Diamond monotone: Difference between revisions

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== Temperaments without diamond monotone tunings ==

While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not. ~~For instance, from~~ the ~~tuning map~~ [{{val| 1 0 5 }}, {{val| 0 1 -2 }}] ~~for the temperament tempering out 45/32 we find that all~~ tunings are of the form {{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} = {{val| 1 ''a'' 5-2''a'' }}. ~~For~~ example, if ''a'' was 7/5, ~~then~~ the map ~~would be~~ {{val|1 (7/5) 5-2(7/5)}} = {{val|1 7/5 25/5-14/5}} = {{val|5 7 11}}~~, and~~ if ''a'' was 4/3 then the map would be {{val|1 (4/3) 5-2(4/3)}} = {{val|1 4/3 15/3-8/3}} = {{val|3 4 7}}. One way to think about preserving the sorting order of the ''p''-odd limit tonality diamond would be to ensure that none of the intervals between its pitches become negative under this temperament. The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3]~~, and the~~ intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5]. We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8]. In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}]. If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3]. Now we need to make sure each of those are not negative, so we get a set of inequalities: ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2. ~~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.

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 }}].

All of this temperaments tunings are some linear combination of these two mapping rows. We could express that idea in the form {{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 {{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 {{val|1 (4/3) 5-2(4/3)}} = {{val|1 4/3 15/3-8/3}} = {{val|3 4 7}}.

One way to think about preserving the sorting order of the ''p''-odd limit tonality diamond would be to ensure that none of the intervals between its pitches become negative under this temperament. Let's work through how to establish that:

# The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3].

# The intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5].

# We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8].

# In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}], respectively.

# If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3].

# Now we need to make sure each of those are not negative, so we get a set of inequalities: ''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.

[[Category:Regular temperament theory]]

[[Category:Diamond]]

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 == Temperaments without diamond monotone tunings ==
-While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not. For instance, from the tuning map [{{val| 1 0 5 }}, {{val| 0 1 -2 }}] for the temperament tempering out 45/32 we find that all tunings are of the form {{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} = {{val| 1 ''a'' 5-2''a'' }}. For example, if ''a'' was 7/5, then the map would be {{val|1 (7/5) 5-2(7/5)}} = {{val|1 7/5 25/5-14/5}} = {{val|5 7 11}}, and if ''a'' was 4/3 then the map would be {{val|1 (4/3) 5-2(4/3)}} = {{val|1 4/3 15/3-8/3}} = {{val|3 4 7}}. One way to think about preserving the sorting order of the ''p''-odd limit tonality diamond would be to ensure that none of the intervals between its pitches become negative under this temperament. The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3], and the intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5]. We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8]. In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}]. If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3]. Now we need to make sure each of those are not negative, so we get a set of inequalities: ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2. 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.
+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 }}].
+All of this temperaments tunings are some linear combination of these two mapping rows. We could express that idea in the form {{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 {{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 {{val|1 (4/3) 5-2(4/3)}} = {{val|1 4/3 15/3-8/3}} = {{val|3 4 7}}.
+One way to think about preserving the sorting order of the ''p''-odd limit tonality diamond would be to ensure that none of the intervals between its pitches become negative under this temperament. Let's work through how to establish that:
+# The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3].
+# The intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5].
+# We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8].
+# In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}], respectively.
+# If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3].
+# Now we need to make sure each of those are not negative, so we get a set of inequalities: ''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.
 [[Category:Regular temperament theory]]
 [[Category:Diamond]]