> yahoo only supports 10 columns. i deleted the heuristic error
> since the RMS error has a visual correlate (distance from
> origin to corresponding line) on the corresponding graph.
Yep; another nice thing about RMS.
But how does heuristic error differ from RMS?
> you're probably asking about heuristic complexity, though,
Yep.
> by which i meant that sorting by denominator is the same as
> sorting by heuristic complexity. so you can now sort by
> heuristic complexity only until scientific notation comes in.
Understood.
-C.
Message: 6033 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 23:58:28
Subject: Re: Mega 11 and 13 limit ets
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> what happened to 504504?
I checked from 3 to 23, and it's not nearly good enough to make the cut in any of them. Are you sure this is the right number?
Message: 6034 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 07:26:20
Subject: Re: Notating Pajara
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:
> As evidence, I offer the minimax generators for Pajara quoted by a
> number of different people earlier in this thread. I believe the same
> value was supplied independently by 2 or 3 people, and I think you
> will find that it is the p-->infinity value.
I've decided to cave, and re-wrote my program to get it to correspond to what other people are doing.
> I think you'd better include p=1 before Paul starts campaigning for
> the limit as p-->0 or p negative. ;-)
So it seems, but I really think we'd better draw a line there.
> Actually I think that the absolutely and ideally perfect optimisation
> will probably occur when p = 2*pi. ;-)
Obviously we need to work pi in somehow.
Message: 6035 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:00:05
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>> If the above is true about commas, then complexity should be
>>> defined in terms of commas, and we could search all sets of
>>> simple commas...
>>
>> What sets of simple commas do you propose to take?
>
> All of them within taxicab radius r on the triangular harmonic
> lattice. T(r), the number of tones within r, then seems to be
> 6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly
> O(r**2). The number of possible commas is than the 2-combin.
> of this, which is again is roughly **2. So O(r**4) at the end
> of the day.
this is silly. all you need to do to get all these commas to come up
as *tones* is to double your radius. so 2*O(r**2) will do -- the O
(r**4) method is extremely redundant.
> Perhaps that first 2 is the number of dimensions on the lattice,
> in which case the 15-limit would be O(r**14).
if we're looking for an efficient search, we'd definitely use a prime-
limit lattice, not one with redundant points!
Message: 6036 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 08:19:28
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>> >he number of possible maps I'm interested in isn't that
>> large. Gene didn't deny that a map uniquely defined its
>> generators (still working through how a list of commas
>> could be more fundamental than a map...).
>
>It is easy to define a canonical set of commas by insisting
>it be TM reduced; and thereby associate something (a set of
>commas) as a unique marker for the temperament. We can also
>do this with maps by for instance Hermite reducing the map.
>No one seems to like Hermite reduction much, so if you want
>to define a canonical map which is better the field of
>opportunity is open.
Well, I don't know what a basis is, let alone a TM reduced
one, let alone a Hermite normal form.
>>>> As far as my combining error and complexity before optimizing
>>>> generators, that was wrong. Moreover, combining them at all
>>>> is not for me. I'm not bound to ask, "What's the 'best' temp.
>>>> in size range x?". Rather, I might ask, "What's the most
>>>> accurate temperament in complexity range x?".
>>>
>>> that's exactly how i've been looking at all this for the entire
>>> history of this list -- witness my comments to dave defending
>>> gene's log-flat badness measures; i took exactly this tack!
>>
>> How could it defend Gene's log-flat badness? It's utterly
>> opposed to it!
>
>Eh? I go with Paul; this is the point of log-flat measures.
Eh? Name your complexity range, and take the x most accurate
temperaments within it. Why do I need badness at all?
-Carl
Message: 6037 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:01:24
Subject: Re: Ultimate 5-limit again
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>> yahoo only supports 10 columns. i deleted the heuristic error
>> since the RMS error has a visual correlate (distance from
>> origin to corresponding line) on the corresponding graph.
>
> Yep; another nice thing about RMS.
>
> But how does heuristic error differ from RMS?
not too much -- less than a factor of 2 in every case that i've seen.
