Subgroup monzos and vals: Difference between revisions

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Given a [[just intonation subgroup]], we can find a canonical form for its ~~generators~~ by means of the [[Normal lists #Normal interval list|normal interval list]] which may be computed from any finite set of generators. In the case of the full ''p''-limit group for any prime ''p'', this consists of the primes from 2 to ''p'' in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and [[Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to ''p''. This generalization we may call the '''subgroup monzos''' and '''subgroup vals''', or '''smonzos''' and '''svals''' for short.

Given a [[just intonation subgroup]], we can find a canonical form for its [[generator]]s by means of the [[Normal lists #Normal interval list|normal interval list]] which may be computed from any finite set of generators. In the case of the full ''p''-limit group for any prime ''p'', this consists of the primes from 2 to ''p'' in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and [[Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to ''p''. This generalization we may call the '''subgroup monzos''' and '''subgroup vals''', or '''smonzos''' and '''svals''' for short.

For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with ~~rows~~ consisting of the monzos for 2, 3, and 13/10. Now take the [[pseudoinverse]] of M, M+. If u is a monzo for an interval in the subgroup, then uM+ gives the corresponding smonzo. For instance, the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and multiplying this by M+ gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 22 3-3 (13/5)2, which is 676/675 as desired.

For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix ''M'' with columns consisting of the monzos for 2, 3, and 13/10. Now take the [[pseudoinverse]] of ''M'', ''M''+. If '''u''' is a monzo for an interval in the subgroup, then ''M''+'''u''' gives the corresponding smonzo. For instance, the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and left-multiplying this by ''M''+ gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 22 3-3 (13/5)2, which is 676/675 as desired.

[[Category:~~Theory~~]]

[[Category:Subgroup]]

[[Category:Val]]

[[Category:Monzo]]

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-Given a [[just intonation subgroup]], we can find a canonical form for its generators by means of the [[Normal lists #Normal interval list|normal interval list]] which may be computed from any finite set of generators. In the case of the full ''p''-limit group for any prime ''p'', this consists of the primes from 2 to ''p'' in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and [[Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to ''p''. This generalization we may call the '''subgroup monzos''' and '''subgroup vals''', or '''smonzos''' and '''svals''' for short.
+Given a [[just intonation subgroup]], we can find a canonical form for its [[generator]]s by means of the [[Normal lists #Normal interval list|normal interval list]] which may be computed from any finite set of generators. In the case of the full ''p''-limit group for any prime ''p'', this consists of the primes from 2 to ''p'' in ascending order. This is precisely the ordered list used to define [[Vals and tuning space|vals]] and [[Monzos and interval space|monzos]], and we may generalize the notation simply by using any normal interval list in place of the ascending primes to ''p''. This generalization we may call the '''subgroup monzos''' and '''subgroup vals''', or '''smonzos''' and '''svals''' for short.
-For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix M with rows consisting of the monzos for 2, 3, and 13/10. Now take the [[pseudoinverse]] of M, M<sup>+</sup>. If u is a monzo for an interval in the subgroup, then uM<sup>+</sup> gives the corresponding smonzo. For instance, the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and multiplying this by M<sup>+</sup> gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 2<sup>2</sup> 3<sup>-3</sup> (13/5)<sup>2</sup>, which is 676/675 as desired.
+For example, consider the subgroup generated by the [[The Archipelago|barbados triad]], 1-13/10-3/2-2. The normal interval list for [13/10, 3/2, 2] is [2, 3, 13/5], and we may use these just like the primes [2, 3, 5] of 5-limit just intonation. We may convert intervals in the subgroup into smonzos by the following procedure: form the matrix ''M'' with columns consisting of the monzos for 2, 3, and 13/10. Now take the [[pseudoinverse]] of ''M'', ''M''<sup>+</sup>. If '''u''' is a monzo for an interval in the subgroup, then ''M''<sup>+</sup>'''u''' gives the corresponding smonzo. For instance, the monzo for 676/675 is {{monzo| 2 -3 2 0 0 2 }}, and left-multiplying this by ''M''<sup>+</sup> gives the smonzo {{monzo| 2 -3 2 }}. We may check this is the correct smonzo from 2<sup>2</sup> 3<sup>-3</sup> (13/5)<sup>2</sup>, which is 676/675 as desired.
-[[Category:Theory]]
+[[Category:Subgroup]]
 [[Category:Val]]
 [[Category:Monzo]]
-{{todo|reduce mathslang}}
+{{Todo| reduce mathslang }}