RE: A gentle introduction to Fokker periodicity blocks, part 3

Paul -

Could you resend parts 1 and 2 again?

Thanks, in advance.

Micah

> -----Original Message-----
> From: Paul H. Erlich [SMTP:PErlich@Acadian-Asset.com]
> Sent: Monday, November 15, 1999 1:12 PM
> To: 'tuning@onelist.com'
> Subject: [tuning] A gentle introduction to Fokker periodicity blocks,
> part 3
>
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> > You may need to switch to a proportional font (such as Courier) to view
> > this correctly:
> >
> > ************************************************************
> > *A gentle introduction to Fokker periodicity blocks, part 3*
> > ************************************************************
> >
> > In this installment we move on to 7-limit just intonation. We will need
> > three dimensions to depict the relationship of the notes: one axis for
> > factors of 3, one for factors of 5, and one for factors of 7. Although
> > it's not too difficult to create the illusion of depth for at least a
> few
> > layers of lattice, showing how three-dimensional space is divided into
> > identical periodicity blocks is beyond my ASCII art abilities. Therefore
> I
> > will display each periodicity block alone, without its identical
> > neighbors, and leave it to the imagination or artistry of the reader to
> > discover how the block repeats to fill space.
> >
> > Let's view a portion of the 7-limit lattice. It contains our
> now-familiar
> > 5-limit system; that system is simply reproduced at transpositions of a
> > 7:4 up ("out of the screen") and down ("into the screen"):
> >
> >
> 1225/1152-----1225/768------1225/1024-----3675/2048----11025/8192
> > ,'| ,'| ,'| ,'| ,'|
> > 175/144|-------175/96|------175/128|------525/512|------675/512|
> > ,'| | ,'| | ,'| | ,'| | ,'| |
> > 25/18|--|-----25/24|--|-----25/16|--|-----75/64|--|---225/128| |
> > ,'| | | ,'| | | ,'| | | ,'| | | ,'| | |
> > 100/63|--|--|--25/21|--|--|--25/14|--|--|--75/56|--|--|225/224| | |
> > | | | | | | | | | | | | | | | | | | | |
> >
> > | | | | | | | | | | | | | | | | | | | |
> >
> > | | | | | | | | | | | | | | | | | | | |
> >
> > | |
> 245/144-------245/192-------245/128-------735/512------2205/2048
> > | | |,'| | | |.'| | | |,'| | | |,'| | | |,'|
> >
> > | |35/18|--------35/24|--------35/32|-------105/64|------315/256|
>
> > | |,'| | | |,'| | | |,'| | | |,'| | | |,'| |
>
> > |10/9-|--|------5/3-|--|------5/4-|--|-----15/8-|--|-----45/32| |
> > |,'| | | |,'| | | |,'| | | |,'| | | |,'| | |
> > 80/63|--|--|--40/21|--|--|--10/7-|--|--|--15/14|--|--|--45/28| | |
> > | | | | | | | | | | | | | | | | | | | |
> > | | | | | | | | | | | | | | | | | | | |
> > | | | | | | | | | | | | | | | | | | | |
> >
> > | |
> |49/36---------49/48---------49/32--------147/128-------441/256
> > | | |,'| | | |,'| | | |,'| | | |,'| | | |,'|
> > | |14/9-|---------7/6-|---------7/4-|--------21/16|--------63/32|
>
> > | |,'| | | |,'| | | |,'| | | |,'| | | |,'| |
> > |16/9-|--|------4/3-|--|------1/1-|--|------3/2-|--|------9/8 | |
> > |,'| | | |,'| | | |,'| | | |,'| | | |,'| | |
> > 64/63|--|--|--32/21|--|--|---8/7-|--|--|--12/7-|--|--|---9/7 | | |
> > | | | | | | | | | | | | | | | | | | | |
> > | | | | | | | | | | | | | | | | | | | |
> > | | | | | | | | | | | | | | | | | | | |
> > | |
