4000/3993 - Revision history

Lucius Chiaraviglio: Add another pair of commas that it separates

2026-06-28T01:42:52Z

Add another pair of commas that it separates

← Older revision		Revision as of 01:42, 28 June 2026
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	'''4000/3993''', the '''wizardharry comma''' or '''pine comma''' is an [[unnoticeable comma\|unnoticeable]] [[11-limit]] (specifically [[2.3.5.11 subgroup\|2.3.5.11-subgroup]]) [[comma]] with a size of roughly 3.03 [[cents]]. It is the amount by which a stack of three [[11/10]] submajor seconds falls short of the [[4/3]] perfect fourth; in other words it is equal to (4/3)/(11/10)<sup>3</sup>.		'''4000/3993''', the '''wizardharry comma''' or '''pine comma''' is an [[unnoticeable comma\|unnoticeable]] [[11-limit]] (specifically [[2.3.5.11 subgroup\|2.3.5.11-subgroup]]) [[comma]] with a size of roughly 3.03 [[cents]]. It is the amount by which a stack of three [[11/10]] submajor seconds falls short of the [[4/3]] perfect fourth; in other words it is equal to (4/3)/(11/10)<sup>3</sup>.

	In terms of commas, it is the difference between [[100/99]] ({{S\|10}}) and [[121/120]] ({{S\|11}}), itself having an [[S-expression]] of S10/S11. It is also the difference between [[540/539]] ([[S-expression\|S12/S14]]) and [[9801/9800]] ({{S\|99}}). It factors into 13-limit commas as ([[1575/1573]])⋅([[2080/2079]]) or ([[625/624]])⋅([[6656/6655]]).		In terms of commas, it is the difference between [[100/99]] ({{S\|10}}) and [[121/120]] ({{S\|11}}), itself having an [[S-expression]] of S10/S11. It is also the difference between [[540/539]] ([[S-expression\|S12/S14]]) and [[9801/9800]] ({{S\|99}}), and between [[1350/1331]] and [[81/80]] ({{S\|9}}). It factors into 13-limit commas as ([[1575/1573]])⋅([[2080/2079]]) or ([[625/624]])⋅([[6656/6655]]).

	== Temperaments ==		== Temperaments ==

FloraC: Undo. It's backwards

2026-06-25T07:44:29Z

Undo. It's backwards

← Older revision		Revision as of 07:44, 25 June 2026
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	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[~~Gravity_family#2.3.5.11_subgroup_(larry)\|~~larry]] in the [[~~Gravity\|~~gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.

	Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.		Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.

Lucius Chiaraviglio: /* Temperaments */ Fix links to Larry and Gravity

2026-06-25T03:30:31Z

Temperaments: Fix links to Larry and Gravity

← Older revision		Revision as of 03:30, 25 June 2026
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	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[Gravity_family#2.3.5.11_subgroup_(larry)\|larry]] in the [[Gravity\|gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.

	Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.		Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.

FloraC: More cleanup

2026-02-13T15:48:57Z

More cleanup

← Older revision		Revision as of 15:48, 13 February 2026
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	}}		}}

	'''4000/3993''', the '''wizardharry comma''' or '''pine comma''' is an [[unnoticeable comma\|unnoticeable]] [[11-limit]] [[comma]] with a size of roughly 3.03 [[cents]]. It is the amount by which a stack of three [[11/10]] submajor seconds falls short of the [[4/3]] perfect fourth~~, therefore~~ it is equal to ([[4/3]])/([[11/10]])<sup>3</sup> ~~= S10/S11~~.		'''4000/3993''', the '''wizardharry comma''' or '''pine comma''' is an [[unnoticeable comma\|unnoticeable]] [[11-limit]] (specifically [[2.3.5.11 subgroup\|2.3.5.11-subgroup]]) [[comma]] with a size of roughly 3.03 [[cents]]. It is the amount by which a stack of three [[11/10]] submajor seconds falls short of the [[4/3]] perfect fourth; in other words it is equal to (4/3)/(11/10)<sup>3</sup>.

