>>>>> >he complexity measures cannot be compared across different
>>>>> dimensionalities, any more than lengths can be compared with
>>>>> areas can be compared with volumes.
>>>>
>>>> I've suggested in the
>>>> past adjusting for it, crudely, by dividing by pi(lim).
>>>
>> 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?
-Carl
Message: 10129 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 20:24:20
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>> ...still trying to understand why the rectangle doesn't
enclose
>>>>> a finite number of temperaments...
>>>>
>>>> Which rectangle?
>>>
>>> The rectangle enclosed by error and complexity bounds.
>>
>> Yes, that would enclose a finite number of temperaments.
>
> Then why the hell do we need a badness bound?
We'd get an awful long list of temperaments without it, especially in
the ET case, if we're insisting on including at least one with
relatively high error and at least one with relatively high
complexity.
> Alternatively, then why doesn't the badness bound alone enclose a
> finite triangle?
Not only is it, like the rectangle, infinite in area on the loglog
plot, since the zero-error line and zero-complexity lines are
infinitely far away, but it actually encloses an infinite number of
temperaments.
> Error units
> ought to be the same!
Yes, an argument could be made for that, though we'd tend to insist
on tighter error bounds for higher limits.
>> Remember, we're dealing with a Pascal's
>> triangle, with one scenario for each number in the triangle, where
>> the number itself tells you the number of elements in the wedgie,
the
>> rrow number is the number of primes, and the column number is the
>> codimension.
>
> I never knew that or forgot it!
Well, the point is that 'limit' is not the only 'dimension' in this
problem; for example in 7-limit, we have ETs, linear temperaments,
and planar temperaments.
Message: 10130 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 21:14:19
Subject: Re: The same page
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
> wrote:
>>
>>> Why not admit both versions of the wedgie in all instances?
>>
>> The wedgie then no longer corresponds 1-1 with temperaments, as
> there
>> are two of them.
>
> So the correspondence is 1-1-1. Why is that a problem?
If I say "here is a wedgie for a 7-limit temperament" you no longer
know which temperament.
Message: 10131 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 06:14:59
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> Could you do me a favor and attempt to speak to me as a human being,
>>> and not deal with me like a chess opponent, trying to look several
>>> moves ahead so that you can defeat me?
>>
>> I washed out of the first round of the US correspondence championship.
>> It's my brother who is the grandmaster.
>
> Is he really a grandmaster?
He's not an OTB grandmaster, he is a correspondence grandmaster and
several times US champion.
Message: 10132 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 16:21:38
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>And we're not suggesting any "goodness" measure which
>is applicable to both 5-limit and 7-limit systems of any respective
>dimensionalities.
Any fundamental reason why not?
>But we are suggesting something similar be used in
>each of the Pascal's triangle of cases, which seems logical.
I'm a bit lost with the Pascal's triangle stuff. Can you populate
a triangle with the things you're associating with it? Such would
be grand, in the Wilson tradition....
>If it's
>wrong, it's wrong, and there goes the premise of our paper. But it's
>a theory paper, not an edict. I think if the criteria we use are
>easily grasped and well justified, we will have done a great job
>publishing something truly pioneering and valuable as fodder for
>experimentation.
We have a choice -- derive badness from first principles or cook
it from a survey of the tuning list, our personal tastes, etc.
-Carl
Message: 10133 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 20:27:59
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>
>>> I'm in the middle of working on an ennealimmal piece now.
Inherent
>>> properties are a major aspect for this kind of thing.
>>
>> You're using a full basis for the kernel? And it's audible? (Real
>> questions, not rhetorical or riddles.)
>
> I'm not sure what your question means. So far I've been sticking to
> the 45 note DE, and obviously doing that makes a clear audible
> difference. However, experience has shown that comma pumps on
> 2401/2400 or 4375/4374 are not so long that they fail to be
> comprehensible. The audibility of the differnce between the starting
> and ending note if you temper is another matter; it's not a hell of
a
> big change, and I don't hear it myself.
I think it's Dave's turn to ponder this.
>>> 612 is a fine
>>> way to tune ennealimmal, though I plan on using TOP for this
one.
>> This
>>> stuff really is practical if you care to practice it.
>>>
>>> In terms of commas, we have a sort of complexity of the harmonic
>>> relationships they imply--distance measured in terms of the
>>> symmetrical lattice norm possibly being more relevant here than
>>> Tenney.
>>
>> How so? You really think a progression by perfect fifths is as
>> complex as a progression by ratios of 7?
>
> It's precisely as complex in terms of the chord relationships
> involved, so long as you stay below the 9-limit.
Why do you say this? Is this some mathematical result, or your
subjective feeling? My ears certainly don't seem to agree.
>>> Past a certain point the equivalencies aren't going to make
>>> any differences to you, and there is another sort of complexity
>> bound
>>> to think about.
>>
>> I thought this was the only kind. Can you elaborate?
>
> If |a b c d> is a 7-limit monzo, the symmetrical lattice norm
> (seminorm, if we are including 2) is
> sqrt(b^2 + c^2 + d^2 + bc + bd + cd), and this may be viewed as its
> complexity in terms of harmonic relationships of 7-limit chords. How
> many consonant intervalsteps at minimum are needed to get there is
> another and related measure.
I think the Tenney lattice is pretty ideal for this, because
progressing by simpler consonances is more comprehensible and thus
allows for longer chord progressions with the same subjective
complexity.
Message: 10134 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 21:16:45
Subject: Re: !
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> The circle rocks, dude. It penalizes temperaments equally for
>>> trading too much of their error for complexity, or complexity
>>> for error. Look
>>> at the plots, and the first things you hit are 19, 12, and 53.
>>> And 22 in the 7-limit. Further, my suggestion that 1cents = zero
>>> should satisfy Dave's micro fears. Or make 0 cents = zero. It
>>> works either way.
>>
>> It does? Look at the graph! How can you make 0 cents = zero when
>> it's infinitely far away?
>
> I thought I had a way to fudge it by adding a constant later,
> but I can't remember it at the moment.
