Sinarabian comma: Difference between revisions

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The '''Sinarabian comma''' is an [[13-limit]] (also 2.3.11.13 subgroup) [[unnoticeable comma]] with a ratio of '''85293/85184''' and a value of approximately 2 cents.

The '''Sinarabian comma''' is an [[13-limit]] (also 2.3.11.13 subgroup) [[unnoticeable comma]] with a ratio of '''85293/85184''' and a value of approximately 2.2 cents.

This comma is identifiable as the amount by which a stack of two [[88/81]] artoneutral seconds falls short of a [[13/11]] Neo-Gothic minor third, and also the amount by which a stack of three [[11/9]] artoneutral thirds fall short of [[117/64]], and even the comma that separates [[1053/1024]] and [[1331/1296]].

This comma is identifiable as the amount by which a stack of two [[88/81]] artoneutral seconds falls short of a [[13/11]] Neo-Gothic minor third, and also the amount by which a stack of three [[11/9]] artoneutral thirds fall short of [[117/64]], the amount by which a stack of three [[27/22]] tendoneutral thirds exceed [[24/13]], and even the comma that separates [[1053/1024]] and [[1331/1296]].

In terms of comma relations, it separates [[243/242]] from [[352/351]], separates [[144/143]] from a stack of two instances of 243/242, and, is the sum of the [[frameshift comma]] and the [[tridecapyth comma]].

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The name of this comma comes from a portmanteau of "Ibn Sina" and "Alpharabian", and was given by [[Aura]] on the basis of a series of connections between intervals that he found and shared with [[Margo Schulter]], who took note of the following implications, here paraphrased in part by Aura.

One of the major commatic relations is between the rastma (243/242) on one hand, which occurs in the tuning of the mode of Zalzal by [[Wikipedia: al-Farabi|al-Farabi]] (c. 870-950) as the distinction between his steps of [[12/11]] (150.637c) and the smaller 88/81 (143.498c), and, on the other hand, the major minthma (352/351, 4.925c) described by [[Wikipedia: Avicenna|Ibn Sina]] (c.980-1037) in noting the "resemblance" between certain complex superpartient ratios and nearby simpler superparticular ratios, e.g. 128/117 (155.562c) and the simpler 12/11, or 88/81 and the simpler 13/12 (138.573c). These comparisons relate to the adjacent intervals included in the [[~~tetrachords~~]] al-Farabi and Ibn Sina favor to realize the 'oudist Mansur Zalzal's tuning favoring a middle third:

One of the major commatic relations is between the rastma (243/242) on one hand, which occurs in the tuning of the mode of Zalzal by [[Wikipedia: al-Farabi|al-Farabi]] (c. 870-950) as the distinction between his steps of [[12/11]] (150.637c) and the smaller 88/81 (143.498c), and, on the other hand, the major minthma (352/351, 4.925c) described by [[Wikipedia: Avicenna|Ibn Sina]] (c.980-1037) in noting the "resemblance" between certain complex superpartient ratios and nearby simpler superparticular ratios, e.g. 128/117 (155.562c) and the simpler 12/11, or 88/81 and the simpler 13/12 (138.573c). These comparisons relate to the adjacent intervals included in the [[tetrachord]]s al-Farabi and Ibn Sina favor to realize the 'oudist Mansur Zalzal's tuning favoring a middle third:

Al-Farabi:

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203.910c--138.573c---155.562c

In effect, the Sinarabian comma serves as a kind of bridge between these two tetrachords and tunings, hence its name. ~~More details~~ to ~~come~~.

In effect, the Sinarabian comma serves as a kind of bridge between these two tetrachords and tunings, hence its name. However, there's more that happens as a result of the relationship that the Sinarabian comma has to both the rastma and the major minthma.

For instance, let's take a look at the first non-commatic relationship defined by the Sinarabian comma from a couple of different angles:

(88/81)^(2) or 7744/6561 (286.996c) versus 13/11 (289.210c)

We know that a just [[13/12]] times 12/11 gives 13/11. Now, on one hand, 88/81 is 352/351 (4.925c) greater than 13/12, but on the other hand, 88/81 is 243/242 (7.139c) smaller than 12/11; these are al-Farabi's two middle second steps in his mode of Zalzal. Thus two 88/81 steps or 7744/6561 must fall short of 13/11 by the difference of these commas, or the Sinarabian comma, 85293/85184 (2.214c). Furthermore, in al-Farabi's tuning, 88/81 and the larger 12/11 (150.637c) together form a just Pythagorean minor third, which is [[32/27]] (294,135c), 352/351 larger than 13/11. Now, 88/81 twice repeated has the first interval matching 88/81, but the second interval falling 243/242 short of 12/11; and therefore, their combined size falls 243/242 short of 32/27, or the difference of 243/242 and 352/351 short of 13/11.

