ||
{| class="wikitable"
|| [21/20, 15/14, 35/32] || [2, 2, 2] || [3, 3, 0] || 1, 3, 2 ||
|-
|| [25/24, 15/14, 35/32] || [1, 2, 3] || [0, 3, 2] || 1, 3, 7 ||
| | Chroma basis
|| [25/24, 21/20, 15/14] || [2, 3, 2] || [0, 2, 3] || 2, 7, 3 ||
| | Major offsets
|| [36/35, 21/20, 15/14] || [0, 3, 3] || [2, 2, 2] || 2, 7, 5 ||
| | Minor offsets
|| [36/35, 25/24, 21/20] || [2, 3, 3] || [3, 2, 2] || 3, 5, 7 ||
| | Dual basis
|| [49/48, 21/20, 15/14] || [2, 2, 3] || [0, 3, 2] || 2, 1, 5 ||
|-
|| [49/48, 21/20, 35/32] || [1, 2, 3] || [0, 3, 2] || 3, 1, 5 ||
| | [21/20, 15/14, 35/32]
|| [49/48, 36/35, 15/14] || [3, 2, 3] || [2, 3, 2] || 7, 1, 5 ||</pre></div>
| | [2, 2, 2]
<h4>Original HTML content:</h4>
| | [3, 3, 0]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Fokker chord&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:4:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:4 --&gt;&lt;!-- ws:start:WikiTextTocRule:5: --&gt;&lt;a href="#x5-limit triads"&gt;5-limit triads&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:5 --&gt;&lt;!-- ws:start:WikiTextTocRule:6: --&gt; | &lt;a href="#x7-limit tetrads"&gt;7-limit tetrads&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:6 --&gt;&lt;!-- ws:start:WikiTextTocRule:7: --&gt;
| | 1, 3, 2
&lt;!-- ws:end:WikiTextTocRule:7 --&gt;&lt;br /&gt;
|-
By a &lt;em&gt;Fokker chord&lt;/em&gt; is meant a chord which is also a strongly epimorphic Fokker block. Such chords belong to an &lt;a class="wiki_link" href="/Fokker%20blocks#First definition of a Fokker block"&gt;arena&lt;/a&gt; of related chords, which includes its various inversions, but other chords besides. Often a Fokker chord is a wakalix, so that it belongs to more than one arena.&lt;br /&gt;
| | [25/24, 15/14, 35/32]
&lt;br /&gt;
| | [1, 2, 3]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x5-limit triads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;5-limit triads&lt;/h1&gt;
| | [0, 3, 2]
The major and minor 5-limit triads are wakalixes; in terms of the &lt;a class="wiki_link" href="/Fokker%20blocks#The Fokblock function and modal UDP notation"&gt;Fokblock function&lt;/a&gt; the major triad in root position can be denoted by Fokblock([16/15, 10/9], [0, 1]), Fokblock([[25/24, 10/9], [2, 0]) or Fokblock([25/24, 16/15], [1, 0]) and the minor triad Fokblock([16/15, 10/9], [1, 0]), Fokblock([25/24, 10/9], [1, 0]), or Fokblock([25/24, 16/15], [0, 0]). Each of these arenas contain the three inversions of both the major and minor triads, plus the inversions of another triad. In the case of the [16/15, 10/9] arena, that's the qunital triad, in its quintal (1-9/8-3/2) and quartal (1-4/3-16/9) forms. [25/24, 10/9] adds the three inversions of the diminished triad, 1-6/5-5/3, and [25/24, 16/15] adds the three inversions of the augmented triad, 1-5/4-8/5.&lt;br /&gt;
| | 1, 3, 7
&lt;br /&gt;
|-
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x7-limit tetrads"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;7-limit tetrads&lt;/h1&gt;
| | [25/24, 21/20, 15/14]
The major (otonal) and minor (utonal) 7-limit tetrads are Fokker blocks in eight different ways. In the table below, the chromas defining each of these arenas are listed in the first columns, and the offsets giving the otonal and utonal tetrads in the second. Hence, from the first row, we can find that the otonal tetrad is Fokblock([21/20, 15/14, 35/32], [2, 2, 2] ) and the utonal tetrad is Fokblock([21/20, 15/14, 35/32], [3, 3, 0]). The dual Fokker group basis is given in the last column in abbreviated form. Each number corresponds to a patent val, and that val, wedged with the patent val for four, gives the Fokker group element. Hence in the first row, 1, 3, 2 is shorthand for 1&amp;amp;4, 3&amp;amp;4, 2&amp;amp;4, and these denote the wedgies &amp;lt;&amp;lt;2 -1 1 -6 -4 5||, &amp;lt;&amp;lt;2 1 -1 -3 -7 -5||, &amp;lt;&amp;lt;0 2 2 3 3 -1||.&lt;br /&gt;
| | [2, 3, 2]
&lt;br /&gt;
| | [0, 2, 3]
| | 2, 7, 3
|-
| | [36/35, 21/20, 15/14]
| | [0, 3, 3]
| | [2, 2, 2]
| | 2, 7, 5
|-
| | [36/35, 25/24, 21/20]
| | [2, 3, 3]
| | [3, 2, 2]
| | 3, 5, 7
|-
| | [49/48, 21/20, 15/14]
| | [2, 2, 3]
| | [0, 3, 2]
| | 2, 1, 5
|-
| | [49/48, 21/20, 35/32]
| | [1, 2, 3]
| | [0, 3, 2]
| | 3, 1, 5
|-
| | [49/48, 36/35, 15/14]
| | [3, 2, 3]
| | [2, 3, 2]
| | 7, 1, 5
|}


