Tuning-Math Digests messages 6455 - 6479

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Message: 6455

Date: Wed, 12 Feb 2003 23:54:03

Subject: Re: vanishing diatonic semitone

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> i'd like to add one more row in this table, before the first row:
> 
> Yahoo groups: /tuning/database? *
> method=reportRows&tbl=10&sortBy=4&sortDir=down&start_at=0&query=
> 
> this row would have 16:15 vanishing, and connect the family of ETs 5, 
> 8, 3.
> 
> who can supply the necessary information?

Aw, c'mon Paul. This isn't a 5-limit temperament, except perhaps in a
musically-irrelevant purely-mathematical sense. This is the thing
where the generator has to act as both the fourth and the major third
(or the fifth and the minor sixth) and of course succeeds in doing
neither.

Next you'll be wanting the one where 9:10 vanishes. ;-)

It's been a stretch for me to accept neutral thirds and pelogic as
5-limit temperaments. I think I have to draw the line at errors
greater than 35 cents. So I have a similar objection to the one where
25:27 vanishes.

But you should find what you want in
http://uq.net.au/~zzdkeena/Music/5LimitTemp.xls.zip - Ok *


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Message: 6456

Date: Wed, 12 Feb 2003 17:20:10

Subject: 5LimitTemp.xls

From: Carl Lumma

Dave,

The degeneracy column seems broken.  I've got Excel 2000.

-Carl


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Message: 6457

Date: Wed, 12 Feb 2003 17:28:39

Subject: Re: vanishing diatonic semitone

From: Gene W Smith

On Thu, 13 Feb 2003 01:27:39 -0000 "wallyesterpaulrus
<wallyesterpaulrus@xxxxx.xxx>" <wallyesterpaulrus@xxxxx.xxx> writes:

> the timbres that people like sethares talk about, even if they don't 
> always say so, start as harmonic and then each harmonic (up to 6, 8, 
> 12, whatever) is "tweaked" toward the nearest et (or whatever) 
> position. therefore, it's an approximation of an approximation of 5-
> limit JI :)

Csound lets you play with these, but I was disappointed to find that
unless the inharmonic partials are close to harmonic, I find the timbres
get on my nerves.


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Message: 6458

Date: Wed, 12 Feb 2003 20:32:19

Subject: Re: 5LimitTemp.xls

From: Carl Lumma

>> The degeneracy column seems broken.  I've got Excel 2000.
>> 
>> -Carl
>
>I've only got Excel 97. Do you have the Analysis Toolpack (or
>whatever) installed so the GCD function works? Look it up in Excel
>Help.

Ah, now it works.  That is, if only rows 17-19, 27-29, 32-34,
44-46 are supposed to be degenerate, and the rest blank.

-Carl


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Message: 6459

Date: Wed, 12 Feb 2003 23:44:28

Subject: Re: poking monz (was: Re: naming temperaments(

From: monz

hi paul,


> From: <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, February 12, 2003 2:40 PM
> Subject: [tuning-math] poking monz (was: Re: naming temperaments(
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> 
> > i'll try to get right on it.
> 
> while we're at it, here's a more complete "small 5-limit intervals" 
> chart to replace the one on your equal temperament page:
> 
> Yahoo groups: /tuning-math/files/Paul/small.gif *


done.



> due to the triangular/hexagonal geometry, it has the magical property 
> that each vector points in exactly the same direction -- that is, has 
> exactly the same slope -- as the green line for the corresponding 
> temperament in the graphs above. check it out!
> 
> moreover, this chart could be used in conjunction with a set of 
> hexagonal bingo cards . . . as you know, i've made quite a few 
> already, and can make more in about 2 seconds apiece . . . so that 
> one can see exactly *how* a given small interval vanishes, or fails 
> to, in a given equal temperament.
> 
> then this chart would have a dual function . . .
> 
> let me know,
> paul



i'd like to include hexagonal graphs for *all* the EDOs on my
"equal temperament" page *and* on the "bingo lattice" page.
it's just a matter of me finding the opportunity to do it.

keep sending me stuff ... i'll incorporate it as i have the chance.