	In terms of commas, it is the difference between [[100/99]] (~~S10~~) and [[121/120]] (S11~~) or~~ between [[540/539]] (S12/S14) and [[9801/9800]] (~~S99~~). It factors into 13-limit commas as ([[1575/1573]])([[2080/2079]]) or ([[625/624]])([[6656/6655]]).		In terms of commas, it is the difference between [[100/99]] ({{S\|10}}) and [[121/120]] ({{S\|11}}), itself having an [[S-expression]] of S10/S11. It is also the difference between [[540/539]] ([[S-expression\|S12/S14]]) and [[9801/9800]] ({{S\|99}}). It factors into 13-limit commas as ([[1575/1573]])⋅([[2080/2079]]) or ([[625/624]])⋅([[6656/6655]]).

	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes [[12/11]] and [[10/9]] equidistant from 11/10.		[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes [[12/11]] and [[10/9]] equidistant from 11/10.

	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making [[porcupine]] as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.

	Another strategy is to take advantage of the size of [[121/120]] (~~{{S\|11}}~~) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/([[11/9]]), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.		Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.

	Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to [[hades]].		Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to [[hades]].

FloraC: /* Temperaments */ hades doesn't "naturally" extend to the 17-limit

2026-02-13T15:40:53Z

Temperaments: hades doesn't "naturally" extend to the 17-limit

← Older revision		Revision as of 15:40, 13 February 2026
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	[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes [[12/11]] and [[10/9]] equidistant from 11/10.		[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes [[12/11]] and [[10/9]] equidistant from 11/10.

	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup porcupine equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making [[porcupine]] as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making [[porcupine]] as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.
Line 18:		Line 18:
	Another strategy is to take advantage of the size of [[121/120]] ({{S\|11}}) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/([[11/9]]), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.		Another strategy is to take advantage of the size of [[121/120]] ({{S\|11}}) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/([[11/9]]), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.

	Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to [[hades]]~~, which extends naturally to the 17-limit by tempering out [[289/288]] = ([[17/12]])/([[24/17]])~~.		Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to [[hades]].

	There is a massive wealth of possible high-accuracy temperaments that temper out this comma, including [[311edo]].		There is a massive wealth of possible high-accuracy temperaments that temper out this comma, including [[311edo]].

FloraC: /* Temperaments */ terminology & + links on S-expressions

2026-02-13T15:37:57Z

Temperaments: terminology & + links on S-expressions

← Older revision		Revision as of 15:37, 13 February 2026
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	[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes [[12/11]] and [[10/9]] equidistant from 11/10.		[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes [[12/11]] and [[10/9]] equidistant from 11/10.

	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11 ~~[[Porcupine family #2.3.5.11~~ subgroup ~~(porkypine)\|~~porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making [[porcupine]] as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering [[385/384]], which ~~[[Porcupine family #Septimal porcupine\|~~septimal porcupine]] also tempers.)		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup porcupine equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making [[porcupine]] as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma ~~S9/S10 =~~ [[8019/8000]] (so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] = ([[3/2]])/([[11/9]])<sup>2</sup> ~~= S9/S11~~ leading to [[~~Gravity family #Larry\|~~larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's [[11-limit]], making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.

	Another strategy is to take advantage of the size of ~~S11~~ so as to equate it with [[144/143]] = ([[16/13]])/([[11/9]]) ~~= S12~~, for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] = S10/S12, a comma with various advantages.		Another strategy is to take advantage of the size of [[121/120]] ({{S\|11}}) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/([[11/9]]), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.

	Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to [[hades]], which extends naturally to the 17-limit by tempering out [[289/288]] = ([[17/12]])/([[24/17]]).		Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to [[hades]], which extends naturally to the 17-limit by tempering out [[289/288]] = ([[17/12]])/([[24/17]]).

