>And how about JI? Or do you just arbitrarily leave that series off
>your Pascal's triangle? If it works for all temperaments of all
>dimensions, it certainly has to work for JI.
Good point.
-Carl
Message: 10227 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:19:51
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> 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?
In some cases at least, it would be comparing lengths vs. areas vs.
volumes.
>> 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....
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
First row -- ?
Second row -- 2-limit (all octaves), ?
Third row -- 3-limit JI, 3-limit temperament, ?
Fourth row -- 5-limit JI, 5-limit LT, 5-limit ET, ?
Fifth row -- 7-limit JI, 7-limit PT, 7-limit LT, 7-limit ET, ?
Haven't really thought about what the '?'s mean -- 1 note?
>> 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.
First principles seems fine insofar as there's nothing arbitrary and
also no flagrant disagreement with known reality. Personally, I don't
advocate using badness at all.
Message: 10228 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:38:01
Subject: Re: loglog!
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>>>>> The complexity measures cannot be compared across
different
>>>>>>>> dimensionalities, any more than lengths can be compared
with
>>>>>>>> areas can be compared with volumes.
>>>>>>>
>>>>>>> Not if it's number of notes, I guess.
>>>>>>
>>>>>> What's number of notes??
>>>>>
>>>>> Complexity units.
>>>>
>>>> It's only that (or very nearly that) in the ET cases.
>>>
>>> Your creepy complexity is giving notes, clearly.
>>
>> Hmm . . .And what do you propose to use for the 5-limit linear and
>> 7-limit planar cases?
>
> 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.
> Second of all, since I do not know how to calculate it (as far as
> I can tell the details are lost in a myriad of messages between
> you and Gene), I was unaware it was undefined for these cases.
Well, I tried applying it to 5-limit linear, and posted a few
results, but even fewer looked 'creepy', unfortunately.
>>>> So it the below
>>>> still relevant?
>>>
>>> Yes! It's a fundamental question about how to view complexity.
>>> I'd be most interested in your answer.
>>
>> 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.
> We just need a way to
> compare a slice with a tube, etc.
Doesn't seem possible to do this in a fair way.
>>>>>>> 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?
> What relations, if any, do we expect, for n
> notes, as lim goes up:
For a given scale? Then this is even more different . . .
> () More dyads or fewer dyads?
Certainly not fewer.
> () More complete chords or less complete chords?
Certainly not more.
Message: 10229 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 12:18:24
Subject: Re: loglog!
From: Carl Lumma
>>>>>>> >he complexity measures cannot be compared across different
>>>>>>> dimensionalities, any more than lengths can be compared with
>>>>>>> areas can be compared with volumes.
>>>>>>
>>>>>> Not if it's number of notes, I guess.
>>>>>
>>>>> What's number of notes??
>>>>
>>>> Complexity units.
>>>
>>> It's only that (or very nearly that) in the ET cases.
>>
>> Your creepy complexity is giving notes, clearly.
>
>Hmm . . .And what do you propose to use for the 5-limit linear and
>7-limit planar cases?
First of all, what is the real name for creepy complexity? L1?
Second of all, since I do not know how to calculate it (as far as
I can tell the details are lost in a myriad of messages between
you and Gene), I was unaware it was undefined for these cases.
>>> So it the below
>>> still relevant?
>>
>> Yes! It's a fundamental question about how to view complexity.
>> I'd be most interested in your answer.
>
>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. We just need a way to
compare a slice with a tube, etc.
>>>>>> 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. What relations, if any, do we expect, for n
notes, as lim goes up:
() More dyads or fewer dyads?
() More complete chords or less complete chords?
-Carl
Message: 10230 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 15:25:23
Subject: preferred moat?
From: Carl Lumma
Yahoo groups: /tuning_files/files/Erlich/7lin2... * [with cont.]
Um, this merely encloses every et 1-23...
-Carl
Message: 10231 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:40:10
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.
>>>
>>> Yet on ET charts like this...
