User:Inthar/Endoparticular extensions - Revision history

Inthar at 03:35, 7 February 2026

2026-02-07T03:35:04Z

← Older revision		Revision as of 03:35, 7 February 2026
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	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.		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. ~~As a corollary~~, an endoparticular extension of a temperament tempering out one given comma to a given extended subgroup is unique if it exists.		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).		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).

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

2026-02-07T03:01:08Z

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

← Older revision		Revision as of 03:01, 7 February 2026
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	== Conjecture: An endoparticular extension of a one-comma temperament is unique ==		== Conjecture: An endoparticular extension of a one-comma temperament is unique ==
	Possible strategy: Since the definition of endoparticular requires consecutive square-particulars, ~~I do have that constraint fortunately and~~ I can~~, say, lower-~~bound the number of consecutive S-commas that must appear in a factorization into consecutive S-commas		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

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

2026-02-07T03:00:30Z

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

← Older revision		Revision as of 03:00, 7 February 2026
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	== Conjecture: An endoparticular extension of a one-comma temperament is unique ==		== Conjecture: An endoparticular extension of a one-comma temperament is unique ==
	Possible strategy: Since the ~~definiiton~~ of endoparticular requires consecutive square-particulars, I do have that constraint fortunately and I can, say, lower-bound the number of consecutive S-commas that must appear in a factorization into consecutive S-commas		Possible strategy: Since the definition of endoparticular requires consecutive square-particulars, I do have that constraint fortunately and I can, say, lower-bound the number of consecutive S-commas that must appear in a factorization into consecutive S-commas

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

2026-02-07T02:59:06Z

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

← Older revision		Revision as of 02:59, 7 February 2026
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	== Conjecture: An endoparticular extension of a one-comma temperament is unique ==		== Conjecture: An endoparticular extension of a one-comma temperament is unique ==
			Possible strategy: Since the definiiton of endoparticular requires consecutive square-particulars, I do have that constraint fortunately and I can, say, lower-bound the number of consecutive S-commas that must appear in a factorization into consecutive S-commas

Inthar: Inthar moved page User:Inthar/Endo- and paraparticular extensions to User:Inthar/Endoparticular extensions without leaving a redirect

2026-02-04T18:56:30Z

Inthar moved page User:Inthar/Endo- and paraparticular extensions to User:Inthar/Endoparticular extensions without leaving a redirect

← Older revision	Revision as of 18:56, 4 February 2026
(No difference)

Inthar at 18:54, 4 February 2026

2026-02-04T18:54:40Z

← Older revision		Revision as of 18:54, 4 February 2026
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	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.		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.		Conjecture: There is at most one S-expression for a comma in a given extended subgroup. As a corollary, 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).		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 ==		== Conjecture: An endoparticular extension of a one-comma temperament is unique ==

Inthar at 18:53, 4 February 2026

2026-02-04T18:53:52Z

← Older revision		Revision as of 18:53, 4 February 2026
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	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.		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: An endoparticular 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)~~.		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).		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 ==		== Conjecture: An endoparticular extension of a one-comma temperament is unique ==

Inthar: /* Conjecture: An endoparticular extension of a codimension-1 temperament is unique */

2026-02-04T18:52:57Z

Conjecture: An endoparticular extension of a codimension-1 temperament is unique

← Older revision		Revision as of 18:52, 4 February 2026
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	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: 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 ~~codimension~~-1 temperament is unique ==		== Conjecture: An endoparticular extension of a one-comma temperament is unique ==

Inthar at 18:52, 4 February 2026

2026-02-04T18:52:45Z

← Older revision		Revision as of 18:52, 4 February 2026
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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: An endoparticular 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).
	~~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: 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 (regardless of the choice of comma basis).~~		== Conjecture: An endoparticular extension of a codimension-1 temperament is unique ==
	~~=== 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)^2S(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 ==

Inthar: Inthar moved page User:Inthar/Formalizing naturalness to User:Inthar/Endo- and paraparticular extensions without leaving a redirect

2026-02-04T18:20:02Z

Inthar moved page User:Inthar/Formalizing naturalness to User:Inthar/Endo- and paraparticular extensions without leaving a redirect

← Older revision	Revision as of 18:20, 4 February 2026
(No difference)