User:Inthar/Endoparticular extensions

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Goal: Formalize Leri's notion of temp extension naturalness or a some notion that is stronger than Leri!naturalness.

Formal definition

Suppose the base temp tempers out a comma that is a product of powers of consecutive S-commas with nonzero exponents. An extension is natural if it additionally tempers out the individual S-commas.

Conjecture: A natural extension of a temperament tempering out one given comma to a given extended subgroup is unique if it exists (since there is at most one S-expression of the comma in a given subgroup).

Stronger conjecture: A natural extension for a given temperament of any rank and a given extended subgroup is unique if it exists (regardless of the choice of comma basis).

Examples

Porcupine

2.3.5 Porc -> 2.3.5.11 Porc is natural:

250/243
= (10/9)^3/(4/3)
= (10/9)^2(11/10)S(10)/(4/3)
= (10/9)(11/10)(12/11)/(4/3)*S(10)^2*S(11)
= S(10)^2*S(11)

And 2.3.5.11 Porc indeed tempers out S10 and S11.

Kleismic

2.3.5 Kleismic -> 2.3.5.13 Kleismic is natural because 15625/15552 = S(25)^2*S(26).