Tuning ranges of regular temperaments: Difference between revisions

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== 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 ''Journal of Mathematics and Music''<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (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 "purer" range was discussed in the technical report ''X_System'' in the Open University’s repository; this was Milne's contribution. Today these are known as [[diamond monotone]] and [[diamond tradeoff]], respectively.

[[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 ''Journal of Mathematics and Music''<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2008). [https://www.researchgate.net/publication/228091827_Tuning_continua_and_keyboard_layouts Tuning continua and keyboard layouts]. Journal of Mathematics and Music, 2(1):1–19.</ref>; according to Milne, this tuning range was Sethares's contribution. Their "purer" range was discussed in the technical report ''X_System'' in the Open University’s repository; this was Milne's contribution. Today these are known as [[diamond monotone]] and [[diamond tradeoff]], respectively.

=== History ===

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===== Diamond tradeoff =====

To find the range of diamond tradeoff tunings, we fix the tuning of the octave to pure, or in other words, we choose one [[eigenmonzo]] (unchanged-interval) to be 2/1. Then we 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 the tuning of the octave to pure, or in other words, we choose one [[eigenmonzo]] (unchanged interval) to be 2/1. Then we 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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The [[mapping]] provided for [[Marvel_family#Undecimal_marvel_.28unimarv.29|undecimal marvel]] is {{rket|{{map|1 0 0 -5 12}} {{map|0 1 0 2 -1}} {{map|0 0 1 2 -3}}}}. We don't know the tuning of our generators yet, so our [[tuning map]] has variables in it: {{bra|1 ''a'' ''b''}}. This means that our first generator (the period) is 1 octave, the second generator is ''a'' octaves, and the third generator is ''b'' octaves. If we left-multiply the mapping by this tuning map, we get a parameterized tuning of {{val| 1 ''a'' ''b'' 2''a''+2''b''-5 12-''a''-3''b'' }} undecimal marvel. Or in other words, all tunings of undecimal marvel are of this form.

Applying this to the steps of the 11-limit tonality diamond, then, 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.

Applying this to the steps of the 11-limit tonality diamond, then, 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 edos 19, 22, and 31. This is the diamond monotone range.

===== Diamond tradeoff =====

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 == 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 ''Journal of Mathematics and Music''<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (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 "purer" range was discussed in the technical report ''X_System'' in the Open University’s repository; this was Milne's contribution. Today these are known as [[diamond monotone]] and [[diamond tradeoff]], respectively.
+[[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 ''Journal of Mathematics and Music''<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2008). [https://www.researchgate.net/publication/228091827_Tuning_continua_and_keyboard_layouts Tuning continua and keyboard layouts]. Journal of Mathematics and Music, 2(1):1–19.</ref>; according to Milne, this tuning range was Sethares's contribution. Their "purer" range was discussed in the technical report ''X_System'' in the Open University’s repository; this was Milne's contribution. Today these are known as [[diamond monotone]] and [[diamond tradeoff]], respectively.
 === History ===
@@ Line 17: / Line 17: @@
 ===== Diamond tradeoff =====
-To find the range of diamond tradeoff tunings, we fix the tuning of the octave to pure, or in other words, we choose one [[eigenmonzo]] (unchanged-interval) to be 2/1. Then we 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 the tuning of the octave to pure, or in other words, we choose one [[eigenmonzo]] (unchanged interval) to be 2/1. Then we 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.
@@ Line 57: / Line 57: @@
 The [[mapping]] provided for [[Marvel_family#Undecimal_marvel_.28unimarv.29|undecimal marvel]] is {{rket|{{map|1 0 0 -5 12}} {{map|0 1 0 2 -1}} {{map|0 0 1 2 -3}}}}. We don't know the tuning of our generators yet, so our [[tuning map]] has variables in it: {{bra|1 ''a'' ''b''}}. This means that our first generator (the period) is 1 octave, the second generator is ''a'' octaves, and the third generator is ''b'' octaves. If we left-multiply the mapping by this tuning map, we get a parameterized tuning of {{val| 1 ''a'' ''b'' 2''a''+2''b''-5 12-''a''-3''b'' }} undecimal marvel. Or in other words, all tunings of undecimal marvel are of this form.
-Applying this to the steps of the 11-limit tonality diamond, then, 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.
+Applying this to the steps of the 11-limit tonality diamond, then, 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 edos 19, 22, and 31. This is the diamond monotone range.
 ===== Diamond tradeoff =====