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Message: 11000 - Contents - Hide Contents

Date: Tue, 01 Jun 2004 23:51:12

Subject: Family commas

From: Gene Ward Smith

One way to sort out the family relationships is to use a comma scheme
which I have intended for some time to discuss, but which I may have
neglected to do. This is defining a linear temperament in terms of a
sequence of commas, each at a succesively higher prime limit, and each
with a minimal Tenney height given that all the previous commas are
fixed. This sort of whatzit reduction, for meantone, would say
meantone is the 81/80-temperament, dominant sevenths the [81/80,
36/35] temperament, septimal meantone the [81/80,
126/125]-temperament, flattone the [81/80, 525/512]-temperament. Then
11-limit meantone is the [81/80, 126/125, 385/384]-temperament and
huygens the [81/80, 126/125, 99/98]-temperament. And so forth.

It should be noted that while this keeps track of the familial
relationships, we don't necessarily get corresponding generators in
these family trees, nor do we necessarily get rid of contorsion.
7-limit ennealimmal is the [ennealimma, breedsma]-temperament, but the
wedge product of this has a common factor of 4. Hemiennealimmal is
then the [ennealimma, breedsma, lehmerisma]-temperament, again with a
common factor of 4, but now with non-corresponding generators.

Message: 11001 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 03:56:10

Subject: The schismic family

From: Gene Ward Smith

schismic 32805/32768 comma
[1200.065, 498.278]

7-limit schismic family

0: <<1 -8 39 -15 59 113|| schismic {4375/4374, 32805/32768}
[1200.075, 498.277]

-41: <<1 -8 -2 -15 -6 -18|| {64/63, 360/343}
[1195.115, 496.240]

-53: <<1 -8 -14 -15 -25 -10|| septischis {225/224, 3125/3087}
[1200.761, 498.119]

11-limit schismic

0: <<1 -8 39 88 -15 59 136 113 232 112|| {441/440, 4375/4374, 32805/32768}
[1200.314, 498.411]

-53: <<1 -8 39 35 -15 59 52 113 109 -37|| {245/242, 896/891, 4375/4374}
[1198.620, 497.673]

-118: <<1 -8 39 -30 -15 59 -51 113 -42 -219|| {385/384, 3388/3375,
4375/4374}
[1200.378, 498.447]

-171: <<1 -8 39 -83 -15 59 -135 113 -165 -368|| {540/539, 4375/4374,
32805/32768}
[1200.005, 498.220]

11-limit septischis

0: <<1 -8 -14 23 -15 -25 33 -10 81 113|| septischis {225/224, 385/384,
2200/2187}
[1200.761, 498.119]

-12: <<1 -8 -14 11 -15 -25 14 -10 53 79|| {55/54, 225/224, 1617/1600}
[1202.076, 497.731]

-29: <<1 -8 -14 -6 -15 -25 -13 -10 14 32|| {45/44, 56/55, 3125/3087}
[1194.570, 495.997]

-41: <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2|| {100/99, 245/243, 1344/1331}
[1201.363, 497.941]

-53: <<1 -8 -14 -30 -15 -25 -51 -10 -42 -36|| {99/98, 176/175, 3125/3087}
[1198.649, 497.794]

Message: 11002 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 22:39:47

Subject: Re: The meantone family

From: Herman Miller

Gene Ward Smith wrote:

> This gives part of the family tree of the meantone family; I don't go
> into cousins such as the various 11-limit versions of flattone or
> dominant sevenths. In each limit, I propose giving the name "meantone"
> to the temperament with the same TOP generators as 5-limit meantone.
> The numbers before the colon give the nexial val with the base
> meantone, plus or minus depending on whether adding or subtracting is
> required. Also, the TM basis and the TOP octave and fourth.
> 
> meantone 81/80 comma 
> [1201.70, 504.13]
> 
> 7-limit meantone family
> 
> 0: <<1 4 10 4 13 12|| meantone {81/80, 126/125}
> [1201.70, 504.13]


Hmm, I must have the brackets backward on my web page; I have meantone 
as [1, 4, 10, 4, 13, 12>. I can never remember which way they go. But is 
there any significance to the double brackets?

