Tuning-Math Digests messages 2025 - 2049

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Message: 2025

Date: Tue, 20 Nov 2001 21:03:16

Subject: Re: LLL reduction pairs revised list

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> the LLL reductions 
> tend to give an interval of equivalence and a generator, which is 
> perfect.

What do you mean, exactly? What do the LLL reductions give that an 
unreduced basis for a linear temperament don't, in the way of an 
interval of equivalence and a generator?
> 
> > At least in the approach I envision in this paper. Another 
> > paper could more specifically address infinite temperaments.
> 
> The scales will presumably be in one of three things: a 
temperament, 
> an et, or a p-limit.

An ET is a temperament -- but, as I said, some might be in a linear 
temperament, some might be in a planar temperament, etc.

> It seems to me you can't get away from 
> addressing one or more of these if you are going to work with 
scales.

Right -- but that doesn't mean that I have to talk about ennealimmal 
temperament if it doesn't give me a scale with a reasonable number of 
notes (as I said, I have to delimit this project somewhere!)


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Message: 2026

Date: Tue, 20 Nov 2001 00:05:48

Subject: Re: "most compact" periodicity block

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Is it clear what I mean when I say "most compact" periodicity 
block 
> > for a particular equivalence class of bases? Won't it, in 
general, 
> be 
> > delimited by a hexagon in the 5-limit, and a rhombic dodecahedron 
> in 
> > the 7-limit? 
> 
> It seems to me that depends on how you define your distance.

You know how I define distance.

> Incidentally, I suspect that instead of using hexagonal blocks, or 
> rrhombic dodecahedra, ellipsoids would work.

No, because then you'd have more than one note within some 
equivalence class, or no notes within some equivalence class.

> I don't think you need a 
> tiling.

You automatically get a tiling if you choose one and only one note 
from each equivalence class!

> Could we use the resulting three or six unison vectors 
> > as a more complete characterization of a given system, rather 
than 
> an 
> > LLL or some such reduced basis, which has some element of 
> > arbitrariness because some "second best" reduction might be 
nearly 
> as 
> > good?
> 
> What's the point? I thought you wanted to produce temperaments, not 
> PBs.

Right, but I think any rule characterizing which unison vectors we 
should use, including relationships they may have with one another, 
should be evaluated with respect to some sort of non-arbitrary 
reduced basis, don't you think?


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Message: 2027

Date: Tue, 20 Nov 2001 21:08:46

Subject: Re: reduced basis for my decatonic

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Gene, what do you get as a reduced basis for my decatonic scale? 
> 
> I'm afraid I get <25/24, 28/27, 49/48>, which seems reasonable to 
me, 
> if not to you. Why did you expect 64/63 or even 225/224, given that 
> they clearly have a higher Tenney height?

Sorry, Gene, I was thinking in terms of commatic vs. chromatic again, 
and you weren't. Of course, the reduced basis stuff we've been 
talking about has no provision for such distinctions. For me, my 
decatonic scale is MOS or altered-MOS by nature, so should be 
associated with two commatic UVs and only one chromatic one. The 
three you gave are all chromatic -- meaning none of them are tempered 
out in my decatonic scale.

> > How about that reduced-basis-for-good-ETs request?
> 
> I'm afraid I've forgotten you made one. Would the idea be to do for 
> other ets what I just did for 10? Do you only want them in the 7-
> limit?

That would be great.


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Message: 2028

Date: Tue, 20 Nov 2001 00:09:28

Subject: Re: LLL reduction pairs revised list

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Actually, we should be taking three at a time, not two. If all 
> three 
> > (in the Minkowski reduced basis, hopefully) are commatic UVs, we 
> get 
> > an ET. If one is chromatic, we get an MOS. If two are chromatic, 
a 
> > planar temperament. If three are chromatic, a JI block.
> 
> This will prevent you from considering a lot of interesting 
> temperaments.

Such as?

