> calculate the numerators and denominators here which came out in
> scientific notation, making it impossible for yahoo to sort by
> denominator:
>
>
Yahoo groups: /tuning/database? * [with cont.]
> method=reportRows&tbl=10&sortBy=4
" large limma", "0 3 -2", ... * [with cont.] (Wayb.)
Graham
Message: 6211 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 08:51:20
Subject: Re: Graham's top 20, with standard vals
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
>> my reasoning in not being particularly interested in this? it's
that,
>> like graham, i don't think the standard val is necessarily the
best
>> val for any ET. try 64-equal in the 5-limit.
>
> Do you have an alternative definition?
how about the choice that minimizes the rms error?
Message: 6212 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 12:07:46
Subject: Calculating geometric complexity II
From: Gene Ward Smith
Here are Maple routines which have the exact coefficients. They are not, of course, computationally effiecient, but it would be easy to
calculate
the logarithms only once if that is a problem, though I havn't found
complexity calculations to be a bottleneck. These should be readily
translatable to Matlab, Python, or anything else.
gc7 := proc(l)
# l is 7-limit wedgie
sqrt(evalf(1/4*(-ln(5)^4+4*ln(5)^2*ln(7)^2)*l[1]^2
+1/4*(-ln(3)^4+4*ln(3)^2*ln(7)^2)*l[2]^2
+1/4*(-ln(3)^4+4*ln(3)^2*ln(5)^2)*l[3]^2
+1/2*(ln(3)^2*ln(5)^2-2*ln(3)^2*ln(7)^2)*l[1]*l[2]
-1/2*ln(3)^2*ln(5)^2*l[1]*l[3]+
1/2*(ln(3)^4-2*ln(3)^2*ln(5)^2)*l[2]*l[3])) end:
gc11 := proc(l)
# l is 11-limit linear wedgie
sqrt(evalf((-1/4*ln(5)^2*ln(7)^4-1/4*ln(11)^2*ln(5)^4+ln(7)^2*ln(11)^2*ln(5)^2)*l[1]^2
+(ln(11)^2*ln(7)^2*ln(3)^2-1/4*ln(3)^4*ln(11)^2-1/4*ln(3)^2*ln(7)^4)*l[2]^2
+(ln(3)^2*ln(5)^2*ln(11)^2-1/4*ln(3)^2*ln(5)^4-1/4*ln(3)^4*ln(11)^2)*l[3]^2
+(ln(7)^2*ln(3)^2*ln(5)^2-1/4*ln(3)^2*ln(5)^4-1/4*ln(7)^2*ln(3)^4)*l[4]^2
+(1/2*ln(3)^2*ln(5)^2*ln(11)^2-ln(11)^2*ln(7)^2*ln(3)^2+1/4*ln(3)^2*ln(7)^4)*l[1]*l[2]
+(-1/2*ln(3)^2*ln(5)^2*ln(11)^2+1/4*ln(7)^2*ln(3)^2*ln(5)^2)*l[1]*l[3]
-1/4*ln(7)^2*ln(3)^2*ln(5)^2*l[1]*l[4]
+(-ln(3)^2*ln(5)^2*ln(11)^2+1/2*ln(7)^2*ln(3)^2*ln(5)^2-1/4*ln(7)^2*ln(3)^4+
1/2*ln(3)^4*ln(11)^2)*l[2]*l[3]+(-1/2*ln(7)^2*ln(3)^2*ln(5)^2+1/4*ln(7)^2*ln(3)^4)*l[2]*l[4]
+(-ln(7)^2*ln(3)^2*ln(5)^2+1/2*ln(3)^2*ln(5)^4+1/4*ln(7)^2*ln(3)^4)*l[3]*l[4]))
end:
gpc11 := proc(l)
# l is 11-limit planar wedgie
sqrt(evalf((-1/4*ln(7)^4+ln(7)^2*ln(11)^2)*l[1]^2
+(-1/4*ln(5)^4+ln(5)^2*ln(11)^2)*l[2]^2
+(-1/4*ln(5)^4+ln(5)^2*ln(7)^2)*l[3]^2
+(-1/4*ln(3)^4+ln(3)^2*ln(11)^2)*l[4]^2
+(-1/4*ln(3)^4+ln(3)^2*ln(7)^2)*l[5]^2
+(-1/4*ln(3)^4+ln(3)^2*ln(5)^2)*l[6]^2
+(-1/2*ln(5)^2*ln(7)^2+ln(5)^2*ln(11)^2)*l[1]*l[2]
+(-1/2*ln(5)^2*ln(7)^2)*l[1]*l[3]
+(-1/2*ln(3)^2*ln(7)^2+ln(3)^2*ln(11)^2)*l[1]*l[4]
+(-1/2*ln(3)^2*ln(7)^2)*l[1]*l[5]
+(1/2*ln(3)^2*ln(5)^2-1/2*ln(3)^2*ln(7)^2)*l[1]*l[6]
+(-1/2*ln(5)^4+ln(5)^2*ln(7)^2)*l[2]*l[3]
+(-1/2*ln(3)^2*ln(5)^2+ln(3)^2*ln(11)^2)*l[2]*l[4]
+(-ln(3)^2*ln(5)^2+ln(3)^2*ln(7)^2)*l[2]*l[5]
+(-1/2*ln(3)^2*ln(5)^2)*l[2]*l[6]
+(-1/2*ln(3)^2*ln(5)^2+ln(3)^2*ln(7)^2)*l[3]*l[5]
+(-1/2*ln(3)^2*ln(5)^2)*l[3]*l[6]
+(-1/2*ln(3)^4+ln(3)^2*ln(7)^2)*l[4]*l[5]
+(-1/2*ln(3)^4+ln(3)^2*ln(5)^2)*l[4]*l[6]
+(-1/2*ln(3)^4+ln(3)^2*ln(5)^2)*l[5]*l[6])) end:
Message: 6213 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 09:17:34
Subject: Re: Graham's Top 20 13-limit temperaments
From: Carl Lumma
> better than any other complexity measure! cool, so you must
> *really* like the heuristic for complexity . . .
