>>> Others, apparently including Wilson, wanted
>>> to include other points, basically out to the power
>>> set (the compound of all CPSs of a given limit).
>>
>> that isn't the case. look over the old discussions on this list
about
>> stellation. the compound of all CPSs of a given limit can be
>> constructed in many ways, but most typically (as d'allessandro) as
an
>> euler genus, and this (or any of the other ways) clearly does not
>> have the symmetry of the original, unstellated CPS, which the
>> stellation must have by definition.
>
> Right, right, you're extending the tones out to a EF Genus in all
> directions.
example?
> A stellated EF Genus is what I called it in that thread.
to my great consternation . . .
> -Carl
Message: 6529 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 10:35:31
Subject: Re: lattice diagram "levels" of complexity
From: Carl Lumma
>> >ight, right, you're extending the tones out to a EF Genus in all
>> directions.
>
>example?
I think that's the way I accounted for a 92-tone structure.
>> A stellated EF Genus is what I called it in that thread.
>
>to my great consternation . . .
What you never showed is how the definition of stellation on
George Hart's site requires the 92-tone, and not the 80-tone
structure.
-Carl
Message: 6530 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 19:23:20
Subject: Re: scala show data
From: Carl Lumma
>> >ith equal 6, I get .2 for Rothenberg stability.
>> How are you getting that? If I delete the 3rd
>> degree, it goes up to .4!
>
> Grrmbl, this is rather shameful, don't know how
> I got so sloppy. Anyhow, the bug krept in because
> of an optimisation, using intermediate variables
> for different calculations. I should test better.
I used to test software for a living, and IMO you've
got more complexity in Scala than any one person
could test in their spare time, even if the data model
was perfect.
By the way, the installer puts a desktop shortcut
even if I tell it not to. ;)
-Carl
Message: 6531 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 19:39:05
Subject: Re: scala show data
From: Carl Lumma
Looks like the Windows version is fixed too (without a
version increment)... thanks, Manuel.
Even *I* don't use the console version anymore, though
it's one of the best command-line interfaces I've used.
I think it's cool that you are still maintaining it.
I'm really impressed with the speed of 2.05. Though it
is 9 megs of widgets, and one can't copy and paste to
the Windows clipboard as was possible with the console
version. . .
. . .Okay, I've installed both. :)
-Carl
Message: 6532 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 19:46:37
Subject: Re: lattice diagram "levels" of complexity
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>> Right, right, you're extending the tones out to a EF Genus in all
>>> directions.
>>
>> example?
>
> I think that's the way I accounted for a 92-tone structure.
how about a much simpler example, with much fewer factors? i'm
looking to understand what you mean by "extending the tones out to a
EF Genus in all directions"
>>> A stellated EF Genus is what I called it in that thread.
>>
>> to my great consternation . . .
>
> What you never showed is how the definition of stellation on
> George Hart's site requires the 92-tone, and not the 80-tone
> structure.
>
> -Carl
i think manuel has got that down.
Message: 6533 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 12:17:25
Subject: Re: lattice diagram "levels" of complexity
From: Carl Lumma
>how about a much simpler example, with much fewer factors?
There really aren't any, since the in the 4-factor case the
"stellated hexany" comes out the same either way. The 5-factor
case is already > 3-D, and doesn't have a symmetrical raw CPS.
The number of tones in a stellated CPS, according to me, is:
> 3 3
> N! (M + (N-M) - N) N!
> ------- + ------------------
> M!(N-M)! (M+1)!(N-M+1)!
I can't remember if this works for 'unsymmetrical' CPSs. In
the case of the dekany, there are 5 triads and 5 tetrads to
complete. That's 15 new notes, 25 notes in all.
Plugging in 2)5 or 3)5 to the above, we get 10 for the first
term and 25 for the second term. So either it doesn't work
on unsymmetrical CPSs, I've missed something in the above
paragraph, or the first term is redundant.
