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Message: 10250 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:45:17
Subject: Re: The same page
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> Is this a start?
>
> Yes, great!!
>
>> ~= 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, and ~= is the dual/complement.
Message: 10251 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:52:50
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.
Message: 10252 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 23:49:40
Subject: Re: The same page
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
>> Apparently these weren't the kinds of questions Gene wants to try
>> answering . . .
>
> I was planning on getting around to it.
You were? Oh, sorry . . .
>Is it urgent for some
>reason?
Yes, I wanted know why the correspondence breaks down beyond the 3-
limit (I suppose that's clear enough mathematically, but . . .) and
what implications that has on how we should consider defining
complexity (and perhaps even error) beyond the 3-limit. Clearly the
exact choice here isn't going to have much impact on how our ET
graphs look and what we choose from them. But in general, my
understanding is not completely crystallized, so every little bit
helps . . .
Message: 10254 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 15:51:21
Subject: Re: The same page
From: Carl Lumma
>>> >rlich 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.
Not sure what you're saying.
monzo-wedgie = reverse(val-wedgie)
or? Can you give the form?
-Carl
Message: 10255 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:47:31
Subject: Re: acceptace regions
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...>
wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith"
<gwsmith@s...>
>>> wrote:
>>>> We might try in analyzing or plotting 7-limit linear
temperaments a
>>>> transformation like this:
>>>>
>>>> u = 4 - ln(complexity) - ln(error)
>>>> v = 12 - 4 ln(complexity) - ln(error)
>>
>> In the terms Paul, Carl and I have been using, this is a cutoff
relation
>>
>> max[ln(complexity)/4 + ln(error)/4,
>> 4 * ln(complexity)/3 + ln(error)/12] < 1
>
> Only if you choose to use it for one. It's a coordinate
> transformation, primarily.
>
>
>> It seems we may be moving towards some kind of agreement. :-)
>
> I've been *trying* to help you and Paul here, with all of this stuff
> about convex hulls and what not which I have been told is useless.
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?
Message: 10257 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 00:50:24
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Carl Lumma
> The symmetrical complexity for codimension one (5-limit linear,
> 7-limit planar) is the symmetrical lattice distance for the
> comma defining it.
Is this the same as the "n" in your "interval count" message,
then? That was what I've been calling "taxicab distance to
the comma".
Between Tenney weighting, which gives log(n*d) for commas n/d
(is this what you're calling L1?), and no weighting (plain
"taxicab", your suggestion that 2401/2400 = 81/80) lies what
I've been calling "number of notes searched before finding
the comma" complexity. This is the volume of everything within
taxicab radius n of the origin, or n^pi(lim). In a sense,
weighting by limit. Clearly this is related to log-flat
badness...
(n-d)log(d)^e / d
...where e is pi(lim) or pi(lim)-1.
-Carl
Message: 10258 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:49:18
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> and the complexity < whatever
>>> you want. 100 notes? 20 notes?
>>
>> Why would you need a complexity bound in addition to the circle?
The
>> circle, being finite, would only extend to a certain maximum
>> complexity anyway . . .
>
> To determine its radius.
Oh, so there isn't an additional complexity bound. Gotcha. I'll try
some circles when I have a chance . . .
Message: 10259 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:01:02
Subject: Re: loglog!
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> First of all, what is the real name for creepy complexity? L1?
>>
>> Yes, it's 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.
>
> Ok.
>
>>>> 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.
>>>>>>>>> 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? It seems like you are, but above you
say "Any scale that is a manifestation of the given temperament", so
I'm not sure what you're getting from what.
Message: 10260 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 23:54:50
Subject: Re: top23
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Paul's Dave-approved list of 23 7-limit temperaments...
>
>> 1. Huygens meantone
>> 2. Pajara
>> 3. Magic
>> 4. Semisixths
>> 5. Dominant Seventh
>> 6. Tripletone
>> 7. Negri
>> 8. Hemifourths
>> 9. Kleismic/Hanson
>> 10. Superpythagorean
>> 11. Injera
>> 12. Miracle
>> 13. Biporky
>> 14. Orwell
>> 15. Diminished
>> 16. Schismic
>> 17. Augmented
>> 18. 1/12 oct. period, 25 cent generator (we discussed this years
ago)
>> 19. Flattone
>> 20. Blackwood
>> 21. Supermajor seconds
>> 22. Nonkleismic
>> 23. Porcupine
>
> This looks reasonable. Let's go back to the top 23 from Gene's
114...
