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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