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Message: 10275 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:59:07
Subject: Re: acceptace regions
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> Gene, I said "this looks interesting, please help me understand what
> it means and what you are doing", not "this is useless". If you can't
> distinguish the two, how is anyone ever going to have any hope of
> communicating with you?
What you said was "this isn't helping", which seemed to me to mean the
same thing. It sounded like I was being shut out of the conversation,
and asked to concentrate on calculating lists instead.
On an unrelated point, changing the font size in Mozilla changes the
font size for this reply message, and changes the font size for
everything outside of the posting, but doesn't change the small font
of the posting itself. I'm tired of squinting and would like a Big
Print font if possible. Carl?
Message: 10276 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 12:59:33
Subject: Re: The same page
From: Carl Lumma
>>> >= will mean "equal when one side is complemented".
>>>
>>> 2 primes:
>>>
>>> >>
>>> 3 primes:
>>>
>>> ()ET:
>>> [monzo> /\ [monzo> ~= <val]
>>> ()LT:
>>> [monzo> ~= <val] /\ <val]
>>>
>>> 4 primes:
>>>
>>> ()ET:
>>> [monzo> /\ [monzo> /\ [monzo> ~= <val]
>>> ()LT:
>>> [monzo> /\ [monzo> ~= <val] /\ <val]
>>> ()PT:
>>> [monzo> ~= <val} /\ <val] /\ <val]
>>>
>>> Hopefully the pattern is clear.
>>
>> I'm missing wedgies here. And maps. And dual/complement.
>
>/\ is the wedgie,
/\ is the wedge product. I mean, you're not showing calculations
where the inputs involve wedgies.
>and ~= is the dual/complement.
So sorry, I read "when one side is completed", or something.
That leaves me:
() What is the form form complement?
() Does dual = complement?
-Carl
Message: 10277 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 14:17:09
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
I can certainly enclose a finite number of plotted points here (as
with the line passing through 12 & 53), and I thought Gene said
badness alone *could* give a finite list, just that it would include
lots of crap (like 1 & 2).
-Carl
Message: 10278 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 13:03:11
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>> >es, but I should think ideally we'd figure out how to normalize
>> in some way to bring this whole business back to scales.
>
>Why must we care that much about scales? Half the time I'm not using
>them myself. For me temperaments are more significant.
Dragnabbit, we've already been through this. I do *not* mean
*diatonics*, I mean *scales*, YOUR definition, pitches.
-Carl
Message: 10279 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 14:19:03
Subject: Re: !
From: Carl Lumma
>>>> > don't know what you meant by "ratio of logs".
>>>
>>> log(n/d)
>>> --------
>>> log(n*d)
>>>
>>> is one log divided by another log, hence a "ratio of logs". It
>>> doesn't matter what base you use, you get the same answer.
>>
>> Of course. I thought you were actually taking real cents error,
>> and then taking the log of that, though.
>
>No, if we use log scaling on the error axis then we're essentially
>taking the log of the above expression.
Right! I thought that's why it's called loglog.
>> Ok. Well however you did it, it seems that middle-of-the-road
>> temperaments like meantone and pajara are very close to 45deg.
>> off the axes.
>
>It's too bad that 45-degree angle is completely arbitrary -- the
>entire graph was scaled by Matlab so that it fit the screen.
Well it'd still be nice to see a circle, out of curiosity.
-Carl
Message: 10280 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:10:03
Subject: Re: A symmetric-based 7-limit temperament list
From: Carl Lumma
>squared symmetric
>complexity, which is an integer. This complexity measure, or else
>whatever we would get as the dual to Hahn taxicab distance,
What is the difference between symmetric complexity and Hahn
taxicab distance?
-Carl
Message: 10281 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 07:24:35
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> For 81/80 this is ceil(sqrt(13))=4 steps, and for
>> 2401/2400 it is ceil(sqrt(11))=4 steps also.
>
> Where do 13 and 11 come from?
81/80 is <-4 4 -1 0|, and 4^2+1^2+0^2+4*(-1)+4*0+(-1)*0=13, and
2401/2400 is <5 -1 -2 4|, from whence (-1)^+(-2)^2+(-4)^2+(-1)*(-2)+
(-1)*4+(-2)*4 = 11. Hence 2401/2400 is closer to the origin in the
symetrical 7-limit lattice we all know and love. It is a small
interval, but not really a complex one.
