User:Inthar/MV3: Difference between revisions
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=== Proof of the alternating-gens (AG) property (except in the case xyxzxyx) === | === Proof of the alternating-gens (AG) property (except in the case xyxzxyx) === | ||
We now prove that except in the case xyxzxyx, if the scale is pairwise well-formed, the scale is generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1'' (AG) Equivalently, the scale can be built by stacking alternating generators. | We now prove that except in the case xyxzxyx, if the scale is pairwise well-formed, the scale is generated by two chains of generators, either both of size ''m'' or one with size ''m'' and one of size ''m-1'' (AG) Equivalently, the scale can be built by stacking alternating generators, either g1 g2 ... g1 g2 g1 g3 or g1 g2 ... g1 g2 g3. | ||
To eliminate xyxzxyx we manually check all words up to length 7... (todo) | To eliminate xyxzxyx we manually check all words up to length 7... (todo) | ||
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* in y, η (η = x or z), with gen g2 | * in y, η (η = x or z), with gen g2 | ||
* in z, ζ (ζ = x or y), with gen g3. | * in z, ζ (ζ = x or y), with gen g3. | ||
Denote their detemperings as G11, G12, G13, G21, G22, G23, G31, G32, G33. | |||
Suppose no gen chain g_i detempers to a chain of alternating generators. | Suppose no gen chain g_i detempers to a chain of alternating generators. | ||