> |49/45---------49/30---------49/40--------147/80--------441/320
> > | | |,' | | |,' | | |,' | | |,'| | | |,'
> > | |56/45---------28/15----------7/5----------21/20---------63/40
> > | |,' | |,' | |,' | |,' | |,'
> > |64/45---------16/15----------8/5-----------6/5-----------9/5
> > |,' |,' |.' |,' |,'
> > 512/315------128/105---------64/35---------48/35---------36/35
> >
> > By now the process of finding unison vectors and constructing the
> > resulting periodicity blocks should be familiar. In the case of one
> > dimension, the operation was almost trivial. In two dimensions, we drew
> a
> > parallelogram (or, in our excursion, a hexagon) constructed from the
> > unison vectors and could make various periodicity blocks by sliding the
> > shape around the lattice (but the number of notes in the block remained
> > unchanged). In three dimensions, the concept is similar -- we can create
> a
> > parallelopiped (the fancy name for a 3-d parallelogram) from three
> unison
> > vectors. The number of notes inside the parallelopiped will always equal
> > the its area. Just like before, the area of the block is equal to the
> > determinant of the matrix formed from the unison vectors. But we need to
> > know how to calculate the determinant of a 3-by-3 matrix. The way I
> > learned in college mechanics is to imagine that the matrix wraps around
> > and the left and right edges are joined together. Then add all products
> of
> > three elements along diagonals that slant downward and to the right, and
> > subtract all the products along diagonals sloping the other way. So:
> >
> > |a b c|
> > |d e f| = a*e*i + b*f*g + c*d*h - c*e*g - a*f*h - b*d*i
> > |g h i|
> >
> > Let's try an example using some simple unison vectors. Looking near 1/1
> in
> > the lattice above, we see 7/5 and 10/7, which are separated by 35 cents.
> > Since going from 7/5 to 10/7 (a move of 50:49) one moves 0 steps along
> the
> > 3-axis, 2 steps up along the 5-axis, and 2 steps into the screen along
> the
> > 7-axis, we may write:
> >
> > (0 2 -2) = 35 cents
> >
> > We need two more unison vectors to define a periodicity block in three
> > dimensions. One can be the interval between 8/7 and 7/6, or 49:48:
> >
> > (-1 0 2) = 36 cents
> >
> > and the other, the interval between 7/4 and 16/9, or 64:63:
> >
> > (-2 0 -1) = 27 cents
> >
> > So the "Fokker matrix" is
> >
> > ( 0 2 -2)
> > ( )
> > (-1 0 2)
> > ( )
> > (-2 0 -1)
> >
> > Let's see how many notes a periodicity block built from these three
> unison
> > vectors would contain. Using the formula above,
> >
> > | 0 2 -2|
> > | |
> > |-1 0 2| = 0 + (-8) + 0 - 0 - 0 - 2 = -10
> > | |
> > |-2 0 -1|
> >
> > As usual, we ignore the sign of the result, so the block has 10 notes.
> >
> > Now, even if we could draw the parallelopiped formed from these unison
> > vectors perfectly in ASCII, we would still have a very hard time
> > determining visually which notes were inside and which were outside. So
> we
> > will resort to mathematics. Technically speaking, the 3-by-3 matrix of
> > unison vectors defines a linear transformation -- a uniform "stretching"
> > or "squashing" -- in three-dimensional space. In particular, the matrix
> of
> > unison vectors takes a 1-by-1-by-1 cube and transforms it to the
> > parallelopiped with unison vectors as edges. Let's show how this works.
