Tuning-Math Digests messages 2950 - 2974

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Message: 2950

Date: Thu, 03 Jan 2002 11:16:19

Subject: Some 8-tone 72-et scales

From: genewardsmith

[0, 5, 12, 19, 35, 42, 58, 65]
[5, 7, 7, 16, 7, 16, 7, 7]
edges   11   17   22   connectivity   2   3   5

[0, 5, 12, 19, 35, 42, 49, 65]
[5, 7, 7, 16, 7, 7, 16, 7]
edges   11   18   21   connectivity   1   3   4

[0, 5, 12, 19, 26, 42, 49, 65]
[5, 7, 7, 7, 16, 7, 16, 7]
edges   9   15   21   connectivity   0   2   4

[0, 5, 12, 19, 26, 42, 58, 65]
[5, 7, 7, 7, 16, 16, 7, 7]
edges   7   13   21   connectivity   0   2   5

[0, 5, 12, 28, 35, 42, 49, 65]
[5, 7, 16, 7, 7, 7, 16, 7]
edges   9   17   20   connectivity   1   3   4

[0, 5, 12, 19, 35, 42, 49, 56]
[5, 7, 7, 16, 7, 7, 7, 16]
edges   8   17   20   connectivity   0   3   4

[0, 5, 12, 19, 26, 42, 49, 56]
[5, 7, 7, 7, 16, 7, 7, 16]
edges   8   16   20   connectivity   0   2   4

[0, 5, 12, 19, 26, 33, 49, 56]
[5, 7, 7, 7, 7, 16, 7, 16]
edges   6   13   19   connectivity   0   1   3

[0, 5, 12, 19, 26, 33, 49, 65]
[5, 7, 7, 7, 7, 16, 16, 7]
edges   5   11   19   connectivity   0   1   3

[0, 5, 12, 28, 35, 42, 49, 56]
[5, 7, 16, 7, 7, 7, 7, 16]
edges   6   14   18   connectivity   0   2   4

[0, 5, 21, 28, 35, 42, 49, 56]
[5, 16, 7, 7, 7, 7, 7, 16]
edges   4   13   18   connectivity   0   2   3

[0, 5, 12, 19, 26, 33, 40, 56]
[5, 7, 7, 7, 7, 7, 16, 16]
edges   3   11   18   connectivity   0   1   3


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Message: 2951

Date: Thu, 3 Jan 2002 13:38 +00

Subject: Re: Some 9-tone 72-et scales

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a1173r+ob05@xxxxxxx.xxx>
gene wrote:

> I'd need to write the code for it, and it isn't a graph property so I'm 
> not going to start with any advantage from the Maple graph theory 
> package. Paul did not think propriety was very important--what's your 
> take on it?

I don't have a definitive answer.  It, or something like it, may be 
important for modality.  Especially for subsets of "comprehensible" ETs.  
The Pythagorean diatonic works fine despite being slightly improper, so 
you shouldn't be over-strict.  For a scale with three step sizes to be 
proper shows that it has a certain level of cohesion.  The extremely 
improper Magic subsets you gave composers, performers and listeners are 
likely to expect the large gaps to be filled in by more notes.  This is a 
general problem with Magic scales of between 3 and 19 notes.

The Decimal MOS has the opposite problem -- its step sizes are so closely 
equal that it doesn't have any shape.  So it's great as a basis for 
notation, but doesn't have any sense of tonal center.  Your top 10 note 
scale might solve this problem, because it's largest interval is almost 
twice the size of its smallest.  And that single 5/72 step could be 
extremely important for leading to the tonic.

10 notes still seems like a lot for a mode.  Perhaps the 6 note [12, 14, 
9, 14, 9, 14] would work.  How is it in terms of connectedness?


                    Graham


> > It's interesting that so many scales came out proper 
> > when that wasn't a criterion in the search.  All the 10-note 72= 
> > scales are strictly proper.
> 
> It's also interesting that the best scores were all proper.
> 
> 
> 
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@xxxxxxxxxxx.xxx
> 
>  
> 
> Your use of Yahoo! Groups is subject to 
> Yahoo! Terms of Service * 
> 
>


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Message: 2952

Date: Thu, 03 Jan 2002 20:19:01

Subject: Re: Some 9-tone 72-et scales

From: clumma

>>You might like to read Rothenberg's original papers on the subject.
>>There's graph stuff in there that none of us have touched (propriety
>>was just a starting point for Rothenberg), plus a fancy algorithm
>>generating all the proper subsets of a scale.
> 
>Where might Rothenberg's papers be found?

Tuning bibliography at:

Tuning & temperament bibliography *

The papers:

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications. Part I: Pitch Structures as Order-Preserving Maps",
Mathematical Systems Theory vol. 11, 1978, pp. 199-234.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part II: The Information Content of Pitch structures",
Mathematical Systems Theory vol. 11, 1978, pp. 353-372.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part III: The Graph Embedding of Pitch Structures",
Mathematical Systems Theory vol. 12, 1978, pp. 73-101.

I'd send you copies, but my copies are locked away in Montana.

-Carl


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Message: 2953

Date: Thu, 03 Jan 2002 20:27:13

Subject: Re: Some 9-tone 72-et scales

From: clumma

>I don't have a definitive answer.  It, or something like it, may be
>important for modality.  Especially for subsets of "comprehensible"
>ETs.

Needless to say, I think it's very important.  There's nothing
against improper scales -- it isn't that kind of criterion.  But
R. shows that certain musical devices rely on proper scales.  These
effects are still available on proper subsets of improper scales,
but if you want some of the effects of traditional diatonic music,
where the entire pitch set is involved with these effects, then you
need propriety over the whole scale.

>The Pythagorean diatonic works fine despite being slightly
>improper, so you shouldn't be over-strict.

Right, which is why R. never uses propriety -- he uses stability.

>For a scale with three step sizes to be proper shows that it has a
>certain level of cohesion.

I'll take this opportunity to stamp out the myth of the importance
of 2nds.  The variety of all the classes of scale intervals are
equally important.  So you need the variety of 2nds, and the
ordering.  To speak of only the variety is scale voodoo.  IMO, the
best measure is simply R.'s mean variety.

>The Decimal MOS has the opposite problem -- its step sizes are so
>closely equal that it doesn't have any shape.

That's an issue of efficiency, not propriety.

-Carl


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Message: 2954

Date: Thu, 03 Jan 2002 20:35:53

Subject: connectivity index?

From: clumma

Gene, have you considered making a general measure out of
connectedness that takes into account the number of notes
and the limit?  Not sure how c looks over the saturated
otonal chords, but certainly things get tougher directly
with the number of notes and inversely with the limit.

-Carl


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Message: 2955

Date: Thu, 03 Jan 2002 21:46:21

Subject: Re: Some 9-tone 72-et scales

From: genewardsmith

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> >The Decimal MOS has the opposite problem -- its step sizes are so
> >closely equal that it doesn't have any shape.
> 
> That's an issue of efficiency, not propriety.

