Temperament addition: Difference between revisions

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For non-min-grade-1 temperaments, arithmetic gets a little trickier. This is discussed in the [[Temperament_arithmetic#Beyond_min-grade-1|beyond min-grade-1 section]] later.

==Visualizing temperament arithmetic==

One way we can visualize temperament arithmetic is on [[projective tuning space]].

[[File:Visualization of temperament arithmetic.png|500px|right|thumb|A visualization of temperament arithmetic on projective tuning space.]]

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' mappings can be expressed with the map for 7-ET as one of their mapping rows.

These are all rank-2 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.

==Conditions on temperament arithmetic==

@@ Line 8: / Line 8: @@
 For non-min-grade-1 temperaments, arithmetic gets a little trickier. This is discussed in the [[Temperament_arithmetic#Beyond_min-grade-1|beyond min-grade-1 section]] later.
+==Visualizing temperament arithmetic==
+One way we can visualize temperament arithmetic is on [[projective tuning space]].
+[[File:Visualization of temperament arithmetic.png|500px|right|thumb|A visualization of temperament arithmetic on projective tuning space.]]
+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' mappings can be expressed with the map for 7-ET as one of their mapping rows.
+These are all rank-2 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.
 ==Conditions on temperament arithmetic==