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Ultimate 5-limit comma list

🔗Gene Ward Smith <genewardsmith@juno.com>

11/26/2002 1:36:03 PM

Not that any list is really ultimate, but with rms error < 40, geometric complexity < 500, and badness < 3500, it covers a lot of ground.

27/25 3.739252 35.60924 1861.731473

135/128 4.132031 18.077734 1275.36536

256/243 5.493061 12.759741 2114.877638

25/24 3.025593 28.851897 799.108711

648/625 6.437752 11.06006 2950.938432

16875/16384 8.17255 5.942563 3243.743713

250/243 5.948286 7.975801 1678.609846

128/125 4.828314 9.677666 1089.323984

3125/3072 7.741412 4.569472 2119.95499

20000/19683 9.785568 2.504205 2346.540676

531441/524288 13.183347 1.382394 3167.444999

81/80 4.132031 4.217731 297.556531

2048/2025 6.271199 2.612822 644.408867

67108864/66430125 15.510107 .905187 3377.402314

78732/78125 12.192182 1.157498 2097.802867

393216/390625 12.543123 1.07195 2115.395301

2109375/2097152 12.772341 .80041 1667.723301

4294967296/4271484375 18.573955 .483108 3095.692488

15625/15552 9.338935 1.029625 838.631548

1600000/1594323 13.7942 .383104 1005.555381

(2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263

(2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204

(2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306

(5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873

(3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562

(2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615

(3)^8*(5)/(2)^15 9.459948 .161693 136.885775

(3)^45/(2)^69/(5) 48.911647 .026391 3088.065497

(2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642

(3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207

(2)*(5)^18/(3)^27 33.653272 .025593 975.428947

(2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149

(3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379

(2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233

(5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284

(2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255

(3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661

(2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786

(3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237

(5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721

(3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642

(2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514

(2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003

(5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867

(3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668

(3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625

(2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527

(2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421

(2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113

(2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823

(3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

12/4/2002 2:05:03 PM

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> Not that any list is really ultimate, but with rms error < 40,
>geometric complexity < 500,

i'll stick with <100; apologies are still due pierre :)

> and badness < 3500, it covers a lot of >ground.
>
> 27/25 3.739252 35.60924 1861.731473
if i recall correctly, this gave some kind of 8-equal monster . . .
bug

> 135/128 4.132031 18.077734 1275.36536
pelogic

> 256/243 5.493061 12.759741 2114.877638
blackwood

> 25/24 3.025593 28.851897 799.108711
dicot

> 648/625 6.437752 11.06006 2950.938432
diminished

> 16875/16384 8.17255 5.942563 3243.743713
negri

> 250/243 5.948286 7.975801 1678.609846
porcupine

> 128/125 4.828314 9.677666 1089.323984
augmented

> 3125/3072 7.741412 4.569472 2119.95499
magic

> 20000/19683 9.785568 2.504205 2346.540676
tetracot

> 531441/524288 13.183347 1.382394 3167.444999
aristoxenean

> 81/80 4.132031 4.217731 297.556531
meantone

> 2048/2025 6.271199 2.612822 644.408867
diaschismic

> 67108864/66430125 15.510107 .905187 3377.402314
carl seems to have missed this before. misty.

> 78732/78125 12.192182 1.157498 2097.802867
semisixths

> 393216/390625 12.543123 1.07195 2115.395301
wuerschmidt

> 2109375/2097152 12.772341 .80041 1667.723301
orwell

> 4294967296/4271484375 18.573955 .483108 3095.692488
this one also seems to have escaped carl's notice. escapade.

> 15625/15552 9.338935 1.029625 838.631548
kleismic

> 1600000/1594323 13.7942 .383104 1005.555381
AMT

> (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263
parakleismic

> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
vulture

> (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306
semisuper

> (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873
enneadecal

> (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562
vavoom

> (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615
tricot

> (3)^8*(5)/(2)^15 9.459948 .161693 136.885775
schismic

> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
turkey

> (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642
semithirds

> (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207
minortone

> (2)*(5)^18/(3)^27 33.653272 .025593 975.428947
ennealimmal

> (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149
astro

> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
crazy

> (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233
whoosh

> (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284
egads

> (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255
monzismic

> (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661
fortune

> (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786
gross

> (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237
senior

> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
pirate

i'll produce the next set of dualzoomers in accordance with this list.

