Tuning ranges of regular temperaments: Difference between revisions

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==== Diamond monotone ====

~~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 [[mapping]] provided for [[Marvel_family#Undecimal_marvel_.28unimarv.29|undecimal marvel]] is {{ket|{{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: {{ket{{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.

==== Diamond tradeoff ====

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 ==== Diamond monotone ====
-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 [[mapping]] provided for [[Marvel_family#Undecimal_marvel_.28unimarv.29|undecimal marvel]] is {{ket|{{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: {{ket{{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.
 ==== Diamond tradeoff ====