This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Previous Next

6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950

6550 - 6575 -

Message: 6575 - Contents - Hide Contents

Date: Thu, 27 Feb 2003 11:13:33

Subject: gd search mk. 4 (was Re: A Property of MOS/DE Scales)

From: Carl Lumma

>> >nybody else interested in reviving this thread?
>> 
>> Relevant messages...
>> 
>> Yahoo groups: /tuning/message/41371 * [with cont.] 
>> Yahoo groups: /tuning/message/41383 * [with cont.] 
>> 
>> Yahoo groups: /tuning-math/message/5132 * [with cont.] 
>> Yahoo groups: /tuning-math/message/5136 * [with cont.] 
>> 
>> -Carl
>

>yes, i am very interested!!


My interest regards a systematic search for scales with
between 5 and 10 notes, in which a single pattern of scale
degrees yields more than one consonant m-ad in a majority
of modes, where the m-ads are approximated at least as
well as the 5-limit in 12-tET (unweighted RMS).

How would one begin such a search?  At the outermost level,
would we step through m?  If so, I suggest 2 <= m <= 6.

Within a given m, we need to identify promising commas.
Probably cents/(n*d) is a good metric for that...

Beyond that, how do we set things up?  Should we limit
ourselves to linear temperaments?  What's the method for
finding which chords/scale pattern will be involved given
a set of commas and the commatic/chromatic designations?
Does this method work when three or four different m-ads
appear on the scale pattern (rather than just two)?

I'm willing to turn English into Java or Scheme...

-Carl

Message: 6576 - Contents - Hide Contents

Date: Thu, 27 Feb 2003 21:13:05

Subject: gd search mk. 4 (was Re: A Property of MOS/DE Scales)

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> Probably cents/(n*d) is a good metric for that...


why do you want cents to be high?

Message: 6577 - Contents - Hide Contents

Date: Fri, 28 Feb 2003 19:24:03

Subject: Re: gd search mk. 4

From: Carl Lumma

>> >robably cents/(n*d) is a good metric for that...
>

>why do you want cents to be high?


Whoops, that should be 1/(n*d)cents.

-C.

Message: 6578 - Contents - Hide Contents

Date: Sat, 1 Mar 2003 09:13:45

Subject: Re: [tuning] Re: Some magic meantones

From: monz

hi Gene,


i've shifted this thread over to here so as
not to bother the regular tuning list members.

as usual, i sense intuitively that the explanation
you've provided for me below has some real insight
buried in it somewhere ... but it's so cryptic
that i have no idea what you're saying.

can you please explain the numbers in brackets?

how do "we see that ... 7/26[-comma meantone]
is marginal for minor triads and really stinks
for major triads"?  based on my understanding of
Woolhouse's work, i thought this was an "optimal"
meantone.  ...?

(i apologize for being stupid, as the cause of
it is simply my lack of reading and following your
many posts here. thanks.)


-monz


----- Original Message ----- 
From: <gwsmith@xxxxx.xxx>
To: <tuning@xxxxxxxxxxx.xxx>
Sent: Friday, February 28, 2003 3:56 PM
Subject: [tuning] Re: Some magic meantones



> --- In tuning@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
>> 

>> i don't know if 3/14-comma is "magic", but it and
>> 7/26-comma are the only fraction-of-a-comma meantones
>> in my list that you left out of yours ... was that
>> simply an oversight, or are those two not "magic"?
> 

> "Magic" is a relative term. We have
> 
> 3/14-comma brat = 15/4 major [15/4, -10/9, -6/15]
> minor [5/2, 3/10, 4/3]
> 
> 7/26-comma brat = -15/4 major [-15/4, 10/21, -14/15]
> minor [-5/2, 7/10, -4/7]
> 
> 4/15-comma brat = -9/2 major [-9/2, 5/12, -8/15]
> minor [-3, -2/3, -1/2]
> 
> Note that if any one of the three, q-comma, major ratios,
> minor ratios, is exact, the other two are approximate.
> 
> We see 4/15-comma is good for minor triads and marginal
> for major ones, 3/14-comma is pretty good for minor triads 
> and fairly awful for major triads, and 7/26 is marginal for
> minor triads and really stinks for major triads.

Message: 6579 - Contents - Hide Contents

Date: Sat, 01 Mar 2003 07:11:28

Subject: Re: gd search mk. 4

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>>> Probably cents/(n*d) is a good metric for that...
>> 

>> why do you want cents to be high?
> 

> Whoops, that should be 1/(n*d)cents.
> 
> -C.


why do you want cents to be low?

Message: 6580 - Contents - Hide Contents

Date: Sat, 01 Mar 2003 21:01:41

Subject: [tuning] Re: Some magic meantones

From: wallyesterpaulrus

monz, gene is talking about the *beat rate ratios* in the close-
voiced root-position triads, since bob brought that up. getting 
different beatings into rhythmic synchrony with one another was *not* 
one of woolhouse's criteria.

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> hi Gene,
> 
> 
> i've shifted this thread over to here so as
> not to bother the regular tuning list members.
> 
> as usual, i sense intuitively that the explanation
> you've provided for me below has some real insight
> buried in it somewhere ... but it's so cryptic
> that i have no idea what you're saying.
> 
> can you please explain the numbers in brackets?
> 
> how do "we see that ... 7/26[-comma meantone]
> is marginal for minor triads and really stinks
> for major triads"?  based on my understanding of
> Woolhouse's work, i thought this was an "optimal"
> meantone.  ...?
> 
> (i apologize for being stupid, as the cause of
> it is simply my lack of reading and following your
> many posts here. thanks.)
> 
> 
> -monz
> 
> 
> ----- Original Message ----- 
> From: <gwsmith@s...>
> To: <tuning@xxxxxxxxxxx.xxx>
> Sent: Friday, February 28, 2003 3:56 PM
> Subject: [tuning] Re: Some magic meantones
> 
> 

>> --- In tuning@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
>>> 

>>> i don't know if 3/14-comma is "magic", but it and
>>> 7/26-comma are the only fraction-of-a-comma meantones
>>> in my list that you left out of yours ... was that
>>> simply an oversight, or are those two not "magic"?
>> 

>> "Magic" is a relative term. We have
>> 
>> 3/14-comma brat = 15/4 major [15/4, -10/9, -6/15]
>> minor [5/2, 3/10, 4/3]
>> 
>> 7/26-comma brat = -15/4 major [-15/4, 10/21, -14/15]
>> minor [-5/2, 7/10, -4/7]
>> 
>> 4/15-comma brat = -9/2 major [-9/2, 5/12, -8/15]
>> minor [-3, -2/3, -1/2]
>> 
>> Note that if any one of the three, q-comma, major ratios,
>> minor ratios, is exact, the other two are approximate.
>> 
>> We see 4/15-comma is good for minor triads and marginal
>> for major ones, 3/14-comma is pretty good for minor triads 
>> and fairly awful for major triads, and 7/26 is marginal for
>> minor triads and really stinks for major triads.

Message: 6581 - Contents - Hide Contents

Date: Sat, 01 Mar 2003 01:59:02

Subject: Re: gd search mk. 4

From: Carl Lumma

>> >hoops, that should be 1/(n*d)cents.
>> 
>> -C.
>

>why do you want cents to be low?


I want the simpler ratios in a given size range to
fare best.

I guess I should figure out if we want the scores
to be periodic as the interval size changes.  The
mean complexity of smaller ratios is higher than
that of larger ratios (for ratios in lowest terms).
We'll have to adjust for that if we do want scores
to be comparable in different size ranges.

I count the number of ratios n/d in lowest terms,
1 < n/d < 2, n > d, for...

(n*d <= 100) = 21
(n*d <= 1000) = 210
(n*d <= 10000) = 2111
(n*d <= 100000) = 21069

For n*d <= 3000, there are 633 such ratios.  They
are in this spreadsheet...

 * [with cont.]  (Wayb.)

...Excel's chart Wizard again completely confounds
me.  I just want to see how (n*d)cents changes with
respect to cents.  Will go to bed instead.

-Carl

Message: 6582 - Contents - Hide Contents

Date: Sat, 1 Mar 2003 21:51:07

Subject: Re: [tuning] Re: Some magic meantones

From: monz

hi paul,


thanks.  i've been following the thread enough
to surmise that those numbers had something to
do with beat ratios, but i'm still very unclear
as to what the numbers are actually representing,
as Gene posted them without any kind of label.


for example, 


>> 4/15-comma brat = -9/2 major [-9/2, 5/12, -8/15]
>> minor [-3, -2/3, -1/2]
>> 
>> ... We see 4/15-comma is good for minor triads 
>> and marginal for major ones




OK, so i can see that the numbers for minor are
both smaller and less complex than those for major.
but what *exactly* does "minor [-3, -2/3, -1/2]"
*signify*?

can i please get a real explanation of that?

and what's the "-9/2"?


Gene and/or paul or anyone else:  since my question
about 2/7-comma meantone the other day, i've gotten
very intrigued by it, and have been doing a lot of
research into it.   can we please use that as an example?


thanks.