(For anyone who might not be on the tuning list, the page I'm referring 
to is Zireen Music * [with cont.]  (Wayb.))

Message: 11003 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 04:00:18

Subject: Re: Family commas

From: Gene Ward Smith

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


> Also, when you say "given that all the previous commas are
> fixed", does this imply any relation to TM-reduction?


It's a different reduction--sequential reduction or something.


> The words that are coming to mind are, temperament n should
> be considered an extension of temperament m if m's TM-reduced
> basis is a subset of n's.  Does that make any sense?


This won't work. It doesn't even work if you replace TM reduction with
sequential reduction, though that is better for this. The nexial
approach does it, however.

Message: 11004 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 04:01:59

Subject: Re: The hanson family

From: Gene Ward Smith

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

>>>>> This family stuff looks awesome.  I wish I understood the
>>>>> half of it.  I'm surprised you're using generator sizes.
>>>>> How do you standardize the generator representation?  Forgive
>>>>> me if this is old stuff, I haven't kept up.
>>>> 

>>>> It's just the TOP tuning for the generators.
>>> 

>>> How do you get a unique set of generators out of the TOP
>>> tuning?
>> 

>> One way is to apply the TOP tuning to a rational number generator
>> which works as a reduced generator. For instance, with meantone that
>> would be 4/3, with miracle 15/14 or 16/15,
> 

> This is apparently not giving unique generators...


Sure it does; miracle(15/14) = [0, 1] = miracle(16/15)

Message: 11005 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 00:50:09

Subject: Re: Family commas

From: Carl Lumma

>> >lso, when you say "given that all the previous commas are
>> fixed", does this imply any relation to TM-reduction?
>

>It's a different reduction--sequential reduction or something.
>

>> The words that are coming to mind are, temperament n should
>> be considered an extension of temperament m if m's TM-reduced
>> basis is a subset of n's.  Does that make any sense?
>

>This won't work. It doesn't even work if you replace TM reduction with
>sequential reduction, though that is better for this. The nexial
>approach does it, however.

Noted.


>>>>>> This family stuff looks awesome.  I wish I understood the
>>>>>> half of it.  I'm surprised you're using generator sizes.
>>>>>> How do you standardize the generator representation?  Forgive
>>>>>> me if this is old stuff, I haven't kept up.
>>>>> 

>>>>> It's just the TOP tuning for the generators.
>>>> 

>>>> How do you get a unique set of generators out of the TOP
>>>> tuning?
>>> 

>>> One way is to apply the TOP tuning to a rational number generator
>>> which works as a reduced generator. For instance, with meantone that
>>> would be 4/3, with miracle 15/14 or 16/15,
>> 

>> This is apparently not giving unique generators...
>

>Sure it does; miracle(15/14) = [0, 1] = miracle(16/15)


Right you are.

-Carl



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Message: 11006 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 00:01:00

Subject: Re: The hanson family

From: Gene Ward Smith

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

>>> This family stuff looks awesome.  I wish I understood the
>>> half of it.  I'm surprised you're using generator sizes.
>>> How do you standardize the generator representation?  Forgive
>>> me if this is old stuff, I haven't kept up.
>> 

>> It's just the TOP tuning for the generators.
> 

> How do you get a unique set of generators out of the TOP
> tuning?


One way is to apply the TOP tuning to a rational number generator
which works as a reduced generator. For instance, with meantone that
would be 4/3, with miracle 15/14 or 16/15, etc. I simply solve for it,
using a close rational approximation of the TOP tuning to finesse
technical problems.