🔗Gene Ward Smith <genewardsmith@juno.com>

12/4/2002 3:18:08 PM

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

> > 27/25 3.739252 35.60924 1861.731473
> if i recall correctly, this gave some kind of 8-equal monster . . .
> bug

I called it limmal. Why bug?

> > 1600000/1594323 13.7942 .383104 1005.555381
> AMT

Amity?

> > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
> vulture

> > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
> turkey

I suppose this makes 53-et the turkey vulture, though putting these two together actually leads to 3-torsion, so maybe not.

Turkey is interesting in that it says 5 ~ 2^(-69) 3^45 =
2^(-24) (3/2)^45. This means that like meantone and schismic, it has a generator of a fifth.

> > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
> crazy

Kwasi.

> > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
> pirate

When *I* tried to name it everyone booed. :(

This is one of the two 4296-et power commas; the other is viking:

2^161 * 3^(-15) * 5^49

Also of note is raider = pirate * viking =

2^71 * 3^(-99) * 5^33

The TM basis for 4296 is <pirate, raider>.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

12/4/2002 3:33:14 PM

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> > > 27/25 3.739252 35.60924 1861.731473
> > if i recall correctly, this gave some kind of 8-equal
monster . . .
> > bug
>
> I called it limmal.

too many definitions of limma -- see http://www.sonic-
arts.org/dict/limma.htm

> Why bug?

a simple, undistinguished animal :)

> > > 1600000/1594323 13.7942 .383104 1005.555381
> > AMT
>
> Amity?

OK.

> > > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
> > vulture
>
> > > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
> > turkey
>
> I suppose this makes 53-et the turkey vulture, though putting these
>two together actually leads to 3-torsion, so maybe not.

a 159-tone periodicity block? for the birds! :)

> Turkey is interesting in that it says 5 ~ 2^(-69) 3^45 =
> 2^(-24) (3/2)^45. This means that like meantone and schismic, it
has a generator of a fifth.
>
> > > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
> > crazy
>
> Kwasi.

oh yeah, kwazy.

> > > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
> > pirate
>
> When *I* tried to name it everyone booed. :(

so sorry :( :( :(

> This is one of the two 4296-et power commas; the other is viking:
>
> 2^161 * 3^(-15) * 5^49
>
> Also of note is raider = pirate * viking =
>
> 2^71 * 3^(-99) * 5^33
>
> The TM basis for 4296 is <pirate, raider>.

u got it. i will up the complexity limit, if that gets these in
there . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

12/4/2002 3:37:44 PM

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

> This is one of the two 4296-et power commas; the other is viking:
>
> 2^161 * 3^(-15) * 5^49
>
> Also of note is raider = pirate * viking =
>
> 2^71 * 3^(-99) * 5^33
>
> The TM basis for 4296 is <pirate, raider>.

if i raise my complexity limit anywhere from 5 to 43 points, i'd be
adding these two and only these two to my list. seems like a sensible
place to stop, at least until our next fit of mathematical
irrelevance! :)

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

12/4/2002 4:22:07 PM

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
>
> > This is one of the two 4296-et power commas; the other is viking:
> >
> > 2^161 * 3^(-15) * 5^49
> >
> > Also of note is raider = pirate * viking =
> >
> > 2^71 * 3^(-99) * 5^33
> >
> > The TM basis for 4296 is <pirate, raider>.
>
> if i raise my complexity limit anywhere from 5 to 43 points, i'd be
> adding these two and only these two to my list. seems like a
sensible
> place to stop, at least until our next fit of mathematical
> irrelevance! :)

wait a minute. i went back to your list with only the 2s exponent and
looked up the complexity based on that. but something's wrong.

2^161 * 3^(-15) * 5^49 = 301200.046966396 cents
2^71 * 3^(-99) * 5^33 = -11145.1925281338 cents

what are they really supposed to be, and what is the geometric
complexity of them?

🔗Gene Ward Smith <genewardsmith@juno.com>

12/4/2002 8:24:09 PM

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

> > This is one of the two 4296-et power commas; the other is viking:
> >
> > 2^161 * 3^(-15) * 5^49
> >
> > Also of note is raider = pirate * viking =
> >
> > 2^71 * 3^(-99) * 5^33
> >
> > The TM basis for 4296 is <pirate, raider>.
>
> u got it. i will up the complexity limit, if that gets these in
> there . . .

4296 is certainly a logical stopping point; all you need to do is up the limit to 105. Viking is 100.53, and raider is 104.66.

🔗Gene Ward Smith <genewardsmith@juno.com>

12/4/2002 8:31:52 PM

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> 2^161 * 3^(-15) * 5^49 = 301200.046966396 cents
> 2^71 * 3^(-99) * 5^33 = -11145.1925281338 cents

Pirate [-90, -15, 49] .046966 cents
Raider [71, -99, 37] .062327 cents
Viking [161, -84, -12] .015361 cents

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

12/5/2002 10:49:30 AM

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

> Viking [161, -84, -12] .015361 cents

this is the difference between 11 pythagorean commas and 12 syntonic
commas. i'm going to call it "atomic" instead, unless someone comes
up with a better name . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

12/5/2002 12:45:49 PM

what if we pushed up the badness limit until ampersand made it in?
the 5-limit comma does have a name, so it must be of some use . . .

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> Not that any list is really ultimate, but with rms error < 40,
geometric complexity < 500, and badness < 3500, it covers a lot of
ground.
>
> 27/25 3.739252 35.60924 1861.731473
>
> 135/128 4.132031 18.077734 1275.36536
>
> 256/243 5.493061 12.759741 2114.877638
>
> 25/24 3.025593 28.851897 799.108711
>
> 648/625 6.437752 11.06006 2950.938432
>
> 16875/16384 8.17255 5.942563 3243.743713
>
> 250/243 5.948286 7.975801 1678.609846
>
> 128/125 4.828314 9.677666 1089.323984
>
> 3125/3072 7.741412 4.569472 2119.95499
>
> 20000/19683 9.785568 2.504205 2346.540676
>
> 531441/524288 13.183347 1.382394 3167.444999
>
> 81/80 4.132031 4.217731 297.556531
>
> 2048/2025 6.271199 2.612822 644.408867
>
> 67108864/66430125 15.510107 .905187 3377.402314
>
> 78732/78125 12.192182 1.157498 2097.802867
>
> 393216/390625 12.543123 1.07195 2115.395301
>
> 2109375/2097152 12.772341 .80041 1667.723301
>
> 4294967296/4271484375 18.573955 .483108 3095.692488
>
> 15625/15552 9.338935 1.029625 838.631548
>
> 1600000/1594323 13.7942 .383104 1005.555381
>
> (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263
>
> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
>
> (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306
>
> (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873
>
> (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562
>
> (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615
>
> (3)^8*(5)/(2)^15 9.459948 .161693 136.885775
>
> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
>
> (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642
>
> (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207
>
> (2)*(5)^18/(3)^27 33.653272 .025593 975.428947
>
> (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149
>
> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
>
> (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233
>
> (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284
>
> (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255
>
> (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661
>
> (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786
>
> (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237
>
> (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721
>
> (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642
>
> (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514
>
> (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003
>
> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
>
> (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668
>
> (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625
>
> (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527
>
> (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421
>
> (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113
>
> (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823
>
> (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

🔗Gene Ward Smith <genewardsmith@juno.com>

12/5/2002 1:51:19 PM

--- In tuning-math@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
>
> > Viking [161, -84, -12] .015361 cents

> this is the difference between 11 pythagorean commas and 12 syntonic
> commas. i'm going to call it "atomic" instead, unless someone comes
> up with a better name . . .