These next two non-commatic relationships are similar to each other, though they're also opposites in a directional sense:

(11/9)^(3) or 1331/729 (1042.224c) versus 117/64 1044.438c)

(27/22)^(3) or 19683/10648 (1063.641c) versus 24/13 (1061.427c)

On one hand, if we take 11/9 (347.408c) thrice, the first instance plus a just 3/2 fifth will yield 11/6 (1048.363c). Now, 11/9 plus 27/22, al-Farabi's two middle thirds differing by 243/242, will yield a just 3/2 (701.955c). Thus, two 11/9 thirds will yield 121/81 (694.816c) narrow of 3/2 by 243/242. Since 117/64 is narrow of 11/6 by 352/351, three 11/9 thirds must fall short of 117/64 by the Sinarabian comma, 85293/85184.

On the other hand, 27/22 is larger than its fifth complement 11/9 by 243/242, and 27/22 combined with 3/2 would yield 81/44 (1056.502c). Thus 729/484 (709.094c) or twice 27/22 is larger than 3/2 by 243/242. Since 24/13 (1061.427c) is 352/351 larger than 81/44, and 19683/10648 (1063.641c) or thrice 27/22 is 243/242 larger than 81/44, thrice 27/22 must be larger than 24/13 by 85293/85184, or the Sinarabian comma.

The last non-commatic relationship is different from the others in some ways:

1053/1024 (48.348c) versus 1331/1296 (46.134c)

An interesting instance of the Sinarabian comma occurs in comparing the Alpharabian parachromatic semilimma at 1331/1296 with Ibn Sina's smaller Zalzalian apotome complement of 1053/1024. To elaborate, the Alpharabian parachromatic semilimma is 243/242 smaller than al-Farabi's smaller Zalzalian apotome complement of 33/32 (53.273c), which is the interval between 32/27 and 11/9 or between 27/22 and 81/64. Al-Farabi's larger Zalzalian apotome complement is 243/242 greater than 33/32, at 729/704 (60.412c). Ibn Sina's Zalzalian apotome complements are 1053/1024 (46.134c) and 27/26 (65.337c), the former of which is the interval between 32/27 and 39/32 or between 81/64 and 16/13, and the latter of which is the interval between 32/27 and 16/13 or between 81/64 and 39/32. The Alpharabian parachromatic semilimma is 243/242 smaller than 33/32, while Ibn Sina's smaller Zalzalian apotome complement at 1053/1024 is 352/351 smaller than 33/32, and thus a Sinarabian comma of 85293/85184 smaller than 1053/1024.

This relationship in particular is interesting in that it connects the systems of al-Farabi and Ibn Sina, as does the 85293/85184 comma named in their honor.

[[Category:Commas named after polymaths]]

