Tuning ranges of regular temperaments: Difference between revisions

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Given a rank-''r'' ''p''-limit regular temperament,

* we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], ~~when this~~ tuning is defined.~~{{Clarify}}~~ This is the '''nice tuning range'''.

* we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the set of all tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined. This is the '''nice tuning range'''.

* We may define another tuning range by requiring that the ''p''-odd limit tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing). This is the '''valid tuning range'''.

* A tuning which is both nice and valid is a ''strict'' tuning and this defines the '''strict tuning range'''.

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[[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] define valid tunings in ''Tuning Continua and Keyboard Layouts'' in the premiere issue of ''Journal of Mathematics and Music''<ref>Andrew Milne, William Sethares & James Plamondon (2008) Tuning continua and keyboard layouts, Journal of Mathematics and Music, 2:1, 1-19, DOI: [https://doi.org/10.1080/17459730701828677 10.1080/17459730701828677]</ref>; they discuss nice tunings in ''X_System'' in the Open University’s repository.

== Example: 5-limit meantone ==

To illustrate the above definitions, 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 "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.

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

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

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.

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:

* tempered 6/5 = 3''g'' - ''p'' > 0

* tempered 25/24 = 3''p'' - 7''g'' > 0

* tempered 16/15 = 5''g'' - 2''p'' > 0

* tempered 9/8 = ''p'' - 2''g'' > 0

These are all homogenous equations, so we can divide through by ''p'' and rearrange to get restrictions on the ratio ''g''/''p'':

* ''g''/''p'' > 1/3

* ''g''/''p'' < 3/7

* ''g''/''p'' > 2/5

* ''g''/''p'' < 1/2

Of these it can be seen that the first and last are redundant, and the overall "valid" tuning range can be summarized as 2/5 < ''g''/''p'' < 3/7, in other words all meantone tunings between [[5edo]] and [[7edo]] (480 to 514.286 cents).

Note that, since the definition of "valid" 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 "valid" meantone tuning, as is ''p'' = 12000 cents and ''g'' = 5000 cents.

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

== References ==

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 Given a rank-''r'' ''p''-limit regular temperament,
-* we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest a set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], when this tuning is defined.{{Clarify}} This is the '''nice tuning range'''.
+* we may define a tuning range by finding the [[Wikipedia: Convex hull|convex hull]] in [[Vals and Tuning Space|tuning space]] of the set of all tunings with one [[Eigenmonzo subgroup|eigenmonzo]] 2 (pure octaves tunings) and the rest of the eigenmonzos any set of ''r'' - 1 members of the ''p''-odd limit [[tonality diamond]], whenever such a tuning is defined. This is the '''nice tuning range'''.
 * We may define another tuning range by requiring that the ''p''-odd limit tonality diamond, when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing). This is the '''valid tuning range'''.
 * A tuning which is both nice and valid is a ''strict'' tuning and this defines the '''strict tuning range'''.
@@ Line 11: / Line 11: @@
 [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] define valid tunings in ''Tuning Continua and Keyboard Layouts'' in the premiere issue of ''Journal of Mathematics and Music''<ref>Andrew Milne, William Sethares & James Plamondon (2008) Tuning continua and keyboard layouts, Journal of Mathematics and Music, 2:1, 1-19, DOI: [https://doi.org/10.1080/17459730701828677 10.1080/17459730701828677]</ref>; they discuss nice tunings in ''X_System'' in the Open University’s repository.
+== Example: 5-limit meantone ==
+To illustrate the above definitions, 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 "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.
+* If 5/4 or 8/5 is the eigenmonzo, the tuning is [2/1, 2/5^(1/4)], or quarter-comma meantone.
+* If 6/5 or 5/3 is the eigenmonzo, the tuning is [2/1, (12/5)^(1/3)], or third-comma meantone.
+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.
+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:
+* tempered 6/5 = 3''g'' - ''p'' > 0
+* tempered 25/24 = 3''p'' - 7''g'' > 0
+* tempered 16/15 = 5''g'' - 2''p'' > 0
+* tempered 9/8 = ''p'' - 2''g'' > 0
+These are all homogenous equations, so we can divide through by ''p'' and rearrange to get restrictions on the ratio ''g''/''p'':
+* ''g''/''p'' > 1/3
+* ''g''/''p'' < 3/7
+* ''g''/''p'' > 2/5
+* ''g''/''p'' < 1/2
+Of these it can be seen that the first and last are redundant, and the overall "valid" tuning range can be summarized as 2/5 < ''g''/''p'' < 3/7, in other words all meantone tunings between [[5edo]] and [[7edo]] (480 to 514.286 cents).
+Note that, since the definition of "valid" 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 "valid" meantone tuning, as is ''p'' = 12000 cents and ''g'' = 5000 cents.
+The "strict" tuning range includes those that are both "nice" and "valid", but in this particular case all of the "nice" tunings are also "valid", so the "strict" range is identical to the "nice" range.
 == References ==