>
>> Number 1 Ennealimmal
>> Number 2 Meantone
>> Number 3 Magic
>> Number 4 Beep
>> Number 5 Augmented
>> Number 6 Pajara
>> Number 7 Dominant Seventh
>> Number 8 Schismic
>> Number 9 Miracle
>> Number 10 Orwell
>> Number 11 Hemiwuerschmidt
>> Number 12 Catakleismic
>> Number 13 Father
>> Number 14 Blackwood
>> Number 15 Semisixths
>> Number 16 Hemififths
>> Number 17 Diminished
>> Number 18 Amity
>> Number 19 Pelogic
>> Number 20 Parakleismic
>> Number 21 {21/20, 28/27}
>> Number 22 Injera
>> Number 23 Dicot
>
> ...also reasonable. Assuming names are synchronized (hemififths=
> hemifourths?,
That can't be the same, but it looks like you didn't assume they were
the same below.
> here's the intersection of
> these lists in Paul order...
>
>> 1. Huygens meantone
>> 2. Pajara
>> 3. Magic
>> 4. Semisixths
>> 5. Dominant Seventh
>> 11. Injera
>> 12. Miracle
>> 14. Orwell
>> 15. Diminished
>> 16. Schismic
>> 17. Augmented
>> 20. Blackwood
>
> Here's the intersection in Gene order...
>
>> Number 2 Meantone
>> Number 3 Magic
>> Number 5 Augmented
>> Number 6 Pajara
>> Number 7 Dominant Seventh
>> Number 8 Schismic
>> Number 9 Miracle
>> Number 10 Orwell
>> Number 14 Blackwood
>> Number 15 Semisixths
>> Number 17 Diminished
>> Number 22 Injera
>
> Agreement is on 12 temperaments, and fairly well on order.
> Schismic, augmented and Blackwood seem to be the greatest
> order disputes. Ennealimmal, beep, tripletone, negri and
> kleismic seem to be the greatest omission disputes.
Sure, and remember, I might have included Ennealimmal but Gene didn't
provide enough data to locate where a moat might go in that vicinity.
When I have time, I'll do my own 7-limit linear temperament search,
though I haven't implemented the linear programming needed (?) to
calculate TOP error yet.
Message: 10261 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 01:41:29
Subject: Another approach to 9-limit
From: Gene Ward Smith
Here is a quite different approach which might prove useful. The An
lattice can be represented in (n+1)-space as [a0,a1,...,an], where
a0+...+an=0. For an A3 lattice of 5,7,9, we can set 5 to be
[-1 1 0 0], 7 [-1 0 1 0], and 9 [-1 0 0 1]. For the 11 limit we would
have 11 as [-1 0 0 0 1] etc. etc. Dot products of one of these basis
vectors with itself gives 2, with another gives 1, so distances have
been scaled up by a factor of sqrt(2), as in the fcc version of A3.
Now suppose we remove the restriction that the coordinates must add to
zero, and set
Note([a3 a5 a7 a9]) = 3^(a3+a5+a7+a9) 5^a5 7^a7 9^a9.
We again have multiple versions of the same note, with
Note([3,0,0,-1]) = 3^2 9^(-1) = 1. Modding out by this is a
possibility. The 11-limit version would be
Note([a3 a5 a7 a9 a11]) = 3^(a3+a5+a7+a11) 5^a5 7^a7 9^a9 11^a11
Still another possibility is the old reliable obvious one, of gluing a
3-lattice half way to the 9.
Message: 10262 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 14:01:22
Subject: Re: !
From: Carl Lumma
>>>>>>> >.S. The relative scaling of the two axes is completely
>>>>>>> arbitrary,
>>>>>>
>>>>>> Howso? They're both base2 logs of fixed units.
>>>>>
>>>>> Actually, the vertical axis isn't base anything, since it's a
>>>>> ratio of logs.
>>>>
>>>> That cents are log seems irrelevant. They're fundamental units!
>>>
>>> ??
>>
>> I 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.
>>> Scaling is one thing, and where you depict the axes intersecting
>>> is another.