> > I think the UVs in the reduced basis have to be in Kees's list or 
> > something like it -- if they aren't, then it seems likely that 
some 
> > step in the scale will be smaller than one of the commatic unison 
> > vectors -- which we shouldn't allow.
> 
> Where is Kees's list? I was thinking of producing a list based on 
> some mathematical conditions, and using that, but I wondered if 
these 
> lists already represent such an effort.

Start with Searching Small Intervals * and follow the link 
to S2357 *. What Kees doesn't tell you 
(he told me) is that the unison vectors not in parentheses are the 
smallest (in octaves or cents) for their level of expressibility 
(i.e., for their length). The ones in parentheses are included so 
that you're guaranteed to have the three smallest for any level of 
expressibility. Perhaps Kees would like to chime in here?


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Message: 2029

Date: Tue, 20 Nov 2001 21:14:12

Subject: Re: LLL reduced 7-limit kernels of some good ets

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> Here's what I got:
> 
> 34: [126/125, 49/48, 6272/6075]

That depends on how you map the 7 in 34.

> 53: [225/224, 1728/1715, 4000/3969]

This is very heartening -- over on the tuning list, I wrote:

"53-tone 7-limit Fokker periodicity block, which approximates the 
full 53-tET rather well, without redundancy, and with the smallest 
ratios of any of the 62 versions I tried . . . the unison vectors 
defining this FPB are 225:224, 1728:1715, and 4000:3969"

> This calculation helps to answer the question of which commas we 
> should list--clearly, 4000/3969 and 2048/2025 are important commas 
> and belong there.

Why is this so clear. Maybe the ETs with 4000/3969 don't satisfy the 
criteria I wish to employ to delimit the project.


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Message: 2030

Date: Tue, 20 Nov 2001 00:40:28

Subject: Re: "most compact" periodicity block

From: genewardsmith@xxxx.xxx

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

> You know how I define distance.

It doesn't work when you want to define blocks, though.

> > Incidentally, I suspect that instead of using hexagonal blocks, 
or 
> > rrhombic dodecahedra, ellipsoids would work.
> 
> No, because then you'd have more than one note within some 
> equivalence class, or no notes within some equivalence class.

I was thinking of using the minimal diameter definition.

> > I don't think you need a 
> > tiling.
> 
> You automatically get a tiling if you choose one and only one note 
> from each equivalence class!

You bet, which is exaclty why you don't need to require that the 
convex figures which result from taking everything less than or equal 
to a certain distance produce a tiling.
> 
> > Could we use the resulting three or six unison vectors 
> > > as a more complete characterization of a given system, rather 
> than 
> > an 
> > > LLL or some such reduced basis, which has some element of 
> > > arbitrariness because some "second best" reduction might be 
> nearly 
> > as 
> > > good?
> > 
> > What's the point? I thought you wanted to produce temperaments, 
not 
> > PBs.
> 
> Right, but I think any rule characterizing which unison vectors we 
> should use, including relationships they may have with one another, 
> should be evaluated with respect to some sort of non-arbitrary 
> reduced basis, don't you think?


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Message: 2031

Date: Tue, 20 Nov 2001 22:32:15

Subject: Re: LLL reduction pairs revised list

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> My last list allowed in some duplicates; the following list of 81 
> pairs is from an improved version of the program:

I'm going to keep the ones where both UVs appear on my original list, 
until we have a firmer basis for making such a list. Commas like 
4000/3969 seem like unlikely choices, since there are three or more 
commas that are both shorter vectors _and_ smaller musical intervals. 
Meanwhile, the superparticulars with smaller numbers than 50:49 lead 
to too much tempering for my taste.

