Preimage - Revision history

Cmloegcmluin: /* See also */ briefly explicitly explain the reason for the connection between this page and the new one linked to in its See Also

2025-03-26T19:26:19Z

See also: briefly explicitly explain the reason for the connection between this page and the new one linked to in its See Also

← Older revision		Revision as of 19:26, 26 March 2025
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	== See also ==		== See also ==
	* [[Detempering]]		* [[Detempering]], replacing tempered pitches with pitches from their preimages

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]

Sintel: see also: detempering

2025-03-26T19:19:06Z

Cmloegcmluin: missed a spot

2022-12-27T18:11:24Z

missed a spot

← Older revision		Revision as of 18:11, 27 December 2022
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	=== A temperament with multiple commas ===		=== A temperament with multiple commas ===

	We could repeat this experiment but with septimal meantone, {{ket\|{{val\|1 0 -4 -13}} {{val\|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it ~~tempers out~~ is [[126/125]] {{vector\|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector\|1 -2 1 0}} and 9/8 {{vector\|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector\|1 -2 1 0}} + {{vector\|1 2 -3 1}} = {{vector\|2 0 -2 1}}, which is 28/25, or {{vector\|-3 2 0 0}} - {{vector\|1 2 -3 1}} = {{vector\|-4 0 3 -1}}, which is 125/112.		We could repeat this experiment but with septimal meantone, {{ket\|{{val\|1 0 -4 -13}} {{val\|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it makes to vanish is [[126/125]] {{vector\|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector\|1 -2 1 0}} and 9/8 {{vector\|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector\|1 -2 1 0}} + {{vector\|1 2 -3 1}} = {{vector\|2 0 -2 1}}, which is 28/25, or {{vector\|-3 2 0 0}} - {{vector\|1 2 -3 1}} = {{vector\|-4 0 3 -1}}, which is 125/112.

	So in this case the full member set of the preimage for any mapped interval of this temperament such as [-1 2} could be arranged as a 2D grid, with members differing by a meantone comma along one axis and members differing by a starling comma along the other axis.		So in this case the full member set of the preimage for any mapped interval of this temperament such as [-1 2} could be arranged as a 2D grid, with members differing by a meantone comma along one axis and members differing by a starling comma along the other axis.

Cmloegcmluin: EBK and terminological updates

2022-12-27T16:52:05Z

EBK and terminological updates

← Older revision		Revision as of 16:52, 27 December 2022
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	For a [[regular temperament]], the '''preimage''' of a mapped interval is the set of all (typically [[JI\|justly intoned]]) intervals that map to it.		For a [[regular temperament]], the '''preimage''' of a mapped interval is the set of all (typically [[JI\|justly intoned]]) intervals that map to it.

	For any interval that's a member of such a preimage, altering it by any one of the [[~~temper out\|tempered-out comma~~]]~~s of the temperament~~ finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest.		For any interval that's a member of such a preimage, altering it by any one of the commas that the temperament [[vanish]]es finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest.

	== Examples ==		== Examples ==
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	=== A temperament with only one comma ===		=== A temperament with only one comma ===

	For the meantone mapping {{~~ket~~\|{{map\|1 1 0}} {{map\|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector\|1 -2 1}} because {{~~ket~~\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector\|-3 2 0}} because {{~~ket~~\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector\|-4 4 -1}}) apart: {{vector\|1 -2 1}} + {{vector\|-4 4 -1}} = {{vector\|-3 2 0}}.		For the meantone mapping {{rket\|{{map\|1 1 0}} {{map\|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector\|1 -2 1}} because {{rket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector\|-3 2 0}} because {{rket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector\|-4 4 -1}}) apart: {{vector\|1 -2 1}} + {{vector\|-4 4 -1}} = {{vector\|-3 2 0}}.

	The meantone comma is the only comma ~~tempered out by~~ meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma basis]] or nullspace contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.		The meantone comma is the only comma that meantone makes vanish (meantone is a [[nullity]]-1 temperament, which means its [[comma basis]] or nullspace contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.

