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 ~~pure~~ 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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* [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 diamond ~~pure~~ 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 diamond ~~pure~~ 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.

These three are the possible extreme points of the diamond purer 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 diamond purer 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 diamond monotone 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 diamond monotone. If we denote the octave period by ''p'' and the perfect fourth generator by ''g'', this yields the equations:

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Note that, since the definition of diamond monotone only depends on the ordering of intervals and not on their absolute size, in theory any amount of stretching or compression is allowed. For example, ''p'' = 12 cents and ''g'' = 5 cents is technically a diamond monotone meantone tuning, as is ''p'' = 12000 cents and ''g'' = 5000 cents.

The diamond ~~strict~~ tuning range includes those that are both diamond ~~pure~~ and diamond monotone, but in this particular case all of the diamond ~~pure~~ tunings are also diamond monotone, so the diamond ~~strict~~ range is identical to the diamond ~~pure~~ range.

The diamond nice tuning range includes those that are both diamond purer and diamond monotone, but in this particular case all of the diamond purer tunings are also diamond monotone, so the diamond nice range is identical to the diamond purer range.

==== 11-limit marvel ====

Consider [[marvel temperament]]. Using the Hermite normal form [[Temperament_Mapping_Matrices|tuning map]] again, we find that all marvel tunings are of the form {{val| 1 ''a'' ''b'' 2''a''+''ab''-5 12-''a''-3''b'' }}. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ ''a'' ≤ 49/31, 2 + ''a''/5 ≤ ''b'' ≤ 4''a'' - 4} with {49/31 ≤ ''a'' ≤ 35/22, 2 + ''a''/5 ≤ ''b'' ≤ 3 - 3''a''/7}, which is the triangular region bounded by the tunings for 19, 22, and 31. This is the diamond monotone range. The diamond ~~pure~~ range is a quadrilateral with vertices (given in terms of frequency ratios rather than log base 2 or cents) [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)], [2, 3, 5, 225/32, 4096/375]]. The three vertices with entirely rational number values for the approximations of 3 and 5 are not in the diamond monotone range, so only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is diamond monotone and hence diamond ~~strict~~. Other examples of diamond ~~strict~~ tunings are 41''p''/41, 53''p''/53, 72''p''/72 etc.; however 19''p''/19, 22''p''/22 and 31''p''/31 are not in the diamond ~~pure~~ range.

Consider [[marvel temperament]]. Using the Hermite normal form [[Temperament_Mapping_Matrices|tuning map]] again, we find that all marvel tunings are of the form {{val| 1 ''a'' ''b'' 2''a''+''ab''-5 12-''a''-3''b'' }}. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ ''a'' ≤ 49/31, 2 + ''a''/5 ≤ ''b'' ≤ 4''a'' - 4} with {49/31 ≤ ''a'' ≤ 35/22, 2 + ''a''/5 ≤ ''b'' ≤ 3 - 3''a''/7}, which is the triangular region bounded by the tunings for 19, 22, and 31. This is the diamond monotone range. The diamond purer range is a quadrilateral with vertices (given in terms of frequency ratios rather than log base 2 or cents) [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)], [2, 3, 5, 225/32, 4096/375]]. The three vertices with entirely rational number values for the approximations of 3 and 5 are not in the diamond monotone range, so only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is diamond monotone and hence diamond nice. Other examples of diamond nice tunings are 41''p''/41, 53''p''/53, 72''p''/72 etc.; however 19''p''/19, 22''p''/22 and 31''p''/31 are not in the diamond purer range.

== Other tuning ranges ==

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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 pure 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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 * [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 diamond pure 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 diamond pure 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.
+These three are the possible extreme points of the diamond purer 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 diamond purer 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 diamond monotone 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 diamond monotone. If we denote the octave period by ''p'' and the perfect fourth generator by ''g'', this yields the equations:
@@ Line 47: / Line 47: @@
 Note that, since the definition of diamond monotone only depends on the ordering of intervals and not on their absolute size, in theory any amount of stretching or compression is allowed. For example, ''p'' = 12 cents and ''g'' = 5 cents is technically a diamond monotone meantone tuning, as is ''p'' = 12000 cents and ''g'' = 5000 cents.
-The diamond strict tuning range includes those that are both diamond pure and diamond monotone, but in this particular case all of the diamond pure tunings are also diamond monotone, so the diamond strict range is identical to the diamond pure range.
+The diamond nice tuning range includes those that are both diamond purer and diamond monotone, but in this particular case all of the diamond purer tunings are also diamond monotone, so the diamond nice range is identical to the diamond purer range.
 ==== 11-limit marvel ====
-Consider [[marvel temperament]]. Using the Hermite normal form [[Temperament_Mapping_Matrices|tuning map]] again, we find that all marvel tunings are of the form {{val| 1 ''a'' ''b'' 2''a''+''ab''-5 12-''a''-3''b'' }}. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ ''a'' ≤ 49/31, 2 + ''a''/5 ≤ ''b'' ≤ 4''a'' - 4} with {49/31 ≤ ''a'' ≤ 35/22, 2 + ''a''/5 ≤ ''b'' ≤ 3 - 3''a''/7}, which is the triangular region bounded by the tunings for 19, 22, and 31. This is the diamond monotone range. The diamond pure range is a quadrilateral with vertices (given in terms of frequency ratios rather than log base 2 or cents) [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)], [2, 3, 5, 225/32, 4096/375]]. The three vertices with entirely rational number values for the approximations of 3 and 5 are not in the diamond monotone range, so only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is diamond monotone and hence diamond strict. Other examples of diamond strict tunings are 41''p''/41, 53''p''/53, 72''p''/72 etc.; however 19''p''/19, 22''p''/22 and 31''p''/31 are not in the diamond pure range.
+Consider [[marvel temperament]]. Using the Hermite normal form [[Temperament_Mapping_Matrices|tuning map]] again, we find that all marvel tunings are of the form {{val| 1 ''a'' ''b'' 2''a''+''ab''-5 12-''a''-3''b'' }}. Applying this to the steps of the 11-limit tonality diamond, we obtain eight inequalities, the solution set of which is the union of {30/19 ≤ ''a'' ≤ 49/31, 2 + ''a''/5 ≤ ''b'' ≤ 4''a'' - 4} with {49/31 ≤ ''a'' ≤ 35/22, 2 + ''a''/5 ≤ ''b'' ≤ 3 - 3''a''/7}, which is the triangular region bounded by the tunings for 19, 22, and 31. This is the diamond monotone range. The diamond purer range is a quadrilateral with vertices (given in terms of frequency ratios rather than log base 2 or cents) [[2, 4096/1375, 5, 524288/75625, 11], [2, 3, 224/45, 1568/225, 30375/2744], [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)], [2, 3, 5, 225/32, 4096/375]]. The three vertices with entirely rational number values for the approximations of 3 and 5 are not in the diamond monotone range, so only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is diamond monotone and hence diamond nice. Other examples of diamond nice tunings are 41''p''/41, 53''p''/53, 72''p''/72 etc.; however 19''p''/19, 22''p''/22 and 31''p''/31 are not in the diamond purer range.
 == Other tuning ranges ==