Tuning ranges of regular temperaments: Difference between revisions

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* [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.

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 505.214 in cents) as the generator.

To find the range of "valid" tunings, we need all the steps in between consecutive members of the tonality diamond to be positive. So 6/5, 25/24, 16/15, and 9/8 must all be positive for the tuning to be "valid". If we denote the octave period by ''p'' and the perfect fourth generator by ''g'', this yields the equations:

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 * [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.
+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 505.214 in cents) as the generator.
 To find the range of "valid" tunings, we need all the steps in between consecutive members of the tonality diamond to be positive. So 6/5, 25/24, 16/15, and 9/8 must all be positive for the tuning to be "valid". If we denote the octave period by ''p'' and the perfect fourth generator by ''g'', this yields the equations: