Yahoo Tuning Groups Ultimate Backup tuning-math Survey VIII

<4375/4374, 3136/3125> Minkowski reduced

Ets: 19, 80, 99, 118

Map:

[3 -2]
[2 3]
[4 2]
[1 11]

The adjusted mappings are pretty horrendous, and this might be used
instead, with generators a' = 46.9967/99 and b' = 20.9953/99.

Adjusted map:

[ 0 1]
[-13 5]
[-14 6]
[-35 12]

Generators: a = .2626420944 (~6/5) = 26.00156735 / 99; b = 1

Errors and 99 et:

3: 0.828 1.975
5: 1.299 1.565
7: 0.206 0.871

<2048/2025, 50/49> Minkowski reduced

Ets: 10, 12, 22

Map (no adjustment)

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

Generators: a = .4093213919 (~4/3) = 9.995070622 / 22; b = 1/2

Errors and 22 et:

3: 6.86 7.14
5: -3.94 -4.50
7: 13.55 12.99

This is, of course, paultone, and it can be seen that we don't get
much milage out of using anything but 12+10 in the 22-et for it.

<3136/3125, 64/63>

Minkowski reduction <3125/3087, 64/63>

Ets: 12,13,25,37,49

Map (no adjustment)

[ 0 1]
[-5 2]
[ 4 2]
[ 10 2]

Generators: a = .08140287107 = 3.01190623 / 37; b = 1

Errors and 37-et

3: 9.63 11.56
5: 4.42 2.88
7: 2.81 4.15

A 12-tone temperament for the adventurous

<4375/4374, 2048/2025> Minkowski reduced

Ets: 46, 80, 126

Map (no adjustment)

[ 0 2]
[ 1 3]
[-2 5]
[15 3]

Generators: a = .08730149627 (~17/16) = 10.99998853 / 126

Errors and 126-et

3: 2.81 2.81
5: 4.16 4.16
7: 2.60 2.60

The difference between this and the 126-et is far below the limits of
perceptibility, so this may as well be called 80+46. Of course the
version in the 46-et is perfectly fine, and 17/16 is so close to
2^(11/126) that it can be used also.

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

In-Reply-To: <9tt3qr+qem5@eGroups.com>
Me:
> > Gene! This is exactly the problem I've been trying to solve since
> before
> > you arrived here! Now, if you look at the example you pointed out
> at the
> > weekend, you'll see even the "right" pair of vectors seem to have
> torsion.

Paul:
> What example was that? I'm guessing that in that example, a power of
> p/q is shorter than r/s.

[-1, 0, -2, 2]
[1, 7, -4, -1].

I can't get rid of the 7 in the second column. I can prove you can't get
rid of the factor of 2 in the third column. For the second and fourth
columns to be even, you have to add an even multiple of the lower vector
to the higher. But in the first column, that means adding an even number
to a multiple of -1, so you need to multiply the higher vector by an even
number as well. Which means both are being multiplied by 2, and
subsequently dividing by 2 doesn't simplify anything.

I don't know what wedge products are or their relevance. Or what p, q, r
and s are supposed to be. But we have two problems: to divide through by
common factors and to know what common factors we don't have to divide
through by.

Graham

🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

In-Reply-To: <9ttaj2+539k@eGroups.com>
In article <9ttaj2+539k@eGroups.com>, paul@stretch-music.com (Paul Erlich)
wrote:

> --- In tuning-math@y..., graham@m... wrote:
>
> > > What example was that? I'm guessing that in that example, a power
> of
> > > p/q is shorter than r/s.
> >
> > [-1, 0, -2, 2]
>
> 49/50
>
> > [1, 7, -4, -1].
>
> 4374/4375
>
> Sure enough, (49/50)^2 = 2401/2500 is simpler than 4374/4375. Hence
> we have a mismatch of unison vectors. I think torsion rules the realm
> outside the "strong Minkowski" condition I proposed.

Ah, but how about 64:63 and 3125:3087?

[0, -2, 5, -3]
[6, -2, 0, -1]

Lots of common factors, but (64:63)^2 = 4096:3969.

I'm not clear about the relevance of torsion here. Really, you should
include the chromatic unison vector to see if the basis has torsion. So
how do we know when we've gone far enough in removing those common factors
if we don't have a chromatic UV?

