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{{Problematic}}

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).

== Conjecture: A natural extension is unique ==