Gencom - Revision history

FloraC: + some actual formulae here

2025-08-30T14:28:44Z

+ some actual formulae here

← Older revision		Revision as of 14:28, 30 August 2025
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	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.		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.

	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~~; the gencom mapping applied to a subgroup basis matrix is a matrix of generator steps in terms of [[smonzos and svals\|subgroup vals (svals)]], which is exactly the sval mapping~~. 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 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.

			In other words, the gencom mapping takes ordinary monzos just as the [[subgroup monzos and vals\|subgroup-val]] mapping takes subgroup monzos:

			$$ V \cdot \vec m = V_G \cdot \vec m_G $$

			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.

			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

			$$ VS = V_G $$

	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.		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.

FloraC: Re-organize

2025-08-30T14:08:28Z

Re-organize

← Older revision		Revision as of 14:08, 30 August 2025
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	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 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]].		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]].

			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.

			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.

			== Gencom mapping ==
	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.		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.

	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; the gencom mapping applied to a subgroup basis matrix is a matrix of generator steps in terms of [[smonzos and svals\|subgroup vals (svals)]], which is exactly the sval mapping. 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 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; the gencom mapping applied to a subgroup basis matrix is a matrix of generator steps in terms of [[smonzos and svals\|subgroup vals (svals)]], which is exactly the sval mapping. 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.

	~~== Invariance ==~~		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.
	Converting a gencom to a [[Normal lists #Normal interval list\|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: 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 [''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.

	== Example ==		== Example ==

Godtone: typo

2025-01-25T18:29:29Z

typo

← Older revision		Revision as of 18:29, 25 January 2025
Line 1:		Line 1:
	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 [[7edo\|7et]].		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 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]].

	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.		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.

FloraC: /* Example */ clarify a bit

2024-10-20T08:42:56Z

Example: clarify a bit

← Older revision		Revision as of 08:42, 20 October 2024
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	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 }}.		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 }}.

	With this mapping 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.		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 }}.

	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 +29 steps ~~as a result~~, ~~but 625~~ is not in the subgroup to begin with.		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.

	[[Category:Generator]]		[[Category:Generator]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

FloraC: /* Example */ there should be a colon in the mapping too

2024-09-28T13:59:44Z

Example: there should be a colon in the mapping too

← Older revision		Revision as of 13:59, 28 September 2024
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	== Example ==		== 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 }}.		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 }}.

	With this mapping 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.		With this mapping 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.

FloraC: So the gencom itself is a subgroup basis matrix

2024-09-28T13:45:11Z

So the gencom itself is a subgroup basis matrix

← Older revision		Revision as of 13:45, 28 September 2024
Line 1:		Line 1:
	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 [[7edo\|7et]].		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 [[7edo\|7et]].

	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 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 ~~[[Wikipedia: Unimodular matrix~~\|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.		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.

	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; the gencom mapping applied to a [[subgroup basis matrix]] is a matrix of generator steps in terms of [[smonzos and svals\|subgroup vals (svals)]], which is exactly the sval mapping. 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 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; the gencom mapping applied to a subgroup basis matrix is a matrix of generator steps in terms of [[smonzos and svals\|subgroup vals (svals)]], which is exactly the sval mapping. 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.

	== Invariance ==		== Invariance ==

FloraC: +a long awaited example section

2024-09-28T13:13:22Z

+a long awaited example section

← Older revision		Revision as of 13:13, 28 September 2024
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	== Invariance ==		== Invariance ==
	Converting a gencom to a [[Normal lists #Normal interval list\|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: 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 [''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.		Converting a gencom to a [[Normal lists #Normal interval list\|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: 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 [''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.

			== 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 }}.

			With this mapping 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.

			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 +29 steps as a result, but 625 is not in the subgroup to begin with.

	[[Category:Generator]]		[[Category:Generator]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

FloraC: Sectioning. +relation to sval mapping

2024-09-28T13:08:27Z

Sectioning. +relation to sval mapping

← Older revision		Revision as of 13:08, 28 September 2024
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	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 [[7edo\|7et]].		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 [[7edo\|7et]].

	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 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 [[Wikipedia: Unimodular matrix\|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 ~~pesudoinverse~~ of the gencom matrix is the extended gencom mapping, and the transpose pseudoinverse of the extended mapping is the gencom matrix.		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 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 [[Wikipedia: Unimodular matrix\|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.

	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 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; the gencom mapping applied to a [[subgroup basis matrix]] is a matrix of generator steps in terms of [[smonzos and svals\|subgroup vals (svals)]], which is exactly the sval mapping. 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.

	Converting a gencom to a [[Normal lists #Normal interval list\|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 [''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.		== Invariance ==
			Converting a gencom to a [[Normal lists #Normal interval list\|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: 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 [''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.

	[[Category:Generator]]		[[Category:Generator]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Fredg999 category edits: Categories

2023-03-02T03:47:09Z

Fredg999 category edits: Moving from Category:Temperament to Category:Regular temperament theory using Cat-a-lot

2022-03-01T02:20:05Z

Moving from Category:Temperament to Category:Regular temperament theory using Cat-a-lot

← Older revision		Revision as of 02:20, 1 March 2022
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	[[Category:Gencom]]		[[Category:Gencom]]
	[[Category:Generator]]		[[Category:Generator]]
	[[Category:~~Temperament~~]]		[[Category:Regular temperament theory]]
	[[Category:Theory]]		[[Category:Theory]]

← Older revision		Revision as of 03:47, 2 March 2023
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	Converting a gencom to a [[Normal lists #Normal interval list\|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 [''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.		Converting a gencom to a [[Normal lists #Normal interval list\|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 [''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.

	~~[[Category:Gencom]]~~
	[[Category:Generator]]		[[Category:Generator]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	~~[[Category:Theory]]~~