A 13-limit comma list

Here is a list of 70 13-limit commas, with epimericity less than 0.25
and size less than 50 cents. There is a very good chance it is complete, since I took it from a similar list of 499 commas with epimericity less than 0.5 after I could add nothing further to it.

[36/35, 77/75, 40/39, 128/125, 45/44, 143/140, 49/48, 50/49, 55/54, 56/55, 64/63, 65/64, 66/65, 78/77, 81/80, 245/242, 91/90, 99/98, 100/99, 105/104,121/120, 245/243, 126/125, 275/273, 144/143, 169/168, 176/175, 896/891, 196/195, 1029/1024, 640/637, 225/224, 1188/1183, 243/242, 1573/1568, 325/324, 351/350, 352/351, 364/363, 385/384, 847/845, 441/440, 1375/1372, 540/539, 4000/3993, 625/624, 676/675, 729/728, 2200/2197, 1575/1573, 5632/5625, 1001/1000, 4459/4455, 10985/10976, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095,
4225/4224, 4375/4374, 6656/6655, 140625/140608, 9801/9800, 10648/10647, 196625/196608, 151263/151250, 1990656/1990625, 123201/123200, 5767168/5767125]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:

> Here is a list of 70 13-limit commas, with epimericity less than
0.25
> and size less than 50 cents.

why do you keep insisting on "size less than 50 cents"? pelogic,
for example relies on a much larger comma; and even negri,
with its very low error, has a comma of 51 cents.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> why do you keep insisting on "size less than 50 cents"? pelogic,
> for example relies on a much larger comma; and even negri,
> with its very low error, has a comma of 51 cents.

I'm not insisting on it; it's just a nice, round number. What would you suggest?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > why do you keep insisting on "size less than 50 cents"? pelogic,
> > for example relies on a much larger comma; and even negri,
> > with its very low error, has a comma of 51 cents.
>
> I'm not insisting on it; it's just a nice, round number. What would
>you suggest?

i'm mystified as to why comma size would either be a convenient, or a
useful, way to set a bound on the search . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> i'm mystified as to why comma size would either be a convenient, or a
> useful, way to set a bound on the search . . .

A comma always introduces its equivalences and sets a limit to accuracy. I don't see much merit in commas which are larger than the smallest consonance of the system, at any rate--do you? One of my "top" systems has a TM basis of [4/3, 7/5, 8/5], which I think is a little over the top for top.

The cutoff I suggested would at least exclude commas larger than
(n+1)/n for odd limit n--151 cents in the 11-limit. Do we need to go that high?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > i'm mystified as to why comma size would either be a convenient,
or a
> > useful, way to set a bound on the search . . .
>
> A comma always introduces its equivalences

?

> and sets a limit to >accuracy.

lots of things set a limit to accuracy.

> I don't see much merit in commas which are larger than the
>smallest consonance of the system, at any rate--do you?

i could rephrase this in terms of the complexity and/or error
(possibly by way of the heuristic), and it would make for a more
convincing "merit" criterion.

i'm just wondering why, when doing the search, you'd bother
calculating the JI size of the commas at all.

btw, did you see my request to revisit the "ultimate 5-limit" list
using the heuristic? perhaps you could guide me as to a good method
to program the search, if you don't want to do it yourself. i think
i'd like the "funky" ones all in there, so i can show them on the "q"
equal temperament graph, the one that includes two instances of 2-
equal.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> i'm just wondering why, when doing the search, you'd bother
> calculating the JI size of the commas at all.

I don't. I use them to generate wedgies.

> btw, did you see my request to revisit the "ultimate 5-limit" list
> using the heuristic? perhaps you could guide me as to a good method
> to program the search, if you don't want to do it yourself.

After getting a comma list it is easy, so are you asking how to get an initial list?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > i'm just wondering why, when doing the search, you'd bother
> > calculating the JI size of the commas at all.
>
> I don't. I use them to generate wedgies.

so why does the size of the comma come it at all?