Message: 6038 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 08:59:23
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>hmm . . . aren't you then just talking about a finite limit
>of some sort?
With a steep weighting and a promise that what we've left
out affects things less than x, sure. Maybe the Zeta
function stuff can do even better...
>>>> We should be able to search map space and assign
>>>> generator values from scratch.
>>>
>>> i don't understand this.
>>
>> The number of possible maps I'm interested in isn't that
>> large. Gene didn't deny that a map uniquely defined its
>> generators
>
>not sure if this concept has been pinned down . . .
Certainly it hasn't. For linear 5-limit temps, if I take
something like this...
2 3 5
gen1
gen2
...and start filling in numbers, either every choice of
numbers converges on a pair of exact sizes for gen1 and
gen2 under an error function of a certain class (say, one
to which RMS, and apparently not minimax, belongs), or it
doesn't. Am I to understand it doesn't?
>> (still working through how a list of commas
>> could be more fundamental than a map...).
>
>given a list of commas, you can determine the
>mapping, no matter how you define your generators.
Are you saying that the same set of commas could vanish
under two different maps, each with different gens? If
so, can you give an example?
>we've been searching by commas, i believe -- i'll let gene
>answer this more fully. graham has been searching by "+"ing
>ET maps to get linear temperament maps -- something i'm not
>sure i can explain right now.
I remember Graham using the +ing ets method. He claimed it
was pretty fast and comprehensive. I remember thinking it
was anything but pretty.
I don't understand wedgies, so...
If the above is true about commas, then complexity should be
defined in terms of commas, and we could search all sets of
simple commas...
>that's not what i mean -- i mean, if you're dealing with a
>planar temperament (which might simply be a linear
>temperament with tweakable octaves)
How can tweaking one of the generators of a linear temperament
turn it into a planar temperament? You need a third gen!
>or something with higher dimension, there's no unique choice
>of the basis of generators -- gene's used things such as
>hermite reduction to make this arbitrary choice for him.
What's a "basis of generators"?
>the idea is that, if you sort by complexity, using a log-
>flat badness criterion guarantees that you'll have a similar
>number of temperaments to look at within each complexity
>range, so the complexity will increase rather smoothly in
>your list.
You mean if I take the twenty "best" 5-limit temperaments
and sort by badness, the resulting list will alse be sorted
by complexity, then accuracy? There didn't seem to be any
exponents that would do this on Dave's spreadsheet, and I
thought I tried the critical exponent for log-flat temps.
>though the mathematics of it is -- naturally -- heuristic
>in nature.
?
AFAIK, a heuristic is an algorithm that attempts to search
only a fraction of a network yet still deliver results one
can have confidence in.
-Carl
Message: 6039 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:03:45
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
> <clumma@y...> wrote:
>>>> If the above is true about commas, then complexity should be
>>>> defined in terms of commas, and we could search all sets of
>>>> simple commas...
>>>
>>> What sets of simple commas do you propose to take?
>>
>> All of them within taxicab radius r on the triangular harmonic
>> lattice. T(r), the number of tones within r, then seems to be
>> 6r + T(r-1); T(0)=1 in the 5-limit. That seems to be roughly
>> O(r**2). The number of possible commas is than the 2-combin.
>> of this, which is again is roughly **2. So O(r**4) at the end
>> of the day.
>
> this is silly. all you need to do to get all these commas to come
up
> as *tones* is to double your radius. so 2*O(r**2) will do -- the O
> (r**4) method is extremely redundant.
and, in fact, kees van prooijen does exactly this search on tones,
here:
Searching Small Intervals * [with cont.] (Wayb.)
and keeps the smallest, second-smallest, and third-smallest commas
for each possible r, and comes up with this list:
S235 * [with cont.] (Wayb.)
Message: 6040 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:13:29
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>> >re you saying that the same set of commas could vanish
>> under two different maps, each with different gens? If
>> so, can you give an example?