>>
>> Yes, I gather. I have no clue what relative scaling is.
>
>Relative scaling would be, for example, what one inch represents on
>one of the axes, vs. what it represents on the other axis.
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.
>>>> Incidentally, I don't see the point of a moat vs. a circle, since
>>>> the moat's 'hole' is apparently empty on your charts
>>>
>>> Don't know what you mean.
>>
>> You should see shortly that I thought the moat was the region of
>> acceptance, not the region of safety.
>
>So what's the 'hole', and what is apparently empty?
Don't worry about this -- I was operating under a false definition
of moat.
-Carl
Message: 10263 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:49:08
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Yes, 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.
Message: 10264 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 01:43:01
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:
>> The symmetrical complexity for codimension one (5-limit linear,
>> 7-limit planar) is the symmetrical lattice distance for the
>> comma defining it.
>
> Is this the same as the "n" in your "interval count" message,
> then? That was what I've been calling "taxicab distance to
> the comma".
No, it's the symmetrical Euclidean lattice distance.
Message: 10265 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 12:50:27
Subject: Re: !
From: Carl Lumma
>>>>> >.S. The relative scaling of the two axes is completely
>>>>> arbitrary,
>>>>
>>>> Howso? They're both base2 logs of fixed units.
>>>
>>> Actually, the vertical axis isn't base anything, since it's a
>>> ratio of logs.
>>
>> That cents are log seems irrelevant. They're fundamental units!
>
>??
I don't know what you meant by "ratio of logs".
>>>> You mean c is arbitrary in y = x + c?
>>>
>>> Not what I meant, but this is the equation of a line, not a circle.
>>
>> Yes, I know. But I wasn't trying to give a circle (IIRC that form
>> is like x**2 + y**2 something something), or a line, but the
>> intersection point of the axes, which is what I thought you meant by
>> relative scaling.
>
>Scaling is one thing, and where you depict the axes intersecting is
>another.
Yes, I gather. I have no clue what relative scaling is.
>> That means I only meant the above to apply when
>> either x or y is zero, I think.
>
>I lost you.
When x or y is zero in the above, you get the intersection point
for the axes.
>> Incidentally, I don't see the point of a moat vs. a circle, since
>> the moat's 'hole' is apparently empty on your charts
>
>Don't know what you mean.
You should see shortly that I thought the moat was the region of
acceptance, not the region of safety.
-Carl
Message: 10266 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:02:43
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>>>>>> why doesn't the badness bound alone
>>>>>>>>> enclose a finite triangle?
>>>>>>>>
>>>>>>>> it actually encloses an infinite number
>>>>>>>> of temperaments.
>>>>>
>>>>> groups.yahoo.com/group/tuning-math/files/Paul/et5loglog.gif
>>>>>
>>>>> ...the region beneath the 7-53 diagonal is empty.
>>>>> Is there stuff there you haven't plotted?
>>>>
>>>> With lower error? No, but you'd never know for sure just from
>>>> looking at the loglog graph.
>>>
>>> Ok, but you're saying there isn't. And your graph goes down to
>>> 1 note, and if the complexity variations are slight that means
>>> the triangle is empty, as opposed to enclosing an infinite number
>>> of temperaments.
>>
>> Which triangle are you talking about? I thought you were talking
>> about the one formed by using one of Gene's log-flat badness
cutoffs
>> by itself, without any complexity or error cutoffs.
>
> That's right.
Well that would enclose an infinite number of temperaments unless
it's so low that it encloses none. But Gene never used such a low
cutoff, since he wanted more than zero temperaments to be included.
Message: 10267 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 01:49:03
Subject: Re: A 9-limit diamond packing
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> The fcc lattice we started with is now everything with even
>> coordinates which sum to 0 mod 4, if we add everything with even
>> coordinates which sum to 2 mod 4, we get the body-centered cubic
>> lattice. We can do this by adding the minor tetrad around [1 1 1]
> to
>> the major one.
> I get 0 mod 4 adding the minor tetrad to the major one...
I meant we have the four notes 1-5-7-9 of a major tetrad surrounding
3, and we also can add 9-9/5-9/7-9/9 around it.
Message: 10268 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:50:52
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>> Having 81/80 in the kernel implies you can harmonize a diatonic
>>>> scale all the way through in consonant thirds. Similar commas
>>>> have similar implications of the kind Carl always seemed to care
>>>> about.
