Diamond monotone: Difference between revisions

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A tuning for a [[rank]]-''r'' [[harmonic 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 {{w|monotonic function|monotone}} increasing (i.e. nondecreasing).

A tuning for a [[rank]]-''r'' [[harmonic 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 {{w|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>~~.

More generally, a temperament tuning is '''monotone''' or '''monotonic''' in an arbitrary set of intervals if it keeps the order of sizes for all the intervals in the set, and analogous to [[consistency limit]], each temperament tuning has a '''monotonicity limit''', which is the maximum of ''q'' such that the ''q''-odd-limit tonality diamond is tuned monotone.

This 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, which 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 the ''p''-odd-limit tonality diamond is an arguably reasonable choice).

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== Temperaments without diamond monotone tunings ==

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

While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not (even though in most practical cases they do occur).

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

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== See also ==

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

* [[Tuning ranges of regular temperaments]] – for examples and other information

* [[Monotonicity limits of small edos]] – the largest odd limit each edo from 1 to 72 is monotonic in

* [[Minimum monotone vals]] – the earliest [[generalized patent val]] monotone in each odd limit from 1 to 127

== References ==

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-A tuning for a [[rank]]-''r'' [[harmonic 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 {{w|monotonic function|monotone}} increasing (i.e. nondecreasing).
+A tuning for a [[rank]]-''r'' [[harmonic 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 {{w|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>.
+More generally, a temperament tuning is '''monotone''' or '''monotonic''' in an arbitrary set of intervals if it keeps the order of sizes for all the intervals in the set, and analogous to [[consistency limit]], each temperament tuning has a '''monotonicity limit''', which is the maximum of ''q'' such that the ''q''-odd-limit tonality diamond is tuned monotone.
 This 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, which 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 the ''p''-odd-limit tonality diamond is an arguably reasonable choice).
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 == Temperaments without diamond monotone tunings ==
-While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not.
+While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not (even though in most practical cases they do occur).
 Let's look at an example: the rank-2 temperament with mapping {{mapping| 1 0 5 | 0 1 -2 }}, tempering out [[45/32]].
@@ Line 25: / Line 25: @@
 == See also ==
-For examples and other information, see the topic page [[Tuning ranges of regular temperaments]].
+* [[Tuning ranges of regular temperaments]] – for examples and other information
+* [[Monotonicity limits of small edos]] – the largest odd limit each edo from 1 to 72 is monotonic in
+* [[Minimum monotone vals]] – the earliest [[generalized patent val]] monotone in each odd limit from 1 to 127
 == References ==