The hanson family

The 5 to 7 limit choice is not as clear as I was thinking, because
Number 91 intrudes itself, so I withdraw my objection to keeping
"hanson" only as a 5-limit name. It would be nice to figure out a
naming scheme which would help sort out these family trees, however.

hanson 15625/15552 comma
[1200.29, 317.07]

7-limit hanson family

0: <<6 5 22 -6 18 37|| catakleismic? {225/224, 4375/4374}
[1200.54, 316.91]

-15: <<6 5 7 -6 -6 2|| {36/35, 375/343}
[1192.24, 314.59]

-19: <<6 5 3 -6 -12 -7|| kleismic? {49/48, 126/125}
[1203.19, 317.83]

-34: <<6 5 -12 -6 -36 -42|| {64/64, 15625/15309}

-53: <<6 5 -31 -6 -66 -86|| "Number 91" {5120/5103, 15625/15552}
[1199.98, 317.09]

11-limit catakleismic?

0: <<6 5 22 -21 -6 18 -54 37 -66 -135|| catakleismic? {225/224,
385/384, 4375/4374}
[1200.54, 316.91]

19: <<6 5 22 -2 -6 18 -24 37 -22 -82|| {100/99, 225/224, 864/847}
[1197.96, 316.75]

53: <<6 5 22 32 -6 18 30 37 57 14|| {99/98, 176/175, 2200/2187}
[1198.27, 316.76]

72: <<6 5 22 51 -6 18 60 37 101 67|| {225/224, 441/440, 4375/4374}
[1200.61, 316.85]

11-limit kleismic?

0: <<6 5 3 -2 -6 -12 -24 -7 -22 -16|| {49/48, 56/55, 100/99}
[1200.82, 318.18]

-4: <<6 5 3 -6 -6 -12 -30 -7 -31 -27|| {33/32, 49/48, 126/125}
[1205.30, 315.59]

15: <<6 5 3 13 -6 -12 0 -7 13 26|| {49/48, 55/54, 77/75}
[1200.98, 318.16]

19: <<6 5 3 17 -6 -12 -6 -7 22 37|| {45/44, 49/48, 126/125}
[1204.38, 315.93]

11-limit Number 91

0: <<6 5 -31 32 -6 -66 30 -86 57 197|| {385/384, 2200/2187, 3388/3375}
[1199.89, 317.15]

> The 5 to 7 limit choice is not as clear as I was thinking,
> because Number 91 intrudes itself, so I withdraw my objection
> to keeping "hanson" only as a 5-limit name. It would be nice
> to figure out a naming scheme which would help sort out these
> family trees, however.

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.

-Carl

--- In tuning-math@yahoogroups.com, "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.

>> 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?

My miserable notes offer...

"""
>>>it's easy to determine that [<1 x y z|, <0 6 -7 -2|]
>>>is a possible mapping of miracle, as is [<x 1 y z|,
>>><-6 0 -25 -20|], but I don't know how to get x, y, and z.
>>>I've been trying to find something like this in the
>>>archives, but I don't know where to look.
>>
>> I don't see that this was ever answered. Did I miss it?
>
>If you know the whole wedgie, finding x, y and z can be done by
>solving a linear system. If you only know the period and
>generator map, you first need to get the rest of the wedgie,
>which will be the one which has a much lower badness than its
>competitors.
>
>For instance, suppose I know the wedgie is <<1 4 10 4 13 12||.
>Then I can set up the equations resulting from
>
><1 x y z| ^ <0 1 4 10| = <<1 4 10 4 13 12||
>
>We have <1 x y z| ^ <0 1 4 10| = <<1 4 10 4x-y 10x-z 10y-4z||
>
>Solving this gives us y=4x-4, z=10x-13; we can pick any integer
>for x so we choose one giving us generators in a range we like.
>Since 3 is represented by [x 1] in terms of octave x and
>generator, if we want 3/2 as a generator we pick x=1.
"""

...which makes it sound as if 'tis predicated on mere fancy.

-Carl

--- In tuning-math@yahoogroups.com, "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.

>> >> 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...

-Carl

--- In tuning-math@yahoogroups.com, "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)