> > Each edge of the 1-by-1-by-1 cube is represented by a vector with one
> "1"
> > and two "0"s. Now to transform a vector using a matrix, you need to use
> > matrix multiplication. In this case, it's defined as:
> >
> >
> > (d e f)
> > (a b c)*(g h i) = (a*d + b*g + c*j a*e + b*h + c*k a*f + b*i +
> c*l)
> > (j k l)
> >
> > Let's try this with our matrix and each of the edges of a unit cube
> (known
> > as "unit vectors":
> >
> > ( 0 2 -2)
> > ( )
> > (1 0 0)*(-1 0 2) = ( 0 2 -2)
> > ( )
> > (-2 0 -1)
> >
> >
> > ( 0 2 -2)
> > ( )
> > (0 1 0)*(-1 0 2) = (-1 0 2)
> > ( )
> > (-2 0 -1)
> >
> >
> > ( 0 2 -2)
> > ( )
> > (0 0 1)*(-1 0 2) = (-2 0 -1)
> > ( )
> > (-2 0 -1)
> >
> > Now in order to determine which notes would be inside the
> parallelopiped,
> > it would be helpful to apply the _inverse_ of this transformation to the
> > whole lattice, because then we would simply have to find which notes are
> > in a 1-by-1-by-1 cube, which is easy to do by just looking at the
> numbers.
> > What is the inverse of a 3-by-3 matrix? Well, this is a bit too much
> > mathematics for us to deal with here, but any decent linear algebra
> > textbook will explain it and any decent math software package should be
> > able to calculate it. For the 3-by-3 "Fokker matrix" above, the inverse
> > is:
> >
> > ( 0 -.2 -.4)
> > ( )
> > ( .5 .4 -.2)
> > ( )
> > ( 0 .4 -.2)
> >
> > We can verify that this matrix transforms each unison vector into a unit
> > vector:
> >
> > ( 0 -.2 -.4)
> > ( )
> > ( 0 2 -2)*( .5 .4 -.2) = (0+1+0 0+.8-.8 0+.4-.4) = (1 0 0)
> > ( )
> > ( 0 .4 -.2)
> >
> >
> > ( 0 -.2 -.4)
> > ( )
> > (-1 0 2)*( .5 .4 -.2) = (0+0+0 .2+0+.8 .4+0-.4) = (0 1 0)
> > ( )
> > ( 0 .4 -.2)
> >
> >
> > ( 0 -.2 -.4)
> > ( )
> > (-2 0 -1)*( .5 .4 -.2) = (0+0+0 .4+0-.4 .8+0+.2) = (0 0 1)
> > ( )
> > ( 0 .4 -.2)
> >
> > OK, so all we have to do is transform a large enough portion of the
> > lattice with this inverse matrix. Then we can just pick a range of
> values
> > that spans exactly one unit along each of the axes, and the transformed
> > lattice points whose components fall into these ranges will be the ones
> in
> > a unit cube. Let's take the lattice above -- it's the set of points
> whose
> > coordinates range from -2 to 2 on the 3-axis, from -1 to 2 on the
> 5-axis,
> > and from -1 to 2 on the 7-axis:
> >
> > -2 -1 -1
> > -2 -1 0
> > -2 -1 1
> > -2 -1 2
> > -2 0 -1
> > -2 0 0
> > -2 0 1
> > -2 0 2
> > -2 1 -1
> > -2 1 0
> > -2 1 1
> > -2 1 2
> > -2 2 -1
> > -2 2 0
> > -2 2 1
> > -2 2 2
> > -1 -1 -1
> > -1 -1 0
> > -1 -1 1
> > -1 -1 2
> > -1 0 -1
> > -1 0 0
> > -1 0 1
> > -1 0 2
> > -1 1 -1
> > -1 1 0
> > -1 1 1
> > -1 1 2
> > -1 2 -1