Message: 11008 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 00:06:58

Subject: Re: Family commas

From: Carl Lumma

> ... defining a linear temperament in terms of a
> sequence of commas, each at a succesively higher prime limit,
> and each with a minimal Tenney height given that all the
> previous commas are fixed. This sort of whatzit reduction,
> for meantone, would say meantone is the 81/80-temperament,
> dominant sevenths the [81/80, 36/35] temperament, septimal
> meantone the [81/80, 126/125]-temperament, flattone the
> [81/80, 525/512]-temperament. Then 11-limit meantone is
> the [81/80, 126/125, 385/384]-temperament and huygens the
> [81/80, 126/125, 99/98]-temperament. And so forth.


This immediately appeals to me more than generators.

I wonder how it relates to Paul's tratios.  They involve
the LCM... I wonder what good that is.

Also, when you say "given that all the previous commas are
fixed", does this imply any relation to TM-reduction?

The words that are coming to mind are, temperament n should
be considered an extension of temperament m if m's TM-reduced
basis is a subset of n's.  Does that make any sense?


> It should be noted that while this keeps track of the familial
> relationships, we don't necessarily get corresponding
> generators in these family trees, nor do we necessarily get
> rid of contorsion. 7-limit ennealimmal is the [ennealimma,
> breedsma]- temperament, but the wedge product of this has a
> common factor of 4.

Rats.

-Carl

Message: 11009 - Contents - Hide Contents

Date: Wed, 02 Jun 2004 00:16:28

Subject: Re: The hanson family

From: Carl Lumma

>>>> >his family stuff looks awesome.  I wish I understood the
>>>> half of it.  I'm surprised you're using generator sizes.
>>>> How do you standardize the generator representation?  Forgive
>>>> me if this is old stuff, I haven't kept up.
>>> 

>>> It's just the TOP tuning for the generators.
>> 

>> How do you get a unique set of generators out of the TOP
>> tuning?
>

>One way is to apply the TOP tuning to a rational number generator
>which works as a reduced generator. For instance, with meantone that
>would be 4/3, with miracle 15/14 or 16/15,


This is apparently not giving unique generators...

-Carl



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Message: 11010 - Contents - Hide Contents

Date: Thu, 03 Jun 2004 18:23:18

Subject: Re: The meantone family

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:


> Hmm, I must have the brackets backward on my web page; I have 
meantone 
> as [1, 4, 10, 4, 13, 12>. I can never remember which way they go. 
But is 
> there any significance to the double brackets?


The double brackets signal that it is a bival; wedgies for linear 
temperaments are always wedge products of two vals.




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Message: 11011 - Contents - Hide Contents

Date: Sat, 05 Jun 2004 20:56:45

Subject: The augmented family

From: Gene Ward Smith

augmented 128/125 comma
[399.02, 93.15]

7-limit augmented family

0: <<3 0 -6 -7 -18 -14|| augmented (old tripletone) {64/63, 126/125}
[399.02, 92.46]

3: <<3 0 -3 -7 -13 -7||  {21/20, 128/125}
[401.71, 124.11]

6: <<3 0 0 -7 -8 0|| {28/25, 35/32}
[409.25, 144.25]

9: <<3 0 3 -7 -4 7|| {15/14, 128/125}
[392.25, 96.15]

12: <<3 0 6 -7 1 -14|| augie (old augmented) {36/35, 128/125}
[399.99, 107.31]

15: <<3 0 9 -7 6 21|| "Number 60" {28/27, 128/125}
[397.81, 76.64]

Message: 11012 - Contents - Hide Contents

Date: Sat, 05 Jun 2004 22:56:11

Subject: The diaschismic family

From: Gene Ward Smith

diaschismic 2048/2025 comma
[599.56, 104.70]

7-limit diaschismic family

0: <<2 -4 -16 -11 -31 -26|| diaschismic {126/125, 2048/2025}
[599.37, 103.79]

12: <<2 -4 -4 -11 -12 2|| pajara {50/49, 64/63}
[598.45, 106.57]