Let me know when you do.
>> And what about the position of the origin on the
>> *complexity* axis??
>
> I already answered that.
Where? I didn't see anything on that, but I could have misunderstood
something.
>> 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. The base of the horizontal axis is arbitrary, so the scaling
between the two is arbitrary. But then the loglog plots take an
additional log of both (I've been showing power-of-10 tick marks),
and in that case, I suppose you're right that the units aren't
arbitrary relative to one another . . . Unfortunately I haven't been
drawing them with the same scaling on both axes, as you can easily
see. So if your circle was based on looking at the graphs, it would
become a highly eccentric ellipse when the two logarithmic axes
actually do use the same scaling.
> You mean c is
> arbitrary in y = x + c?
Not what I meant, but this is the equation of a line, not a circle.
Message: 10135 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 06:49:56
Subject: acceptace regions
From: Gene Ward Smith
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)
We can obtain a fine list simply by taking everything in the first
quadrant and leaving the rest. Morover, while the cornet here is not
sharp, if we want to smooth it we can easily accomodate such a desire
by taking everything above a hyperpola uv = constant in the first
quadrant--in other words, use uv as a goodness function, and insist
on a goodness higher than zero.
Think the resulting list is too small? Try moving the origin
elsewhere, by setting
u' = A - ln(complexity) - ln(error)
v' = B - 4 ln(complexity) - ln(error)
Still unhappy? I think the slopes of -1 and -4 I use work well, but
you could try changing slopes *and* origins in order to better get
what you think is a moat, or are willing to claim is one.
I think a uv plot of 7-limit linears would be interesting. I'd also
like some kind of feedback, so I don't get the feeling I am talking
to myself here.
Message: 10136 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 20:29:39
Subject: Re: The same page
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>> Anybody have a handy asci 'units' table for popular wedge
>>>>> products in ket notation? ie,
>>>>>
>>>>> [ val > ^ [ val > -> [[ wedgie >>
>>>>> < monzo ] ^ < monzo ] -> ?
>>>>
>>>> >>> [monzo> ^ [monzo> -> ||bimonzo>>
>>>
>>> Great, so what happens when the monzos are commas being
>>> tempered out?
>>
>> That's what they always represent here.
>
> Yes of course, but in that case, what does the bimonzo give
> us? Anything musical?
Sure; in the 5-limit it gives the periodicity block, and so on.
>>> A chart running over comma useful things would help our
>>> endeavor tremendously.
>>
>> What would you like to see?
>
> A dummy chart for what I need to wedge in order to get what
> I care about about temperaments.
Can I see an example of what you have in mind?
Message: 10137 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 21:26:18
Subject: Re: The same page
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>>>> Anybody have a handy asci 'units' table for popular wedge
>>>>>>> products in ket notation? ie,
>>>>>>>
>>>>>>> [ val > ^ [ val > -> [[ wedgie >>
>>>>>>> < monzo ] ^ < monzo ] -> ?
>>>>>>
>>>>>> >>>>> [monzo> ^ [monzo> -> ||bimonzo>>
>>>>>
>>>>> Great, so what happens when the monzos are commas being
>>>>> tempered out?
>>>>
>>>> That's what they always represent here.
>>>
>>> Yes of course, but in that case, what does the bimonzo give
>>> us? Anything musical?
>>
>> Sure; in the 5-limit it gives the periodicity block, and so on.
>>
>>>>> A chart running over comma useful things would help our
>>>>> endeavor tremendously.
>>>>
>>>> What would you like to see?
>>>
>>> A dummy chart for what I need to wedge in order to get what
>>> I care about about temperaments.
>>
>> Can I see an example of what you have in mind?
>
> Above! For all operations one would want to do. With templates
> for dual and every other damn thing that can be done to a vector by
> flipping signs, rearranging elements, and other trivial operations.
> If I could do any better than this I'd make the thing myself!
Is this a start? ~= will mean "equal when one side is complemented".
2 primes:
<val] ~= [monzo>
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.
Message: 10138 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 16:30:04
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>>>> >'m not.
>>>
>>> Then why are you suddenly silent on all this?
>>
>> Huh? I've been posting at a record rate.
>
>Not on this subject of cognitive limits that used to occupy you so.
Those apply to scales, not tunings. Ideally the paper would show
how to use the tools of temperament to find both. But that's up
to you guys. Dave doesn't seem to want the macros which would
be necessary for the scale-building stuff.
>>>> It is well known that Dave, for example, is far more
>>>> micro-biased than I!
>>>
>>> ?
>>
>> What's your question?
>
>What does micro-biased mean, on what basis do you say this about you
>vs. Dave, and what is its relevance here?
Micro-biased means biased in favor of microtemperaments. I've
historically fought for macros vs. Dave. But in general if it
ever appears that I'm taking a side on any of these lists, please
stop adn consider that I rarely do so -- I sometimes appear to
do so if a position hasn't been _explored_ to my satisfaction.
-Carl
Message: 10139 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 18:18:05
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>>> >umans seem to find a particular region of complexity and error
>>> attractive and have a certain approximate function relating error and
>>> complexity to usefulness. Extra-terrestrial music-makers (or humpback
>>> whales) may find completely different regions attractive.
>>
>> This seems to be the key statement of this thread. I don't think
>> this has been established. If it had, I'd be all for it. But it
>> seems instead that whenever you cut out temperament T, somebody
>> could come along and do something with T that would make you wish
>> you hadn't have cut it. Therefore it seems logical to use something
>> that allows a comparison of temperaments in any range (like logflat).
>
>So Carl. You really think it's possible that some human musician
>could find the temperament where 3/2 vanishes to be a useful
>approximation of 5-limit JI (but hey at least the complexity is
>0.001)? And likewise for some temperament where the number of
>generators to each prime is around a google (but hey at least the
>error is 10^-99 cents)?
This is a false dilemma. The size of this thread shows how hard
it is to agree on the cutoffs.
>> Then no matter what T is, we can say...