> > -1 2 0
> > -1 2 1
> > -1 2 2
> > 0 -1 -1
> > 0 -1 0
> > 0 -1 1
> > 0 -1 2
> > 0 0 -1
> > 0 0 0
> > 0 0 1
> > 0 0 2
> > 0 1 -1
> > 0 1 0
> > 0 1 1
> > 0 1 2
> > 0 2 -1
> > 0 2 0
> > 0 2 1
> > 0 2 2
> > 1 -1 -1
> > 1 -1 0
> > 1 -1 1
> > 1 -1 2
> > 1 0 -1
> > 1 0 0
> > 1 0 1
> > 1 0 2
> > 1 1 -1
> > 1 1 0
> > 1 1 1
> > 1 1 2
> > 1 2 -1
> > 1 2 0
> > 1 2 1
> > 1 2 2
> > 2 -1 -1
> > 2 -1 0
> > 2 -1 1
> > 2 -1 2
> > 2 0 -1
> > 2 0 0
> > 2 0 1
> > 2 0 2
> > 2 1 -1
> > 2 1 0
> > 2 1 1
> > 2 1 2
> > 2 2 -1
> > 2 2 0
> > 2 2 1
> > 2 2 2
> >
> > Transforming these using the inverse matrix, we get:
> >
> > -0.5000 -0.4000 1.2000
> > -0.5000 0 1.0000
> > -0.5000 0.4000 0.8000
> > -0.5000 0.8000 0.6000
> > 0 0 1.0000
> > 0 0.4000 0.8000
> > 0 0.8000 0.6000
> > 0 1.2000 0.4000
> > 0.5000 0.4000 0.8000
> > 0.5000 0.8000 0.6000
> > 0.5000 1.2000 0.4000
> > 0.5000 1.6000 0.2000
> > 1.0000 0.8000 0.6000
> > 1.0000 1.2000 0.4000
> > 1.0000 1.6000 0.2000
> > 1.0000 2.0000 0
> > -0.5000 -0.6000 0.8000
> > -0.5000 -0.2000 0.6000
> > -0.5000 0.2000 0.4000
> > -0.5000 0.6000 0.2000
> > 0 -0.2000 0.6000
> > 0 0.2000 0.4000
> > 0 0.6000 0.2000
> > 0 1.0000 0
> > 0.5000 0.2000 0.4000
> > 0.5000 0.6000 0.2000
> > 0.5000 1.0000 0
> > 0.5000 1.4000 -0.2000
> > 1.0000 0.6000 0.2000
> > 1.0000 1.0000 0
> > 1.0000 1.4000 -0.2000
> > 1.0000 1.8000 -0.4000
> > -0.5000 -0.8000 0.4000
> > -0.5000 -0.4000 0.2000
> > -0.5000 0 0
> > -0.5000 0.4000 -0.2000
> > 0 -0.4000 0.2000
> > 0 0 0
> > 0 0.4000 -0.2000
> > 0 0.8000 -0.4000
> > 0.5000 0 0
> > 0.5000 0.4000 -0.2000
> > 0.5000 0.8000 -0.4000
> > 0.5000 1.2000 -0.6000
> > 1.0000 0.4000 -0.2000
> > 1.0000 0.8000 -0.4000
> > 1.0000 1.2000 -0.6000
> > 1.0000 1.6000 -0.8000
> > -0.5000 -1.0000 0
> > -0.5000 -0.6000 -0.2000
> > -0.5000 -0.2000 -0.4000
> > -0.5000 0.2000 -0.6000
> > 0 -0.6000 -0.2000
> > 0 -0.2000 -0.4000
> > 0 0.2000 -0.6000
> > 0 0.6000 -0.8000
> > 0.5000 -0.2000 -0.4000
> > 0.5000 0.2000 -0.6000
> > 0.5000 0.6000 -0.8000
> > 0.5000 1.0000 -1.0000
> > 1.0000 0.2000 -0.6000
> > 1.0000 0.6000 -0.8000
> > 1.0000 1.0000 -1.0000
> > 1.0000 1.4000 -1.2000
> > -0.5000 -1.2000 -0.4000
> > -0.5000 -0.8000 -0.6000
> > -0.5000 -0.4000 -0.8000
> > -0.5000 0 -1.0000
> > 0 -0.8000 -0.6000
> > 0 -0.4000 -0.8000
> > 0 0 -1.0000
> > 0 0.4000 -1.2000
> > 0.5000 -0.4000 -0.8000
> > 0.5000 0 -1.0000
> > 0.5000 0.4000 -1.2000
> > 0.5000 0.8000 -1.4000
> > 1.0000 0 -1.0000
> > 1.0000 0.4000 -1.2000
> > 1.0000 0.8000 -1.4000
> > 1.0000 1.2000 -1.6000
> >
> > Which of these are in a unit cube? Let's take the points where all three
> > transformed coordinates are greater than or equal to 0 and less than 1.