22: <<2 -4 6 -11 4 25|| {28/27, 525/512}
[599.74, 115.71]

24: <<2 -4 8 -11 7 30||  {36/35, 2048/2025}
[598.71, 97.51]

34: <<2 -4 18 -11 23 53|| {875/864, 2048/2025}
[599.27, 107.35]

11-limit diaschismic family

0: <<2 -4 -16 -24 -11 -31 -45 -26 -42 -12|| diaschismic {126/125,
176/175, 896/891}
[599.37, 103.79]

12: <<2 -4 -16 -12 -11 -31 -26 -26 -14 22|| {56/55, 100/99, 2048/2025}
[598.18, 103.58]

46: <<2 -4 -16 22 -11 -31 28 -26 65 117|| {126/125, 385/384, 1232/1215}
[599.74, 104.26]

58: <<2 -4 -16 34 -11 -31 47 -26 93 151|| {126/125, 540/539, 2048/2025}
[599.30, 103.47]

11-limit pajara family

0: <<2 -4 -4 -12 -11 -12 -26 2 -14 -20|| pajara {50/49, 64/63, 99/98}
[598.45, 106.57]

12: <<2 -4 -4 0 -11 -12 -7 2 14 14|| {45/44, 50/49, 56/55}
[596.50, 101.36]

22: <<2 -4 -4 10 -11 -12 9 2 37 42|| {50/49, 55/54, 64/63}
[599.27, 108.86]

Message: 11013 - Contents - Hide Contents

Date: Sun, 06 Jun 2004 08:27:37

Subject: The porcupine family

From: Gene Ward Smith

porcupine 250/243 comma
[1196.91, 162.32]

7-limit porcupine family

0: <<3 5 -6 1 -18 -28|| porcupine {64/63, 250/243}
[1196.91, 162.32]

7: <<3 5 1 1 -7 -12|| hystrix {36/35, 160/147}
[1187.93, 161.10]

8: <<3 5 2 1 -5 -9|| {21/20, 175/162}
[1218.24, 168.54]

15: <<3 5 9 1 6 7|| "Number 59" {28/27, 126/125}
[1193.42, 158.15]

22: <<3 5 16 1 17 23|| {225/224, 250/243}
[1198.78, 163.50]

11-limit porcupine family

0: <<3 5 -6 4 1 -18 -4 -28 -8 32|| {55/54, 64/63, 100/99}
[1198.23, 168.15]

7: <<3 5 -6 11 1 -18 7 -28 8 52|| {45/44, 64/63, 250/243}
[1198.67, 165.23]

-15: <<3 5 -6 -11 1 -18 -28 -28 -43 -10|| {56/55, 64/63, 250/243}
[1195.98, 161.21]

-22: <<3 5 -6 -18 1 -18 -39 -28 -59 -30|| {64/63, 99/98, 250/243}
[1198.65, 163.42]





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Message: 11014 - Contents - Hide Contents

Date: Mon, 07 Jun 2004 05:56:38

Subject: The semisixths family

From: Gene Ward Smith

semisixths 78732/78125 comma
[1199.59, 442.98]

7-limit semisixths family

0: <<7 9 13 -2 1 5|| semisixths {126/125, 245/243}
[1198.39, 443.16]

19: <<7 9 32 -2 31 49|| semimedium {225/224, 78732/78125}
[1200.80, 443.07] 

11-limit semisixths family 

0: <<7 9 13 31 -2 1 25 5 41 42|| semisixths {126/125, 176/175, 245/243}
[1198.39, 443.16]

-46: <<7 9 13 -15 -2 1 -48 55 -66 -87|| {126/125, 245/243, 385/384}
[1199.44, 443.30]

11-limit semimedium family

0: <<7 9 32 -34 -2 31 -78 49 -110 -206|| semimedium {225/224, 385/384,
78732/78125}
[1200.81, 443.07]

19: <<7 9 32 -15 -2 31 -48 49 -66 -153|| {225/224, 1944/1925, 2560/2541}
[1199.77, 442.82]