> {81/80, 126/125}
> {64/63, 245/243}
> {50/49, 81/80}
> {64/63, 126/125}
> {1728/1715, 3136/3125}
> {81/80, 3136/3125}
> {1029/1024, 4375/4374}
> {3136/3125, 4375/4374}
> {225/224, 1029/1024}
> {50/49, 64/63}
> {126/125, 245/243}
> {50/49, 245/243}
> {50/49, 6144/6125}
> {245/243, 225/224}
> {50/49, 4375/4374}
> {126/125, 1029/1024}
> {225/224, 4375/4374}
> {81/80, 6144/6125}
> {81/80, 1728/1715}
> {64/63, 4375/4374}
> {49/48, 225/224}
> {81/80, 225/224}
> {245/243, 1029/1024}
> {245/243, 3136/3125}
> {1029/1024, 3136/3125}
> {2401/2400, 3136/3125}
> {81/80, 2401/2400}
> {2401/2400, 6144/6125}
> {126/125, 1728/1715}
> {225/224, 1728/1715}
> {64/63, 225/224}
> {64/63, 3136/3125}
> {4375/4374, 6144/6125}
> {2401/2400, 4375/4374}

Still a whopping 34 possibilities! Who can be quickest to the draw to 
give the generator, period, and 7-limit complexity for all 34 linear 
temperaments? If Gene did this right, they should all be 
distinct . . . and this very well might be all the "interesting" ones 
for my present purposes.


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Message: 2032

Date: Tue, 20 Nov 2001 00:45:44

Subject: Re: "most compact" periodicity block

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > You know how I define distance.
> 
> It doesn't work when you want to define blocks, though.

What do you mean?
> 
> > > Incidentally, I suspect that instead of using hexagonal blocks, 
> or 
> > > rrhombic dodecahedra, ellipsoids would work.
> > 
> > No, because then you'd have more than one note within some 
> > equivalence class, or no notes within some equivalence class.
> 
> I was thinking of using the minimal diameter definition.

I could tell.
> 
> > > I don't think you need a 
> > > tiling.
> > 
> > You automatically get a tiling if you choose one and only one 
note 
> > from each equivalence class!
> 
> You bet, which is exaclty why you don't need to require that the 
> convex figures which result from taking everything less than or 
equal 
> to a certain distance produce a tiling.

??? I don't want to simply produce convex figures which result from 
taking everything less than or equal to certain distance! I want to 
define equivalence relations (i.e. a kernel), and _then_ use a block 
(possibly not unique up to reflections and such small changes) which, 
given that there's one and only one note from each equivalence class, 
is as compact as possible. Then, there should generally be three or 
six operative unison vectors, right?

> > 
> > > Could we use the resulting three or six unison vectors 
> > > > as a more complete characterization of a given system, rather 
> > than 
> > > an 
> > > > LLL or some such reduced basis, which has some element of 
> > > > arbitrariness because some "second best" reduction might be 
> > nearly 
> > > as 
> > > > good?
> > > 
> > > What's the point? I thought you wanted to produce temperaments, 
> not 
> > > PBs.
> > 
> > Right, but I think any rule characterizing which unison vectors 
we 
> > should use, including relationships they may have with one 
another, 
> > should be evaluated with respect to some sort of non-arbitrary 
> > reduced basis, don't you think?

You didn't answer.


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Message: 2033

Date: Tue, 20 Nov 2001 22:34:12

Subject: Re: LLL reduction pairs revised list

From: Paul Erlich

I wrote,

> If Gene did this right, they should all be 
> distinct . . . 

Well, the least-squares generators should be distinct . . . some of 
the minimax generators might turn out to be the same (?)


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Message: 2034

Date: Tue, 20 Nov 2001 01:36:04

Subject: Re: LLL reduction pairs revised list

From: genewardsmith@xxxx.xxx

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

> > This will prevent you from considering a lot of interesting 
> > temperaments.

> Such as?

If you take 2401/2400 and 4375/4374 by themselves, you get 
ennealimmal temperament. If you try to add 25/24, 28/27, 36/35 you 
get garbage. Besides, there's no point in it that I can see--why add 
anything when two commas are all you need to define a 7-limit linear 
temperament?


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Message: 2035

Date: Tue, 20 Nov 2001 01:46:12

Subject: Re: LLL reduction pairs revised list

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., genewardsmith@j... wrote:
> 
> > > This will prevent you from considering a lot of interesting 
> > > temperaments.
> 
> > Such as?
> 
> If you take 2401/2400 and 4375/4374 by themselves, you get 
> ennealimmal temperament.

