Equivalence continuum: Difference between revisions
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NB: Not all tuning continua are projective spaces, because not all Grassmannians are; for example, '''Gr'''(2, 4) is not a projective space. | NB: Not all tuning continua are projective spaces, because not all Grassmannians are; for example, '''Gr'''(2, 4) is not a projective space. | ||
=== Example (7-limit rank-2 temperaments in 31edo) === | === Example (7-limit rank-2 temperaments in 31edo) === | ||
Let '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> = 81/80, 126/125, 1029/1024 be the basis for the kernel of 7-limit [[31edo]]. | Let's look at where some well-known 7-limit rank-2 temperaments supported by [[31edo]] live in the equivalence continuum. Let '''u'''<sub>''x''</sub>, '''u'''<sub>''y''</sub>, '''u'''<sub>''z''</sub> = 81/80, 126/125, 1029/1024 be the basis for the kernel of 7-limit [[31edo]]. Then: | ||
* [[septimal meantone]] tempers out 81/80 = '''u'''<sub>''x''</sub> = (1, 0, 0) and 126/125 = '''u'''<sub>''y''</sub> = (0, 1, 0), thus corresponds to the plane ''z'' = 0. This corresponds to '''v''' = (0, 0, 1). | * [[septimal meantone]] tempers out 81/80 = '''u'''<sub>''x''</sub> = (1, 0, 0) and 126/125 = '''u'''<sub>''y''</sub> = (0, 1, 0), thus corresponds to the plane ''z'' = 0. This corresponds to '''v''' = (0, 0, 1). | ||
* [[hemithirds]] tempers out 1029/1024 = '''u'''<sub>''z''</sub> = (0, 0, 1) and 3136/3125 = 2'''u'''<sub>''x''</sub> + '''u'''<sub>''y''</sub> = (2, 1, 0). This corresponds to '''v''' = (1, −2, 0). | * [[hemithirds]] tempers out 1029/1024 = '''u'''<sub>''z''</sub> = (0, 0, 1) and 3136/3125 = 2'''u'''<sub>''x''</sub> + '''u'''<sub>''y''</sub> = (2, 1, 0). This corresponds to '''v''' = (1, −2, 0). | ||