-monz






> From: <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, March 01, 2003 1:01 PM
> Subject: [tuning-math] [tuning] Re: Some magic meantones
>
>
> monz, gene is talking about the *beat rate ratios* in the close-
> voiced root-position triads, since bob brought that up. getting 
> different beatings into rhythmic synchrony with one another was *not* 
> one of woolhouse's criteria.
> 
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
>> hi Gene,
>> 
>> 

>> i've shifted this thread over to here so as
>> not to bother the regular tuning list members.
>> 
>> as usual, i sense intuitively that the explanation
>> you've provided for me below has some real insight
>> buried in it somewhere ... but it's so cryptic
>> that i have no idea what you're saying.
>> 
>> can you please explain the numbers in brackets?
>> 
>> how do "we see that ... 7/26[-comma meantone]
>> is marginal for minor triads and really stinks
>> for major triads"?  based on my understanding of
>> Woolhouse's work, i thought this was an "optimal"
>> meantone.  ...?
>> 
>> (i apologize for being stupid, as the cause of
>> it is simply my lack of reading and following your
>> many posts here. thanks.)
>> 
>> 
>> -monz
>> 
>> 
>> ----- Original Message ----- 
>> From: <gwsmith@s...>
>> To: <tuning@xxxxxxxxxxx.xxx>
>> Sent: Friday, February 28, 2003 3:56 PM
>> Subject: [tuning] Re: Some magic meantones
>> 
>> 

>>> --- In tuning@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
>>>> 

>>>> i don't know if 3/14-comma is "magic", but it and
>>>> 7/26-comma are the only fraction-of-a-comma meantones
>>>> in my list that you left out of yours ... was that
>>>> simply an oversight, or are those two not "magic"?
>>> 

>>> "Magic" is a relative term. We have
>>> 
>>> 3/14-comma brat = 15/4 major [15/4, -10/9, -6/15]
>>> minor [5/2, 3/10, 4/3]
>>> 
>>> 7/26-comma brat = -15/4 major [-15/4, 10/21, -14/15]
>>> minor [-5/2, 7/10, -4/7]
>>> 
>>> 4/15-comma brat = -9/2 major [-9/2, 5/12, -8/15]
>>> minor [-3, -2/3, -1/2]
>>> 
>>> Note that if any one of the three, q-comma, major ratios,
>>> minor ratios, is exact, the other two are approximate.
>>> 
>>> We see 4/15-comma is good for minor triads and marginal
>>> for major ones, 3/14-comma is pretty good for minor triads 
>>> and fairly awful for major triads, and 7/26 is marginal for
>>> minor triads and really stinks for major triads.

Message: 6583 - Contents - Hide Contents

Date: Sun, 02 Mar 2003 13:07:37

Subject: Brats, beats, and 2/7-comma meantone

From: Gene Ward Smith

This got bounced as email by Yahoo, on the grounds that I don't have
an account. Does anyone know how to cure that?

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:


>>> 4/15-comma brat = -9/2 major [-9/2, 5/12, -8/15]
>>> minor [-3, -2/3, -1/2]
>>> 
>>> ... We see 4/15-comma is good for minor triads 
>>> and marginal for major ones
> 
> 
> 

> OK, so i can see that the numbers for minor are
> both smaller and less complex than those for major.
> but what *exactly* does "minor [-3, -2/3, -1/2]"
> *signify*?
> 
> can i please get a real explanation of that?
> 
> and what's the "-9/2"?


The ratios are, for a closed position major *or* minor triad:

[minor/major, fifth/minor, major/fifth]

These are (signed) ratios of beats. The "-9/2", which I've been
calling the "brat", is minor/major in the major triad, or

brat = (6t-5f)/(4t-5)

where t is the major third and f is the minor third. These definitions
are general, and have nothing specific to do with meantone. For
meantone, or any other linear temperament, we have a specific function
which goes from the fraction of a comma together with the canonical
5-limit generator to the brat. In the case of 2/7-comma meantone, that 
gives a brat of -1.497, or nearly -1.5. The major beats are

[-3/2, 5/6, -4/5]

the minor beats

[-1, 1, -1]

The synchronized beating of minor triads in 2/7-comma meantone is
striking. To get the same beats for major triads, we can use 
5/17-comma meantone, a system worth investigating.

Message: 6584 - Contents - Hide Contents

Date: Sun, 02 Mar 2003 08:56:26

Subject: Re: Brats, beats, and 2/7-comma meantone

From: Carl Lumma

>This got bounced as email by Yahoo, on the grounds that I don't have
>an account. Does anyone know how to cure that?


You have to send from the address that you're subscribed under.

-Carl

Message: 6585 - Contents - Hide Contents

Date: Sun, 02 Mar 2003 09:01:23

Subject: Re: Brats, beats, and 2/7-comma meantone

From: Carl Lumma

>The ratios are, for a closed position major *or* minor triad:


So what happens as one inverts and re-voices these chords?


>The synchronized beating of minor triads in 2/7-comma meantone is
>striking.


All the beats happen at the same rate?  Does that also mean they'll
happen at the same *time* (wouldn't this depend on phase)?

Bob says he's empirically verified that beat *ratios* of 2/1 or
3/2 are pleasant.  Is the same true for 1/1?

Is 1/1 what is traditionally called "equal beating"?

-Carl

Message: 6586 - Contents - Hide Contents

Date: Mon, 03 Mar 2003 08:58:04

Subject: Re: Brats, beats, and 2/7-comma meantone

From: Carl Lumma

>I sent it from my primary email addess. How do I change the one I am
>subscribed under?


Go to Sign In - * [with cont.]  (Wayb.) and click 'e-mail prefs' and
make sure the address is listed there.  Then click 'edit my groups'
and assign that address to the affected groups.

-Carl

Message: 6587 - Contents - Hide Contents

Date: Mon, 03 Mar 2003 04:03:52

Subject: Re: Brats, beats, and 2/7-comma meantone

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>> This got bounced as email by Yahoo, on the grounds that I don't 
have
>> an account. Does anyone know how to cure that?
> 

> You have to send from the address that you're subscribed under.


I sent it from my primary email addess. How do I change the one I am
subscribed under?

Message: 6588 - Contents - Hide Contents

Date: Tue, 04 Mar 2003 04:55:48

Subject: Re: Brats, beats, and 2/7-comma meantone

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

>> I sent it from my primary email addess. How do I change the one I am
>> subscribed under?
> 

> Go to Sign In - * [with cont.]  (Wayb.) and click 'e-mail prefs' and
> make sure the address is listed there.  Then click 'edit my groups'
> and assign that address to the affected groups.


Did that some time back. With all of the amazing computer problems I
have been having, it's small potates anyway aqt this point.

Message: 6589 - Contents - Hide Contents

Date: Wed, 05 Mar 2003 08:13:43

Subject: An RI version of Wendell Well

From: Gene Ward Smith

If we look at the brat ratios of Wendell Well, they are 
approximately [r, 3/2, 3/2, r, 3/2, 3/2, 3/2, r, 3/2, q, r, r] 
where both r and q are nearly 2, but r is a little less and q is  
a little more. If we solve for these under the condition that  
the product of the fifths comes to 128 (octave closure) then q is 
 
320*(-27*r+18*r^2+40)/r/(2187*r^4-5184*r^3-7728*r+5760+5216*r^2) 
 
We can relate r and q in various ways; for instance making q exactly 
2, or having the average of r and q be 2. The simplest way however 
is to make r exactly 2, which makes q to be 580/293. This gives us 
the following scale: 
 
[1, 1215/1144, 321/286, 68187/57200, 180/143, 3823/2860, 405/286, 
214/143, 22729/14300, 963/572, 5107/2860, 270/143] 
 
It is a rational intonation version of Robert's temperament.

Message: 6590 - Contents - Hide Contents

Date: Wed, 5 Mar 2003 12:21:35

Subject: need definition of "brat"

From: monz

Gene or whomever,


can i please get a good, complete definition
(with examples) of "brat", for the Tuning Dictionary?

thanks.



i've decided to create a page about Wendell
well-temperaments.  this is the incipient version:
404 Not Found * [with cont.]  Search for http://sonic-arts.org/dict/wendell.htm in Wayback Machine

... i'm having trouble following much of the thread,
but i hope to be able to expand this into a nice
comprehensive page about this interesting category
of tunings.



-monz

Message: 6591 - Contents - Hide Contents

Date: Wed, 5 Mar 2003 10:37:51

Subject: Re: An RI version of Wendell Well

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:


>320*(-27*r+18*r^2+40)/r/(2187*r^4-5184*r^3-7728*r+5760+5216*r^2) 


                      ^ ^ two divisions?

I was wondering about the result of r = q. So it wasn't as
hairy as I thought!


>It is a rational intonation version of Robert's temperament. 


Where 1/1 = A.

Manuel

Message: 6592 - Contents - Hide Contents

Date: Wed, 05 Mar 2003 21:29:08

Subject: Re: An RI version of Wendell Well

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:
>> Gene wrote:
>> 
>>> 320*(-27*r+18*r^2+40)/r/(2187*r^4-5184*r^3-7728*r+5760+5216*r^2) 
>> 

>> ^ ^ two divisions?
> 
> Blame Maple.
> 

>> I was wondering about the result of r = q. So it wasn't as
>> hairy as I thought!
> 

> What's really hairy, it turns out, is trying to sort out all the
> possibilities. I have more than 41 but less than 510 (depending on 
how
> many duplications) examples of RI scales *all* of whose brats are
> either exactly 2 or exactly 3/2, and with either 6 or 7 pure fifths.
> How do I evaluate these?


please ask bob wendell. some of these may end up being published 
right alongside his!

Message: 6593 - Contents - Hide Contents

Date: Wed, 05 Mar 2003 10:55:05

Subject: Re: An RI version of Wendell Well

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:

> Gene wrote:
> 
>> 320*(-27*r+18*r^2+40)/r/(2187*r^4-5184*r^3-7728*r+5760+5216*r^2) 
> 

>                       ^ ^ two divisions?

Blame Maple.

> I was wondering about the result of r = q. So it wasn't as
> hairy as I thought!


What's really hairy, it turns out, is trying to sort out all the
possibilities. I have more than 41 but less than 510 (depending on how
many duplications) examples of RI scales *all* of whose brats are
either exactly 2 or exactly 3/2, and with either 6 or 7 pure fifths.
How do I evaluate these?

Message: 6594 - Contents - Hide Contents

Date: Wed, 5 Mar 2003 12:18:27

Subject: Re: An RI version of Wendell Well

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:

>Blame Maple.


Can it be written out as a polynomial? Then I could find
the result for r = q myself.