In case you haven't caught up with the simplification process, I'll work
through your example of 2048:2025 and 50:49

[11 -4 -2 0]
[ 1 0 2 -2]

All you have to do is add them

[11 -4 -2 0]
+[1 0 2 -2]
=
[12 -4 0 -2]

and divide the result through by 2

[6 -2 0 -1]

which is 64:63. If [12 -4 0 -2] is a unison, so must [6 -2 0 -1] be.

It also sounds like a cop-out to say the UVs are "mismatched". 49/50
and 4374/4375 remain the simplest way of producing the mapping [(2, 0),
(2, 5), (3, 7), (4, 7)] does that mean the temperament itself is a
mismatch? Incidentally, the bad vectors I got for this, 17496:16807
and 9765625:9565938, pass the square test. I'm having difficulty seeing
any relevance to that.

Graham

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ttaj2+539k@e...>
> In article <9ttaj2+539k@e...>, paul@s... (Paul Erlich)
> wrote:
>
> > --- In tuning-math@y..., graham@m... wrote:
> >
> > > > What example was that? I'm guessing that in that example, a
power
> > of
> > > > p/q is shorter than r/s.
> > >
> > > [-1, 0, -2, 2]
> >
> > 49/50
> >
> > > [1, 7, -4, -1].
> >
> > 4374/4375
> >
> > Sure enough, (49/50)^2 = 2401/2500 is simpler than 4374/4375.
Hence
> > we have a mismatch of unison vectors. I think torsion rules the
realm
> > outside the "strong Minkowski" condition I proposed.
>
> Ah, but how about 64:63 and 3125:3087?
>
> [0, -2, 5, -3]
> [6, -2, 0, -1]
>
> Lots of common factors, but (64:63)^2 = 4096:3969.

So this is a good one. Why do you bring this example up? Isn't it
just a normal example?

> It also sounds like a cop-out to say the UVs are "mismatched".
49/50
> and 4374/4375 remain the simplest way of producing the mapping [(2,
0),
> (2, 5), (3, 7), (4, 7)] does that mean the temperament itself is a
> mismatch?

That's what I'm suggesting we may wish to say. Look at it --
complexity of 14, worst error of 17.5 cents. The 4374/4375 is
powerless to do any good in the presence of 49/50.

🔗graham@microtonal.co.uk

In-Reply-To: <9ttf31+gk9@eGroups.com>
Me:
> > Ah, but how about 64:63 and 3125:3087?
> >
> > [0, -2, 5, -3]
> > [6, -2, 0, -1]
> >
> > Lots of common factors, but (64:63)^2 = 4096:3969.

Paul:
> So this is a good one. Why do you bring this example up? Isn't it
> just a normal example?

I don't know, what is "normal"? You can't combine these vectors to get a
1 in any column but the last. That has something to do with torsion, but
I don't think it's the real problem.

> > It also sounds like a cop-out to say the UVs are "mismatched".
> 49/50
> > and 4374/4375 remain the simplest way of producing the mapping [(2,
> 0),
> > (2, 5), (3, 7), (4, 7)] does that mean the temperament itself is a
> > mismatch?
>
> That's what I'm suggesting we may wish to say. Look at it --
> complexity of 14, worst error of 17.5 cents. The 4374/4375 is
> powerless to do any good in the presence of 49/50.

50:49 and 245:243 give a complexity of 10 and a worst error of 17.5 cents.
But there's no problem with them. Exactly how complex do you have to get
to be a mismatch? The rule sounds arbitrary. I thought it was supposed
to show that the vectors hadn't been reduced properly, but it doesn't work
in all cases.

Another example:

63:64 and 1024:1029 give 256:243 and 16807:15552 from my program.

That is

[-6, 2, 0, 1]
[10, -1, 0, -3]

comes out as

[8, -5, 0, 0]
[-6, -5, 0, 5]

when we should have got

[-6, 2, 0, 1]
[ 4, 1, 0, -2]

49:48 and 63:64. But (256:243)^2=65536:59049 which is more complex than
16807:15552. So the test tells us nothing.

Another example, 125:126 and 245:243 give 78732:78125
1647086:1594323. But 1647086:1594323 is simpler than
6198727824:6103515625.

And again, 125:126 and 4374:4375 give 78732:78125 and
1647086:1594323.

