A brief introduction to Regular Temperament Theory: Difference between revisions
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When the generation of tunings is formulated in this way, the tools of linear algebra can be applied (or the slightly more powerful but less well-known and less available tools of a related algebra called exterior algebra, Grassman algebra or geometric algebra). | When the generation of tunings is formulated in this way, the tools of linear algebra can be applied (or the slightly more powerful but less well-known and less available tools of a related algebra called exterior algebra, Grassman algebra or geometric algebra). | ||
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.</ref> This is called the temperament's [[Temperament_Mapping_Matrices_(M-maps)|mapping]], and can be represented as a matrix. | 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. | ||
Alternatively, you can avoid wedge products, by agreeing on rules that determine a canonical set of generators. For example, the first generator alone should generate the lowest prime using a positive count. Then the first and second generators alone should generate the next prime using a positive count of the second generator, and so on. This is equivalent to requiring the mapping matrix to be in a certain form. I don't know if this form has a name. I note that an RT mapping matrix is never a square matrix (otherwise it would be JI).</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]]. | 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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