>>
>> "You could have used U, which is in the same range but better."
>>
>> ...or...
>>
>> "T's the best in that range. Bravo!"
>>
>> ...The worst that could happen would seem to be...
>>
>> "T falls outside the range we established for our paper, sorry."
>>
>> ...in which case the reader could perform his own analysis in the
>> above way. With a cooked acceptance region, however, the following
>> could happen...
>>
>> "Oh, T. It didn't meet our guesses about human cognition, but YMMV."
>
>I don't understand why you think log-flat is a magic bullet in this
>regard. If you use log flat badness and include the same number of
>temperaments as Paul and I and Gene are considering (around 20), then
>exactly the same scenario is possible, only this time it will be
>temperaments with moderate amounts of both error and complexity that
>are omitted and the objecting musician won't be fictitious, he'll be
>Herman Miller.
Can you name the temperaments that fell outside of the top 20 on
Gene's 114 list?
-Carl
Message: 10140 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 20:31:07
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> The rectangle enclosed by error and complexity bounds. You
answered
>>> that the axes were infinitely far away, but the badness line AB
>>> doesn't seem to be helping that.
>>
>> If you simply bound complexity alone, you get a finite number of
>> temperaments. Most are complete crap.
>
> Above I suggest a rectangle which bounds complexity and error, not
> complexity alone.
>
> In the circle suggestion I suggest a circle plus a complexity bound
> is sufficient.
Can you give an example of the latter?
Message: 10141 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 21:27:15
Subject: 310 7-limit linear temperaments in the u-v plane
From: Gene Ward Smith
Below I list the wedgie, u and v for the 310 temperaments on my list of
32000-odd possibles for which u>-2 and v>-2. Here if C is the
complexity and E is the error, I set x=ln(C) and y=ln(E). Then u=4-x-y
and v=12-4x-y. It would in my view be very useful to see a plot of this.
The first coordinate of the pair, which I have given as [u, v], is
unforgiving of error. The second is unforgiving of complexity. I could
have sorted the list according to either, but instead used
g = 2/3 u + 1/3 v. This is a goodness measure of the infamous log-flat
variety; we have g = 20/3 - 2x - y, so exp(-g) = K (E C^2) where K =
exp(-20/3).
[18, 27, 18, 1, -22, -34] [3.629230331, .575465612]
[37, 46, 75, -13, 15, 45] [3.406220860, -1.874710583]
[22, -5, 3, -59, -57, 21] [2.714891805, -.970728663]
[19, 19, 57, -14, 37, 79] [2.908019068, -1.601883155]
[4, -32, -15, -60, -35, 55] [2.722577966, -1.258540154]
[1, -8, 39, -15, 59, 113] [2.633249246, -1.257146728]
[1, 4, 10, 4, 13, 12] [1.005097996, 1.609665600]
[6, -7, -2, -25, -20, 15] [1.411065061, .2629785497]
[16, 2, 5, -34, -37, 6] [1.736866439, -.585529481]
[1, 4, -2, 4, -6, -16] [.363511386, 2.141744135]
[0, 5, 0, 8, 0, -14] [.152491081, 2.548674345]
[5, 1, 12, -10, 5, 25] [1.012524503, .7830372729]
[2, 3, 1, 0, -4, -6] [-.205382324, 3.133144155]
[0, 0, 3, 0, 5, 7] [-.589897843, 3.859219804]
[2, 1, 3, -3, -1, 4] [-.308576499, 3.236590508]
[2, -4, -4, -11, -12, 2] [.524469574, 1.498444023]
[7, 9, 13, -2, 1, 5] [.8521893702, .8383344292]
[6, 5, 22, -6, 18, 37] [1.398998141, -.2728789107]
[0, 2, 2, 3, 3, -1] [-.666311581, 3.828729871]
[0, 0, 2, 0, 3, 5] [-1.114694760, 4.583757375]
[4, 4, 4, -3, -5, -2] [.160875338, 1.953852436]
[1, -8, -14, -15, -25, -10] [1.080908962, .5026039173e-1]
[7, -3, 8, -21, -7, 27] [1.060730636, .7657743908e-1]
[2, 25, 13, 35, 15, -40] [1.630631612, -1.092923122]
[1, -1, 3, -4, 2, 10] [-.297159006, 2.756450471]
[1, 2, 1, 1, -1, -3] [-.933260604, 3.972326987]
[1, 1, 0, -1, -3, -3] [-1.067288298, 4.222744655]
[1, 2, 3, 1, 2, 1] [-.742852499, 3.554638596]
[5, 13, -17, 9, -41, -76] [1.646019337, -1.276833941]
[3, 0, 6, -7, 1, 14] [.113185821, 1.762756409]
[13, 14, 35, -8, 19, 42] [1.661188320, -1.374233657]
[1, -1, -2, -4, -6, -2] [-.540654341, 2.936334354]
[1, -1, 0, -4, -3, 3] [-.872793868, 3.560962162]
[3, 0, -6, -7, -18, -14] [.426316706, .940456192]
[1, 4, 3, 4, 2, -4] [-.374818183, 2.471764627]
