Diamond monotone: Difference between revisions

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A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond monotone''', or '''diamond valid''', if it satisfies the following condition: the [[Odd-limit|''p''-odd limit]] [[tonality diamond]], when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing).

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] – and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "valid" tuning range<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2007). [https://www.researchgate.net/publication/228091824_Isomorphic_Controllers_and_Dynamic_Tuning_Invariant_Fingering_over_a_Tuning_Continuum Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum]. Computer Music Journal, 31(4):15–32.</ref><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>.

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]]—and to some extent on the wiki and in the regular temperament community—this tuning range was referred to simply as the "valid" tuning range<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2007). [https://www.researchgate.net/publication/228091824_Isomorphic_Controllers_and_Dynamic_Tuning_Invariant_Fingering_over_a_Tuning_Continuum Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum]. Computer Music Journal, 31(4):15–32.</ref><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>.

The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege – privileging those in p-limit tonality diamond is an arguably reasonable choice).

The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege—privileging those in p-limit tonality diamond is an arguably reasonable choice).

The "empirical" range is likely to fall somewhere between diamond monotone and diamond tradeoff. Though, when one is using tempered spectra to match the tuning, it is possible the empirical range can be made wider.

==Temperaments without diamond monotone tunings==

== Temperaments without diamond monotone tunings ==

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}}.

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}}}}.

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:

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#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].

#If we map those using {{val| 1 ''a'' 5−2''a'' }} we obtain the tempered sizes [{{nowrap|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.

#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.

==See also==

== See also ==

For examples and other information, see the topic page [[Tuning ranges of regular temperaments]].

==References==

== References ==

[[Category:Tuning]]

[[Category:Regular temperament theory]]

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[[Category:Math]]

[[Category:Diamond]]

~~<references />~~

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 A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond monotone''', or '''diamond valid''', if it satisfies the following condition: the [[Odd-limit|''p''-odd limit]] [[tonality diamond]], when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing).
-In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] – and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "valid" tuning range<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2007). [https://www.researchgate.net/publication/228091824_Isomorphic_Controllers_and_Dynamic_Tuning_Invariant_Fingering_over_a_Tuning_Continuum Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum]. Computer Music Journal, 31(4):15–32.</ref><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>.
+In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]]&mdash;and to some extent on the wiki and in the regular temperament community&mdash;this tuning range was referred to simply as the "valid" tuning range<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2007). [https://www.researchgate.net/publication/228091824_Isomorphic_Controllers_and_Dynamic_Tuning_Invariant_Fingering_over_a_Tuning_Continuum Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum]. Computer Music Journal, 31(4):15–32.</ref><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>.
-The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege – privileging those in p-limit tonality diamond is an arguably reasonable choice).
+The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege&mdash;privileging those in p-limit tonality diamond is an arguably reasonable choice).
 The "empirical" range is likely to fall somewhere between diamond monotone and diamond tradeoff. Though, when one is using tempered spectra to match the tuning, it is possible the empirical range can be made wider.
-==Temperaments without diamond monotone tunings==
+== Temperaments without diamond monotone tunings ==
 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}}.
+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}}}}.
 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:
@@ Line 21: / Line 20: @@
 #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].
+#If we map those using {{val| 1 ''a'' 5&minus;2''a'' }} we obtain the tempered sizes [{{nowrap|3''a'' &minus; 4|7 &minus; 5''a''|''a'' &minus; 1|2''a'' &minus; 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.
+#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.
-==See also==
+== See also ==
 For examples and other information, see the topic page [[Tuning ranges of regular temperaments]].
-==References==
+== References ==
+<references />
 [[Category:Tuning]]
 [[Category:Regular temperament theory]]
@@ Line 35: / Line 36: @@
 [[Category:Math]]
 [[Category:Diamond]]
-<references />