>How do I evaluate these?


Brute force? I find it a little easier to use the beat
rate quotient of 3/2 and 5/4 which is 0 and ~1/5 (say 1/m) 
for 6 tempered fifths. What would be the m nearest to an
integer for 5 tempered fifths in any order, or 7? 
Is Robert's order the best order for 6 tempered fifths? 
(I suppose so).

Manuel

Message: 6595 - Contents - Hide Contents

Date: Wed, 05 Mar 2003 20:57:20

Subject: Re: need definition of "brat"

From: Carl Lumma

>Why not a page about the more general concept of well temperaments
>with synchronized beating? Unless that's what you meant...


Yes; I for one have no idea why I the Maj3rd/Min3rd ratio should
be so important, how to parse the Scala output Paul and Manuel
have been posting, how Paul's approach compares to Bob's/Gene's,
etc.

-Carl

Message: 6596 - Contents - Hide Contents

Date: Wed, 05 Mar 2003 13:44:40

Subject: 114 temperaments in the big parade

From: Gene Ward Smith

Here are the temperaments I spoke of. I give the scale degrees, the
circle of fifths, the major thirds and the brats, in that order. Note
that the brats are all exact!

If anyone can explain how to evaluate these, I'm interested. Saying
brute force doesn't help, since I don't know what I should try to force.


Six pure fifths


[1, 540/511, 572/511, 1215/1022, 1287/1022, 4146795/3104836, 720/511,
3823/2555, 810/511, 858/511, 3645/2044, 963/511]
[3823/2555, 5720/3823, 3/2, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2,
6826/4557, 5110/3413]
[1287/1022, 4815/3823, 180/143, 180/143, 180/143, 135/107, 81/64,
30717/24304, 1095/868, 3823/3038, 5720/4557, 4290/3413]
[2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2, 2]


[1, 540/511, 572/511, 1215/1022, 1287/1022, 4146795/3104836,
2889/2044, 3823/2555, 810/511, 858/511, 3645/2044, 963/511]
[3823/2555, 5720/3823, 3/2, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2,
6826/4557, 5110/3413]
[1287/1022, 4815/3823, 2889/2288, 180/143, 180/143, 135/107, 135/107,
30717/24304, 1095/868, 3823/3038, 5720/4557, 4290/3413]
[2, 2, 3/2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2]


[1, 14445/13652, 3820/3413, 4050/3413, 8595/6826, 18225/13652,
19305/13652, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 6435/3413]
[5107/3413, 7640/5107, 3/2, 3/2, 286/191, 3/2, 214/143, 160/107, 3/2,
3/2, 3/2, 6826/4559]
[8595/6826, 6435/5107, 3861/3056, 963/764, 240/191, 180/143, 180/143,
135/107, 92151/72944, 45963/36472, 5730/4559, 5730/4559]
[2, 2, 3/2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 14445/13652, 3823/3413, 4050/3413, 4290/3413, 18225/13652,
4815/3413, 5110/3413, 5400/3413, 11469/6826, 6075/3413, 6435/3413]
[5110/3413, 3823/2555, 3/2, 5720/3823, 3/2, 214/143, 3/2, 160/107,
3/2, 3/2, 3/2, 68260/45599]
[4290/3413, 1287/1022, 4815/3823, 4815/3823, 180/143, 180/143,
135/107, 135/107, 460755/364792, 114975/91198, 57345/45599, 57345/45599]
[2, 2, 3/2, 2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2]


[1, 7200/6823, 7640/6823, 8100/6823, 8580/6823, 18225/13646,
9630/6823, 10214/6823, 10800/6823, 11460/6823, 12150/6823, 12870/6823]
[10214/6823, 7640/5107, 3/2, 286/191, 3/2, 214/143, 160/107, 3/2, 3/2,
3/2, 3/2, 13646/9115]
[8580/6823, 6435/5107, 963/764, 240/191, 180/143, 180/143, 135/107,
81/64, 184221/145840, 45963/36460, 2292/1823, 2292/1823]
[2, 2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2]


[1, 75/71, 1275/1136, 675/568, 715/568, 6075/4544, 100/71, 425/284,
225/142, 955/568, 2025/1136, 535/284]
[425/284, 3/2, 382/255, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2,
3/2, 1136/759]
[715/568, 107/85, 64/51, 240/191, 180/143, 135/107, 81/64, 81/64,
639/506, 1275/1012, 1275/1012, 955/759]
[2, 3/2, 2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2]


[1, 5400/5107, 5720/5107, 6075/5107, 6420/5107, 41449725/31019918,
14445/10214, 7646/5107, 8100/5107, 8580/5107, 18225/10214, 9630/5107]
[7646/5107, 5720/3823, 3/2, 214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2,
13646/9111, 10214/6823]
[6420/5107, 4815/3823, 2889/2288, 180/143, 135/107, 135/107, 135/107,
61407/48592, 15321/12148, 3823/3037, 11440/9111, 8580/6823]
[2, 2, 3/2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2]


[1, 14445/13652, 3823/3413, 4050/3413, 4290/3413, 18225/13652,
19305/13652, 5110/3413, 5400/3413, 11469/6826, 6075/3413, 6435/3413]
[5110/3413, 3823/2555, 3/2, 5720/3823, 3/2, 3/2, 214/143, 160/107,
3/2, 3/2, 3/2, 68260/45599]
[4290/3413, 1287/1022, 19305/15292, 4815/3823, 180/143, 180/143,
180/143, 135/107, 460755/364792, 114975/91198, 57345/45599, 57345/45599]
[2, 2, 3/2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 5400/5107, 5720/5107, 6075/5107, 6420/5107, 41449725/31019918,
7200/5107, 7646/5107, 8100/5107, 8580/5107, 18225/10214, 9630/5107]
[7646/5107, 5720/3823, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2,
13646/9111, 10214/6823]
[6420/5107, 4815/3823, 180/143, 180/143, 135/107, 135/107, 81/64,
61407/48592, 15321/12148, 3823/3037, 11440/9111, 8580/6823]
[2, 2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 2]


[1, 3600/3413, 3820/3413, 4050/3413, 8595/6826, 18225/13652,
4815/3413, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 6435/3413]
[5107/3413, 7640/5107, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 3/2, 6826/4559]
[8595/6826, 6435/5107, 963/764, 240/191, 240/191, 180/143, 135/107,
81/64, 92151/72944, 45963/36472, 5730/4559, 5730/4559]
[2, 2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2]


[1, 28890/27313, 160/143, 32400/27313, 180/143, 36450/27313, 270/191,
214/143, 43200/27313, 240/143, 48600/27313, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 286/191]
[180/143, 135/107, 3861/3056, 963/764, 240/191, 240/191, 180/143,
135/107, 3861/3056, 963/764, 240/191, 240/191]
[2, 2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 16200/15301, 160/143, 18225/15301, 180/143, 34809750/26057603,
21600/15301, 214/143, 24300/15301, 240/143, 46473750/26057603, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2, 2550/1703,
382/255, 286/191]
[180/143, 135/107, 135/107, 135/107, 135/107, 135/107, 34425/27248,
8595/6812, 165/131, 2140/1703, 64/51, 240/191]
[2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 2]


[1, 14445/13652, 3820/3413, 4050/3413, 8595/6826, 18225/13652,
4815/3413, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 6435/3413]
[5107/3413, 7640/5107, 3/2, 3/2, 286/191, 214/143, 3/2, 160/107, 3/2,
3/2, 3/2, 6826/4559]
[8595/6826, 6435/5107, 963/764, 963/764, 240/191, 180/143, 135/107,
135/107, 92151/72944, 45963/36472, 5730/4559, 5730/4559]
[2, 2, 3/2, 3/2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2]


[1, 540/511, 572/511, 1215/1022, 1287/1022, 62201925/46602178,
720/511, 3823/2555, 810/511, 858/511, 41467950/23301089, 963/511]
[3823/2555, 5720/3823, 3/2, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2,
68260/45599, 3/2, 5110/3413]
[1287/1022, 4815/3823, 180/143, 180/143, 180/143, 135/107,
460755/364792, 460755/364792, 114975/91198, 57345/45599, 4290/3413,
4290/3413]
[2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2]


[1, 14445/13646, 7656/6823, 8100/6823, 8595/6823, 18225/13646,
9630/6823, 10208/6823, 10800/6823, 11460/6823, 12150/6823, 12870/6823]
[10208/6823, 3/2, 955/638, 3/2, 286/191, 214/143, 3/2, 160/107, 3/2,
3/2, 3/2, 13646/9115]
[8595/6823, 585/464, 1605/1276, 963/764, 240/191, 180/143, 135/107,
135/107, 184221/145840, 11484/9115, 11484/9115, 2292/1823]
[2, 3/2, 2, 3/2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2]


[1, 7200/6823, 7656/6823, 8100/6823, 8595/6823, 18225/13646,
9630/6823, 10208/6823, 10800/6823, 11460/6823, 12150/6823, 12870/6823]
[10208/6823, 3/2, 955/638, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 3/2, 13646/9115]
[8595/6823, 585/464, 1605/1276, 240/191, 240/191, 180/143, 135/107,
81/64, 184221/145840, 11484/9115, 11484/9115, 2292/1823]
[2, 3/2, 2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2]


[1, 4050/3823, 4290/3823, 207248625/174267632, 4815/3823,
465375375/348535264, 5400/3823, 5720/3823, 6075/3823, 6420/3823,
621745875/348535264, 7200/3823]
[5720/3823, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 34115/22792,
3/2, 10214/6823, 7646/5107]
[4815/3823, 180/143, 180/143, 135/107, 135/107, 921105/729344,
921105/729344, 229815/182336, 57345/45584, 8580/6823, 8580/6823,
6420/5107]
[2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2]