Squib: simplify details about porcupine (those belong on the porcupine page) & add extension to prime 7

2026-02-12T15:57:02Z

simplify details about porcupine (those belong on the porcupine page) & add extension to prime 7

← Older revision		Revision as of 15:57, 12 February 2026
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	== Temperaments ==		== Temperaments ==
	[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes 12/11 and 10/9 equidistant from 11/10.		[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes [[12/11]] and [[10/9]] equidistant from 11/10.

	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three [[10/9]]'s instead ~~and it is for these reasons that~~ 2.3.5.11 [[Porcupine family #2.3.5.11 subgroup (porkypine)\|porcupine]] equates 11/10 ~~with the [[superparticular]]s adjacent to it of~~ 12/11 ~~and 10/9, tempering out S10 and S11 as the porcupine comma is S10<sup>2</sup> × S11~~ so ~~through the extension to~~ the 11~~-limit we make~~ [[porcupine]] as efficient and elegant as it can reasonably be.		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11 [[Porcupine family #2.3.5.11 subgroup (porkypine)\|porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making [[porcupine]] as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering [[385/384]], which [[Porcupine family #Septimal porcupine\|septimal porcupine]] also tempers.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma S9/S10 = [[8019/8000]] (so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] = ([[3/2]])/([[11/9]])<sup>2</sup> = S9/S11 leading to [[Gravity family #Larry\|larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's [[11-limit]], making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma S9/S10 = [[8019/8000]] (so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] = ([[3/2]])/([[11/9]])<sup>2</sup> = S9/S11 leading to [[Gravity family #Larry\|larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's [[11-limit]], making it an excellent way to extend schismic to include prime 11.

Xenllium at 14:47, 31 January 2026

2026-01-31T14:47:23Z

← Older revision		Revision as of 14:47, 31 January 2026
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	{{Infobox ~~Interval~~		{{Infobox interval
	\| Name = wizardharry comma, pine comma		\| Name = wizardharry comma, pine comma
	\| Color name = 1u<sup>3</sup>y<sup>3</sup>1, triluyo unison,<br>Triluyo comma		\| Color name = 1u<sup>3</sup>y<sup>3</sup>1, triluyo unison,<br>Triluyo comma
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	''Pine comma'' was given by [[User:Godtone\|Godtone]] in 2023, for its relevance and importance to the pine ([[7L 1s]]) scale.		''Pine comma'' was given by [[User:Godtone\|Godtone]] in 2023, for its relevance and importance to the pine ([[7L 1s]]) scale.

			== See also ==
			* [[Wizardharry clan]]

	== Notes ==		== Notes ==

FloraC: Hades is unrelated to tempering out 325/324

2025-06-14T13:01:17Z

Hades is unrelated to tempering out 325/324

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	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma S9/S10 = [[8019/8000]] (so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] = ([[3/2]])/([[11/9]])<sup>2</sup> = S9/S11 leading to [[Gravity family #Larry\|larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's [[11-limit]], making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma S9/S10 = [[8019/8000]] (so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] = ([[3/2]])/([[11/9]])<sup>2</sup> = S9/S11 leading to [[Gravity family #Larry\|larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's [[11-limit]], making it an excellent way to extend schismic to include prime 11.

	Another strategy is to take advantage of the size of S11 so as to equate it with [[144/143]] = ([[16/13]])/([[11/9]]) = S12, for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] = S10/S12, a comma with various advantages. Observing that 4000/3993 = (540/539)/(9801/9800) ~~then~~ shows a natural path, if one is willing to split the octave in half, leading to [[hades]], which extends naturally to the 17-limit by tempering out [[289/288]] = ([[17/12]])/([[24/17]]).		Another strategy is to take advantage of the size of S11 so as to equate it with [[144/143]] = ([[16/13]])/([[11/9]]) = S12, for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] = S10/S12, a comma with various advantages.