I know you love that one.

> If you try to add 25/24, 28/27, 36/35 you 
> get garbage.

These are not the only canditates. You left out 49/48. Also, 50/49 
and 64/63 can be used as _either_ commatic _or_ chromatic -- I don't 
know if I mentioned that but I think so. Other candidates may emerge 
once we have a solid foundation for all this.

Anyway, I'd just like to set some reasonable bounds within which we 
can flesh out the possibilities. If someone wants to use 36/35 as a 
commatic unison vector, I'm all for that, but I'd like to start with 
a digestible array of possibilities just for the sake of presentation.

> Besides, there's no point in it that I can see--why add 
> anything when two commas are all you need to define a 7-limit 
linear 
> temperament?

The point is not so much infinite temperaments but rather finite 
scales. At least in the approach I envision in this paper. Another 
paper could more specifically address infinite temperaments.


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Message: 2036

Date: Tue, 20 Nov 2001 03:27:23

Subject: reduced basis for my decatonic

From: Paul Erlich

Gene, what do you get as a reduced basis for my decatonic scale? 
Looking at the most compact lattice arrangement, it appears that 
(64/63, 50/49, 49/48) should win, though 225/224 could replace 63/64 
without too much damage, and 28/27 and 25/24, while both weak 
replacements for 49/48, are almost equally good in that role.

How about that reduced-basis-for-good-ETs request?


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Message: 2037

Date: Tue, 20 Nov 2001 04:37:11

Subject: Re: LLL reduction pairs revised list

From: genewardsmith@xxxx.xxx

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

> The point is not so much infinite temperaments but rather finite 
> scales.

Given two generators, you are set to make scales; the LLL reductions 
tend to give an interval of equivalence and a generator, which is 
perfect. What I like about starting from two commas is your idea to 
use this as a canonical scheme for classification, but certainly one 
can go on to scales.

 At least in the approach I envision in this paper. Another 
> paper could more specifically address infinite temperaments.

The scales will presumably be in one of three things: a temperament, 
an et, or a p-limit. It seems to me you can't get away from 
addressing one or more of these if you are going to work with scales. 
Scales are also less easy to classify than temperaments, because 
there are more reasonable possibilities. Moreover, if you are willing 
to restrict yourself to et scales, my a;n+m notation already does 
classify them.


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Message: 2038

Date: Tue, 20 Nov 2001 05:10:08

Subject: Re: reduced basis for my decatonic

From: genewardsmith@xxxx.xxx

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

> Gene, what do you get as a reduced basis for my decatonic scale? 

I'm afraid I get <25/24, 28/27, 49/48>, which seems reasonable to me, 
if not to you. Why did you expect 64/63 or even 225/224, given that 
they clearly have a higher Tenney height?

> How about that reduced-basis-for-good-ETs request?

I'm afraid I've forgotten you made one. Would the idea be to do for 
other ets what I just did for 10? Do you only want them in the 7-
limit?


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Message: 2039

Date: Tue, 20 Nov 2001 08:19:16

Subject: LLL reduced 7-limit kernels of some good ets

From: genewardsmith@xxxx.xxx

Here's what I got:

10: [49/48, 28/27, 25/24]
12: [64/63, 50/49, 36/35]
15: [126/125, 49/48, 28/27]
19: [126/125, 81/80, 49/48]
22: [225/224, 245/243, 64/63]
27: [126/125, 245/243, 64/63]
31: [225/224, 1728/1715, 81/80]
34: [126/125, 49/48, 6272/6075]
41: [225/224, 4000/3969, 245/243]
46: [1029/1024, 126/125, 245/243]
53: [225/224, 1728/1715, 4000/3969]
58: [1728/1715, 126/125, 2048/2025]
68: [4000/3969, 245/243, 2048/2025]
72: [4375/4374, 225/224, 1029/1024]
99: [4375/4374, 6144/6125, 3136/3125]
171: [4375/4374, 2401/2400, 32805/32768]

This calculation helps to answer the question of which commas we 
should list--clearly, 4000/3969 and 2048/2025 are important commas 
and belong there.