[1, 5400/5107, 5730/5107, 6075/5107, 6435/5107, 24869835/18620122,
7200/5107, 7640/5107, 8100/5107, 8580/5107, 16579890/9310061, 9630/5107]
[7640/5107, 3/2, 286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2,
13646/9115, 3/2, 10214/6823]
[6435/5107, 963/764, 240/191, 180/143, 180/143, 135/107,
184221/145840, 184221/145840, 45963/36460, 2292/1823, 8595/6823,
8580/6823]
[2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2]


[1, 14445/13652, 15321/13652, 4050/3413, 8595/6826, 18225/13652,
19305/13652, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 6435/3413]
[5107/3413, 3/2, 7640/5107, 3/2, 286/191, 3/2, 214/143, 160/107, 3/2,
3/2, 3/2, 6826/4559]
[8595/6826, 6435/5107, 6435/5107, 963/764, 240/191, 180/143, 180/143,
135/107, 92151/72944, 45963/36472, 45963/36472, 5730/4559]
[2, 3/2, 2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 405/382, 429/382, 2071575/1740392, 240/191, 1162755/870196,
270/191, 286/191, 1215/764, 321/191, 6214725/3480784, 360/191]
[286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 1705/1139, 3/2,
232/155, 955/638]
[240/191, 180/143, 180/143, 135/107, 81/64, 46035/36448, 46035/36448,
2871/2278, 2865/2278, 39/31, 39/31, 1605/1276]
[2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2]


[1, 405/382, 429/382, 2071575/1740392, 240/191, 1162755/870196,
270/191, 286/191, 1215/764, 321/191, 387585/217549, 360/191]
[286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 1705/1139,
232/155, 3/2, 955/638]
[240/191, 180/143, 180/143, 135/107, 81/64, 46035/36448, 2871/2278,
2871/2278, 2865/2278, 39/31, 585/464, 1605/1276]
[2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2, 2, 3/2, 2]


[1, 14445/13652, 7665/6826, 4050/3413, 4290/3413, 18225/13652,
19305/13652, 5110/3413, 5400/3413, 11469/6826, 6075/3413, 6435/3413]
[5110/3413, 3/2, 3823/2555, 5720/3823, 3/2, 3/2, 214/143, 160/107,
3/2, 3/2, 3/2, 68260/45599]
[4290/3413, 1287/1022, 1287/1022, 4815/3823, 180/143, 180/143,
180/143, 135/107, 460755/364792, 114975/91198, 114975/91198, 57345/45599]
[2, 3/2, 2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 405/382, 429/382, 3645/3056, 240/191, 11745/8786, 270/191,
286/191, 1215/764, 321/191, 62775/35144, 360/191]
[286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 310/207,
232/155, 955/638]
[240/191, 180/143, 180/143, 135/107, 81/64, 81/64, 465/368, 29/23,
955/759, 260/207, 39/31, 1605/1276]
[2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2]


[1, 14445/13646, 15321/13646, 8100/6823, 8580/6823, 18225/13646,
9630/6823, 10214/6823, 10800/6823, 11469/6823, 12150/6823, 12840/6823]
[10214/6823, 3/2, 7646/5107, 5720/3823, 214/143, 3/2, 3/2, 160/107,
3/2, 3/2, 3/2, 34115/22792]
[8580/6823, 6420/5107, 6420/5107, 4815/3823, 180/143, 135/107,
135/107, 135/107, 921105/729344, 229815/182336, 229815/182336,
57345/45584]
[2, 3/2, 2, 2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2]


[1, 360/341, 174/155, 405/341, 39/31, 3645/2728, 963/682, 232/155,
540/341, 573/341, 1215/682, 642/341]
[232/155, 3/2, 955/638, 286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2,
3/2, 1705/1139]
[39/31, 1605/1276, 1605/1276, 240/191, 180/143, 135/107, 135/107,
81/64, 46035/36448, 2871/2278, 2871/2278, 2865/2278]
[2, 3/2, 2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2]


[1, 1215/1144, 160/143, 95499/80080, 180/143, 139239/104104, 405/286,
214/143, 3645/2288, 240/143, 185895/104104, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 262/175, 2550/1703,
382/255, 286/191]
[180/143, 135/107, 81/64, 81/64, 81/64, 3537/2800, 459/364, 573/455,
44/35, 2140/1703, 64/51, 240/191]
[2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 2]


[1, 4050/3823, 4290/3823, 207248625/174267632, 4815/3823,
465375375/348535264, 5400/3823, 5720/3823, 6075/3823, 6420/3823,
155125125/87133816, 7200/3823]
[5720/3823, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 34115/22792,
10214/6823, 3/2, 7646/5107]
[4815/3823, 180/143, 180/143, 135/107, 135/107, 921105/729344,
229815/182336, 229815/182336, 57345/45584, 8580/6823, 6435/5107,
6420/5107]
[2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2, 2]


[1, 14445/13652, 15321/13652, 4050/3413, 8595/6826, 18225/13652,
19305/13652, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 25785/13652]
[5107/3413, 3/2, 7640/5107, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2,
3/2, 3/2, 6826/4559]
[8595/6826, 25785/20428, 6435/5107, 963/764, 240/191, 240/191,
180/143, 135/107, 92151/72944, 45963/36472, 45963/36472, 5730/4559]
[2, 3/2, 2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 5400/5107, 11469/10214, 6075/5107, 6420/5107, 621745875/465594976,
14445/10214, 7646/5107, 8100/5107, 8580/5107, 18225/10214, 9630/5107]
[7646/5107, 3/2, 5720/3823, 214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2,
34115/22792, 10214/6823]
[6420/5107, 4815/3823, 4815/3823, 180/143, 135/107, 135/107, 135/107,
921105/729344, 229815/182336, 57345/45584, 57345/45584, 8580/6823]
[2, 3/2, 2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2]


[1, 62775/59228, 321/286, 35235/29614, 180/143, 3480975/2606032,
405/286, 214/143, 188325/118456, 240/143, 1160325/651508, 270/143]
[214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 1705/1139, 3/2, 232/155,
955/638, 3/2, 286/191]
[180/143, 135/107, 135/107, 46035/36448, 46035/36448, 2871/2278,
2865/2278, 1719/1364, 39/31, 1605/1276, 963/764, 240/191]
[2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2, 3/2, 2]


[1, 1215/1144, 321/286, 35235/29614, 180/143, 3480975/2606032,
405/286, 214/143, 188325/118456, 240/143, 1160325/651508, 270/143]
[214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2, 1705/1139, 232/155,
955/638, 3/2, 286/191]
[180/143, 135/107, 135/107, 81/64, 46035/36448, 2871/2278, 2865/2278,
2865/2278, 39/31, 1605/1276, 963/764, 240/191]
[2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2, 2]


[1, 1215/1144, 321/286, 41425425/34750144, 180/143, 23224725/17375072,
405/286, 214/143, 13808475/8687536, 963/572, 31025025/17375072, 270/143]
[214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2, 11365/7594, 3/2,
10214/6819, 7646/5107, 5720/3823]
[180/143, 135/107, 135/107, 135/107, 306855/243008, 306855/243008,
76605/60752, 19115/15188, 2860/2273, 8560/6819, 6420/5107, 4815/3823]
[2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2, 2]


[1, 405/382, 214/191, 3645/3056, 240/191, 1162755/869432, 270/191,
286/191, 1215/764, 321/191, 776385/434716, 360/191]
[286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 852/569,
319/213, 955/638]
[240/191, 180/143, 135/107, 135/107, 81/64, 81/64, 5751/4552,
2871/2276, 2865/2276, 715/569, 535/426, 1605/1276]
[2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2]


[1, 405/382, 214/191, 2069145/1738864, 240/191, 4647375/3477728,
270/191, 286/191, 1215/764, 320/191, 6207435/3477728, 360/191]
[286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 1703/1138, 3/2,
2550/1703, 382/255]
[240/191, 180/143, 135/107, 81/64, 81/64, 45981/36416, 45981/36416,
11475/9104, 2865/2276, 165/131, 2140/1703, 64/51]
[2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2]


[1, 575100/542971, 321/286, 1293975/1085942, 180/143, 5801625/4343768,
405/286, 214/143, 862650/542971, 240/143, 176175/98722, 270/143]
[214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 5680/3797, 3/2, 3/2, 319/213,
955/638, 286/191]
[180/143, 135/107, 135/107, 9585/7594, 9585/7594, 9585/7594,
4785/3797, 2865/2272, 715/568, 535/426, 1605/1276, 240/191]
[2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 2, 2, 2]


[1, 62775/59228, 160/143, 35235/29614, 180/143, 3480975/2606032,
405/286, 214/143, 188325/118456, 240/143, 1160325/651508, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 1705/1139, 3/2, 232/155,
955/638, 3/2, 286/191]
[180/143, 135/107, 81/64, 46035/36448, 46035/36448, 2871/2278,
2865/2278, 1719/1364, 39/31, 1605/1276, 240/191, 240/191]
[2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2, 3/2, 2]


[1, 5167800/4879093, 120/107, 5801625/4879093, 135/107,
13030875/9758186, 6890400/4879093, 160/107, 7751700/4879093, 180/107,
8687250/4879093, 9211050/4879093]
[160/107, 3/2, 3/2, 3/2, 68230/45599, 10208/6823, 3/2, 3/2, 955/638,
286/191, 3/2, 214/143]
[135/107, 921105/729584, 57420/45599, 57420/45599, 57420/45599,
8595/6823, 585/464, 585/464, 1605/1276, 240/191, 180/143, 180/143]
[2, 3/2, 3/2, 3/2, 2, 2, 3/2, 3/2, 2, 2, 3/2, 2]


[1, 405/382, 214/191, 57510/48323, 240/191, 11745/8786, 270/191,
286/191, 1215/764, 321/191, 86265/48323, 360/191]
[286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2, 1136/759, 3/2,
319/213, 955/638]
[240/191, 180/143, 135/107, 135/107, 81/64, 639/506, 639/506, 29/23,
955/759, 715/568, 535/426, 1605/1276]
[2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2]