>
>81:80
>
>in terms of octave and fifth, the map is [1,0] [1,1] [4,0]
>
>in terms of octave and twelfth, the map is [1,0] [0,1] [4,-4]
Hmm. How bad are such cases? Could we take them out at the
end?
>> If the above is true about commas, then complexity should be
>> defined in terms of commas,
>
>it's nice when there's only one comma. then the log of the
>numbers in the comma (say, the log of the odd limit) is an
>excellent estimate of complexity (it's what i call the
>heuristic complexity).
That's what I call taxicab complexity, I think.
Isn't it also the case that the same temperament may be defined
by different lists of commas?
>if there's more than one comma being tempered out, we need
>a notion of the "angle" between the commas . . .
Please explain.
>> How can tweaking one of the generators of a linear temperament
>> turn it into a planar temperament? You need a third gen!
>
> it's a planar temperament in the sense that there are two
> independently tweakable generators, two independent dimensions
> needed in order to define each pitch of the tuning.
You assume there was an untweakable one in the 5-limit case?
Bah!
And in the 'now it's planar' case, you would no longer have an
untweakable one. Something's got to give!
>> What's a "basis of generators"?
>
>for the case of meantone, one example would be octave and fifth.
>another example would be octave and twelfth. another example
>would be fifth and fourth. another example would be major second
>and minor second. each pair comprises a complete basis for the
>vector space of pitches in the tuning.
Thanks. There is also, I gather, such a thing as a basis of
commas? TM (TM stands for?) reduction applies to commas only,
right?
>> You mean if I take the twenty "best" 5-limit temperaments
>> and sort by badness, the resulting list will alse be sorted
>> by complexity, then accuracy?
>
> no. you use a badness cutoff simply to define the list of
> temperaments in the first place.
That's the same as taking the 20 "best" temperaments.
>*then* you sort by complexity.
Aha! This isn't better than taking the 20 simplest temperaments
and sorting by accuracy.
-Carl
Message: 6041 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:18:01
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
Man, it's been a while since I looked at Kees' site. This
is a goldmine!
>and keeps the smallest, second-smallest, and third-smallest
>commas for each possible r, and comes up with this list:
>
>
S235 * [with cont.] (Wayb.)
That's badass.
-C.
Message: 6042 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:28:25
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>>> Are you saying that the same set of commas could vanish
>>> under two different maps, each with different gens? If
>>> so, can you give an example?
>>
>> 81:80
>>
>> in terms of octave and fifth, the map is [1,0] [1,1] [4,0]
>>
>> in terms of octave and twelfth, the map is [1,0] [0,1] [4,-4]
>
> Hmm. How bad are such cases? Could we take them out at the
> end?
i have no idea what you're asking. bad?
>>> If the above is true about commas, then complexity should be
>>> defined in terms of commas,
>>
>> it's nice when there's only one comma. then the log of the
>> numbers in the comma (say, the log of the odd limit) is an
>> excellent estimate of complexity (it's what i call the
>> heuristic complexity).
>
> That's what I call taxicab complexity, I think.
not quite. for one thing, read this:
lattice orientation * [with cont.] (Wayb.)
including the link to my observations.
> Isn't it also the case that the same temperament may be defined
> by different lists of commas?
>
>> if there's more than one comma being tempered out, we need
>> a notion of the "angle" between the commas . . .
>
> Please explain.
search for "straightess" in these archives . . .
>>> How can tweaking one of the generators of a linear temperament
>>> turn it into a planar temperament? You need a third gen!
>>
>> it's a planar temperament in the sense that there are two
>> independently tweakable generators, two independent dimensions
>> needed in order to define each pitch of the tuning.
>
> You assume there was an untweakable one in the 5-limit case?
> Bah!
are you saying the octave should never be assumed to be exactly 1200
cents?
> And in the 'now it's planar' case, you would no longer have an
> untweakable one. Something's got to give!
i'm not following you.
>>> What's a "basis of generators"?
>>
>> for the case of meantone, one example would be octave and fifth.