I don't know what "efficiency" means
in this connection, but it sounds to me like what you were talking
about--variety. I haven't jumped on the Miracle bandwagon precisely
because the pudding-like sameness of the Decimal put me off; but the
72-et scales I've been cooking up are just the sort of thing I like.


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Message: 2956

Date: Thu, 03 Jan 2002 22:03:53

Subject: Re: Some 9-tone 72-et scales

From: clumma

>>>The Decimal MOS has the opposite problem -- its step sizes are so
>>>closely equal that it doesn't have any shape.
>> 
>>That's an issue of efficiency, not propriety.
>
>I don't know what "efficiency" means in this connection, but it
>sounds to me like what you were talking about--variety. I haven't
>jumped on the Miracle bandwagon precisely because the pudding-like
>sameness of the Decimal put me off; but the 72-et scales I've been
>cooking up are just the sort of thing I like.

Try message number 4044 on the main list for a general overview
of and quotes from the Rothenberg papers, including the formula
for efficiency.

-Carl


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Message: 2957

Date: Thu, 03 Jan 2002 22:09:37

Subject: Re: Some 9-tone 72-et scales

From: clumma

I wrote...

>Try message number 4044 on the main list for a general overview
>of and quotes from the Rothenberg papers, including the formula
>for efficiency.

Yahoo has removed the tabs from my message, and least on
the web, so you can tell where the citations start and stop..
Egregious.

-C.


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Message: 2958

Date: Thu, 03 Jan 2002 22:17:31

Subject: Re: Some 9-tone 72-et scales

From: clumma

>>Try message number 4044 on the main list for a general overview
>>of and quotes from the Rothenberg papers, including the formula
>>for efficiency.
> 
>Yahoo has removed the tabs from my message, and least on
>the web, so you can tell where the citations start and stop..
>Egregious.

That's supposed to be _can't_ tell.

-Carl


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Message: 2959

Date: Thu, 3 Jan 2002 22:22:30

Subject: Re: coordinates from unison-vectors

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Thursday, January 03, 2002 9:37 PM
> Subject: [tuning-math] Re: coordinates from unison-vectors
>
>
> Here's my Matlab code for doing this. I arbitrarily start with a 101-
> by-101 square of lattice points. Rye is the 2-by-2 matrix of unison 
> vectors. t is the set of points inside the PB.
> 
> 
> 
> for a=-50:50;
> l(a+51,:)=a*ones(1,101);
> m(:,a+51)=a*ones(101,1);
> end;
> p=[l(:) m(:)];
> s=p*inv(rye);
> s(find(s(:,1)>.50000001),:)=[];
> s(find(s(:,2)>.50000001),:)=[];
> s(find(s(:,2)<-.49999999),:)=[];
> s(find(s(:,1)<-.49999999),:)=[];
> t=s*rye



Thanks, Paul!  This should help me a bit.



-monz

 



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Message: 2960

Date: Thu, 03 Jan 2002 03:14:43

Subject: Some 8-note 7-limit scales

From: genewardsmith

These are based on (21/20)^3 (15/14) (10/9)^2 (8/7)^2 = 2

[1, 21/20, 9/8, 5/4, 21/16, 3/2, 63/40, 7/4]
[21/20, 15/14, 10/9, 21/20, 8/7, 21/20, 10/9, 8/7]
edges   16   connectivity   3

[1, 21/20, 7/6, 49/40, 7/5, 147/100, 49/30, 7/4]
[21/20, 10/9, 21/20, 8/7, 21/20, 10/9, 15/14, 8/7]
edges   16   connectivity   2

[1, 21/20, 9/8, 5/4, 21/16, 3/2, 63/40, 9/5]
[21/20, 15/14, 10/9, 21/20, 8/7, 21/20, 8/7, 10/9]
edges   15   connectivity   2

[1, 21/20, 9/8, 5/4, 21/16, 3/2, 5/3, 7/4]
[21/20, 15/14, 10/9, 21/20, 8/7, 10/9, 21/20, 8/7]
edges   15   connectivity   2

[1, 21/20, 9/8, 9/7, 27/20, 3/2, 63/40, 9/5]
[21/20, 15/14, 8/7, 21/20, 10/9, 21/20, 8/7, 10/9]
edges   15   connectivity   1

[1, 21/20, 9/8, 9/7, 27/20, 3/2, 12/7, 9/5]
[21/20, 15/14, 8/7, 21/20, 10/9, 8/7, 21/20, 10/9]
edges   14   connectivity   2

[1, 21/20, 9/8, 9/7, 10/7, 3/2, 12/7, 9/5]
[21/20, 15/14, 8/7, 10/9, 21/20, 8/7, 21/20, 10/9]
edges   14   connectivity   2

[1, 21/20, 9/8, 189/160, 21/16, 3/2, 63/40, 7/4]
[21/20, 15/14, 21/20, 10/9, 8/7, 21/20, 10/9, 8/7]
edges   14   connectivity   1

[1, 21/20, 9/8, 189/160, 21/16, 3/2, 63/40, 9/5]
[21/20, 15/14, 21/20, 10/9, 8/7, 21/20, 8/7, 10/9]
edges   14   connectivity   1

[1, 21/20, 9/8, 5/4, 10/7, 3/2, 5/3, 7/4]
[21/20, 15/14, 10/9, 8/7, 21/20, 10/9, 21/20, 8/7]
edges   14   connectivity   1

[1, 21/20, 7/6, 5/4, 25/18, 35/24, 5/3, 7/4]
[21/20, 10/9, 15/14, 10/9, 21/20, 8/7, 21/20, 8/7]
edges   14   connectivity   1

[1, 21/20, 441/400, 63/50, 7/5, 147/100, 63/40, 7/4]
[21/20, 21/20, 8/7, 10/9, 21/20, 15/14, 10/9, 8/7]
edges   13   connectivity   2

[1, 21/20, 9/8, 5/4, 21/16, 3/2, 12/7, 9/5]
[21/20, 15/14, 10/9, 21/20, 8/7, 8/7, 21/20, 10/9]
edges   13   connectivity   2

[1, 21/20, 9/8, 5/4, 10/7, 3/2, 12/7, 9/5]
[21/20, 15/14, 10/9, 8/7, 21/20, 8/7, 21/20, 10/9]
edges   13   connectivity   2

[1, 21/20, 441/400, 49/40, 7/5, 147/100, 63/40, 7/4]
[21/20, 21/20, 10/9, 8/7, 21/20, 15/14, 10/9, 8/7]
edges   13   connectivity   2

[1, 21/20, 441/400, 49/40, 21/16, 3/2, 63/40, 7/4]
[21/20, 21/20, 10/9, 15/14, 8/7, 21/20, 10/9, 8/7]
edges   13   connectivity   1