Then shouldn't pirate and raider be electron and proton? Good spot, by the way.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

12/5/2002 2:22:56 PM

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
> >
> > > Viking [161, -84, -12] .015361 cents
>
> > this is the difference between 11 pythagorean commas and 12
syntonic
> > commas. i'm going to call it "atomic" instead, unless someone
comes
> > up with a better name . . .
>
> Then shouldn't pirate and raider be electron and proton?

if you say so, but which is which? then again, speaking
of "electronic music" might get confusing . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

12/18/2002 12:57:42 PM

still hoping for an answer to this . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> what if we pushed up the badness limit until ampersand made it in?
> the 5-limit comma does have a name, so it must be of some use . . .
>
> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
> > Not that any list is really ultimate, but with rms error < 40,
> geometric complexity < 500, and badness < 3500, it covers a lot of
> ground.
> >
> > 27/25 3.739252 35.60924 1861.731473
> >
> > 135/128 4.132031 18.077734 1275.36536
> >
> > 256/243 5.493061 12.759741 2114.877638
> >
> > 25/24 3.025593 28.851897 799.108711
> >
> > 648/625 6.437752 11.06006 2950.938432
> >
> > 16875/16384 8.17255 5.942563 3243.743713
> >
> > 250/243 5.948286 7.975801 1678.609846
> >
> > 128/125 4.828314 9.677666 1089.323984
> >
> > 3125/3072 7.741412 4.569472 2119.95499
> >
> > 20000/19683 9.785568 2.504205 2346.540676
> >
> > 531441/524288 13.183347 1.382394 3167.444999
> >
> > 81/80 4.132031 4.217731 297.556531
> >
> > 2048/2025 6.271199 2.612822 644.408867
> >
> > 67108864/66430125 15.510107 .905187 3377.402314
> >
> > 78732/78125 12.192182 1.157498 2097.802867
> >
> > 393216/390625 12.543123 1.07195 2115.395301
> >
> > 2109375/2097152 12.772341 .80041 1667.723301
> >
> > 4294967296/4271484375 18.573955 .483108 3095.692488
> >
> > 15625/15552 9.338935 1.029625 838.631548
> >
> > 1600000/1594323 13.7942 .383104 1005.555381
> >
> > (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263
> >
> > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
> >
> > (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306
> >
> > (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873
> >
> > (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562
> >
> > (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615
> >
> > (3)^8*(5)/(2)^15 9.459948 .161693 136.885775
> >
> > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
> >
> > (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642
> >
> > (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207
> >
> > (2)*(5)^18/(3)^27 33.653272 .025593 975.428947
> >
> > (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149
> >
> > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
> >
> > (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233
> >
> > (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284
> >
> > (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255
> >
> > (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661
> >
> > (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786
> >
> > (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237
> >
> > (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721
> >
> > (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642
> >
> > (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514
> >
> > (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003
> >
> > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
> >
> > (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668
> >
> > (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625
> >
> > (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527
> >
> > (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421
> >
> > (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113
> >
> > (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823
> >
> > (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