[4, 2, 2, -6, -8, -1] [-.74238576e-1, 1.810219121]
[2, 1, -1, -3, -7, -5] [-.475239786, 2.576382050]
[2, 8, 1, 8, -4, -20] [.283811920, 1.034875923]
[4, -3, 2, -14, -8, 13] [.345586529, .859876933]
[10, 9, 7, -9, -17, -9] [.8297942242, -.2058606098]
[6, 5, 3, -6, -12, -7] [.322424097, .767227201]
[2, 1, 6, -3, 4, 11] [-.218755495, 1.845931213]
[1, 9, -2, 12, -6, -30] [.4535440254, .4454275914]
[2, 8, 8, 8, 7, -4] [.245867248, .811824633]
[0, 0, 7, 0, 11, 16] [-.248247752, 1.739146134]
[9, 5, -3, -13, -30, -21] [.8424961778, -.4842422682]
[1, -3, -4, -7, -9, -1] [-.280612685, 1.704441303]
[6, 10, 10, 2, -1, -5] [.287044794, .548744901]
[3, 17, -1, 20, -10, -50] [.8955276833, -.7518398037]
[12, 34, 20, 26, -2, -49] [1.462830432, -1.914845007]
[24, 32, 40, -5, -4, 3] [1.483539340, -1.995207689]
[0, 0, 4, 0, 6, 9] [-.922056519, 2.805750119]
[15, -2, -5, -38, -50, -6] [1.191031885, -1.439618129]
[8, 18, 11, 10, -5, -25] [.8374157707, -.7456108813]
[3, 12, -1, 12, -10, -36] [.5552853406, -.1895850224]
[0, 0, 5, 0, 8, 12] [-.677810269, 2.251962805]
[0, 12, 24, 19, 38, 22] [1.055447382, -1.226619823]
[0, 0, 12, 0, 19, 28] [.178510293, .520800106]
[1, 4, -9, 4, -17, -32] [.3364294610, .1379787620]
[3, -12, -30, -26, -56, -36] [1.323782432, -1.954253378]
[2, 6, 6, 5, 4, -3] [-.291711764, 1.217976987]
[2, -1, 1, -6, -4, 5] [-.787640236, 2.183807521]
[8, 13, 23, 2, 14, 17] [.7514403927, -.9215592083]
[2, -4, -16, -11, -31, -26] [.6436652400, -.7139691630]
[2, -2, 1, -8, -4, 8] [-.562262654, 1.696035879]
[3, 5, 9, 1, 6, 7] [-.181870640, .926592561]
[2, 1, -4, -3, -12, -12] [-.349909310, 1.240369407]
[3, -24, -1, -45, -10, 65] [1.109082908, -1.726264787]
[1, -1, 1, -4, -1, 5] [-1.257426652, 2.987275426]
[12, -2, 20, -31, -2, 52] [.9685827999, -1.466539439]
[11, 13, 17, -5, -4, 3] [.5451289967, -.6327346883]
[3, 5, -6, 1, -18, -28] [.176167209, .93101647e-1]
[12, 22, -4, 7, -40, -71] [1.168736181, -1.916947188]
[6, 0, 0, -14, -17, 0] [-.144103110, .552331535]
[11, 18, 5, 3, -23, -39] [.7393125860, -1.304148322]
[2, -2, -2, -8, -9, 1] [-.648390366, 1.435403570]
[17, 6, 15, -30, -24, 18] [.9225671470, -1.740042662]
[4, 16, 9, 16, 3, -24] [.4104041316, -.7691098524]
[4, 21, -3, 24, -16, -66] [.8279921578, -1.650503107]
[1, -3, -2, -7, -6, 4] [-.796717686, 1.588083020]
[3, 5, 1, 1, -7, -12] [-.580823747, 1.147950612]
[12, 10, -9, -12, -48, -49] [.9283293931, -1.875845700]
[8, 6, 6, -9, -13, -3] [.68663257e-1, -.172409567]
[5, -11, -12, -29, -33, 3] [.6475907742, -1.340799742]
[6, -2, -2, -17, -20, 1] [-.2200063e-2, -.50795881e-1]
[3, 12, 11, 12, 9, -8] [.1733958065, -.4114685585]
[1, 4, 5, 4, 5, 0] [-.842313783, 1.614549170]
[6, 0, 3, -14, -12, 7] [-.186135940, .253731850]
[2, -9, -4, -19, -12, 16] [.110765143, -.367098214]
[9, 10, -3, -5, -30, -35] [.4841383122, -1.114817471]
[3, 2, 4, -4, -2, 4] [-.878126390, 1.586446158]
[4, -8, 14, -22, 11, 55] [.5972396400, -1.382584001]
[1, -8, -2, -15, -6, 18] [-.167272250, .143772795]
[6, 10, 3, 2, -12, -21] [.4101414e-2, -.242117955]
[13, 19, 23, 0, 0, 0] [.4556158489, -1.149903551]
[1, -3, 5, -7, 5, 20] [-.474868918, .678598615]
[5, 3, 7, -7, -3, 8] [-.383860087, .493175395]
[8, 1, 18, -17, 6, 39] [.4064523106, -1.107722833]
[3, 0, -3, -7, -13, -7] [-.584010116, .871684734]
[12, 10, 25, -12, 6, 30] [.6281053654, -1.574944233]
[2, 3, 6, 0, 4, 6] [-.906409329, 1.473314402]
[3, 0, 9, -7, 6, 21] [-.333176615, .321368661]
[3, -7, -8, -18, -21, 1] [.71689763e-1, -.518553272]
[1, -3, 3, -7, 2, 15] [-.762855535, 1.135262755]
[1, -1, -5, -4, -11, -9] [-.745754672, 1.098651594]
[6, -12, 10, -33, -1, 57] [.6101417586, -1.638299992]
[2, -6, 1, -14, -4, 19] [-.348997990, .241608334]
[1, -13, -2, -23, -6, 32] [.1433815335, -.7600710565]
[3, -5, -6, -15, -18, 0] [-.167754823, -.149512117]
[1, 6, 5, 7, 5, -5] [-.655893330, .823648016]
[2, 8, -11, 8, -23, -48] [.3204117006, -1.129079582]
[5, 8, 2, 1, -11, -18] [-.285071625, .62437310e-1]
[0, 0, 19, 0, 30, 44] [.1665485838, -.8594312212]
[2, 5, 3, 3, -1, -7] [-.961974750, 1.377881885]
[5, 1, -7, -10, -25, -19] [.826121e-3, -.556383277]
[5, 8, 14, 1, 8, 10] [-.142165912, -.304220656]
[7, 9, 32, -2, 31, 49] [.6797859680, -1.962055240]
