Vanishing tratios

Before I finalize my paper, I'd like to explore the following idea.

What if I take a 3-term ratio ("tratio"?) and have it vanish?

Let's say 125:126:128.

So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 vanishes.

Any two of these three 'commas' of course would be enough to give you
the result: in 7-limit multibreed/multival/wedgie form,

<<3, 0, -6, -7, -18, -14]],

the temperament formerly known as Tripletone.

Other examples would seem to be:

243:252:256 for 7-limit Blackwood
245:252:256 for Dominant Sevenths
343:350:360 for 7-limit Diminished
441:448:450 for Pajara

Does anyone know a way to find the simplest (lowest numbers) tratio
for a given codimension-two temperament? How about for the 7-
limit 'linear' temperaments listed here:

/tuning-math/message/10266
?

And, salivating, I ask, is there a straightforward calculation to go
from the vanishing tratio to the TOP error and/or complexity -- like
there is for vanishing ratios in the codimension-1 case?

For a single vanishing ratio n:d, the TOP error is proportional to

log(n/d)/log(n*d),

and complexity [= 'L1 norm' of the wedgie] is proportional to

log(n*d).

Since I just mentioned Negri on the MakeMicroMusic list, I think it's

672:675:686

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Before I finalize my paper, I'd like to explore the following idea.
>
> What if I take a 3-term ratio ("tratio"?) and have it vanish?
>
> Let's say 125:126:128.
>
> So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125 vanishes.
>
> Any two of these three 'commas' of course would be enough to give
you
> the result: in 7-limit multibreed/multival/wedgie form,
>
> <<3, 0, -6, -7, -18, -14]],
>
> the temperament formerly known as Tripletone.
>
> Other examples would seem to be:
>
> 243:252:256 for 7-limit Blackwood
> 245:252:256 for Dominant Sevenths
> 343:350:360 for 7-limit Diminished
> 441:448:450 for Pajara
>
> Does anyone know a way to find the simplest (lowest numbers) tratio
> for a given codimension-two temperament? How about for the 7-
> limit 'linear' temperaments listed here:
>
> /tuning-math/message/10266
> ?
>
> And, salivating, I ask, is there a straightforward calculation to
go
> from the vanishing tratio to the TOP error and/or complexity --
like
> there is for vanishing ratios in the codimension-1 case?
>
> For a single vanishing ratio n:d, the TOP error is proportional to
>
> log(n/d)/log(n*d),
>
> and complexity [= 'L1 norm' of the wedgie] is proportional to
>
> log(n*d).

7-limit Miracle -- 7168:7200:7203?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Since I just mentioned Negri on the MakeMicroMusic list, I think
it's
>
> 672:675:686
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > Before I finalize my paper, I'd like to explore the following
idea.
> >
> > What if I take a 3-term ratio ("tratio"?) and have it vanish?
> >
> > Let's say 125:126:128.
> >
> > So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125
vanishes.
> >
> > Any two of these three 'commas' of course would be enough to give
> you
> > the result: in 7-limit multibreed/multival/wedgie form,
> >
> > <<3, 0, -6, -7, -18, -14]],
> >
> > the temperament formerly known as Tripletone.
> >
> > Other examples would seem to be:
> >
> > 243:252:256 for 7-limit Blackwood
> > 245:252:256 for Dominant Sevenths
> > 343:350:360 for 7-limit Diminished
> > 441:448:450 for Pajara
> >
> > Does anyone know a way to find the simplest (lowest numbers)
tratio
> > for a given codimension-two temperament? How about for the 7-
> > limit 'linear' temperaments listed here:
> >
> > /tuning-math/message/10266
> > ?
> >
> > And, salivating, I ask, is there a straightforward calculation to
> go
> > from the vanishing tratio to the TOP error and/or complexity --
> like
> > there is for vanishing ratios in the codimension-1 case?
> >
> > For a single vanishing ratio n:d, the TOP error is proportional to
> >
> > log(n/d)/log(n*d),
> >
> > and complexity [= 'L1 norm' of the wedgie] is proportional to
> >
> > log(n*d).

5-limit 12-equal -- 625:640:648?