			Observing that 4000/3993 = (540/539)/(9801/9800) also shows a natural path, if one is willing to split the octave in half, leading to [[hades]], which extends naturally to the 17-limit by tempering out [[289/288]] = ([[17/12]])/([[24/17]]).

	There is a massive wealth of possible high-accuracy temperaments that temper out this comma, including [[311edo]].		There is a massive wealth of possible high-accuracy temperaments that temper out this comma, including [[311edo]].

FloraC: Cleanup

2025-06-14T13:00:47Z

Cleanup

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	}}		}}

	'''4000/3993''', the '''wizardharry comma''' or '''pine comma''' is an [[unnoticeable comma\|unnoticeable]] [[11-limit]] [[comma]] with a size of roughly 3.03 [[cents]]. It is the amount by which a stack of three [[11/10]] submajor seconds falls short of the [[4/3]] perfect fourth, therefore it is equal to ([[4/3~~\|12/9~~]])/([[11/10]])<sup>3</sup> = S10/S11.		'''4000/3993''', the '''wizardharry comma''' or '''pine comma''' is an [[unnoticeable comma\|unnoticeable]] [[11-limit]] [[comma]] with a size of roughly 3.03 [[cents]]. It is the amount by which a stack of three [[11/10]] submajor seconds falls short of the [[4/3]] perfect fourth, therefore it is equal to ([[4/3]])/([[11/10]])<sup>3</sup> = S10/S11.

	In terms of commas, it is ~~trivially~~ the difference between ~~S10 =~~ [[100/99]] and ~~S11 =~~ [[121/120]] or ~~less trivially~~ between ~~S12/S14 =~~ [[540/539]] and ~~S99 =~~ [[9801/9800]]. It factors into 13-limit commas as ([[1575/1573]])([[2080/2079]]) or ([[625/624]])([[6656/6655]]).		In terms of commas, it is the difference between [[100/99]] (S10) and [[121/120]] (S11) or between [[540/539]] (S12/S14) and [[9801/9800]] (S99). It factors into 13-limit commas as ([[1575/1573]])([[2080/2079]]) or ([[625/624]])([[6656/6655]]).

	== Temperaments ==		== Temperaments ==
	Tempering it out means the fourth is divided into an equal stepped tetrachord, the step of which is a "trienfourth" (from "1/3 of a fourth") or "submajor second" and because it is an [[ultraparticular]] it makes 12/11 and 10/9 equidistant from 11/10.		[[Tempering out]] this comma means the fourth is divided into an equal stepped tetrachord, the step of which is a ''trienfourth'' (from 1/3 of a fourth) or ''submajor second'' and because it is an [[ultraparticular]] it makes 12/11 and 10/9 equidistant from 11/10.

	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three [[10/9]]'s instead and it is for these reasons that 2.3.5.11 [[~~Porcupine_family~~#2.3.5.~~11_subgroup_.28porkypine.29~~\|porcupine]] equates 11/10 with the [[superparticular]]s adjacent to it of 12/11 and 10/9, tempering S10 and S11 as the porcupine comma is ~~[[250/243]] =~~ S10<sup>2</sup> * S11 so through the extension to 11 we make [[porcupine]] as efficient and elegant as it can reasonably be.		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three [[10/9]]'s instead and it is for these reasons that 2.3.5.11 [[Porcupine family #2.3.5.11 subgroup (porkypine)\|porcupine]] equates 11/10 with the [[superparticular]]s adjacent to it of 12/11 and 10/9, tempering out S10 and S11 as the porcupine comma is S10<sup>2</sup> × S11 so through the extension to the 11-limit we make [[porcupine]] as efficient and elegant as it can reasonably be.

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma S9/S10 = [[8019/8000]] (so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] = ([[3/2]])/([[11/9]])<sup>2</sup> = S9/S11 leading to [[Gravity family #Larry\|larry]] in the [[gravity family]]. Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's [[11-limit]], making it an excellent way to extend schismic to include prime 11.

