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===The unchanged-interval basis=== | ===The unchanged-interval basis=== | ||
The '''unchanged-interval basis''' of a tuning is the [[basis]] for all of its [[unchanged-interval]]s. | |||
Obtaining the unchanged-interval basis <math>\textrm{U}</math> means to obtain the unchanged-intervals <math>\textbf{u}_1, \textbf{u}_2, …</math> of <math>P</math> or in other words any <math>\textbf{u}_i</math> for which <math>P\textbf{u}_i = \textbf{u}_i</math>. There are many ways to find these, but one way stands out for its clarity. We can rewrite this equation by subtracting <math>\textbf{u}_i</math> from both sides to get: | Obtaining the unchanged-interval basis <math>\textrm{U}</math> means to obtain the unchanged-intervals <math>\textbf{u}_1, \textbf{u}_2, …</math> of <math>P</math> or in other words any <math>\textbf{u}_i</math> for which <math>P\textbf{u}_i = \textbf{u}_i</math>. There are many ways to find these, but one way stands out for its clarity. We can rewrite this equation by subtracting <math>\textbf{u}_i</math> from both sides to get: | ||