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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | A '''gencom''' is a list of [[generator]]s for a [[regular temperament|temperament]] followed by [[comma]]s for the temperament, in a specific order. The generators are [[transversal generators]], meaning rational intervals belonging to the JI group the temperament tempers, which it tempers to generators for the temperament. The gencom is denoted [generator list; comma list], with a semicolon between the generators and the commas. For instance, [16/15 25/24; 81/80] is a gencom for [[5-limit]] meantone. On the other hand, the exact same intervals with a different placement of the semicolon is a gencom for 5-limit JI: [16/15 25/24 81/80;], and another, [16/15; 25/24 81/80] gives 5-limit [[7edo|7et]]. |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-06-14 15:51:15 UTC</tt>.<br>
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| : The original revision id was <tt>345428762</tt>.<br>
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| : The revision comment was: <tt></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">A //gencom// is a list of generators for a temperament followed by commas for the temperament, in a specific order. The generators are [[transversal generators]], meaning rational intervals belonging to the JI group the temperament tempers, which it tempers to generators for the temperment. The gencom is denoted [generator list; comma list], with a semicolon between the generators and the commas. For instance, [16/15, 25/24; 81/80] is a gencom for 5-limit meantone. On the other hand, the exact same intervals with a different placement of the semicolon is a gencom for 5-limit JI: [16/15, 25/24, 81/80;], and another, [16/15; 25/24, 81/80] gives 5-limit 7-equal.
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| The reason for putting the generators together with the commas is that notating the gencom as a list of [[monzos]] allows it to be treated as a matrix. The group of intervals generated by the gencom is the same no matter how we place the semicolon, and so is the matrix. When this group is a full p-limit group, as in the example above, the matrix is a [[http://en.wikipedia.org/wiki/Unimodular_matrix|unimodular matrix]]. Inverting and transposing it gives a matrix whose rows are vals; if r is the rank of the temperament then the first r rows are the mapping matrix corresponding to the generator transversal. More interesting is the case where the gencom generates a [[Just intonation subgroups|JI subgroup]] of some p-limit. In all cases the transpose of the [[Tenney-Euclidean Tuning#The pseudoinverse|pseudoinverse]] of the matrix of monzos gives a matrix of vals whose first r rows we call the gencom mapping, and which in its entirety we call the extended gencom mapping. The extended gencom mapping is only a unimodular matrix, and the inversion ordinary matrix inversion, in the case of the full p-limit. However in all cases the transpose pesudoinverse of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix.
| | A gencom uniquely characterizes a temperament in that it specifies the unique temperament on the given group tempering out the given commas. In other words, converting a gencom to a [[normal forms #Normal forms for commas|normal interval list]] gives a canonical form for the subgroup, which is an {{w|invariant (mathematics)|invariant}} of the temperament; doing the same to just the commas produces another invariant, and together these determine the temperament. |
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| The rows of the extended gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.