[1, 4050/3823, 4280/3823, 18225/15292, 4815/3823, 93075075/69632122,
5400/3823, 5720/3823, 6075/3823, 6420/3823, 124276275/69632122, 7200/3823]
[5720/3823, 214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2,
13638/9107, 10214/6819, 7646/5107]
[4815/3823, 180/143, 135/107, 135/107, 135/107, 81/64, 184113/145712,
45963/36428, 11469/9107, 11440/9107, 8560/6819, 6420/5107]
[2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2]


[1, 360/341, 382/341, 405/341, 39/31, 3645/2728, 480/341, 232/155,
540/341, 573/341, 1215/682, 642/341]
[232/155, 955/638, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2,
3/2, 1705/1139]
[39/31, 1605/1276, 240/191, 240/191, 180/143, 135/107, 81/64, 81/64,
46035/36448, 2871/2278, 2865/2278, 2865/2278]
[2, 2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2]


[1, 28890/27313, 214/191, 32400/27313, 240/191, 36450/27313, 270/191,
286/191, 43200/27313, 320/191, 48600/27313, 360/191]
[286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 214/143, 160/107, 3/2, 3/2,
3/2, 382/255]
[240/191, 180/143, 135/107, 2889/2288, 180/143, 180/143, 180/143,
135/107, 1719/1360, 429/340, 107/85, 64/51]
[2, 2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 18/17, 286/255, 183924/154819, 64/51, 413829/309638, 24/17,
382/255, 27/17, 428/255, 275886/154819, 32/17]
[382/255, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 13624/9107,
3/2, 3/2, 2550/1703]
[64/51, 240/191, 180/143, 135/107, 81/64, 45981/36428, 45981/36428,
45981/36428, 11475/9107, 8595/6812, 165/131, 2140/1703]
[2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 2]


[1, 1382265/1303874, 160/143, 773550/651937, 180/143, 3480975/2607748,
921510/651937, 214/143, 2068335/1303874, 240/143, 1160325/651937, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 6826/4559, 3/2, 5107/3413,
7640/5107, 3/2, 3/2, 286/191]
[180/143, 135/107, 92151/72944, 92151/72944, 45963/36472, 5730/4559,
8595/6826, 8595/6826, 6435/5107, 963/764, 240/191, 240/191]
[2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 2]


[1, 920565/868868, 160/143, 516375/434434, 180/143, 1160325/868868,
306855/217217, 214/143, 344250/217217, 240/143, 1549125/868868, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 2273/1519, 3/2, 3400/2273, 3/2, 3/2,
382/255, 286/191]
[180/143, 135/107, 61371/48608, 61371/48608, 3825/3038, 3825/3038,
11475/9092, 2865/2273, 429/340, 107/85, 64/51, 240/191]
[2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 3/2, 3/2, 2, 2]


[1, 13816575/13041314, 160/143, 704700/592787, 180/143,
17404875/13041314, 9211050/6520657, 214/143, 939600/592787, 240/143,
11603250/6520657, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 68230/45599, 3/2, 10208/6823, 3/2,
955/638, 3/2, 286/191]
[180/143, 135/107, 921105/729584, 921105/729584, 57420/45599,
57420/45599, 8595/6823, 8595/6823, 585/464, 1605/1276, 240/191, 240/191]
[2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 3/2, 2, 3/2, 2]


[1, 172125/162533, 120/107, 1549125/1300264, 135/107, 868725/650132,
920565/650132, 160/107, 516375/325066, 180/107, 1160325/650132, 405/214]
[160/107, 3/2, 3/2, 3/2, 3/2, 2273/1519, 3400/2273, 3/2, 3/2, 382/255,
286/191, 214/143]
[135/107, 81/64, 61371/48608, 3825/3038, 3825/3038, 3825/3038,
2865/2273, 429/340, 107/85, 64/51, 240/191, 180/143]
[2, 3/2, 3/2, 3/2, 3/2, 2, 2, 3/2, 3/2, 2, 2, 2]


[1, 14445/13646, 7656/6823, 8100/6823, 8595/6823, 18225/13646,
19305/13646, 10208/6823, 10800/6823, 11484/6823, 12150/6823, 25785/13646]
[10208/6823, 3/2, 3/2, 955/638, 3/2, 286/191, 214/143, 160/107, 3/2,
3/2, 3/2, 68230/45599]
[8595/6823, 25785/20416, 585/464, 1605/1276, 240/191, 240/191,
180/143, 135/107, 921105/729584, 57420/45599, 57420/45599, 57420/45599]
[2, 3/2, 3/2, 2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 387585/364792, 9/8, 868725/729584, 921105/729584, 1950075/1459168,
129195/91198, 3/2, 1160325/729584, 27/16, 650025/364792, 86130/45599]
[3/2, 3/2, 3/2, 68230/45599, 10208/6823, 3/2, 3/2, 955/638, 286/191,
214/143, 3/2, 160/107]
[921105/729584, 57420/45599, 57420/45599, 57420/45599, 8595/6823,
585/464, 1605/1276, 1605/1276, 240/191, 180/143, 135/107, 135/107]
[3/2, 3/2, 3/2, 2, 2, 3/2, 3/2, 2, 2, 2, 3/2, 2]


[1, 2889/2728, 9/8, 405/341, 1719/1364, 3645/2728, 351/248, 3/2,
540/341, 261/155, 1215/682, 5157/2728]
[3/2, 3/2, 232/155, 955/638, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 775/518]
[1719/1364, 1719/1364, 39/31, 1605/1276, 240/191, 240/191, 180/143,
135/107, 20925/16576, 20925/16576, 20925/16576, 1305/1036]
[3/2, 3/2, 2, 2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 4815/4544, 9/8, 675/568, 11475/9088, 6075/4544, 6435/4544, 3/2,
225/142, 3825/2272, 2025/1136, 8595/4544]
[3/2, 3/2, 425/284, 3/2, 382/255, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 4544/3037]
[11475/9088, 2865/2272, 715/568, 107/85, 64/51, 240/191, 180/143,
135/107, 3834/3037, 3834/3037, 3834/3037, 3825/3037]
[3/2, 3/2, 2, 3/2, 2, 2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 75/71, 1275/1136, 675/568, 2865/2272, 6075/4544, 1605/1136, 3/2,
225/142, 3825/2272, 2025/1136, 2145/1136]
[3/2, 425/284, 3/2, 382/255, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2,
3/2, 4544/3037]
[2865/2272, 715/568, 107/85, 64/51, 240/191, 180/143, 135/107, 81/64,
3834/3037, 3834/3037, 3825/3037, 3825/3037]
[3/2, 2, 3/2, 2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2]


[1, 2889/2728, 9/8, 405/341, 1719/1364, 3645/2728, 351/248, 3/2,
540/341, 261/155, 1215/682, 117/62]
[3/2, 3/2, 232/155, 955/638, 286/191, 3/2, 214/143, 160/107, 3/2, 3/2,
3/2, 775/518]
[1719/1364, 39/31, 39/31, 1605/1276, 240/191, 180/143, 180/143,
135/107, 20925/16576, 20925/16576, 20925/16576, 1305/1036]
[3/2, 3/2, 2, 2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2]


[1, 2889/2728, 174/155, 405/341, 1719/1364, 3645/2728, 963/682, 3/2,
540/341, 261/155, 1215/682, 117/62]
[3/2, 232/155, 3/2, 955/638, 286/191, 214/143, 3/2, 160/107, 3/2, 3/2,
3/2, 775/518]
[1719/1364, 39/31, 1605/1276, 1605/1276, 240/191, 180/143, 135/107,
135/107, 20925/16576, 20925/16576, 1305/1036, 1305/1036]
[3/2, 2, 3/2, 2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2]


[1, 675/638, 2865/2552, 6075/5104, 585/464, 14124375/10575488,
450/319, 3/2, 2025/1276, 195/116, 4708125/2643872, 4815/2552]
[3/2, 955/638, 286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 775/518,
3/2, 232/155]
[585/464, 1605/1276, 240/191, 180/143, 180/143, 135/107, 20925/16576,
20925/16576, 1305/1036, 1305/1036, 1719/1364, 39/31]
[3/2, 2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2]



Seven pure fifths


[1, 14445/13652, 7665/6826, 4050/3413, 4290/3413, 18225/13652,
19305/13652, 5110/3413, 5400/3413, 11469/6826, 6075/3413, 6435/3413]
[5110/3413, 3/2, 3823/2555, 5720/3823, 3/2, 3/2, 214/143, 160/107,
3/2, 3/2, 3/2, 3/2]
[4290/3413, 1287/1022, 1287/1022, 4815/3823, 180/143, 180/143,
180/143, 135/107, 81/64, 68985/54608, 68985/54608, 34407/27304]
[2, 3/2, 2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 405/382, 429/382, 6205005/5215064, 240/191, 18225/13652, 270/191,
286/191, 1215/764, 321/191, 18615015/10430128, 360/191]
[286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 5107/3413, 3/2,
7640/5107, 3/2]
[240/191, 180/143, 180/143, 135/107, 81/64, 137889/109216,
137889/109216, 8595/6826, 8595/6826, 6435/5107, 6435/5107, 963/764]
[2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 2, 3/2]


[1, 405/382, 429/382, 3645/3056, 240/191, 18225/13646, 270/191,
286/191, 1215/764, 321/191, 18615015/10425544, 360/191]
[286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 10214/6823,
7640/5107, 3/2]
[240/191, 180/143, 180/143, 135/107, 81/64, 81/64, 137889/109168,
8595/6823, 8595/6823, 8580/6823, 6435/5107, 963/764]
[2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 3/2]


