Tuning ranges of regular temperaments: Difference between revisions
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===== Diamond tradeoff ===== | ===== Diamond tradeoff ===== | ||
To find the range of diamond tradeoff tunings, we fix the tuning of the octave to pure, or in other words, we choose one [[eigenmonzo]] (unchanged | To find the range of diamond tradeoff tunings, we fix the tuning of the octave to pure, or in other words, we choose one [[eigenmonzo]] (unchanged interval) to be 2/1. Then we iterate through the 5-limit tonality diamond for what to use for the other eigenmonzo. (If the rank were more than 2, we would be iterating over subsets of the tonality diamond of size ''r'' - 1, but since ''r'' = 2 we are iterating over single ratios.) | ||
* 1/1 is the 0-vector monzo and so it is always a (trivial) eigenmonzo of any tuning. We have to use a non-1/1 interval as the other eigenmonzo besides 2/1 to define a tuning. | * 1/1 is the 0-vector monzo and so it is always a (trivial) eigenmonzo of any tuning. We have to use a non-1/1 interval as the other eigenmonzo besides 2/1 to define a tuning. | ||
* If 4/3 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean. | * If 4/3 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean. | ||