[1, 21/20, 441/400, 49/40, 7/5, 147/100, 49/30, 7/4]
[21/20, 21/20, 10/9, 8/7, 21/20, 10/9, 15/14, 8/7]
edges   13   connectivity   1

[1, 21/20, 441/400, 49/40, 7/5, 3/2, 63/40, 7/4]
[21/20, 21/20, 10/9, 8/7, 15/14, 21/20, 10/9, 8/7]
edges   13   connectivity   1

[1, 21/20, 7/6, 49/40, 49/36, 343/240, 49/30, 7/4]
[21/20, 10/9, 21/20, 10/9, 21/20, 8/7, 15/14, 8/7]
edges   12   connectivity   2

[1, 21/20, 9/8, 189/160, 27/20, 3/2, 63/40, 7/4]
[21/20, 15/14, 21/20, 8/7, 10/9, 21/20, 10/9, 8/7]
edges   12   connectivity   2

[1, 21/20, 9/8, 9/7, 27/20, 3/2, 63/40, 7/4]
[21/20, 15/14, 8/7, 21/20, 10/9, 21/20, 10/9, 8/7]
edges   12   connectivity   2

[1, 21/20, 9/8, 5/4, 21/16, 35/24, 5/3, 7/4]
[21/20, 15/14, 10/9, 21/20, 10/9, 8/7, 21/20, 8/7]
edges   12   connectivity   1

[1, 21/20, 441/400, 189/160, 21/16, 3/2, 63/40, 7/4]
[21/20, 21/20, 15/14, 10/9, 8/7, 21/20, 10/9, 8/7]
edges   12   connectivity   1

[1, 21/20, 7/6, 49/40, 49/36, 35/24, 5/3, 7/4]
[21/20, 10/9, 21/20, 10/9, 15/14, 8/7, 21/20, 8/7]
edges   12   connectivity   1

[1, 21/20, 9/8, 9/7, 10/7, 3/2, 5/3, 7/4]
[21/20, 15/14, 8/7, 10/9, 21/20, 10/9, 21/20, 8/7]
edges   11   connectivity   2

[1, 21/20, 441/400, 189/160, 21/16, 3/2, 63/40, 9/5]
[21/20, 21/20, 15/14, 10/9, 8/7, 21/20, 8/7, 10/9]
edges   11   connectivity   1

[1, 21/20, 441/400, 189/160, 21/16, 441/320, 63/40, 7/4]
[21/20, 21/20, 15/14, 10/9, 21/20, 8/7, 10/9, 8/7]
edges   11   connectivity   1

[1, 21/20, 441/400, 49/40, 21/16, 3/2, 63/40, 9/5]
[21/20, 21/20, 10/9, 15/14, 8/7, 21/20, 8/7, 10/9]
edges   11   connectivity   1

[1, 21/20, 441/400, 49/40, 1029/800, 147/100, 63/40, 7/4]
[21/20, 21/20, 10/9, 21/20, 8/7, 15/14, 10/9, 8/7]
edges   11   connectivity   1

[1, 21/20, 9/8, 5/4, 21/16, 35/24, 49/32, 7/4]
[21/20, 15/14, 10/9, 21/20, 10/9, 21/20, 8/7, 8/7]
edges   11   connectivity   1

[1, 21/20, 441/400, 49/40, 21/16, 441/320, 63/40, 7/4]
[21/20, 21/20, 10/9, 15/14, 21/20, 8/7, 10/9, 8/7]
edges   11   connectivity   1

[1, 21/20, 9/8, 189/160, 21/16, 441/320, 63/40, 7/4]
[21/20, 15/14, 21/20, 10/9, 21/20, 8/7, 10/9, 8/7]
edges   11   connectivity   1

[1, 21/20, 441/400, 189/160, 27/20, 3/2, 63/40, 9/5]
[21/20, 21/20, 15/14, 8/7, 10/9, 21/20, 8/7, 10/9]
edges   11   connectivity   1

[1, 21/20, 441/400, 49/40, 7/5, 147/100, 63/40, 9/5]
[21/20, 21/20, 10/9, 8/7, 21/20, 15/14, 8/7, 10/9]
edges   11   connectivity   1

[1, 21/20, 441/400, 63/50, 1323/1000, 147/100, 63/40, 7/4]
[21/20, 21/20, 8/7, 21/20, 10/9, 15/14, 10/9, 8/7]
edges   11   connectivity   1

[1, 21/20, 441/400, 63/50, 27/20, 3/2, 63/40, 7/4]
[21/20, 21/20, 8/7, 15/14, 10/9, 21/20, 10/9, 8/7]
edges   11   connectivity   1

[1, 21/20, 9/8, 9/7, 27/20, 54/35, 12/7, 9/5]
[21/20, 15/14, 8/7, 21/20, 8/7, 10/9, 21/20, 10/9]
edges   11   connectivity   1

[1, 21/20, 441/400, 49/40, 1029/800, 147/100, 49/30, 7/4]
[21/20, 21/20, 10/9, 21/20, 8/7, 10/9, 15/14, 8/7]
edges   10   connectivity   1

[1, 21/20, 9/8, 9/7, 27/20, 3/2, 5/3, 7/4]
[21/20, 15/14, 8/7, 21/20, 10/9, 10/9, 21/20, 8/7]
edges   10   connectivity   1

[1, 21/20, 9/8, 9/7, 72/49, 54/35, 12/7, 9/5]
[21/20, 15/14, 8/7, 8/7, 21/20, 10/9, 21/20, 10/9]
edges   10   connectivity   1

[1, 21/20, 441/400, 189/160, 27/20, 3/2, 63/40, 7/4]
[21/20, 21/20, 15/14, 8/7, 10/9, 21/20, 10/9, 8/7]
edges   10   connectivity   1

[1, 21/20, 441/400, 49/40, 1029/800, 441/320, 63/40, 7/4]
[21/20, 21/20, 10/9, 21/20, 15/14, 8/7, 10/9, 8/7]
edges   9   connectivity   1

[1, 21/20, 9/8, 189/160, 27/20, 567/400, 63/40, 7/4]
[21/20, 15/14, 21/20, 8/7, 21/20, 10/9, 10/9, 8/7]
edges   9   connectivity   1

[1, 21/20, 441/400, 189/160, 27/20, 567/400, 63/40, 7/4]
[21/20, 21/20, 15/14, 8/7, 21/20, 10/9, 10/9, 8/7]
edges   8   connectivity   1

[1, 21/20, 441/400, 9261/8000, 1323/1000, 147/100, 63/40, 7/4]
[21/20, 21/20, 21/20, 8/7, 10/9, 15/14, 10/9, 8/7]
edges   8   connectivity   1

[1, 21/20, 441/400, 189/160, 3969/3200, 567/400, 63/40, 7/4]
[21/20, 21/20, 15/14, 21/20, 8/7, 10/9, 10/9, 8/7]
edges   7   connectivity   1