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Message: 2040

Date: Tue, 20 Nov 2001 08:48:32

Subject: Re: LLL reduced 7-limit kernels of some good ets

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., genewardsmith@j... wrote:

Some of these temperaments derive from a comma shared by two reduced 
bases:

10&15: [49/48, 28/27]
15&34: [49/48, 126/125]
22&27: [64/63, 245/243]
41&53: [225/224, 4000/3969]
41&68: [245/243, 4000/3969]

This might be one place to start.


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Message: 2041

Date: Tue, 20 Nov 2001 12:45 +0

Subject: Re: LLL definitions

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9t7fj7+odag@xxxxxxx.xxx>
Gene wrote:

> Starting from an ordered lattice basis [b_1, b_2, ..., b_m] we can 
> obtain another basis [g_1, g_2, ..., g_m] by the Gram-Schmidt 
> orthogonalization process. This will span the same linear subspace, 
> but *not*, normally, the same lattice. We also obtain Gram-Schmidt 
> coefficients:

What's an ordered basis?

> c[i,j] = <b_i, g_j>/<g_j, g_j>

I've implemented this in Python code:

import operator

def dotprod(x,y):
    return reduce(operator.add,
        map(operator.mul, x, y))

def makec(b,g):
    c = []
    for i in range(len(g)):
        row = []
        for j in range(len(g)):
            row.append(float(dotprod(b[i],g[j]))
                / dotprod(g[j], g[j]))
        c.append(row)
    return c

Please say if I'm going wrong anywhere.  One thing that worries me is that 
I'm getting floating point results where you stay with integers.


> We do this by the standard Gram-Schmidt recursion from linear algebra:
> 
> g_1 = b_1
> g_i = b_i - sum_{j = 1 to i - 1} c[i,j] g_j

Again, I've implemented that

def LLL(b, g):
    c = makec(b,g)
    newg = []
    for i in range(len(g)):
        row = []
        for k in range(len(g)):
            thing = b[i][k]
            for j in range(i):
                thing -= c[i][j] * j[j][k]
            row.append(thing)
        newg.append(row)
    return newg

I'm assuming that repeatedly applying this will give the right result, but 
it clearly doesn't.  For one thing, the first ratio can never change.  
Also, you haven't said what seed value to use for g.  I'm using the 
initial value of b.

If you can supply an alternative algorithm in some kind of procedural 
code, that would be cood too.  I've found LiDIA, and it comes with the 
source code but an imperfect license.  I'll have a look at it sometimt.



                 Graham


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Message: 2042

Date: Wed, 21 Nov 2001 19:42:39

Subject: Re: LLL reduction pairs revised list

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > I'm going to keep the ones where both UVs appear on my original 
> list, 
> > until we have a firmer basis for making such a list. Commas like 
> > 4000/3969 seem like unlikely choices, since there are three or 
more 
> > commas that are both shorter vectors _and_ smaller musical 
> intervals. 
> 
> However, 4000/3969 shows up as part of the reduced basis for 41, 53 
> and 68, which are all important ets.

Maybe not for the conditions I have in mind.

> I conclude therefore it's 
> significant, and in any case this gives us a way of picking them.
> 
> The alternative approach, of course, is to start from pairs of ets; 
I 
> think perhaps we should do it both ways as a check.
> 
> > Meanwhile, the superparticulars with smaller numbers than 50:49 
> lead 
> > to too much tempering for my taste.
> 
> I think drawing the line between 50/49 and 49/48 is a little 
absurd; 
> why not between 49/48 and 36/35?

I wouldn't have a big problem with that -- I'm trying to keep the set 
of outcomes as small as possible, without introducing a larger number 
of conditions than is reasonable.


