Wedgie/Archived version: Difference between revisions

Line 1:

An alternating {{w|multilinear map}} which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value ~~we may call an '''''n''-map'''~~. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.

An '''n-map''' is an alternating {{w|multilinear map}} which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.

The simplest kind of ''n''-map is the 1-map, or [[val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a {{w|group homomorphism}} and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are {{w|Linear map|linear}}: If you take the product of two ''p''-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|f(''u'', ''v'') {{=}} −f(''v'', ''u'')}}.

@@ Line 1: / Line 1: @@
 {{texmap}}
 {{inacc}}
-An alternating {{w|multilinear map}} which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value we may call an '''''n''-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.
+An '''n-map''' is an alternating {{w|multilinear map}} which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.
 The simplest kind of ''n''-map is the 1-map, or [[val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a {{w|group homomorphism}} and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are {{w|Linear map|linear}}: If you take the product of two ''p''-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|f(''u'', ''v'') {{=}} −f(''v'', ''u'')}}.