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Message: 2961

Date: Thu, 03 Jan 2002 23:10:12

Subject: Some 11-tone 72-et scales

From: genewardsmith

[0, 5, 14, 19, 28, 33, 42, 47, 56, 61, 70]
[5, 9, 5, 9, 5, 9, 5, 9, 5, 9, 2]
edges   13   28   41   connectivity   0   4   7

[0, 5, 14, 19, 28, 37, 42, 47, 56, 61, 70]
[5, 9, 5, 9, 9, 5, 5, 9, 5, 9, 2]
edges   13   26   40   connectivity   0   3   6

[0, 5, 14, 19, 28, 37, 42, 51, 56, 61, 70]
[5, 9, 5, 9, 9, 5, 9, 5, 5, 9, 2]
edges   12   26   40   connectivity   0   3   6

[0, 9, 14, 23, 28, 37, 46, 51, 56, 65, 70]
[9, 5, 9, 5, 9, 9, 5, 5, 9, 5, 2]
edges   12   26   40   connectivity   0   3   6

[0, 5, 14, 19, 28, 33, 42, 51, 56, 61, 70]
[5, 9, 5, 9, 5, 9, 9, 5, 5, 9, 2]
edges   12   26   40   connectivity   0   3   6

[0, 5, 10, 19, 28, 33, 42, 47, 56, 61, 70]
[5, 5, 9, 9, 5, 9, 5, 9, 5, 9, 2]
edges   12   25   40   connectivity   0   2   6

[0, 9, 14, 23, 28, 37, 46, 51, 60, 65, 70]
[9, 5, 9, 5, 9, 9, 5, 9, 5, 5, 2]
edges   11   25   40   connectivity   0   3   6

[0, 9, 14, 19, 28, 37, 42, 51, 56, 61, 70]
[9, 5, 5, 9, 9, 5, 9, 5, 5, 9, 2]
edges   13   24   40   connectivity   0   2   6

[0, 9, 14, 19, 28, 33, 42, 47, 56, 61, 70]
[9, 5, 5, 9, 5, 9, 5, 9, 5, 9, 2]
edges   12   24   40   connectivity   0   1   6

[0, 5, 14, 19, 28, 33, 42, 51, 56, 65, 70]
[5, 9, 5, 9, 5, 9, 9, 5, 9, 5, 2]
edges   11   26   39   connectivity   0   3   6

[0, 5, 14, 19, 28, 33, 42, 47, 56, 65, 70]
[5, 9, 5, 9, 5, 9, 5, 9, 9, 5, 2]
edges   12   26   39   connectivity   0   2   6

[0, 5, 14, 23, 28, 37, 42, 47, 56, 61, 70]
[5, 9, 9, 5, 9, 5, 5, 9, 5, 9, 2]
edges   12   24   39   connectivity   0   3   6

[0, 5, 14, 23, 28, 33, 42, 47, 56, 61, 70]
[5, 9, 9, 5, 5, 9, 5, 9, 5, 9, 2]
edges   12   24   39   connectivity   0   2   6

[0, 9, 14, 23, 32, 37, 42, 51, 56, 65, 70]
[9, 5, 9, 9, 5, 5, 9, 5, 9, 5, 2]
edges   12   24   39   connectivity   0   2   5

[0, 5, 14, 23, 28, 37, 42, 51, 56, 61, 70]
[5, 9, 9, 5, 9, 5, 9, 5, 5, 9, 2]
edges   11   24   39   connectivity   0   2   6

[0, 5, 14, 19, 24, 33, 38, 47, 56, 61, 70]
[5, 9, 5, 5, 9, 5, 9, 9, 5, 9, 2]
edges   11   24   39   connectivity   0   3   5

[0, 9, 14, 23, 32, 37, 46, 51, 60, 65, 70]
[9, 5, 9, 9, 5, 9, 5, 9, 5, 5, 2]
edges   10   23   39   connectivity   0   2   5

[0, 9, 14, 19, 28, 37, 42, 47, 56, 61, 70]
[9, 5, 5, 9, 9, 5, 5, 9, 5, 9, 2]
edges   12   22   39   connectivity   0   1   6

[0, 9, 14, 23, 28, 37, 42, 47, 56, 61, 70]
[9, 5, 9, 5, 9, 5, 5, 9, 5, 9, 2]
edges   11   22   39   connectivity   0   2   6

[0, 5, 14, 23, 28, 33, 42, 47, 56, 65, 70]
[5, 9, 9, 5, 5, 9, 5, 9, 9, 5, 2]
edges   13   24   38   connectivity   0   3   6

[0, 5, 14, 19, 28, 37, 42, 47, 56, 65, 70]
[5, 9, 5, 9, 9, 5, 5, 9, 9, 5, 2]
edges   12   24   38   connectivity   0   2   6

[0, 9, 18, 23, 28, 37, 42, 51, 56, 65, 70]
[9, 9, 5, 5, 9, 5, 9, 5, 9, 5, 2]
edges   12   23   38   connectivity   0   1   5

[0, 9, 14, 19, 28, 37, 46, 51, 56, 65, 70]
[9, 5, 5, 9, 9, 9, 5, 5, 9, 5, 2]
edges   10   23   38   connectivity   0   2   5

[0, 5, 10, 19, 28, 33, 42, 47, 52, 61, 70]
[5, 5, 9, 9, 5, 9, 5, 5, 9, 9, 2]
edges   13   22   38   connectivity   0   2   6

[0, 9, 18, 23, 28, 37, 46, 51, 60, 65, 70]
[9, 9, 5, 5, 9, 9, 5, 9, 5, 5, 2]
edges   12   22   38   connectivity   0   2   6

[0, 5, 14, 23, 28, 33, 42, 51, 56, 61, 70]
[5, 9, 9, 5, 5, 9, 9, 5, 5, 9, 2]
edges   11   22   38   connectivity   0   2   6

[0, 9, 18, 23, 32, 37, 46, 51, 60, 65, 70]
[9, 9, 5, 9, 5, 9, 5, 9, 5, 5, 2]
edges   11   22   38   connectivity   0   1   5

[0, 5, 10, 19, 28, 37, 42, 47, 56, 61, 70]
[5, 5, 9, 9, 9, 5, 5, 9, 5, 9, 2]
edges   11   22   38   connectivity   0   1   5

[0, 5, 10, 19, 28, 33, 42, 51, 56, 61, 70]
[5, 5, 9, 9, 5, 9, 9, 5, 5, 9, 2]
edges   11   21   38   connectivity   0   1   6

[0, 5, 10, 19, 28, 33, 38, 47, 56, 61, 70]
[5, 5, 9, 9, 5, 5, 9, 9, 5, 9, 2]
edges   11   21   38   connectivity   0   2   5

