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Recursive structure of MOS scales: Difference between revisions

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  w3:  [... sXL ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)
  w3:  [... sXL ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)


(both 1.3 and 1.4 imply w₃ has one more s).
(both 1.3 and 1.4 imply w₃ has one more s than w₂).


Case 2
Case 2
Line 455: Line 455:


Case 2.1:
Case 2.1:
  w2: <L ... sXL ... sXL ... sXL ... sXL ... s> (complete chunks only)
  w₂: <L ... sXL ... sXL ... sXL ... sXL ... s> (complete chunks only)
  w3:              [sXL ... sXL ... sXL ... sXL ... s> (s followed by complete chunks)
  w₃:              [sXL ... sXL ... sXL ... sXL ... s> (s followed by complete chunks)


Truncate the strings w1, w2, w3 as follows, to get three distinct K-steps in w:
Truncate the strings w₁, w₂, w₃ as follows, to get three distinct K-steps in w:
  w1':  L ... [lop off the s at the beginning of w1, so w1' has k s's]
  w₁':  L ... [lop off the s at the beginning of w₁, so w₁' has k s's]
  w2': <L ... sXL ... sXL ... sXL ... ] [lop off one s at the end of w2, so w2' has one fewer s than w3']
  w₂': <L ... sXL ... sXL ... sXL ... ] [lop off one s at the end of w₂, so w₂' has one fewer s than w₃']
  w3':        <L ... sXL ... sXL ... sXL ... s> [lop off an L at the beginning of w3, so w3' has at most k-1 s's]
  w₃':        <L ... sXL ... sXL ... sXL ... s> [lop off an L at the beginning of w₃, so w₃' has at most k-1 s's]
so this contradicts our original scale being a mos.
so this contradicts our original scale being a mos.


Case 2.2:
Case 2.2:
  w2: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)
  w₂: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)
  w3:          [ ... sXL ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)
  w₃:          [ ... sXL ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)


Case 2.3:
Case 2.3:
  w2: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)
  w₂: <L ... sXL ... sXL ... sXL ... sXL ... sXL ...] (complete chunks followed by one or more L's)
  w3:                <L ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)
  w₃:                <L ... sXL ... sXL ... sXL ... s> (one or more L's followed by an s, followed by complete chunks)


Both 2.2 and 2.3 are contradictions: The number of letters added to the last incomplete chunk to complete it is at least as long as λ. But since the first chunk is λ, it has to be at least as big as the last chunk.
Both 2.2 and 2.3 are contradictions: The number of letters added to the last incomplete chunk to complete it is at least as long as λ. But since the first chunk is λ, it has to be at least as big as the last chunk.
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