	~~Tempering~~ it ~~out~~ with ~~the trimitone comma S9~~/~~S10~~ = [[~~8019~~/~~8000~~]] (~~so that three~~ [[10/9]]~~'s are also an~~ [[11/8]]) ~~allows us~~ to ~~also temper~~ the [[~~semiparticular~~]] [[~~243~~/~~242~~]] = ([[3/2\|12/8]])/([[11/9]])~~<sup>2</sup> = S9/S11 leading to [[Gravity_family#Larry\|Larry]] in the [[Gravity family]]~~.		Another strategy is to take advantage of the size of S11 so as to equate it with [[144/143]] = ([[16/13]])/([[11/9]]) = S12, for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] = S10/S12, a comma with various advantages. Observing that 4000/3993 = (540/539)/(9801/9800) then shows a natural path, if one is willing to split the octave in half, leading to [[hades]], which extends naturally to the 17-limit by tempering out [[289/288]] = ([[17/12]])/([[24/17]]).

	~~Tempering it out with both the [[schisma]] and [[trimitone comma]] gives a description of [[65edo]] in the no-7's [[11-limit]], making it an excellent way to extend schismic to include prime 11.~~		There is a massive wealth of possible high-accuracy temperaments that temper out this comma, including [[311edo]].

	Another strategy is to take advantage of the size of [[121/120\|S11]] so as to equate it with [[144/143\|S12 = 144/143]] = ([[16/13]])/([[11/9]]), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering [[325/324\|S10/S12 = S25*S26 = 325/324]], a comma with various advantages. Observing that [[4000/3993\|S10/S11]] = ([[540/539\|S12/S14]])/([[9801/9800\|S33/S35 = S99]]) then shows a natural path, if one is willing to split the octave in half, leading to [[hades]], which extends naturally to the 17-limit by tempering [[289/288\|S17]] = ([[17/12]])/([[24/17]]).

	~~These are just a few examples, but there~~ is a massive wealth of possible high-accuracy temperaments that temper out this comma, including ~~the [[41-limit]] temperament~~ [[311edo]].

	== Etymology ==		== Etymology ==

← Older revision		Revision as of 07:44, 25 June 2026
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	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[~~Gravity_family#2.3.5.11_subgroup_(larry)\|~~larry]] in the [[~~Gravity\|~~gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.

	Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.		Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.

← Older revision		Revision as of 03:30, 25 June 2026
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	On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)		On the low-accuracy end, this may be reminiscent of how the [[250/243\|porcupine comma]] splits 4/3 into three 10/9's instead. In fact, 2.3.5.11-subgroup [[porcupine]] equates 10/9 with 11/10 and 12/11 so that the equally-split fourth represents 9:10:11:12, making porcupine as efficient and elegant as it can reasonably be. (As a [[rank-3]] detemperament of porcupine, one way to extend it to include prime 7 is by tempering out [[385/384]], which septimal porcupine also tempers out.)

	Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[larry]] in the [[gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.		Tempering it out along with the [[schisma]] results in the rank-2 [[tertiaschis]] temperament. Tempering it out with the trimitone comma [[8019/8000]] ([[S-expression\|S9/S10]], so that three [[10/9]]'s are also an [[11/8]]) implies also tempering out the [[semiparticular]] [[243/242]] ([[S-expression\|S9/S11]]) = ([[3/2]])/([[11/9]])<sup>2</sup> leading to [[Gravity_family#2.3.5.11_subgroup_(larry)\|larry]] in the [[Gravity\|gravity family]]. Tempering it out with both the schisma and trimitone comma gives a description of [[65edo]] in the no-7's 11-limit, making it an excellent way to extend schismic to include prime 11.

	Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.		Another strategy is to take advantage of the size of 121/120 (S11) so as to equate it with [[144/143]] ({{S\|12}}) = ([[16/13]])/(11/9), for those seeking to keep the undecimal and tridecimal neutral thirds distinct, thus tempering out the marveltwin comma, [[325/324]] ([[S-expression\|S10/S12]]), a comma with various advantages.