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| ====== PMOS implies AG (except in the case xyxzxyx) (WIP) ====== | | ====== PMOS implies AG (except in the case xyxzxyx) (WIP) ====== |
| We now prove that except in the case xyxzxyx, if the scale is pairwise MOS, then it is AG.
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| To eliminate xyxzxyx we manually check all words up to length 7... (todo)
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| Now assume len(S) >= 8.
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| PMOS -> Consider mos temperings
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| * in x, ξ (ξ = y or z), with gen g1 -> g1g1...g1g1' (g1' = imperfect gen)
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| * in y, η (η = x or z), with gen g2
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| * in z, ζ (ζ = x or y), with gen g3.
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| Denote their detemperings as G11, G12, G13, G21, G22, G23, G31, G32, G33.
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| ''To be continued...''
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| <!--
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| A gen chain g1...g1g1' [assuming this word is not multiperiod] detempers to G1i(1)G1i(2)...G1i(n-1)G13, where i(t) is in {1,2} and n = len(S). This word must be MV3, since otherwise the original scale wouldn't be MV3. Similarly, g2...g2g2' detempers to G2j(1)...G2j(n-1)G23, and g3...g3g3' detempers to G3k(1)...G3k(n-1)G33.-->
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| ====== AG implies "ax by bz" ====== | | ====== AG implies "ax by bz" ====== |