12/23/2002 10:13:39 PM

grasping, yelping for an answer . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> still hoping for an answer to this . . .
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > what if we pushed up the badness limit until ampersand made it
in?
> > the 5-limit comma does have a name, so it must be of some
use . . .
> >
> > --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...>
> wrote:
> > > Not that any list is really ultimate, but with rms error < 40,
> > geometric complexity < 500, and badness < 3500, it covers a lot
of
> > ground.
> > >
> > > 27/25 3.739252 35.60924 1861.731473
> > >
> > > 135/128 4.132031 18.077734 1275.36536
> > >
> > > 256/243 5.493061 12.759741 2114.877638
> > >
> > > 25/24 3.025593 28.851897 799.108711
> > >
> > > 648/625 6.437752 11.06006 2950.938432
> > >
> > > 16875/16384 8.17255 5.942563 3243.743713
> > >
> > > 250/243 5.948286 7.975801 1678.609846
> > >
> > > 128/125 4.828314 9.677666 1089.323984
> > >
> > > 3125/3072 7.741412 4.569472 2119.95499
> > >
> > > 20000/19683 9.785568 2.504205 2346.540676
> > >
> > > 531441/524288 13.183347 1.382394 3167.444999
> > >
> > > 81/80 4.132031 4.217731 297.556531
> > >
> > > 2048/2025 6.271199 2.612822 644.408867
> > >
> > > 67108864/66430125 15.510107 .905187 3377.402314
> > >
> > > 78732/78125 12.192182 1.157498 2097.802867
> > >
> > > 393216/390625 12.543123 1.07195 2115.395301
> > >
> > > 2109375/2097152 12.772341 .80041 1667.723301
> > >
> > > 4294967296/4271484375 18.573955 .483108 3095.692488
> > >
> > > 15625/15552 9.338935 1.029625 838.631548
> > >
> > > 1600000/1594323 13.7942 .383104 1005.555381
> > >
> > > (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263
> > >
> > > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
> > >
> > > (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306
> > >
> > > (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873
> > >
> > > (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562
> > >
> > > (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615
> > >
> > > (3)^8*(5)/(2)^15 9.459948 .161693 136.885775
> > >
> > > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
> > >
> > > (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642
> > >
> > > (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207
> > >
> > > (2)*(5)^18/(3)^27 33.653272 .025593 975.428947
> > >
> > > (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149
> > >
> > > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
> > >
> > > (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233
> > >
> > > (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284
> > >
> > > (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255
> > >
> > > (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661
> > >
> > > (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786
> > >
> > > (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237
> > >
> > > (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721
> > >
> > > (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642
> > >
> > > (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514
> > >
> > > (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003
> > >
> > > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
> > >
> > > (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668
> > >
> > > (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625
> > >
> > > (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527
> > >
> > > (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421
> > >
> > > (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113
> > >
> > > (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823
> > >
> > > (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

🔗Gene Ward Smith <genewardsmith@juno.com> <genewardsmith@juno.com>

12/24/2002 5:54:22 AM

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> grasping, yelping for an answer . . .

I guess I gotta do it, but recall that last time you wanted the limit upped, you told me what I did was all you wanted. :)

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

12/24/2002 6:15:32 AM

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > grasping, yelping for an answer . . .
>
> I guess I gotta do it, but recall that last time you wanted the
>limit upped, you told me what I did was all you wanted. :)

well, you're kinda too late -- see the tuning list post that should
have just appeared . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

12/24/2002 6:36:33 AM

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

> well, you're kinda too late -- see the tuning list post that should
> have just appeared . . .

well, it didn't . . . but i've done these:

http://www.stretch-music.com/_wsn/page2.html

/tuning/database?
method=reportRows&tbl=10&sortBy=9&sortDir=up

🔗Carl Lumma <clumma@yahoo.com> <clumma@yahoo.com>

12/24/2002 11:15:54 AM

>>well, you're kinda too late -- see the tuning list post that
>>should have just appeared . . .
>
>well, it didn't . . . but i've done these:
>
>http://www.stretch-music.com/_wsn/page2.html

Good, good.

>/tuning/database?
>method=reportRows&tbl=10&sortBy=9&sortDir=up

Aha! Now all we need is RMS optimum generators, and a blurb
on what the heck heuristic error and complexity are.

-C.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

12/24/2002 2:31:05 PM

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>well, you're kinda too late -- see the tuning list post that
> >>should have just appeared . . .
> >
> >well, it didn't . . . but i've done these:
> >
> >http://www.stretch-music.com/_wsn/page2.html
>
> Good, good.
>
> >/tuning/database?
> >method=reportRows&tbl=10&sortBy=9&sortDir=up
>
> Aha! Now all we need is RMS optimum generators, and a blurb
> on what the heck heuristic error and complexity are.
>
> -C.

as you know, error and complexity can both be defined in various
ways -- rms/minimax/mad, weightedthisway/weightedthatway/unweighted,
euclidean/taxicab, etc.

what i've come up with are heuristics for error and complexity, which
are approximately correct no matter what definition you use.

they're calculated as follows:

d = odd limit of comma ratio; s = numerator minus denominator
ln(d) = heuristic complexity
|s|/(d*ln(d)) = heuristic error (actually i multiplied by 1200/ln(2)?)

what i would like to figure out is what natural choice of metric,
weighting, loss function, etc. makes these heuristics *exactly*
correct. then report *those* optimal generators . . .