Apparently so.
> my best recollection, off the top of my head:
>
> log-flat badness < 3500, rms error < 50 cents, geometric
> complexity < 104-151 (doesn't matter where you draw the
> line in this range).
Ok, but two small nits:
() Is that geometric complexity as Gene defines it?
() Being that badness is just a combination error and
complexity, why is it needed / how can it change the
bounds on the list?
-Carl
Message: 6214 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 12:34:21
Subject: Re: Calculating geometric complexity II
From: Graham Breed
Gene Ward Smith wrote:
> Here are Maple routines which have the exact coefficients. They are not, of course, computationally effiecient, but it would be easy to
> calculate
the logarithms only once if that is a problem, though I havn't found
complexity calculations to be a bottleneck. These should be readily
translatable to Matlab, Python, or anything else.
How are you indexing your wedgies? I use tuples, so that when
multiplying 1-vectors,
z[i, j] = x[i,] + y[j,]
where x[0,] is the octave coefficient, x[1,] the 3:1 and so on. Can
you
provide conversion tables between your [i] and my [i,j]?
I think I've got the idea of vals as well. A dual isn't the same as a
complement! Using ^ for the wedge product, and ~ for the complement,
we
have
h12^~comma = ~comma^h12 = {}
where "h12" is the val for 5-limit 12-equal, "comma" is the unison
vector for 81:80 and {} is the empty wedgie.
Vals and unison vectors are both 1-vectors. For duality, I'll have to
add a flag to each object. So the complement operation also inverts
the
flag. A unison vector has the dual flag set to 0 and a val has the
dual
flag set to 1.
To compute a wedge product, both dual flags has to agree. So when you
ask to calculate h12^comma, the function can look at the two dual
flags,
see they aren't the same, and take the complement of the second
element
to make it so. That means, asking for
h12^comma
means you get
h12^~comma
and it doesn't matter if you meant
~h12^comma
because the dual flag gets set again for the next step of the
calculation. And in this case the result is the same anyway.
You can also say that
h12^h7 == comma
because h12^h7 will have its dual flag set, and the comparison
function
knows it really has to return
h12^h7 == ~comma
and the routine for calculating a linear temperament knows it needs to
start with a wedgie that has its dual flag cleared, and so if you feed
it h12^h7 it converts it to ~(h12^h7).
I still don't know how to store a multivector so that it's its own
dual
which seems to be what you're doing.
Graham
Message: 6215 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 09:37:43
Subject: Re: Graham's Top 20 13-limit temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> better than any other complexity measure! cool, so you must
>> *really* like the heuristic for complexity . . .
>
> Apparently so.
>
>> my best recollection, off the top of my head:
>>
>> log-flat badness < 3500, rms error < 50 cents, geometric
>> complexity < 104-151 (doesn't matter where you draw the
>> line in this range).
>
> Ok, but two small nits:
>
> () Is that geometric complexity as Gene defines it?
yes.
> () Being that badness is just a combination error and
> complexity, why is it needed
because otherwise you'd have a huge number of temperaments, and not
the same number in each complexity range. for example, imagine how
many possible temperaments there must be with rms error < 50 cents
and complexity between, say, 74 and 104. some huge number.
Message: 6216 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 13:16:03
Subject: Re: Calculating geometric complexity II
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>> Here are Maple routines which have the exact coefficients. They are not, of course, computationally effiecient, but it would be easy to
>> calculate
the logarithms only once if that is a problem, though I havn't found
complexity calculations to be a bottleneck. These should be readily
translatable to Matlab, Python, or anything else.
