Tuning ranges of regular temperaments: Difference between revisions

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also add a note about "diamond pure"
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revise page to cover tuning ranges in general, with the Milne, Sethares, and Plamodon definitions restricted to a specific approach; extract details about these tuning ranges to new dedicated pages
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There are various methods which have been suggested for defining '''tuning ranges appropriate to a given [[regular temperament]]'''.
There are various methods which have been suggested for defining '''tuning ranges appropriate to a given [[regular temperament]]'''.


Given a rank-''r'' ''p''-limit regular temperament,
== Diamond tuning ranges ==
* 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'''.


While nice tunings are always guaranteed to occur, valid tunings are not. For instance, from the tuning map [{{val| 1 0 5 }}, {{val| 0 1 -2 }}] for the temperament tempering out 45/32 we find that all tunings are of the form {{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} = {{val| 1 ''a'' 5-2''a'' }}. For example, if ''a'' was 7/5, then the map would be {{val|1 (7/5) 5-2(7/5)}} = {{val|1 7/5 25/5-14/5}} = {{val|5 7 11}}, and if ''a'' was 4/3 then the map would be {{val|1 (4/3) 5-2(4/3)}} = {{val|1 4/3 15/3-8/3}} = {{val|3 4 7}}. One way to think about preserving the sorting order of the ''p''-odd limit tonality diamond would be to ensure that none of the intervals between its pitches become negative under this temperament. The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3], and the intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5]. We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8]. In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}]. If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3]. Now we need to make sure each of those are not negative, so we get a set of inequalities: ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2. These inequalities have no solution: there's no way ''a'' can be both greater or equal to 1.5 and less than or equal to 1.4. Hence there are no valid tunings of this temperament.
[[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] defined some important tuning ranges. Their "valid" range was defined 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>; according to Milne, this tuning range was Sethares's contribution. Their "nice" range was discussed in ''X_System'' in the Open University’s repository.  


For a more typical example, consider marvel temperament. Using the Hermite normal form 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 valid range. The nice tuning 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 all rational number values for the approximate 3 and 5 are not in the valid range, so that only the [2, 1620/539, (4/3)×sqrt (14), 291600/41503, (44/15)×sqrt (14)] tuning is valid and hence strict. Other examples of 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 nice range.
In their writings these two tuning ranges are referred to simply as "valid" and "nice", and for some time and to some extent on the wiki and in the regular temperament community these names were used as-is. In May 2021 Milne agreed with a community effort to give them the more specific names "diamond valid" and "diamond nice" (or "diamond monotone" and "diamond pure", respectively), due to the fact that they are based on the effect the temperament has on a relevant tonality diamond. "Diamond strict" is the combination of both conditions.  


[[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] define the above valid tuning range 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. According to Milne, the "VTR" was Sethares's contribution. This particular validity range, though it has historical momentum, does not preclude definition of other validity ranges for the tuning of temperaments. The topic of tuning ranges is subjective, so referring to these properties as simply "valid" and "nice" may be misleading, or cause contention; a tuning that has Sethares's validity property may seem invalid to someone else, or vice versa, a tuning that does not have his validity property may seem valid to someone else. In light of this, it has been proposed to provide this validity property with a more specific name, such as "diamond valid" or "diamond monotone", because of its basis on the effects tempering has on the tonality diamond. Similarly, the "nice" property could be referred to as "diamond pure".
* [[diamond monotone]]
* [[diamond pure]]
* [[diamond strict]]


== Example: 5-limit meantone ==
=== Examples ===


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.
==== 5-limit meantone ====


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.)
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.)
* 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.
* 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.
* 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, 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
* [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 505.214 in cents) as the generator.
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.


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:
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:
* tempered 6/5 = 3''g'' - ''p'' > 0
* tempered 6/5 = 3''g'' - ''p'' > 0
* tempered 25/24 = 3''p'' - 7''g'' > 0
* tempered 25/24 = 3''p'' - 7''g'' > 0
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* ''g''/''p'' > 2/5
* ''g''/''p'' > 2/5
* ''g''/''p'' < 1/2
* ''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).
Of these it can be seen that the first and last are redundant, and the overall diamond monotone 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 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.
 
==== 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.
 
== Other tuning ranges ==
 
The diamond tuning ranges, though they have historical momentum, do not preclude definition of other validity ranges for the tuning of temperaments. The topic of tuning ranges is relatively subjective.


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.
[[Paul Erlich]] has proposed other tuning ranges, such as the set of regular tunings in which the temperament has up to 2×, 5×, 10× etc. its optimal damage under some metric (such as [[Kees_height|KE]]), or a set of absolute cutoffs on damage applied across all temperaments, though there could be no objective value for such cutoffs that would be both amenable to the entire community as well as useful for the entire set of regular temperaments (including extreme cases like [[macrotemperaments]] and [[microtemperaments]]).


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.
Others have proposed the [[step ratio spectrum]] as a helpful way of thinking about tuning ranges.


== References ==
== References ==

Revision as of 21:28, 26 May 2021

There are various methods which have been suggested for defining tuning ranges appropriate to a given regular temperament.

Diamond tuning ranges

Andrew Milne, Bill Sethares and James Plamondon defined some important tuning ranges. Their "valid" range was defined in Tuning Continua and Keyboard Layouts in the premiere issue of Journal of Mathematics and Music[1]; according to Milne, this tuning range was Sethares's contribution. Their "nice" range was discussed in X_System in the Open University’s repository.

In their writings these two tuning ranges are referred to simply as "valid" and "nice", and for some time and to some extent on the wiki and in the regular temperament community these names were used as-is. In May 2021 Milne agreed with a community effort to give them the more specific names "diamond valid" and "diamond nice" (or "diamond monotone" and "diamond pure", respectively), due to the fact that they are based on the effect the temperament has on a relevant tonality diamond. "Diamond strict" is the combination of both conditions.

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 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.)

  • 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.
  • 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 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.

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:

  • tempered 6/5 = 3g - p > 0
  • tempered 25/24 = 3p - 7g > 0
  • tempered 16/15 = 5g - 2p > 0
  • tempered 9/8 = p - 2g > 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 diamond monotone 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 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.

11-limit marvel

Consider marvel temperament. Using the Hermite normal form tuning map again, we find that all marvel tunings are of the form ⟨1 a b 2a+ab-5 12-a-3b]. 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 ≤ 4a - 4} with {49/31 ≤ a ≤ 35/22, 2 + a/5 ≤ b ≤ 3 - 3a/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 41p/41, 53p/53, 72p/72 etc.; however 19p/19, 22p/22 and 31p/31 are not in the diamond pure range.

Other tuning ranges

The diamond tuning ranges, though they have historical momentum, do not preclude definition of other validity ranges for the tuning of temperaments. The topic of tuning ranges is relatively subjective.

Paul Erlich has proposed other tuning ranges, such as the set of regular tunings in which the temperament has up to 2×, 5×, 10× etc. its optimal damage under some metric (such as KE), or a set of absolute cutoffs on damage applied across all temperaments, though there could be no objective value for such cutoffs that would be both amenable to the entire community as well as useful for the entire set of regular temperaments (including extreme cases like macrotemperaments and microtemperaments).

Others have proposed the step ratio spectrum as a helpful way of thinking about tuning ranges.

References

  1. ↑ Andrew Milne, William Sethares & James Plamondon (2008) Tuning continua and keyboard layouts, Journal of Mathematics and Music, 2:1, 1-19, DOI: 10.1080/17459730701828677
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