| | However, the normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3. |
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| Converting a gencom to a [[Normal lists#x-Normal interval lists|normal interval list]] gives a canonical form for the subgroup, which is an invariant of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3. However, the extended gencom mapping can be used to determine if an interval q is in the group of the temperament. Suppose [c1 c2 ... cn] is a gencom and [v1 v2 ... vn] is the corresponding extended mapping. Then each of v1(q), v2(q) ... vn(q) must be an integer, and moreover we must have q = c1^v1(q) * c2^v2(q) ... cn^vn(q). This provides sufficient coditions as well as necessary ones.</pre></div>
| | == Gencom mapping == |
| <h4>Original HTML content:</h4>
| | The reason for putting the generators together with the commas is that notating the gencom as a list of [[monzo]]s allows it to be treated as a [[subgroup basis matrix]]: the group of intervals generated by the gencom is the same no matter how we place the semicolon, and so is the matrix. When this group is a [[harmonic limit|full ''p''-limit group]], as in the example above, the matrix is a {{w|unimodular matrix}}. Inverting and transposing it gives a matrix whose rows are [[val]]s; if ''r'' is the [[rank]] of the temperament, then the first ''r'' rows are the [[mapping|mapping matrix]] corresponding to the generator transversal. More interesting is the case where the gencom generates a [[Just intonation subgroups|JI subgroup]] of some ''p''-limit. In all cases the transpose of the [[pseudoinverse]] of the matrix of monzos gives a matrix of vals whose first ''r'' rows we call the '''gencom mapping''', and which in its entirety we call the '''extended gencom mapping'''. The extended gencom mapping is only a unimodular matrix, and the inversion ordinary matrix inversion, in the case of the full ''p''-limit. However in all cases the transpose pseudoinverse of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix. |
| <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>Gencom</title></head><body>A <em>gencom</em> is a list of generators for a temperament followed by commas for the temperament, in a specific order. The generators are <a class="wiki_link" href="/transversal%20generators">transversal generators</a>, meaning rational intervals belonging to the JI group the temperament tempers, which it tempers to generators for the temperment. The gencom is denoted [generator list; comma list], with a semicolon between the generators and the commas. For instance, [16/15, 25/24; 81/80] is a gencom for 5-limit meantone. On the other hand, the exact same intervals with a different placement of the semicolon is a gencom for 5-limit JI: [16/15, 25/24, 81/80;], and another, [16/15; 25/24, 81/80] gives 5-limit 7-equal.<br />
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| | The rows of the extended gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup. |
| The reason for putting the generators together with the commas is that notating the gencom as a list of <a class="wiki_link" href="/monzos">monzos</a> allows it to be treated as a matrix. The group of intervals generated by the gencom is the same no matter how we place the semicolon, and so is the matrix. When this group is a full p-limit group, as in the example above, the matrix is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Unimodular_matrix" rel="nofollow">unimodular matrix</a>. Inverting and transposing it gives a matrix whose rows are vals; if r is the rank of the temperament then the first r rows are the mapping matrix corresponding to the generator transversal. More interesting is the case where the gencom generates a <a class="wiki_link" href="/Just%20intonation%20subgroups">JI subgroup</a> of some p-limit. In all cases the transpose of the <a class="wiki_link" href="/Tenney-Euclidean%20Tuning#The pseudoinverse">pseudoinverse</a> of the matrix of monzos gives a matrix of vals whose first r rows we call the gencom mapping, and which in its entirety we call the extended gencom mapping. The extended gencom mapping is only a unimodular matrix, and the inversion ordinary matrix inversion, in the case of the full p-limit. However in all cases the transpose pesudoinverse of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix.<br /> | | |
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| | In other words, the gencom mapping takes ordinary monzos just as the [[subgroup monzos and vals|subgroup-val]] mapping takes subgroup monzos: |
| The rows of the extended gencom mapping are in general fractional vals, meaning the coefficients are allowed to be rational numbers. When applied to elements of the subgroup generated by the gencom, these fractional vals always return an integer value, which gives the number of times the corresponding generator or comma appears in the expression of the interval in terms of the gencom. However, the converse is not the case: if the gencom mapping returns integer values, it does not mean the interval must belong to the gencom subgroup.<br /> | | |
