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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Beginner| Mathematical theory of Fokker blocks}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{Wikipedia| Fokker periodicity block }} |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-06-30 03:29:05 UTC</tt>.<br>
| | A '''Fokker block''' (or '''periodicity block''') is a [[periodic scale|periodic]] [[scale]] that can be thought of as a tile on a [[lattice]] of [[pitch class]]es (of a [[JI subgroup]] or a [[regular temperament]]) shaped as a parallelogram, parallelepiped or higher-dimensional analog. It comprises those intervals in the lattice which fall inside the tile after moving the tile on the lattice to a place where no lattice point is on its boundary. (Different positions of the tile can create scales which are not rotations of each other.) The scale repeats at the [[interval of equivalence]], which lies on the [[1/1|unison]] in the lattice of pitch classes. |
| : The original revision id was <tt>151066255</tt>.<br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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">The **Fokker block** is one of the most notable inventions of the physicist and music theorist [[http://en.wikipedia.org/wiki/Adriaan_Fokker|Adriaan Fokker]]. While the idea generalizes easily to [[just intonation subgroups]], for ease of exposition we will suppose that we are in a [[Harmonic Limit|p-limit]] situation with n=pi(p) primes up to an including p.
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| Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep>, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the [[Vals and Tuning Space|val]] v = <v2, v3, ... vp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if v2<0 we reverse sign, and we have a val v which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 81/80, 128/125 and 64/63, the above procedure gives us the val v = <12 19 28 34|, and we will be looking at a 12-note scale in the 7-limit.
| | All Fokker blocks are weakly [[epimorphic]], which means that there is a [[val]] that maps each note of the Fokker block onto its own equal temperament scale step and leaves no equal temperament scale step without a mapping towards it. (In other words, this val provides a bijection between the Fokker block and the equal temperament.) If a Fokker block is epimorphic, which means that the val preserves the order of the steps, it is a ''strong'' Fokker block; otherwise it is a ''weak'' Fokker block. The expression "Fokker block" without any qualifier generally denotes a strong block. |
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| Now choose a "chroma" for the Fokker block, which is a p-limit interval c such that v(c) = 1; that is, if m is the monzo for c, then <v|m>=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 81/80, 128/125 and 64/63, we could for instance choose 28/27, 25/24, 21/20, 16/15 or 15/14. Having selected a chroma, form the n by n matrix whose first row is the monzo for the chroma c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be +-1. It is therefore a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]], that is, a square matrix with coefficients which are integers and with determinant +-1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the [[http://en.wikipedia.org/wiki/Kronecker_delta|Kronecker delta]]. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.
| | The concept of the Fokker block was developed by the physicist and music theorist [[Adriaan Fokker]]. |
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| These unimodular matricies define a [[http://en.wikipedia.org/wiki/Change_of_basis|change of basis]] for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such
| | == Theory == |
| interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as
| | Fokker blocks have a shape which {{w|tessellation|tiles}} the lattice; an interval between pitches which lie across an edge of two Fokker blocks within the tiling will be altered from its normal value by an interval corresponding to an edge of the parellelepiped. This edge turns out to be the difference between intervals that span the same number of steps in the scale, and so it is called a [[chroma]]. (Fokker called them ''unison vectors'' in his original text.) |
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| q = c1^v1(q) * c2^v2(q) ... cn^vn(q)
| | If we place the coordinate vectors of the chromas into a matrix, the number of notes in the Fokker block is equal to the determinant of that matrix. It is also equal to the volume of the parallelepiped formed by the chromas, which is called the '''fundamental domain'''. |
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| | The rank of a Fokker block is the rank of the underlying lattice of pitches including the interval of equivalence. A rank-''n'' Fokker block has {{nowrap| ''n'' - 1 }} chromas: a consequence of this is that a Fokker block of rank ''n'' has [[maximum variety]] at most 2<sup style="white-space: nowrap;">(''n'' − 1)</sup> (since that's the number of combinations of chromas a note can be altered by). For example, a rank-2 Fokker block has max variety at most 2 (hence is a [[MOS scale|mos]]), and a rank-3 Fokker block has max variety at most 4. These results are true for strong and weak Fokker blocks only if we allow negative steps; otherwise they are only true for strong Fokker blocks. In this way, Fokker blocks generalize mos scales. |
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| | A Fokker block can be made [[constant structure]] (with negative steps allowed) by moving the generator sizes by an arbitrarily small amount. If the logarithmic sizes of the generators are linearly independent (as happens in JI, for example), the generator sizes need not be moved. The constant structure will have no negative steps if and only if the Fokker block is strong. |
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| | A Fokker '''arena''' contains all the periodic scales that can be constructed as Fokker blocks from the same list of chromas. |
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| | == Examples == |
| | === Ptolemy's intense diatonic === |
| | [[File:Fokker_block_zarlino.png|400px|thumb|Fokker block corresponding to the just diatonic scale. The gray grid is the interval lattice, and the black lines show the sublattice generated by the chromas. The fundamental domain is colored in blue.]]Let's take [[5-limit]] just intonation, and pick the [[25/24|just chromatic semitone]] (25/24) and the [[syntonic comma]] (81/80) as our chromas. |
| | The octave equivalent lattice is generated by fifths and just major thirds. |
