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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-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).