Preimage: Difference between revisions

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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 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 ==

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=== 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.

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=== 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.

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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.

== See also ==

* [[Detempering]], replacing tempered pitches with pitches from their preimages

[[Category:Regular temperament theory]]

[[Category:Terms]]

[[Category:Math]]

@@ Line 1: / Line 1: @@
 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 ==
@@ Line 9: / Line 9: @@
 === 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.
@@ Line 19: / Line 19: @@
 === 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.
@@ Line 32: / Line 32: @@
 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]], replacing tempered pitches with pitches from their preimages
 [[Category:Regular temperament theory]]
 [[Category:Terms]]
 [[Category:Math]]