>i'm looking to understand what you mean by "extending the tones
>out to a EF Genus in all directions"
If we imagine completing the triads of a hexany, we see that the
4-factor EF Genus does this for two only two of the triads. If
you look at the mating pattern for "the bottom line" (fig.20
D'Alessandro), and imagine rotating it around the lines of the
eikosany, I believe you get fig.20b.
-Carl
Message: 6534 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 20:39:43
Subject: Re: lattice diagram "levels" of complexity
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> i'm looking to understand what you mean by "extending the tones
>> out to a EF Genus in all directions"
>
> If we imagine completing the triads of a hexany, we see that the
> 4-factor EF Genus does this for two only two of the triads.
right . . .
> If
> you look at the mating pattern for "the bottom line" (fig.20
> D'Alessandro), and imagine rotating it around the lines of the
> eikosany, I believe you get fig.20b.
ok, but the original "EF Genus"ness is irrelevant. there are simpler
structures that, when rotated to all possible positions, will do the
job, and you're kind of mixing apples and oranges in
that "directions" in the EF world would really just refer to the
primes, while in the CPS world there are more operative "directions".
Message: 6535 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 06:08:26
Subject: Re: lattice diagram "levels" of complexity
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> i don't really get the appeal of this (what would level 3 be?), but
> it's somewhat similar to paul hahn's diameter measure. a *single*
> tetrad has a diameter of 1, while *either* a hexany *or* a diamond
> would have a diameter of 2, since you'd need no more than 2
consonant
> intervals to connect any two pitches . . .
This is graph theory language.
Message: 6536 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 22:14:24
Subject: updated 5-limit commas and temperaments table for monz
From: wallyesterpaulrus
see below. i've just updated the table, as well as the zoom-100 graph
Yahoo groups: /tuning-math/files/Paul/zooms.gif * [with cont.]
and the zoom-1000 graph
Yahoo groups: /tuning-math/files/Paul/zoomt.gif * [with cont.]
to include "counterschismic" -- the family that includes three of
monz's favorites, 53, 306, and 730 . . .
the table below not only needs to replace the table on the equal
temperament page, but would also be (and so would the new small 5-
limit intervals hexagonal graph) a fine addition to the small 5-limit
intervals page and to the linear temperament page . . .
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
>>> can you please post the link to the table again?
>>> i'll try to get right on it.
>>>
>>>
>>> -monz
>>
>> the table can be sorted any way you please, but i think most
> helpful
>> for the user would be alphabetically by temperament name, so that
> one
>> can see a green line on the graph and immediately find the
relevant
>> info:
>>
>>
Yahoo groups: /tuning/database? * [with cont.]
>> method=reportRows&tbl=10&sortBy=6
>>
>> my second choice would be sorting by the "odd limit" of the
> vanishing
>> comma's ratio:
>>
>>
Yahoo groups: /tuning/database? * [with cont.]
>> method=reportRows&tbl=10&sortBy=4
>>
>> feel free to replace the 60+-digit numerators and denominators
with
>> the scientific notation on your current table, if you wish.
>>
>> thanks again and let me know when you have time again to revisit
> your
>> wonderful Dictionary, so that i may help improve it.
>>
>> -paul
>
> hi monz,
>
> i see you've incorporated my other corrections for your equal
> temperament page -- THANKS!! -- but haven't updated the table
> discussed above.
>
> is there something i can do to the format to make it easier for you?
>
> a third useful way of sorting the table would be according to
> decreasing rms error -- how close the green line comes to the
origin:
>
>
Yahoo groups: /tuning/database? * [with cont.]
> method=reportRows&tbl=10&sortBy=10&sortDir=up
>
> hopefully one of these three sorting methods will be satisfactory
for
> you.
>
> btw, credit for this table (which i compiled) belongs not only to
> carl lumma, but to pretty much the entire tuning-math team, and
> primarily gene.