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| | $$ V \cdot \vec m = V_G \cdot \vec m_G $$ |
| Converting a gencom to a <a class="wiki_link" href="/Normal%20lists#x-Normal interval lists">normal interval list</a> gives a canonical form for the subgroup, which is an invariant of the temperament. Doing the same to just the commas produces another invariant, and together these determine the temperament: it is the unique temperament on the given group tempering out the given commas. The normal list defined by the generators alone is not an invariant of the temperament, since the generators give only a transversal for the tempered intervals of the temperament, not the full set of intervals being tempered. Hence, for instance, [2 40/27; 81/80] and [2 3/2; 81/80] both define 5-limit meantone, but the normal list for [2 40/27] is 2.27/5 and for [2 3/2] is 2.3. However, the extended gencom mapping can be used to determine if an interval q is in the group of the temperament. Suppose [c1 c2 ... cn] is a gencom and [v1 v2 ... vn] is the corresponding extended mapping. Then each of v1(q), v2(q) ... vn(q) must be an integer, and moreover we must have q = c1^v1(q) * c2^v2(q) ... cn^vn(q). This provides sufficient coditions as well as necessary ones.</body></html></pre></div>
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| | where ''V'' and ''V''<sub>''G''</sub> are gencom and subgroup-val mappings of the same temperament in subgroup ''G'', and '''m''' and '''m'''<sub>''G''</sub> are ordinary and subgroup monzos of the same interval in ''G'', respectively. |
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| | It follows that the gencom mapping applied to the subgroup basis matrix of the temperament is a matrix of generator steps in terms of subgroup vals, which is exactly the subgroup-val mapping. If ''S'' is the subgroup basis matrix, then |
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| | $$ VS = V_G $$ |
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| | The extended gencom mapping can also be used to determine if an interval ''q'' is in the group of the temperament. Suppose [''c''<sub>1</sub> ''c''<sub>2</sub> … ''c''<sub>n</sub>] is a gencom and [''v''<sub>1</sub> ''v''<sub>2</sub> … ''v''<sub>''n''</sub>] is the corresponding extended mapping. Then each of ''v''<sub>1</sub> (''q''), ''v''<sub>2</sub> (''q'') … ''v''<sub>''n''</sub> (''q'') must be an integer, and moreover we must have ''q'' = ''c''<sub>1</sub>^''v''<sub>1</sub> (''q'') · ''c''<sub>2</sub>^''v''<sub>2</sub> (''q'') · … · ''c''<sub>''n''</sub>^''v''<sub>''n''</sub> (''q''). This provides sufficient conditions as well as necessary ones. |
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| | == Example == |
| | Consider [[baldy]], the temperament tempering out 225/224, 325/324, and 640/637 in the 2.9.5.7.13 subgroup. This is every other step of [[garibaldi|garibaldi/cassandra]] in the 13-limit, without prime 11. With normalized generators ~2 and ~9, the gencom is [2 9; 225/224 325/324 640/637]. Converting it to a matrix of monzos, we get [{{monzo| 1 0 0 0 0 0 }}, {{monzo| 0 2 0 0 0 0 }}; {{monzo| -5 2 2 -1 0 0 }}, {{monzo| -2 -4 2 0 0 1 }}, {{monzo| 7 0 1 -2 0 -1 }}]. Taking the pseudoinverse and canonicalizing it, the extended gencom mapping is found to be [{{val| 1 0 15 25 0 28 }}, {{val| 0 1/2 -4 -7 0 10 }}; {{val| 0 0 2 3 0 4 }}, {{val| 0 0 -1 -2 0 3 }}, {{val| 0 0 -1 -2 0 2 }}]. Since this is a rank-2 temperament, the gencom mapping is the first two row thereof, {{mapping| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}. |
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| | With this mapping we can insert the monzo of an arbitrary interval to see its number of generator steps. For instance we can insert the monzo of 9, {{monzo| 0 2 }}, to the mapping and see it is represented by +1 generator step. Further, we can see prime 3, not in the subgroup, must be "1/2" generator step. Through the same process we find prime 5 is -4 steps and prime 7 is -7 steps, which correspond to -8 and -14 steps of garibaldi. Prime 11 is not in the temperament so it is signified by "0" steps. Finally, prime 13 is +10 steps, corresponding to +20 steps of garibaldi/cassandra. Collecting the numbers of generator steps of all the monzos in the subgroup basis we can convert the gencom mapping to the sval mapping: {{mapping| 1 0 15 25 -28 | 0 1 -4 -7 10 }}. |
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| | For a more complicated case let us consider [[edson]], the temperament tempering out 196/195, 352/351, and 364/363 in the 2.3.7/5.11/5.13/5 subgroup. Using normalized generators ~2 and ~3, its gencom mapping is {{mapping| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}. Like before, we can see 7/5 is mapped to -6 generator steps by inserting its monzo {{monzo| 0 0 -1 1 }} to the mapping. However, if we insert the monzo of 625, {{monzo| 0 0 4 }}, it will also return a seemingly meaningful result of +29 steps, which should ''not'' be so interpreted because the interval is not in the subgroup to begin with. |
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| | [[Category:Generator]] |
| | [[Category:Regular temperament theory]] |