25 best weighted generator steps 5-limit temperaments

Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5) and
1/ln(5) times the steps to 3, 5, and 5/3. To get it to more or less correspond to the previous g, I adjusted by a factor of

(1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln(5))

This isn't the only adjustment you might think of, so your milage may vary. I did a search for badness (adjusted) less than 750, rms less
than 35 (as suggested by Dave) and adjusted g less than 20. Here's the
result:

135/128

map [[0, -1, 3], [1, 2, 1]]

generators 522.8623453 1200

badness 347.2957065 rms 18.07773298 weighted_g 2.678254330

256/243

map [[0, 0, 1], [5, 8, 10]]

generators 395.3362123 240

badness 612.7594631 rms 12.75974156 weighted_g 3.634818387

25/24

map [[0, 2, 1], [1, 1, 2]]

generators 350.9775007 1200

badness 134.9517606 rms 28.85189698 weighted_g 1.672379084

648/625

map [[0, -1, -1], [4, 8, 11]]

generators 505.8656425 300

badness 536.5364242 rms 11.06006024 weighted_g 3.647096372

16875/16384

map [[0, -4, 3], [1, 2, 2]]

generators 126.2382718 1200

badness 716.2095928 rms 5.942562596 weighted_g 4.939565964

250/243

map [[0, -3, -5], [1, 2, 3]]

generators 162.9960265 1200

badness 363.8269146 rms 7.975800816 weighted_g 3.573058894

128/125

map [[0, -1, 0], [3, 6, 7]]

generators 491.2018553 400

badness 198.0597092 rms 9.677665980 weighted_g 2.735322279

3125/3072

map [[0, 5, 1], [1, 0, 2]]

generators 379.9679494 1200

badness 368.6051706 rms 4.569472316 weighted_g 4.320809550

20000/19683

map [[0, 4, 9], [1, 1, 1]]

generators 176.2822703 1200

badness 565.4935338 rms 2.504205191 weighted_g 6.089559954

81/80

map [[0, -1, -4], [1, 2, 4]]

generators 503.8351546 1200

badness 81.02783490 rms 4.217730124 weighted_g 2.678254330

2048/2025

map [[0, -1, 2], [2, 4, 3]]

generators 494.5534684 600

badness 167.3579339 rms 2.612822498 weighted_g 4.001094414

78732/78125

map [[0, 7, 9], [1, -1, -1]]

generators 442.9792974 1200

badness 412.2839142 rms 1.157498146 weighted_g 7.088571030

393216/390625

map [[0, 8, 1], [1, -1, 2]]

generators 387.8196732 1200

badness 373.3878885 rms 1.071950166 weighted_g 7.036043951

2109375/2097152

map [[0, 7, -3], [1, 0, 3]]

generators 271.5895996 1200

badness 340.7350960 rms .8004099292 weighted_g 7.522602841

4294967296/4271484375

map [[0, -9, 7], [1, 2, 2]]

generators 55.27549315 1200

badness 687.5758107 rms .4831084292 weighted_g 11.24843211

15625/15552

map [[0, 6, 5], [1, 0, 1]]

generators 317.0796754 1200

badness 146.7501955 rms 1.029625097 weighted_g 5.223559222

1600000/1594323

map [[0, -5, -13], [1, 3, 6]]

generators 339.5088258 1200

badness 252.5491335 rms .3831037874 weighted_g 8.703150208

1224440064/1220703125

map [[0, -13, -14], [1, 5, 6]]

generators 315.2509133 1200

badness 497.4863561 rms .2766026501 weighted_g 12.16115842

10485760000/10460353203

map [[0, 4, 21], [1, 0, -6]]

generators 475.5422335 1200

badness 441.3527601 rms .1537673823 weighted_g 14.21152037

6115295232/6103515625

map [[0, 7, 3], [2, -3, 2]]

generators 528.8539366 600

badness 313.0746567 rms .1940181460 weighted_g 11.72920349

19073486328125/19042491875328

map [[0, 1, 1], [19, 24, 38]]

generators 386.2396202 1200/19

badness 544.7739924 rms .1047837215 weighted_g 17.32370777

32805/32768

map [[0, -1, 8], [1, 2, -1]]

generators 498.2724869 1200

badness 39.20134011 rms .1616904714 weighted_g 6.235512615

274877906944/274658203125

map [[0, -15, 2], [1, 4, 2]]

generators 193.1996149 1200

badness 178.5465796 rms .6082244804e-1 weighted_g 14.31844579

7629394531250/7625597484987

map [[0, -2, -3], [9, 19, 28]]

generators 315.6754868 400/3

badness 203.0321497 rms .2559261582e-1 weighted_g 19.94420419

9010162353515625/9007199254740992

map [[0, -8, 5], [2, 9, 1]]

generators 437.2581077 600

badness 116.1747349 rms .1772520822e-1 weighted_g 18.71429401

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5)
and
> 1/ln(5) times the steps to 3, 5, and 5/3. To get it to more or less
correspond to the previous g, I adjusted by a factor of
>
> (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2 sqrt(1/ln
(5))

thanks for doing this, gene.