-monz


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Message: 6460

Date: Wed, 12 Feb 2003 10:35:09

Subject: Re: notational specificity cont'd

From: David C Keenan

Aaron wrote:

>Hi Dave.  Thanks for your patience.  Could one say that "comma inflected" 
>is a fairly accurate way to describe the notation?

Yes! But of course it only describes one aspect of the notation.

>   I've seen the uploaded samples of the notation, and seen posts 
> referring to assignments of letter names and staff placements of 
> notes...  I wonder: how many notes of various inflections - or, said 
> another way, how many discrete pitches - may conceivably occupy the same 
> position on a five line staff?

We haven't counted them yet, and the more obscure ones using schisma 
accents are still in flux. But assuming the single or double symbol 
versions of the notation (considering any schisma accent marks to be part 
of a single symbol) we can go from double-flat to double-sharp in steps 
which are not more than 2 cents wide relative to just fifths, therefore we 
can do _at_least_ 233 discrete pitches, but I believe the actual figure is 
more like 400.

However most of what anyone will ever want to do with the notation can be 
done with only 12 symbols (and their inversions) in conjunction with 
existing sharp and flat symbols.

>   Is it simply a matter of symbol combinatorics?

In the single-symbol version (using the multi-shaft and X-shaft arrows) 
there are no combinations of symbols required. In the double-symbol version 
one uses only the single-shaft arrows in combination with conventional 
sharps and flats and their doubles. So the answer to your question is "No" 
for these versions of the notation. It's simply a matter of the number of 
discrete symbols (including any which appear as an arrow with a schisma 
accent mark).

The symbols themselves have been derived as combinations of 8 flags or half 
arrowheads (at most two at a time), 2 accent marks (at most one used), 4 
shafts (exactly one used) and 2 directions (up or down). But not all 
combinations are necessary or valid.

There is also the possibility of using multiple symbols against a single 
note in a one-symbol-per-prime manner. This is what I've called the 
multi-symbol version of the notation. We would discourage this as being 
_much_ harder to read. This could involve up to 10 symbols to determine a 
single pitch! And would give 98,415 (= 5*3^9) discrete pitches on a single 
staff line! About 98,000 of which would be utterly indistinguishable from 
their neighbours by even the most expert listener.

>   How are positional boundaries determined?

Again, I'm not sure what you mean, but I'll assume you mean how does one 
determine which staff line or space to place a note on. This is really not 
much different from the situation _without_ comma inflection, and is often 
referred to as the issue of "correct spelling". This is usually based on 
the structure of the scale or the structure of any chords the note might be 
part of. But in the absence of such context the default solution would be 
to use the position that requires the least pitch deviation from the 
natural note (in which case double-flats or sharps would not be used).

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6462

Date: Wed, 12 Feb 2003 15:28:04

Subject: Re: A common notation for JI and ETs

From: David C Keenan

>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
><d.keenan@u...> wrote:
>I think that the term "comma" has been used in a broad sense to
>denote smaller intervals (which we now call kleisma and schisma) more
>often than larger ones,

Possibly more often. But I expect it _has_ been used to cover larger ones 
often enough.

>  inasmuch as the term "diesis" has been used
>for the latter since at least the 14th century.  So I would be
>inclined not to use the term "comma" for anything above ~37 cents,
>even in a broader sense.

What term do you suggest we use for all these intervals typically less than 
a scale step, from schisminas to small semitones?

Here's what my Shorter Oxford (1959) has to say:

Comma ...
3. Mus. A minute interval or difference of pitch 1597.
...

Diesis ...
1. Mus. a. In ancient Gr. music, the pythagorean semitone (ratio 
243:256).  b. Now, the interval equal to the difference between three major 
thirds and an octave (ratio 125:128); usually called enharmonic diesis.
...