> > btw, did you see my request to revisit the "ultimate 5-limit"
list
> > using the heuristic? perhaps you could guide me as to a good
method
> > to program the search, if you don't want to do it yourself.
>
> After getting a comma list it is easy,

it is? how would i be sure that there isn't some comma with lower
heuristic badness than the one in your list with highest heuristic
badness?

> so are you asking how to get an initial list?

that too.

Gene Ward Smith wrote:
> Here is a list of 70 13-limit commas, with epimericity less than 0.25
> and size less than 50 cents. There is a very good chance it is complete, since I took it from a similar list of 499 commas with epimericity less than 0.5 after I could add nothing further to it.
> > [36/35, 77/75, 40/39, 128/125, 45/44, 143/140, 49/48, 50/49, 55/54, 56/55, 64/63, 65/64, 66/65, 78/77, 81/80, 245/242, 91/90, 99/98, 100/99, 105/104,121/120, 245/243, 126/125, 275/273, 144/143, 169/168, 176/175, 896/891, 196/195, 1029/1024, 640/637, 225/224, 1188/1183, 243/242, 1573/1568, 325/324, 351/350, 352/351, 364/363, 385/384, 847/845, 441/440, 1375/1372, 540/539, 4000/3993, 625/624, 676/675, 729/728, 2200/2197, 1575/1573, 5632/5625, 1001/1000, 4459/4455, 10985/10976, 1716/1715, 2080/2079, 2401/2400, 3025/3024, 4096/4095,
> 4225/4224, 4375/4374, 6656/6655, 140625/140608, 9801/9800, 10648/10647, 196625/196608, 151263/151250, 1990656/1990625, 123201/123200, 5767168/5767125]

I left a script running overnight to generate linear temperaments from all combinations (there are nearly a million) of these columns. The results are below. It took 7.8 hours in pure Python (using wedge products, so no Numeric libraries). You could probably knock 2 orders of magnitude off that if you rewrote the whole thing in C, so it'd only be about 5 minutes. Which is still a lot slower than the search by equal temperaments, but that isn't comprehensive yet because I'm not taking all versions of inconsistent temperaments. I might look at it sometime.

It's difficult to recognise the temperaments without the ETs as hints, so I don't know if there's anything new or missing. But here they are...

0/1, 15.8 cent generator

basis:
(0.034482758620689655, 0.013188156675199889)

mapping by period and generator:
[(29, 0), (46, 0), (67, 1), (81, 1), (100, 1), (107, 1)]

mapping by steps:
[(29, 0), (46, 0), (-1, 1), (13, 1), (32, 1), (39, 1)]

highest interval width: 1
complexity measure: 29 (58 for smallest MOS)
highest error: 0.003552 (4.263 cents)
unique

1/2, 351.6 cent generator

basis:
(1.0, 0.29301432925652815)

mapping by period and generator:
[(1, 0), (1, 2), (-5, 25), (-1, 13), (2, 5), (4, -1)]

mapping by steps:
[(1, 1), (-1, 1), (-30, -5), (-14, -1), (-3, 2), (5, 4)]

highest interval width: 26
complexity measure: 26 (41 for smallest MOS)
highest error: 0.006546 (7.855 cents)

1/6, 183.2 cent generator

basis:
(0.5, 0.15268768272511829)

mapping by period and generator:
[(2, 0), (5, -6), (8, -11), (5, 2), (6, 3), (8, -2)]

mapping by steps:
[(10, 2), (1, -1), (-4, -3), (33, 7), (42, 9), (32, 6)]

highest interval width: 15
complexity measure: 30 (46 for smallest MOS)
highest error: 0.005815 (6.978 cents)

