Lumma stability - Revision history

TallKite at 09:25, 20 October 2024

2024-10-20T09:25:20Z

← Older revision		Revision as of 09:25, 20 October 2024
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	'''Lumma stability''' is a concept in the theory of musical [[scale]]s developed by [[Carl Lumma]]. It is the portion of a scale's period which is not covered by its [[Interval class\|interval classes]]. The "Lumma impropriety factor" is the portion which is more than singly-covered. [[Scala]] will report these for any scale via its "show data" command.		'''Lumma stability''' is a concept in the theory of musical [[scale]]s developed by [[Carl Lumma]]. It is the portion of a scale's period which is not covered by its [[Interval class\|interval classes]]. The "Lumma impropriety factor" is the portion which is more than singly-covered. [[Scala]] will report these for any scale via its "show data" command.

	[[Rothenberg propriety\|Rothenberg]] assumes a process whereby a listener hears melodic [[Interval\|intervals]] between sounds and sorts them by size. This produces an "[[interval matrix]]" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, he may remember it for later use.		[[Rothenberg propriety\|Rothenberg]] assumes a process whereby a listener hears melodic [[Interval\|intervals]] between sounds and sorts them by size. This produces an "[[interval matrix]]" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, they may remember it for later use.

	The IM for a scale is simply a list of all its intervals ~~([[Dyad\|dyads]])~~ grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.		The IM for a scale is simply a list of all its intervals grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.

	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered—where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered—where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.

ArrowHead294 at 21:42, 23 May 2024

2024-05-23T21:42:35Z

← Older revision		Revision as of 21:42, 23 May 2024
Line 5:		Line 5:
	The IM for a scale is simply a list of all its intervals ([[Dyad\|dyads]]) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.		The IM for a scale is simply a list of all its intervals ([[Dyad\|dyads]]) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.

	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered{{mdash}}where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered—where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.

	== External links ==		== External links ==

ArrowHead294 at 13:45, 23 May 2024

2024-05-23T13:45:05Z

← Older revision		Revision as of 13:45, 23 May 2024
Line 5:		Line 5:
	The IM for a scale is simply a list of all its intervals ([[Dyad\|dyads]]) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.		The IM for a scale is simply a list of all its intervals ([[Dyad\|dyads]]) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.

	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered — where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered{{mdash}}where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.

	== External links ==		== External links ==

BudjarnLambeth: Added links (as requested by the to:do notice)

2023-04-20T07:26:36Z

Added links (as requested by the to:do notice)

← Older revision		Revision as of 07:26, 20 April 2023
Line 1:		Line 1:
	'''Lumma stability''' is a concept in the theory of musical [[scale]]s developed by [[Carl Lumma]]. It is the portion of a scale's period which is not covered by its interval classes. The "Lumma impropriety factor" is the portion which is more than singly-covered. Scala will report these for any scale via its "show data" command.		'''Lumma stability''' is a concept in the theory of musical [[scale]]s developed by [[Carl Lumma]]. It is the portion of a scale's period which is not covered by its [[Interval class\|interval classes]]. The "Lumma impropriety factor" is the portion which is more than singly-covered. [[Scala]] will report these for any scale via its "show data" command.

	Rothenberg assumes a process whereby a listener hears melodic intervals between sounds and sorts them by size. This produces an "interval matrix" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, he may remember it for later use.		[[Rothenberg propriety\|Rothenberg]] assumes a process whereby a listener hears melodic [[Interval\|intervals]] between sounds and sorts them by size. This produces an "[[interval matrix]]" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, he may remember it for later use.

	The IM for a scale is simply a list of all its intervals (dyads) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.		The IM for a scale is simply a list of all its intervals ([[Dyad\|dyads]]) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.

	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 cents). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered — where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered — where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.

	== External links ==		== External links ==
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	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]

	~~{{todo\|clarify\|add examples\|add links}}~~

Fredg999 category edits: Removing from Category:Measure using Cat-a-lot

2023-02-26T06:34:33Z

Removing from Category:Measure using Cat-a-lot

← Older revision		Revision as of 06:34, 26 February 2023
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	* [http://www.tonalsoft.com/enc/l/lumma-impropriety.aspx "Lumma impropriety"] on [[Tonalsoft encyclopedia]]		* [http://www.tonalsoft.com/enc/l/lumma-impropriety.aspx "Lumma impropriety"] on [[Tonalsoft encyclopedia]]

	~~[[Category:Measure]]~~
	[[Category:Scale]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]

	{{todo\|clarify\|add examples\|add links}}		{{todo\|clarify\|add examples\|add links}}

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

2022-02-26T16:32:51Z

Moving from Category:Scale theory to Category:Scale using Cat-a-lot

← Older revision		Revision as of 16:32, 26 February 2022
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	[[Category:Measure]]		[[Category:Measure]]
	[[Category:Scale ~~theory~~]]		[[Category:Scale]]
	[[Category:Terms]]		[[Category:Terms]]

	{{todo\|clarify\|add examples\|add links}}		{{todo\|clarify\|add examples\|add links}}

Fredg999: Added another link from Rothenberg propriety

2022-01-01T21:19:06Z

Added another link from Rothenberg propriety

← Older revision		Revision as of 21:19, 1 January 2022
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	== External links ==		== External links ==
	* [http://www.tonalsoft.com/enc/l/lumma-stability.aspx "Lumma stability"] on [[Tonalsoft encyclopedia]]		* [http://www.tonalsoft.com/enc/l/lumma-stability.aspx "Lumma stability"] on [[Tonalsoft encyclopedia]]
			* [http://www.tonalsoft.com/enc/l/lumma-impropriety.aspx "Lumma impropriety"] on [[Tonalsoft encyclopedia]]

	[[Category:Measure]]		[[Category:Measure]]

Fredg999: Bolded "impropriety factor"

2022-01-01T21:17:44Z

Bolded "impropriety factor"

← Older revision		Revision as of 21:17, 1 January 2022
Line 5:		Line 5:
	The IM for a scale is simply a list of all its intervals (dyads) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.		The IM for a scale is simply a list of all its intervals (dyads) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.

