Fokker chord: Difference between revisions
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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= | == 5-limit triads == | ||
The major and minor 5-limit triads are wakalixes; in terms of the [[ | 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. | ||
== 7-limit tetrads == | |||
{{Todo|rework|inline=1|text=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||. | 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||. | ||
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The eight overlapping but not identical arenas above contain a total of 54 | 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]]. | ||
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 marvel (225/224): 1-5/4-7/5-7/4, 1-5/4-7/5-8/5, 1-5/4-10/7-8/5 | ||
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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} | 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:Fokker block]] | ||
[[Category:Wakalixes]] | [[Category:Wakalixes]] | ||
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
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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}