Sinarabian comma
| Interval information |
(Shannon, [math]\sqrt{nd}[/math])
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, 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.
Etymology and Implications
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 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 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:
Al-Farabi:
0.0c---203.910c--354.547c----498.045c
1/1--------9/8-------27/22-------4/3
9:8--------12:11--------88:81
203.910c-----150.637c---143.498
Ibn Sina:
0.0c------203.910c. 342.483c 498.045c
1/1----------9/8-------39/32------4/3
9:8---------13:12------128:117
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. 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.