0/1, 103.8 cent generator

basis:
(0.5, 0.086488824362764394)

mapping by period and generator:
[(2, 0), (3, 1), (5, -2), (7, -8), (9, -12), (10, -15)]

mapping by steps:
[(2, 0), (-1, 1), (13, -2), (39, -8), (57, -12), (70, -15)]

highest interval width: 17
complexity measure: 34 (46 for smallest MOS)
highest error: 0.005094 (6.113 cents)
unique

1/7, 186.0 cent generator

basis:
(0.5, 0.15499541683101503)

mapping by period and generator:
[(2, 0), (1, 7), (0, 15), (5, 2), (6, 3), (4, 11)]

mapping by steps:
[(12, 2), (-1, 1), (-15, 0), (28, 5), (33, 6), (13, 4)]

highest interval width: 15
complexity measure: 30 (32 for smallest MOS)
highest error: 0.005555 (6.666 cents)

0/1, 497.9 cent generator

basis:
(1.0, 0.41489358489291139)

mapping by period and generator:
[(1, 0), (2, -1), (-1, 8), (-3, 14), (13, -23), (12, -20)]

mapping by steps:
[(1, 0), (-1, 1), (23, -8), (39, -14), (-56, 23), (-48, 20)]

highest interval width: 37
complexity measure: 37 (41 for smallest MOS)
highest error: 0.004468 (5.362 cents)
unique

1/10, 310.3 cent generator

basis:
(1.0, 0.25859402576420476)

mapping by period and generator:
[(1, 0), (-1, 10), (0, 9), (1, 7), (-3, 25), (5, -5)]

mapping by steps:
[(9, 1), (1, -1), (9, 0), (16, 1), (-2, -3), (40, 5)]

highest interval width: 30
complexity measure: 30 (31 for smallest MOS)
highest error: 0.006590 (7.908 cents)

0/1, 475.7 cent generator

basis:
(1.0, 0.39641218196468747)

mapping by period and generator:
[(1, 0), (0, 4), (-6, 21), (4, -3), (-12, 39), (-7, 27)]

mapping by steps:
[(1, 0), (-4, 4), (-27, 21), (7, -3), (-51, 39), (-34, 27)]

highest interval width: 42
complexity measure: 42 (43 for smallest MOS)
highest error: 0.003409 (4.090 cents)
unique

2/5, 339.4 cent generator

basis:
(1.0, 0.28284398720617654)

mapping by period and generator:
[(1, 0), (3, -5), (6, -13), (-2, 17), (6, -9), (2, 6)]

mapping by steps:
[(3, 2), (-1, 1), (-8, -1), (28, 13), (0, 3), (18, 10)]

highest interval width: 30
complexity measure: 30 (32 for smallest MOS)
highest error: 0.006663 (7.995 cents)

1/3, 166.1 cent generator

basis:
(0.5, 0.13840115997461999)

mapping by period and generator:
[(2, 0), (4, -3), (-2, 24), (7, -5), (0, 25), (11, -13)]

mapping by steps:
[(4, 2), (-1, 1), (68, 22), (-1, 2), (75, 25), (-17, -2)]

highest interval width: 38
complexity measure: 76 (94 for smallest MOS)
highest error: 0.000971 (1.165 cents)
unique

0/1, 83.0 cent generator

basis:
(0.33333333333333331, 0.06917176981320311)

mapping by period and generator:
[(3, 0), (6, -6), (8, -5), (8, 2), (11, -3), (14, -14)]

mapping by steps:
[(3, 0), (-6, 6), (-2, 5), (12, -2), (5, 3), (-14, 14)]

highest interval width: 16
complexity measure: 48 (57 for smallest MOS)
highest error: 0.002358 (2.830 cents)
unique

0/1, 15.8 cent generator

basis:
(0.027777777777777776, 0.013143428289896716)

mapping by period and generator:
[(36, 0), (57, 0), (83, 1), (101, 0), (124, 1), (133, 0)]

mapping by steps:
[(36, 0), (57, 0), (-1, 1), (101, 0), (40, 1), (133, 0)]

highest interval width: 1
complexity measure: 36 (72 for smallest MOS)
highest error: 0.005995 (7.194 cents)