[0, 5, 10, 19, 28, 33, 42, 47, 56, 65, 70]
[5, 5, 9, 9, 5, 9, 5, 9, 9, 5, 2]
edges   11   23   37   connectivity   0   2   5

[0, 5, 14, 23, 28, 37, 46, 51, 56, 65, 70]
[5, 9, 9, 5, 9, 9, 5, 5, 9, 5, 2]
edges   11   22   37   connectivity   0   1   5

[0, 9, 18, 23, 32, 37, 42, 51, 56, 65, 70]
[9, 9, 5, 9, 5, 5, 9, 5, 9, 5, 2]
edges   11   21   37   connectivity   0   2   5

[0, 9, 18, 23, 28, 37, 46, 51, 56, 65, 70]
[9, 9, 5, 5, 9, 9, 5, 5, 9, 5, 2]
edges   11   21   37   connectivity   0   1   5

[0, 9, 14, 23, 32, 37, 42, 51, 56, 61, 70]
[9, 5, 9, 9, 5, 5, 9, 5, 5, 9, 2]
edges   11   21   37   connectivity   0   1   4

[0, 5, 14, 19, 24, 33, 38, 47, 52, 61, 70]
[5, 9, 5, 5, 9, 5, 9, 5, 9, 9, 2]
edges   10   21   37   connectivity   0   2   5

[0, 5, 10, 19, 28, 33, 38, 47, 52, 61, 70]
[5, 5, 9, 9, 5, 5, 9, 5, 9, 9, 2]
edges   12   20   37   connectivity   0   1   5

[0, 9, 14, 19, 28, 37, 46, 51, 56, 61, 70]
[9, 5, 5, 9, 9, 9, 5, 5, 5, 9, 2]
edges   10   20   37   connectivity   0   1   4

[0, 9, 14, 19, 28, 37, 46, 51, 60, 65, 70]
[9, 5, 5, 9, 9, 9, 5, 9, 5, 5, 2]
edges   9   20   37   connectivity   0   1   5

[0, 5, 14, 19, 28, 37, 42, 51, 60, 65, 70]
[5, 9, 5, 9, 9, 5, 9, 9, 5, 5, 2]
edges   10   21   36   connectivity   0   1   5

[0, 5, 14, 23, 28, 37, 46, 51, 60, 65, 70]
[5, 9, 9, 5, 9, 9, 5, 9, 5, 5, 2]
edges   10   21   36   connectivity   0   1   4

[0, 5, 14, 23, 32, 37, 42, 51, 56, 65, 70]
[5, 9, 9, 9, 5, 5, 9, 5, 9, 5, 2]
edges   10   21   36   connectivity   0   1   4

[0, 5, 14, 19, 28, 37, 46, 51, 56, 65, 70]
[5, 9, 5, 9, 9, 9, 5, 5, 9, 5, 2]
edges   9   21   36   connectivity   0   1   4

[0, 9, 18, 23, 28, 37, 42, 51, 56, 61, 70]
[9, 9, 5, 5, 9, 5, 9, 5, 5, 9, 2]
edges   11   19   36   connectivity   0   1   4

[0, 5, 14, 23, 28, 33, 38, 47, 56, 61, 70]
[5, 9, 9, 5, 5, 5, 9, 9, 5, 9, 2]
edges   10   19   36   connectivity   0   1   4

[0, 5, 14, 23, 28, 37, 46, 51, 56, 61, 70]
[5, 9, 9, 5, 9, 9, 5, 5, 5, 9, 2]
edges   9   19   36   connectivity   0   1   4

[0, 5, 14, 23, 28, 33, 42, 47, 52, 61, 70]
[5, 9, 9, 5, 5, 9, 5, 5, 9, 9, 2]
edges   11   20   35   connectivity   0   1   4

[0, 5, 10, 19, 28, 37, 42, 47, 56, 65, 70]
[5, 5, 9, 9, 9, 5, 5, 9, 9, 5, 2]
edges   10   20   35   connectivity   0   1   5

[0, 5, 14, 23, 32, 37, 42, 51, 56, 61, 70]
[5, 9, 9, 9, 5, 5, 9, 5, 5, 9, 2]
edges   9   20   35   connectivity   0   1   3

[0, 9, 18, 23, 32, 41, 46, 51, 60, 65, 70]
[9, 9, 5, 9, 9, 5, 5, 9, 5, 5, 2]
edges   11   19   35   connectivity   0   1   4

[0, 9, 14, 23, 32, 41, 46, 51, 60, 65, 70]
[9, 5, 9, 9, 9, 5, 5, 9, 5, 5, 2]
edges   9   19   35   connectivity   0   1   3

[0, 5, 10, 19, 28, 37, 42, 47, 52, 61, 70]
[5, 5, 9, 9, 9, 5, 5, 5, 9, 9, 2]
edges   11   18   35   connectivity   0   1   4

[0, 9, 14, 23, 32, 37, 42, 47, 56, 61, 70]
[9, 5, 9, 9, 5, 5, 5, 9, 5, 9, 2]
edges   9   18   35   connectivity   0   1   3

[0, 9, 14, 23, 32, 37, 46, 55, 60, 65, 70]
[9, 5, 9, 9, 5, 9, 9, 5, 5, 5, 2]
edges   8   18   35   connectivity   0   1   4

[0, 9, 18, 23, 32, 37, 42, 51, 56, 61, 70]
[9, 9, 5, 9, 5, 5, 9, 5, 5, 9, 2]
edges   10   18   34   connectivity   0   1   4

[0, 9, 18, 23, 28, 37, 42, 47, 56, 61, 70]
[9, 9, 5, 5, 9, 5, 5, 9, 5, 9, 2]
edges   10   18   34   connectivity   0   1   3

[0, 5, 10, 19, 28, 33, 38, 47, 56, 65, 70]
[5, 5, 9, 9, 5, 5, 9, 9, 9, 5, 2]
edges   9   18   34   connectivity   0   1   4

[0, 9, 18, 23, 28, 37, 46, 55, 60, 65, 70]
[9, 9, 5, 5, 9, 9, 9, 5, 5, 5, 2]
edges   9   18   34   connectivity   0   1   4

[0, 5, 10, 19, 24, 33, 42, 51, 56, 65, 70]
[5, 5, 9, 5, 9, 9, 9, 5, 9, 5, 2]
edges   8   18   34   connectivity   0   2   4

[0, 9, 18, 27, 32, 37, 46, 51, 60, 65, 70]
[9, 9, 9, 5, 5, 9, 5, 9, 5, 5, 2]
edges   9   17   34   connectivity   0   1   4

[0, 5, 14, 23, 28, 33, 38, 47, 52, 61, 70]
[5, 9, 9, 5, 5, 5, 9, 5, 9, 9, 2]
edges   9   17   33   connectivity   0   1   4

