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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Dear Monz,

Thank you for your fantastic data and tables. All that is nearly "encyclopedic"
and I could retrieve therein a few old acquaintances.
You must have a pretty fast computer, and, besides, a sophisticated and very
suitable, appropriate software.

Some "goodies" out of my box:

wohltemperirt C-E-G "best" and smallest integer approximation: 123,154,184
sum 4-6-1: 41=J.S.BACH numerologically centered upon 6, primus numerus perfectus

wohltemperirt B-d#-f# "best" integer approximation: 492,621,736
sum 1-84-9 =43*43: 43=CREDO; the 19 tuning steps numerologically centered
upon 84. 84=6*14: 6, primus numerus perfectus, 14=BACH

496: First accolade (system) of B-major (prelude, of course) in Bach's
autograph of WTC (12 systems are on the lacking page, but auxiliary systems
Bach had only set at the END of pieces. On the missing page no piece ends!)

84: page, where B-major in the autograph starts!

(B-major starts on bar 1913: 19 tuning steps, + juxtaopsition UNITAS-TRINITAS)

In his calligraphic manuscript, Bach thus has simultaneously coordinated the
count of bars, accolades and pages towards B-major, the tempering tonality.

In the pieces, I found several examples where he controls the keystrokes:
The B-major prelude comprises an overall of 417 keystrokes (=3*139); but
there are 4 ordinary fifths "wohltemperirt", 1 tempering fifth and 7 perfect
ones; let alone the decompte of keystrokes along the piece.
139: the 19 tuningsteps numerologically centered upon the 3=TRINITY of his
(=Werckmeister's!!) tri-unitarian musical temperament.

monz schrieb:
> 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
*******
123 & 184 have 154 as their third to make up the C-E-G

> 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"
>Dear Monz,
Again thanks for your passionating calculations and data!!

Herbert-Anton
>
>
>
>
> _________________________________________________________
>

--- In tuning@y..., ha.kellner@t... wrote:
/tuning/topicId_26108.html#26121

> Dear Monz,
>
> Thank you for your fantastic data and tables. All that
> is nearly "encyclopedic" and I could retrieve therein a
> few old acquaintances. You must have a pretty fast computer,
> and, besides, a sophisticated and very suitable, appropriate
> software.

600 MHz, and all my calculations were done on a Microsoft
Excel spreadsheet I created for doing prime-factored
interval conversions. Not really very sophisticated...
I draw graphs of the approximating fractions and look
at the deviations from the integer values to find which
ones are the closest rational approximation.

Thanks for all the numerology lore on Bach.

(I'm still trying to finish the post I started yesterday
on ancient Greek use of 11 and higher primes for
"superparticular" reasons. I got sidetracked earlier
today with all this factoring and approximating stuff.)

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

Dear Monz,
I tested by my actualized NORTON the attached EXCEL.
It is straightforward; should you ever try to use it, never change
the read only-feature.
At the obvious insertion point you paste in the VALUE of the
(irrational) number to be approximated.

I do not know whether this routine could be of any use to you -
I wonder.

Kind regards,
Herbert-Anton

monz schrieb:
> --- In tuning@y..., ha.kellner@t... wrote:
> /tuning/topicId_26108.html#26121
>
> > Dear Monz,
> >
> > Thank you for your fantastic data and tables. All that
> > is nearly "encyclopedic" and I could retrieve therein a
> > few old acquaintances. You must have a pretty fast computer,
> > and, besides, a sophisticated and very suitable, appropriate
> > software.
>
> 600 MHz, and all my calculations were done on a Microsoft
> Excel spreadsheet I created for doing prime-factored
> interval conversions. Not really very sophisticated...
> I draw graphs of the approximating fractions and look
> at the deviations from the integer values to find which
> ones are the closest rational approximation.
>
> Thanks for all the numerology lore on Bach.
>
> (I'm still trying to finish the post I started yesterday
> on ancient Greek use of 11 and higher primes for
> "superparticular" reasons. I got sidetracked earlier
> today with all this factoring and approximating stuff.)
>
>
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
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