some factorizations of commas and their divisions

Just for fun, I decided to calculate the closest
superparticular rational approximations for other
divisions of a Pythagorean Comma (whose ratio will
be called "P") and Syntonic Comma (with ratio "S").
Here is a table of the first dozen of each:

P = (2^-19)*(3^12) = ~74/73

P^(1/2) = ~148/147
P^(1/3) = ~222/221
P^(1/4) = ~296/295
P^(1/5) = ~369/368
P^(1/6) = ~443/442
P^(1/7) = ~517/516
P^(1/8) = ~591/590
P^(1/9) = ~665/664
P^(1/10) = ~738/737
P^(1/11) = ~812/811
P^(1/12) = ~886/885

S = (2^-4)*(3^4)*(5^-1) = exactly 81/80

S^(1/2) = ~162/161
S^(1/3) = ~242/241
S^(1/4) = ~322/321
S^(1/5) = ~403/402
S^(1/6) = ~483/482
S^(1/7) = ~564/563
S^(1/8) = ~644/643
S^(1/9) = ~725/724
S^(1/10) = ~805/804
S^(1/11) = ~886/885
S^(1/12) = ~966/965

Notice that in this set of approximations, tempering by
P^(1/12), S^(1/11), and 886/885 all give the same result,
because P^(1/12) ~= S^(1/11).

P^(1/12) is the interval measurement known as a "grad",
and it is very close in size to S/P, which is the "skhisma".
AFAIK, S^(1/11) does not have a name other than "1/11-comma".
Below is a comparison.

grads skhismas cents

S^(1/11) ~1.000059525 ~1.000714763 ~1.955117236
P^(1/12) 1.0 ~1.0006552 ~1.955000865.
S/P ~0.999345229 1.0 ~1.953720788

prime-factorizations:

2^ 3^ 5^

P^(1/12) = | -19/12 1 0 |
S^(1/11) = | -4/11 4/11 -1/11 |
S/P = | -15 8 1 |

Note that 887/886 gives the closest superparticular rational
approximation to the skhisma. 2^(1/614) is a good EDO
approximation of all three of these intervals.

P^(1/5) can be factored as 2^(-19/5) * 3^(12/5), so
the Bach/wohltemperirt tempered "5th" of (3/2) / P^(1/5) can
be factored as 2^(14/5) * 3^-(7/5) . The lowest-integer
ratio that comes close to it is 184/123, and 2395/1601 is
much closer.

The (4/3)^(1/30) version of moria can be factored as
2^(1/15) * 3^-(1/30) . If the exponents of the Bach/wohltemperirt
"5th" are multiplied so that its denominators match these,
that interval is expressed as 2^(42/15) * 3^-(42/30), which
therefore shows that it is equal to exactly 42 of these morias,
or 4:3 "+" 12 morias. [Note that there is another type of
moria which is 2^(1/72) ].

P^(1/4) can be factored as 2^(-19/4) * 3^3, so the
Werckmeister III tempered "5th" of (3/2) / P^(1/4) can
be factored as 2^(15/4) * 3^-2 . It is very close to the 50-EDO
"5th" of 2^(29/50) [a more exact figure: 2^(29.00374993/50) ],
which was pointed out by Woolhouse as being nearly identical
with his "optimal" 7/26-comma meantone. See:
http://www.ixpres.com/interval/monzo/woolhouse/essay.htm

-monz
http://www.monz.org
"All roads lead to n^0"

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