[8, 1, -13, -17, -43, -33] [.5073592316, -1.641189793]
[3, 5, 16, 1, 17, 23] [-.23980789e-1, -.587567616]
[11, -6, 10, -35, -15, 40] [.5818775046, -1.825094348]
[4, 2, 9, -6, 3, 15] [-.516138894, .237011021]
[3, 2, -1, -4, -10, -8] [-.903474981, .983810996]
[3, 7, -1, 4, -10, -22] [-.467662952, .84355164e-1]
[4, -8, -20, -22, -43, -24] [.5251916022, -1.910857084]
[14, 11, 9, -15, -25, -10] [.3765010353, -1.620633894]
[4, 4, 0, -3, -11, -11] [-.737971277, .597488784]
[0, 0, 5, 0, 8, 11] [-1.299404555, 1.716508494]
[8, -11, -6, -36, -32, 17] [.4773026098, -1.873039923]
[4, 9, 26, 5, 30, 35] [.3885253755, -1.726927868]
[4, 9, 19, 5, 19, 19] [.938231575e-1, -1.139670180]
[5, -4, -10, -18, -30, -12] [.903299045e-1, -1.171003260]
[2, 13, 1, 16, -4, -34] [-.129567301, -.746767354]
[6, 7, 5, -3, -9, -8] [-.481343559, -.93481826e-1]
[1, 7, 3, 9, 2, -13] [-.760448506, .453124399]
[0, 0, 8, 0, 13, 19] [-.867601868, .647228967]
[3, 12, 18, 12, 20, 8] [.730840107e-1, -1.261518978]
[0, 5, 10, 8, 16, 9] [-.492128352, -.155852317]
[11, 6, 15, -16, -7, 18] [.1122979282, -1.365027872]
[1, -6, 3, -12, 2, 24] [-.594540504, .32941526e-1]
[9, -2, 14, -24, -3, 38] [.1476387417, -1.454725281]
[6, 0, 15, -14, 7, 35] [-.67879467e-1, -1.036328496]
[4, 2, 5, -6, -3, 6] [-.930494064, .687971897]
[0, 5, -5, 8, -8, -26] [-.582128162, -.10439184e-1]
[4, 9, -8, 5, -24, -44] [.527903825e-1, -1.321161350]
[2, -4, 8, -11, 7, 30] [-.366759238, -.490652025]
[6, -12, -2, -33, -20, 29] [.2396727699, -1.711253700]
[4, 9, -15, 5, -35, -60] [.3295592714, -1.891195103]
[10, 2, 2, -20, -25, -1] [-.135958369, -.983381730]
[4, 2, -1, -6, -13, -8] [-.840404798, .399233499]
[9, 0, 9, -21, -11, 21] [-.88241812e-1, -1.117777876]
[3, -19, -1, -37, -10, 51] [.3203051393, -1.938482989]
[2, -6, -6, -14, -15, 3] [-.490643293, -.324972878]
[2, -11, 1, -22, -4, 33] [-.208605058, -.928487618]
[4, 7, 2, 2, -8, -15] [-.729973553, .105553053]
[10, 14, 14, -1, -6, -7] [-.130239572, -1.120392559]
[6, 5, 7, -6, -6, 2] [-.669182904, -.52572019e-1]
[10, 2, 5, -20, -20, 6] [-.144935409, -1.102210677]
[13, 9, 1, -16, -35, -23] [.2040258001, -1.811428592]
[4, 4, -8, -3, -24, -30] [-.250244066, -.905880539]
[0, 12, 12, 19, 19, -6] [-.153602558, -1.110036858]
[4, 11, 2, 8, -8, -26] [-.392976455, -.698339145]
[0, 0, 10, 0, 16, 23] [-.798950098, .94142260e-1]
[0, 0, 9, 0, 14, 21] [-.910594332, .312296123]
[10, 7, 5, -12, -20, -8] [-.171649456, -1.172003014]
[0, 5, 5, 8, 8, -2] [-1.069386426, .613463954]
[4, 4, 16, -3, 14, 26] [-.293166681, -.946892724]
[3, -7, 11, -18, 9, 45] [-.613254898e-1, -1.420879511]
[5, -6, 7, -21, -3, 33] [-.203779439, -1.141852673]
[4, 14, 14, 13, 11, -7] [-.159713909, -1.238289909]
[4, -8, 2, -22, -8, 27] [-.264479353, -1.033889617]
[9, -7, 4, -32, -19, 29] [.1269925897, -1.819072876]
[2, -4, 18, -11, 23, 53] [.546965533e-1, -1.683034542]
[3, 10, 9, 9, 6, -7] [-.543080986, -.518248827]
[1, 9, 3, 12, 2, -18] [-.712851570, -.179354638]
[5, -4, 2, -18, -11, 16] [-.469100176, -.673676890]
[3, -2, 1, -10, -7, 8] [-1.050697330, .484233616]
[2, -4, 6, -11, 4, 25] [-.655813167, -.314681978]
[4, -1, 9, -11, 3, 24] [-.608340692, -.417591882]
[6, 3, 5, -9, -9, 3] [-.746637222, -.147651739]
[5, 1, -10, -10, -30, -26] [-.200994740, -1.276223819]
[6, 3, 1, -9, -15, -6] [-.729419044, -.226572839]
[9, 0, -3, -21, -30, -7] [-.221482197, -1.248814995]
[7, 4, 10, -10, -4, 12] [-.513527003, -.669199578]
[9, 15, 19, 3, 5, 2] [-.181219698, -1.336445983]
[9, 17, 14, 6, -3, -15] [-.128303853, -1.481158664]
[1, 4, -21, 4, -36, -60] [.934456477e-1, -1.926243230]
[2, 8, 13, 8, 15, 8] [-.469057709, -.807412748]
[8, 8, 4, -6, -16, -13] [-.407897003, -.931833274]
[7, 2, 15, -13, 4, 29] [-.279250412, -1.190689962]
[5, 18, 17, 17, 13, -11] [.146447901e-1, -1.781586163]
[7, 4, -2, -10, -23, -16] [-.407040381, -.940462937]
[6, 12, 12, 5, 2, -6] [-.485397722, -.795045536]
[2, 13, -4, 16, -12, -46] [-.149758098, -1.475182572]
[8, 13, 4, 2, -16, -27] [-.245101239, -1.284705873]
[5, 20, 0, 20, -14, -56] [.1050152716, -1.990665292]
[1, -6, -2, -12, -6, 12] [-.925443902, .55637649e-1]
[3, -3, 1, -12, -7, 11] [-.957172730, .116395575]
[13, 14, 16, -8, -11, -2] [-.586887779e-1, -1.701323570]
[5, 1, 9, -10, 0, 18] [-.775406912, -.296839099]
