A brief introduction to Regular Temperament Theory: Difference between revisions

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 These two extremes were well explored prior to RTT. What RTT did was open up a vast middle ground between JI and ET, where the number of generators is greater than one but less than the number of primes being approximated. These are called regular temperaments (RT). Only a very small region of that middle ground had been explored prior to RTT, namely the "[[meantone]]" region that approximates primes 2, 3 and 5, using two generators which are an octave (prime 2) and a slightly narrow fifth (approximate 2:3).
-When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.<ref>or our homegrown variety of multilinear algebra, which includes two copies of exterior algebra and uses an extended bra-ket notation, for which, unfortunately, tools are not readily available.</ref>
+When the generation of tunings is formulated in this way, the tools of linear algebra can be applied.
-The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Wedgies_and_Multivals|wedge product]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.
+The defining thing about a regular temperament is the the count of each generator required to approximate each prime number.<ref>Strictly speaking, it is the [[Canonical_form|canonical form]] of the rows of the mapping matrix that defines the temperament, because you can replace any set of generators with linear combinations of those generators, and change the mapping accordingly, to obtain the same temperament.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)|mapping]], and can be represented as a matrix.
-Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators, and therefore a canonical form for the mapping matrix. For example, the first generator, called the period, can be given as a unit fraction of the lowest prime, usually the octave (2). The second generator can be given as a unit fraction of the simplest ratio that allows it to be smaller than the period, and so on.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)|mapping]], and can be represented as a matrix.
 We can then institute computer searches to find optimum mappings, with our desired balance of error versus complexity. Many such searches have been done and [[Tour_of_Regular_Temperaments|many resulting temperaments named and catalogued]].
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