[1, 540/511, 572/511, 1215/1022, 1287/1022, 18225/13652, 2889/2044,
3823/2555, 810/511, 858/511, 3645/2044, 963/511]
[3823/2555, 5720/3823, 3/2, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2,
5110/3413, 3/2]
[1287/1022, 4815/3823, 2889/2288, 180/143, 180/143, 135/107, 135/107,
68985/54608, 68985/54608, 34407/27304, 4290/3413, 1287/1022]
[2, 2, 3/2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 14445/13652, 3820/3413, 4050/3413, 8595/6826, 18225/13652,
19305/13652, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 6435/3413]
[5107/3413, 7640/5107, 3/2, 3/2, 286/191, 3/2, 214/143, 160/107, 3/2,
3/2, 3/2, 3/2]
[8595/6826, 6435/5107, 3861/3056, 963/764, 240/191, 180/143, 180/143,
135/107, 81/64, 137889/109216, 8595/6826, 8595/6826]
[2, 2, 3/2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 14445/13652, 3823/3413, 4050/3413, 4290/3413, 18225/13652,
4815/3413, 5110/3413, 5400/3413, 11469/6826, 6075/3413, 6435/3413]
[5110/3413, 3823/2555, 3/2, 5720/3823, 3/2, 214/143, 3/2, 160/107,
3/2, 3/2, 3/2, 3/2]
[4290/3413, 1287/1022, 4815/3823, 4815/3823, 180/143, 180/143,
135/107, 135/107, 81/64, 68985/54608, 34407/27304, 34407/27304]
[2, 2, 3/2, 2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 75/71, 1275/1136, 675/568, 715/568, 6075/4544, 100/71, 425/284,
225/142, 955/568, 2025/1136, 535/284]
[425/284, 3/2, 382/255, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2,
3/2, 3/2]
[715/568, 107/85, 64/51, 240/191, 180/143, 135/107, 81/64, 81/64,
81/64, 11475/9088, 11475/9088, 2865/2272]
[2, 3/2, 2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2]


[1, 5400/5107, 5720/5107, 6075/5107, 6420/5107, 18225/13646,
14445/10214, 7646/5107, 8100/5107, 8580/5107, 18225/10214, 9630/5107]
[7646/5107, 5720/3823, 3/2, 214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2,
10214/6823, 3/2]
[6420/5107, 4815/3823, 2889/2288, 180/143, 135/107, 135/107, 135/107,
137889/109168, 137889/109168, 34407/27292, 8580/6823, 6435/5107]
[2, 2, 3/2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 14445/13652, 3823/3413, 4050/3413, 4290/3413, 18225/13652,
19305/13652, 5110/3413, 5400/3413, 11469/6826, 6075/3413, 6435/3413]
[5110/3413, 3823/2555, 3/2, 5720/3823, 3/2, 3/2, 214/143, 160/107,
3/2, 3/2, 3/2, 3/2]
[4290/3413, 1287/1022, 19305/15292, 4815/3823, 180/143, 180/143,
180/143, 135/107, 81/64, 68985/54608, 34407/27304, 34407/27304]
[2, 2, 3/2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 5400/5107, 5720/5107, 6075/5107, 6420/5107, 18225/13646,
7200/5107, 7646/5107, 8100/5107, 8580/5107, 18225/10214, 9630/5107]
[7646/5107, 5720/3823, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2,
10214/6823, 3/2]
[6420/5107, 4815/3823, 180/143, 180/143, 135/107, 135/107, 81/64,
137889/109168, 137889/109168, 34407/27292, 8580/6823, 6435/5107]
[2, 2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 5400/5107, 5720/5107, 6075/5107, 6420/5107, 182250/136547,
7200/5107, 7646/5107, 8100/5107, 8580/5107, 243000/136547, 9630/5107]
[7646/5107, 5720/3823, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2,
204280/136547, 3/2, 3/2]
[6420/5107, 4815/3823, 180/143, 180/143, 135/107, 135/107,
689445/546188, 689445/546188, 689445/546188, 172035/136547, 6435/5107,
6435/5107]
[2, 2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2]


[1, 3600/3413, 3820/3413, 4050/3413, 8595/6826, 18225/13652,
4815/3413, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 6435/3413]
[5107/3413, 7640/5107, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 3/2, 3/2]
[8595/6826, 6435/5107, 963/764, 240/191, 240/191, 180/143, 135/107,
81/64, 81/64, 137889/109216, 8595/6826, 8595/6826]
[2, 2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2]


[1, 28890/27313, 160/143, 32400/27313, 180/143, 36450/27313, 270/191,
214/143, 43200/27313, 240/143, 48600/27313, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 3/2]
[180/143, 135/107, 3861/3056, 963/764, 240/191, 240/191, 180/143,
135/107, 81/64, 2889/2288, 180/143, 180/143]
[2, 2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 16200/15301, 160/143, 18225/15301, 180/143, 91125/68266,
21600/15301, 214/143, 24300/15301, 240/143, 17404875/9762038, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2, 955/638,
286/191, 3/2]
[180/143, 135/107, 135/107, 135/107, 135/107, 135/107, 25785/20416,
585/464, 585/464, 1605/1276, 240/191, 180/143]
[2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2]


[1, 14445/13652, 3820/3413, 4050/3413, 8595/6826, 18225/13652,
4815/3413, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 6435/3413]
[5107/3413, 7640/5107, 3/2, 3/2, 286/191, 214/143, 3/2, 160/107, 3/2,
3/2, 3/2, 3/2]
[8595/6826, 6435/5107, 963/764, 963/764, 240/191, 180/143, 135/107,
135/107, 81/64, 137889/109216, 8595/6826, 8595/6826]
[2, 2, 3/2, 3/2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 540/511, 572/511, 1215/1022, 1287/1022, 182250/136607, 720/511,
3823/2555, 810/511, 858/511, 243000/136607, 963/511]
[3823/2555, 5720/3823, 3/2, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2,
204400/136607, 3/2, 3/2]
[1287/1022, 4815/3823, 180/143, 180/143, 180/143, 135/107,
344925/273214, 344925/273214, 344925/273214, 172035/136607, 1287/1022,
1287/1022]
[2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2]


[1, 14445/13646, 7656/6823, 8100/6823, 8595/6823, 18225/13646,
9630/6823, 10208/6823, 10800/6823, 11460/6823, 12150/6823, 12870/6823]
[10208/6823, 3/2, 955/638, 3/2, 286/191, 214/143, 3/2, 160/107, 3/2,
3/2, 3/2, 3/2]
[8595/6823, 585/464, 1605/1276, 963/764, 240/191, 180/143, 135/107,
135/107, 81/64, 8613/6823, 8613/6823, 8595/6823]
[2, 3/2, 2, 3/2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 7200/6823, 7656/6823, 8100/6823, 8595/6823, 18225/13646,
9630/6823, 10208/6823, 10800/6823, 11460/6823, 12150/6823, 12870/6823]
[10208/6823, 3/2, 955/638, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2,
3/2, 3/2, 3/2]
[8595/6823, 585/464, 1605/1276, 240/191, 240/191, 180/143, 135/107,
81/64, 81/64, 8613/6823, 8613/6823, 8595/6823]
[2, 3/2, 2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2]


[1, 4050/3823, 4290/3823, 620865000/522248561, 4815/3823,
182250/136607, 5400/3823, 5720/3823, 6075/3823, 6420/3823,
931297500/522248561, 7200/3823]
[5720/3823, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 204400/136607,
3/2, 3823/2555, 3/2]
[4815/3823, 180/143, 180/143, 135/107, 135/107, 344925/273214,
344925/273214, 172035/136607, 172035/136607, 1287/1022, 1287/1022,
4815/3823]
[2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 3/2]


[1, 5400/5107, 5730/5107, 6075/5107, 6435/5107, 18225/13652,
7200/5107, 7640/5107, 8100/5107, 8580/5107, 6075/3413, 9630/5107]
[7640/5107, 3/2, 286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2,
5107/3413, 3/2, 3/2]
[6435/5107, 963/764, 240/191, 180/143, 180/143, 135/107,
137889/109216, 137889/109216, 137889/109216, 8595/6826, 25785/20428,
6435/5107]
[2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2]


[1, 14445/13646, 15321/13646, 8100/6823, 8580/6823, 18225/13646,
9630/6823, 10214/6823, 10800/6823, 11469/6823, 12150/6823, 12840/6823]
[10214/6823, 3/2, 7646/5107, 5720/3823, 214/143, 3/2, 3/2, 160/107,
3/2, 3/2, 3/2, 3/2]
[8580/6823, 6420/5107, 6420/5107, 4815/3823, 180/143, 135/107,
135/107, 135/107, 81/64, 137889/109168, 137889/109168, 34407/27292]
[2, 3/2, 2, 2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 360/341, 174/155, 405/341, 39/31, 3645/2728, 963/682, 232/155,
540/341, 573/341, 1215/682, 642/341]
[232/155, 3/2, 955/638, 286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2,
3/2, 3/2]
[39/31, 1605/1276, 1605/1276, 240/191, 180/143, 135/107, 135/107,
81/64, 81/64, 783/620, 783/620, 1719/1364]
[2, 3/2, 2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 3/2]


[1, 4050/3823, 4290/3823, 18225/15292, 4815/3823, 18225/13652,
5400/3823, 5720/3823, 6075/3823, 6420/3823, 46564875/26095798, 14445/7646]
[5720/3823, 3/2, 214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2, 5110/3413,
3823/2555, 3/2]
[4815/3823, 2889/2288, 180/143, 135/107, 135/107, 135/107,
68985/54608, 34407/27304, 34407/27304, 4290/3413, 1287/1022, 4815/3823]
[2, 3/2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2]


[1, 1215/1144, 160/143, 21141/17732, 180/143, 3645/2728, 405/286,
214/143, 3645/2288, 240/143, 696195/390104, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 232/155, 955/638,
286/191, 3/2]
[180/143, 135/107, 81/64, 81/64, 81/64, 783/620, 1719/1364, 39/31,
39/31, 1605/1276, 240/191, 180/143]
[2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2]


