4000/3993: Difference between revisions

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{{Infobox ~~Interval~~

{{Infobox interval

| Name = wizardharry comma, pine comma

| Color name = 1u3y31, triluyo unison, Triluyo comma

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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]])3 ~~= 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)3.

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 ==

[[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 S102 × 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-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]])2 ~~= 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]])2 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 (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]]~~, 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]].

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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.

== See also ==

* [[Wizardharry clan]]

== Notes ==

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-{{Infobox Interval
+{{Infobox interval
 | Name = wizardharry comma, pine comma
 | Color name = 1u<sup>3</sup>y<sup>3</sup>1, triluyo unison,<br>Triluyo comma
@@ Line 5: / Line 5: @@
 }}
-'''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 ==
-[[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-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 (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]], 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]].
@@ Line 26: / Line 26: @@
 ''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 ==