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Message: 2043

Date: Wed, 21 Nov 2001 19:46:04

Subject: Re: Start of survey

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> I did the first three pairs on my list, and got the following. (All 
> turned out to be Minkowski reduced according to Tenney height.)
> 
> <1728/1715, 2048/2025>
> 
> ets: 14, 22, 58, 80
> 
> LLL reduced map:
> 
> [ 0  2]
> [-3  4]
> [ 6  3]
> [-5  7]
> 
> Generators: a = 0.1376381046 = 11.01104837 / 80; b = 1/2
> 
> Appromimately 58+22 in the 80-et.
> 
> Errors: 
> 
> 3: 2.55
> 5: 4.68
> 7: 5.35
> 
> Extension of map to the 11-limit:
> 
> [ 0  2]
> [-3  4]
> [ 6  3]
> [-5  7]
> [ 7  5]
> 
> <225/224, 49/48>
> 
> ets: 9, 10, 19, 29
> 
> LLL-reduced map:
> 
> [-1  1]
> [-2 -2]
> [-2  5]
> [-3  1]
> 
> Adjusted map:
> 
> [ 0  1]
> [-4  2]
> [ 3  2]
> [-2  3]
> 
> Generator a = 0.1045573299 = 1.986589268 / 19
> 
> This system is closely related to 10+9 in the 19-et, and also 
related 
> to 19+10.
> 
> Errors:
> 
> 3: -3.83
> 5: -9.91
> 7: -19.76
> 
> <245/243, 50/49>
> 
> Map:
> 
> [-2 -2]
> [-1  5]
> [-1  9]
> [-2  8]
> 
> Adjusted map:
> 
> [0 2]
> [3 1]
> [5 1]
> [5 2]
> 
> Generator: 0.3629853525 = 7.985677755 / 22
> 
> Errors:
> 
> 3:  4.79
> 5: -8.40
> 7:  9.09
> 
> This one may as well be taken as the generator 8/22 in the 22-et; 
> this is a supermajor third (9/7), and we have two parallel chains 
> separated by sqrt(2).

Then shouldn't you have said

a = 0.3629853525 = 7.985677755 / 22, b = 1/2

above, similar to what you did for the first example?


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Message: 2044

Date: Wed, 21 Nov 2001 19:50:10

Subject: Re: Start of survey

From: genewardsmith@xxxx.xxx

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

> Then shouldn't you have said
> 
> a = 0.3629853525 = 7.985677755 / 22, b = 1/2
> 
> above, similar to what you did for the first example?

I got lazy, but I suppose I'd better do it systematically. I also 
left out b=1 when that was a generator.


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Message: 2045

Date: Wed, 21 Nov 2001 23:11:00

Subject: Re: LLL definitions

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., graham@m... wrote:
> Oh dear.  I wanted something like this:
> 
> (2, -1, 2, -1, 0)
> (0, -3, 1, 1, 1)
> (-3, 1, -1, -1, 1)
> (-3, 0, 0, 1, -1)

I can certainly LLL reduce this, after adding in the 2s; I get
<196/195, 364/363, 441/440, 1575/1573>

> So it looks like this off the shelf LLL algorithm isn't what I want 
at 
> all.  Do you know of any way of doing that kind of reduction?  Or, 
more 
> specifically, of getting a simple set of unison vectors for the 
consistent 
> 29+58 temperament?

The above does it; however you did most of the work, so I presume you 
must have some idea how to proceed. One can brute force it by first 
getting a 13-limit notation with a basis of about the right size, 
dual to a set of ets containing 29 and 58, and then searching for 
elements of the kernel of 29&58--one should not need exponents beyond 
+-2, so a search would be feasible. If we find something of rank 4 
which can be extended to make a basis for the kernel of both 29 and 
58, we are ready to LLL reduce it. Since you did the above 
calculation, however, perhaps you have another idea.