225:224 and 245:243 give 3125:3072 and 537824:531441

225:224 and 1715:1728 give 839808:823543 and
2109375:2097152

245:243 and 4374:4375 give and 78732:78125
1647086:1594323

4374:4375 and 1715:1728 give 390625000:387420489 and
2038431744:1977326743

4374:4375 and 3125:3136 give 1224440064:1220703125 and
50797745488265216:50031545098999707

1715:1728 and 3125:3136 give 782757789696:762939453125 and
240734712102912:232630513987207

In all these cases, the bad vectors pass the square test. It looks like
the special thing about the 2048:2025 and 50:49 example is that one of the
vectors is correct. So does that mean Gene already has a way of handling
bad pairs of vectors? As I don't know what algorithm he uses, I'm
assuming it happens to get bad cases less often, but doesn't avoid them
completely.

Graham

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ttf31+gk9@e...>
> Me:
> > > Ah, but how about 64:63 and 3125:3087?
> > >
> > > [0, -2, 5, -3]
> > > [6, -2, 0, -1]
> > >
> > > Lots of common factors, but (64:63)^2 = 4096:3969.
>
> Paul:
> > So this is a good one. Why do you bring this example up? Isn't it
> > just a normal example?
>
> I don't know, what is "normal"?

Regular, ordinary.

> You can't combine these vectors to get a
> 1 in any column but the last.

So? Why would you want to?

> That has something to do with torsion, but
> I don't think it's the real problem.

So what exactly _is_ the problem with this system? Is there one?

> > > It also sounds like a cop-out to say the UVs are "mismatched".
> > 49/50
> > > and 4374/4375 remain the simplest way of producing the mapping
[(2,
> > 0),
> > > (2, 5), (3, 7), (4, 7)] does that mean the temperament itself
is a
> > > mismatch?
> >
> > That's what I'm suggesting we may wish to say. Look at it --
> > complexity of 14, worst error of 17.5 cents. The 4374/4375 is
> > powerless to do any good in the presence of 49/50.
>
> 50:49 and 245:243 give a complexity of 10 and a worst error of 17.5
cents.
> But there's no problem with them. Exactly how complex do you have
to get
> to be a mismatch? The rule sounds arbitrary.

The idea is that if you were willing to temper 50:49, surely you'd
also be willing to temper some interval in the lattice that you found
sooner than 4374:4375.

> I thought it was supposed
> to show that the vectors hadn't been reduced properly,

_If_ the rule is satisfied, _then_ the vectors are reduced properly.
The converse doesn't hold, but the contrapositive does.
>
> 63:64 and 1024:1029 give 256:243 and 16807:15552 from my program.
>
> That is
>
> [-6, 2, 0, 1]
> [10, -1, 0, -3]
>
> comes out as
>
> [8, -5, 0, 0]
> [-6, -5, 0, 5]
>
> when we should have got
>
> [-6, 2, 0, 1]
> [ 4, 1, 0, -2]
>
> 49:48 and 63:64.

I don't know what your program is doing, or why. I wasn't talking
about your program anyway.

> But (256:243)^2=65536:59049 which is more complex than
> 16807:15552. So the test tells us nothing.

Graham, graham -- this is not a correct application of the test!
256:243 and 65536:59049 is _not_ a Minkowski-reduced basis for this
system, so 256:243 and 65536:59049 don't pass "weak Minkowski", let
alone "strong Minkowski"!! 49:48 and 63:64 is the Minkowski-reduced
basis. _Those_ are the vectors that are supposed to satisfy
the "strong Minkowski" condition.

> Another example, 125:126 and 245:243 give 78732:78125
> 1647086:1594323.

Forget about that. Your program is doing something weird. 125:126 and
245:243 is already Minkoski-reduced!

🔗graham@microtonal.co.uk

In-Reply-To: <9ttk1g+e71q@eGroups.com>
Paul wrote:

> The idea is that if you were willing to temper 50:49, surely you'd
> also be willing to temper some interval in the lattice that you found
> sooner than 4374:4375.

Well, that's up to you. I don't see what we gain by enforcing the square
rule.

> > I thought it was supposed
> > to show that the vectors hadn't been reduced properly,
>
> _If_ the rule is satisfied, _then_ the vectors are reduced properly.
> The converse doesn't hold, but the contrapositive does.

What's a contrapositive? I've given an example that's fine when the rule
isn't satisfied, and lots that are fine even though it isn't.