	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 cents). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The impropriety factor is the portion that's more than singly covered — where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 cents). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered — where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.

	== External links ==		== External links ==

Fredg999: Added external link from Rothenberg propriety, misc. edits

2022-01-01T21:10:21Z

Added external link from Rothenberg propriety, misc. edits

← Older revision		Revision as of 21:10, 1 January 2022
Line 1:		Line 1:
	Lumma stability is the portion of a scale's period which is not covered by its interval classes. The "Lumma impropriety factor" is the portion which is more than singly-covered. Scala will report these for any scale via its "show data" command.		'''Lumma stability''' is a concept in the theory of musical [[scale]]s developed by [[Carl Lumma]]. It is the portion of a scale's period which is not covered by its interval classes. The "Lumma impropriety factor" is the portion which is more than singly-covered. Scala will report these for any scale via its "show data" command.

	Rothenberg assumes a process whereby a listener hears melodic intervals between sounds and sorts them by size. This produces an "interval matrix" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, he may remember it for later use.		Rothenberg assumes a process whereby a listener hears melodic intervals between sounds and sorts them by size. This produces an "interval matrix" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, he may remember it for later use.

	The IM for a scale is simply a list of all its intervals (dyads) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.		The IM for a scale is simply a list of all its intervals (dyads) grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.

	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 cents). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The impropriety factor is the portion that's more than singly covered -- where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 cents). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The impropriety factor is the portion that's more than singly covered — where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.

			== External links ==
			* [http://www.tonalsoft.com/enc/l/lumma-stability.aspx "Lumma stability"] on [[Tonalsoft encyclopedia]]

	[[Category:Measure]]		[[Category:Measure]]

Xenwolf: recat,todo

2021-01-02T00:42:14Z

recat,todo

← Older revision		Revision as of 00:42, 2 January 2021
Line 7:		Line 7:
	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 cents). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The impropriety factor is the portion that's more than singly covered -- where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 cents). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The impropriety factor is the portion that's more than singly covered -- where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.

	[[Category:~~measure~~]]		[[Category:Measure]]
	[[Category:~~scales~~]]		[[Category:Scale theory]]
	[[Category:Terms]]		[[Category:Terms]]
	~~[[Category:theory]]~~
	~~[[Category:~~todo~~:add_examples]]~~		{{todo\|clarify\|add examples\|add links}}
	~~[[Category:todo:definition]]~~

← Older revision		Revision as of 09:25, 20 October 2024
Line 1:		Line 1:
	'''Lumma stability''' is a concept in the theory of musical [[scale]]s developed by [[Carl Lumma]]. It is the portion of a scale's period which is not covered by its [[Interval class\|interval classes]]. The "Lumma impropriety factor" is the portion which is more than singly-covered. [[Scala]] will report these for any scale via its "show data" command.		'''Lumma stability''' is a concept in the theory of musical [[scale]]s developed by [[Carl Lumma]]. It is the portion of a scale's period which is not covered by its [[Interval class\|interval classes]]. The "Lumma impropriety factor" is the portion which is more than singly-covered. [[Scala]] will report these for any scale via its "show data" command.

	[[Rothenberg propriety\|Rothenberg]] assumes a process whereby a listener hears melodic [[Interval\|intervals]] between sounds and sorts them by size. This produces an "[[interval matrix]]" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, he may remember it for later use.		[[Rothenberg propriety\|Rothenberg]] assumes a process whereby a listener hears melodic [[Interval\|intervals]] between sounds and sorts them by size. This produces an "[[interval matrix]]" (IM) in the listener's mind. The listener continually refines this IM as new sounds arrive, and if a sound source conforms to a single IM long enough, they may remember it for later use.

	The IM for a scale is simply a list of all its intervals ~~([[Dyad\|dyads]])~~ grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.		The IM for a scale is simply a list of all its intervals grouped by the modes in which they appear. Scala will display it with "show/line intervals". Every fixed scale corresponds to one and only one IM. So if we assume that listeners will eventually perfect their picture of a scale's IM, we can infer things about the scale from its IM.

	Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered—where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.		Imagine a log-frequency ruler whose total length is the interval of equivalence of our periodic scale (e.g. 1200 [[Cent\|cents]]). Mark off each interval in the IM on this ruler. Now we'll draw line segments on the ruler with colored pencil, using the marks as endpoints. We'll connect all marks belonging to the same interval class with a single line, using a different color for each interval class. Lumma stability is the portion of the ruler that has no pencil on it. The '''impropriety factor''' is the portion that's more than singly covered—where different colors overlap. The idea being that when two interval classes overlap, listeners will not be able to distinguish them in all cases. Lumma stability measures how easily distinguishable the non-overlapping classes will be. Rothenberg stability is very similar, except it counts the number of overlaps, so it can't distinguish a gross overlap (in cents) from a small one, or detect when two interval classes are arbitrarily close to overlapping.