| | Since <math>25/24 = 2^{-3} \cdot 3^{-1} \cdot 5^2</math>, it has coordinates <math>(-1, 2)</math> in the octave-equivalent lattice. |
| | The syntonic comma is reached by stacking 4 fifths and going down a major third, so it has coordinates <math>(4, -1)</math>. |
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| </pre></div>
| | The corresponding Fokker block is [[Ptolemy's intense diatonic]], also known as Zarlino, specifically the lydian mode. |
| <h4>Original HTML content:</h4>
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| <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"><html><head><title>Fokker blocks</title></head><body>The <strong>Fokker block</strong> is one of the most notable inventions of the physicist and music theorist <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow">Adriaan Fokker</a>. While the idea generalizes easily to <a class="wiki_link" href="/just%20intonation%20subgroups">just intonation subgroups</a>, for ease of exposition we will suppose that we are in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> situation with n=pi(p) primes up to an including p. <br />
| | Tempering out either of the two chromas gives a MOS scale related to the temperament. |
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| | * Tempering out the syntonic comma gives the [[diatonic scale]] LLLsLLs, in [[meantone]]. |
| Suppose we have n-1 commas, and we form an n by n matrix, the top row of which are n indeterminate elements |e2 e3 e5 ... ep&gt;, and the other rows of which are the monzos corresponding to our chosen commas. If we take the determinant of this matrix, we get v2*e2+v3*e3+...+vp*ep where the v2, v3 ... vp are integers. We interpret this as the <a class="wiki_link" href="/Vals%20and%20Tuning%20Space">val</a> v = &lt;v2, v3, ... vp|. If this is a zero vector the commas are not independent, and if the there exists a common divisor we have what is known as a torsion problem, and we discard the comma set. Otherwise, if v2&lt;0 we reverse sign, and we have a val v which tells us what equal temperament our Fokker block will be approximating. For example, starting with the commas 81/80, 128/125 and 64/63, the above procedure gives us the val v = &lt;12 19 28 34|, and we will be looking at a 12-note scale in the 7-limit.<br />
| | * Tempering out the chromatic semitone gives the [[mosh]] LsLsLss (a 7-note neutral scale), in [[dicot]]. |
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| | If we temper out both 25/24 and 81/80, we get [[7edo|7 equal temperament]], which we can interpret as an equalized diatonic scale. |
| Now choose a &quot;chroma&quot; for the Fokker block, which is a p-limit interval c such that v(c) = 1; that is, if m is the monzo for c, then &lt;v|m&gt;=1. Precisely which interval with this property we choose doesn't actually matter, so if our commas are 81/80, 128/125 and 64/63, we could for instance choose 28/27, 25/24, 21/20, 16/15 or 15/14. Having selected a chroma, form the n by n matrix whose first row is the monzo for the chroma c, and whose other rows are the monzos of the n-1 commas. Because we have chosen c so that v(c)=1, the determinant of this matrix will be +-1. It is therefore a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>, that is, a square matrix with coefficients which are integers and with determinant +-1. Such a matrix is invertible, and the inverse matrix is also unimodular. If we call c c1, and label the commas c2, c3, ... cn; and if we consider the columns of the inverse matrix to be vals and call them v1, v2, ... vn, then by the definition of the inverse of a matrix, vi(cj) = delta(i,j), where delta(i,j) is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Kronecker_delta" rel="nofollow">Kronecker delta</a>. Stated another way, vi(cj) is 0 unless i equals j, in which case vi(ci) = 1.<br />
| | This scale is a Fokker block in multiple ways: it is also possible to arrive at the same set of notes using [[135/128]] together with either 81/80 or 25/24 as the chromas. |
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| These unimodular matricies define a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Change_of_basis" rel="nofollow">change of basis</a> for the p-limit system of musical intervals: just as every p-limit interval can be written as a product of primes up to p with integer exponents, every such <br />
| | === Duodene and 12 equal temperament === |
| interval is a product of c1, c2, ... cn with integer exponents. To determine the exponents, we use v1, v2, ... vn, so that if q is a p-limit rational number, we may write it as<br />
| | [[File:Fokker_block_duodene.png|400px|thumb|Duodene as a Fokker block. Note that the fundamental domain is shifted from the origin to obtain the familiar scale.]]Let's now use the [[128/125|diesis]] (128/125) and the [[syntonic comma]] (81/80) as our chromas, in 5-limit JI as above. |
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| | The diesis is the difference between an octave and three major thirds, so it has coordinates <math>(0, -3)</math>. |
| q = c1^v1(q) * c2^v2(q) ... cn^vn(q)</body></html></pre></div>
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| | The number of notes in the tile will be 12, since the determinant is: |
| | :<math> |
| | \begin{vmatrix} |
| | 0 & -3 \\ |
| | 4 & -1 |
| | \end{vmatrix} |
| | = (0 \cdot -1) - (-3 \cdot 4) |
| | = 12 |
| | </math> |
| | This should not be surprising, as [[12edo|12 equal temperament]] tempers out exactly these two commas, so each note in the resulting scale will correspond to an interval of 12et. |
| | The resulting just scale is known as [[duodene]]. |
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| | == See also == |
| | * [[User:Hkm/Fokker block code|This Python code]] can be used to find Fokker blocks. |
| | * [[Catalog of Fokker blocks]] |
| | * [[List of weak Fokker blocks]] |
| | * [[Fokker chord]] |
| | * [[Minkowski block]] |
| | * [[Minkowski reduced bases for Fokker groups of certain vals]] |
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| | == Further reading == |
| | * [http://www.huygens-fokker.org/docs/fokkerpb.html Unison Vectors and Periodicity Blocks] by [[Adriaan Fokker|A.D. Fokker]] |
| | * [http://www.tonalsoft.com/enc/f/fokker-gentle-1.aspx A gentle introduction to Fokker periodicity blocks], by [[Paul Erlich]] |
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| | [[Category:Fokker block| ]] <!-- main article --> |
| | [[Category:Pitch space]] |
| | {{Todo| add illustration | improve linking | review }} <!-- add beginner section that explains how to build Fokker blocks either by hand or using common software, along with visualizations. --> |