Tuning ranges of regular temperaments: Difference between revisions

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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.

[[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.

[[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.

== 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.

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 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.
-[[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.
+[[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.
 == 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.