User:Akselai/On the infinite division of the octave: Difference between revisions
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=== ∞edo as a temperament === | === ∞edo as a temperament === | ||
The axioms above only specify ∞edo as a ''tuning''. To turn ∞edo into a ''temperament'', we have the following: | |||
'''4) There exists a mapping ''V'' from a JI subgroup ''I'' to ∞edo such that the regular temperament property holds, i.e. ''V''(''α'') ''V''(''β'') = ''V''(''αβ'') for all ''α'', ''β'' ∈ ''I''.''' | '''4) There exists a mapping ''V'' from a JI subgroup ''I'' to ∞edo such that the regular temperament property holds, i.e. ''V''(''α'') ''V''(''β'') = ''V''(''αβ'') for all ''α'', ''β'' ∈ ''I''.''' | ||
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∞edo by this construction is a flexible object. Not all ∞edos are the same, even if the temperament structure is discarded. Some have defined ∞edo as simply the union of all edos, which is actually supported by this construction. At the ''h''-th level of the tower with ''m''edo, we only need to adjoin (''mh'')edo to obtain the (''h''+1)-th level, and we would have encompassed all integer factors along the tower and hence all edos. (Though, the intervals of an arbitrary subset edo do not follow a val mapping.) | ∞edo by this construction is a flexible object. Not all ∞edos are the same, even if the temperament structure is discarded. Some have defined ∞edo as simply the union of all edos, which is actually supported by this construction. At the ''h''-th level of the tower with ''m''edo, we only need to adjoin (''mh'')edo to obtain the (''h''+1)-th level, and we would have encompassed all integer factors along the tower and hence all edos. (Though, the intervals of an arbitrary subset edo do not follow a val mapping.) | ||
On the other hand, ∞edo can also be built from, say 5<sup>''n''</sup>edos. Then it would not contain 2edo, among other edos that are not powers of 5. | On the other hand, ∞edo can also be built from, say 5<sup>''n''</sup>edos. Then it would not contain 2edo, among other edos that are not powers of 5. This tuning is practically isomorphic to the [https://en.wikipedia.org/wiki/Pr%C3%BCfer_group Prüfer ''p''-group]. | ||
== Implementation == | == Implementation == | ||