Message 2347
From:  "monz"  
Date:  Mon Dec 31, 2001  3:52 pm
Subject:  Re: [tuning-math] Re: coordinates from unison-vectors
 

Hi J!


> From: unidala 
> To: 
> Sent: Monday, December 31, 2001 8:05 AM
> Subject: [tuning-math] Re: coordinates from unison-vectors
>
>
> ...
>
> J Gill: So your (dynamite presentation!) XLS Spreadsheet
> will allow determination of mean-tone "coefficients" which
> "best fit" any 5-limit periodicity block?



[Click this link to download my spreadsheet relating Duodene with -2/9- and -1/4-comma meantones.]



Well... the spreadsheet *allows* determination of the meantone
coordinates and axis... but it still can't *make* the determination!
The user still has to do some of the work.

The spreadsheet automatically figures out where to close
the meantone chain, at (+/- 1/2 determinant) generators. But
I haven't yet "coordinated" [groan pun intended] the mathematics
to determine the "best-fit" meantone automatically. So far, you
can only try out different values and see what they look like.



> Not being very adept with this PB stuff you guys apply,


Have patience, J... I've spent the night working on a terrific
graphic which will go into a Dictionary entry webpage for
"Transformation". That should clear up a lot of the fog for you.


> what are the mean-tone "coefficients" which represent
> a "best-fit" for Ellis' "Duodene", with 12 pitches at:
>
> 1/1--16/15--9/8--6/5--5/4--4/3--45/32--3/2--8/5--5/3--9/5--15/8
> and
> 16/15--9/8--6/5--5/4--4/3--45/32--3/2--8/5--5/3--9/5--15/8--2/1
>
> and for
> 1/1--16/15--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--9/5--15/8
> and
> 16/15--9/8--6/5--5/4--4/3--7/5--3/2--8/5--5/3--9/5--15/8--2/1
>
>
> Curiously, J Gill



Well, as I said, I don't have mathematics to do the job, so
I just try a few by eye and intuition, and see what looks good.

First of all, the first two of your Duodenes are equivalent to
each other on my lattice, because it's 2-dimensional for
prime-factors 3 and 5, with 2 ignored.

Secondly, I could use only that first pair, because the
second pair are 7-limit, making the lattice 3-dimensional,
and my spreadsheet doesn't handle that ... yet ...



So:


Ellis's Duodene is a 12-tone set defined by the
unison-vectors 128:125 = [0 3] (the diesis) and
81:80 = [4 -1] (the syntonic comma), as Paul shows in his
"Gentle Introduction to Fokker Periodicity Blocks, part 2".



JI pitch-classes enclosed within the parallelogram:


 coordinates  ratio    ~cents

  ( 0  0 )     1/1      0
  (-1 -1 )    16/15   111.7312853
  ( 2  0 )     9/8    203.9100017
  ( 1 -1 )     6/5    315.641287
  ( 0  1 )     5/4    386.3137139
  (-1  0 )     4/3    498.0449991
  ( 2  1 )    45/32   590.2237156
  ( 1  0 )     3/2    701.9550009
  ( 0 -1 )     8/5    813.6862861
  (-1  1 )     5/3    884.358713
  ( 2 -1 )     9/5   1017.596288
  ( 1  1 )    15/8   1088.268715


-1/4-comma meantone pitch-classes enclosed within the parallelogram:

generator  coordinates    ~cents

   -6       ( 0 -3/2)   620.5294292
   -5       ( 0 -5/4)   117.1078577
   -4       ( 0 -1  )   813.6862861
   -3       ( 0 -3/4)   310.2647146
   -2       ( 0 -1/2)  1006.843143
   -1       ( 0 -1/4)   503.4215715
    0       ( 0  0  )     0
    1       ( 0  1/4)   696.5784285
    2       ( 0  1/2)   193.1568569
    3       ( 0  3/4)   889.7352854
    4       ( 0  1  )   386.3137139
    5       ( 0  5/4)  1082.892142
    6       ( 0  3/2)   579.4705708



Below is a graphic of the -1/4-comma meantone lattice.
The reference pitch is C = n0.  Note that it is
based on the unit-square, expressed in matrix form
as

  [1 0]
  [0 1] 

of the array or grid of prime-factors 3 and 5, with
3 on the horizontal axis and 5 on the vertical.  Expressed
as "8ve"-equivalent ratios, the unit-square matrix is:

  [ 3/2  1/1 ]
  [ 1/1  5/4 ]

This makes it different from my usual lattice formula.








Legend:

* pink:  outlines the parallelogram boundaries of the
periodicity-block according to the shape defined by the
pair of unison-vectors, centered on (0,0) = 1:1 ratio;

* blue:  plots the coordinates of the twelve 5-limit JI
pitch-classes of Ellis's "Duodene" which lie within the
boundaries of the periodicity-block, and shows where
the JI "wolves" lie (this actually didn't work here...
I have to fix that);

* yellow:  plots the more-or-less arbitrarily chosen
meantones, whose chains are closed according to the
interval spanned by one of the unison-vectors, symmetrical
on either side of (0,0) = 1/1;

* green:  connects each meantone pitch-class with its
closest JI relative within the periodicity-block,
closeness measured in pitch-height.



(I drew those green lines in by hand... so if you change
the meantone in the spreadsheet, the lines won't adjust.)


Note that this periodicity-block has three pitch-classes
which fall right on the eastern boundary: (2,-1) = 9/5,
(2,0) = 9/8 and (2,1) = 45/32. All three of these thus
have alternates a comma lower -- in other words, by the
 -[4,-1] = 80:81 unison-vector--, and the alternate
pitch-classes fall on the western boundary: (-2,0) = 16/9,
(-2,1) = 10/9, and (-2,2) = 25/18, respectively.

Also, since (-2,2) = 25/18 and (2,1) = 45/32 happen to
fall right on the northwest and northeast *corners* of the
boundary (respectively), they also have lower alternates at
the distance of the *other* unison-vector -[0 3] = 64:125,
which would place the alternates at (-2,-1) = 64/45 and
(2,-2) = 36/25, respectively.

With a total of 17 possible pitch-classes which may be used
to define this periodicity-block, the only meantone which goes
right down the middle of all of them is -1/4-comma. That's
why I chose to use that for the graphic.


Below is another example of the same Duodene periodicity-block,
this time with a "best fit meantone" of -2/9-comma.









If one keeps the JI pitches in place, and moves the bounding
unison-vectors and the meantone 1/2-step to the right along
the 3-axis, so that the boundaries enclose only 12
pitches (the minimal set for this pair of unison-vectors)
and the meantone is exactly centered and symmetrical to those
12 pitches, the entire system is centered and symmetrical
around the ratio 3(1/2), or in terms of "8ve"-equivalent
ratios, the square-root of 3/2 :








In fact this structure more accurately portrays what the
meantone really represents: because of the disappearance
of the syntonic comma unison-vector, this flat lattice
should be imagined to wrap around as a cylinder, so that
the right and left edges connect.  Thus the centered
meantone may imply either of any pair of pitches which
would be separated by a comma on the flat lattice, and
each of those pairs of points on the flat lattice map to the
same point on a cylinder .


-monz



See also: my Lattice diagrams comparing rational implications 
of various meantone chains.