> > But (256:243)^2=65536:59049 which is more complex than
> > 16807:15552. So the test tells us nothing.
>
> Graham, graham -- this is not a correct application of the test!
> 256:243 and 65536:59049 is _not_ a Minkowski-reduced basis for this
> system, so 256:243 and 65536:59049 don't pass "weak Minkowski", let
> alone "strong Minkowski"!! 49:48 and 63:64 is the Minkowski-reduced
> basis. _Those_ are the vectors that are supposed to satisfy
> the "strong Minkowski" condition.

Why aren't they Minkowski reduced? Okay, my algorithm isn't perfect, but
even if it didn't do the reduction perfectly, it'll never get to 49:48 and
63:64 using

{(p/q)^i (r/s)^j}

assuming i and j are integers. It's the cases where they aren't we're
having trouble with. And if they're allowed, the test becomes redundant
anyway. Gene did say in the original case "Neither LLL nor Minkowski got
rid of this problem" so I assumed they did have to be integers. If i and
j are allowed to be irrational, you can get anything you like, so shall we
say they have to be rational?

> > Another example, 125:126 and 245:243 give 78732:78125
> > 1647086:1594323.
>
> Forget about that. Your program is doing something weird. 125:126 and
> 245:243 is already Minkoski-reduced!

It doesn't matter where they came from. They aren't properly reduced but
they pass your square test.

Oh, while I'm posting, the 64:63 and 3125:3087 system is interesting
because 3125:3136 is one of the original vectors. So there's a
superparticular ratio that isn't one of the simplest pair.

Graham

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ttk1g+e71q@e...>
> Paul wrote:
>
> > The idea is that if you were willing to temper 50:49, surely
you'd
> > also be willing to temper some interval in the lattice that you
found
> > sooner than 4374:4375.
>
> Well, that's up to you. I don't see what we gain by enforcing the
square
> rule.

It's not a square rule. But what we gain is simplicity.

> > > I thought it was supposed
> > > to show that the vectors hadn't been reduced properly,
> >
> > _If_ the rule is satisfied, _then_ the vectors are reduced
properly.
> > The converse doesn't hold, but the contrapositive does.
>
> What's a contrapositive? I've given an example that's fine when
the rule
> isn't satisfied, and lots that are fine even though it isn't.

Right -- but I'm claiming you won't find an example that's _not_ fine
when the rule is satisfied.

> > > But (256:243)^2=65536:59049 which is more complex than
> > > 16807:15552. So the test tells us nothing.
> >
> > Graham, graham -- this is not a correct application of the test!
> > 256:243 and 65536:59049 is _not_ a Minkowski-reduced basis for
this
> > system, so 256:243 and 65536:59049 don't pass "weak Minkowski",
let
> > alone "strong Minkowski"!! 49:48 and 63:64 is the Minkowski-
reduced
> > basis. _Those_ are the vectors that are supposed to satisfy
> > the "strong Minkowski" condition.
>
> Why aren't they Minkowski reduced? Okay, my algorithm isn't
perfect, but
> even if it didn't do the reduction perfectly, it'll never get to
49:48 and
> 63:64 using
>
> {(p/q)^i (r/s)^j}
>
> assuming i and j are integers.

Yes, they are integers -- I thought you said 49:48 and 63:64 was
equivalent to this, and I took "your" word for it. So what _do_ you
get when you Minkowski-reduce this system? Gene?
>
> > > Another example, 125:126 and 245:243 give 78732:78125
> > > 1647086:1594323.
> >
> > Forget about that. Your program is doing something weird. 125:126
and
> > 245:243 is already Minkoski-reduced!
>
> It doesn't matter where they came from. They aren't properly
reduced but
> they pass your square test.

No, no, no! Again, my test _isn't_ a square test! It's a "strong
Minkowski" condition, which includes Gene's Minkowski condition
within it!

My test, in case this wasn't clear, says that the shorter member of
the reduced basis is the _only_ member of the kernel that is shorter
than the longer member of the reduced basis. Is that clear? It's
_not_ a square test -- though in some cases, the test may fail
_because_ the square of the shorter vector is shorter than the longer
vector. But that's not the only condition under which the test can
fail!

> Oh, while I'm posting, the 64:63 and 3125:3087 system is
interesting
> because 3125:3136 is one of the original vectors. So there's a
> superparticular ratio that isn't one of the simplest pair.