>> another example would be octave and twelfth. another example
>> would be fifth and fourth. another example would be major second
>> and minor second. each pair comprises a complete basis for the
>> vector space of pitches in the tuning.
>
> Thanks. There is also, I gather, such a thing as a basis of
> commas? TM (TM stands for?)
tenney-minkowski. tenney is the metric being minimized, and minkowski
provided a basis-reduction algorithm applicable to such a case.
> reduction applies to commas only,
> right?
right.
>>> You mean if I take the twenty "best" 5-limit temperaments
>>> and sort by badness, the resulting list will alse be sorted
>>> by complexity, then accuracy?
>>
>> no. you use a badness cutoff simply to define the list of
>> temperaments in the first place.
>
> That's the same as taking the 20 "best" temperaments.
well, if your badness cutoff, extreme error cutoff, and extreme
complexity cutoff leave you with 20 inside, and if such a clunky
tripartite criterion is what you define as "best".
>> *then* you sort by complexity.
>
> Aha! This isn't better than taking the 20 simplest temperaments
> and sorting by accuracy.
of course it is. most of the 20 simplest temperaments are garbage.
don't you want the several best temperaments in each complexity
range, going up to complexity higher than you could ever use?
Message: 6043 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:31:24
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> Man, it's been a while since I looked at Kees' site. This
> is a goldmine!
>
>> and keeps the smallest, second-smallest, and third-smallest
>> commas for each possible r, and comes up with this list:
>>
>>
S235 * [with cont.] (Wayb.)
>
> That's badass.
>
> -C.
Message: 6044 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:32:41
Subject: Re: Nonoctave scales and linear temperaments
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> search for "straightess" in these archives . . .
oops -- i meant "straightness"!
Message: 6045 - Contents - Hide Contents
Date: Thu, 16 Jan 2003 22:46:08
Subject: Re: Nonoctave scales and linear temperaments
From: Graham Breed
Carl Lumma wrote:
> Perhaps that first 2 is the number of dimensions on the lattice,
> in which case the 15-limit would be O(r**14).
Yes, it's the volume of a hypersphere. So it climbs dramatically with
the number of dimensions, which is one less the number of primes (or odd
numbers if you're being masochistic). For 8 dimensions, you still get
O((r**7)**7) = O(r**49) candidate wedgies.
>> The files I have here show equivalences between second-order
>> tonality diamonds.
>
> Which are? Tonality diamonds of the pitches in a tonality
> diamond?
Yes
> Ah yeah, that's a lot of commas, and it seems a rather ad hoc
> way to select them.
The advantage of commas taken from nth order tonality diamond is that
you can predict how good a temperament can be involved in. The size of
the comma in cents divided by the order of the tonality diamond is the
best possible minimax error.
> There must be many ways to reduce the complexity of the
> method I suggest. For example, rather than finding all
> the pitches and taking combinations, we might set a max
> comma size (as an interval) and step through ratios
> involving up to r compoundings of the allowed factors
> that are smaller than that bound.
Yes, you could, for example, take only commas less than 10 cents. That
means only 10/1200 or 1/120 of the commas stay. So you save two orders
of magnitude from you set of commas, but no more. I've found the code
for my search, and I was using this kind of optimization.
The big reduction in complexity would come from reducing the number of
combinations of commas you need to take. Ideally some way of predicting
that a small set of commas (all good on their own) will never give a
good temperament by adding more.
There are certainly a lot of commas that can be relevant. Below are all
42 the equivalences from my list of 19-limit temperaments (generated
from 20 equal temperaments). Some may not be unique unison vectors, but
whatever, they aren't sufficient to generate most of the linear
temperaments anyway. There are 27 million combinations of 7 from 42. 7
from 28 already gives over a million combinations.