>
> thanks,
> paul
Message: 6537 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 06:12:03
Subject: Re: lattice diagram "levels" of complexity
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> I said "lattice region". Gene has used the term
> "ball", and I think that's a convex hull, plus
> everything inside. zthat right, Gene?
That would be a closed ball in the particular metric that convex
region and origin defined.
Message: 6538 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 22:22:57
Subject: Re: Diatonics (Was: Huron Voice Leading)
From: Graham Breed
Carl Lumma wrote:
> Must not be the right linear temperament then?
I think it works with Orwell, but the Orwell generator is half the
generator of the Rothenberg diatonic. Otherwise, you can't get 6:5,
8:7, 11:8 and 11:9 all at the same time. It's like the diatonic Mark
proposed which is half-Magic. In both cases 3:2 isn't produced by the
diatonic generator.
Graham
Message: 6539 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 22:10:04
Subject: bingo cards for monz
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
>
>> i'd like to include hexagonal graphs for *all* the EDOs on my
>> "equal temperament" page *and* on the "bingo lattice" page.
>
>>
>> keep sending me stuff ... i'll incorporate it as i have the
chance.
>
> i've done them for all ets from 7 through 80 which are both 5-limit
> consistent and "non-torsional".
>
> they're the first 50 or so files on this page:
>
>
Yahoo groups: /tuning-math/files/Paul/ * [with cont.]
>
> enjoy!
Message: 6540 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 09:43:58
Subject: Re: scala show data
From: manuel.op.de.coul@xxxxxxxxxxx.xxx
Carl wrote:
>With equal 6, I get .2 for Rothenberg stability.
>How are you getting that? If I delete the 3rd
>degree, it goes up to .4!
Grrmbl, this is rather shameful, don't know how
I got so sloppy. Anyhow, the bug krept in because
of an optimisation, using intermediate variables
for different calculations. I should test better.
Manuel
Message: 6541 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 15:19:16
Subject: Re: lattice diagram "levels" of complexity
From: monz
hi paul, Gene, Carl, and others in this thread,
> From: <gwsmith@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, February 18, 2003 10:08 PM
> Subject: [tuning-math] Re: lattice diagram "levels" of complexity
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
>> i don't really get the appeal of this (what would
>> level 3 be?), but it's somewhat similar to paul hahn's
>> diameter measure. a *single* tetrad has a diameter
>> of 1, while *either* a hexany *or* a diamond would
>> have a diameter of 2, since you'd need no more than
>> 2 consonant intervals to connect any two pitches . . .
>
> This is graph theory language.
well ... since "level 2" results from building
complete otonalities and utonalities on all the
ratios which lie on the outside of the "level 1"
structure ...
i suppose "level 3" would be the structure which
results from building complete otonalities and
utonalities on the ratios which lie on the outside
of the "level 2" structure.
no?
so anyway, is "diameter" more-or-less accepted as
the standard terminology for this kind of thing?
i have my reasons for needing this ... it has to
do with my software project.
-monz
Message: 6542 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 10:49:01
Subject: Diatonics (Was: Huron Voice Leading)
From: Graham Breed
Carl Lumma wrote:
> While many of their criteria were the same, many were not. So it
must
> be considered a strange coincidence that R. and B. wind up
recommending
> the same scale, R in 31-tET, and B in 20-tET. B missed the 31-tET
> version because it didn't have all the other nonsense properties
that
> he was so fascinated with (such as the product of the sizes of the
3rds
> giving the number of notes in the embedding et). Dan Stearns
further
> independently suggested this scale in 20-tET for his own reasons.
But
> AFAIK I'm the first to notice its excellent approximations to 5:3
and
> 7:4 nicely interleaved on its 8ths. The 31-tET version gets closer
to
> JI, but the 20-tET version has higher Lumma stability and already
gets
> you closer than 12-tET to these intervals.
3 3 5 3 3 3 5 3 3 =31
2 2 3 2 2 2 3 2 2 =20
Where does Rothenberg give this scale? I've got the three
Mathematical Systems Theory papers in Manuel's biblography, and can't
find it.
