Temperament addition: Difference between revisions
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=Visualizing temperament arithmetic= | =Visualizing temperament arithmetic= | ||
[[File:Sum diff and wedge.png|thumb| | [[File:Sum diff and wedge.png|thumb|left|300px|A and B are vectors representing temperaments. They could be maps or prime count vectors. A∧B is their wedge product and gives a higher-[[grade]] temperament that [[merge]]s (sometimes called "meets" or "joins") both A and B. A+B and A-B give the sum and difference, respectively.]] | ||
==Versus the wedge product== | ==Versus the wedge product== | ||
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==Tuning and tone space== | ==Tuning and tone space== | ||
[[File:Visualization of temperament arithmetic.png|300px|right|thumb|A visualization of temperament arithmetic on projective tuning space.]] | |||
One way we can visualize temperament arithmetic is on [[projective tuning space]]. | One way we can visualize temperament arithmetic is 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' [[mapping]]s can be expressed with the map for 7-ET as one of their mapping-rows. | 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. | ||
[[File:Visualization of temperament arithmetic on projective tone space.png|200px|thumb|right|A visualization of temperament arithmetic on projective tone space.]] | |||
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. | ||
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. | 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. | ||