> > > Unfortunately, the
> > > particular dieses that we're using the o and m characters for are
> > > both in the para category.
> >
> > I wouldn't place too much importance on this. But I note that in the
> > three categories we have these symbols.
> >
> > small   dq   /|~  (|(  ~|\  //|  |~)
> > middle  unv  /|)  (|~  /|\  |))  (/|
> > large   owm  |\)  (|)  (|\
> >
> > But I find there is not much hope of making our prefixes match up
>with
> > any of these, except possibly in the large category.
> >
> > > Perhaps we could use meta for the largest
> > > group (the meaning, "beyond," would still apply) and find a
>couple of
> > > other prefixes that wouldn't conflict with (and might even tie in
> > > with) the letters q and n for the small and middle ranges.
> >
> > Good luck!
> >
> > There aren't very many prefixes starting with q. The only one that
>is
> > even slightly appropriate is "quasi-" but that means "almost but not
> > quite" and would be better used for those things that have
> > historically been called dieses but are smaller than 36.93 cents.
>
>I wasn't expecting to find anything appropriate for q, anyway.  I'm
>just trying to avoid names that might cause confusion.
>
> > "meso-" is _the_ Greek prefix meaning middle.
>
>Yes, I thought of that one, but would rather not use it, since it
>begins with m.

Given that meso- is such an obvious greek prefix for the job and it starts 
with the same letter as the English words middle, medium and mean, I don't 
feel we should avoid using it merely because the limtations of ASCII (which 
may not be relevant in a few years time) and the absence of a proper font, 
cause us to use the letter m to represent, in email, something which is not 
in the middle category.

Someone might come up with a reason tomorrow that would cause us to change 
our single-character ASCII assignments. ASCII will never appear on the 
staff. And I should hope that the single-character ASCII approximations 
would never be used in teaching or explaining the notation.

> > It is used with various
> > other Greek pairs such as:
> >
> > hypo-  under
> > meso-  middle
> > hyper- over
> >
> > endo- inside
> > meso- middle
> > ecto- outside
> >
> > proto- (or pro-) earlier or to the front
> > meso-  middle
> > meta-  later or to the rear
> >
> > lepto- fine small thin delicate
> > meso-  middle
> > hadro- thick stout
>
>I also found intra- (within or inside), neo- (new), and peri- (close
>at hand, near, adjacent).  In evaluating all of these, I tried to
>identify what I would call the prototypical diesis in each group:

Shouldn't you instead be looking at the primary interpretation of the most 
commonly ocurring sagittal symbol in each group?

>37-45 cents -- 125:128, the meantone diesis, is not only in the group
>with the *smallest size*, but is also the diesis by which three 4:5s
>*fall short* of (i.e., on the near side of) an octave.  So I thought
>that peri- or intra- might be appropriate.  Of these two I prefer
>peri-.  But proto- is also good, for a couple of reasons:  it is
>similar in meaning to peri-, and it is the opposite of meta- (should
>we use that term for the large group).  Besides, 125:128, which is
>probably the best-known of any diesis in any group (and thus, on
>account of its prominence, the one with the strongest claim to the
>label proto-diesis), would validate an additional shade of meaning by
>which the term could be applied to this group.

But the minor diesis 125:128 is rarely used in the sagittal notation, 
having symbol .//|.

By far the most common in this range will be the 25-<small>diesis //|. I 
can't find anywhere this has been previously named, presumably because it 
is simply a double syntonic comma. So, considered as a "comma" in its own 
right it is almost as "neo-" as the 11 and 13 commas below. And there are 
other commas in this group which are probably newer.

>45-57 cents -- 32:33, the unidecimal diesis (or quartertone),
>introduces some of the *strangest new* harmonies encountered in
>alternative tunings.  I thought neo- might be more descriptive of an
>interval such as this, rather than some nondescript label (such as
>meso-) that suggests that it might be average or middlin'.

But it _is_ average as far as size goes, and that's what these prefixes are 
supposed to be about.

>   Even the
>13 diesis (1024:1053, the second most prominent member of the group,
>and the one that's actually symbolized by an "n") is new and strange.

But their complements in the large-diesis group are just as new and 
strange. And anyway, how long does something remain "new"?

Also, I should think that if 125:128 is prototypical of the small group 
then 243:250 would be that for the medium group. But again this is not a 
common comma to want to notate. It might be notated as /|) or (|~ .

The 11-<medium>diesis /|\ will certainly be the most common in this group.

>57-69 cents -- 625:648, besides being a *large* diesis (~27:28, or
>1deg19) is also the amount by which four 5:6s *exceed* (i.e., go
>beyond) an octave.

I agree that the prototypical diesis in the large group is the major diesis 
625:648, again not something we'd commonly use since it is '(|) .