[0, 5, 14, 19, 24, 29, 38, 47, 52, 61, 70]
[5, 9, 5, 5, 5, 9, 9, 5, 9, 9, 2]
edges   9   17   33   connectivity   0   1   3

[0, 5, 14, 19, 24, 33, 38, 43, 52, 61, 70]
[5, 9, 5, 5, 9, 5, 5, 9, 9, 9, 2]
edges   8   17   32   connectivity   0   1   3

[0, 9, 14, 23, 32, 41, 46, 55, 60, 65, 70]
[9, 5, 9, 9, 9, 5, 9, 5, 5, 5, 2]
edges   7   16   32   connectivity   0   1   3

[0, 9, 18, 23, 32, 37, 42, 47, 56, 61, 70]
[9, 9, 5, 9, 5, 5, 5, 9, 5, 9, 2]
edges   8   16   31   connectivity   0   2   3

[0, 5, 14, 19, 24, 29, 38, 43, 52, 61, 70]
[5, 9, 5, 5, 5, 9, 5, 9, 9, 9, 2]
edges   7   15   29   connectivity   0   1   3


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Message: 2962

Date: Thu, 3 Jan 2002 20:05:08

Subject: updated definition: transformation

From: monz

I've corrected and updated my Dictionary entry
for "transformation":
Definitions of tuning terms: transformation, (c) 2001 by Joe Monzo *

I'd appreciate errata, criticism, etc.,
before I post an announcement on the big list.


In particular:

 - is is correct to use the word "translation"? 
    (at the beginning of the definition)


 - When I look at the final (bottom) diagram on the
    "transformation" webpage (the one that has the d,q
    unit-vectors divided into 1/7ths), my spatial imagination
    tells me that the correct algebraic solution is:

      x = (di + cj) / n
      y = (bi + aj) / n

   I can see by looking at the diagram that this is
    the way the coordinates transpose according to the
    matrix determinant, in a sort of criss-cross pattern.

   But the formula I derive (about 1/5 of the way down
    the page) is

      x = (di - bj) / n
      y = (aj - ci) / n

   They're not the same.        ... ?

   Is the first set correct generally, or only for this
    particular example?  Or is it the second set that's
    only correct in this particular example?



-monz


 




_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


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Message: 2963

Date: Fri, 04 Jan 2002 05:37:11

Subject: Re: coordinates from unison-vectors

From: paulerlich

Here's my Matlab code for doing this. I arbitrarily start with a 101-
by-101 square of lattice points. Rye is the 2-by-2 matrix of unison 
vectors. t is the set of points inside the PB.



for a=-50:50;
l(a+51,:)=a*ones(1,101);
m(:,a+51)=a*ones(101,1);
end;
p=[l(:) m(:)];
s=p*inv(rye);
s(find(s(:,1)>.50000001),:)=[];
s(find(s(:,2)>.50000001),:)=[];
s(find(s(:,2)<-.49999999),:)=[];
s(find(s(:,1)<-.49999999),:)=[];
t=s*rye


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Message: 2964

Date: Fri, 04 Jan 2002 08:38:40

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>No. It just assumes that the overtones are pretty close to harmonic,
>because they will then lead to the same ratio-intepretations for the
>fundamentals as the fundamentals by themselves. If they're 50 cents
>from harmonic, they will lead to a larger s value for the resulting
>harmonic entropy curve, but that's about it.

s represents the blur of the spectral components coming in.  How
could an inharmonic timbre change that?

>You can synthesize inharmonic sounds, yes?

No, that's the problem.

>You can use a high-limit JI scale that sounds like a pelog scale,
>yes?

The scale isn't stuck in JI, the timbre is.
 
>>>The gamelan scales sound like they contain a rough major
>>>triad and a rough minor triad, forming a very rough major
>>>seventh chord together, plus one extra note -- don't they?
>> 
>>Yes, to me, pelog sounds like a I and a III with a 4th in the
>>middle.  But the music seems to use a fixed tonic, with not
>>much in the way of triadic structure.
>
> How about 5-limit intervals?

Not sure what you're asking.

>> Okay, let's take a
>> journey...
>> 
>> "Instrumental music of Northeast Thailand"
>> 
>> Characteristic stop rhythm.  Harmonium and marimba-sounding
>> things play major pentatonic on C# (A=440) or relative minor
>> on A#.
> 
>This is clearly not a pelog tuning!

Right, it's the chinese pentatonic.  I threw it in for
completeness.

>>I suppose there is some argument for triadic structure here
>>too, but if I hadn't heard the last disc beforehand, I'd
>>say they were just doing the 'start the figure on different
>>scale members' thing, as in the first disc.  I don't know
>>Paul, this is not life as we know it (or hear it).
> 
>What on earth does that mean?

It's easy for me to hear triadic structure.  I'll bend over
backwards to do it.  It's easy for me to hear pelog as a
subset of the diatonic scale, too.  Indonesians might hear
it differently.

>>I still say there's nothing here that would turn up an optimized
>>5-limit temperament!
>
>Forget the optimization. All you need is the mapping -- that
>chains of three fifths make a major third and that chains of
>four fifths make a minor third. This seems to be a definite
>characteristic of pelog! Just as much as the "opposite" is a
>characteristic of Western music, regardless of whether strict
>JI, optimized meantone, 12-tET, or whatever is used.

Western music uses progressions of four fifths and expects to
wind up on a major third.  I didn't notice anything like this
for the [1 -3] map (right?) on the cited discs.

>>I guess it all depends if you consider these tonic changes
>>or just points of symmetry in a melisma (sp?).
>
>Why does that matter?

One's a harmonic device, the other melodic.  All other things
being equal, it wouldn't matter.  But I think a lot of the
other stuff that goes along with harmonic music is missing
from this music.  Western music requires meantone.  The pelog
5-limit map is far more extreme, but what suffers in this
music as we change the tuning from 5-of- 7, to 23, to 16, all
the way to strict JI?  I think the tuning on these discs is
closer to JI than 23-tET, and I don't hear them avoiding a
disjoint interval.  Do you?

Incidentally, I think Wilson agrees with your point of view
here.  While he does caution against eager interps. of his
ethno music theory, I think he thinks that harmonic mapping
is inevitable, and atomic in music.  I'm not sure I agree.
Not sure I disagree.

-Carl


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Message: 2965

Date: Fri, 04 Jan 2002 05:41:54

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Ask a musician, e.g Paul. I don't think I've ever seen them before. 
>I 
> wouldn't miss them. But I do think they look better than pelogic. 

As a musician, I have to say pelogic is one of my favorites. Perhaps 
I'm really advocating a gens^4 (weighted, of course) times mistuning 
badness measure here . . . but I'm happy as long as pelogic is in 
there. Try it with a marimba or other appropriate, inharmonic sound --
 it instantly transports you to Bali -- big time! My keyboardist 
friend was improvising on a 12-tone mapping of this generator -- now 
that was some awesome music!