22:19 =~ 15:13
16:15 =~ 17:16
15:11 =~ 26:19
135:128 =~ 19:18
65:48 =~ 19:14
20:19 =~ 19:18
24:19 =~ 19:15
39:32 =~ 128:105
14:13 =~ 128:119
45:32 =~ 128:91
256:195 =~ 21:16
256:221 =~ 22:19
65:64 =~ 64:63
128:117 =~ 35:32
17:13 =~ 64:49
13:10 =~ 64:49
21:20 =~ 20:19
40:33 =~ 17:14
48:35 =~ 256:187
18:17 =~ 128:121
40:39 =~ 49:48
128:117 =~ 12:11
11:9 =~ 39:32
256:221 =~ 22:19
25:24 =~ 133:128
22:17 =~ 128:99
128:91 =~ 7:5
64:57 =~ 28:25
32:27 =~ 13:11
256:209 =~ 11:9
19:18 =~ 128:121
28:25 =~ 143:128
18:17 =~ 17:16
9:8 =~ 64:57
171:128 =~ 4:3
28:27 =~ 133:128
128:95 =~ 27:20
32:27 =~ 19:16
24:19 =~ 81:64
135:128 =~ 20:19
165:128 =~ 128:99
135:128 =~ 19:18 =~ 128:121
Graham
Message: 6046 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 00:59:54
Subject: Re: Nonoctave scales and linear temperaments
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:
>> --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...>
wrote:
>
>>> Oops! That should be O((r**7)**6) or O(r**42) and 6 from 42
only
>> gives
>>> 5 million or so combinations.
>>
>> still vastly redundant.
>
> One possibility would be to choose a comma list with a badness
>cutoff, where the badness was the badness of the corresponding
>codimension one, or one-comma, temperament.
you'd get a lot of temperaments that are worse than a lot you woudn't
get, because of the straightness thing.
Message: 6047 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 10:30:03
Subject: Re: heuristic and straightness
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:
> wallyesterpaulrus wrote:
>
>> insufficient? how can a set of unison vectors be insufficient?
>
> For an arbitrary search to be practicable,
> there has to be a way of rejecting sets of three unison vectors
because
> you know they can't give a good linear temperament.
then you need a 3-d generalization of "straightness". i bet if i went
and learned grassmann algebra i'd be able to get a better grasp on
all this.
Message: 6048 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 11:31:56
Subject: Re: heuristic and straightness
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>
>> I'm not sure what you want here--if all of the commas point in
>> about the same direction, do you mean with or without 2?
>
> i'm thinking without.
One method which might come to the same thing as "straightness" in
effect is to take two commas, and combine to get a codimension 2
wedgie. Produce a list of these by taking the best (here you run into
geometric badness) and then wedge these with another comma, and so
forth.
Message: 6049 - Contents - Hide Contents
Date: Fri, 17 Jan 2003 01:37:37
Subject: Re: Nonoctave scales and linear temperaments
From: Carl Lumma
>> >an I identify the duplicate temperaments?
>
> duplicate? any temperament can be generated from an infinite
> number of possible basis vectors.
"The results of a search of all possible maps is bound to return
pairs, trios, etc. of maps that represent the same temperament.
Can we find them in the mess of results?" Was all I was asking!
>> Assuming 2:1 reduction makes me squirm in my chair, is all.
>> Plentiful near-2:1s should emerge from the search if the criteria
>> are right.
>
>if the criteria include mapping to, and minimizing error from,
>2:1, then of course a near 2:1 will emerge in each temperament.
Yup, that's what I've been saying alright.
>> Oh. Now I get it! You're right. But doesn't the same
>> problem occur with different commatic representations, when
>> defining complexity off the commas?
>
> not if you define complexity right!
Aha!
>>>> Are you saying a badness cutoff is not sufficient to give a
>>>> finite list of temperaments?
>>>
>>> exactly. in *every* complexity range you have about the same
>>> number of temperaments with log-flat badness lower than some
>>> cutoff -- and there are an infinite number of non-overlapping
>>> complexity ranges.
>>
>> Oh. I guess I need some examples, then, of most of the simple
>> temperaments that are garbage...
>
> what are the 20 simplest 5-limit intervals? now set each of
> these to be the commatic unison vector, and what temperaments
> do you get?
Perhaps we could enforce "validity", and maybe also Kees'
'complexity validity'.
-Carl
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