The generator approximates 11:8. That ties in with the 5:3 and 7:4 if
you temper out 385:384. I can get a linear temperament from the best
(not nearest prime) approximations of 31- and 20-equal:
23/51, 541.9 cent generator
basis:
(1.0, 0.45158026468779938)
mapping by period and generator:
[(1, 0), (7, -12), (10, -17), (1, 4), (3, 1)]
mapping by steps:
[(31, 20), (49, 32), (72, 47), (87, 56), (107, 69)]
highest interval width: 28
complexity measure: 28 (31 for smallest MOS)
highest error: 0.009059 (10.871 cents)
unique
It should come up in a search for pairs of the simplest 30 ETs with a
consistency cutoff of 0.8. There are lots of other consonances you
get in the 31-equal version of the scale that don't match this
> The lower stability of Rothenberg's scale kicks it down to position
6
> on my gd spreadsheet -- Balzano's version is at position 3, just
below
> the pentatonic and diatonic scales...
>
>
* [with cont.] (Wayb.)
> All the scales are available as scala files...
>
>
* [with cont.] (Wayb.)
Graham
Message: 6543 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 23:25:41
Subject: Re: lattice diagram "levels" of complexity
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi paul, Gene, Carl, and others in this thread,
>
>
>> From: <gwsmith@s...>
>> To: <tuning-math@xxxxxxxxxxx.xxx>
>> Sent: Tuesday, February 18, 2003 10:08 PM
>> Subject: [tuning-math] Re: lattice diagram "levels" of complexity
>>
>>
>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
>> >
>>> i don't really get the appeal of this (what would
>>> level 3 be?), but it's somewhat similar to paul hahn's
>>> diameter measure. a *single* tetrad has a diameter
>>> of 1, while *either* a hexany *or* a diamond would
>>> have a diameter of 2, since you'd need no more than
>>> 2 consonant intervals to connect any two pitches . . .
>>
>> This is graph theory language.
>
>
> well ... since "level 2" results from building
> complete otonalities and utonalities on all the
> ratios which lie on the outside of the "level 1"
> structure ...
um . . . not exactly . . . the way you did it was to build
otonalities on *some* of them, and utonalities on *some* others.
> i suppose "level 3" would be the structure which
> results from building complete otonalities and
> utonalities on the ratios which lie on the outside
> of the "level 2" structure.
>
> no?
it doesn't sound like you have a precisely defined concept here. i
hope you'll prove me wrong!
> so anyway, is "diameter" more-or-less accepted as
> the standard terminology for this kind of thing?
yes, but be precise. examples of a diameter-1 structure in the 7-
limit lattice would include the otonal tetrad and the utonal tetrad,
but not the agglomeration of both stuck together. examples of
diameter-2 include the hexany and the diamond. your original "level-
1" structure also has a diameter of 2, since to get from any note in
the utonality to any note in the otonality, assuming neither is 1/1,
you have to pass through 1/1, and thus make 2 steps. there are of
course many other diameter-2 structures possible.
Message: 6544 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 15:43:56
Subject: Re: lattice diagram "levels" of complexity
From: Carl Lumma
>ok, but the original "EF Genus"ness is irrelevant.
I agree. But I say the same is then true of the 92-tone
structure being a "stellated eikosany". Pending otherwise
via the 'official definition of stellation'. . .
-Carl
Message: 6545 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 23:49:16
Subject: Re: lattice diagram "levels" of complexity
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> ok, but the original "EF Genus"ness is irrelevant.
>
> I agree. But I say the same is then true
how is the same true? the 92-tone structure has the right symmetry to
begin with.
> of the 92-tone
> structure being a "stellated eikosany". Pending otherwise
> via the 'official definition of stellation'. . .
>
> -Carl
Message: 6546 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 23:50:08
Subject: Re: lattice diagram "levels" of complexity
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>> ok, but the original "EF Genus"ness is irrelevant.