Message: 11015 - Contents - Hide Contents

Date: Mon, 07 Jun 2004 22:52:35

Subject: The semisuper family

From: Gene Ward Smith

semisuper |23 6 -14> comma
[599.974, 71.146]

7-limit semisuper family

0: <<14 6 74 -23 78 155|| semisuper {4375/4374, 29360128/29296875}
[599.960, 71.108]

-118: <<14 6 -44 -23 -109 -119|| {65625/65536, 321489/320000}
[600.114, 71.266]

11-limit semisuper

0: <<14 6 74 52 -23 78 34 155 100 -110|| semisuper {3025/3024,
4375/4374, 5632/5625}
[599.949, 71.097]

Message: 11016 - Contents - Hide Contents

Date: Mon, 07 Jun 2004 08:19:05

Subject: The parakleismic family

From: Gene Ward Smith

parakleismic 1224440064/1220703125 comma
[1199.912, 315.211]

7-limit parakleismic family

0: <<13 14 35 -8 19 42|| parakleismic {3136/3125, 4375/4374}
[1199.738, 315.108]

-19: <<14 14 16 =8 -11 -2|| {126/125, 12005/11664}
[1202.327, 315.875]

11-limit parakleismic family

0: <<13 14 35 -36 -8 19 -102 42 -132 -222|| {385/384, 3136/3125,
4375/4374}
[1200.317, 315.223]

19: <<13 14 35 -18 -8 19 -72 42 -88 -169|| {540/539, 896/891, 3136/3125}
[1199.308, 314.884]

99: <<13 14 35 63 -8 19 55 42 98 56|| {176/175, 1375/1372, 2200/2187}
[1199.265, 314.866]

118: <<13 14 35 82 -8 19 85 42 142 109|| {441/440, 3136/3125, 4375/4374}
[1200.102, 315.242]




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Message: 11017 - Contents - Hide Contents

Date: Mon, 07 Jun 2004 21:27:08

Subject: Re: The orwell family

From: Herman Miller

Gene Ward Smith wrote:


> 11-limit orwell family
> 
> 0: <<7 -3 8 2 -21 -7 -21 27 15 -22|| {99/98, 121/120, 176/175}
> [1201.25, 271.43]


This is an 11-limit version of orwell that I've been playing with from 
time to time, although I've mostly been using 7-limit scales recently. 
You can fine-tune it to get a 22-note MOS that represents all 11-limit 
intervals consistently (anything between 1/1 and 2/1 with a numerator of 
11 or less). That's a characteristic that most of the good 7-limit 
temperaments seem to have, including meantone, miracle, kleismic, 
porcupine, pajara, injera, and even some of the more exotic ones like 
#82 (which I've been calling "lemba").

Message: 11018 - Contents - Hide Contents

Date: Tue, 08 Jun 2004 23:49:38

Subject: Atomic notation

From: Gene Ward Smith

If A = 2^(1/12) (semitone) and B = (648/625)^(1/32) (diminished
schisma) we get the minimax tuning of atomic by setting

3 ~ A^19 B
5 ~ A^28 B^(-7)

Then 5/3 ~ A^9/B^8, which is 5/3 exactly. 

We can define a notation system by taking twelve nominals and a symbol
for the (diminished) schisma. Then the major third and the fifth are
both sharp by 0.00016 cents, which I would hope would be enough
accuracy for even the most intransigent JI proponent, but of course in
higher limits we get a lot less accuracy:

A^34 B^(-16) ~ 7, 0.11 cents flat

A^42 B^(-25) ~ 11, 0.20 cents flat

Message: 11019 - Contents - Hide Contents

Date: Tue, 08 Jun 2004 00:45:04

Subject: The orwell family

From: Gene Ward Smith

orwell 2109375/2097152 comma 
[1200.24, 271.65]