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Message: 2046

Date: Wed, 21 Nov 2001 00:08:11

Subject: Two versions of 12&34 in the 11-limit

From: genewardsmith@xxxx.xxx

From 12 and 34, we can obtain a unique generator/period for the 
46=12+34 et, which I denote by "34+12", by the following procedure:

(1) Find the penultimate convergent to 12/34, obtaining 1/3

(2) Take the mediant of 1/12 and 3/34, obtaining 4/46 = 2/23

(3) Find the mapping to primes of 46, obtaining [46, 73, 107, 129, 
159]. Note that this does *not* require us to even look at mappings 
for 12 or 34, much less worry about validity!

(4) Taking our period of 23 steps and our generator of 4/46, we 
calculate generator steps:

73/4 = 1 mod 23
107/4 = -2 mod 23
129/4 = -8 mod 23
159/4 = 11 mod 23

(5) Our other generator is 1/2, and we find the corresponding number 
of steps for it:

73/46 - 2/23 = 3/2
107/46 + 2(2/23) = 5/2
129/46 + 8(2/23) = 7/2
159/46 - 11(2/23) = 9/2

(6) Since the mapping to number of generator steps from an interval 
is a val, and we represent vals by column vectors, we can put our 
results together in a 5x2 matrix:

[ 0 2]
[ 1 3]
[-2 5]
[-8 7]
[11 5]

(7) We now have all we need so far as the 46-et goes; however we may 
also detemper using the above map and linear programming or least 
squares to find an optimal tuning. Using least squares in the 11-
limit gives a generator a1 = .08700594368 = 4.002273409 / 46; this is 
not much different from the 46-et and gives similar tuning errors. 
However, the map to primes was only unique mod 23, and we might have 
used instead -12 = 11 mod 23 for the number of steps we mapped 11 to.
We obtain instead the map:

[  0  2]
[  1  3]
[ -2  5]
[ -8  7]
[-12  9]

 Since the other maps to primes have a negative tendency, this seems 
like it is probably the best plan. If we adopt it, we get instead 
a2 = .08648628 = 3.97836888 / 46 as our generator, which is quite a 
bit farther from 46-et than the other system. A comparison of tunings 
shows:

3:   2.39  2.45 1.83
5:   4.99  4.87 6.12
7:  -3.61 -4.08 0.91
11: -3.49 -2.84 3.28


Here the first column is the 46-et, the second our first detempering, 
and the third the alternative--which looks pretty good!


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Message: 2047

Date: Wed, 21 Nov 2001 05:04:25

Subject: Re: LLL reduction pairs revised list

From: genewardsmith@xxxx.xxx

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

> I'm going to keep the ones where both UVs appear on my original 
list, 
> until we have a firmer basis for making such a list. Commas like 
> 4000/3969 seem like unlikely choices, since there are three or more 
> commas that are both shorter vectors _and_ smaller musical 
intervals. 

However, 4000/3969 shows up as part of the reduced basis for 41, 53 
and 68, which are all important ets. I conclude therefore it's 
significant, and in any case this gives us a way of picking them.

The alternative approach, of course, is to start from pairs of ets; I 
think perhaps we should do it both ways as a check.

> Meanwhile, the superparticulars with smaller numbers than 50:49 
lead 
> to too much tempering for my taste.

I think drawing the line between 50/49 and 49/48 is a little absurd; 
why not between 49/48 and 36/35? The schisma showed up in the reduced 
basis for 171; perhaps we should include that, then look at 130 and 
140 and call it a day?


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Message: 2048

Date: Wed, 21 Nov 2001 05:25:03

Subject: Re: LLL reduction pairs revised list

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., genewardsmith@j... wrote:

> I think drawing the line between 50/49 and 49/48 is a little 
absurd; 
> why not between 49/48 and 36/35? The schisma showed up in the 
reduced 
> basis for 171; perhaps we should include that, then look at 130 and 
> 140 and call it a day?