[3, 12, 6, 12, 1, -20] [-.531950950, -.802817614]
[0, 0, 15, 0, 24, 35] [-.513636132, -.851260538]
[4, 16, 21, 16, 22, 4] [-.78596493e-2, -1.873118974]
[4, 4, 12, -3, 8, 17] [-.679180114, -.540549884]
[8, -1, 4, -20, -16, 12] [-.444783364, -1.078038217]
[5, 6, -3, -2, -19, -24] [-.617424790, -.801516088]
[0, 10, 10, 16, 16, -5] [-.542587957, -.967067536]
[1, 14, -2, 20, -6, -44] [-.312833355, -1.432919432]
[6, 5, -12, -6, -36, -42] [-.950792365e-1, -1.912639830]
[1, -3, -11, -7, -20, -17] [-.673850885, -.767068718]
[1, 4, 22, 4, 32, 40] [-.1693814110, -1.778259566]
[0, 7, 7, 11, 11, -3] [-.943824367, -.252435548]
[5, 11, 7, 6, -3, -15] [-.680975735, -.803381348]
[2, 11, 6, 13, 4, -17] [-.688064282, -.847196060]
[0, 7, 14, 11, 22, 13] [-.524527117, -1.183832259]
[12, 0, 3, -28, -29, 7] [-.249740543, -1.763772806]
[3, 5, -13, 1, -29, -44] [-.346032864, -1.648490772]
[12, 10, 10, -12, -18, -5] [-.291080286, -1.763755419]
[2, -2, 4, -8, 1, 15] [-1.347075171, .323104394]
[9, 15, 4, 3, -19, -33] [-.295423014, -1.793259243]
[8, 8, 16, -6, 3, 15] [-.533863321, -1.335099102]
[3, 0, 21, -7, 25, 49] [-.273424726, -1.858509536]
[2, 6, -4, 5, -12, -26] [-.889032202, -.655693165]
[5, -9, -10, -26, -30, 2] [-.274449954, -1.895682348]
[0, 0, 22, 0, 35, 51] [-.326185152, -1.800558870]
[10, -5, 0, -31, -28, 14] [-.255187679, -1.950767071]
[7, 2, 18, -13, 9, 36] [-.379591474, -1.755590519]
[5, 6, 15, -2, 10, 18] [-.696144582, -1.130503207]
[0, 7, -7, 11, -11, -36] [-.709612161, -1.118256842]
[3, 0, -18, -7, -37, -42] [-.295882202, -1.948339436]
[7, 9, -6, -2, -29, -39] [-.358001449, -1.825655743]
[4, -1, 6, -11, -2, 17] [-1.104659762, -.335771190]
[9, 7, 11, -10, -8, 6] [-.622976626, -1.301922259]
[9, -2, -5, -24, -33, -6] [-.385338344, -1.815095725]
[8, 18, 18, 10, 6, -9] [-.338613285, -1.953887409]
[0, 5, 15, 8, 24, 21] [-.640548326, -1.358909077]
[0, 3, 6, 5, 10, 6] [-1.480169567, .298132908]
[0, 10, -5, 16, -8, -40] [-.645134209, -1.377252546]
[3, -14, -1, -29, -10, 37] [-.376708599, -1.921119003]
[3, 0, -9, -7, -23, -21] [-.841397568, -1.042826585]
[5, 3, -2, -7, -17, -13] [-1.022478254, -.688601470]
[2, -16, 1, -30, -4, 47] [-.368807442, -1.997698949]
[4, 6, 11, 0, 6, 9] [-1.115853383, -.508740081]
[2, -4, -6, -11, -15, -3] [-1.100447199, -.540671689]
[7, 11, 3, 1, -15, -24] [-.719265915, -1.313960440]
[8, -4, -4, -25, -29, 2] [-.512686428, -1.760131799]
[1, -3, 12, -7, 16, 36] [-.752771862, -1.290666263]
[10, 14, 22, -1, 7, 12] [-.426849901, -1.943606890]
[3, 10, -6, 9, -18, -42] [-.544035928, -1.716057489]
[2, -14, -4, -27, -12, 30] [-.463119391, -1.915165091]
[10, 12, 12, -4, -9, -6] [-.630242442, -1.584551407]
[1, -8, 3, -15, 2, 29] [-.972667832, -.915282941]
[5, 11, 15, 6, 10, 4] [-.759758953, -1.384960687]
[5, 15, 0, 12, -14, -42] [-.540714112, -1.876606374]
[6, 10, 17, 2, 10, 11] [-.732738351, -1.506453601]
[3, 8, 7, 6, 3, -6] [-1.262056591, -.471165680]
[9, 0, -6, -21, -35, -14] [-.507327921, -1.982866831]
[9, 0, 6, -21, -16, 14] [-.708661547, -1.583606112]
[6, -5, 3, -22, -12, 21] [-.730995567, -1.549129140]
[3, -5, 9, -15, 6, 35] [-.777601343, -1.456330247]
[12, 8, 8, -15, -21, -4] [-.524433549, -1.966617243]
[0, 14, 7, 22, 11, -23] [-.658590100, -1.720084168]
[2, -6, 8, -14, 7, 35] [-.825941518, -1.402635443]
[7, -3, -1, -21, -21, 6] [-.827922635, -1.399236329]
[4, -8, 4, -22, -5, 32] [-.735636348, -1.632194308]
[6, 0, -9, -14, -31, -21] [-.661554479, -1.792866842]
[1, -3, -9, -7, -17, -12] [-1.165916763, -.788936577]
[3, 10, 11, 9, 9, -3] [-1.004923763, -1.138316014]
[3, -9, -1, -21, -10, 23] [-.852450521, -1.457373275]
[9, 5, 12, -13, -6, 14] [-.774419582, -1.628957306]
[8, -2, 1, -22, -21, 8] [-.844690583, -1.547280400]
[4, -6, -6, -19, -21, 3] [-.881851174, -1.483439323]
[10, 11, 9, -6, -14, -10] [-.735143805, -1.805689360]
[9, 12, 6, -2, -16, -20] [-.755596135, -1.771344462]
[5, -1, 1, -13, -12, 5] [-1.369221501, -.545931199]
[6, 9, 7, 0, -6, -9] [-1.207564144, -.875583127]
[0, 10, 15, 16, 24, 7] [-.753152571, -1.809326057]
[1, -8, 10, -15, 13, 46] [-.699344503, -1.930288508]
[6, 10, 11, 2, 1, -2] [-1.205596539, -.920142499]
[5, 1, 15, -10, 10, 32] [-.843892655, -1.645820618]
