Meet and join: Difference between revisions
Wikispaces>genewardsmith **Imported revision 535825630 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 535876738 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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Given two temperaments A and B defined in terms of normal val lists, the join A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again via the [[dual list]] function, this defines both join and meet as operations on normal val lists. | Given two temperaments A and B defined in terms of normal val lists, the join A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again via the [[dual list]] function, this defines both join and meet as operations on normal val lists. | ||
There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it <JI>; and in the temperament defined by G^, nothing is tempered out, and we may also call it <1>. | There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it <JI>; and in the temperament defined by G^, nothing is tempered out, and we may also call it <1>. A≤B may be expressed by "A is supported by B". | ||
In mathematical order theory, meet and join are denoted by ∨ and ∧. We avoid doing that for two reasons; the first is to avoid confusion with the interior and wedgects of multivals. The second is that meet and join are operations on abstract temperaments; ordering by increasing size of the group of commas and decreasing size of the group of vals is regarded and notated as the same. | |||
=Examples= | =Examples= | ||
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Given two temperaments A and B defined in terms of normal val lists, the join A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again via the <a class="wiki_link" href="/dual%20list">dual list</a> function, this defines both join and meet as operations on normal val lists.<br /> | Given two temperaments A and B defined in terms of normal val lists, the join A⋏B is the reduction to a normal val list of the concatination of A and B, which is to say, the Hermite reduction of the list of vals of A with the vals of B (with the obvious extension to svals if G is not a full p-limit group.) The join in terms of vals is the subgroup of G^ generated by A and B, and in terms of intervals is defined by intersection of the commas of A and B. The meet A⋎B is defined by the intersection of the group of vals generated by A with the group of vals generated by B. If A and B are defined by normal interval lists, then the meet A⋎B is defined by concatinating A and B, and reducing the result to a normal interval list. Since temperaments expressed as normal val lists can be converted to temperaments expressed as normal interval lists and back again via the <a class="wiki_link" href="/dual%20list">dual list</a> function, this defines both join and meet as operations on normal val lists.<br /> | ||
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There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it &lt;JI&gt;; and in the temperament defined by G^, nothing is tempered out, and we may also call it &lt;1&gt;.<br /> | There is a partial order on the temperaments of G, given by A≤B iff A⋏B = A and A≤B iff A⋎B = B. Since A⋎G = G, G is the maximal temperament, and since A⋏G^ = G^, G^ is the minimal temperament. In the temperament defined by G, everything is tempered out, and we may also call it &lt;JI&gt;; and in the temperament defined by G^, nothing is tempered out, and we may also call it &lt;1&gt;. A≤B may be expressed by &quot;A is supported by B&quot;.<br /> | ||
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In mathematical order theory, meet and join are denoted by ∨ and ∧. We avoid doing that for two reasons; the first is to avoid confusion with the interior and wedgects of multivals. The second is that meet and join are operations on abstract temperaments; ordering by increasing size of the group of commas and decreasing size of the group of vals is regarded and notated as the same.<br /> | |||
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