	First we find {{vector\|-3 2 0}} + {{vector\|-4 4 -1}} = {{vector\|-7 6 -1}} which is 729/640; then from there we find {{vector\|-7 6 -1}} + {{vector\|-4 4 -1}} = {{vector\|-11 10 -2}} = 59049/51200; then we find {{vector\|-11 10 -2}} + {{vector\|-4 4 -1}} = {{vector\|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.		First we find {{vector\|-3 2 0}} + {{vector\|-4 4 -1}} = {{vector\|-7 6 -1}} which is 729/640; then from there we find {{vector\|-7 6 -1}} + {{vector\|-4 4 -1}} = {{vector\|-11 10 -2}} = 59049/51200; then we find {{vector\|-11 10 -2}} + {{vector\|-4 4 -1}} = {{vector\|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.

Cmloegcmluin: prefer nullspace to null-space

2022-12-15T00:18:19Z

prefer nullspace to null-space

← Older revision		Revision as of 00:18, 15 December 2022
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	For the meantone mapping {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector\|1 -2 1}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector\|-3 2 0}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector\|-4 4 -1}}) apart: {{vector\|1 -2 1}} + {{vector\|-4 4 -1}} = {{vector\|-3 2 0}}.		For the meantone mapping {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector\|1 -2 1}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector\|-3 2 0}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector\|-4 4 -1}}) apart: {{vector\|1 -2 1}} + {{vector\|-4 4 -1}} = {{vector\|-3 2 0}}.

	The meantone comma is the only comma tempered out by meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma-basis]] or ~~null-space~~ contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.		The meantone comma is the only comma tempered out by meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma basis]] or nullspace contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.

	First we find {{vector\|-3 2 0}} + {{vector\|-4 4 -1}} = {{vector\|-7 6 -1}} which is 729/640; then from there we find {{vector\|-7 6 -1}} + {{vector\|-4 4 -1}} = {{vector\|-11 10 -2}} = 59049/51200; then we find {{vector\|-11 10 -2}} + {{vector\|-4 4 -1}} = {{vector\|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.		First we find {{vector\|-3 2 0}} + {{vector\|-4 4 -1}} = {{vector\|-7 6 -1}} which is 729/640; then from there we find {{vector\|-7 6 -1}} + {{vector\|-4 4 -1}} = {{vector\|-11 10 -2}} = 59049/51200; then we find {{vector\|-11 10 -2}} + {{vector\|-4 4 -1}} = {{vector\|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.

Cmloegcmluin: /* Image */ better to speak of the domain and range than to encourage these generalizations of the idea

2022-10-23T18:24:42Z

Image: better to speak of the domain and range than to encourage these generalizations of the idea

← Older revision		Revision as of 18:24, 23 October 2022
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	A [[mapping]] in RTT always maps any given interval to only a single other interval. The '''image''' for an interval is that one interval that it's mapped to by a (regular) temperament.		A [[mapping]] in RTT always maps any given interval to only a single other interval. The '''image''' for an interval is that one interval that it's mapped to by a (regular) temperament.

	~~We can also speak of the "image of a temperament", in general. This means the set of all intervals that can be mapped to.~~

	== Terminology ==		== Terminology ==

Cmloegcmluin: /* Image */ simplify and forestall misreadings

2022-10-18T13:54:41Z

Image: simplify and forestall misreadings

← Older revision		Revision as of 13:54, 18 October 2022
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	== Image ==		== Image ==

	If "preimage" is ~~a thing, then~~ "image" ~~must be a thing~~, ~~too~~.		The "preimage" is literally "that which came before the image", or in other words, the set of possible things that the image might have been projected from.

	~~The general mathematical definition of "function" is something that for any given input always returns the same output.~~ A [[mapping]] in RTT ~~thereby qualifies as a function; it~~ always maps any given interval to only a single other interval. ~~In fact, this is essentially the meaning of the word "regular" in the term "regular temperament." And so, the~~ '''image''' for an interval ~~under a (regular) temperament~~ is that one interval that it's mapped to.		A [[mapping]] in RTT always maps any given interval to only a single other interval. The '''image''' for an interval is that one interval that it's mapped to by a (regular) temperament.

	We can also speak of the "image of a temperament", in general. This means the set of all intervals that can be mapped to.		We can also speak of the "image of a temperament", in general. This means the set of all intervals that can be mapped to.