>
> How are you indexing your wedgies? I use tuples, so that when
> multiplying 1-vectors,
>
> z[i, j] = x[i,] + y[j,]
>
> where x[0,] is the octave coefficient, x[1,] the 3:1 and so on. Can
you
> provide conversion tables between your [i] and my [i,j]?
I'm not sure if we are speaking the same language, but I'm using
lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would
be followed by z[1,2]...z[1,n] and so forth. This gives a linear
temperament wedgie as the product of two vals, and puts the 2-part,
which is related to the generators column of the period-generator
matrix, at the beginning.
> I think I've got the idea of vals as well. A dual isn't the same as
a
> complement! Using ^ for the wedge product, and ~ for the
complement, we
> have
>
> h12^~comma = ~comma^h12 = {}
>
> where "h12" is the val for 5-limit 12-equal, "comma" is the unison
> vector for 81:80 and {} is the empty wedgie.
Are you still using empty wedgies for zero vectors? I hope this isn't
giving problems. In any case, the above is definitional; ~comma is the
3-product such that h ^ ~comma = h(comma), so that we can identify
compliments with duals.
> Vals and unison vectors are both 1-vectors. For duality, I'll have
to
> add a flag to each object. So the complement operation also inverts
the
> flag. A unison vector has the dual flag set to 0 and a val has the
dual
> flag set to 1.
Makes sense.
> I still don't know how to store a multivector so that it's its own
dual
> which seems to be what you're doing.
I'm simply being unsophisticated about it--I store the wedgies as
lists, and reverse the ordering when I compute from commas, etc. in
order to get the lists to be the same.
Message: 6217 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 10:55:35
Subject: Re: Graham's Top 20 13-limit temperaments
From: Carl Lumma
>because otherwise you'd have a huge number of temperaments, and not
>the same number in each complexity range. for example, imagine how
>many possible temperaments there must be with rms error < 50 cents
>and complexity between, say, 74 and 104. some huge number.
Right on. -C.
Message: 6218 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 14:52:55
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
>
>> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS, HERE'S A
>> QUESTION: What other ETs above 494 besides 612 and 624 would you
>> want to notate -- ones in which the 5' comma (a.k.a, historical 5-
>> schisma, 32768:32805) is either a single degree of the ET or
vanishes?
>
> 665, 684, 730, 742 and 836.
Thanks, Gene. I'll also add 653 to that list.
But we won't be able to notate 684, because the 5' comma vanishes and
no other symbol in the sagittal notation would represent a single
degree, either.
--George
Message: 6219 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 15:15:59
Subject: Re: hi everybodyyyyyyyyyyyyyyyyyyyyyyyyy
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> on this list,
>
> one group of people is talking about equal temperaments which
belong
> to important families of tunings (each family being where a
> particular set of unison vectors vanishes),
>
> another group is concerned with notating equal temperaments
according
> to where the potential unison vectors lie relative to their chains
of
> fifths,
>
> and the two groups are not talking to one another.
>
> am i perceiving the situation correctly?
Hi, Paul. Thanks for checking up on us.
I did participate in the discussions about hemififth and kleismic
temperaments (but Gene eventually realized that I was really
addressing catakleismic). Nothing in these discussions resulted in
any changes to any ET notations that Dave and I have already agreed
on. The catakleismic discussion did cover some larger divisions that
we have not yet addressed, so I can't say yet how the results using
these two approaches might differ. Another factor in all of this is
our recent introduction of the 5' comma (traditional 5-schisma) into
the notation, which is going to affect how some of these things are
done.
So there has been a limited amount of communication about these
things, but we've been pretty busy working on our separate
approaches, and eventually we're going to have to bring it all
together and compare notes.
--George
Message: 6220 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 15:33:53
Subject: Re: hi everybodyyyyyyyyyyyyyyyyyyyyyyyyy
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
>
>> Gene has contributed to the Common notation thread regarding
notating
>> linear temperaments. But I agree he seems to have been following
the
>> idea that a temperament can be notated adequately using the
notation
>> for its most representative ET, and apparently assuming that
George
>> and I have already found the best notation for that ET (within the
>> constraints we have imposed upon ourselves). neither of which may
e true.
>
> Oh. Well, darn.
>
>> Although chains of fifths are the backbone of the sagittal
notation,
>> this does not prevent it from notating ETs in LT-specific ways,
so the
>> same ET can be notated differently depending on which LT you are
>> considering it as. I recently gave some examples in a "Notating
Linear
>> Temperaments" thread (or some such), but no one responded or
carried
>> it forward.
>
> I can't follow these things when they involve the symbols, not at
least unless I have a key handy (which I don't.)
