Tuning ranges of regular temperaments: Difference between revisions

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To find the range of "nice" 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.)

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

* 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 5/4 ~~or 8~~/5 is the eigenmonzo, the tuning is [2/1, 2/5^(1/4)], or quarter-comma meantone.

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

* If 6/5 or 5/3 is the eigenmonzo, the ~~tuning is~~ [2/1, (12/5)^(1/3)], or third-comma meantone.

* If 3/2 is the eigenmonzo, that is equivalent to 4/3 being an eigenmonzo because if 2/1 and 4/3 are tuned pure, then 3/2 is automatically tuned pure also.

* If 5/4 is the eigenmonzo, then 5/1 is also tuned pure so the fifth is 5^(1/4) and the generator is 2/5^(1/4). Therefore the tuning is [2/1, 2/5^(1/4)], or quarter-comma meantone.

* If 8/5 is the eigenmonzo, that's equivalent to 5/4 being the eigenmonzo and leads to the same tuning.

* If 6/5 is the eigenmonzo, then 12/5 is also tuned pure so the fourth (the generator) is (12/5)^(1/3). Therefore the tuning is [2/1, (12/5)^(1/3)], or third-comma meantone.

* If 5/3 is the eigenmonzo, that's equivalent to 6/3 being the eigenmonzo.

These lead to three distinct tunings:

* [2/1, 4/3] or Pythagorean - 4/3 and 3/2 are pure

* [2/1, 2/5^(1/4)], or quarter-comma meantone - 5/4 and 8/5 are pure

* [2/1, (12/5)^(1/3)], or third-comma meantone - 6/5 and 5/3 are pure

These three are the possible extreme points of the "nice" tuning range, so to describe the whole range, we must take their convex hull. In this case it is easy because they are all collinear, and Pythagorean and third-comma are the two endpoints of the line segment. So the "nice" tuning range of 5-limit meantone consists of exactly those tunings with a pure 2/1 as the period and anywhere from 4/3 to (12/5)^(1/3) (498.045 to 515.214 in cents) as the generator.

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 To find the range of "nice" 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.)
-* If 4/3 or 3/2 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean.
+* 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 5/4 or 8/5 is the eigenmonzo, the tuning is [2/1, 2/5^(1/4)], or quarter-comma meantone.
+* If 4/3 is the eigenmonzo, the tuning is [2/1, 4/3] or Pythagorean.
-* If 6/5 or 5/3 is the eigenmonzo, the tuning is [2/1, (12/5)^(1/3)], or third-comma meantone.
+* If 3/2 is the eigenmonzo, that is equivalent to 4/3 being an eigenmonzo because if 2/1 and 4/3 are tuned pure, then 3/2 is automatically tuned pure also.
+* If 5/4 is the eigenmonzo, then 5/1 is also tuned pure so the fifth is 5^(1/4) and the generator is 2/5^(1/4). Therefore the tuning is [2/1, 2/5^(1/4)], or quarter-comma meantone.
+* If 8/5 is the eigenmonzo, that's equivalent to 5/4 being the eigenmonzo and leads to the same tuning.
+* If 6/5 is the eigenmonzo, then 12/5 is also tuned pure so the fourth (the generator) is (12/5)^(1/3). Therefore the tuning is [2/1, (12/5)^(1/3)], or third-comma meantone.
+* If 5/3 is the eigenmonzo, that's equivalent to 6/3 being the eigenmonzo.
+These lead to three distinct tunings:
+* [2/1, 4/3] or Pythagorean - 4/3 and 3/2 are pure
+* [2/1, 2/5^(1/4)], or quarter-comma meantone - 5/4 and 8/5 are pure
+* [2/1, (12/5)^(1/3)], or third-comma meantone - 6/5 and 5/3 are pure
 These three are the possible extreme points of the "nice" tuning range, so to describe the whole range, we must take their convex hull. In this case it is easy because they are all collinear, and Pythagorean and third-comma are the two endpoints of the line segment. So the "nice" tuning range of 5-limit meantone consists of exactly those tunings with a pure 2/1 as the period and anywhere from 4/3 to (12/5)^(1/3) (498.045 to 515.214 in cents) as the generator.