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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 ==
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