Cmloegcmluin: /* Image */ avoid misreading

2022-10-15T13:31:40Z

Image: avoid misreading

← Older revision		Revision as of 13:31, 15 October 2022
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	The general mathematical definition of "function" is something that for any given input always returns the same output. A [[mapping]] in RTT thereby qualifies as a function; it always maps any given interval to only a single other interval. In fact, this is essentially the meaning of the word "regular" in the term "regular temperament." And so, the '''image''' for an interval under a (regular) temperament is that one interval that it's mapped to.		The general mathematical definition of "function" is something that for any given input always returns the same output. A [[mapping]] in RTT thereby qualifies as a function; it always maps any given interval to only a single other interval. In fact, this is essentially the meaning of the word "regular" in the term "regular temperament." And so, the '''image''' for an interval under a (regular) temperament is that one interval that it's mapped to.

	We can also speak of the "image of a temperament", in general. This means ~~its image ''over all intervals in its domain'' (its [[interval basis\|interval subspace]]), or in other words,~~ the set of all ~~of its~~ mapped ~~intervals~~.		We can also speak of the "image of a temperament", in general. This means the set of all intervals that can be mapped to.

	== Terminology ==		== Terminology ==

Cmloegcmluin: correct sections nesting

2022-01-25T04:13:32Z

correct sections nesting

← Older revision		Revision as of 04:13, 25 January 2022
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	For any interval that's a member of such a preimage, altering it by any one of the [[temper out\|tempered-out comma]]s of the temperament finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest.		For any interval that's a member of such a preimage, altering it by any one of the [[temper out\|tempered-out comma]]s of the temperament finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest.

	== ~~Example~~ ==		== Examples ==

	This section uses curly brackets for [[tmonzos and tvals\|generator-count vector]]s representing mapped intervals. This helps distinguish them from the angle brackets used for [[prime-count vector]]s representing (unmapped) intervals.		This section uses curly brackets for [[tmonzos and tvals\|generator-count vector]]s representing mapped intervals. This helps distinguish them from the angle brackets used for [[prime-count vector]]s representing (unmapped) intervals.

	== A temperament with only one comma ==		=== A temperament with only one comma ===

	For the meantone mapping {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector\|1 -2 1}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector\|-3 2 0}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector\|-4 4 -1}}) apart: {{vector\|1 -2 1}} + {{vector\|-4 4 -1}} = {{vector\|-3 2 0}}.		For the meantone mapping {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector\|1 -2 1}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector\|-3 2 0}} because {{ket\|{{map\|1 1 0}} {{map\|0 1 4}}}}{{vector\|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector\|-4 4 -1}}) apart: {{vector\|1 -2 1}} + {{vector\|-4 4 -1}} = {{vector\|-3 2 0}}.
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	We could also proceed along this series of meantone-comma-separated members of [-1 2}'s preimage in the other direction, by repeatedly ''subtracting'' meantone commas from 9/8. But we won't work through that because we know a similar effect will happen: the intervals found will grow steadily more complex and with greater error under the temperament.		We could also proceed along this series of meantone-comma-separated members of [-1 2}'s preimage in the other direction, by repeatedly ''subtracting'' meantone commas from 9/8. But we won't work through that because we know a similar effect will happen: the intervals found will grow steadily more complex and with greater error under the temperament.

	== A temperament with multiple commas ==		=== A temperament with multiple commas ===

	We could repeat this experiment but with septimal meantone, {{ket\|{{val\|1 0 -4 -13}} {{val\|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it tempers out is [[126/125]] {{vector\|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector\|1 -2 1 0}} and 9/8 {{vector\|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector\|1 -2 1 0}} + {{vector\|1 2 -3 1}} = {{vector\|2 0 -2 1}}, which is 28/25, or {{vector\|-3 2 0 0}} - {{vector\|1 2 -3 1}} = {{vector\|-4 0 3 -1}}, which is 125/112.		We could repeat this experiment but with septimal meantone, {{ket\|{{val\|1 0 -4 -13}} {{val\|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it tempers out is [[126/125]] {{vector\|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector\|1 -2 1 0}} and 9/8 {{vector\|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector\|1 -2 1 0}} + {{vector\|1 2 -3 1}} = {{vector\|2 0 -2 1}}, which is 28/25, or {{vector\|-3 2 0 0}} - {{vector\|1 2 -3 1}} = {{vector\|-4 0 3 -1}}, which is 125/112.

Cmloegcmluin: Created page with "For a regular temperament, the '''preimage''' of a mapped interval is the set of all (typically justly intoned) intervals that map to it. For any interval that's..."