> This isn't the only adjustment you might think of, so your milage
may vary. I did a search for badness (adjusted) less than 750, rms
less
> than 35 (as suggested by Dave) and adjusted g less than 20. Here's
the
> result:

>
> 256/243
>
> map [[0, 0, 1], [5, 8, 10]]
>
> generators 395.3362123 240
>
> badness 612.7594631 rms 12.75974156 weighted_g 3.634818387

blackwood's decatonic system. this is pretty essential, really. i
can't believe we forgot all about it all this time. i'd like to see
it in the paper.

> 16875/16384
>
> map [[0, -4, 3], [1, 2, 2]]
>
> generators 126.2382718 1200
>
> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964

someone tell me why i should care about this one.

> 4294967296/4271484375
>
> map [[0, -9, 7], [1, 2, 2]]
>
> generators 55.27549315 1200
>
> badness 687.5758107 rms .4831084292 weighted_g 11.24843211

ditto. perhaps a weighted badness cutoff of 650, then.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Yet another "best" list. This one uses weights of 1/ln(3), 1/ln(5)
and
> 1/ln(5) times the steps to 3, 5, and 5/3.

Thanks Gene.

> To get it to more or less
correspond to the previous g, I adjusted by a factor of
>
> (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2
sqrt(1/ln(5))
>
> This isn't the only adjustment you might think of, so your milage
may vary.

Indeed. I see no justification for a weighted rms (or any kind of
average or mean) that can produce a result which is _outside_ the
range of its inputs. Feeding it 1, 1, 1 should give you 1 no matter
what the weights are.

>I did a search for badness (adjusted) less than 750, rms
less
> than 35 (as suggested by Dave) and adjusted g less than 20. Here's
the
> result:

The fact that you uncovered for the first time that quintuple thirds
temperament (256/243), that Paul tells us has actually been used by
Blackwood, is I think justification enough for using Paul's weighting
of the numbers of generators to favour fifths.

Your list contains all the 5-limit temperaments that I think are of
some interest (except of course it is missing the degenerates).
However there are five temperaments in your list that I think are of
no interest whatsoever. If you were to reduce your complexity cutoff
so it is between that of parakleismic and 10485760000/10460353203 we
would agree on a best 20. In your terms the g value would need to be
between 12.2 and 14.2. In terms of the true weighted rms complexity it
needs to be between 11.7 and 13.5.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> blackwood's decatonic system. this is pretty essential, really. i
> can't believe we forgot all about it all this time. i'd like to see
> it in the paper.

So would I. According to my badness it rates better than septathirds
(4294967296/4271484375) or parakleismic (1224440064/1220703125).

>
> > 16875/16384
> >
> > map [[0, -4, 3], [1, 2, 2]]
> >
> > generators 126.2382718 1200
> >
> > badness 716.2095928 rms 5.942562596 weighted_g
4.939565964
>
> someone tell me why i should care about this one.

I'm calling this tertiathirds (was quadrafourths). By my badness it is
better than:
diminished, pelogic, twin-tertiatenths (semisuper), quintuple-thirds
(Blackwood's decatonic) and parakleismic.

> > 4294967296/4271484375
> >
> > map [[0, -9, 7], [1, 2, 2]]
> >
> > generators 55.27549315 1200
> >
> > badness 687.5758107 rms .4831084292 weighted_g
11.24843211
>
> ditto.

I'm calling this septathirds. By my badness it is marginally better
than parakleismic. They both have rms error less than 0.5 c but
tertiathirds needs one less generator (in weghted rms terms).

> perhaps a weighted badness cutoff of 650, then.

That won't work for me. The only way Gene and I can agree, when he is
using sharp cutoffs and I'm using gentle rolloffs, is if there doesn't
happen to be anything near the 3 upper/outer corners of Gene's
"acceptance brick". I'm visualising here points in a 3D space where
the horizontal dimensions are error and complexity and the vertical
dimension is Gene's badness.

There are very few choices for Gene's badness, error and complexity
cutoffs, that agree with a single badness cutoff for me.