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-The '''Sinarabian comma''' is an [[13-limit]] (also 2.3.11.13 subgroup) [[unnoticeable comma]] with a ratio of '''85293/85184''' and a value of approximately 2 cents.
+The '''Sinarabian comma''' is an [[13-limit]] (also 2.3.11.13 subgroup) [[unnoticeable comma]] with a ratio of '''85293/85184''' and a value of approximately 2.2 cents.
-This comma is identifiable as the amount by which a stack of two [[88/81]] artoneutral seconds falls short of a [[13/11]] Neo-Gothic minor third, and also the amount by which a stack of three [[11/9]] artoneutral thirds fall short of [[117/64]], and even the comma that separates [[1053/1024]] and [[1331/1296]].
+This comma is identifiable as the amount by which a stack of two [[88/81]] artoneutral seconds falls short of a [[13/11]] Neo-Gothic minor third, and also the amount by which a stack of three [[11/9]] artoneutral thirds fall short of [[117/64]], the amount by which a stack of three [[27/22]] tendoneutral thirds exceed [[24/13]], and even the comma that separates [[1053/1024]] and [[1331/1296]].
 In terms of comma relations, it separates [[243/242]] from [[352/351]], separates [[144/143]] from a stack of two instances of 243/242, and, is the sum of the [[frameshift comma]] and the [[tridecapyth comma]].
@@ Line 15: / Line 15: @@
 The name of this comma comes from a portmanteau of "Ibn Sina" and "Alpharabian", and was given by [[Aura]] on the basis of a series of connections between intervals that he found and shared with [[Margo Schulter]], who took note of the following implications, here paraphrased in part by Aura.
-One of the major commatic relations is between the rastma (243/242) on one hand, which occurs in the tuning of the mode of Zalzal by [[Wikipedia: al-Farabi|al-Farabi]] (c. 870-950) as the distinction between his steps of [[12/11]] (150.637c) and the smaller 88/81 (143.498c), and, on the other hand, the major minthma (352/351, 4.925c) described by [[Wikipedia: Avicenna|Ibn Sina]] (c.980-1037) in noting the "resemblance" between certain complex superpartient ratios and nearby simpler superparticular ratios, e.g. 128/117 (155.562c) and the simpler 12/11, or 88/81 and the simpler 13/12 (138.573c).  These comparisons relate to the adjacent intervals included in the [[tetrachords]] al-Farabi and Ibn Sina favor to realize the 'oudist Mansur Zalzal's tuning favoring a middle third:
+One of the major commatic relations is between the rastma (243/242) on one hand, which occurs in the tuning of the mode of Zalzal by [[Wikipedia: al-Farabi|al-Farabi]] (c. 870-950) as the distinction between his steps of [[12/11]] (150.637c) and the smaller 88/81 (143.498c), and, on the other hand, the major minthma (352/351, 4.925c) described by [[Wikipedia: Avicenna|Ibn Sina]] (c.980-1037) in noting the "resemblance" between certain complex superpartient ratios and nearby simpler superparticular ratios, e.g. 128/117 (155.562c) and the simpler 12/11, or 88/81 and the simpler 13/12 (138.573c).  These comparisons relate to the adjacent intervals included in the [[tetrachord]]s al-Farabi and Ibn Sina favor to realize the 'oudist Mansur Zalzal's tuning favoring a middle third:
 Al-Farabi:
@@ Line 29: / Line 29: @@
 .910c--138.573c---155.562c
-In effect, the Sinarabian comma serves as a kind of bridge between these two tetrachords and tunings, hence its name.  More details to come.
+In effect, the Sinarabian comma serves as a kind of bridge between these two tetrachords and tunings, hence its name.  However, there's more that happens as a result of the relationship that the Sinarabian comma has to both the rastma and the major minthma.
+For instance, let's take a look at the first non-commatic relationship defined by the Sinarabian comma from a couple of different angles:
+              (88/81)^(2) or 7744/6561 (286.996c) versus 13/11 (289.210c)
+We know that a just [[13/12]] times 12/11 gives 13/11.  Now, on one hand, 88/81 is 352/351 (4.925c) greater than 13/12, but on the other hand, 88/81 is 243/242 (7.139c) smaller than 12/11; these are al-Farabi's two middle second steps in his mode of Zalzal. Thus two 88/81 steps or 7744/6561 must fall short of 13/11 by the difference of these commas, or the Sinarabian comma, 85293/85184 (2.214c).  Furthermore, in al-Farabi's tuning, 88/81 and the larger 12/11 (150.637c) together form a just Pythagorean minor third, which is [[32/27]] (294,135c), 352/351 larger than 13/11.  Now, 88/81 twice repeated has the first interval matching 88/81, but the second interval falling 243/242 short of 12/11; and therefore, their combined size falls 243/242 short of 32/27, or the difference of 243/242 and 352/351 short of 13/11.
+These next two non-commatic relationships are similar to each other, though they're also opposites in a directional sense:
+              (11/9)^(3) or 1331/729 (1042.224c) versus 117/64 1044.438c)
+              (27/22)^(3) or 19683/10648 (1063.641c) versus 24/13 (1061.427c)
+On one hand, if we take 11/9 (347.408c) thrice, the first instance plus a just 3/2 fifth will yield 11/6 (1048.363c).  Now, 11/9 plus 27/22, al-Farabi's two middle thirds differing by 243/242, will yield a just 3/2 (701.955c).  Thus, two 11/9 thirds will yield 121/81 (694.816c) narrow of 3/2 by 243/242. Since 117/64 is narrow of 11/6 by 352/351, three 11/9 thirds must fall short of 117/64 by the Sinarabian comma, 85293/85184.
+On the other hand, 27/22 is larger than its fifth complement 11/9 by 243/242, and 27/22 combined with 3/2 would yield 81/44 (1056.502c).  Thus 729/484 (709.094c) or twice 27/22 is larger than 3/2 by 243/242.  Since 24/13 (1061.427c) is 352/351 larger than 81/44, and 19683/10648 (1063.641c) or thrice 27/22 is 243/242 larger than 81/44, thrice 27/22 must be larger than 24/13 by 85293/85184, or the Sinarabian comma.
+The last non-commatic relationship is different from the others in some ways:
+/1024 (48.348c) versus 1331/1296 (46.134c)
+An interesting instance of the Sinarabian comma occurs in comparing the Alpharabian parachromatic semilimma at 1331/1296 with Ibn Sina's smaller Zalzalian apotome complement of 1053/1024.  To elaborate, the Alpharabian parachromatic semilimma is 243/242 smaller than al-Farabi's smaller Zalzalian apotome complement of 33/32 (53.273c), which is the interval between 32/27 and 11/9 or between 27/22 and 81/64.  Al-Farabi's larger Zalzalian apotome complement is 243/242 greater than 33/32, at 729/704 (60.412c). Ibn Sina's Zalzalian apotome complements are 1053/1024 (46.134c) and 27/26 (65.337c), the former of which is the interval between 32/27 and 39/32 or between 81/64 and 16/13, and the latter of which is the interval between 32/27 and 16/13 or between 81/64 and 39/32.  The Alpharabian parachromatic semilimma is 243/242 smaller than 33/32, while Ibn Sina's smaller Zalzalian apotome complement at 1053/1024 is 352/351 smaller than 33/32, and thus a Sinarabian comma of 85293/85184 smaller than 1053/1024.
+This relationship in particular is interesting in that it connects the systems of al-Farabi and Ibn Sina, as does the 85293/85184 comma named in their honor.
 [[Category:Commas named after polymaths]]