Clearly the 11-<large>diesis (|) will be the most common here.

>I believe that we agree that meta- is a good
>prefix for this group.

Well, no. Only that it applies to this group better than it does to the 
medium group.

The use of "meta-" to mean "beyond" is a recent departure from the Greek 
usage. As the Shorter Oxford puts it:

"In supposed analogy to 'Metaphysics' (misaprehended as meaning 'the 
science of that which transcends the physical'), meta- has been prefixed to 
the name of a science, to form a designation of a higher science of the 
same nature but dealing with ulterior problems."

But why not use prefixes that are a valid description of _all_ the commas 
in the group, rather than just ones that may be typical in any sense? i.e. 
ones that relate to size.

>   Need I say more?

I'm afraid so. :-)

My main objection is that neo- tells one nothing about the size.

And if one adopts the modern sense of meta- one might take a meta-diesis to 
be a difference between dieses, in the same way that a diesis is a 
difference between other intervals. For example, we might well have used 
the term meta-comma to describe the differences between commas that we 
instead called schismas and now schisminas.

If one takes the biological meanings of proto- front and meta- rear (of 
organisms) it is unclear that there is any correspondence with small and 
large. If one takes the temporal meaning of proto- before or early or 
primitive and meta- after or late or advanced, then it is only slightly 
more clear.

In regard to having the right _meaning_, the best Greek set I can find are
  hypo-
  meso-
  hyper-

If we were to depart from the Greek
  minor
  neutral
  major
would be obvious enough, and so would
  small
  medium (or mean)
  large

It is unfortunate that the word "diesis" already has two more syllables 
than we'd like it to have. This is presumably why we feel compelled to 
shorten any prefix we might add to it, down to a single syllable. We might 
instead shorten "diesis" to "di" for convenience when spoken (say in 
rehearsals) and then not need to shorten the prefixes.

>With these labels, the boundaries (in cents) would then be:
>
>0
>schismina
>0.98
>schisma
>4.50
>kleisma
>13.47
>comma
>36.93
>protodiesis
>45.11
>neodiesis
>56.84
>metadiasis
>68.57

Boundaries good. Labels still need work.

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6467

Date: Thu, 13 Feb 2003 11:29:55

Subject: scala show data

From: Carl Lumma

Manuel,

With Scala 2.05f, I observe...

equal 6
show data

strictly proper
roth stability 0
lumma stability 1

show data

strictly proper
roth stability 423799.833333
lumma stability 1

show data

strictly proper
roth stability 67041792.766666
lumma stability 1

The goofy stability value seems to max out at the 67 value
despite further show data commands.

-Carl


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Message: 6469

Date: Thu, 13 Feb 2003 11:58:57

Subject: Re: scala show data

From: Carl Lumma

Manuel,

I also notice that "Lumma stability" is the title of the value
in the show data output, but "Lumma instability" is the title
in the help for show data.

Also in the help, the return type is given as n>1.  But if it
really is stability you're returning, it would be 0 <= n <= 1,
right?  Since it's the *portion* of the interval of equivalence
not covered by the spans of the interval classes...

-Carl


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Message: 6470

Date: Thu, 13 Feb 2003 14:08:32

Subject: "Ultimate" 5-limit again

From: Gene W Smith

I returned to this, and added names, poptimal generators (this time using
everyone's favorite defintion of the minimax generator) and "extensions".
These are defined in a way which is very strict and perhaps a little
arbitrary, but the results seem of some interest. I took the poptimal
generator, found the corresponding val with the lowest badness, extended
it in the way which gave lowest badness, and looked for the lowest
badness 7-limit temperament compatible with this val and the 5-limit
comma. I only go up to Monzimic with this list, which really seems far
enough. To make up for that and make Paul happy, I tacked Miracle on the
end despite the fact that as a 5-limit temperament it's nothing to get
excited about.

