Diamond monotone - Revision history

Overthink: move to see also; add another

2026-03-29T00:51:29Z

move to see also; add another

← Older revision		Revision as of 00:51, 29 March 2026
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	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 "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 ==		== Temperaments without diamond monotone tunings ==
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	== See also ==		== 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 ==		== References ==

Overthink: /* Temperaments without diamond monotone tunings */ grammar

2025-12-15T03:06:07Z

Temperaments without diamond monotone tunings: grammar

← Older revision		Revision as of 03:06, 15 December 2025
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	== Temperaments without diamond monotone tunings ==		== Temperaments without diamond monotone tunings ==
	While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not (even though in most practical cases they do).		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]].		Let's look at an example: the rank-2 temperament with mapping {{mapping\| 1 0 5 \| 0 1 -2 }}, tempering out [[45/32]].

Overthink: /* Temperaments without diamond monotone tunings */ note

2025-12-15T03:05:49Z

Temperaments without diamond monotone tunings: note

← Older revision		Revision as of 03:05, 15 December 2025
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	== Temperaments without diamond monotone tunings ==		== 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).

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

FloraC: + link

2025-08-07T11:51:51Z

+ link

← Older revision		Revision as of 11:51, 7 August 2025
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	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 "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 ==		== Temperaments without diamond monotone tunings ==

FloraC: Expand to arbitrary sets of intervals

2025-08-06T13:52:08Z

Expand to arbitrary sets of intervals

← Older revision		Revision as of 13:52, 6 August 2025
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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).		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).

FloraC: Misc. cleanup

2025-08-06T13:51:22Z

Misc. cleanup

← Older revision		Revision as of 13:51, 6 August 2025
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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).		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>.

	~~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).		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 "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.
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	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.

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

	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}}}}.		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:		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 us 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 [{{nowrap\| 3''a'' − 4 }}, {{nowrap\| 7 − 5''a'' }}, {{nowrap\| ''a'' − 1 }}, {{nowrap\| 2''a''− 3 }}].
			# Now we need to make sure each of those are not negative, so we get a set of inequalities: {{nowrap\| ''a'' > 4/3 }}, {{nowrap\| ''a'' < 7/5 }}, {{nowrap\| ''a'' > 1 }}, {{nowrap\| ''a'' > 3/2 }}.

	~~#The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3].~~		We can see that these inequalities have no solution: there is 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; any tuning must necessarily either map 6/5 wider than 5/4 or 4/3 wider than 3/2.
	~~#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 [{{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: {{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; any tuning must necessarily either map 6/5 wider than 5/4 or 4/3 wider than 3/2.

	== See also ==		== See also ==

Meta04: my 1st non-userspace edit; this page's example seemed a little... abstract to me at first

2025-01-14T21:54:11Z

my 1st non-userspace edit; this page's example seemed a little... abstract to me at first

← Older revision		Revision as of 21:54, 14 January 2025
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	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.

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

	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}}}}.		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}}}}.
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	#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}}.		#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.		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; any tuning must necessarily either map 6/5 wider than 5/4 or 4/3 wider than 3/2.

	== See also ==		== See also ==

ArrowHead294: Formatting

2024-11-05T16:20:59Z

Formatting

← Older revision		Revision as of 16:20, 5 November 2024
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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).		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.		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.		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 }}].		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:		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].		#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.		#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.		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]].		For examples and other information, see the topic page [[Tuning ranges of regular temperaments]].

	==References==		== References ==
			<references />

	[[Category:Tuning]]		[[Category:Tuning]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Diamond]]		[[Category:Diamond]]

	~~<references />~~

AndyMilne at 02:40, 10 June 2024

2024-06-10T02:40:52Z

← Older revision		Revision as of 02:40, 10 June 2024
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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).		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). Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum. Computer Music Journal, 31(4):15–32.</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).
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	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 "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.		While diamond tradeoff tunings are always guaranteed to occur, diamond monotone tunings are not.
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	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:		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 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].		#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].		#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.		#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 [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: ''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.		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]].		For examples and other information, see the topic page [[Tuning ranges of regular temperaments]].

	== References ==		==References==
	[[Category:Tuning]]		[[Category:Tuning]]
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Diamond]]		[[Category:Diamond]]

			<references />

AndyMilne: Move text for intro paragraph to new "See also" header to match parallel page on Diamond Tradeoff

2024-06-10T02:20:50Z

Move text for intro paragraph to new "See also" header to match parallel page on Diamond Tradeoff

← Older revision		Revision as of 02:20, 10 June 2024
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	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 "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.

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

	== Temperaments without diamond monotone tunings ==		== Temperaments without diamond monotone tunings ==
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	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.		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 ==
			For examples and other information, see the topic page [[Tuning ranges of regular temperaments]].

	== References ==		== References ==