Normal forms: Difference between revisions

There is some redundancy in the statement of these conditions, but that does no harm.

Given a list of ''p''-limit intervals, we can convert it to a normal list by the following procedure:

Normal interval lists can also be used to characterize the [just intonation subgroups]] on which subgroup temperaments are defined and using which subgroup scales may be constructed. On the pages [[chromatic pairs]], [[subgroup temperaments]] and [[just intonation subgroups]] can be found many examples; the subgroup lists are given in a form where generators of the subgroup are separated by periods so as to flag the fact that the list defines a subgroup. An example would be the Barbados subgroup, 2.3.13/5.

If L is a list of ''n'' vals, we may write it as an ''n''×''m'' matrix, where the rows of the matrix are the vals, and ''m'' = π (''p''), where ''p'' is the prime limit. To get the normal val list, we do the following:

The point of steps two and three is that now the vals on the list correspond to a list of generators which are all positive (written additively) or equivalently greater than 1 (written multiplicatively). Just as a normal comma list can be used to classify an [[abstract regular temperament]], so can a normal val list. The val list is what on [[Graham Breed]]'s [http://x31eq.com/temper/ web site] is called a "mapping", put into a canonical form. The "Maps" (though not the "Map to lattice") listed on temperament pages of this wiki are all normal val lists; an example would be [{{val| 1 0 -4 -13 }}, {{val| 0 1 4 10 }}], the normal val list for septimal meantone. Using this as input, the output is 1199.25¢, 1899.45¢ (tempered 2/1, tempered 3/1).

<pre>