Tuning-Math Digests messages 6525 - 6549

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Message: 6525

Date: Tue, 18 Feb 2003 23:44:49

Subject: monz page: new 5-limit names (was: A common notation...)

From: monz

> From: <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, February 18, 2003 3:25 PM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> Onelist Tuning Digest # 483 message 26, (c)2000 by Joe Monzo *
>
>
> thanks monz. shall we move on to the next correction now?



yes, please do.




-monz


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Message: 6526

Date: Wed, 19 Feb 2003 17:39:27

Subject: Re: scala show data

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

I've made an update to 1.84 of the command line version
in case anyone still uses it. Show data is also fixed
of course so Rothenberg stabilitity = 1 for ETs.

http://www.xs4all.nl/~huygensf/software/scala18win.zip - Ok *

Manuel


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Message: 6527

Date: Wed, 19 Feb 2003 09:51:29

Subject: Re: Diatonics (Was: Huron Voice Leading)

From: Carl Lumma

>Where does Rothenberg give this scale?  I've got the three
>Mathematical Systems Theory papers in Manuel's biblography, and can't
>find it.

I don't actually know, but John Chalmers told me about it, and it's
in the Scala scale archive.

>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 

Must not be the right linear temperament then?

-Carl


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Message: 6529

Date: Wed, 19 Feb 2003 10:35:31

Subject: Re: lattice diagram "levels" of complexity

From: Carl Lumma

>> 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.

>> 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


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Message: 6530

Date: Wed, 19 Feb 2003 19:23:20

Subject: Re: scala show data

From: Carl Lumma

>>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.

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


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Message: 6531

Date: Wed, 19 Feb 2003 19:39:05

Subject: Re: scala show data

From: Carl Lumma

> I've made an update to 1.84 of the command line version
> in case anyone still uses it. Show data is also fixed
> of course so Rothenberg stabilitity = 1 for ETs.
> 
> http://www.xs4all.nl/~huygensf/software/scala18win.zip - Ok *
> 
> Manuel

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


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Message: 6533

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


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Message: 6538

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


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Message: 6540

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


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Message: 6541

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


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Message: 6542

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...
> 
> http://lumma.org/stuff/gd.xls *

For those who can't read Excel, here's the ranking:

05_pentatonic
07_diatonic
09_balzano-20
08_octatonic
10_subset-13
09_rothenberg
10_blackwood
10_pent-major
06_hexatonic
09_trichordal-433
10_sym-major
08_nova
08_star
07_qm(2)
09_quadrafourths
07_hungarian-minor
06_super7
07_porcupine
09_trichordal-334
05_sss9-31
10_qm(3)
07_harmonic-minor
07_melodic-minor
10_quadrafourths
10_lumma-x2-1
06_sss9-22
08_kleismic
07_neutral-thirds-b-31
10_miracle
09_orwell
07_hungarian-major
10_gamelion
09_tetra-major
06_sss9-31
10_lumma-x2-2
06_mode6

> All the scales are available as scala files...
> 
> http://lumma.org/stuff/gd-scl.zip *


                   Graham


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Message: 6544

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


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Message: 6547

Date: Wed, 19 Feb 2003 15:51:40

Subject: Re: lattice diagram "levels" of complexity

From: Carl Lumma

>> 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 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 *

I don't think it's really related to your levels.

-Carl


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Message: 6548

Date: Wed, 19 Feb 2003 16:35:06

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
>
>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


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