I'm going tratio-wild, but I have to go! :(

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> 7-limit Miracle -- 7168:7200:7203?
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > Since I just mentioned Negri on the MakeMicroMusic list, I think
> it's
> >
> > 672:675:686
> >
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > Before I finalize my paper, I'd like to explore the following
> idea.
> > >
> > > What if I take a 3-term ratio ("tratio"?) and have it vanish?
> > >
> > > Let's say 125:126:128.
> > >
> > > So 128:125 vanishes, 128:126 = 64:63 vanishes, and 126:125
> vanishes.
> > >
> > > Any two of these three 'commas' of course would be enough to
give
> > you
> > > the result: in 7-limit multibreed/multival/wedgie form,
> > >
> > > <<3, 0, -6, -7, -18, -14]],
> > >
> > > the temperament formerly known as Tripletone.
> > >
> > > Other examples would seem to be:
> > >
> > > 243:252:256 for 7-limit Blackwood
> > > 245:252:256 for Dominant Sevenths
> > > 343:350:360 for 7-limit Diminished
> > > 441:448:450 for Pajara
> > >
> > > Does anyone know a way to find the simplest (lowest numbers)
> tratio
> > > for a given codimension-two temperament? How about for the 7-
> > > limit 'linear' temperaments listed here:
> > >
> > > /tuning-math/message/10266
> > > ?
> > >
> > > And, salivating, I ask, is there a straightforward calculation
to
> > go
> > > from the vanishing tratio to the TOP error and/or complexity --
> > like
> > > there is for vanishing ratios in the codimension-1 case?
> > >
> > > For a single vanishing ratio n:d, the TOP error is proportional
to
> > >
> > > log(n/d)/log(n*d),
> > >
> > > and complexity [= 'L1 norm' of the wedgie] is proportional to
> > >
> > > log(n*d).

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Before I finalize my paper, I'd like to explore the
> following idea.
>
> What if I take a 3-term ratio ("tratio"?) and have it vanish?
>
> Let's say 125:126:128.
>
> So 128:125 vanishes, 128:126 = 64:63 vanishes, and
>126:125 vanishes.

traditionally, "ratio" has been used for 2-term comparisons,
and for >2-term, "proportion". just nit-picking on terminology.

but that's a cool idea you posted about.

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> 5-limit 12-equal -- 625:640:648?
>
> I'm going tratio-wild, but I have to go! :(

Let's define the function

weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b)

5-limit ETs and lowest-weird tratios (by inspection)

ET........tratio............weird
03-equal: 45:48:50......... 3600
04-equal: 24:25:27......... 5400
05-equal: 75:80:81......... 32400
07-equal: 384:400:405...... 259200
("""""""""240:243:250...... 486000)
09-equal: 125:128:135...... 432000
10-equal: 729:768:800...... 4665600
12-equal: 625:640:648...... 6480000
15-equal: 243:250:256...... 7776000
16-equal: 3072:3125:3240... 259200000
19-equal: 15360:15552:15625 3888000000
22-equal: 6075:6144:6250... 1555200000

The monotonic pattern seems to break here. Did I miss any lower-weird
and/or simpler tratios?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

Speaking of vanishing, I was going to respond to a post of yours about
tratios and yantras, and it seems to have vanished!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > 5-limit 12-equal -- 625:640:648?
> >
> > I'm going tratio-wild, but I have to go! :(
>
> Let's define the function
>
> weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b)

What's wrong with lcm(a,b,c) = abc/gcd(ab,ac,bc)?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> Speaking of vanishing, I was going to respond to a post of yours
about
> tratios and yantras, and it seems to have vanished!

I figured out what you meant.

>>Speaking of vanishing, I was going to respond to a post
>>of yours about tratios and yantras, and it seems to have
>>vanished!
>
>I figured out what you meant.

I appreciate the desire to focus the discussion, but thought
I'd mention that your readers might not have figured it out...

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> >
> > > 5-limit 12-equal -- 625:640:648?
> > >
> > > I'm going tratio-wild, but I have to go! :(
> >
> > Let's define the function
> >
> > weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b)
>
> What's wrong with lcm(a,b,c) = abc/gcd(ab,ac,bc)?

You're right -- all these formulae seem to produce the same result
(that's what you meant, right?) . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > > wrote:
> > >
> > > > 5-limit 12-equal -- 625:640:648?
> > > >
> > > > I'm going tratio-wild, but I have to go! :(
> > >
> > > Let's define the function
> > >
> > > weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b)
> >
> > What's wrong with lcm(a,b,c) = abc/gcd(ab,ac,bc)?
>
> You're right -- all these formulae seem to produce the same result
> (that's what you meant, right?) . . .

