Ternary scale theorems: Difference between revisions

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# Classify all twisted SV3 scales, thereby completing the classification of all abstractly SV3 scales.

# Conjecture: If a twisted MV3 is not SV3, then it is constructed from ''ka'''''X'''''kb'''''Z''' where ''k'' is composite.

=== Conjecture ("MV3 Sequences") ===

Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the '''MV3 sequence''' for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.

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# Given any arbitrary MOS scale with at least three notes per period, is there *always* a MV3 generator-offset 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 generator-offset scale LmsLmLs as a detempering. Does a similar MV3 detempering exist for every possible DE scale with at least three notes per period, or at least for strict MOS's with one period per octave (e.g. well-formed scales)?

# 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.

# 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 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 generator-offset-MV3 scales?

# 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?

[[Category:Math]]

@@ Line 325: / Line 325: @@
 # Classify all twisted SV3 scales, thereby completing the classification of all abstractly SV3 scales.
 # Conjecture: If a twisted MV3 is not SV3, then it is constructed from ''ka'''''X'''''kb'''''Z''' where ''k'' is composite.
 === Conjecture ("MV3 Sequences") ===
 Given any two generators, we can iterate them to any number of notes and see what the maximum-variety of the resulting scale is. In particular, we can look at those scale sizes which are MV3, and thus compute the '''MV3 sequence''' for the pair of generators (similar to the "MOS sequence" one can compute for one generator). Thus, for any pair of generators, we can form the associated sequence of increasingly large MV3 scales.
@@ Line 337: / Line 338: @@
 # Given any arbitrary MOS scale with at least three notes per period, is there *always* a MV3 generator-offset 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 generator-offset scale LmsLmLs as a detempering. Does a similar MV3 detempering exist for every possible DE scale with at least three notes per period, or at least for strict MOS's with one period per octave (e.g. well-formed scales)?
 # 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.
-# 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 L/s &lt; 2) or "hard" (with 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?
+# 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?
 [[Category:Math]]