6656/6561 - Revision history

Overthink: /* Temperaments */ remove extraneous links

2026-03-17T19:18:49Z

Temperaments: remove extraneous links

← Older revision		Revision as of 19:18, 17 March 2026
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	=== Tetris ===		=== Tetris ===
	If tempered out only on the 2.3.13 subgroup, you get tetris (while tetric would fit the -ic convention better, it is already used for a MOS pattern). [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good [[~]][[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is [[46edo]] since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related leapfrog temperament; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.		If tempered out only on the 2.3.13 subgroup, you get tetris (while tetric would fit the -ic convention better, it is already used for a MOS pattern). [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good ~[[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is [[46edo]] since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related leapfrog temperament; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.

	[[Subgroup]]: 2.3.13		[[Subgroup]]: 2.3.13
Line 26:		Line 26:

	=== Leapfrog ===		=== Leapfrog ===
	As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering out [[896/891]]) and [[~]][[13/11]] as the minor third (tempering out [[352/351]]) as well as tempering out the tetris comma, [[6656/6561]]. We may also find [[~]][[23/16]] as the tritone, tempering out [[736/729]]. This results in a temperament called leapfrog, which is the no-5's version of [[leapday]].		As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding ~[[14/11]] as the major third (tempering out [[896/891]]) and ~[[13/11]] as the minor third (tempering out [[352/351]]) as well as tempering out the tetris comma, [[6656/6561]]. We may also find ~[[23/16]] as the tritone, tempering out [[736/729]]. This results in a temperament called leapfrog, which is the no-5's version of [[leapday]].

	== Etymology ==		== Etymology ==

FloraC: Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

2025-06-06T16:39:11Z

Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

← Older revision		Revision as of 16:39, 6 June 2025
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	{{Optimal ET sequence\|legend=1\| 5, 12, 17, 63, 80, 97, 114, 131, 245b }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 63, 80, 97, 114, 131, 245b }}

	[[Badness]] (~~Dirichlet~~): 0.522		[[Badness]] (Sintel): 0.522

	=== Leapfrog ===		=== Leapfrog ===

BudjarnLambeth at 23:06, 15 November 2024

2024-11-15T23:06:04Z

← Older revision		Revision as of 23:06, 15 November 2024
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	== Etymology ==		== Etymology ==
	This comma was named by [[User:Jerdle\|Jerdle]] and [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".		This comma was named by [[User:Jerdle\|Jerdle]] and [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".

			[[Category:Commas named by combining multiple temperament names]]

Jerdle: Added two comma relations.

2024-10-28T19:10:01Z

Added two comma relations.

← Older revision		Revision as of 19:10, 28 October 2024
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	}}		}}
	The '''6656/6561''', the '''tetris comma''', is a [[small comma\|small]] [[13-limit]] [[comma]]. It is the amount by which [[13/8]] exceeds ([[9/8]])<sup>4</sup>, that is, the [[tetratone]].		The '''6656/6561''', the '''tetris comma''', is a [[small comma\|small]] [[13-limit]] [[comma]]. It is the amount by which [[13/8]] exceeds ([[9/8]])<sup>4</sup>, that is, the [[tetratone]].

			It is a [[schisma]] flat of [[65/64]], and a [[Pythagorean comma]] flat of [[1053/1024]].

	== Temperaments ==		== Temperaments ==

FloraC: Correction (tempering -> tempering out). -links in the section title and see also (already properly linked in the text)

2024-10-23T08:57:59Z

Correction (tempering -> tempering out). -links in the section title and see also (already properly linked in the text)

← Older revision		Revision as of 08:57, 23 October 2024
Line 9:		Line 9:

	=== Tetris ===		=== Tetris ===
	If tempered out only on the 2.3.13 subgroup, you get tetris (while tetric would fit the -ic convention better, it is already used for a MOS pattern). [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good [[~]][[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is [[46edo]] since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related [[leapfrog]] temperament; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.		If tempered out only on the 2.3.13 subgroup, you get tetris (while tetric would fit the -ic convention better, it is already used for a MOS pattern). [[17edo]] is a tone-efficient tetris tuning, though it is significantly sharper than ideal, as ideally you want [[13/8]] to be tuned flat so that the fifths need not be sharpened more than actually necessary for the equivalence. Nonetheless, [[34edo]] may be of interest for extending the subgroup so as to find more 13-limit harmonies than present in 17edo, though 17edo does reasonably well enough with the 2.3.13 subgroup alone, as it has an accurate [[~]][[13/9]] and still good [[~]][[13/12]]. By contrast, [[29edo]] is close to the just-3's tuning, still tempering the fifth in the right direction (as contrasted to 12edo) but with virtually all the error on 13 at 13{{cent}} flat. Therefore through the addition of [[val]]s we can deduce that the smallest reasonably optimized tuning is [[46edo]] since 17 + 29 = 46, which we can verify has a sharp 3 and a flat 13, so fits our basic requirements, though interestingly this does not appear in the optimal ET sequence here. Notably tetris prefers sharper tunings of the fifth than the related leapfrog temperament; this corresponds to having larger edos in the [[optimal ET sequence]]. Perhaps more amazingly is that adding all primes except 5 through parapythic results in a temperament with even lower badness than the pure 2.3.13 version.