>> In fact, both can be reached in four steps in only one way, up
>> to commuitivity; we have
>>
>> 81/80 = (6/5)(3/2)^3 (1/4)
>> 2401/2400 = (7/6)(7/5)^2(7/4) (1/4)
>
> This sure looks like taxicab, but what are the "(1/4)" terms?
They make the math come out right. I don't really care about the
octaves, but those are them. You could write
49/48 = (7/6)(7/8), 49/50 = (7/5)(7/10), and then, look Ma!, no
octaves:
2401/2400 = (7/6)(7/8)(7/5)(7/10)
Message: 10282 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:03:27
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Lost me. The only unweighted lattice types I'm aware of are
> rectangular, where all the angles are 90deg,
= cubic = Zn, Z3 in three dimensions.
and triangular,
> where I believe all the angles are 60deg,
= A2 in two dimensions, and An in n dimensions
at least through FCC.
FCC = A3 = D3
Message: 10284 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 08:54:26
Subject: Still another
From: Gene Ward Smith
Another way of viewing this 9-and-up lattice question is to start by
noticing that the quadratic form for unweighted rms optimization in
the 5 and 7 limits is the dual or mapping form for An*, and that in
this way we can get the symmetrical lattice of 5 and 7 limit notes.
We now can take the dual lattice either of the way Graham and I have
been doing it, or Paul's preferred way, and get a corresponding
lattice in the 9, 11 etc limits. While this works better than the
attempt to find a good Euclidean version of the Tenney norm did, it
doesn't give us something as symmetrical as we might prefer.
By the way, from the above comment rms error is the same as
symmetrical lattice error in the 5 and 7 limits, and I could put that
together with the formula for symmetrical lattice error of 7-limit
linears to do a badness listing for symmetrical error, complexity and
(log flat or other) badness, which could provide a different
perspective on the question of "best". I don't believe in Dave's
quantization of agony theory, and don't think there is one single
notion of "good". This would be another "good", with its own
character and forms of usefulness.
Message: 10285 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 13:05:43
Subject: Re: The same page
From: Carl Lumma
>() What is the form form complement?
Form *for*, sorry. :) -C.
Message: 10286 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:21:21
Subject: Re: The same page
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>> () What is the form form complement?
>>>
>>> Form *for*, sorry. :) -C.
>>
>> The complement reverses the order of the entries and changes some
of
>> the signs. Dave Keenan and others made extensive posts here
detailing
>> this. I haven't yet had to worry about the change of signs, as the
>> measures I've looked at so far take the absolute values anyway.
>
> How am I ever going to find these posts of Dave's to get to
>
> | a b c > ~= | -b c a >
>
> or whatever?
Shouldn't be too hard, it was a fairly recent discussion here.
> And I'm still missing things to do with wedgies. For example,
> your L1 complexity. Gene gives
>
>> Erlich magic L1 norm; if
>>
>> <<a1 a2 a3 a4 a5 a6||
>>
>> is the wedgie, then complexity is
>>
>> |a1/p3|+|a2/p5|+|a3/p7|+|a4/p3p5|+|a5/p3p7|+|a6/p5p7|
>
> Where wedgie is val-wedgie. But apparently there's a monzo-wedgie
> formualation...
Simply reverse the order of the entries.
>> the L1 norm of the *monzo-wedgie* in the Tenney lattice.
>> In other words, it's the 'taxicab area' of the (nontorsional)
>> vanishing bivector, something which seems to give three times
>> the number of notes in the 5-limit ET case.
>
> C'mon guys, help me publish this stuff in one place. I'm
> begging you.
We should be thanking you. The 5-limit ET case (12 was used as the
example) was referred to a number of times; I'll see if I can locate
that post . . .
Message: 10287 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 01:14:44
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Carl Lumma
>>> >or 81/80 this is ceil(sqrt(13))=4 steps, and for
>>> 2401/2400 it is ceil(sqrt(11))=4 steps also.
>>
>> Where do 13 and 11 come from?
>
>81/80 is <-4 4 -1 0|, and 4^2+1^2+0^2+4*(-1)+4*0+(-1)*0=13,
So is this
|| <a b c d] || = b^2+c^2+d^2+bc+bd+cd
?
Wait, this looks familiar from a few mails back...
>The symmetrical complexity for codimension one (5-limit linear,
>7-limit planar) is the symmetrical lattice distance for the comma
>defining it. For a 7-limit linear lattice, we have
>
>Symcomp( <a b c d e f| ) = sqrt(3 (a^2+b^2+c^2) - 2(ab+ac+bc))
Ah, so this is new. It might be nice to surround new terms with
*asterisks* or something, to distinguish them from existing
definitions (such as you gracefully gave for Voronoi and Delaunay
cells in another post).
