Diamond monotone: Difference between revisions

Line 10:

While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not.

Let's look at an example: the temperament with mapping [{{val| 1 0 5 }}, {{val| 0 1 -2 }}].

Let's look at an example: the rank-2 temperament with mapping [{{val| 1 0 5 }}, {{val| 0 1 -2 }}], tempering out [[45/32]].

All of this temperaments tunings are some linear combination of these two mapping rows. We could express that idea in the form {{nowrap|{{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} {{=}} {{val| 1 ''a'' 5-2''a'' }}}}. So one example tuning would be if this ''a'' variable was 7/5, which would give us the map {{nowrap|{{val|1 (7/5) 5-2(7/5)}} {{=}} {{val|1 7/5 25/5-14/5}} {{=}} {{val|5 7 11}}}}. Another example tuning would be if ''a'' was 4/3; then the map would be {{nowrap|{{val|1 (4/3) 5-2(4/3)}} {{=}} {{val|1 4/3 15/3-8/3}} {{=}} {{val|3 4 7}}}}.

Line 23:

#Now we need to make sure each of those are not negative, so we get a set of inequalities: {{nowrap|''a'' ≥ 4/3|''a'' ≤ 7/5|''a'' ≥ 1|''a'' ≥ 3/2}}.

We can see that 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 diamond monotone tunings of this temperament.

We can see that 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 diamond monotone tunings of this temperament; any tuning must necessarily either map 6/5 wider than 5/4 or 4/3 wider than 3/2.

== See also ==

@@ Line 10: / Line 10: @@
 While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not.
-Let's look at an example: the temperament with mapping [{{val| 1 0 5 }}, {{val| 0 1 -2 }}].
+Let's look at an example: the rank-2 temperament with mapping [{{val| 1 0 5 }}, {{val| 0 1 -2 }}], tempering out [[45/32]].
 All of this temperaments tunings are some linear combination of these two mapping rows. We could express that idea in the form {{nowrap|{{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} {{=}} {{val| 1 ''a'' 5-2''a'' }}}}. So one example tuning would be if this ''a'' variable was 7/5, which would give us the map {{nowrap|{{val|1 (7/5) 5-2(7/5)}} {{=}} {{val|1 7/5 25/5-14/5}} {{=}} {{val|5 7 11}}}}. Another example tuning would be if ''a'' was 4/3; then the map would be {{nowrap|{{val|1 (4/3) 5-2(4/3)}} {{=}} {{val|1 4/3 15/3-8/3}} {{=}} {{val|3 4 7}}}}.
@@ Line 23: / Line 23: @@
 #Now we need to make sure each of those are not negative, so we get a set of inequalities: {{nowrap|''a'' &ge; 4/3|''a'' &le; 7/5|''a'' &ge; 1|''a'' &ge; 3/2}}.
-We can see that 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 diamond monotone tunings of this temperament.
+We can see that 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 diamond monotone tunings of this temperament; any tuning must necessarily either map 6/5 wider than 5/4 or 4/3 wider than 3/2.
 == See also ==