[1, 4050/3823, 4290/3823, 620865000/522248561, 4815/3823,
182250/136607, 5400/3823, 5720/3823, 6075/3823, 6420/3823,
243000/136607, 7200/3823]
[5720/3823, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 204400/136607,
3823/2555, 3/2, 3/2]
[4815/3823, 180/143, 180/143, 135/107, 135/107, 344925/273214,
172035/136607, 172035/136607, 172035/136607, 1287/1022, 19305/15292,
4815/3823]
[2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2, 3/2]


[1, 540/511, 11469/10220, 1215/1022, 1287/1022, 182250/136607,
2889/2044, 3823/2555, 810/511, 858/511, 3645/2044, 3861/2044]
[3823/2555, 3/2, 5720/3823, 3/2, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2,
204400/136607, 3/2]
[1287/1022, 19305/15292, 4815/3823, 180/143, 180/143, 180/143,
135/107, 344925/273214, 344925/273214, 172035/136607, 172035/136607,
1287/1022]
[2, 3/2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 14445/13652, 15321/13652, 4050/3413, 8595/6826, 18225/13652,
19305/13652, 5107/3413, 5400/3413, 5730/3413, 6075/3413, 25785/13652]
[5107/3413, 3/2, 7640/5107, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2,
3/2, 3/2, 3/2]
[8595/6826, 25785/20428, 6435/5107, 963/764, 240/191, 240/191,
180/143, 135/107, 81/64, 137889/109216, 137889/109216, 8595/6826]
[2, 3/2, 2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 405/382, 321/286, 32400/27313, 180/143, 36450/27313, 405/286,
214/143, 43335/27313, 240/143, 48600/27313, 270/143]
[214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2,
3/2, 3/2]
[180/143, 135/107, 135/107, 3861/3056, 963/764, 240/191, 240/191,
180/143, 135/107, 2889/2288, 2889/2288, 180/143]
[2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2]


[1, 1215/1144, 321/286, 4644945/3904472, 180/143, 18225/13652,
405/286, 214/143, 3104325/1952236, 240/143, 6075/3413, 270/143]
[214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2, 5110/3413, 3823/2555,
5720/3823, 3/2, 3/2]
[180/143, 135/107, 135/107, 81/64, 68985/54608, 34407/27304,
4290/3413, 4290/3413, 1287/1022, 4815/3823, 2889/2288, 180/143]
[2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2]


[1, 405/382, 214/191, 3645/3056, 240/191, 3645/2728, 270/191, 286/191,
1215/764, 321/191, 21141/11842, 360/191]
[286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 232/155,
955/638, 3/2]
[240/191, 180/143, 135/107, 135/107, 81/64, 81/64, 783/620, 1719/1364,
1719/1364, 39/31, 1605/1276, 963/764]
[2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 3/2]


[1, 46413/43648, 9/8, 9477/7936, 81/64, 3645/2728, 7047/4960, 3/2,
139239/87296, 27/16, 78003/43648, 243/128]
[3/2, 3/2, 3/2, 3/2, 3/2, 232/155, 955/638, 3/2, 286/191, 214/143,
160/107, 3/2]
[81/64, 81/64, 783/620, 1719/1364, 1719/1364, 39/31, 1605/1276,
240/191, 240/191, 180/143, 135/107, 81/64]
[3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 3/2, 2, 2, 2, 3/2]


[1, 15483150/14616949, 120/107, 17374500/14616949, 135/107,
182250/136607, 20644200/14616949, 160/107, 23224725/14616949, 180/107,
243000/136607, 27594000/14616949]
[160/107, 3/2, 3/2, 3/2, 204400/136607, 3823/2555, 3/2, 3/2,
5720/3823, 214/143, 3/2, 3/2]
[135/107, 344925/273214, 172035/136607, 172035/136607, 172035/136607,
1287/1022, 4815/3823, 4815/3823, 4815/3823, 180/143, 135/107, 135/107]
[2, 3/2, 3/2, 3/2, 2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2]


[1, 5400/5107, 11469/10214, 6075/5107, 6420/5107, 182250/136547,
14445/10214, 7646/5107, 8100/5107, 8580/5107, 18225/10214, 9630/5107]
[7646/5107, 3/2, 5720/3823, 214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2,
204280/136547, 3/2]
[6420/5107, 4815/3823, 4815/3823, 180/143, 135/107, 135/107, 135/107,
689445/546188, 689445/546188, 172035/136547, 172035/136547, 6435/5107]
[2, 3/2, 2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 1215/1144, 321/286, 18615015/15611024, 180/143, 18225/13646,
405/286, 214/143, 3645/2288, 240/143, 13934835/7805512, 270/143]
[214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 10214/6823,
7646/5107, 5720/3823, 3/2]
[180/143, 135/107, 135/107, 81/64, 81/64, 137889/109168, 34407/27292,
8580/6823, 8580/6823, 6420/5107, 4815/3823, 180/143]
[2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2]


[1, 28890/27313, 160/143, 32400/27313, 180/143, 36450/27313, 270/191,
214/143, 43335/27313, 240/143, 48600/27313, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 286/191, 214/143, 3/2, 160/107, 3/2,
3/2, 3/2]
[180/143, 135/107, 3861/3056, 963/764, 963/764, 240/191, 180/143,
135/107, 135/107, 2889/2288, 180/143, 180/143]
[2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2]


[1, 929475/871936, 9/8, 40095/33536, 81/64, 18225/13624, 729/512, 3/2,
696195/435968, 27/16, 390015/217984, 243/128]
[3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 2550/1703, 382/255, 286/191, 214/143,
160/107, 3/2]
[81/64, 81/64, 81/64, 34425/27248, 8595/6812, 165/131, 2140/1703,
64/51, 240/191, 180/143, 135/107, 81/64]
[3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 2, 2, 3/2]


[1, 4050/3823, 4280/3823, 18225/15292, 4815/3823, 18225/13646,
5400/3823, 5720/3823, 6075/3823, 6420/3823, 93075075/52168658, 7200/3823]
[5720/3823, 214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2,
10214/6823, 7646/5107, 3/2]
[4815/3823, 180/143, 135/107, 135/107, 135/107, 81/64, 137889/109168,
34407/27292, 34407/27292, 8580/6823, 6420/5107, 4815/3823]
[2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 2, 3/2]


[1, 4050/3823, 4290/3823, 620865000/522248561, 4815/3823,
182250/136607, 5400/3823, 5720/3823, 6075/3823, 6435/3823,
243000/136607, 7200/3823]
[5720/3823, 3/2, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 204400/136607,
3823/2555, 3/2, 3/2]
[4815/3823, 180/143, 180/143, 180/143, 135/107, 344925/273214,
172035/136607, 172035/136607, 172035/136607, 1287/1022, 19305/15292,
19305/15292]
[2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2, 3/2]


[1, 28890/27313, 214/191, 32400/27313, 240/191, 36450/27313, 270/191,
286/191, 43200/27313, 320/191, 48600/27313, 360/191]
[286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 214/143, 160/107, 3/2, 3/2,
3/2, 3/2]
[240/191, 180/143, 135/107, 2889/2288, 180/143, 180/143, 180/143,
135/107, 81/64, 3861/3056, 963/764, 240/191]
[2, 2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 21600/20437, 214/191, 24300/20437, 240/191, 91125/68266, 270/191,
286/191, 32400/20437, 320/191, 36450/20437, 360/191]
[286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2,
955/638, 3/2]
[240/191, 180/143, 135/107, 135/107, 135/107, 135/107, 135/107,
25785/20416, 25785/20416, 585/464, 1605/1276, 240/191]
[2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 14445/13646, 7646/6823, 8100/6823, 8580/6823, 18225/13646,
9630/6823, 10214/6823, 10800/6823, 11440/6823, 12150/6823, 12840/6823]
[10214/6823, 7646/5107, 5720/3823, 3/2, 214/143, 3/2, 3/2, 160/107,
3/2, 3/2, 3/2, 3/2]
[8580/6823, 6420/5107, 4815/3823, 2889/2288, 180/143, 135/107,
135/107, 135/107, 81/64, 137889/109168, 34407/27292, 8580/6823]
[2, 2, 2, 3/2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 3600/3413, 3823/3413, 4050/3413, 4290/3413, 18225/13652,
4815/3413, 5110/3413, 5400/3413, 5720/3413, 6075/3413, 6435/3413]
[5110/3413, 3823/2555, 5720/3823, 3/2, 3/2, 214/143, 160/107, 3/2,
3/2, 3/2, 3/2, 3/2]
[4290/3413, 1287/1022, 4815/3823, 180/143, 180/143, 180/143, 135/107,
81/64, 81/64, 68985/54608, 34407/27304, 4290/3413]
[2, 2, 2, 3/2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2]


[1, 675/638, 65/58, 6075/5104, 1605/1276, 18225/13646, 450/319,
955/638, 2025/1276, 535/319, 12150/6823, 600/319]
[955/638, 286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 3/2,
10208/6823, 3/2, 3/2]
[1605/1276, 240/191, 180/143, 135/107, 135/107, 81/64, 8613/6823,
8613/6823, 8613/6823, 8595/6823, 585/464, 1605/1276]
[2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 3/2]


[1, 675/638, 65/58, 40500/34133, 1605/1276, 91125/68266, 450/319,
955/638, 2025/1276, 535/319, 60750/34133, 600/319]
[955/638, 286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 160/107, 3/2,
3/2, 3/2]
[1605/1276, 240/191, 180/143, 135/107, 135/107, 135/107, 135/107,
135/107, 135/107, 25785/20416, 585/464, 1605/1276]
[2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 3/2]


