Ternary scale theorems: Difference between revisions

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#* Yes. For an ''a'''''x'''''b'''''y''' MOS with gcd(''a'', ''b'') = 1, if one of ''a'' and ''b'' is even, detemper '''x''' resp. '''y''' into two step sizes. The result is a 1-period odd-regular MV3. If neither is even, assume ''a'' > ''b''. Then use {{nowrap|(''a'' - ''b'')'''x'''''b'''''y'''''b'''''z'''}}, which is a 1-period even-regular MV3 since {{nowrap|gcd(''a'' - ''b'', ''b'') {{=}} gcd(''a'', ''b'') {{=}} 1.}}

# 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 generator-offset-MV3 scales, where given some word representing a generator-offset-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 generator-offset-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.

#* Yes, there is a scale triangle for step size ratios (and for generator/offset tunings if applicable), where the degenerate cases corresponds to the boundary of the triangle. However, the theory of taking mediants in higher dimensions is nowhere nearly as nice as for continued fractions.

#* Yes, there is a scale triangle for step size ratios (and for generator/offset tunings if applicable), where the degenerate cases corresponds to the boundary of the triangle. However, the theory of taking mediants in higher dimensions is nowhere nearly as nice as for continued fractions, so I doubt there's an analogue for Brun's algorithm.

# In the theory of MOS, there is a second [[MOS Scale Family Tree|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 {{nowrap|L/s < 2}}) or "hard" (with {{nowrap|L/s > 2}}). For instance, LsLss has the two children LLsLLLs and ssLsssL. Does a similar scale plane exist for these generator-offset-MV3 scales?

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 #* Yes. For an ''a'''''x'''''b'''''y''' MOS with gcd(''a'', ''b'') = 1, if one of ''a'' and ''b'' is even, detemper '''x''' resp. '''y''' into two step sizes. The result is a 1-period odd-regular MV3. If neither is even, assume ''a'' > ''b''. Then use {{nowrap|(''a'' - ''b'')'''x'''''b'''''y'''''b'''''z'''}}, which is a 1-period even-regular MV3 since {{nowrap|gcd(''a'' - ''b'', ''b'') {{=}} gcd(''a'', ''b'') {{=}} 1.}}
 # 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 generator-offset-MV3 scales, where given some word representing a generator-offset-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 generator-offset-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.
-#* Yes, there is a scale triangle for step size ratios (and for generator/offset tunings if applicable), where the degenerate cases corresponds to the boundary of the triangle. However, the theory of taking mediants in higher dimensions is nowhere nearly as nice as for continued fractions.
+#* Yes, there is a scale triangle for step size ratios (and for generator/offset tunings if applicable), where the degenerate cases corresponds to the boundary of the triangle. However, the theory of taking mediants in higher dimensions is nowhere nearly as nice as for continued fractions, so I doubt there's an analogue for Brun's algorithm.
 # In the theory of MOS, there is a second [[MOS Scale Family Tree|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 {{nowrap|L/s &lt; 2}}) or "hard" (with {{nowrap|L/s &gt; 2}}). For instance, LsLss has the two children LLsLLLs and ssLsssL. Does a similar scale plane exist for these generator-offset-MV3 scales?