What I actually meant was that I thought what you probably wanted is
the least common multiple, and that abc/gcd(ab,ac,bc) is a formula for
it in terms of the gcd. Weird(a,b,c) is a different arthmetic
function, unknown to me.

weird(2,4,6) = 2*4*6/gcd(2,4)/gcd(2,6)/gcd(4,6) = 48/8 = 6
lcm(2,4,6) = 2*4*6/gcd(8,12,24) = 48/4 = 12

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > > > wrote:
> > > >
> > > > > 5-limit 12-equal -- 625:640:648?
> > > > >
> > > > > I'm going tratio-wild, but I have to go! :(
> > > >
> > > > Let's define the function
> > > >
> > > > weird(a,b,c) = a*b*c/gcd(a,b)/gcd(a,c)/gcd(a,b)
> > >
> > > What's wrong with lcm(a,b,c) = abc/gcd(ab,ac,bc)?
> >
> > You're right -- all these formulae seem to produce the same
result
> > (that's what you meant, right?) . . .
>
> What I actually meant was that I thought what you probably wanted is
> the least common multiple, and that abc/gcd(ab,ac,bc) is a formula
for
> it in terms of the gcd. Weird(a,b,c) is a different arthmetic
> function, unknown to me.
>
> weird(2,4,6) = 2*4*6/gcd(2,4)/gcd(2,6)/gcd(4,6) = 48/8 = 6
> lcm(2,4,6) = 2*4*6/gcd(8,12,24) = 48/4 = 12

Hmm . . . the two formulae seem to give the same result as long as gcd
(a,b,c) = 1, which was the case for all the tratios in question.
Right?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Hmm . . . the two formulae seem to give the same result as long as gcd
> (a,b,c) = 1, which was the case for all the tratios in question.
> Right?

Right.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Hmm . . . the two formulae seem to give the same result as long
as gcd
> > (a,b,c) = 1, which was the case for all the tratios in question.
> > Right?
>
> Right.

So how can we express the tratio, or the lcm, in terms of the wedgie?

As we've seen, in this case genie comes out to about 3 times the
number of notes per octave. So the comparison is essentially with
number of notes.

Again:

> ET........tratio............lcm.............
> 03-equal: 45:48:50......... 3,600...........
> 04-equal: 24:25:27......... 5,400...........
> 05-equal: 75:80:81......... 32,400..........
> 07-equal: 384:400:405...... 259,200.........
> ("""""""""240:243:250...... 486,000).......(
> 09-equal: 125:128:135...... 432,000.........
> 10-equal: 729:768:800...... 4,665,600.......
> 12-equal: 625:640:648...... 6,480,000.......
> 15-equal: 243:250:256...... 7,776,000.......
> 16-equal: 3072:3125:3240... 259,200,000.....
> 19-equal: 15360:15552:15625 3,888,000,000
> 22-equal: 6075:6144:6250... 1,555,200,000
>
> The monotonic pattern seems to break here. Did I miss any lower-
weird
> and/or simpler tratios?

The next step is to draw a picture.

Sorry, wrong subject line.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> As we've seen, in this case genie comes out to about 3 times the
> number of notes per octave. So the comparison is essentially with
> number of notes.
>
> Again:
>
> > ET........tratio............lcm.............
> > 03-equal: 45:48:50......... 3,600...........
> > 04-equal: 24:25:27......... 5,400...........
> > 05-equal: 75:80:81......... 32,400..........
> > 07-equal: 384:400:405...... 259,200.........
> > ("""""""""240:243:250...... 486,000).......(
> > 09-equal: 125:128:135...... 432,000.........
> > 10-equal: 729:768:800...... 4,665,600.......
> > 12-equal: 625:640:648...... 6,480,000.......
> > 15-equal: 243:250:256...... 7,776,000.......
> > 16-equal: 3072:3125:3240... 259,200,000.....
> > 19-equal: 15360:15552:15625 3,888,000,000
> > 22-equal: 6075:6144:6250... 1,555,200,000
> >
> > The monotonic pattern seems to break here. Did I miss any lower-
> weird
> > and/or simpler tratios?
>
> The next step is to draw a picture.