>>>
>>> Don't you mean 25:24?
>>
>> No, 81;80. 25;24 in the kernel doesn't give you either a diatonic
>> scale or 'consonant thirds'.
>
> Oh, in the kernel means tempered out (right?)
Yeah, well, what I meant was in the kernel of the temperament.
> giving neutral
> thirds. So it isn't immediately obvious why 81:80 throws 5:4
> and 6:5 on the same scale degree (always thirds?). . .
81:80 alone doesn't define a single finite scale, but the whole
family that it does imply includes the diatonic scale as a member.
Message: 10269 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 22:05:17
Subject: Re: !
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>>>>> P.S. The relative scaling of the two axes is completely
>>>>>>>> arbitrary,
>>>>>>>
>>>>>>> Howso? They're both base2 logs of fixed units.
>>>>>>
>>>>>> Actually, the vertical axis isn't base anything, since it's a
>>>>>> ratio of logs.
>>>>>
>>>>> That cents are log seems irrelevant. They're fundamental
units!
>>>>
>>>> ??
>>>
>>> I 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.
>>>> Scaling is one thing, and where you depict the axes
intersecting
>>>> is another.
>>>
>>> Yes, I gather. I have no clue what relative scaling is.
>>
>> Relative scaling would be, for example, what one inch represents
on
>> one of the axes, vs. what it represents on the other axis.
>
> 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.
Message: 10271 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 12:56:54
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Carl Lumma
>> >symmetrical lattice distance' returns nil at Google and mathworld.
>
>People on tuning-math already knew the symmetrical 7-limit lattice of
>note-classes when I got here.
But they didn't call it that!
>They didn't seem to know the formula for
>calculating lattice distance, but clearly would have understood there
>had to be one, so I don't regard this as a new topic. Anyway I've
>talked about it endlessly in the last few years.
Have you ever called it "symmetrical lattice distance"?
>>> 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 you know the whole wedgie, finding x, y and z can be done by
>solving a linear system. If you only know the period and generator
>map, you first need to get the rest of the wedgie, which will be the
>one which has a much lower badness than its competitors.
>
>For instance, suppose I know the wedgie is <<1 4 10 4 13 12||. Then
>I can set up the equations resulting from
>
><1 x y z| ^ <0 1 4 10| = <<1 4 10 4 13 12||
>
>We have <1 x y z| ^ <0 1 4 10| = <<1 4 10 4x-y 10x-z 10y-4z||
>
>Solving this gives us y=4x-4, z=10x-13; we can pick any integer for x
>so we choose one giving us generators in a range we like. Since 3 is
>represented by [x 1] in terms of octave x and generator, if we want
>3/2 as a generator we pick x=1.
This is huge. Processing. . .
>> Ok. So I'm at a loss to describe how this is different from
>> taxicab distance.
>
>It's clearly not taxicab Tenney distance,
That's weighted. I'm saying it looks like unit/unweighted taxicab.
>It's taxicab distance with Fifth Element style flying
>taxicabs,
:)
>and routes which form an A3=D3 lattice.
Lost me. The only unweighted lattice types I'm aware of are
rectangular, where all the angles are 90deg, and triangular,
where I believe all the angles are 60deg, at least through FCC.
-Carl
Message: 10272 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 14:10:59
Subject: Re: The same page
From: Carl Lumma
>>> >) 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?
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...
>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.
-Carl
Message: 10273 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 05:04:20
Subject: Old reliable
From: Gene Ward Smith
If you project the (n+1)-dimensional system down to n dimensions so
that there is a 1-1 relationship between lattice points and notes,
you get what I called the old reliable and obvious system, which
still seems like the best one. In the 11-limit case, for instance,
you can use the note-class norm
|| (a3 a5 a7 a11) || = sqrt(a3^2+4a5^2+4a7^2+4a11^2+2a3a5+2a3a7+2a3a11
+4a5a7+4a5a11+4a7a11)
I've never invesitaged this for theta series, scales, etc; maybe I'll
get around to it.
Message: 10274 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 14:13:46
Subject: Re: loglog!
From: Carl Lumma
>>>>> >gain, 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.
>>>>>>>>>> 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.
-Carl
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