[1, 18/17, 286/255, 81/68, 64/51, 6075/4546, 24/17, 382/255, 27/17,
428/255, 4050/2273, 32/17]
[382/255, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2,
3400/2273, 3/2, 3/2]
[64/51, 240/191, 180/143, 135/107, 81/64, 81/64, 11475/9092,
11475/9092, 11475/9092, 2865/2273, 429/340, 107/85]
[2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 3/2]


[1, 18/17, 286/255, 2160/1819, 64/51, 2430/1819, 24/17, 382/255,
27/17, 428/255, 3240/1819, 32/17]
[382/255, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 160/107, 3/2,
3/2, 3/2]
[64/51, 240/191, 180/143, 135/107, 81/64, 135/107, 135/107, 135/107,
135/107, 1719/1360, 429/340, 107/85]
[2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 3/2]


[1, 18/17, 286/255, 2160/1819, 64/51, 2430/1819, 24/17, 382/255,
2880/1819, 428/255, 3240/1819, 32/17]
[382/255, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2, 160/107, 3/2, 3/2,
3/2, 3/2]
[64/51, 240/191, 180/143, 135/107, 135/107, 135/107, 135/107, 135/107,
81/64, 1719/1360, 429/340, 107/85]
[2, 2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 2400/2273, 7646/6819, 2700/2273, 8560/6819, 6075/4546, 3210/2273,
10214/6819, 3600/2273, 11440/6819, 4050/2273, 4280/2273]
[10214/6819, 7646/5107, 5720/3823, 214/143, 3/2, 3/2, 160/107, 3/2,
3/2, 3/2, 3/2, 3/2]
[8560/6819, 6420/5107, 4815/3823, 180/143, 135/107, 135/107, 135/107,
81/64, 81/64, 45963/36368, 11469/9092, 2860/2273]
[2, 2, 2, 2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 3/2, 3/2]


[1, 405/382, 160/143, 32400/27313, 180/143, 36450/27313, 270/191,
214/143, 43335/27313, 240/143, 48600/27313, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 286/191, 3/2, 214/143, 160/107, 3/2,
3/2, 3/2]
[180/143, 135/107, 3861/3056, 3861/3056, 963/764, 240/191, 180/143,
180/143, 135/107, 2889/2288, 180/143, 180/143]
[2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2]


[1, 2068335/1952236, 160/143, 1160325/976118, 180/143, 18225/13652,
689445/488059, 214/143, 773550/488059, 240/143, 3480975/1952236, 270/143]
[214/143, 160/107, 3/2, 3/2, 3/2, 5107/3413, 3/2, 7640/5107, 3/2, 3/2,
286/191, 3/2]
[180/143, 135/107, 137889/109216, 137889/109216, 8595/6826, 8595/6826,
25785/20428, 6435/5107, 3861/3056, 963/764, 240/191, 180/143]
[2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 3/2, 3/2, 2, 3/2]


[1, 3645/3424, 120/107, 1160325/972416, 135/107, 6075/4544, 1215/856,
160/107, 387585/243104, 180/107, 868725/486208, 405/214]
[160/107, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 319/213, 955/638, 286/191,
214/143, 3/2]
[135/107, 81/64, 81/64, 81/64, 2871/2272, 2865/2272, 715/568, 535/426,
1605/1276, 240/191, 180/143, 135/107]
[2, 3/2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 2, 3/2]


[1, 1215/1144, 321/286, 4644945/3904472, 180/143, 18225/13652,
405/286, 214/143, 3104325/1952236, 963/572, 6075/3413, 270/143]
[214/143, 3/2, 3/2, 160/107, 3/2, 3/2, 3/2, 5110/3413, 3823/2555,
5720/3823, 3/2, 3/2]
[180/143, 135/107, 135/107, 135/107, 68985/54608, 34407/27304,
4290/3413, 4290/3413, 1287/1022, 4815/3823, 2889/2288, 2889/2288]
[2, 3/2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2]


[1, 675/638, 65/58, 6075/5104, 1605/1276, 3645/2728, 450/319, 955/638,
2025/1276, 195/116, 18225/10208, 600/319]
[955/638, 286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2, 3/2,
232/155, 3/2]
[1605/1276, 240/191, 180/143, 180/143, 135/107, 81/64, 81/64, 783/620,
783/620, 1719/1364, 39/31, 585/464]
[2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 4050/3823, 4290/3823, 18225/15292, 4815/3823, 18225/13652,
5400/3823, 5720/3823, 6075/3823, 6435/3823, 46564875/26095798, 14445/7646]
[5720/3823, 3/2, 3/2, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 5110/3413,
3823/2555, 3/2]
[4815/3823, 2889/2288, 180/143, 180/143, 135/107, 135/107,
68985/54608, 34407/27304, 34407/27304, 4290/3413, 1287/1022, 19305/15292]
[2, 3/2, 3/2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 2, 3/2]


[1, 14445/13646, 7656/6823, 8100/6823, 8595/6823, 18225/13646,
19305/13646, 10208/6823, 10800/6823, 11484/6823, 12150/6823, 25785/13646]
[10208/6823, 3/2, 3/2, 955/638, 3/2, 286/191, 214/143, 160/107, 3/2,
3/2, 3/2, 3/2]
[8595/6823, 25785/20416, 585/464, 1605/1276, 240/191, 240/191,
180/143, 135/107, 81/64, 8613/6823, 8613/6823, 8613/6823]
[2, 3/2, 3/2, 2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 3/2]


[1, 5400/5107, 5730/5107, 6075/5107, 25785/20428, 18225/13652,
14445/10214, 3/2, 8100/5107, 8595/5107, 18225/10214, 19305/10214]
[3/2, 7640/5107, 3/2, 3/2, 286/191, 214/143, 160/107, 3/2, 3/2, 3/2,
5107/3413, 3/2]
[25785/20428, 6435/5107, 963/764, 240/191, 240/191, 180/143, 135/107,
137889/109216, 137889/109216, 137889/109216, 8595/6826, 25785/20428]
[3/2, 2, 3/2, 3/2, 2, 2, 2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 31025025/29221058, 120/107, 69674175/58442116, 135/107,
182250/136547, 20683350/14610529, 3/2, 93075075/58442116, 180/107,
26061750/14610529, 405/214]
[3/2, 160/107, 3/2, 3/2, 3/2, 204280/136547, 3/2, 3/2, 7646/5107,
5720/3823, 214/143, 3/2]
[135/107, 135/107, 689445/546188, 689445/546188, 689445/546188,
172035/136547, 6435/5107, 6420/5107, 6420/5107, 4815/3823, 180/143,
135/107]
[3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 3/2, 2, 2, 2, 3/2]


[1, 675/638, 65/58, 6075/5104, 1605/1276, 18225/13646, 450/319,
955/638, 2025/1276, 195/116, 12150/6823, 600/319]
[955/638, 286/191, 3/2, 214/143, 160/107, 3/2, 3/2, 3/2, 3/2,
10208/6823, 3/2, 3/2]
[1605/1276, 240/191, 180/143, 180/143, 135/107, 81/64, 8613/6823,
8613/6823, 8613/6823, 8595/6823, 585/464, 585/464]
[2, 2, 3/2, 2, 2, 3/2, 3/2, 3/2, 3/2, 2, 3/2, 3/2]


[1, 675/638, 2865/2552, 6075/5104, 585/464, 18225/13646, 14445/10208,
3/2, 2025/1276, 8595/5104, 18225/10208, 4815/2552]
[3/2, 955/638, 3/2, 286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2,
10208/6823, 3/2]
[585/464, 1605/1276, 963/764, 240/191, 180/143, 135/107, 135/107,
8613/6823, 8613/6823, 8613/6823, 8595/6823, 25785/20416]
[3/2, 2, 3/2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2]


[1, 405/382, 429/382, 6205005/5215064, 963/764, 18225/13652, 270/191,
3/2, 1215/764, 321/191, 18615015/10430128, 360/191]
[3/2, 286/191, 214/143, 3/2, 160/107, 3/2, 3/2, 3/2, 5107/3413, 3/2,
7640/5107, 3/2]
[963/764, 240/191, 180/143, 135/107, 135/107, 137889/109216,
137889/109216, 8595/6826, 8595/6826, 25785/20428, 6435/5107, 963/764]
[3/2, 2, 2, 3/2, 2, 3/2, 3/2, 3/2, 2, 3/2, 2, 3/2]

Message: 6597 - Contents - Hide Contents

Date: Wed, 05 Mar 2003 13:47:21

Subject: Re: An RI version of Wendell Well

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e... wrote:

> Gene wrote:
>> Blame Maple.
> 

> Can it be written out as a polynomial? Then I could find
> the result for r = q myself.


Maple thinks these associate left to right; that is,

A/B/C = (A/B)/C = A/(B*C)


> Is Robert's order the best order for 6 tempered fifths? 
> (I suppose so).


It's hardly clear. You tell me how to decide.

Message: 6598 - Contents - Hide Contents

Date: Wed, 5 Mar 2003 18:19:53

Subject: Re: An RI version of Wendell Well

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>Maple thinks these associate left to right; that is,

>A/B/C = (A/B)/C = A/(B*C)


Ok, when I solve for q=r I get r=1.99780788361.


>> Is Robert's order the best order for 6 tempered fifths? 
>> (I suppose so).


>It's hardly clear. You tell me how to decide.


Yeah, that was a vague question, never mind.

Manuel

Message: 6599 - Contents - Hide Contents

Date: Wed, 5 Mar 2003 18:38:33

Subject: Re: 114 temperaments in the big parade

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Gene wrote:

>Here are the temperaments I spoke of. I give the scale degrees, the
>circle of fifths, the major thirds and the brats, in that order. Note
>that the brats are all exact!


Impressive. Do rational scales also exist with exact beat ratios P5/M3
of 0 and 1/5?

Manuel

Previous Next

6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950

6550 - 6575 -