names for 3^-41 and 3^-306 "commas" (unison-vectors)

While ruminating on pythagorean intonation today,
i came up with some naming questions ...

We have "mercator-comma" for 3^53

2,3-monzo .......... ratio .......... ~cents
[ -84 53, > .. 8418691 / 8401130 .. ~3.615045866

and "satanic-comma" (a name i love) for interval
represented by 3^665

2,3-monzo ............. ratio .......... ~cents
[ -1054 665, > .. 6574553 / 6574266 .. 0.075575483

Has anyone given a name to this small interval:

2,3-monzo .......... ratio .......... ~cents
[ 65 -41, > .. 7191344 / 7109381 .. 19.84496452

?

This is the one which, when tempered out of a pythagorean
chain-of-5ths, results in 41-edo.

How about this one:

2,3-monzo ............ ratio .......... ~cents
[ 485 -306, > .. 6131836 / 6125571 .. 1.769735191

?

This one produces 306-edo.

(The ratios i posted here for this last one and for
the satanic-comma are probably not totally accurate ...
i think the numbers are too big for Excel to calculate
properly.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Monz.

> (The ratios i posted here for this last one and for
> the satanic-comma are probably not totally accurate ...
> i think the numbers are too big for Excel to calculate
> properly.)

Actually, I'm afraid none of them is. As 2^32 is more than 4 billion, I
rather don't understand where Excel manages to find these numbers. Sorry to
be so open but I'm just "developing" your idea.

Petr

Hi Petr,

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:

> Hi Monz.
>
> > (The ratios i posted here for this last one and for
> > the satanic-comma are probably not totally accurate ...
> > i think the numbers are too big for Excel to calculate
> > properly.)
>
> Actually, I'm afraid none of them is. As 2^32 is more
> than 4 billion, I rather don't understand where Excel manages
> to find these numbers. Sorry to be so open but I'm just
> "developing" your idea.

The version of Excel i'm running can handle powers of 3
up to 3^646, which is renders as "1.6609E+308", that is,
1.6609 * 10^308.

Trying to get it to calculate 3^647 produces the result
"Num!", which, according to Excel Help, is Excel secret-code
for "i can only calculate expressions which have a value
between -1*(10^307) and 1*(10^307)". Why it's able to
give me a value with 10^308, i don't know.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> and "satanic-comma" (a name i love) for interval
> represented by 3^665

AKA "the Comma of the Beast". It's the amount by which 666 fifths,
reduced to the octave, differs from a fifth.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> While ruminating on pythagorean intonation today,
> i came up with some naming questions ...
>
>
> We have "mercator-comma" for 3^53
>
> 2,3-monzo .......... ratio .......... ~cents
> [ -84 53, > .. 8418691 / 8401130 .. ~3.615045866

I'd call that a schisma.

>
>
> and "satanic-comma" (a name i love) for interval
> represented by 3^665
>
> 2,3-monzo ............. ratio .......... ~cents
> [ -1054 665, > .. 6574553 / 6574266 .. 0.075575483

And that a schismina.

> Has anyone given a name to this small interval:
>
> 2,3-monzo .......... ratio .......... ~cents
> [ 65 -41, > .. 7191344 / 7109381 .. 19.84496452

Only "complex pythagorean comma"

> How about this one:
>
> 2,3-monzo ............ ratio .......... ~cents
> [ 485 -306, > .. 6131836 / 6125571 .. 1.769735191

No but it would also be a schismina.

Hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > While ruminating on pythagorean intonation today,
> > i came up with some naming questions ...
> >
> >
> > We have "mercator-comma" for 3^53
> >
> > 2,3-monzo .......... ratio .......... ~cents
> > [ -84 53, > .. 8418691 / 8401130 .. ~3.615045866
>
> I'd call that a schisma.

The name "mercator-comma" or "Comma of Mercator" is
already fairly well-established, but i agree with you,
it ought to be "mercator-schisma".