2022-01-24T22:54:16Z

Created page with "For a regular temperament, the '''preimage''' of a mapped interval is the set of all (typically justly intoned) intervals that map to it. For any interval that's..."

New page

For a [[regular temperament]], the '''preimage''' of a mapped interval is the set of all (typically [[JI|justly intoned]]) intervals that map to it.

For any interval that's a member of such a preimage, altering it by any one of the [[temper out|tempered-out comma]]s of the temperament finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest.

== Example ==

This section uses curly brackets for [[tmonzos and tvals|generator-count vector]]s representing mapped intervals. This helps distinguish them from the angle brackets used for [[prime-count vector]]s representing (unmapped) intervals.

== A temperament with only one comma ==

For the meantone mapping {{ket|{{map|1 1 0}} {{map|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector|1 -2 1}} because {{ket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector|-3 2 0}} because {{ket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector|-4 4 -1}}) apart: {{vector|1 -2 1}} + {{vector|-4 4 -1}} = {{vector|-3 2 0}}.

The meantone comma is the only comma tempered out by meantone (meantone is a [[nullity]]-1 temperament, which means its [[comma-basis]] or null-space contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.

First we find {{vector|-3 2 0}} + {{vector|-4 4 -1}} = {{vector|-7 6 -1}} which is 729/640; then from there we find {{vector|-7 6 -1}} + {{vector|-4 4 -1}} = {{vector|-11 10 -2}} = 59049/51200; then we find {{vector|-11 10 -2}} + {{vector|-4 4 -1}} = {{vector|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.

We could also proceed along this series of meantone-comma-separated members of [-1 2}'s preimage in the other direction, by repeatedly ''subtracting'' meantone commas from 9/8. But we won't work through that because we know a similar effect will happen: the intervals found will grow steadily more complex and with greater error under the temperament.

== A temperament with multiple commas ==

We could repeat this experiment but with septimal meantone, {{ket|{{val|1 0 -4 -13}} {{val|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it tempers out is [[126/125]] {{vector|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector|1 -2 1 0}} and 9/8 {{vector|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector|1 -2 1 0}} + {{vector|1 2 -3 1}} = {{vector|2 0 -2 1}}, which is 28/25, or {{vector|-3 2 0 0}} - {{vector|1 2 -3 1}} = {{vector|-4 0 3 -1}}, which is 125/112.

So in this case the full member set of the preimage for any mapped interval of this temperament such as [-1 2} could be arranged as a 2D grid, with members differing by a meantone comma along one axis and members differing by a starling comma along the other axis.

== Image ==

If "preimage" is a thing, then "image" must be a thing, too.

The general mathematical definition of "function" is something that for any given input always returns the same output. A [[mapping]] in RTT thereby qualifies as a function; it always maps any given interval to only a single other interval. In fact, this is essentially the meaning of the word "regular" in the term "regular temperament." And so, the '''image''' for an interval under a (regular) temperament is that one interval that it's mapped to.

We can also speak of the "image of a temperament", in general. This means its image ''over all intervals in its domain'' (its [[interval basis|interval subspace]]), or in other words, the set of all of its mapped intervals.

== Terminology ==

The terms "[[Wikipedia:Image_(mathematics)|image]]" and "[https://mathworld.wolfram.com/Preimage.html preimage]" come from general mathematics where they are used regarding mathematical functions and their inputs (their domain) and outputs (their range). They are used in the same sense in regular temperament theory; in our application, [[mappings]] are the functions, intervals are the inputs, and mapped intervals are the outputs.

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]

← Older revision		Revision as of 19:19, 26 March 2025
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	The terms "[[Wikipedia:Image_(mathematics)\|image]]" and "[https://mathworld.wolfram.com/Preimage.html preimage]" come from general mathematics where they are used regarding mathematical functions and their inputs (their domain) and outputs (their range). They are used in the same sense in regular temperament theory; in our application, [[mappings]] are the functions, intervals are the inputs, and mapped intervals are the outputs.		The terms "[[Wikipedia:Image_(mathematics)\|image]]" and "[https://mathworld.wolfram.com/Preimage.html preimage]" come from general mathematics where they are used regarding mathematical functions and their inputs (their domain) and outputs (their range). They are used in the same sense in regular temperament theory; in our application, [[mappings]] are the functions, intervals are the inputs, and mapped intervals are the outputs.

			== See also ==
			* [[Detempering]]

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]
	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]