'symmetrical lattice distance' returns nil at Google and mathworld.
The difference between these two forms might be due to that you're
using wedgies for one and a comma-monzo for the other. But wait,
now I can check the "What the numbers mean" form!
I believe you meant "symcomp( <<a b c d e f || )". This means
there are a generators in 3/2, b in 5/2, c in 7/2, d in 5/3,
e in 7/3 and f in 7/5. Turning that into a 2&g map...
...whoops, looks like I'm hitting the same snag as Herman...
>> 3&g: [<1 1 3 3|, <-6 0 -25 -20|] g ~ 7/72 ~ 11/114
>>
>> 5&g: [<-2 -8 1 4|, <7 25 0 15|] g ~ 58/72 ~ 135/167
>>
>> 7&g: [<1 7 -4 1|, <-2 -20 15 0|] g ~ 65/72 ~ 182/202
>
>That's useful to know. I can see where the second part of the maps
>come from, but how do you get the first part? It's clear that the
>element corresponding to the period is always 1 in this example,
>which makes sense, but is there any easy way to get the other three
>numbers other than trying a few until you find one that works? In
>other words, 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 I could turn a generic wedgie into a 2&g map, I might be able
to find a corresponding comma and see if b^2+c^2+d^2+bc+bd+cd is
the symcomp for a monzo.
>and
>2401/2400 is <5 -1 -2 4|, from whence (-1)^+(-2)^2+(-4)^2+(-1)*(-2)+
>(-1)*4+(-2)*4 = 11. Hence 2401/2400 is closer to the origin in the
>symetrical 7-limit lattice we all know and love. It is a small
>interval, but not really a complex one.
Yes, I gather you were saying that.
>>> In fact, both can be reached in four steps in only one way, up
>>> to commuitivity; we have
>>>
>>> 81/80 = (6/5)(3/2)^3 (1/4)
>>> 2401/2400 = (7/6)(7/5)^2(7/4) (1/4)
>>
>> This sure looks like taxicab, but what are the "(1/4)" terms?
>
>They make the math come out right. I don't really care about the
>octaves, but those are them. You could write
>
>49/48 = (7/6)(7/8), 49/50 = (7/5)(7/10), and then, look Ma!, no
>octaves:
>
>2401/2400 = (7/6)(7/8)(7/5)(7/10)
Ok. So I'm at a loss to describe how this is different from
taxicab distance.
-Carl
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Message: 10288 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 13:09:10
Subject: Re: acceptace regions
From: Carl Lumma
>On an unrelated point, changing the font size in Mozilla changes the
>font size for this reply message, and changes the font size for
>everything outside of the posting, but doesn't change the small font
>of the posting itself. I'm tired of squinting and would like a Big
>Print font if possible. Carl?
My advice is to have the list delivered by e-mail, where you can
control this and many other things to your heart's content. You
do miss out a bit on the threading, is the only drawback.
If you insist on using the web interface, I'm not sure anything can
be done if yahoo is using absolute font sizes in their html. You
might try your user options to see if there is a 'display formatted
message / display plain text message' option, and toggle it to see
if that helps.
-Carl
Message: 10289 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:23:33
Subject: Re: loglog!
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>>> Again, I view complexity as a measure of length, area,
>>>>>> volume . . . in the Tenney lattice with taxicab metric. We're
>>>>>> measuring the size of the finite dimensions of the
periodicity
>>>>>> slice, periodicity tube, periodicity block . . .
>>>>>
>>>>> The units in all cases should be notes.
>>>>
>>>> I disagree, since I feel the Tenney lattice is much more
>>>> appropriate than the symmetrical cubic lattice.
>>>
>>> Why would that make any difference?
>>
>> Tenney makes the lower primes simpler than the higher primes.
>
> You're still enclosing notes.
An infinite number of them, except for the ET case, where (and only
where) cubic and Tenney will agree -- the cells are shaped
differently, but there are always the same number of them.
>>>>>>>>>>> I've suggested in the
>>>>>>>>>>> past adjusting for it, crudely, by dividing by pi
(lim).
>>>>>>>>>>
>>>>>>>>>> Huh? What's that?
>>>>>>>>>
>>>>>>>>> If we're counting dyads, I suppose higher limits ought
to
>> do
>>>>>>>>> better with constant notes.