>
> I agree. But I say the same is then true of the 92-tone
> structure being a "stellated eikosany". Pending otherwise
> via the 'official definition of stellation'. . .
>
> -Carl
there is more than one way to stellate something. check this out:
Stellations of the Rhombic Triacontahedron * [with cont.] (Wayb.)
Message: 6547 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 15:51:40
Subject: Re: lattice diagram "levels" of complexity
From: Carl Lumma
>> >his is graph theory language.
>
>
>well ... since "level 2" results from building
>complete otonalities and utonalities on all the
>ratios which lie on the outside of the "level 1"
>structure ...
I think Gene meant that "diameter", not "level",
is graph theory lang.
>i suppose "level 3" would be the structure which
>results from building complete otonalities and
>utonalities on the ratios which lie on the outside
>of the "level 2" structure.
>
>no?
That's your call! Have you verified that such a
structure is the collection of lattice points
within a radius of 3 from a given point?
>so anyway, is "diameter" more-or-less accepted as
>the standard terminology for this kind of thing?
No, diameter is diameter...
Graph Diameter -- from MathWorld * [with cont.]
I don't think it's really related to your levels.
-Carl
Message: 6548 - Contents - Hide Contents
Date: Wed, 19 Feb 2003 16:35:06
Subject: Re: lattice diagram "levels" of complexity
From: Carl Lumma
>>> >k, but the original "EF Genus"ness is irrelevant.
>>
>> I agree. But I say the same is then true
>
>how is the same true? the 92-tone structure has the
>right symmetry to begin with.
Note that Wilson's 20b is not called "stellated eikosany".
[Oh, by the way, it's obvious that "stellated EF Genus" is awful.
I won't push that anymore.]
Here's some quotes from the old thread...
>a CPS is supposed to be a fancy subset of a tonespace, not a gross
>chunk of it. I find it much more natural to think of stellation as
>simply completing all the chords I had in my original structure.
>Using them all is enough of a challenge, without extras besides.
>As I've been saying, we're running up against adjacent pentadekanies
>here, and completing the pentads to hexads (while ignoring the
>incomplete triads, I might add). So if, as I suggested way back in
>this thread, we add 1 to m at each iteration, stellation terminates
>when m=n. For the hebdomekontany, we must add three tones to each
>chord at the first iteration, then two, and then 1 (again, ignoring
>the "lesser" deficient chords in the adjacent CPSs -- else, three,
>two and six, one and seven).
>
>All this isn't worth the trouble. Might as well just say, "I'm going
>to use the entire lattice". The idea of the CPS as a structure
>providing "special" access to the relations in a tonespace looses all
>meaning.
-Carl
Message: 6549 - Contents - Hide Contents
Date: Thu, 20 Feb 2003 10:21:20
Subject: Re: Reduced generators and special commas
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> *The* special comma -- couldn't there be more than one? In
meantone,
> i=4 gives 81:80, i=12 the Pythagorean comma, etc. Right?
I defined it so that there is only one. Sometimes that may be a bit
arbitrary, as in the case of Amity, where there are other reductions
of the generator with height not too much higher, leading to
different commas. However, in your case 4 is less than 12, so 12 is
out of it; moreover, 12 isn't actually mapped to a 7-limit consonance
anyway.
By my definition, i=3 is the first value to consider, and
we get (12/5)/(4/3)^3 = 81/80 as the special comma. We get the same
for i=4 and (16/5)/(4/3)^4, but i=6 gives us (4/3)^6/(28/5) =
5120/5103, i=9 leads to (96/7)/(4/3)^9 = 59049/57344, and finally
i=10 to (128/7)/(4/3)^10 = 59049/57344. Hence, septimal meantone has
special commas [81/80, 5120/5103, 59049/57344] in that order, if we
want to extend the definition to secondary and tertiary special
commas (and etc.)
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