I get

130: <2401/2400, 3136/3125, 19683/19600>
140: <2401/2400, 5120/5103, 15625/15552>


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Message: 2049

Date: Wed, 21 Nov 2001 06:25:17

Subject: Yet another revised list

From: genewardsmith@xxxx.xxx

This started from the commas

49/48, 50/49, 64/63, 81/80, 2048/2025, 245/243, 126/125, 4000/3969, 
1728/1715, 1029/1024, 225/224, 3136/3125, 5120/5103, 6144/6125, 
2401/2400, 4375/4374

I obtained the following 72 reduced pairs; the number following the 
pair is the ratio between the largest and the smallest comma. I think 
some bound needs to be placed on this, as the [4375/4374, 50/49] 
system is obviously a little absurd.

[1728/1715, 2048/2025]   1.495580025
[225/224, 49/48]   4.629021956
[245/243, 50/49]   2.464716366
[5103/5000, 49/48]   1.011210863
[2401/2400, 3136/3125]   8.434916361
[49/48, 28/27]   1.763768279
[5120/5103, 1728/1715]   2.270583084
[64/63, 50/49]   1.282845376
[4000/3969, 245/243]   1.053543673
[3136/3125, 245/243]   2.332723121
[126/125, 49/48]   2.587706812
[3645/3584, 50/49]   1.197064798
[6144/6125, 81/80]   4.010835974
[245/243, 64/63]   1.921288732
[225/224, 1728/1715]   1.695329007
[126/125, 245/243]   1.028688881
[126/125, 81/80]   1.559018011
[2401/2400, 2048/2025]   27.11124152
[4375/4374, 6144/6125]   13.54885592
[1029/1024, 686/675]   3.318654575
[3136/3125, 49/48]   5.868055561
[225/224, 4000/3969]   1.746649139
[81/80, 128/125]   1.909155841
[4375/4374, 225/224]   19.48552996
[1728/1715, 81/80]   1.645020488
[64/63, 686/675]   1.026452250
[1029/1024, 245/243]   1.682792874
[4375/4374, 2401/2400]   1.822327650
[50/49, 525/512]   1.241102862
[875/864, 50/49]   1.596910924
[4000/3969, 2048/2025]   1.451636818
[2048/2025, 50/49]   1.788798893
[4375/4374, 3136/3125]   15.37118131
[3136/3125, 64/63]   4.481834648
[4375/4374, 2048/2025]   49.40556506
[225/224, 64/63]   3.535500094
[225/224, 1029/1024]   1.093522071
[2401/2400, 5120/5103]   7.983665859
[2401/2400, 81/80]   29.82023498
[2401/2400, 6144/6125]   7.434917601
[1029/1024, 126/125]   1.635861828
[875/864, 64/63]   1.244819488
[49/48, 2240/2187]   1.161297413
[4375/4374, 64/63]   68.89109299
[81/80, 50/49]   1.626297004
[81/80, 875/864]   1.018401828
[3136/3125, 1029/1024]   1.386221179
[245/243, 2048/2025]   1.377861075
[6144/6125, 5120/5103]   1.073806905
[4375/4374, 50/49]   88.37662008
[126/125, 64/63]   1.976408356
[49/48, 250/243]   1.377325721
[49/48, 25/24]   1.979796637
[225/224, 245/243]   1.840171149
[1728/1715, 126/125]   1.055164518
[6144/6125, 4000/3969]   2.511974761
[49/48, 6272/6075]   1.547739961
[3136/3125, 81/80]   3.535332623
[49/48, 392/375]   2.150210807
[4375/4374, 1029/1024]   21.30785708
[64/63, 36/35]   1.788813730
[4000/3969, 875/864]   1.626068363
[225/224, 81/80]   2.788850951
[50/49, 128/125]   1.173928155
[81/80, 49/48]   1.659831249
[1728/1715, 4000/3969]   1.030271488
[126/125, 2048/2025]   1.417390368
[36/35, 21/20]   1.731936266
[50/49, 36/35]   1.394411021
[3136/3125, 1728/1715]   2.149111606
[6144/6125, 50/49]   6.522810530
[5120/5103, 3136/3125]   1.056521717


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