[4, -3, 11, -14, 6, 34] [-.887790747, -1.575956489]
[4, 4, -12, -3, -30, -39] [-.694993988, -1.976473736]
[4, 11, -3, 8, -16, -38] [-.816278177, -1.792813061]
[0, 0, 16, 0, 25, 37] [-.980434480, -1.479020967]
[4, -1, -5, -11, -19, -9] [-1.220423797, -1.000632331]
[0, 10, 5, 16, 8, -17] [-1.126564346, -1.218812557]
[1, 7, -2, 9, -6, -24] [-1.361435968, -.774343958]
[9, 15, 12, 3, -6, -14] [-.803857576, -1.897789038]
[2, -6, 3, -14, -1, 24] [-1.304397911, -.911505494]
[3, -9, 6, -21, 1, 39] [-.865897490, -1.825989668]
[1, 7, 11, 9, 15, 6] [-1.201373732, -1.182376976]
[3, -7, -6, -18, -18, 6] [-1.035465007, -1.517700429]
[10, 7, 9, -12, -14, 1] [-.915901457, -1.763460124]
[3, 14, 13, 15, 12, -9] [-.835318975, -1.926654079]
[7, 12, 3, 3, -15, -27] [-.936140254, -1.766554710]
[7, 1, 5, -15, -12, 9] [-1.187636720, -1.267808925]
[3, -9, -3, -21, -13, 18] [-.996214423, -1.704890805]
[8, 4, 1, -12, -21, -9] [-1.107189953, -1.514968038]
[4, 4, -4, -3, -18, -21] [-1.297607496, -1.161464948]
[2, -6, -13, -14, -26, -13] [-.920819337, -1.925723954]
[2, -7, 6, -16, 4, 34] [-1.116778791, -1.594522890]
[9, 2, -1, -18, -27, -8] [-.999716722, -1.846002663]
[3, -3, 2, -12, -5, 13] [-1.619763945, -.609195091]
[0, 5, -10, 8, -16, -37] [-1.125392889, -1.653364917]
[6, 13, 8, 7, -4, -18] [-1.124890570, -1.761890308]
[8, -2, 4, -22, -16, 15] [-1.057295717, -1.923373901]
[6, 8, -2, -1, -20, -27] [-1.217476604, -1.690891501]
[7, 2, 12, -13, -1, 22] [-1.241598007, -1.685048822]
[5, -4, 3, -18, -9, 18] [-1.334427835, -1.580298079]
[3, -9, 4, -21, -2, 34] [-1.192996665, -1.956894453]
[6, 2, -4, -11, -23, -15] [-1.436567969, -1.852273705]
Message: 10142 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 00:02:50
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> Unfortunately, we can't publish an infinitely long paper.
You don't get an infinite list in this way for slopes less than the
critical exponent.
I've been
> working extremely hard to wrap my brain around your nearly
> impenetrable posts for years, going way out on a limb to defend you
> against your detractors, and hailing you as the savior of our cause.
In the last week, however, you've demoted me to being an annoying
idiot. Can you think about my hyperbola plan and tell me if I should
bother?
Message: 10143 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 06:52:28
Subject: Re: Gene: contact monz
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi Gene,
>
>
> i'm only writing here because i know that you'll
> read it before you read your regular emails ...
> apologies to other list members.
>
> please contact me ASAP.
> monz@t...
Hi Joe. If you sent me email when you posted this, it didn't get to
me.
________________________________________________________________________
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Message: 10144 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 16:34:51
Subject: Re: The same page
From: Carl Lumma
>>>>>>> >nybody have a handy asci 'units' table for popular wedge
>>>>>>> products in ket notation? ie,
>>>>>>>
>>>>>>> [ val > ^ [ val > -> [[ wedgie >>
>>>>>>> < monzo ] ^ < monzo ] -> ?
>>>>>>
>>>>>> >>>>> [monzo> ^ [monzo> -> ||bimonzo>>
>>>>>
>>>>> Great, so what happens when the monzos are commas being
>>>>> tempered out?
>>>>
>>>> That's what they always represent here.
>>>
>>> Yes of course, but in that case, what does the bimonzo give
>>> us? Anything musical?
>>
>> Sure; in the 5-limit it gives the periodicity block, and so on.
>>
>>>>> A chart running over comma useful things would help our
>>>>> endeavor tremendously.
>>>>
>>>> What would you like to see?
>>>
>>> A dummy chart for what I need to wedge in order to get what
>>> I care about about temperaments.
>>
>> Can I see an example of what you have in mind?
>
>Above! For all operations one would want to do. With templates
>for dual and every other damn thing that can be done to a vector by
>flipping signs, rearranging elements, and other trivial operations.
>If I could do any better than this I'd make the thing myself!
Sorry to be so hasty here. Like any true addict, I was making
myself late for my interview. I managed to arrive on time, thanks
to favorable traffic conditions, and it went well.
