Tuning ranges of regular temperaments: Difference between revisions

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To illustrate the diamond tuning ranges, let's consider 5-limit [[meantone]]. This is a 5-limit temperament so the tonality diamond is {1, 3, 5, 1/3, 5/3, 1/5, 3/5}, or octave reduced, {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. It is also a rank-2 temperament, so any particular tuning can be specified by tunings of the two generators, which we take to be the tempered 2/1 and 4/3.

To find the range of diamond purer tunings, we fix one eigenmonzo as 2/1 and 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.)

To find the range of diamond purer tunings, we fix one [[eigenmonzo]] as 2/1 and 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.

* If 4/3 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean.

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 To illustrate the diamond tuning ranges, let's consider 5-limit [[meantone]]. This is a 5-limit temperament so the tonality diamond is {1, 3, 5, 1/3, 5/3, 1/5, 3/5}, or octave reduced, {1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3}. It is also a rank-2 temperament, so any particular tuning can be specified by tunings of the two generators, which we take to be the tempered 2/1 and 4/3.
-To find the range of diamond purer tunings, we fix one eigenmonzo as 2/1 and 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.)
+To find the range of diamond purer tunings, we fix one [[eigenmonzo]] as 2/1 and 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.
 * If 4/3 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean.