User:Inthar/Endoparticular extensions: Difference between revisions
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Goal: Formalize Leri's notion of temp extension naturalness or a some notion that is stronger than Leri!naturalness. | 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. | 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 subgroup tempers out one given comma to a given extended subgroup is unique. | |||
Stronger conjecture: A natural extension for a given temperament of any rank and a given extended subgroup, is unique if it exists. | |||
=== Examples === | === Examples === | ||
==== Porcupine ==== | ==== Porcupine ==== | ||
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==== Kleismic ==== | ==== Kleismic ==== | ||
2.3.5 Kleismic -> 2.3.5.13 Kleismic is natural because 15625/15552 = S(25)^2*S(26). | 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 == | |||
Revision as of 15:36, 4 February 2026
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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 subgroup tempers out one given comma to a given extended subgroup is unique.
Stronger conjecture: A natural extension for a given temperament of any rank and a given extended subgroup, is unique if it exists.
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).