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Message: 2966

Date: Fri, 04 Jan 2002 08:49:36

Subject: Re: tetrachordality

From: clumma

>>So obviously, these two scales will come out
>>the same.  But you've view -- and I remember
>>doing some listening experiments that back you
>>up (the low efficiency of the symmetrical
>>version was the other theory there) -- is that
>>the symmetrical version is not tetrachordal.
>>
>>So what's going on here?  Where's the error
>>in tetrachordality = similarity at transposition
>>by a 3:2? 
>
>An octave species is homotetrachordal if it has identical melodic
>structure within two 4:3 spans, separated by either a 4:3 or a 3:2.
>In the pentachordal scale, _all_ of the octave species are
>homotetrachordal (some in more than one way). In the symmetrical
>scale, _none_ of the octave species are homotetrachordal.

That's the def. in your paper.  But:

() I never understood how it reflects symmetry at the 3:2.

() "homotetrachordal" is a new term on me.  Are there precise
defs. of homo- vs. omni- around?  How did you choose these
prefixes?

() We agreed a bit ago that 'the number of notes that change
when a scale is transposed by 3:2 index its omnitetrachordality',
right?  My current approach is just a re-scaling of this.  So
do we want to revise this agreement?

-Carl


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Message: 2967

Date: Fri, 04 Jan 2002 05:43:31

Subject: Re: Some 12-tone meantone scales/temperaments

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here are up to isomorphism by mode and inversion all of the 
>meantone scales of twelve tones which have a 7-limit edge-
>connectivity greater than two. While the usual meantone scale (with 
>a connectivity of six) wins, it does not dominate, and the other 
>scales/temperaments are worth considering.

One of them ought to be the Keenan scale, which is the meantonized 
Lumma/Fokker scale. Which one? I'm suprised no one said anything.

>While the results are given in terms of the 31-et, they do not 
>depend on the precise tuning, and are generic meantone results.
> 
> I am not aware if this sort of thing has ever been investigated, 
>but it certainly seems worth pursuing.

Oh yes.
> 
> [0, 2, 5, 8, 10, 13, 15, 18, 20, 23, 26, 28]
> [2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3]   6
> 
> [0, 3, 6, 8, 11, 13, 16, 19, 21, 23, 26, 29]
> [3, 3, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2]   5
> 
> [0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 29]
> [3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2]   5
> 
> [0, 3, 6, 8, 10, 13, 16, 19, 21, 23, 26, 29]
> [3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2]   5
> 
> [0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28]
> [3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3]   4
> 
> [0, 3, 6, 8, 11, 13, 16, 18, 21, 24, 26, 28]
> [3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3]   4
> 
> [0, 2, 5, 8, 10, 13, 15, 17, 20, 23, 25, 28]
> [2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3]   4
> 
> [0, 3, 6, 8, 11, 13, 15, 18, 21, 23, 26, 28]
> [3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3]   3
> 
> [0, 3, 6, 8, 11, 13, 16, 18, 20, 23, 26, 28]
> [3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3]   3
> 
> [0, 3, 6, 9, 11, 13, 15, 18, 21, 24, 26, 28]
> [3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 2, 3]   3
> 
> [0, 3, 6, 9, 11, 13, 15, 17, 19, 22, 25, 28]
> [3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3]   3
> 
> [0, 2, 5, 8, 10, 12, 15, 18, 20, 22, 25, 28]
> [2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3]   3
> 
> [0, 2, 5, 8, 11, 13, 15, 17, 20, 23, 26, 28]
> [2, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3]   3
> 
> [0, 3, 5, 7, 10, 13, 15, 18, 20, 22, 25, 28]
> [3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 3]   3


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Message: 2968

Date: Fri, 04 Jan 2002 09:07:52

Subject: Some types of 46-et scales

From: genewardsmith

These were derived from the 126/125 planar temperament as interpreted by 46-et.

6 tones

[12, 4, 3]
[3, 1, 2]

7 tones

[8, 4, 7]
[3, 2, 2]

9 tones

[3, 9, 4]
[5, 3, 1]

10 tones

[7, 1, 4]
[5, 3, 2]

[4, 4, 7]
[5, 3, 2]

12 tones

[1, 7, 3]
[5, 5, 2]

[4, 4, 3]
[5, 5, 2]

[4, 4, 3]
[7, 3, 2]

[3, 6, 4]
[8, 3, 1]

13 tones

[3, 6, 1]
[9, 3, 1]

15 tones

[3, 4, 1]
[5, 7, 3]

[6, 1, 4]
[5, 8, 2]

17 tones

[1, 4, 3]
[5, 5, 7]

19 tones

[1, 3, 4]
[7, 9, 3]

20 tones

[2, 4, 1]
[5, 7, 8]


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Message: 2969

Date: Fri, 4 Jan 2002 04:07:19

Subject: Some definitions

From: Pierre Lamothe

Reduced set of short definitions about chordoid and gammier structures permitting to see their relations

--------------------------------------------------------------------------------


Gammier structure

Gammier structure is
  Gammoid structure with
     Fertility axiom
Gammoid structure is
  Harmoid structure with
    Regularity axiom
    Contiguity axiom
    Congruity axiom
Harmoid structure is
  Chordoid structure on
    rational numbers with
      standard multiplication
      standard order
      finite chordoid congruence modulo 2
Chordoid structure
  See Chordoid structure
   
  It is sufficient to know at this level that any finite set of odd numbers
    A = <k1 k2 ... kn>
  generates a finite chordoid of classes modulo 2 with the matrix
    A\A = [aij]
  where the generic element is
    aij = kj/ki
  and a corresponding harmoid with the set
    {2xaij}
  where the x are relative integers. Inversely, for any harmoid there exist
  a such set of minimal odd values generating it and so called its minimal
  harmonic generator.

  The minimal genericity is the rank of that minimal generator.

 Atom definition in an harmoid
  a is an atom if
    a > u (where u is the unison) and
    xy = a has no solution where both (u < x < a) and (u < y < a)
Regularity axiom is
  a < 2/a for any atom a
Contiguity axiom is
  any interval k is divisible by an atom
    or there exist an atom a such that ax = k has a solution
Congruity axiom is
  for any interval k there exist a stable number D of atoms
  in any variant of a complete atomic decomposition of k
Degree function definition in gammoid
  number D(X) of atoms in an interval X
Octave periodicity definition in gammoid
  number D(X) where X is the octave
Fertility axiom is
  octave periodicity > minimal genericity

--------------------------------------------------------------------------------


Chordoid structure

Chordoid structure is
  Simploid structure with
    Right associativity axiom
    Commutativity axiom
    Chordicity axiom
Simploid structure is
  set of elements with
    partial binary law
    Right simplicity axiom
Right simplicity axiom is
  ak = ak' Þ k = k'
Lemme 1 in simploid
  ab = c Þ b = a\c

  behind
    the reverse law \
    the interval a\b
    the interval domain A\B
      which is all x\y where x in A and y in B
Right associativity axiom is
  ak = (ab)c Þ k = bc
Commutativity axiom is
  k = ab Þ k = ba
Chordicity axiom is
  There exist a subset A in E such that E = A\A

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Message: 2970

Date: Fri, 04 Jan 2002 05:44:58

Subject: Re: The 7-limit connectivity of 7-tone meantone scales

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> I might as well give this:
> 
> [0 5 10 15 18 23 28]
> 5553553 c = 4
> 
> [0 5 10 15 18 23 26]
> 5553535 c = 3
> 
> [0 5 10 15 18 21 26]
> 5553355 c = 3

Yes, that augmented sixth in the last scale gives you a nice 7-limit 
consonance that doesn't occur in the other scales.