7-limit orwell

0: <<7 -3 8 -21 -7 27|| orwell {225/224, 1728/1715}
[1199.533, 271.494]

11-limit orwell family

0: <<7 -3 8 2 -21 -7 -21 27 15 -22|| {99/98, 121/120, 176/175}
[1201.25, 271.43]

9: <<7 -3 8 11 -21 -7 -7 -27 36 3|| {45/44, 56/55, 1728/1715}
[1195.02, 270.47]

-22: <<7 -3 8 -20 -21 -7 -56 27 -36 -84|| {100/99, 225/224, 1728/1715}
[1199.23, 271.82]

31: <<7 -3 8 33 -21 -7 28 27 87 65|| {225/224, 441/440, 1728/1715}
[1200.56, 271.44]

Message: 11020 - Contents - Hide Contents

Date: Tue, 08 Jun 2004 06:20:54

Subject: The amity family

From: Gene Ward Smith

amity 1600000/1594323 comma
[1199.851, 339.472]

7-limit amity

0: <<5 13 -17 9 -41 -76|| amity {4375/4374, 5120/5103}
[1199.724, 339.356]

11-limit amity family

0: <<5 13 -17 62 9 -41 81 -76 99 233|| amity {540/539, 4375/4374,
5120/5103}
[1199.656, 339.359]

-53: <<5 13 -17 9 9 -41 -3 -76 -24 84|| {121/120, 176/175, 2200/2187}
[1200.624, 339.610]

-99: <<5 13 -17 -37 9 -41 -76 -76 -131 -45|| {441/440, 896/891, 4375/4374}
[1199.524, 339.190]

-152: <<5 13 -17 -90 9 -41 -160 -76 -254 -194|| {1375/1372, 4375/4374,
5120/5103}
[1199.648, 339.359]

Message: 11021 - Contents - Hide Contents

Date: Tue, 08 Jun 2004 08:59:31

Subject: Fading families

From: Gene Ward Smith

As the 5-limit commas get more complex, the associated families fade
away--you get stuck with a single 7-limit version of the temperament,
with all the others ruled out in badness terms because if we have a
large nexial adjustment, it leads to large complexities. As the
complexity of the 5-limit comma gets even higher, this begans to work
the same way for higher limits also. To give an example of this, I
took a look at the atomic family, with starting point the atom of
Kirnberger. In the 5-limit, we have the absurdly precise (and quite
possibly theoretically useful) atomic temperament mapping [<12 19 28|,
<0 1 -7|], with period a semitone and generator a schisma. In the
7-limit, the only choice which makes much sense is 12&612, which
extends the mapping to [<12 19 28 34|, <0 1 -7 -16|], still very
accurate but not the crazed super-accuracy of the 5-limit, where we
have an rms error of 0.00012 cents, as opposed to 0.03484 cents in the
7-limit.

Also part of the pattern is that the way to get better badness numbers
is to subdivide the period, the octave, or both. For example, the
mapping [<24 37 63 68|, <0 5 -35 -3|], which does both, gets us down
to 0.00063 for an rms error.



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Message: 11022 - Contents - Hide Contents

Date: Wed, 09 Jun 2004 21:33:36

Subject: Re: Atomic projection

From: Gene Ward Smith

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

> We can define what I've called a "notation"--a pair of unimodular
> matries inverse to each other, one of which is a row matrix of monzos
> and another a column matrix of vals--for atomic in various ways. The
> simplest is 
> 
> [|-67 35 5>, |-15 8 1>, |161 -84 -12>]
> [<12 19 28|, <0 -1 7|, <5 8 11|]


It doesn't give the atomic mapping, but if we only want the projection 
a much simpler choice would be 135/128 in place of |-67 35 5>. This
gives us

[|-7 3 1>, |-15 8 1>, |161 -84 -12>]
[<12 19 28|, <48 77 105|, <5 8 11|]

It is interesting that 135/128 and the schisma allow us to represent
the 5-limit up to atomic equivalence. We can also relate this to an
Agmon/meantone system, since <48 77 105| = 11<5 8 11|-<7 11 16|. Since
meantone is a*h12 + b*h7, it is a*[1,0,0]+b*[0,-1,11] = 
[a,-b,11b] in these coordinates.