Bug 7/31 Extends 15/14 to Bug ([2, 3, 5, 0, 2, 3])
27/25 [[1, 2, 3], [0, -2, -3]] [1200., 268.056438833948093748427143263]
3.739252 35.609240 1861.731473

Pelogic 10/23 Extends 36/35 to Pelogic (aka Hexadecimal)
135/128 [[1, 2, 1], [0, -1, 3]] [1200., 522.862345874111793591855751693]
4.132031 18.077734 1275.365360

Blackwood Universal
256/243 [[5, 8, 12], [0, 0, -1]] [240., 84.6637865678588914278600509674]
5.493061 12.759741 2114.877638

Dicot Universal
25/24 [[1, 1, 2], [0, 2, 1]] [1200., 350.977500432693708872243366367]
3.025593 28.851897 799.108711

Diminished Universal
648/625 [[4, 6, 9], [0, 1, 1]] [300., 94.1343573651111175944350240576]
6.437752 11.060060 2950.938432

Negri 2/19 Extends 49/48 to Negri (aka Tertiathirds)
16875/16384 [[1, 2, 2], [0, -4, 3]] [1200.,
126.238272015257926746682149917]
8.172550 5.942563 3243.743713

Porcupine 11/81 Extends 64/63 to Porcupine
250/243 [[1, 2, 3], [0, -3, -5]] [1200., 162.996026370546548951179738408]
5.948286 7.975801 1678.609846

Augmented Universal
128/125 [[3, 5, 7], [0, -1, 0]] [400., 91.2018560670299909777049249654]
4.828314 9.677666 1089.323984

Magic 19/60 Extends 225/224 to Magic
3125/3072 [[1, 0, 2], [0, 5, 1]] [1200., 379.967949195094816842076920201]
7.741412 4.569472 2119.954990

Quadrafifths 26/177 Extends 245/243 to Octafifths
20000/19683 [[1, 1, 1], [0, 4, 9]] [1200.,
176.282270436412295298990817071]
9.785568 2.504205 2346.540676

Pythagoric Universal
531441/524288 [[12, 19, 28], [0, 0, -1]] [100.,
14.6637865678588914278600509674]
13.183347 1.382394 3167.444999

Meantone 34/81 Extends 126/125 to Meantone
81/80 [[1, 2, 4], [0, -1, -4]] [1200., 503.835154026035812053011163756]
4.132031 4.217731 297.556531

Diaschismic 10/114 Extends 245/243 to Shrutar
2048/2025 [[2, 3, 5], [0, 1, -2]] [600., 105.446531009812541696859310996]
6.271199 2.612822 644.408867

Tertiary 23/285 to Extends 3136/3125 to Tertiary ([3, -12, -30, -26, -56,
-36])
67108864/66430125 [[3, 5, 6], [0, -1, 4]] [400.,
96.7879385616949726317268914802]
15.510107 .905187 3377.402314

Hemisixths 55/149 Various extensions, none much good
78732/78125 [[1, -1, -1], [0, 7, 9]] [1200.,
442.979297439105373735900374126]
12.192182 1.157498 2097.802867

Wuerschmidt 53/164 Various extensions, none much good
393216/390625 [[1, -1, 2], [0, 8, 1]] [1200.,
387.819673068349143521938606127]
12.543123 1.071950 2115.395301

Orwell 43/190 Extends 1029/1024 to Trifokker ([21, -9, -7, -63, -70, 9])
2109375/2097152 [[1, 0, 3], [0, 7, -3]] [1200.,
271.589599585245148575185388331]
12.772341 .800410 1667.723301

Septathirds 31/673 No good extensions
4294967296/4271484375 [[1, 2, 2], [0, -9, 7]] [1200.,
55.2754932571412314963954609732]
18.573955 .483108 3095.692488

Kleismic 65/246 Extends 5120/5103 to Countercatakleismic ([6, 5, -31, -6,
-66, -86])
15625/15552 [[1, 0, 1], [0, 6, 5]] [1200.,
317.079675185758890225628070818]
9.338935 1.029625 838.631548

Amity 58/205 Extends 5120/5103 to Amity
1600000/1594323 [[1, 3, 6], [0, -5, -13]] [1200.,
339.508825625715624367834924710]
13.794200 .383104 1005.555381

Parakleismic 31/118 Extends 3136/3125 to Parakleismic ([13, 14, 35, -8,
19, 42])
1224440064/1220703125
[[1, 5, 6], [0, -13, -14]] [1200., 315.250913337821936408197840098]
21.322672 .276603 2681.521263