0/1, 104.9 cent generator

basis:
(0.5, 0.087412649330450343)

mapping by period and generator:
[(2, 0), (3, 1), (5, -2), (3, 15), (5, 11), (6, 8)]

mapping by steps:
[(2, 0), (-1, 1), (13, -2), (-57, 15), (-39, 11), (-26, 8)]

highest interval width: 17
complexity measure: 34 (46 for smallest MOS)
highest error: 0.006039 (7.247 cents)

1/3, 234.5 cent generator

basis:
(1.0, 0.19540011144114833)

mapping by period and generator:
[(1, 0), (1, 3), (-1, 17), (3, -1), (6, -13), (8, -22)]

mapping by steps:
[(2, 1), (-1, 1), (-19, -1), (7, 3), (25, 6), (38, 8)]

highest interval width: 39
complexity measure: 39 (41 for smallest MOS)
highest error: 0.005231 (6.277 cents)
unique

1/3, 165.8 cent generator

basis:
(0.5, 0.13817544367976164)

mapping by period and generator:
[(2, 0), (4, -3), (11, -23), (7, -5), (13, -22), (11, -13)]

mapping by steps:
[(4, 2), (-1, 1), (-47, -12), (-1, 2), (-40, -9), (-17, -2)]

highest interval width: 23
complexity measure: 46 (58 for smallest MOS)
highest error: 0.003280 (3.935 cents)
unique

4/11, 263.7 cent generator

basis:
(1.0, 0.21974602053976744)

mapping by period and generator:
[(1, 0), (4, -11), (1, 6), (5, -10), (5, -7), (7, -15)]

mapping by steps:
[(8, 3), (-1, 1), (26, 9), (10, 5), (19, 8), (11, 6)]

highest interval width: 28
complexity measure: 28 (32 for smallest MOS)
highest error: 0.008185 (9.822 cents)

0/1, 565.9 cent generator

basis:
(1.0, 0.47161343493195518)

mapping by period and generator:
[(1, 0), (3, -3), (16, -29), (8, -11), (11, -16), (7, -7)]

mapping by steps:
[(1, 0), (-3, 3), (-42, 29), (-14, 11), (-21, 16), (-7, 7)]

highest interval width: 29
complexity measure: 29 (36 for smallest MOS)
highest error: 0.010144 (12.173 cents)

1/6, 116.8 cent generator

basis:
(1.0, 0.097316259752627809)

mapping by period and generator:
[(1, 0), (1, 6), (3, -7), (3, -2), (2, 15), (0, 38)]

mapping by steps:
[(5, 1), (-1, 1), (22, 3), (17, 3), (-5, 2), (-38, 0)]

highest interval width: 45
complexity measure: 45 (72 for smallest MOS)
highest error: 0.003454 (4.145 cents)
unique

1/4, 175.9 cent generator

basis:
(1.0, 0.14656139906015953)

mapping by period and generator:
[(1, 0), (1, 4), (1, 9), (-1, 26), (2, 10), (4, -2)]

mapping by steps:
[(3, 1), (-1, 1), (-6, 1), (-29, -1), (-4, 2), (14, 4)]

highest interval width: 28
complexity measure: 28 (34 for smallest MOS)
highest error: 0.009313 (11.176 cents)

0/1, 476.0 cent generator

basis:
(1.0, 0.39667339152715836)

mapping by period and generator:
[(1, 0), (0, 4), (17, -37), (4, -3), (11, -19), (16, -31)]

mapping by steps:
[(1, 0), (-4, 4), (54, -37), (7, -3), (30, -19), (47, -31)]

highest interval width: 45
complexity measure: 45 (48 for smallest MOS)
highest error: 0.003774 (4.529 cents)
unique

Graham