3125:3136 is not superparticular!
>
>
> Graham

🔗graham@microtonal.co.uk

In-Reply-To: <9ttq3j+5tck@eGroups.com>
Me:
> > Well, that's up to you. I don't see what we gain by enforcing the
> square
> > rule.

Paul:
> It's not a square rule. But what we gain is simplicity.

How? And why is this worth losing perfectly valid results?

Me:
> > What's a contrapositive? I've given an example that's fine when
> the rule
> > isn't satisfied, and lots that are fine even though it isn't.

Paul:
> Right -- but I'm claiming you won't find an example that's _not_ fine
> when the rule is satisfied.

Sorry, typo.

Me:
> > {(p/q)^i (r/s)^j}
> >
> > assuming i and j are integers.

Paul:
> Yes, they are integers -- I thought you said 49:48 and 63:64 was
> equivalent to this, and I took "your" word for it. So what _do_ you
> get when you Minkowski-reduce this system? Gene?

Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced then.
63:64 and 49:48 describe the same temperament.

Incidentally, you did say 256:243 and 65536:59049 before, but my comments
were for 256:243 and 16807:15552, although I didn't point that out.
65536:59049 is (256:243)^2. So 256:243 and 16807:15552 aren't Minkowski
reduced, but there's no particular reason why they should be.

The vectors are

[8, -5, 0, 0]
[-6, -5, 0, 5]

How could you combine them to get anything simpler? My program's already
checking the simplest cases. [14, 0, 0, -5] being the obvious one.

>>> 16384*16807
275365888
>>> 16807*15552
261382464

I'm certain these are Minkowski reduced, if I've got the definition right.

Me:
> > It doesn't matter where they came from. They aren't properly
> reduced but
> > they pass your square test.

Paul:
> No, no, no! Again, my test _isn't_ a square test! It's a "strong
> Minkowski" condition, which includes Gene's Minkowski condition
> within it!

Well, whatever you call it, they pass.

> My test, in case this wasn't clear, says that the shorter member of
> the reduced basis is the _only_ member of the kernel that is shorter
> than the longer member of the reduced basis. Is that clear? It's
> _not_ a square test -- though in some cases, the test may fail
> _because_ the square of the shorter vector is shorter than the longer
> vector. But that's not the only condition under which the test can
> fail!

But without that condition, it becomes Gene's Minkowski, which these pairs
pass. So how about Graham's Strong Minkowski test, where we start with
Gene's definition, but allow i and j to be rationals? Keeping p/q, r/s
and t/u as rationals.

You could also replace (p/q) with (p/q)^i to make your test pass, which
definitely isn't right.

> > Oh, while I'm posting, the 64:63 and 3125:3087 system is
> interesting
> > because 3125:3136 is one of the original vectors. So there's a
> > superparticular ratio that isn't one of the simplest pair.
>
> 3125:3136 is not superparticular!

oops

Graham

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ttq3j+5tck@e...>
> Me:
> > > Well, that's up to you. I don't see what we gain by enforcing
the
> > square
> > > rule.
>
> Paul:
> > It's not a square rule. But what we gain is simplicity.
>
> How?

Because instead of the rule for Minkowski reduction, we use an even
simpler rule.

> And why is this worth losing perfectly valid results?

You can also get additional valid results by increasing the number of
unison vectors in our list. But shouldn't there be
some "reasonableness" criteria applied? Otherwise, you'll get an
infinite list of linear temperaments!

>
> Me:
> > > What's a contrapositive? I've given an example that's fine
when
> > the rule
> > > isn't satisfied, and lots that are fine even though it isn't.
>
> Paul:
> > Right -- but I'm claiming you won't find an example that's _not_
fine
> > when the rule is satisfied.
>
> Sorry, typo.
>
> Me:
> > > {(p/q)^i (r/s)^j}
> > >
> > > assuming i and j are integers.
>
> Paul:
> > Yes, they are integers -- I thought you said 49:48 and 63:64 was
> > equivalent to this, and I took "your" word for it. So what _do_
you
> > get when you Minkowski-reduce this system? Gene?
>
> Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced
then.

Let's call this statement G. See below.

> 63:64 and 49:48 describe the same temperament.

Hmm . . . so this is one of the torsion-spawning cases? Can you show
how this works?

> Incidentally, you did say 256:243 and 65536:59049 before, but my
comments
> were for 256:243 and 16807:15552, although I didn't point that out.