Vulture 128/323 Extends 4375/4374 to Vulture ([4, 21, -56, 24, -100,
-189])
10485760000/10460353203 
[[1, 0, -6], [0, 4, 21]] [1200., 475.542233398945960632986914825]
21.733049 .153767 1578.433204

Semisuper 30/506 Nothing much good
6115295232/6103515625
[[2, 4, 5], [0, -7, -3]] [600., 71.1460635722374759764193142621]
21.207625 .194018 1850.624306

Enneadecal Universal
19073486328125/19042491875328 
[[19, 30, 44], [0, 1, 1]] [63.1578947368421052631578947368,
7.29225210195322285759291880280]
30.579320 .104784 2996.244873

Semitonic 95/1019 No good extensions
295578376007080078125/295147905179352825856 
[[1, 0, 4], [0, 17, -18]] [1200., 111.875426120872633513689333181]
38.845486 .058853 3449.774562

Tricot 233/494 Extends 4375/4374 to Tricot ([3, 29, -95, 39, -159, -302])
68719476736000/68630377364883 
[[1, 3, 16], [0, -3, -29]] [1200., 565.988014913065527948022354197]
30.550812 .057500 1639.596150

Schismic 120/289 Extends 4375/4374 to Infraschismic ([1, -8, 39, -15, 59,
113])
32805/32768 
[[1, 2, -1], [0, -1, 8]] [1200., 498.272487171563819993901705714]
9.459948 .161693 136.885775

Counterschismic 237/571 No good extensions
2954312706550833698643/2951479051793528258560
[[1, 2, 21], [0, -1, -45]] [1200., 498.082318148218414995068857757]
48.911647 .026391 3088.065497

Hemithird 232/1441 Extends 4375/4374 to Infrahemithird ([15, -2, 113,
-38, 137, 268])
274877906944/274658203125
[[1, 4, 2], [0, -15, 2]] [1200., 193.199614933859969427837273253]
24.977022 .060822 947.732642

Minortone 196/1289 Extends 2460375/2458624 to Hemiminortone
([30, 70, 129, 32, 109, 103]. Minortone is [17, 35, -21, 16, -81, -147])
50031545098999707/50000000000000000 
[[1, -1, -3], [0, 17, 35]] [1200., 182.466089137089694182158775289]
38.845486 .025466 1492.763207

Ennnealimmal 68/1665 Extends 2401/2400 to Ennealimmal
7629394531250/7625597484987
[[9, 15, 22], [0, -2, -3]] [133.333333333333333333333333333,
49.0088197863290461293795242156]
33.653272 .025593 975.428947

Glum 103/935 Nothing much good
2475880078570760549798248448/2474715001881122589111328125
[[1, 5, 1], [0, -31, 12]] [1200., 132.194510561451335831533197063]
55.785793 .014993 2602.883149

Kwasy 182/1342 Extends 4375/4374 to Hemikwasy ([16, -10, 152, -53, 196,
381])
9010162353515625/9007199254740992
[[2, 1, 6], [0, 8, -5]] [600., 162.741892126380267669129153916]
31.255737 .017725 541.228379

Mum 341/730 Extends 4375/4374 to Mum ([33, 25, 131, -37, 115, 234])
116450459770592056836096/116415321826934814453125 
[[1, 17, 14], [0, -33, -25]] [1200., 560.546969532517954992081849041]
50.788153 .012388 1622.898233

Bum 237/901 Extends 4375/4374 to Bum ([51, 52, 149, -36, 93, 200])
444089209850062616169452667236328125/444002166576103304796646509039845376
[[1, 15, 16], [0, -51, -52]] [1200., 315.647874693157629813083932838]
82.462759 .004660 2613.109284

Monzimic 116/559 Extends 4375/4374 to Monzimic ([2, 37, -134, 54, -218,
-415])
450359962737049600/450283905890997363
[[1, 2, 10], [0, -2, -37]] [1200., 249.018447894645757478665305415]
39.665603 .005738 358.125500

...

Miracle 41/422 Extends 225/224 to Miracle 
34171875/33554432
[[1, 1, 3], [0, 6, -7]]  [1200., 116.578231256479]
14.2507126003310  1.98070788903097  5732.31669654049


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