> > That leaves:
> >
> > 0 0.4000 0.8000
> > 0 0.8000 0.6000
> > 0.5000 0.4000 0.8000
> > 0.5000 0.8000 0.6000
> > 0 0.2000 0.4000
> > 0 0.6000 0.2000
> > 0.5000 0.2000 0.4000
> > 0.5000 0.6000 0.2000
> > 0 0 0
> > 0.5000 0 0
> >
> > Thankfully, this is 10 points, so we included enough notes in the
> original
> > lattice. Which notes are they? Well, we can transform them back to the
> > lattice using the original Fokker matrix. That gives:
> >
> > -2 0 0
> > -2 0 1
> > -2 1 -1
> > -2 1 0
> > -1 0 0
> > -1 0 1
> > -1 1 -1
> > -1 1 0
> > 0 0 0
> > 0 1 -1
> >
> > Let's transpose this one unit to the right along the 3-axis to center it
> > better around 1/1:
> >
> > -1 0 0
> > -1 0 1
> > -1 1 -1
> > -1 1 0
> > 0 0 0
> > 0 0 1
> > 0 1 -1
> > 0 1 0
> > 1 0 0
> > 1 1 -1
> >
> > That represents the following scale:
> >
> >
> >
> > 5/3-----------5/4
> > ,'| ,'|
> > 40/21|--------10/7-|--------15/14
> > | |
> > | |
> > | |
> > | |
> > | |
> > | 7/6-----------7/4
> > |,' |,'
> > 4/3-----------1/1-----------3/2
> >
> >
> > If you have particularly good spatial imagination, you can imagine how
> > this scale tiles with copies of itself transposed by the unit vectors
> > ( 0 2 -2), (-1 0 2), and (-2 0 -1) to fill the 3-d lattice.
> >
> > This scale resembles one of my decatonic scales in 22-tET (see
> > http://www-math.cudenver.edu/~jstarret/22ALL.pdf); it would be
> > 0 2 5 7 9 11 13 16 18 20, or the Dynamic Major mode of the symmetrical
> > decatonic scale.
> >
> > If we shift the unit cube slightly, so that the first two coordinates
> are
> > greater than or equal to 0 and less than 1, and the third coordinate is
> > between -.5 and .5, we get the following set of points:
> >
> > 0 0.2000 0.4000
> > 0 0.6000 0.2000
> > 0.5000 0.2000 0.4000
> > 0.5000 0.6000 0.2000
> > 0 0 0
> > 0 0.4000 -0.2000
> > 0 0.8000 -0.4000
> > 0.5000 0 0
> > 0.5000 0.4000 -0.2000
> > 0.5000 0.8000 -0.4000
> >
> > which, when transformed back to the lattice, gives
> >
> > -1 0 0
> > -1 0 1
> > -1 1 -1
> > -1 1 0
> > 0 0 0
> > 0 0 1
> > 0 0 2
> > 0 1 -1
> > 0 1 0
> > 0 1 1
> >
> > Let's transpose this one unit to the right along the 3-axis:
> >
> > 0 0 0
> > 0 0 1
> > 0 1 -1
> > 0 1 0
> > 1 0 0
> > 1 0 1
> > 1 0 2
> > 1 1 -1
> > 1 1 0
> > 1 1 1
> >
> >
> > 105/64
> > ,'|
> > 5/4----------15/8 |
> > ,' | ,'| |
> > 10/7--|--------15/14| |
> > | | |
> > | | |
> > | | |
> > | | 147/128
> > | | |,'
> > | 7/4-----------21/16
> > |,' |.'