>>>
>>>
Yahoo groups: /tuning- * [with cont.]
math/files/Paul/et5loglog.gif
>>>
>>> ...the region beneath the 7-53 diagonal is empty.
>>
>> Your point?
>>
>>> 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 we've gone 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.
Message: 10232 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:24:30
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>> I'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.
Macros?
Anyway, since you brought up the inevitability of scales in the
thread earlier, I'd encourage you to continue with that line of
thought.
>>>>> 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.
Well hopefully you're aware of Dave's current position.
Is that what you meant by macros above? I think Dave is saying the
opposite now, saying that temperaments should be included only if we
use enough of the relevant pitches to really necessitate the
particular temperament, and that seems to imply the use of scales
associated specifically with that temperament, at least if we use
your open-ended idea of "scales".
Message: 10233 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:42:58
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> It's clearly not taxicab Tenney distance, which is what Paul has
been
> calling that.
I also call Paul Hahn's complexity in the symmetrical triangular
lattice 'taxicab'.
> It's taxicab distance with Fifth Element style flying
> taxicabs, and routes which form an A3=D3 lattice.
What do you mean by 'Element style flying taxicabs' here? I thought
the cabs drive only on the routes, as normal.
Message: 10234 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 23:31:13
Subject: Re: preferred moat?
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
These are clearly not ETs, since the complexity of an ET is nearly
identical to the number of notes.
As the name indicates, these are 7-limit linear temperaments, 23 of
them within a 'moat', and the red line is formed by raising each of
complexity and error (after some scaling) to the 2/3 power, adding,
and setting equal to a constant. Dave and I had already looked at ETs
and 5-limit linear temperaments . . .
How did you find this graph? My original post which mentioned it
named the linear temperaments that these numbers index, and then I
later gave the first three components of their wedgies for Gene.
Message: 10235 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 12:28:44
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>>> >nd 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?
>
>In some cases at least, it would be comparing lengths vs. areas vs.
>volumes.
Yes, but I should think ideally we'd figure out how to normalize
in some way to bring this whole business back to scales.
>>> 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....
>
>1
>1 1
>1 2 1
>1 3 3 1
>1 4 6 4 1
>
>First row -- ?
>Second row -- 2-limit (all octaves), ?
>Third row -- 3-limit JI, 3-limit temperament, ?
>Fourth row -- 5-limit JI, 5-limit LT, 5-limit ET, ?
>Fifth row -- 7-limit JI, 7-limit PT, 7-limit LT, 7-limit ET, ?
Great! Don't forget to mention that the number tells the number
of elements in the wedgie. This should go on form chart.
-Carl
Message: 10236 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:47:51
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.
>>>>> 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.
Relative scaling would be, for example, what one inch represents on
one of the axes, vs. what it represents on the other axis.
>>> 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.
What's "the above", exactly? y = x + c? This is the equation of a
line, which intersects the y-axis at c, and the x-axis at -c.
>>> 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?
Message: 10237 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 23:33:25
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> 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?)
They diverge everywhere except n=3.
> By the way, in 1999, Paul Hahn gave the following:
Thanks!
Message: 10238 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:28:56
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>>> 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.
>
With lower error? No, but you'd never know for sure just from looking
at the loglog graph.
> Wait -- and how can ETs appear more than once -- different maps?
Yes.
> That might explain different errors, but they are appearing at
> different complexities too... baffling.
I explained what I was doing with the complexity stuff, but these
complexity differences are very minor. My questions about this were
never answered, but you can look again at "Attn: Gene 2" to see how
it makes sense in the 3-limit (where complexity is proportional to
the number of TOP notes per acoustical octave, so the map matters
slightly).
Message: 10239 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:49:26
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> and triangular,
> where I believe all the angles are 60deg, at least through FCC.
That's the one!
Message: 10240 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 23:42:19
Subject: Re: preferred moat?
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
>> Um, this merely encloses every et 1-23...
>>
>> -Carl
>
> These are clearly not ETs, since the complexity of an ET is nearly
> identical to the number of notes.
