Tuning ranges of regular temperaments: Difference between revisions
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* [[diamond tradeoff]] | * [[diamond tradeoff]] | ||
== Examples == | |||
=== 5-limit meantone === | |||
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 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. | ||
==== Diamond tradeoff ==== | |||
To find the range of diamond tradeoff 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 tradeoff 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.) | ||
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* [2/1, (12/5)^(1/3)], or third-comma meantone - 6/5 and 5/3 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 tradeoff 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 tradeoff 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 tradeoff 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 tradeoff 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. | ||
==== Diamond monotone ==== | |||
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: | 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. | 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. | ||
==== Diamond tradeoff and diamond monotone ==== | |||
In this particular case all of the diamond tradeoff tunings are also diamond monotone, so the diamond tradeoff range is entirely inside the diamond monotone range. | |||
=== 11-limit marvel === | |||
==== Diamond monotone ==== | |||
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. | |||
==== Diamond tradeoff ==== | |||
The diamond tradeoff 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]]. | |||
==== | ==== Diamond tradeoff and diamond monotone ==== | ||
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 both diamond tradeoff and diamond monotone. Other examples of tunings that are both diamond tradeoff and diamond monotone 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 tradeoff range. | |||
== Other tuning ranges == | == Other tuning ranges == | ||