>
> >
> >
> > and "satanic-comma" (a name i love) for interval
> > represented by 3^665
> >
> > 2,3-monzo ............. ratio .......... ~cents
> > [ -1054 665, > .. 6574553 / 6574266 .. 0.075575483
>
> And that a schismina.
>
> > Has anyone given a name to this small interval:
> >
> > 2,3-monzo .......... ratio .......... ~cents
> > [ 65 -41, > .. 7191344 / 7109381 .. 19.84496452
>
> Only "complex pythagorean comma"
>
> > How about this one:
> >
> > 2,3-monzo ............ ratio .......... ~cents
> > [ 485 -306, > .. 6131836 / 6125571 .. 1.769735191
>
> No but it would also be a schismina.

I'm surprised to see this one and the "satanic-comma"
both falling under the "schismina" category, as the
"satanic-comma" is orders of magnitude smaller.

Isn't ~0.076 cent an "atom"?

In any case, in talking about pseudo-pythagorean systems,
it would be good if these last two had nice memorable names.
Any suggestions?

-monz
http://tonalsoft.com
Tonescape microtonal music software

monz wrote:
> > --- In tuning@yahoogroups.com, Petr Par�zek <p.parizek@...> wrote:>>>properly.)
>>
>>Actually, I'm afraid none of them is. As 2^32 is more
>>than 4 billion, I rather don't understand where Excel manages
>>to find these numbers. Sorry to be so open but I'm just
>>"developing" your idea.
> > The version of Excel i'm running can handle powers of 3
> up to 3^646, which is renders as "1.6609E+308", that is,
> 1.6609 * 10^308.

Petr is assuming that Excel uses 32 bit integers. When you're calculating the sizes of small commas, it's important that you keep the full integer precision. For numbers up to 1e308, it's using floating point instead.

It may be that Excel can use larger, or even arbitrary-sized integers. It certainly should as Microsoft must already have the libraries for implementing encryption. But if not, you'll get more accurate results by doing the work with prime-factor notation and not worrying about large integers.

> Trying to get it to calculate 3^647 produces the result
> "Num!", which, according to Excel Help, is Excel secret-code
> for "i can only calculate expressions which have a value
> between -1*(10^307) and 1*(10^307)". Why it's able to
> give me a value with 10^308, i don't know.

It's probably using 64-bit IEEE754 floats with a maximum value of approximately 1.8e308. 3^647 is around 5.0e308.

Graham

Hi Graham,

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Petr is assuming that Excel uses 32 bit integers. When
> you're calculating the sizes of small commas, it's
> important that you keep the full integer precision.
> For numbers up to 1e308, it's using floating point
> instead.
>
> It may be that Excel can use larger, or even arbitrary-sized
> integers. It certainly should as Microsoft must already have
> the libraries for implementing encryption.

I am using a very old version, i think it's from 1997.

> But if not, you'll get more accurate results by doing
> the work with prime-factor notation and not worrying
> about large integers.

Of course! After all, *i'm* the guy who likes using
prime-factor notation so much ... precisely because of
its accuracy in dealing with very large numbers ...

... along with its ability to convey the lattice data
which gives me a clear picture of how the numbers
all relate to each other.

Thanks for explaining what Excel is doing.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
in
/tuning/topicId_63911.html#63935
the pythagorean 3-limit interval 3^-41:
was baptized there as:
and the "cocomma":
"%" := 2^65/3^41 ~19.8..cents (to be read as interval, not percent)
.........
"the cocomma 2^65/3^41, is
leading to the 53-comma"
and labeled with the symbol "%"
in order to dinstinguish it from the ordinary PC
3^12/2^19 in Bosanquet-Helmholtz notation: "/".

> monz wrote:
... Excel i'm running can handle powers of 3
> > up to 3^646, which is renders as "1.6609E+308", that is,
> > 1.6609 * 10^308.

