Mercator's comma

19383245667680079896796723
--------------------------
19342813113834066795298816

big number.

But, IIRC, you have to write the 5/4 as 4/3 flat. i.e. E=407.55cents 53, but
Fb =384.91cents, so the chord is written C Fb G.

> From: tuning@yahoogroups.com
> Reply-To: tuning@yahoogroups.com
> Date: 26 Oct 2002 03:33:29 -0000
> To: tuning@yahoogroups.com
> Subject: [tuning] Digest Number 2279
>
> Mercator's comma, which is a ratio with hideously large
> numbers which i prefer to write simply as 3^53, and it's about
> 3.615045866 cents.

hi Mark,

> From: "Mark Gould" <mark.gould@argonet.co.uk>
> To: "Tuning List" <tuning@yahoogroups.com>
> Sent: Saturday, October 26, 2002 9:34 AM
> Subject: [tuning] Mercator's comma
>
>
> 19383245667680079896796723
> --------------------------
> 19342813113834066795298816
>
> big number.

exactly. isn't 3^53 much better?

> But, IIRC, you have to write the 5/4 as 4/3 flat.
> i.e. E=407.55 cents 53,

why is the "53" in there? looks like it was a typo.

> but Fb =384.91cents, so the chord is written C Fb G.

that's correct. you just didn't explicitly state
your assumption that C=1/1.

the entire 53edo gives an extremely good approximation
of the Pythagorean chain 3^(-26...+26), the maximum
error (at the two bounding notes) being ~1.77 cents,
just under 1/2 of a Mercator-comma.

since it emulates Pythagorean tuning, 53edo gives
two possibilites for the "major-3rd", as you state.
one can use either the Pythagorean version, which is
+4 generators (i.e., the four "5ths" upward C:G:D:A:E)
or the "schismic" version, which is -8 generators
(i.e., 8 "5ths" downward or 8 "4ths" upward:
C:F:Bb:Eb:Ab:Db:Gb:Cb:Fb).

however, if it is assumed that the skhisma (~2 cents)
is below the margin-of-error in tuning (usually claimed
as ~5 cents), then both Pythagorean tuning itself and
53edo can be viewed as a schismic temperament, in which case,
where C=1/1, the -8 generator can just be written as "E".

but since the tuning also includes the other, higher "E",
this one needs some other marking to distinguish it.
i propose a minus sign in my HEWM notation (E-), others who
are also using a 72edo-based notation use a down-arrow
symbolized in ASCII by "v" (Ev), Helmholtz used a subscript 1
... all these are to indicate a lowering by 1 syntonic comma
(or the EDO's approximation of it) from the Pythagorean E.

regarding the name: someone just asked if this Mercator
was the same as the famous cartographer. i just read
somewhere recently (on the internet) that it wasn't.
but it was while i was on "tuning vacation" and so i
didn't keep the reference.

-monz
"all roads lead to n^0"

--- In tuning@y..., "monz" <monz@a...> wrote:

/tuning/topicId_40188.html#40189

>
> but since the tuning also includes the other, higher "E",
> this one needs some other marking to distinguish it.
> i propose a minus sign in my HEWM notation (E-), others who
> are also using a 72edo-based notation use a down-arrow
> symbolized in ASCII by "v" (Ev), Helmholtz used a subscript 1
> ... all these are to indicate a lowering by 1 syntonic comma
> (or the EDO's approximation of it) from the Pythagorean E.
>

***Hi Monz!

Isn't this "higher E" also written as Fb by some?? I vaguely
remember that.

>
> regarding the name: someone just asked if this Mercator
> was the same as the famous cartographer. i just read
> somewhere recently (on the internet) that it wasn't.
> but it was while i was on "tuning vacation" and so i
> didn't keep the reference.
>

***Yes, that was me. I guess there is no relationship...

Joe Pehrson

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> /tuning/topicId_40188.html#40189
>
> >
> > but since the tuning also includes the other, higher "E",
> > this one needs some other marking to distinguish it.
> > i propose a minus sign in my HEWM notation (E-), others
who
> > are also using a 72edo-based notation use a down-arrow
> > symbolized in ASCII by "v" (Ev), Helmholtz used a subscript 1
> > ... all these are to indicate a lowering by 1 syntonic comma
> > (or the EDO's approximation of it) from the Pythagorean E.
> >
>
> ***Hi Monz!
>
> Isn't this "higher E" also written as Fb by some?? I vaguely
> remember that.

joseph -- Fb is the *lower* E. in fact, it's only 2 cents different
from the "E-" monz speaks of above.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_40188.html#40197

wrote:
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> > --- In tuning@y..., "monz" <monz@a...> wrote:
> >
> > /tuning/topicId_40188.html#40189
> >
> > >
> > > but since the tuning also includes the other, higher "E",
> > > this one needs some other marking to distinguish it.
> > > i propose a minus sign in my HEWM notation (E-), others
> who
> > > are also using a 72edo-based notation use a down-arrow
> > > symbolized in ASCII by "v" (Ev), Helmholtz used a subscript 1
> > > ... all these are to indicate a lowering by 1 syntonic comma
> > > (or the EDO's approximation of it) from the Pythagorean E.
> > >
> >
> > ***Hi Monz!
> >
> > Isn't this "higher E" also written as Fb by some?? I vaguely
> > remember that.
>
> joseph -- Fb is the *lower* E. in fact, it's only 2 cents different
> from the "E-" monz speaks of above.

Heh. I meant, of course the E- and instead I got an F- ... :)

Thanks, Paul!

JP