tratios and yantras

I took all the 7-limit integers less than 2^20 (recall this is a
yantra when reduced to an octave) and found the smallest instance of
three successive 7-limit integers which were mapped to the same val
when wedged with an 7-limit wedgie, in order to get tratios for some
of the most important 7-limit linear temperaments. Here's the results:

meantone 1120:1125:1134
miracle 7168:7200:7203
ennealimmal 419904:420000:420175
magic 6048:6075:6125
pajara 441:448:450
(septimal) schismic 27783:28000:28125
orwell 12005:12096:12150

I'm adding LCMs below, though i still don't know if the numbers
and/or LCMs are smallest possible. It's interesting that the LCMs
seem to give a much better ranking of complexity than the size of the
numbers. Of course we're essentially using LCM (=n*d) already in the
one-comma case. We should be able to visualize this, as well as the
L1 (and maybe L-inf??) complexity measures, as specific sorts
of "area" on the lattice, though 4-D is kind of hard. Need help,
please . . .

Of course if you divide the lcm by the numbers in the tratio you get
another tratio with the same LCM, representing the same temperament,
but I think always with larger numbers (if you start with one of
these):

7-limit Blackwood 243:252:256 lcm 435,456
Dominant Sevenths 245:252:256 lcm 564,480
7-limit Diminished 343:350:360 lcm 617,400
Pajara 441:448:450 lcm 705,600
Semifourths 240:243:245 lcm 952,560
Tripletone 125:126:128 lcm 1,008,000
Injera 392:400:405 lcm 1,587,600
meantone 1120:1125:1134 lcm 2,268,000
7-limit Augmented:
1125:1152:1176 lcm 7,056,000
BUT ALSO
1568:1575:1620 lcm 3,175,200
(any simpler tratio?)
OldKleismic 1000:1008:1029 lcm 6,174,000
Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000
Negri 672:675:686 lcm 7,408,800
semisixths 3375:3402:3430 lcm 20,837,250
Superpythagorean 1701:1715:1728 lcm 26,671,680
magic 6048:6075:6125 lcm 47,628,000
miracle 7168:7200:7203 lcm 553,190,400
orwell 12005:12096:12150 lcm 933,508,800
(septimal) schismic 27783:28000:28125 lcm 2,778,300,000
ennealimmal 419904:420000:420175 lcm 4,410,829,080,000

What's bigger -- the lcm of ennealimmal or the national debt?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> I'm adding LCMs below, though i still don't know if the numbers
> and/or LCMs are smallest possible. It's interesting that the LCMs
> seem to give a much better ranking of complexity than the size of
the
> numbers. Of course we're essentially using LCM (=n*d) already in
the
> one-comma case. We should be able to visualize this, as well as the
> L1 (and maybe L-inf??) complexity measures, as specific sorts
> of "area" on the lattice, though 4-D is kind of hard. Need help,
> please . . .
>
> Of course if you divide the lcm by the numbers in the tratio you
get
> another tratio with the same LCM, representing the same
temperament,
> but I think always with larger numbers (if you start with one of
> these):
>
>
> 7-limit Blackwood 243:252:256 lcm 435,456
> Dominant Sevenths 245:252:256 lcm 564,480
> 7-limit Diminished 343:350:360 lcm 617,400
> Pajara 441:448:450 lcm 705,600
> Semifourths 240:243:245 lcm 952,560
> Tripletone 125:126:128 lcm 1,008,000
> Injera 392:400:405 lcm 1,587,600
> meantone 1120:1125:1134 lcm 2,268,000
> 7-limit Augmented:
> 1125:1152:1176 lcm 7,056,000
> BUT ALSO
> 1568:1575:1620 lcm 3,175,200
> (any simpler tratio?)
> OldKleismic 1000:1008:1029 lcm 6,174,000
> Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000
> Negri 672:675:686 lcm 7,408,800
> semisixths 3375:3402:3430 lcm 20,837,250
> Superpythagorean 1701:1715:1728 lcm 26,671,680
> magic 6048:6075:6125 lcm 47,628,000
> miracle 7168:7200:7203 lcm 553,190,400
> orwell 12005:12096:12150 lcm 933,508,800
> (septimal) schismic 27783:28000:28125 lcm 2,778,300,000
> ennealimmal 419904:420000:420175 lcm 4,410,829,080,000
>
> What's bigger -- the lcm of ennealimmal or the national debt?

