Tuning ranges of regular temperaments: Difference between revisions

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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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 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 =====