Gene's already got badness and error cutoffs that work for me, all he
needs to do now is lower his complexity cutoff from 20 to 14 (or 13 if
he fixes up his weighted rms calculation).

And "best 20" is a nice round number. Here they are by name, in order
of weighted complexity.

neutral thirds
meantone
pelogic
augmented
semiminorthirds (porcupine)
quintuple thirds (Blackwood's decatonic)
diminished
diaschismic
magic
tertiathirds (quadrafourths)
kleismic
quartafifths (minimal diesic)
schismic
wuerschmidt
semisixths (tiny diesic)
subminor thirds (orwell)
quintelevenths (AMT)
septathirds (4294967296/4271484375)
twin tertiatenths (semisuper)
parakleismic (1224440064/1220703125)

The fact that no-one's written any music in, or even heard of the last
four (or is it the last six?) as 5-limit temperaments until recently,
should indicate that we have gone far enough in the direction of
increased complexity.

As usual, you can experiment with agreement between Gene's and my
lists, updated with Gene's latest temperaments, in
http://dkeenan.com/Music/5LimitTemp.xls.zip

On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@stretch-music.com>
wrote:

>--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>> 16875/16384
>>
>> map [[0, -4, 3], [1, 2, 2]]
>>
>> generators 126.2382718 1200
>>
>> badness 716.2095928 rms 5.942562596 weighted_g 4.939565964
>
>someone tell me why i should care about this one.

It could have melodic uses in 19-ET, dividing major thirds into three equal
parts. A 9-note scale is a pretty useful size for a basic scale.

C Db D# E F F# G Ab A# B C

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > Yet another "best" list. This one uses weights of 1/ln(3), 1/ln
(5)
> and
> > 1/ln(5) times the steps to 3, 5, and 5/3.
>
>
> Thanks Gene.
>
> > To get it to more or less
> correspond to the previous g, I adjusted by a factor of
> >
> > (1/sqrt(3) + 1/sqrt(3) + 1/sqrt(3)) / (sqrt(1/ln(3) + 2
> sqrt(1/ln(5))
> >
> > This isn't the only adjustment you might think of, so your milage
> may vary.
>
> Indeed. I see no justification for a weighted rms (or any kind of
> average or mean) that can produce a result which is _outside_ the
> range of its inputs. Feeding it 1, 1, 1 should give you 1 no matter
> what the weights are.
>
> >I did a search for badness (adjusted) less than 750, rms
> less
> > than 35 (as suggested by Dave) and adjusted g less than 20.
Here's
> the
> > result:
>
> The fact that you uncovered for the first time that quintuple
thirds
> temperament (256/243), that Paul tells us has actually been used by
> Blackwood, is I think justification enough for using Paul's
weighting
> of the numbers of generators to favour fifths.
>
> Your list contains all the 5-limit temperaments that I think are of
> some interest (except of course it is missing the degenerates).
> However there are five temperaments in your list that I think are
of
> no interest whatsoever. If you were to reduce your complexity
cutoff
> so it is between that of parakleismic and 10485760000/10460353203
we
> would agree on a best 20. In your terms the g value would need to
be
> between 12.2 and 14.2. In terms of the true weighted rms complexity
it
> needs to be between 11.7 and 13.5.

a lower complexity seems desirable now but we'll regret it next year
when we're working on the 19-limit.

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@s...>
> wrote:
>
> >--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> >> 16875/16384
> >>
> >> map [[0, -4, 3], [1, 2, 2]]
> >>
> >> generators 126.2382718 1200
> >>
> >> badness 716.2095928 rms 5.942562596 weighted_g
4.939565964
> >
> >someone tell me why i should care about this one.
>
> It could have melodic uses in 19-ET, dividing major thirds into
three equal
> parts. A 9-note scale is a pretty useful size for a basic scale.
>
> C Db D# E F F# G Ab A# B C

dave and you both? ok, then i think we should keep this in the paper -
- it's john negri's system in 19-equal (looks a little better in the
7-limit).

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> a lower complexity seems desirable now but we'll regret it next year
> when we're working on the 19-limit.

I have always assumed we would allow higher complexities for higher
primes. We might well be looking at stuff with up to 25 rms generators
for 19-limit. But 13 is plenty for 5-limit.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:
> > On Thu, 21 Mar 2002 20:47:00 -0000, "paulerlich" <paul@s...>
> > wrote:
> > >someone tell me why i should care about this one. [tertiathirds]
> >
> > It could have melodic uses in 19-ET, dividing major thirds into
> three equal
> > parts. A 9-note scale is a pretty useful size for a basic scale.
> >
> > C Db D# E F F# G Ab A# B C
>
> dave and you both? ok, then i think we should keep this in the paper
-
> - it's john negri's system in 19-equal (looks a little better in the
> 7-limit).

