RE: [tuning] doubling phenomenon (also, transposing notes by unis on vectors)

Carl wrote,

>I wish I had better tools to work with blocks,

Here's a Matlab program to get the ratios and coordinates of the 5-d Fokker
PB given a matrix of unison vectors (called "rye"):
--------------------------------------
function [out,points]=perio5d(rye)
for a=-5:5;
k(a+6,:,:,:,:)=a*ones(1,11,11,11,11);
l(:,a+6,:,:,:)=a*ones(11,1,11,11,11);
m(:,:,a+6,:,:)=a*ones(11,11,1,11,11);
n(:,:,:,a+6,:)=a*ones(11,11,11,1,11);
o(:,:,:,:,a+6)=a*ones(11,11,11,11,1);
end;
p=[k(:) l(:) m(:) n(:) o(:)];
s=p*inv(rye);
s(find(s(:,1)>.50000001),:)=[];
s(find(s(:,2)>.50000001),:)=[];
s(find(s(:,3)>.50000001),:)=[];
s(find(s(:,4)>.50000001),:)=[];
s(find(s(:,5)>.50000001),:)=[];
s(find(s(:,5)<-.49999999),:)=[];
s(find(s(:,4)<-.49999999),:)=[];
s(find(s(:,3)<-.49999999),:)=[];
s(find(s(:,2)<-.49999999),:)=[];
s(find(s(:,1)<-.49999999),:)=[];
t=s*rye;
r=(3.^t(:,1).*5.^t(:,2).*7.^t(:,3).*11.^t(:,4).*13.^t(:,5));
r=r./2.^(floor(log(r)/log(2)));
[y,i]=sort(r);
out=cents(y);
points=t(i,:)
-------------------------------------
And here's one for listing and popularity-ranking all the cardinalities of
the 5-d periodicity blocks that can come from a given array of unison
vectors:
---------------------------------------
function r=thirteen(fac)
res=[];
n=size(fac,1);
for a=5:n;
for b=4:a-1;
for c=3:b-1;
for d=2:c-1;
for e=1:d-1
tmp=abs(det([fac(a,:);fac(b,:);fac(c,:);fac(d,:);fac(e,:)]));
res=[res;tmp];
end
end
end
end
end
res=abs(res);
mx=max(res);
mn=min(res);
h=hist(res,mx-mn+1);
s=[mn:mx]';
[y,i]=sort(h);
r=[y' s(i)];
--------------------------------------

>but could the doubling
>phenomenon be caused when one unison vector more closely approximates one
>of the unit vectors (i.e. 3:2) than it does the unison?

All the unison vectors we've looked at are far closer to a unison than to
any of the unit vectors.

>And in general,
>what happens when you build blocks by, say, approximating a unison on one
>axis, and approximating the unit vector of the first axis on the second?

I don't know. I'm not too interested in using large unison vectors -- 'cause
then any "closed"-like properties of the PB are gone.

I wrote,

>>Well, that is what the existing "excursion" is about -- in a special case.

Carl wrote,

Really? I had a feeling the two were related (or rather, questions about
what happens when notes are transposed by unison vectors came up when I was
thinking about the doubling phenomenon). Perhaps you could clarify.

The gist of the excursion is, when you transpose notes by unison vectors,
you just get variants of the scale you started with: diatonic remains
diatonic, chromatic gamut remains chromatic gamut. I've also mentioned how
srutis remain the same srutis when you transpose them by unison vectors --
Framjee considers 512/405 (406¢) to be a version of sruti 7 (normally 5/4 =
386¢) and not sruti 8 (408¢), a fact not unrelated to the diaschisma unison
vector separating 512/405 and 5/4.