🔗Carl Lumma <clumma@yahoo.com> <clumma@yahoo.com>

12/24/2002 7:23:16 PM

>>>/tuning/database?
>>>method=reportRows&tbl=10&sortBy=9&sortDir=up
>>
>>Aha! Now all we need is RMS optimum generators, and a blurb
>>on what the heck heuristic error and complexity are.
>>
>>-C.
>
>as you know, error and complexity can both be defined in
>various ways -- rms/minimax/mad,
>weightedthisway/weightedthatway/unweighted,
>euclidean/taxicab, etc.
>
>what i've come up with are heuristics for error and complexity,
>which are approximately correct no matter what definition you
>use.

Sounds like a good idea.

>they're calculated as follows:
>
>d = odd limit of comma ratio; s = numerator minus denominator
>ln(d) = heuristic complexity
>|s|/(d*ln(d)) = heuristic error
>(actually i multiplied by 1200/ln(2)?)
>
>what i would like to figure out is what natural choice of metric,
>weighting, loss function, etc. makes these heuristics *exactly*
>correct. then report *those* optimal generators . . .

Why would you assume the heuristics are more ideal than any
of the current formulations they approximate? Do they require
fewer assumptions? Can you plot unweighted Graham complexity
against heuristic complexity for me?

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

12/25/2002 5:31:25 PM

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>>/tuning/database?
> >>>method=reportRows&tbl=10&sortBy=9&sortDir=up
> >>
> >>Aha! Now all we need is RMS optimum generators, and a blurb
> >>on what the heck heuristic error and complexity are.
> >>
> >>-C.
> >
> >as you know, error and complexity can both be defined in
> >various ways -- rms/minimax/mad,
> >weightedthisway/weightedthatway/unweighted,
> >euclidean/taxicab, etc.
> >
> >what i've come up with are heuristics for error and complexity,
> >which are approximately correct no matter what definition you
> >use.
>
> Sounds like a good idea.
>
> >they're calculated as follows:
> >
> >d = odd limit of comma ratio; s = numerator minus denominator
> >ln(d) = heuristic complexity
> >|s|/(d*ln(d)) = heuristic error
> >(actually i multiplied by 1200/ln(2)?)
> >
> >what i would like to figure out is what natural choice of metric,
> >weighting, loss function, etc. makes these heuristics *exactly*
> >correct. then report *those* optimal generators . . .
>
> Why would you assume the heuristics are more ideal than any
> of the current formulations they approximate?

a bit of "numerology" one might say . . .

> Do they require
> fewer assumptions?

hmm . . . you might say that, since the formulas are simpler.
especially if you're given the ratios and have to factorize them
yourself.

> Can you plot unweighted Graham complexity
> against heuristic complexity for me?

the only thing i can do on this computer is surf yahoo and a few
other websites. i'm sitting in a convenience store . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