If you keep this one (tertiathirds) you're also gonna have to keep the
other one you asked about, septathirds (4294967296/4271484375), since
septathirds is better than tertiathirds by Gene's badness.

But I agree that it's extremely boring melodically, being essentially
22-tET.

In-Reply-To: <a7eg9l+omel@eGroups.com>
Paul:
> > a lower complexity seems desirable now but we'll regret it next year
> > when we're working on the 19-limit.

You don't plan on getting to the 19-limit until next year?

Dave K:
> I have always assumed we would allow higher complexities for higher
> primes. We might well be looking at stuff with up to 25 rms generators
> for 19-limit. But 13 is plenty for 5-limit.

31&62 has an RMS complexity of 22.5 and it's the best one I've got below
25. It's got a worst error of 11 cents (or RMS of 5 cents) so I'm not
sure it counts as 19-limit anything. There's also 31&72 with RMS
complexity of 24.8 and 8.8 cent minimax, and 62&72 with a 23.5 complexity
and 7.5 cent minimax.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a7eg9l+omel@e...>
> Paul:
> > > a lower complexity seems desirable now but we'll regret it next
year
> > > when we're working on the 19-limit.
>
> You don't plan on getting to the 19-limit until next year?
>
> Dave K:
> > I have always assumed we would allow higher complexities for
higher
> > primes. We might well be looking at stuff with up to 25 rms
generators
> > for 19-limit. But 13 is plenty for 5-limit.
>
> 31&62 has an RMS complexity of 22.5 and it's the best one I've got
below
> 25. It's got a worst error of 11 cents (or RMS of 5 cents) so I'm
not
> sure it counts as 19-limit anything. There's also 31&72 with RMS
> complexity of 24.8 and 8.8 cent minimax, and 62&72 with a 23.5
complexity
> and 7.5 cent minimax.

Thanks. So we'll definitely be going _beyond_ weighted rms complexity
of 25 for 19-limit. Or maybe it just says that 19-limit really isn't
worth the trouble.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> neutral thirds
> meantone
> pelogic
> augmented
> semiminorthirds (porcupine)
> quintuple thirds (Blackwood's decatonic)
> diminished
> diaschismic
> magic
> tertiathirds (quadrafourths)
> kleismic
> quartafifths (minimal diesic)
> schismic
> wuerschmidt
> semisixths (tiny diesic)
> subminor thirds (orwell)
> quintelevenths (AMT)
> septathirds (4294967296/4271484375)
> twin tertiatenths (semisuper)
> parakleismic (1224440064/1220703125)

I find that I can also agree with Gene on a best 17, he just needs to
change his complexity cutoff to 10 and we lose the last three off the
list above, including septathirds which Paul objected to, and I agreed
was boring. I would prefer this list of 17, to the list of 20. There
is quite a big gap in complexity between quintelevenths and
septathirds.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > neutral thirds
> > meantone
> > pelogic
> > augmented
> > semiminorthirds (porcupine)
> > quintuple thirds (Blackwood's decatonic)
> > diminished
> > diaschismic
> > magic
> > tertiathirds (quadrafourths)
> > kleismic
> > quartafifths (minimal diesic)
> > schismic
> > wuerschmidt
> > semisixths (tiny diesic)
> > subminor thirds (orwell)
> > quintelevenths (AMT)
> > septathirds (4294967296/4271484375)
> > twin tertiatenths (semisuper)
> > parakleismic (1224440064/1220703125)
>
> I find that I can also agree with Gene on a best 17, he just needs
to
> change his complexity cutoff to 10 and we lose the last three off
the
> list above, including septathirds which Paul objected to, and I
agreed
> was boring. I would prefer this list of 17, to the list of 20.
There
> is quite a big gap in complexity between quintelevenths and
> septathirds.

let's use the name 'negri' for negri's system, shall we?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> > - it's john negri's system in 19-equal (looks a little better in
the
> > 7-limit).
>
> If you keep this one (tertiathirds) you're also gonna have to keep
the
> other one you asked about, septathirds (4294967296/4271484375),
since
> septathirds is better than tertiathirds by Gene's badness.
>
> But I agree that it's extremely boring melodically, being
essentially
> 22-tET.

hey, how could i argue with any reference to 22-equal? :)