	[[Subgroup]]: 2.3.13		[[Subgroup]]: 2.3.13
Line 21:		Line 21:
	{{Optimal ET sequence\|legend=1\| 5, 12, 17, 63, 80, 97, 114, 131, 245b }}		{{Optimal ET sequence\|legend=1\| 5, 12, 17, 63, 80, 97, 114, 131, 245b }}

	Badness (Dirichlet): 0.522		[[Badness]] (Dirichlet): 0.522

	=== [[Leapfrog~~]] (no-5's leapday)~~ ===		=== Leapfrog ===
	As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering [[896/891]]) and [[~]][[13/11]] as the minor third (tempering [[352/351]]) as well as tempering the tetris comma, [[6656/6561]]. (We may also find [[~]][[23/16]] as the tritone, tempering [[736/729]]) This results in a temperament called [[leapfrog]], which is the no-5's version of [[leapday]].		As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering out [[896/891]]) and [[~]][[13/11]] as the minor third (tempering out [[352/351]]) as well as tempering out the tetris comma, [[6656/6561]]. We may also find [[~]][[23/16]] as the tritone, tempering out [[736/729]]. This results in a temperament called leapfrog, which is the no-5's version of [[leapday]].

	~~== See also ==~~
	* [[Leapfrog]]
	* [[Leapday]]

	== Etymology ==		== Etymology ==
	This comma was named by [[User:Jerdle\|Jerdle]] and [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".		This comma was named by [[User:Jerdle\|Jerdle]] and [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".

Jerdle: /* Leapfrog (no-5's leapday) */Leapfrog also tempers out 736/729.

2024-10-23T08:39:25Z

Leapfrog (no-5's leapday): Leapfrog also tempers out 736/729.

← Older revision		Revision as of 08:39, 23 October 2024
Line 24:		Line 24:

	=== [[Leapfrog]] (no-5's leapday) ===		=== [[Leapfrog]] (no-5's leapday) ===
	As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering [[896/891]]) and [[~]][[13/11]] as the minor third (tempering [[352/351]]) as well as tempering the tetris comma, [[6656/6561]]. (We may also find [[~]][[23/16]] as the tritone.) This results in a temperament called [[leapfrog]], which is the no-5's version of [[leapday]].		As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering [[896/891]]) and [[~]][[13/11]] as the minor third (tempering [[352/351]]) as well as tempering the tetris comma, [[6656/6561]]. (We may also find [[~]][[23/16]] as the tritone, tempering [[736/729]]) This results in a temperament called [[leapfrog]], which is the no-5's version of [[leapday]].

	== See also ==		== See also ==

Godtone: /* Leapfrog (no-5's leapday) */

2024-10-22T19:57:28Z

Leapfrog (no-5's leapday)

← Older revision		Revision as of 19:57, 22 October 2024
Line 24:		Line 24:

	=== [[Leapfrog]] (no-5's leapday) ===		=== [[Leapfrog]] (no-5's leapday) ===
	As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering [[896/891]]) and [[~]][[13/11]] as the minor third (tempering [[352/351]]) as well as tempering the tetris comma, [[6656/6561]]. This results in a temperament called [[leapfrog]], which is the no-5's version of [[leapday]].		As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering [[896/891]]) and [[~]][[13/11]] as the minor third (tempering [[352/351]]) as well as tempering the tetris comma, [[6656/6561]]. (We may also find [[~]][[23/16]] as the tritone.) This results in a temperament called [[leapfrog]], which is the no-5's version of [[leapday]].

	== See also ==		== See also ==

Godtone: /* Temperaments */ link to leapfrog and note leapfrog and leapday in see also

2024-10-22T19:56:47Z

Temperaments: link to leapfrog and note leapfrog and leapday in see also

← Older revision		Revision as of 19:56, 22 October 2024
Line 22:		Line 22:

	Badness (Dirichlet): 0.522		Badness (Dirichlet): 0.522

			=== [[Leapfrog]] (no-5's leapday) ===
			As the fifth is tuned sharp in tetris, we can find an efficient extension to the no-5's 13-limit by finding [[~]][[14/11]] as the major third (tempering [[896/891]]) and [[~]][[13/11]] as the minor third (tempering [[352/351]]) as well as tempering the tetris comma, [[6656/6561]]. This results in a temperament called [[leapfrog]], which is the no-5's version of [[leapday]].

			== See also ==
			* [[Leapfrog]]
			* [[Leapday]]

	== Etymology ==		== Etymology ==
	This comma was named by [[User:Jerdle\|Jerdle]] and [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".		This comma was named by [[User:Jerdle\|Jerdle]] and [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".

Jerdle: /* Etymology */Gave myself credit, as I actually came up with the name in the Discord, but kept Godtone's credit for coming up with the connection in a discussion about the comma.

2024-10-22T10:51:13Z

Etymology: Gave myself credit, as I actually came up with the name in the Discord, but kept Godtone's credit for coming up with the connection in a discussion about the comma.

← Older revision		Revision as of 10:51, 22 October 2024
Line 24:		Line 24:

	== Etymology ==		== Etymology ==
	This comma was named by [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".		This comma was named by [[User:Jerdle\|Jerdle]] and [[User:Godtone\|Godtone]] in 2024 as a contraction of "tetratone" and "tridecimal".

Jerdle: /* Temperaments */Fixed a link and used the name leapfrog.

2024-10-22T10:48:55Z

Temperaments: Fixed a link and used the name leapfrog.

← Older revision		Revision as of 10:48, 22 October 2024
Line 6:		Line 6:

	== Temperaments ==		== Temperaments ==
	When [[tempering out\|tempered out]], it implies a sharp fifth or a doubly as sharp [[9/8\|tone]], and it features as the important comma that reduces rank-3 [[parapythic]] to [[~~#No-5's leapday\|~~no-~~5's~~ leapday]], which is notable as having much lower [[badness]]~~, as discussed there on this page~~.		When [[tempering out\|tempered out]], it implies a sharp fifth or a doubly as sharp [[9/8\|tone]], and it features as the important comma that reduces rank-3 [[parapythic]] to [[leapfrog]] (aka no-fives leapday), which is notable as having much lower [[badness]].

	=== Tetris ===		=== Tetris ===