>>>>>>>>> If we're counting complete chords,
>>>>>>>>> they ought to do worse. Yes/no?
>>>>>>
>>>>>> Still have no idea how to approach this questioning, or what
>> the
>>>>>> thinking behind it is . . .
>>>>>
>>>>> Think scales.
>>>>
>>>> Well that's different. What kind of scales? ET? DE? JI? Other?
>>>
>>> Any scale that is a manifestation of the given temperament.
>>>
>>>>> What relations, if any, do we expect, for n
>>>>> notes, as lim goes up:
>>>>
>>>> For a given scale? Then this is even more different . . .
>>>
>>> Ultimately if we can't show a relation to notes in scales
>>> we've gone off the deep end.
>>
>> That's a separate issue. I mean are you comparing a given scale
>> across *different* temperaments?
>
> Eventually I hope to characterize temperaments by the kinds of
> scales they can manifest. But before that, things like Graham
> complexity and consistency range need to be cultured to fix
> a relationship between a single temperament and its scales.
You're only talking about *linear* temperaments here, right?
Message: 10290 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:47:13
Subject: Dicot and "Number 56"
From: Herman Miller
This pair has the same problem as Pelogic and Hexidecimal.
Number 23 Dicot
[2, 1, 6, -3, 4, 11] [[1, 1, 2, 1], [0, 2, 1, 6]]
TOP tuning [1204.048158, 1916.847810, 2764.496143, 3342.447113]
TOP generators [1204.048159, 356.3998255]
bad: 42.920570 comp: 2.137243 err: 9.396316
Number 56
[2, 1, -4, -3, -12, -12] [[1, 1, 2, 4], [0, 2, 1, -4]]
TOP tuning [1204.567524, 1916.451342, 2765.076958, 3394.502460]
TOP generators [1204.567524, 355.9419091]
bad: 66.522610 comp: 2.696901 err: 9.146173
With the TOP tuning for Number 23, the [4, -4] approximation of 7:1
(3390.59 cents) is better than the [1, 6] approximation. Because the [1,
6] approximation is 26 cents flat, the 4-cent sharp octaves actually
make the 7:4 worse.
I'm wondering why "number 56" and "hexidecimal" are so far down the
list, if they're clearly better than the ones labeled "dicot" and
"pelogic"? Try tuning these up in Scala; it's clear that the "number 56"
and "hexadecimal" mappings are better. The "4:5:6:7" in number 23
mapping actually sounds more like a "1/6:5:4:7"!
Message: 10291 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:12:28
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Dragnabbit, we've already been through this. I do *not* mean
> *diatonics*, I mean *scales*, YOUR definition, pitches.
A discrete set of notes? A periodic discrete set of notes? Some other
definition?
Message: 10292 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:26:55
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> I can certainly enclose a finite number of plotted points here (as
> with the line passing through 12 & 53),
Just 1, 1, and 3, right? But then there might be more to the right of
the graph's range.
> and I thought Gene said
> badness alone *could* give a finite list, just that it would include
> lots of crap (like 1 & 2).
Right, but that wasn't log-flat badness, that was a badness measure
which uses any higher exponent on complexity than what you would use
to get log-flat badness.
Message: 10293 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:18:11
Subject: Re: Rhombic dodecahedron scale
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Could you tell me please how this relates
> to the 14 points of a rhombic dodecahedron and how that is based on
> 6 hexanies and 8 tetrads? Thanks. Paul Hj
Paul, did you look at my paper as I suggested? All this stuff is only
3-dimensional, so many of us can best understand it through pictures.
Message: 10294 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:28:59
Subject: Re: The same page
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
Apparently these weren't the kinds of questions Gene wants to try
answering . . .
Message: 10295 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 13:29:08
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Carl Lumma
>> >ost me. The only unweighted lattice types I'm aware of are
>> rectangular, where all the angles are 90deg,
>
>= cubic = Zn, Z3 in three dimensions.
Thank you, thank you, thank you.
> and triangular,
>> where I believe all the angles are 60deg,
>
>= A2 in two dimensions, and An in n dimensions
Thank you.
> at least through FCC.
>
>FCC = A3 = D3
Then again, I think we're talking about the same thing!
You're counting the 'rungs' on the shortest path to the
target, no? (And where do An and Dn diverge?)
By the way, in 1999, Paul Hahn gave the following:
>Given a Fokker-style interval vector (I1, I2, . . . In):
>
>1. Go to the rightmost nonzero exponent; add the product of its
>absolute value with the log of its base to the total.
