Generator-offset property: Difference between revisions
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It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large N, far beyond the scale sizes we typically use in music theory, but it would be good to have a proof. | It is pretty easy to see this behavior is true if we simply compute the MV3 sequences up to any very large N, far beyond the scale sizes we typically use in music theory, but it would be good to have a proof. | ||
== Open | == Open questions == | ||
This heading has those open questions for which no conjecture has yet been formed either way. (These can be updated as necessary) | This heading has those open questions for which no conjecture has yet been formed either way. (These can be updated as necessary) | ||
# Given any arbitrary MOS (or DE, etc) scale, is there *always* a MV3 GO scale which can be derived as a "detempering" of that scale? Or is this only true for some MOS's? For instance, the MOS LLsLLLs has the MV3 GO scale LmsLmLs as a detempering. Does a similar MV3 detempering exist for every possible | # Given any arbitrary MOS (or DE, etc) scale, is there *always* a MV3 GO scale which can be derived as a "detempering" of that scale? Or is this only true for some MOS's? For instance, the MOS LLsLLLs has the MV3 GO scale LmsLmLs as a detempering. Does a similar MV3 detempering exist for every possible well-formed scale (single-period mos)? | ||
# The scale tree is a great way to analyze MOS scales. For any generator, we can compute the various MOS's it forms if we simply look at the scale tree, and indeed MOS "words" like LLsLLLs can be identified with regions on the scale tree (in this situation the interval between 4/7 and 3/5). A similar "scale plane" should exist for GO-MV3 scales, where given some word representing a GO-MV3 scale, we can look at the set of points on the generator plane which generates it; these seem to often be triangles, with the lines corresponding to MOS's and the vertices corresponding to EDOs (though is this always true?). What is the big picture of this scale plane? Can we use Viggo Brun's algorithm for this, generalizing the theory of continued fractions? Is there some simple formula we can use to predict, given some GO-MV3 scale, which region on the scale plane it corresponds to? Can we plot simple generator-size-proportions as points in this space? And so on. | # The scale tree is a great way to analyze MOS scales. For any generator, we can compute the various MOS's it forms if we simply look at the scale tree, and indeed MOS "words" like LLsLLLs can be identified with regions on the scale tree (in this situation the interval between 4/7 and 3/5). A similar "scale plane" should exist for GO-MV3 scales, where given some word representing a GO-MV3 scale, we can look at the set of points on the generator plane which generates it; these seem to often be triangles, with the lines corresponding to MOS's and the vertices corresponding to EDOs (though is this always true?). What is the big picture of this scale plane? Can we use Viggo Brun's algorithm for this, generalizing the theory of continued fractions? Is there some simple formula we can use to predict, given some GO-MV3 scale, which region on the scale plane it corresponds to? Can we plot simple generator-size-proportions as points in this space? And so on. | ||
# In the theory of MOS, there is a second scale tree that is less frequently talked about, which Erv Wilson calls the "Rabbit Sequence" ([http://www.anaphoria.com/RabbitSequence.pdf Erv Wilson's original version], [https://mikebattagliamusic.com/MOSTree/MOSTreeab.html interactive version 1], [https://mikebattagliamusic.com/MOSTree/MOSTreeLs.html interactive version 2]). This is a tree for which each MOS word has two children, depending on if the MOS is "soft" (with L/s < 2) or "hard" (with L/s > 2). For instance, LsLss has the two children LLsLLLs and ssLsssL. Does a similar scale plane exist for these GO-MV3 scales? | # In the theory of MOS, there is a second scale tree that is less frequently talked about, which Erv Wilson calls the "Rabbit Sequence" ([http://www.anaphoria.com/RabbitSequence.pdf Erv Wilson's original version], [https://mikebattagliamusic.com/MOSTree/MOSTreeab.html interactive version 1], [https://mikebattagliamusic.com/MOSTree/MOSTreeLs.html interactive version 2]). This is a tree for which each MOS word has two children, depending on if the MOS is "soft" (with L/s < 2) or "hard" (with L/s > 2). For instance, LsLss has the two children LLsLLLs and ssLsssL. Does a similar scale plane exist for these GO-MV3 scales? | ||
== Falsified conjectures == | == Falsified conjectures == | ||
=== All mosses have an MV3 detempering === | |||
LsLsLsLsLs does ''not'' have an MV3 detempering: Wolog we “detemper” L to L and M. | |||
LsMsLsMsLs is not MV3: LsM LsL sMs sLs | |||
LsMsLsMsMs and can be rotated to MsLsMsLsMs which is not MV3 by symmetry with LsMsLsMsLs. | |||
LsMsMsMsMs is not MV3: LsM MLM sMs sLs | |||
MsLsLsLsLs not MV3 for the same reason as LsMsMsMsMs | |||
[[Category:GO scales| ]]<!--Main article--> | [[Category:GO scales| ]]<!--Main article--> | ||
[[Category:Theory]] | [[Category:Theory]] | ||