Temperament addition: Difference between revisions

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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.]]

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 rank-2 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 20: / Line 20: @@
 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.]]
+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 rank-2 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==