>
> As the name indicates, these are 7-limit linear temperaments, 23 of
> them within a 'moat', and the red line is formed by raising each of
> complexity and error (after some scaling) to the 2/3 power, adding,
> and setting equal to a constant. Dave and I had already looked at
ETs
> and 5-limit linear temperaments . . .
>
> How did you find this graph? My original post which mentioned it
> named the linear temperaments that these numbers index,
Yahoo groups: /tuning-math/message/9317 * [with cont.]
Message: 10241 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 12:34:00
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>>>>>> >hy doesn't the badness bound alone
>>>>>> enclose a finite triangle?
>>>>>
>>>>> it actually encloses an infinite number
>>>>> of temperaments.
>>
>> 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 we've gone 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.
>> Wait -- and how can ETs appear more than once -- different maps?
>
>Yes.
>
>> That might explain different errors, but they are appearing at
>> different complexities too... baffling.
>
>I explained what I was doing with the complexity stuff, but these
>complexity differences are very minor. My questions about this were
>never answered, but you can look again at "Attn: Gene 2" to see how
>it makes sense in the 3-limit (where complexity is proportional to
>the number of TOP notes per acoustical octave, so the map matters
>slightly).
Ok...
-Carl
Message: 10242 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 13:49:59
Subject: Re: loglog!
From: Carl Lumma
>> >irst 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?
>> We just need a way to
>> compare a slice with a tube, etc.
>
>Doesn't seem possible to do this in a fair way.
Well, sure, but after a coupla margaritas... :)
>>>>>>>> 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.
With so-called Graham complexity, as you grow the scale
you get nothing, nothing, nothing, then boom, 2 complete
chords. Then every note you add after than gives you
another pair of chords. Thus, Graham complexity seems
important.
Likewise, Herman was onto something with his consintency
range thing.
>> () More dyads or fewer dyads?
>
>Certainly not fewer.
>
>> () More complete chords or less complete chords?
>
>Certainly not more.
Thank you.
-Carl
Message: 10243 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 23:41:47
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Gene Ward Smith
--- 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), and I thought Gene said
> badness alone *could* give a finite list, just that it would include
> lots of crap (like 1 & 2).
It can and will, so long as you go past the critical, log-flat
exponent.
Message: 10244 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:41:11
Subject: Re: !
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>> 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.
>
> Sorry; you use 1 cent and 1 note as zeros.
>
>>>> 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!
??
>>> 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.
> That means I only meant the above to apply when
> either x or y is zero, I think.
I lost you.
> If a circle is just so unsatisfactory, please instead consider my
> suggestion to be that we equally penalize temperaments for trading
> too much of their error for comp., or too much of their comp for
> error.
Have I ever *not* done this?
> 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.
> -- but I
> guess the moat is only meant for linear-linear, or?
It looks a little different on loglog but loglog's not a total deal-
stopper.
Message: 10245 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 21:50:49
Subject: Re: The same page
From: Paul Erlich
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>> ~= 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?
Form-form complement? Never heard of it.
> () Does dual = complement?
I think so.
Message: 10246 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 23:44:59
Subject: Re: The same page
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>
wrote:
I was planning on getting around to it. Is it urgent for some reason?
Message: 10247 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 13:52:24
Subject: Re: 23 "pro-moated" 7-limit linear temps
From: Carl Lumma
>>>>>>>> >hy 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.
-Carl
Message: 10248 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 15:48:10
Subject: top23
From: Carl Lumma
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?, meantone=huygens?, etc), 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.
-Carl
Message: 10249 - Contents - Hide Contents
Date: Fri, 13 Feb 2004 20:44:06
Subject: Re: Symmetrical complexity for 5 and 7 limit temperaments
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> '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. 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.
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.
> Ok. So I'm at a loss to describe how this is different from
> taxicab distance.
It's clearly not taxicab Tenney distance, which is what Paul has been
calling that. It's taxicab distance with Fifth Element style flying
taxicabs, and routes which form an A3=D3 lattice.
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