I meant to add that scenarios for different dimensionalities
should be included. For example, do I need to wedge three vals
to get a 7-limit codimenision 1 temperament, or...? etc.
-Carl
Message: 10145 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 20:35:10
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>
>>> so the whole
>>> premise we've all been operating under can be questioned by
someone
>>> who is interested in the character of the commas in the kernel,
not
>>> what complexity they give.
>>
>> Please elaborate on this point of view -- I'm not seeing it.
>
> You can look at meantone as something which gives nice triads, as a
> superior system because it has fifths for generators, as a nice deal
> because of a low badness figure. Or, you can say, wow, it has 81/80,
> 126/125 and 225/224 all in the kernel, and look what that implies.
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.
> The
> last point of view has nothing directly to do with error and
> complexity, though the relationship is a close one when analyzed. As
> you move to lower-error systems, your interest in the error per se
> falls off, and complexity from the point of view of a vast
conceptual
> keyboard not too interesting--but oh, those commas!
Example?
> That's where the
> action is in some ways, and more so as we increase the prime limit
and
> we have commas up the wazoo. Ennealimmal is not just low-error, it
has
> commas which are still in the useful range.
It has 4375:4374 in the kernel. Look what that implies. Umm . . .
what (musically useful thing)?
Message: 10146 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 21:29:13
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
>> I rely on you for that. Can you possibly believe my track record for
>> working out the logic of a proposal is not a bad one, and that if I
> am
>> saying something it might be worth thinking about?
>
> I've been thinking about it for years, and mostly supporting it. It's
> just that I think Dave and Graham should both be in on this, and we
> were going to lose Dave entirely if we didn't at least try to address
> his objections. I'm hoping this process will continue, whenever Dave
> gets back.
Hey Paul, I assume your recent change of mind on this stuff wasn't
just so you wouldn't "lose" me. I certainly never made any threats of
that kind.
Part of the reason I suggested recently that you go ahead without me,
was the difficulty I sometimes have communicating with Gene in a civil
manner, but also because I'm starting some new (paying) work where I'm
going to have a lot less time for this list, and won't be able to
continue to follow discussions as closely as the current one. I'm
already working two days a week at this new job (Wed & Fri) and will
be doing it four days a week starting next week (Tue to Fri).
Gene, I feel I've been bending over backwards to accomodate you
recently and I don't think I can keep it up. I have found it
extraordinary how often you have been unable to find plots or figure
out their axes, even after I have posted that information in response
to questions by Carl (but I agree Paul should have labelled the axes
unless the software made it difficult). And yet you seem to think it's
fine to post lists of numbers with no column headings, and that it's
easy to figure them out. And we have been considering your cutoff
proposals for years. I can't for the life of me understand why you say
we haven't.
Why can't you understand that the mathematical fact that temperaments
come out with a straight edge on a log-log plot has absolutely no
bearing on which of them will be found musically useful to humans. It
is a beautiful mathematical fact and no more.
Humans seem to find a particular region of complexity and error
attractive and have a certain approximate function relating error and
complexity to usefulness. Extra-terrestrial music-makers (or humpback
whales) may find completely different regions attractive.
These are _not_ facts of mathematics, but of psychology and
physiology. We don't have much data on them, but we are far from
having _none_ as Carl hyperbolically insists.
I can certainly understand Paul's impatience. But he has agreed to
work thru it with you again from the start, so lets' see what happens
there.
Message: 10147 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 00:05:19
Subject: Re: The seven-limit lattices
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>>
>
>>> 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
>>> Could you please give an example of a hexany...
>>
>> I'm not sure what your question means re the cubeoctahedron;
>
> I guess I don't understand what "shallow holes" and "deep holes" are
A hole is centered at a point which is a local maximum for distance
from the lattice; a deep hole is a global maximum.
>>> 4. Paragraph 7. Where do the negatives in the (1/2, 1/2, 1/2)
> terms
>>> come from?
>>
>> Those are the basis elements for the lattice of mappings.
>
> Okay. Thanks. What is the significance of these forming a dot product
> with (1,4,10) and equalling "1"?
(1 4 10) is the meantone mapping to generator steps, the 1 is 1
generator step to get to a fifth.
Message: 10148 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 16:38:26
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>>>> >lternatively, then why doesn't the badness bound alone enclose a
>>>> finite triangle?
>>>
>>> Not only is it, like the rectangle, infinite in area on the loglog
>>> plot, since the zero-error line and zero-complexity lines are
>>> infinitely far away, but it actually encloses an infinite number
>>> of temperaments.
Yet on ET charts like this...
Yahoo groups: /tuning-math/files/Paul/et5loglo... * [with cont.]
...the region beneath the 7-53 diagonal is empty. Is there stuff
there you haven't plotted?
Wait -- and how can ETs appear more than once -- different maps?
That might explain different errors, but they are appearing at
different complexities too... baffling.
-Carl
Message: 10149 - Contents - Hide Contents
Date: Wed, 11 Feb 2004 18:39:23
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>>> >'d like to know what you mean by micro-biased. It may well be true,
>>> but I'd like to know.
>>
>> Of all the amazing things I've seen on these lists, the failure of
>> both you and Paul to understand the meaning of "micro-biased" is
>> possibly the most amazing.
>
>You misjudge. It wasn't failure to understand, it was carefulness in
>checking for possible misunderstandings, rather than immediately
>telling someone they are wrong. Something that surely we'd all like to
>see more of.
It's a careful balance, but in this case I suppose you did the
right thing.
>>> I don't want to include either the very high error low
>>> complexity or very high complexity low error temperaments that a
>>> log-flat cutoff alone would include.
>>
>> Yes, you are apparently centrally biased. You should like circles
>> in that case. :)
>
>Yes, I do, so far. Haven't you read that?
Yes.
>For me there are three candidates on the table at the moment. log-log
>circles or ellipses, log-log hyperbolae, and linear-linear
>nearly-straight-lines.
Can we keep log-flat on the table for the moment?