> It's probably using 64-bit IEEE754 floats with a maximum value of
> approximately 1.8e308. 3^647 is around 5.0e308.
>
if one has only pocket calculator precision available
then try instead of:

2^485/3^306
simply
(2^242.5/3^153)^2
=((~7.06738826...×10^72)/(~9.989689...×10^72))^2
in order to keep the powers in the exponents ratio below 100
or in terms of
www.google.com
inherent arithmetics:
(1 200 * ln((2^485) / (3^306))) / ln(2) = ~1.76973519 Cents

Remarks on "Moritz Wilhelm Drobisch's" 3^665 comma:
http://www.leipzig-lexikon.de/PERSONEN/18020816.htm

Correctly spoken: that's the Pythagorean ratio: 3^665/2^1054
(he obtained that interval already in the 19.th century
by continued fraction expansion of ln3/ln2 series)

Attend:
665 allows the prime factor decomposition: 19*7*5
That enables to more handy reprensentations of the 665-comma
in terms of Pythagorean intervals like:
apotome, limma & Comma(PC)

(1)
with help of the apotome (2187/2048):=3^7/2^11
and 95=19*5 by the exponent-product-law we obtain:
(2 187 / 2 048)^95 = 512.022351....
that becomes if divided by 9 octaves
(2 187/2 048)^95 / 512 = 1.00004366....
deliverning the tiny reminder of only ~0.0755754826..Cents
by having used the decomposition
3^665/2^1054:=(3^7/2^11)^95/2^9, short 665=7*95 conciesely

(2)
or similar we obtain applying limma
(256/243):=2^8/3^5
likewise
(256/243)^133 = 1 023.9553....
which delivers also the same result as already above by.
1 024 / ((256 / 243)^133) = 1.00004366....
or again ~0.0755754826..Cents also too at the end as already in (1)
using the alternative decomposition
3^665/2^1054:=2^10/(2^8/3^5)^133, short 665=5*133 conciesely

(3)
with the PC*apotome=(3^12/2^19)*(3^7/2^11)=3^19/2^30

((3^19) / (2^30))^35 = 16.0006985...........
using 19*35=665 as an other possible exponent decomposition.
Remark:
Therefore 665et contains the popular choice 19et just 35 times.

(4)
55*12+5=660+5=665
constituting the ratio of:
(3^12/2^19)^55 = 2.10708787......Pythagorean_Comma^55
and 512 / 243 = 2.10699588.......octaved limma 2^9/3^5
in demanding only the space of amounting an octave*limma,
in order to reply on the tuners fundamentalistic question:

What's "satanic-sharp" in devils ears?
Response:
That's PC^55 above an octaved limma.

Find yourself further alternative possible
representations of 3^665 too, in terms of
apotome, limma & comma, however you want!

CONCLUSION
What a dickens demonic
Remind the devils mnemonic:
for counting our "old nick"
"lord-Harry" fiendly
tamed arithmetically:

(1) apotome^95 = 512.022351....=512*satanic-c, the 9 oct. view.
(2) limma^133 = 1023.9553......=1024/satanic-c, the 10 oct. view.
(3) (PC*apotome)^35 = 16.0006985....=16*satanic-c, the 4 oct. view.
or
(4) PC^55/2/limma = satanic-c, the represantation within an minor 9th

Beelzebub becomes so easy available determinated,
even on scientific pocket-calculators instead
overfloating EXCELs limited capabilities, all done simply
due to the prime defactorization trick 665=19*7*5
or splitting that into the sum 55*12+5.

have a lot of fun in veryfing yourself that numerical jokes
by playing on the keys on your own private claculator

kind regards
yours sincerely
A.S.
p.s:
But who except "satan" himself will be yet able to discriminate an
sharp 5th of 3^666/2^1055=~1.50006548... versus the ordinary 3/2=1.5
one, amounting about only 1/13 cents different in seize?