The eight overlapping but not identical arenas above contain a total of 54 [[dome]]s. Of these six are chords of 9-[[odd-limit]] just intonation: the major tetrad, 1-5/4-3/2-7/4, the minor tetrad, 1-6/5-3/2-12/7, the supermajor tetrad, 1-9/7-3/2-9/5, the subminor tetrad, 1-7/6-3/2-5/3, the added sixth tetrad, 1-5/4-3/2-5/3, and the swiss tetrad, 1-7/6-3/2-7/4. In the 15-odd-limit we may add 1-5/4-3/2-15/8. They also contain a number of [[essentially tempered chord|essentially tempered tetrads]].


&lt;table class="wiki_table"&gt;
7-limit marvel (225/224): 1-5/4-7/5-7/4, 1-5/4-7/5-8/5, 1-5/4-10/7-8/5
    &lt;tr&gt;
        &lt;td&gt;Chroma basis&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Major offsets&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Minor offsets&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Dual basis&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[21/20, 15/14, 35/32]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[2, 2, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[3, 3, 0]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1, 3, 2&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[25/24, 15/14, 35/32]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[1, 2, 3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[0, 3, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1, 3, 7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[25/24, 21/20, 15/14]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[2, 3, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[0, 2, 3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2, 7, 3&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[36/35, 21/20, 15/14]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[0, 3, 3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[2, 2, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2, 7, 5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[36/35, 25/24, 21/20]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[2, 3, 3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[3, 2, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3, 5, 7&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[49/48, 21/20, 15/14]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[2, 2, 3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[0, 3, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2, 1, 5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[49/48, 21/20, 35/32]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[1, 2, 3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[0, 3, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3, 1, 5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;[49/48, 36/35, 15/14]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[3, 2, 3]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;[2, 3, 2]&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;7, 1, 5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
7-limit starling (126/125): 1-5/4-3/2-9/5, 1-5/4-7/5-5/3, 1-5/4-7/5-7/4, 1-6/5-3/2-5/3, 1-6/5-10/7-8/5
 
diminished {1-6/5-10/7-5/3, 1-6/5-7/5-5/3, 1-6/5-25/18-5/3, 1-7/6-7/5-5/3}
 
keenanismic (385/384): 1-5/4-3/2-12/7, 1-6/5-3/2-7/4, 1-6/5-7/5-7/4, 1-7/6-7/5-8/5, 1-7/6-35/24-8/5, 1-8/7-35/24-5/3
 
werckismic (441/440): 1-5/4-10/7-7/4
 
swetismic (540/539): 1-6/5-7/5-12/7
 
zeus: 1-5/4-35/24-8/5, 1-5/4-25/16-12/7, 1-6/5-35/24-7/4
 
minerva: 1-7/6-3/2-12/7, 1-9/7-3/2-7/4, 1-5/4-7/5-7/4, 1-5/4-7/5-8/5, 1-5/4-10/7-8/5
 
jove: 1-7/6-10/7-7/4, 1-8/7-7/5-12/7
 
orwell: 1-7/6-7/5-12/7, 1-7/6-10/7-12/7, {1-5/4-35/24-12/7, 1-7/6-48/35-12/7, 1-7/6-35/24-12/7, 1-7/6-49/36-12/7}
 
[[Category:Fokker block]]
[[Category:Wakalixes]]
[[Category:Lists of chords]]
[[Category:Chords]]
[[Category:5-limit]]
[[Category:7-limit]]

Latest revision as of 19:06, 28 July 2025

A Fokker chord is a chord which is also a strong Fokker block. Such chords belong to an arena of related chords, which includes its various inversions, but other chords besides. Often a Fokker chord is a wakalix, so that it belongs to more than one arena.