Message: 11024 - Contents - Hide Contents

Date: Wed, 09 Jun 2004 18:22:14

Subject: Atomic projection

From: Gene Ward Smith

We can define what I've called a "notation"--a pair of unimodular
matries inverse to each other, one of which is a row matrix of monzos
and another a column matrix of vals--for atomic in various ways. The
simplest is 

[|-67 35 5>, |-15 8 1>, |161 -84 -12>]
[<12 19 28|, <0 -1 7|, <5 8 11|]

While the most common way to project intervals and vals is to toss
octaves, we can do an atomic projection for ets by tossing result of
mapping the atom with an et, and an atomic projection for intervals by
ignoring the results of the <5 8 11| val. Below I give 5-limit ets and
intervals in the atomic coordinates; leaving off the last coordinate
does the projection. The interval |-67 35 5> is of course
approximately 100 cents; more precisely, 99.9936 cents.

Just as with projecting away the octave, we can plot the results of
projecting away the atom on a plane diagram; anyone who wants to
ponder the results of that is welcome to do so. Of course we get lines
on both kinds of diagrams, with either ets or intervals as points; for
example 55-43-31-19 is [0 -1 11]-[-1 -1 11]-[-2 -1 11]-[-3 -1 11].


5-limit ets in atomic

1 <13 3 -31|
2 <-4 -1 10|
3 <9 2 -21|
4 <-13 -3 32|
5 <5 1 -11|
6 <18 4 -42|
7 <-4 -1 11|
8 <14 3 -32|
9 <-8 -2 21|
10 <5 1 -10|
12 <1 0 0|
15 <10 2 -21|
19 <-3 -1 11|
22 <6 1 -10|
31 <-2 -1 11|
34 <7 1 -10|
46 <8 1 -10|
53 <4 0 1|
65 <5 0 1|
84 <2 -1 12|
87 <11 1 -9|
99 <12 1 -9|
118 <9 0 2|
152 <16 1 -8|
171 <13 0 3|
205 <20 1 -7|
236 <18 0 4|
289 <22 0 5|
323 <29 1 -5|
441 <38 1 -3|
559 <47 1 -1|
612 <51 1 0|
730 <60 1 2|


5-limit commas in atomic

|161 -84 -12> [0 0 1]
|71 -99 37> [7 -358 -30]
|-90 -15 49> [7 -358 -31]
|-17 62 -35> [-6 307 26]
|144 -22 -47> [-6 307 27]
|-107 47 14> [1 -51 -5]
|54 -37 2> [1 -51 -4]
|-36 -52 51> [8 -409 -35]
|37 25 -33> [-5 256 22]
|-53 10 16> [2 -102 -9]
|91 -12 -31> [-4 205 18]
|1 -27 18> [3 -153 -13]
|-16 35 -17> [-3 154 13]
|38 -2 -15> [-2 103 9]
|-52 -17 34> [5 -255 -22]
32805/32768 [0 1 0]
19073486328125/19042491875328 [3 -152 -13]
6115295232/6103515625 [-2 104 9]
1224440064/1220703125 [-2 105 9]
1600000/1594323 [1 -48 -4]
15625/15552 [1 -47 -4]
2109375/2097152 [1 -46 -4]
393216/390625 [-1 57 5]
78732/78125 [-1 58 5]
2048/2025 [0 10 1]
81/80 [0 11 1]
3125/3072 [1 -36 -3]
128/125 [0 21 2]
250/243 [1 -26 -2]
648/625 [0 32 3]
25/24 [1 -15 -1]
135/128 [1 -4 0]
16/15 [1 6 1]
27/25 [1 17 2]




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