Douglas Blumeyer's RTT How-To: Difference between revisions

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The latter is sometimes called the “musician’s form” of the temperament, because it’s easy to reason about from a musical perspective. But it turns out there’s not a particularly clean function for consistently getting to it, or even defining it.

Another form you might want the mapping in is the type Graham Breed's temperament finder puts them in, where all values in a mapping row may be negative, but this is in the service of the generator being less than half the size of the period. For example, for meantone, we'd want the fourth instead of the fifth, and we can see that

<math>

\left[ \begin{array} {rrr}

1 & 2 & 4 \\

0 & -1 & -4

\end{array} \right]

</math>

maps the fourth (4/3, {{monzo|2 -1 0 }}) to {{monzo|0 1}}.

It’s often the case that a temperament’s nullity is greater than 1 or its rank is greater than 1, and therefore we have an infinitude of equivalent ways of expressing the comma basis or the mapping. This can be problematic, if we want to efficiently communicate about and catalog temperaments. It’s good to have a standardized form in these cases. The approach RTT takes here is to get these matrices into '''“normal” form'''. In plain words, this just means: we have a function which takes in a matrix and spits out a matrix of the same shape, and no matter which matrix we input from a set of matrices which we consider all to be equivalent to each other, it will spit out the same result. This output is what we call the “normalized” matrix, and it can therefore uniquely identify a temperament.

@@ Line 1,028: / Line 1,028: @@
 The latter is sometimes called the “musician’s form” of the temperament, because it’s easy to reason about from a musical perspective. But it turns out there’s not a particularly clean function for consistently getting to it, or even defining it.
+Another form you might want the mapping in is the type Graham Breed's temperament finder puts them in, where all values in a mapping row may be negative, but this is in the service of the generator being less than half the size of the period. For example, for meantone, we'd want the fourth instead of the fifth, and we can see that
+<math>
+\left[ \begin{array} {rrr}
+& 2 & 4 \\
+& -1 & -4
+    \end{array} \right]
+</math>
+maps the fourth (4/3, {{monzo|2 -1 0 }}) to {{monzo|0 1}}.
 It’s often the case that a temperament’s nullity is greater than 1 or its rank is greater than 1, and therefore we have an infinitude of equivalent ways of expressing the comma basis or the mapping. This can be problematic, if we want to efficiently communicate about and catalog temperaments. It’s good to have a standardized form in these cases. The approach RTT takes here is to get these matrices into '''“normal” form'''. In plain words, this just means: we have a function which takes in a matrix and spits out a matrix of the same shape, and no matter which matrix we input from a set of matrices which we consider all to be equivalent to each other, it will spit out the same result. This output is what we call the “normalized” matrix, and it can therefore uniquely identify a temperament.