1/13/2003 12:12:10 PM

could i ask for periods and rms-optimal generators for each?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@j...> wrote:
> Not that any list is really ultimate, but with rms error < 40,
geometric complexity < 500, and badness < 3500, it covers a lot of
ground.
>
> 27/25 3.739252 35.60924 1861.731473
>
> 135/128 4.132031 18.077734 1275.36536
>
> 256/243 5.493061 12.759741 2114.877638
>
> 25/24 3.025593 28.851897 799.108711
>
> 648/625 6.437752 11.06006 2950.938432
>
> 16875/16384 8.17255 5.942563 3243.743713
>
> 250/243 5.948286 7.975801 1678.609846
>
> 128/125 4.828314 9.677666 1089.323984
>
> 3125/3072 7.741412 4.569472 2119.95499
>
> 20000/19683 9.785568 2.504205 2346.540676
>
> 531441/524288 13.183347 1.382394 3167.444999
>
> 81/80 4.132031 4.217731 297.556531
>
> 2048/2025 6.271199 2.612822 644.408867
>
> 67108864/66430125 15.510107 .905187 3377.402314
>
> 78732/78125 12.192182 1.157498 2097.802867
>
> 393216/390625 12.543123 1.07195 2115.395301
>
> 2109375/2097152 12.772341 .80041 1667.723301
>
> 4294967296/4271484375 18.573955 .483108 3095.692488
>
> 15625/15552 9.338935 1.029625 838.631548
>
> 1600000/1594323 13.7942 .383104 1005.555381
>
> (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263
>
> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
>
> (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306
>
> (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873
>
> (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562
>
> (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615
>
> (3)^8*(5)/(2)^15 9.459948 .161693 136.885775
>
> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
>
> (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642
>
> (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207
>
> (2)*(5)^18/(3)^27 33.653272 .025593 975.428947
>
> (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149
>
> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
>
> (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233
>
> (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284
>
> (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255
>
> (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661
>
> (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786
>
> (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237
>
> (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721
>
> (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642
>
> (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514
>
> (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003
>
> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
>
> (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668
>
> (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625
>
> (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527
>
> (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421
>
> (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113
>
> (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823
>
> (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

1/15/2003 10:37:51 AM

well, could i?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> could i ask for periods and rms-optimal generators for each?
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@j...> wrote:
> > Not that any list is really ultimate, but with rms error < 40,
> geometric complexity < 500, and badness < 3500, it covers a lot of
> ground.
> >
> > 27/25 3.739252 35.60924 1861.731473
> >
> > 135/128 4.132031 18.077734 1275.36536
> >
> > 256/243 5.493061 12.759741 2114.877638
> >
> > 25/24 3.025593 28.851897 799.108711
> >
> > 648/625 6.437752 11.06006 2950.938432
> >
> > 16875/16384 8.17255 5.942563 3243.743713
> >
> > 250/243 5.948286 7.975801 1678.609846
> >
> > 128/125 4.828314 9.677666 1089.323984
> >
> > 3125/3072 7.741412 4.569472 2119.95499
> >
> > 20000/19683 9.785568 2.504205 2346.540676
> >
> > 531441/524288 13.183347 1.382394 3167.444999
> >
> > 81/80 4.132031 4.217731 297.556531
> >
> > 2048/2025 6.271199 2.612822 644.408867
> >
> > 67108864/66430125 15.510107 .905187 3377.402314
> >
> > 78732/78125 12.192182 1.157498 2097.802867
> >
> > 393216/390625 12.543123 1.07195 2115.395301
> >
> > 2109375/2097152 12.772341 .80041 1667.723301
> >
> > 4294967296/4271484375 18.573955 .483108 3095.692488
> >
> > 15625/15552 9.338935 1.029625 838.631548
> >
> > 1600000/1594323 13.7942 .383104 1005.555381
> >
> > (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263
> >
> > (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204
> >
> > (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306
> >
> > (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873
> >
> > (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562
> >
> > (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615
> >
> > (3)^8*(5)/(2)^15 9.459948 .161693 136.885775
> >
> > (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497
> >
> > (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642
> >
> > (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207
> >
> > (2)*(5)^18/(3)^27 33.653272 .025593 975.428947
> >
> > (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149
> >
> > (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379
> >
> > (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233
> >
> > (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284
> >
> > (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255
> >
> > (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661
> >
> > (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786
> >
> > (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237
> >
> > (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721
> >
> > (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642
> >
> > (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514
> >
> > (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003
> >
> > (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867
> >
> > (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668
> >
> > (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625
> >
> > (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527
> >
> > (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421
> >
> > (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113
> >
> > (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823
> >
> > (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

1/15/2003 10:50:06 AM

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...>
wrote:
>
> > Viking [161, -84, -12] .015361 cents
>
> this is the difference between 11 pythagorean commas and 12
syntonic
> commas. i'm going to call it "atomic" instead, unless someone comes
> up with a better name . . .

was it kirnberger who proposed foreshortening each fifth by a schisma
to approximate 12-equal? a chain of 12 such fifths would fail to
close on itself by a mere "atom", or 0.015 cents . . .