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Message: 2971

Date: Fri, 04 Jan 2002 05:48:03

Subject: Re: 7 and 10 note Magic scales

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> These are 7-connected scales in the Magic temperament 
> [5 1 12 25 -5 -10]. They are presented in terms of the 41-et, but 
>are generic 7-limit Magic scales. It is striking that in both the 7 
>and 10 note cases, there is a non-MOS scale with the same 
>connectivity as the MOS scale.

This would be very striking if the 7- and 10-tone scales are really 
intended to be used for 7-limit harmony. I thought we really had to 
go to 13 notes. Anyway, I'd like to see lattices -- repeating ones to 
show the cylidricality of Magic. Maybe I should make some myself -- 
someone please remind me sometime.


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Message: 2972

Date: Fri, 04 Jan 2002 09:51:54

Subject: Re: Some 9-tone 72-et scales

From: clumma

>>It's interesting that so many scales came out proper
>>when that wasn't a criterion in the search.  All the
>>10-note 72= scales are strictly proper.
> 
>It's also interesting that the best scores were all proper.

The max connectedness c of a scale of cardinality k is k-1,
right?  To actually get in octave-equivalent scales at
odd-limit n, you have to have k <= (n+1)/2, right?

For scales with a high c for k > n, some n-limit interval(s)
will have to appear between more than one pair of scale
members.  Can we get propriety from this?

It seems that any scale with one interval class always
consonant will be connected.  Given that the sizes of the
consonances are fairly well distributed across the octave. . .

Oh well.  Something interesting is going on here, that's
for sure.

-Carl


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Message: 2973

Date: Fri, 04 Jan 2002 05:49:12

Subject: Re: Nine tone Orwell scales

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here are the 7-limit connectivities of the nine note Orwells--I> 
wish I had known about these excellent non-MOS scales when I was 
>doing my Orwell piece!

Any of these show greater tetrachordality than the original MOS?


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Message: 2974

Date: Fri, 4 Jan 2002 12:10 +00

Subject: Re: Optimal 5-Limit Generators For Dave

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a13gbn+gl7s@xxxxxxx.xxx>
Me:
> > There is some ambiguity, but if you mean the 
> > half-fifth system, isn't that Vicentino's enharmonic?  That's 31&24 
> or 
> > [(1, 0), (1, 2), (0, 8)].  Two meantone scales, only 5-limit 
> consonances 
> > recognize, but neutral intervals used in melody.  It may not be a 
> > temperament, but does have a history of both theory and music, so 
> don't 
> > write it off so lightly.

Paul:
> I doubt this reflects Vicentino's practice well at all. For instance, 
> he didn't base any consonant harmonies on the second meantone scale, 
> did he?

How do you mean?  The two meantones fit snugly on the two different 
keyboards, and chords in the enharmonic genus typically alternate between 
them.  As most chords are consonances, there's no other way of getting the 
enharmonic melodies right.  For you to ask this question suggests either I 
didn't understand you, or you don't have a copy of Vicentino's book.  It 
is worth reading.  I thought you had it because you recommended it to 
somebody else.

> > The half-fifth system is 24&19 or [(1, 0), (2, -2), (4, -8)].
> 
> You mean half-fourth system?

Looks like it.


Me:
> > There's 
> > also a half-octave system, [(2, 0), (3, 1), (4, 4)].  That's the 
> one my 
> > program would deduce from the octave-equivalent mapping [2 8].

Paul:
> >From that unison vector? If so, I think you're confusion torsion 
> with "contorsion".

This has nothing directly to do with unison vectors.

Me:
> > If I had 
> > such a program.  If anybody cares, is it possible to write one?  
> Where 
> > torsion's present, we'll have to assume it means divisions of the 
> octave 
> > for uniqueness.

Paul:
> Huh? Clearly this doesn't work in the Monz sruti 24 case.

No, that can't be expressed in this particular octave equivalent system.  
It may be possible to include it later, but let's deal with the simple 
cases first.

> > Gene said it isn't possible, but I'm not convinced.  How 
> > could [1 4] be anything sensible but meantone?
> 
> Not sure what the connection is.

[1 4] is a definition of meantone: 4 fifths are equivalent to a major 
third.  Is that a unique definition, or do we have to add "plus two 
octaves"?


> > Perhaps the first step is to find an interval that's only one 
> generator 
> > step, take the just value, period-reduce it and work everything 
> else out 
> > from that.
> 
> If the half-fifth is the generator, what's the just value?

Well, it could be either 5:4 or 6:5.  Or 11:9 or 27:22.  Or 49:40 or 
60:49.  But if you mean the case where all consonances are specified in 
terms of fifths, but the generator is a half-fifth, I thought I defined 
those out of existence above.  If not, you can take the square root.

Me:
> > But there may be some cases where the optimal value should 
> > cross a period boundary.

Paul:
> ??

Say you have a system that divides the octave into two equal parts, and 
7:5 is a single generator steps.  It may happen that 7:5 approximates best 
to be larger than a half octave, so taking its just value for calculating 
the mapping will get the wrong results.  This may be a real problem when 
the octave is divided into 41 equal parts, like one of the higher-limit 
temperaments I came up with, and the generator is a fairly complex 
interval.

Me:
> > If you think it can't be done, show a counter-example: an 
> > octave-equivalent mapping without torsion that can lead to two 
> different 
> > but equally good temperaments.

Paul:
> Equally good? Under what criteria? Look, why do we care about the 
> octave-equivalent mapping? Certainly we can't object to asking the 
> mapping to be octave-specific, can we?

It should be fairly obvious if you get the mapping right because the 
errors will be small.  If you can find an example that depends on the 
choice of reasonable criteria, that'll do as a counterexample.

You were the one originally pushing for octave-equivalent calculations.  
If you aren't bothered any more, I'm not; I was only trying to answer your 
questions.  But it would be elegant to describe systems in the simplest 
possible way, and one consistent with Fokker.  It's up to you if you don't 
want the paper to cover that.


                       Graham


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