. . . on Sept. 30th, 1993?

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

> 7-limit Blackwood 243:252:256 lcm 435,456

I just found this for Blackwood:

189:192:196 lcm 84,672

I guess "by inspection" doesn't cut it.

I'll have to do this systematically.

Here's my Matlab program to convert from tratios to wedgies:

function out=trat2wedg(a1,a2,a3);
j=factor(a1);
a12=sum(j==2);
a13=sum(j==3);
a15=sum(j==5);
a17=sum(j==7);
j=factor(a2);
a22=sum(j==2);
a23=sum(j==3);
a25=sum(j==5);
a27=sum(j==7);
j=factor(a3);
a32=sum(j==2);
a33=sum(j==3);
a35=sum(j==5);
a37=sum(j==7);
w=([det([a22-a12 a23-a13;a22-a32 a23-a33]) det([a22-a12 a25-a15;a22-
a32 a25-a35]) det([a22-a12 a27-a17;a22-a32 a27-a37]) det([a23-a13 a25-
a15;a23-a33 a25-a35]) det([a23-a13 a27-a17;a23-a33 a27-a37]) det([a25-
a15 a27-a17;a25-a35 a27-a37])]);
g=gcd(w(1),gcd(w(2),gcd(w(3),gcd(w(4),gcd(w(5),w(6))))));
out=w/g;

> Dominant Sevenths 245:252:256 lcm 564,480
> 7-limit Diminished 343:350:360 lcm 617,400
> Pajara 441:448:450 lcm 705,600
> Semifourths 240:243:245 lcm 952,560
> Tripletone 125:126:128 lcm 1,008,000
> Injera 392:400:405 lcm 1,587,600
> meantone 1120:1125:1134 lcm 2,268,000
> 7-limit Augmented:
> 1125:1152:1176 lcm 7,056,000
> BUT ALSO
> 1568:1575:1620 lcm 3,175,200
> (any simpler tratio?)
> OldKleismic 1000:1008:1029 lcm 6,174,000
> Catler -- same as 12-equal -- 625:648:640 lcm 6,480,000
> Negri 672:675:686 lcm 7,408,800
> semisixths 3375:3402:3430 lcm 20,837,250
> Superpythagorean 1701:1715:1728 lcm 26,671,680
> magic 6048:6075:6125 lcm 47,628,000
> miracle 7168:7200:7203 lcm 553,190,400
> orwell 12005:12096:12150 lcm 933,508,800
> (septimal) schismic 27783:28000:28125 lcm 2,778,300,000
> ennealimmal 419904:420000:420175 lcm 4,410,829,080,000
>
> What's bigger -- the lcm of ennealimmal or the national debt?

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

> I'll have to do this systematically.

But now I must sleep . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I took all the 7-limit integers less than 2^20 (recall this is a
> yantra when reduced to an octave) and found the smallest instance of
> three successive 7-limit integers which were mapped to the same val
> when wedged with an 7-limit wedgie, in order to get tratios for some
> of the most important 7-limit linear temperaments. Here's the
results:
>
> meantone 1120:1125:1134
> miracle 7168:7200:7203
> ennealimmal 419904:420000:420175
> magic 6048:6075:6125
> pajara 441:448:450
> (septimal) schismic 27783:28000:28125
> orwell 12005:12096:12150

I did something a little more straightforward. I calculated the
wedgie for every triplet of consecutive 7-limit integers, through
99999. Then I just looked for the earliest occurence of each wedgie.
I found a lot more, though a few wedgies apparently don't show up at
all. There are a couple of improvements over the results I posted in