5-limit triads

The major and minor 5-limit triads are wakalixes; in terms of the Fokblock function the major triad in root position can be denoted by Fokblock([16/15, 10/9], [0, 1]), Fokblock([[25/24, 10/9], [2, 0]) or Fokblock([25/24, 16/15], [1, 0]) and the minor triad Fokblock([16/15, 10/9], [1, 0]), Fokblock([25/24, 10/9], [1, 0]), or Fokblock([25/24, 16/15], [0, 0]). Each of these arenas contain the three inversions of both the major and minor triads, plus the inversions of another triad. In the case of the [16/15, 10/9] arena, that's the qunital triad, in its quintal (1-9/8-3/2) and quartal (1-4/3-16/9) forms. [25/24, 10/9] adds the three inversions of the diminished triad, 1-6/5-5/3, and [25/24, 16/15] adds the three inversions of the augmented triad, 1-5/4-8/5.

7-limit tetrads

Todo: rework

Replace wedgies

The major (otonal) and minor (utonal) 7-limit tetrads are Fokker blocks in eight different ways. In the table below, the chromas defining each of these arenas are listed in the first columns, and the offsets giving the otonal and utonal tetrads in the second. Hence, from the first row, we can find that the otonal tetrad is Fokblock([21/20, 15/14, 35/32], [2, 2, 2] ) and the utonal tetrad is Fokblock([21/20, 15/14, 35/32], [3, 3, 0]). The dual Fokker group basis is given in the last column in abbreviated form. Each number corresponds to a patent val, and that val, wedged with the patent val for four, gives the Fokker group element. Hence in the first row, 1, 3, 2 is shorthand for 1&4, 3&4, 2&4, and these denote the wedgies <<2 -1 1 -6 -4 5||, <<2 1 -1 -3 -7 -5||, <<0 2 2 3 3 -1||.

Chroma basis Major offsets Minor offsets Dual basis
[21/20, 15/14, 35/32] [2, 2, 2] [3, 3, 0] 1, 3, 2
[25/24, 15/14, 35/32] [1, 2, 3] [0, 3, 2] 1, 3, 7
[25/24, 21/20, 15/14] [2, 3, 2] [0, 2, 3] 2, 7, 3
[36/35, 21/20, 15/14] [0, 3, 3] [2, 2, 2] 2, 7, 5
[36/35, 25/24, 21/20] [2, 3, 3] [3, 2, 2] 3, 5, 7
[49/48, 21/20, 15/14] [2, 2, 3] [0, 3, 2] 2, 1, 5
[49/48, 21/20, 35/32] [1, 2, 3] [0, 3, 2] 3, 1, 5
[49/48, 36/35, 15/14] [3, 2, 3] [2, 3, 2] 7, 1, 5

The eight overlapping but not identical arenas above contain a total of 54 domes. Of these six are chords of 9-odd-limit just intonation: the major tetrad, 1-5/4-3/2-7/4, the minor tetrad, 1-6/5-3/2-12/7, the supermajor tetrad, 1-9/7-3/2-9/5, the subminor tetrad, 1-7/6-3/2-5/3, the added sixth tetrad, 1-5/4-3/2-5/3, and the swiss tetrad, 1-7/6-3/2-7/4. In the 15-odd-limit we may add 1-5/4-3/2-15/8. They also contain a number of essentially tempered tetrads.

7-limit marvel (225/224): 1-5/4-7/5-7/4, 1-5/4-7/5-8/5, 1-5/4-10/7-8/5

7-limit starling (126/125): 1-5/4-3/2-9/5, 1-5/4-7/5-5/3, 1-5/4-7/5-7/4, 1-6/5-3/2-5/3, 1-6/5-10/7-8/5

diminished {1-6/5-10/7-5/3, 1-6/5-7/5-5/3, 1-6/5-25/18-5/3, 1-7/6-7/5-5/3}

keenanismic (385/384): 1-5/4-3/2-12/7, 1-6/5-3/2-7/4, 1-6/5-7/5-7/4, 1-7/6-7/5-8/5, 1-7/6-35/24-8/5, 1-8/7-35/24-5/3

werckismic (441/440): 1-5/4-10/7-7/4

swetismic (540/539): 1-6/5-7/5-12/7

zeus: 1-5/4-35/24-8/5, 1-5/4-25/16-12/7, 1-6/5-35/24-7/4

minerva: 1-7/6-3/2-12/7, 1-9/7-3/2-7/4, 1-5/4-7/5-7/4, 1-5/4-7/5-8/5, 1-5/4-10/7-8/5

jove: 1-7/6-10/7-7/4, 1-8/7-7/5-12/7

orwell: 1-7/6-7/5-12/7, 1-7/6-10/7-12/7, {1-5/4-35/24-12/7, 1-7/6-48/35-12/7, 1-7/6-35/24-12/7, 1-7/6-49/36-12/7}

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