I know, my typo.

> 65536:59049 is (256:243)^2.

OK, forget my typo, please.

> So 256:243 and 16807:15552 aren't Minkowski
> reduced,

Now you're saying not G. You've claimed both G and not G, but you
haven't given evidence for either.

> but there's no particular reason why they should be.

>
> The vectors are
>
> [8, -5, 0, 0]
> [-6, -5, 0, 5]
>
> How could you combine them to get anything simpler? My program's
already
> checking the simplest cases. [14, 0, 0, -5] being the obvious one.
>
> >>> 16384*16807
> 275365888
> >>> 16807*15552
> 261382464
>
> I'm certain these are Minkowski reduced, if I've got the definition
right.

OK, so now you're saying G again, and you've given a reason why.

🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

In-Reply-To: <9tttj7+v8k6@eGroups.com>
Paul wrote:

> > Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced
> then.
> > 63:64 and 49:48 describe the same temperament.
>
> Hmm . . . so this is one of the torsion-spawning cases? Can you show
> how this works?

How I generated <http://x31eq.com/survey.out>:

I went through each pair of unison vectors in your list. (Some list, Gene
seems to have a longer one.) I worked out the linear temperament
consistent with each pair of unison vectors, and optimized it. Then I
used the method that Temperament objects already have to return the
simplest unison vectors I could find.

That's what <http://x31eq.com/makeSurvey.py> does.

Unfortunately, that method sometimes throws up unisons that are way too
complex, and it's related to torsion. So I went through and listed them.

256:243 and 16807:15552 are [8, -5, 0, 0] and [-6, -5, 0, 5] in vector
form. 63:64 and 49:48 become [-6, 2, 0, 1] and [-4, -1, 0, 2].

Both pairs give the same mapping by period and generator: [(5, 0), (8, 0),
(10, 1), (14, 0)]. We can verify this

[ 8 -5 0 0][ 5 0] [0 0]
[-6 -5 0 5][ 8 0] = [0 0]
[-6 2 0 1][10 1] [0 0]
[-4 -1 0 2][14 0] [0 0]

We can also transform the first pair into the second pair

([-6, -5, 0, 5]-3*[8, -5, 0, 0])/5 = [-6 2 0 1]

This would be i=-3/5, j=1/5

(2*[-6, -5, 0, 5]-[8, -5, 0, 0])/5 = [-4 -1 0 2]

Which is i=-1/5, j=2/5.

However, the Minkowski criterion as we currently understand it, and my
reduction method, don't allow for fractional values of i and j. The
practical difficulty in implementing this is that the vectors have to get
bigger before they can be simplified by dividing through by a common
factor. That makes the search harder.

A brute force approach would work, trying every possible vector and seeing
if it approximates to a unison. I'll look at this if I find the time.
Another idea would be to list all members of the second-order odd limit
that approximate to be the same. Any more complex unison vectors probably
won't be that interesting anyway.

Graham

🔗genewardsmith@juno.com

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > Okay, we'll assume 256:243 and 16807:15552 are Minkowski reduced
> then.

> Let's call this statement G. See below.

It's true.

> > 63:64 and 49:48 describe the same temperament.
>
> Hmm . . . so this is one of the torsion-spawning cases?

Indeed it is. I just wrote a wedge product routine, and here is what
I get:

16807/15552^256/243 = [70, 0, -40, 0, 25, 0]
64^63^49/48 = [-14, 0, 8, 0, -5,0]

The first has coefficients with a gcd of 5, telling us there is
5-torsion; the second has a gcd of 1, telling us there is no torsion.

🔗genewardsmith@juno.com

--- In tuning-math@y..., graham@m... wrote:

> > > Ah, but how about 64:63 and 3125:3087?
> > >
> > > [0, -2, 5, -3]
> > > [6, -2, 0, -1]
> > >
> > > Lots of common factors, but (64:63)^2 = 4096:3969.

> Paul:
> > So this is a good one. Why do you bring this example up? Isn't it
> > just a normal example?
>
> I don't know, what is "normal"? You can't combine these vectors to
get a
> 1 in any column but the last. That has something to do with
torsion, but
> I don't think it's the real problem.

I get 64/63^3125/3087 = [-12, 30, -18, -10, 4, 5], and so no torsion.
I'm with Paul--this seems quite normal to me.