I mentioned in message #5403 that I made a quick-reference table for
the most common symbols in a file when I had to answer your question
about "what the 11-diesis is":
Yahoo groups: /tuning- * [with cont.]
math/files/secor/notation/quickref.txt
The actual symbols may be seen in these files:
Yahoo groups: /tuning- * [with cont.]
math/files/secor/notation/AdaptJI.gif
Yahoo groups: /tuning- * [with cont.]
math/files/secor/notation/Symbols6.gif
I hope that this is good enough for now, until we have a decent
explanation of the notation available.
--George
Message: 6221 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 08:08:37
Subject: Re: A common notation for JI and ETs
From: monz
> From: <gdsecor@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, January 27, 2003 6:52 AM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
> <gdsecor@y...> wrote:
>>
>>> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS,
>>> HERE'S A QUESTION: What other ETs above 494 besides
>>> 612 and 624 would you want to notate -- ones in which
>>> the 5' comma (a.k.a, historical 5-schisma, 32768:32805)
>>> is either a single degree of the ET or vanishes?
>>
>> 665, 684, 730, 742 and 836.
>
> Thanks, Gene. I'll also add 653 to that list.
>
> But we won't be able to notate 684, because the 5' comma
> vanishes and no other symbol in the sagittal notation
> would represent a single degree, either.
>
> --George
how about 768? ... because it's the tuning resolution for a
number of popular electronic instruments.
-monz
Message: 6222 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 16:29:31
Subject: Re: Calculating geometric complexity II
From: Graham Breed
Gene Ward Smith wrote:
> I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would
> be
followed by z[1,2]...z[1,n] and so forth. This gives a linear
temperament wedgie as the product of two vals, and puts the 2-part,
which is related to the generators column of the period-generator
matrix, at the beginning.
Oh, that's good. It should be the same as my invariant. But are
7-limit wedge products taken from vectors or vals?
I get 7-limit meantone as 21.97, 11-limit meantone as 31.72 and
h12^h19^h22 in the 11-limit as 29.52. The planar temperament with
441:440 and 225:224 is 34.44.
> Are you still using empty wedgies for zero vectors? I hope this
isn't giving problems. In any case, the above is definitional; ~comma
is the
> 3-product such that h ^ ~comma = h(comma), so that we can identify
> compliments with duals.
Empty wedgies are empty wedgies. I haven't had any trouble with them.
> I'm simply being unsophisticated about it--I store the wedgies as
lists, and reverse the ordering when I compute from commas, etc. in
order to get the lists to be the same.
That sounds like taking the complement. I thought you said you didn't
have to because you were using duality. And how can you be sure that
reversing the list will do the trick? Some of the coefficients should
be negated if you aren't using a special ordering.
Graham
Message: 6223 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 16:38:22
Subject: Re: Calculating geometric complexity II
From: Graham Breed
Gene Ward Smith wrote:
> I'm not sure if we are speaking the same language, but I'm using lexicographical order; that is, z[0,1], z[0,2] .... z[0,n] would
> be
followed by z[1,2]...z[1,n] and so forth. This gives a linear
temperament wedgie as the product of two vals, and puts the 2-part,
which is related to the generators column of the period-generator
matrix, at the beginning.
Oh, and I expect you're indexing from 1 as well. In which case I get
7-limit meantone 23.76
11-limit meantone 23.85
h12^h19^h22 22.77
441:440 ^ 225:224 28.57
Graham
Message: 6224 - Contents - Hide Contents
Date: Mon, 27 Jan 2003 19:38:03
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
>> >>
>>>> ***** HEY IF ANYBODY ELSE OUT THERE IS READING THIS,
>>>> HERE'S A QUESTION: What other ETs above 494 besides
>>>> 612 and 624 would you want to notate -- ones in which
>>>> the 5' comma (a.k.a, historical 5-schisma, 32768:32805)
>>>> is either a single degree of the ET or vanishes?
>>>
>>> 665, 684, 730, 742 and 836.
>>
>> Thanks, Gene. I'll also add 653 to that list.
>>
>> But we won't be able to notate 684, because the 5' comma
>> vanishes and no other symbol in the sagittal notation
>> would represent a single degree, either.
>>
>> --George
>
> how about 768? ... because it's the tuning resolution for a
> number of popular electronic instruments.
>
> -monz
Sorry, that one can't be done with the commas that we have. The 5'
comma (32768:32805) is actually -1 degrees, so it would be too
confusing to use it. And while the 19 comma (512:513) could be used
as 1 degree, there's nothing for 2, 3, and 4 degrees.
Anyway, 768 isn't even 1,3,9-consistent, hence not very desirable
musically.
But after taking a quick look at 653, 665, 730, 742 and 836, I think
those five are all do-able.
--George
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