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).

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.

The page [[Monotonicity limits of small edos]] shows the largest odd limit that a given edo is monotonic in.

== Temperaments without diamond monotone tunings ==

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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).
 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.
+The page [[Monotonicity limits of small edos]] shows the largest odd limit that a given edo is monotonic in.
 == Temperaments without diamond monotone tunings ==