User:Inthar/Endoparticular extensions: Difference between revisions

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{{Problematic}}
{{Problematic}}


Goal: Formalize Leri's notion of temp extension naturalness or a some notion that is stronger than Leri!naturalness.
Suppose the base temp tempers out a comma that is a product of powers of consecutive S-commas with nonzero exponents. An extension is ''endoparticular'' if it additionally tempers out the individual S-commas.


== Formal definition ==
Conjecture: There is at most one S-expression for a comma in a given extended subgroup. Moreover, an endoparticular extension of a temperament tempering out one given comma to a given extended subgroup is unique if it exists.
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 if it exists.
Stronger conjecture: An endoparticular 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).


Stronger conjecture: A natural extension for a given temperament of any rank and a given extended subgroup, is unique if it exists.
== Conjecture: An endoparticular extension of a one-comma temperament is unique ==
=== Examples ===
Possible strategy: Since the definition of endoparticular requires consecutive square-particulars, maybe I can bound the number of consecutive S-commas that must appear in a factorization into consecutive S-commas
==== 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 ==

Latest revision as of 03:35, 7 February 2026

This is a problematic page or section. It lacks sufficient justification, content, or organization, and is subject to future overhaul or deletion.

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

Conjecture: There is at most one S-expression for a comma in a given extended subgroup. Moreover, an endoparticular extension of a temperament tempering out one given comma to a given extended subgroup is unique if it exists.

Stronger conjecture: An endoparticular 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).

Conjecture: An endoparticular extension of a one-comma temperament is unique

Possible strategy: Since the definition of endoparticular requires consecutive square-particulars, maybe I can bound the number of consecutive S-commas that must appear in a factorization into consecutive S-commas

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