>
>2. Use that exponent to cancel out as many exponents of the opposite
>sign as possible, starting to its immediate left and working right;
>discard anything remaining of that exponent.
>
> Example: starting with, say, (4 2 -3), we would add 3 lg(7) to
> our total, then cancel the -3 against the 2, then the remaining
> -1 against the 4, leaving (3 0 0). OTOH, starting with
> (-2 3 5), we would add 5 lg(7) to our total, then cancel 2 of
> the 5 against the -2 and discard the remainder, leaving (0 3 0).
>
>3. If any nonzero exponents remain, go back to step one, otherwise
>stop.
This gives taxicab distance on an unweighted odd-limit An lattice,
IIRC.
-Carl
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Message: 10296 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 14:30:33
Subject: Re: loglog!
From: Carl Lumma
>>>>>>> >e're measuring the size of the finite dimensions of the
>>>>>>> periodicity slice, periodicity tube, periodicity block . . .
>>>>>>
>>>>>> The units in all cases should be notes.
>>>>>
>>>>> I disagree, since I feel the Tenney lattice is much more
>>>>> appropriate than the symmetrical cubic lattice.
>>>>
>>>> Why would that make any difference?
>>>
>>> Tenney makes the lower primes simpler than the higher primes.
>>
>> You're still enclosing notes.
>
>An infinite number of them, except for the ET case, where (and
>only where) cubic and Tenney will agree -- the cells are shaped
>differently, but there are always the same number of them.
Yes but units are still notes.
>>>>>>>>>>>> I've suggested in the
>>>>>>>>>>>> past adjusting for it, crudely, by dividing by
>>>>>>>>>>>> pi(lim).
>>>>>>>>>>>
>>>>>>>>>>> Huh? What's that?
>>>>>>>>>>
>>>>>>>>>> If we're counting dyads, I suppose higher limits
>>>>>>>>>> ought to do
>>>>>>>>>> better with constant notes.
>>>>>>>>>> If we're counting complete chords,
>>>>>>>>>> they ought to do worse. Yes/no?
//
>>>>>> What relations, if any, do we expect, for n
>>>>>> notes, as lim goes up:
//
>>>> Ultimately if we can't show a relation to notes in scales
>>>> we've gone off the deep end.
>>>
>>> That's a separate issue. I mean are you comparing a given scale
>>> across *different* temperaments?
>>
>> Eventually I hope to characterize temperaments by the kinds of
>> scales they can manifest. But before that, things like Graham
>> complexity and consistency range need to be cultured to fix
>> a relationship between a single temperament and its scales.
>
>You're only talking about *linear* temperaments here, right?
No, I hope for a single complexity measure for all temperaments.
I've thought about PTs in terms of a lattice of generators, and
wracked my brain against the issue of scales coming from that.
-Carl
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Message: 10297 - Contents - Hide Contents
Date: Sat, 14 Feb 2004 12:28:44
Subject: symmetrical
From: Carl Lumma
Is "symmetrical lattice" synonymous with "unweighted lattice"?
-Carl
Message: 10298 - Contents - Hide Contents
Date: Sat, 14 Feb 2004 21:19:59
Subject: Re: A symmetric-based 7-limit temperament list
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> This complexity measure, or else
> whatever we would get as the dual to Hahn taxicab distance,
Wouldn't the complexity measure here be the Hahn taxicab distance
itself? Or at least a Euclidean version of it? Where does duality
come into play?
> seem to be
> the logical ones to use when we are using symmetric, octave
equivalent
> rms error.
Which ones -- taxicab and Euclidean?
Message: 10299 - Contents - Hide Contents
Date: Sat, 14 Feb 2004 21:22:25
Subject: Re: A symmetric-based 7-limit temperament list
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> squared symmetric
>>> complexity, which is an integer. This complexity measure, or else
>>> whatever we would get as the dual to Hahn taxicab distance,
>>
>> What is the difference between symmetric complexity and Hahn
>> taxicab distance?
>
> They aren't measuring the same thing. You need to compare symmetric
> distance and Hahn distance, or symmetric complexity and Hahn-dual
> complexity.
You mean the complexity applies to the wedgie? Then how is the dual
to the distance, which applies to monzos? Wouldn't the dual to the
distance be something that operates on vals?
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10150
10200
10250
10300
10350
10400
10450
10500
10550
10600
10650
10700
10750
10800
10850
10900
10950
10250 -
10275 -