Temperament addition: Difference between revisions

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This shows both the sum and the difference of porcupine and meantone. All four temperaments — the two input temperaments, porcupine and meantone, as well as the sum, tetracot, and the diff, dicot — can be seen to intersect at 7-ET. This is because all four temperaments' [[mapping]]s can be expressed with the map for 7-ET as one of their mapping-rows.

These are all <math>r=2</math> temperaments, so their mappings each have one other row besides the one reserved for 7-ET. Any line that we draw across these four temperament lines will strike four ETs whose maps have a sum and difference relationship. On this diagram, two such lines have been drawn. The first one runs through 5-ET, 20-ET, 15-ET, and 10-ET. We can see that 5 + 15=20, which corresponds to the fact that 20-ET is the ET on the line for tetracot, which is the sum of porcupine and meantone, while 5-ET and 15-ET are the ETs on their lines. Similarly, we can see that 15 - 5=10, which corresponds to the fact that 10-ET is the ET on the line for dicot, which is the difference of porcupine and meantone.

These are all <math>r=2</math> temperaments, so their mappings each have one other row besides the one reserved for 7-ET. Any line that we draw across these four temperament lines will strike four ETs whose maps have a sum and difference relationship. On this diagram, two such lines have been drawn. The first one runs through 5-ET, 20-ET, 15-ET, and 10-ET. We can see that 5 + 15 = 20, which corresponds to the fact that 20-ET is the ET on the line for tetracot, which is the sum of porcupine and meantone, while 5-ET and 15-ET are the ETs on their lines. Similarly, we can see that 15 - 5 = 10, which corresponds to the fact that 10-ET is the ET on the line for dicot, which is the difference of porcupine and meantone.

The other line runs through the ETs 12, 41, 29, and 17, and we can see again that 12 + 29=41 and 29 - 12=17.

The other line runs through the ETs 12, 41, 29, and 17, and we can see again that 12 + 29 = 41 and 29 - 12 = 17.

[[File:Visualization of temperament arithmetic on projective tone space.png|300px|thumb|right|A visualization of temperament arithmetic on projective tone space.]]

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We can also visualize temperament arithmetic on [[projective tone space]]. Here relationships are inverted: points are lines, and lines are points. So all four temperaments are found along the line for 7-ET.

Note that when viewed in tuning space, the sum is found between the two input temperaments, and the difference is found on the outside of them, to one side or the other. While in tone space, it's the difference that's found between the two input temperaments, and its the sum that's found outside. In either situation when a temperament is on the outside and may be on one side or the other, the explanation for this can be inferred from behavior of the scale tree on any temperament line, where e.g. if 5-ET and 7-ET support a <math>r=2</math> temperament, then so will 5 + 7=12-ET, and then so will 5 + 12 and 7 + 12 in turn, and so on and so on recursively; when you navigate like this, what we could call ''down'' the scale tree, children are always found between their parents. But when you try to go back ''up'' the scale tree, to one or the other parent, you may not immediately know which side of the child to go.

Note that when viewed in tuning space, the sum is found between the two input temperaments, and the difference is found on the outside of them, to one side or the other. While in tone space, it's the difference that's found between the two input temperaments, and its the sum that's found outside. In either situation when a temperament is on the outside and may be on one side or the other, the explanation for this can be inferred from behavior of the scale tree on any temperament line, where e.g. if 5-ET and 7-ET support a <math>r=2</math> temperament, then so will 5 + 7 = 12-ET, and then so will 5 + 12 and 7 + 12 in turn, and so on and so on recursively; when you navigate like this, what we could call ''down'' the scale tree, children are always found between their parents. But when you try to go back ''up'' the scale tree, to one or the other parent, you may not immediately know which side of the child to go.

=Conditions on temperament arithmetic=

@@ Line 169: / Line 169: @@
 This shows both the sum and the difference of porcupine and meantone. All four temperaments — the two input temperaments, porcupine and meantone, as well as the sum, tetracot, and the diff, dicot — can be seen to intersect at 7-ET. This is because all four temperaments' [[mapping]]s can be expressed with the map for 7-ET as one of their mapping-rows.
-These are all <math>r=2</math> temperaments, so their mappings each have one other row besides the one reserved for 7-ET. Any line that we draw across these four temperament lines will strike four ETs whose maps have a sum and difference relationship. On this diagram, two such lines have been drawn. The first one runs through 5-ET, 20-ET, 15-ET, and 10-ET. We can see that 5 + 15=20, which corresponds to the fact that 20-ET is the ET on the line for tetracot, which is the sum of porcupine and meantone, while 5-ET and 15-ET are the ETs on their lines. Similarly, we can see that 15 - 5=10, which corresponds to the fact that 10-ET is the ET on the line for dicot, which is the difference of porcupine and meantone.
+These are all <math>r=2</math> temperaments, so their mappings each have one other row besides the one reserved for 7-ET. Any line that we draw across these four temperament lines will strike four ETs whose maps have a sum and difference relationship. On this diagram, two such lines have been drawn. The first one runs through 5-ET, 20-ET, 15-ET, and 10-ET. We can see that 5 + 15 = 20, which corresponds to the fact that 20-ET is the ET on the line for tetracot, which is the sum of porcupine and meantone, while 5-ET and 15-ET are the ETs on their lines. Similarly, we can see that 15 - 5 = 10, which corresponds to the fact that 10-ET is the ET on the line for dicot, which is the difference of porcupine and meantone.
-The other line runs through the ETs 12, 41, 29, and 17, and we can see again that 12 + 29=41 and 29 - 12=17.
+The other line runs through the ETs 12, 41, 29, and 17, and we can see again that 12 + 29 = 41 and 29 - 12 = 17.
 [[File:Visualization of temperament arithmetic on projective tone space.png|300px|thumb|right|A visualization of temperament arithmetic on projective tone space.]]
@@ Line 177: / Line 177: @@
 We can also visualize temperament arithmetic on [[projective tone space]]. Here relationships are inverted: points are lines, and lines are points. So all four temperaments are found along the line for 7-ET.
-Note that when viewed in tuning space, the sum is found between the two input temperaments, and the difference is found on the outside of them, to one side or the other. While in tone space, it's the difference that's found between the two input temperaments, and its the sum that's found outside. In either situation when a temperament is on the outside and may be on one side or the other, the explanation for this can be inferred from behavior of the scale tree on any temperament line, where e.g. if 5-ET and 7-ET support a <math>r=2</math> temperament, then so will 5 + 7=12-ET, and then so will 5 + 12 and 7 + 12 in turn, and so on and so on recursively; when you navigate like this, what we could call ''down'' the scale tree, children are always found between their parents. But when you try to go back ''up'' the scale tree, to one or the other parent, you may not immediately know which side of the child to go.
+Note that when viewed in tuning space, the sum is found between the two input temperaments, and the difference is found on the outside of them, to one side or the other. While in tone space, it's the difference that's found between the two input temperaments, and its the sum that's found outside. In either situation when a temperament is on the outside and may be on one side or the other, the explanation for this can be inferred from behavior of the scale tree on any temperament line, where e.g. if 5-ET and 7-ET support a <math>r=2</math> temperament, then so will 5 + 7 = 12-ET, and then so will 5 + 12 and 7 + 12 in turn, and so on and so on recursively; when you navigate like this, what we could call ''down'' the scale tree, children are always found between their parents. But when you try to go back ''up'' the scale tree, to one or the other parent, you may not immediately know which side of the child to go.
 =Conditions on temperament arithmetic=