/tuning-math/message/10408

Meantone 1120:1125:1134
Magic 6048:6075:6125
Pajara 441:448:450
Semisixths 3375:3402:3430
Dominant Seventh 315:320:324
-- larger numbers than 245:252:256, but lcm much smaller at 181,440
Injera 392:400:405
OldKleismic 864:875:882
-- smaller numbers than 1000:1008:1029, lcm smaller at 5,292,000
Semifourths 240:243:245
Negri 672:675:686
Tripletone 125:126:128
Schismic 27783:28000:28125
Superpythagorean 1701:1715:1728
Orwell 12005:12096:12150
Augmented NOT FOUND
Porcupine NOT FOUND
<<6, 10, 10, 2, -1, -5]] 243:245:250
Supermajor seconds 5120:5145:5184
Flattone 2560:2592:2625
Diminished 245:250:252
<<6, 10, 3, 2, -12, -21]] NOT FOUND
Catler NOT FOUND
Gawel 10800:10935:10976
Nonkleismic 12000:12005:12096
Miracle 7168:7200:7203
Beatlemania 4725:4800:4802
<<6, -2, -2, -17, -20, 1]] 1024:1029:1050
<<8, 6, 6, -9, -13, -3]] 1715:1728:1750
Blackwood 189:192:196

Now for 5-limit ETs, one semi-improvement over

/tuning-math/message/10404

3-equal 27:30:32
-- smaller numbers than 45:48:50
4-equal 24:25:27
5-equal 75:80:81
7-equal 240:243:250
9-equal 125:128:135
10-equal 16200:16384:16875 YUCK!
12-equal 625:640:648
15-equal 243:250:256
16-equal NOT FOUND
19-equal 15360:15552:15625
22-equal 6075:6144:6250
New ones:
27-equal 78125:78732:80000 lcm 196,830,000,000
29-equal 32768:32805:33750 lcm 134,369,280,000

hi Gene and Paul,

definitions of tratio and yantra, please.
thanks.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Gene and Paul,
>
>
> definitions of tratio and yantra, please.
> thanks.
>
>
>
> -monz

Hi Monz,

Gene defined yantra here:

/tuning-math/message/10095

It's just the first N integers with no prime factors above P, for a
given N and P.

Tratio was something you posted to tell me I should use the
terminology "proportion" for -- remember? If a temperament has
codimension 1, it can be described by a vanishing ratio. If a
temperament has codimension 2, it can be described by a vanishing
tratio (a three-term proportion, like 625:640:648).

Apparently, many, but not all, of the important codimension-1
temperaments can be described by vanishing tratios of three
consective terms in the yantra (for which P is either 5 or 7, and N
is infinite or simply "large enough").

hi Paul,

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi Gene and Paul,
> >
> >
> > definitions of tratio and yantra, please.
> > thanks.
> >
> >
> >
> > -monz
>
> Hi Monz,
>
> Gene defined yantra here:
>
> /tuning-math/message/10095
>
> It's just the first N integers with no prime factors above P, for a
> given N and P.
>
> Tratio was something you posted to tell me I should use the
> terminology "proportion" for -- remember? If a temperament has
> codimension 1, it can be described by a vanishing ratio. If a
> temperament has codimension 2, it can be described by a vanishing
> tratio (a three-term proportion, like 625:640:648).
>
> Apparently, many, but not all, of the important codimension-1
> temperaments can be described by vanishing tratios of three
> consective terms in the yantra (for which P is either 5 or 7, and N
> is infinite or simply "large enough").

thanks.

Ernest McClain's definition of yantra (which i believe
was Gene's source) always includes a terminating
maximum value below which all the included integers
must lie.

References
----------

McClain, Ernest.
_The Myth of Invariance_

McClain, Ernest.
_The Pythagorean Plato_

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> Ernest McClain's definition of yantra (which i believe
> was Gene's source) always includes a terminating
> maximum value below which all the included integers
> must lie.

I think my definition is preferable, as it gives a 1-1 mapping. In any
case, they are equivalent.