> > 1/1-----------3/2
> >
> > This is another rational interpretation of the Dynamic Major mode of the
> > symmetrical decatonic scale.
> >
> > If we shift the unit cube slightly, so that the first coordinate is
> > greater than 0 and less than or equal to 1, the second coordinate is
> > between .5 and 1.5, and the third coordinate is between -.5 and .5, we
> get
> > the following set of points:
> >
> > 0.5000 1.2000 0.4000
> > 1.0000 1.2000 0.4000
> > 0.5000 0.6000 0.2000
> > 0.5000 1.0000 0
> > 0.5000 1.4000 -0.2000
> > 1.0000 0.6000 0.2000
> > 1.0000 1.0000 0
> > 1.0000 1.4000 -0.2000
> > 0.5000 0.8000 -0.4000
> > 1.0000 0.8000 -0.4000
> >
> > Transforming back:
> >
> > -2 1 1
> > -2 2 0
> > -1 1 0
> > -1 1 1
> > -1 1 2
> > -1 2 -1
> > -1 2 0
> > -1 2 1
> > 0 1 1
> > 0 2 0
> >
> > and transposing two units to the right along the 3-axis and two units
> down
> > along the 5-axis:
> >
> > 0 -1 1
> > 0 0 0
> > 1 -1 0
> > 1 -1 1
> > 1 -1 2
> > 1 0 -1
> > 1 0 0
> > 1 0 1
> > 2 -1 1
> > 2 0 0
> >
> > 21/16
> > ,'|
> > 1/1-----------3/2-----------9/8
> > ,' | |
> > 12/7 | |
> > | |
> > | |
> > | |
> > | 147/80
> > | |.'
> > 7/5---------|21/20----------63/40
> > |,'
> > 6/5
> >
> > In 22tET, this would be 0 2 4 6 9 11 13 15 17 20, the Static Minor mode
> of
> > the symmetrical decatonic scale.
> >
> > OK, I'm going to start being a little more concise here; hopefully you
> > will be able to follow me. If not, re-read what we've done so far. Let's
> > move on to a different set of unison vectors. We'll keep our smallest
> one,
> > 64:63 or
> >
> > (-2 0 -1) = 27 cents
> >
> > We'll replace the other two with 81:80 or
> >
> > ( 4 -1 0) = 21.5 cents
> >
> > and 126:125 or
> >
> > ( 2 -3 1) = 13.8 cents.
> >
> > Now how many notes would our periodicity block contain?
> >
> > |-2 0 -1|
> > | |
> > | 4 -1 0| = 2 + 0 + 12 - 2 - 0 - 0 = 12
> > | |
> > | 2 -3 1|
> >
> > Two examples:
> >
> > 1.
> >
> > -1 0 0
> > -1 1 0
> > -1 2 0
> > 0 -1 0
> > 0 0 0
> > 0 1 0
> > 1 -1 0
> > 1 0 0
> > 1 1 0
> > 2 -1 0
> > 2 0 0
> > 2 1 0
> >
> >
> > 25/24
> > |
> > |
> > |
> > |
> > |
> > |
> > |
> > 5/3-----5/4----15/8----45/32
> > | | | |
> > | | | |
> > | | | |
> > | | | |
> > | | | |
> > | | | |
> > | | | |
> > 4/3-----1/1-----3/2-----9/8
> > | | |
> > | | |
> > | | |
> > | | |
> > | | |
> > | | |
> > | | |
> > 8/5-----6/5-----9/5
> >
> > 2.
> >
> > -2 1 0
> > -1 1 0
> > 0 -1 1
> > 0 0 0
> > 0 0 1
> > 0 1 0
> > 1 -1 1
> > 1 0 0
> > 1 0 1
> > 1 1 0
> > 2 -1 1
> > 3 -1 1
> >
> > 10/9-----5/3-----5/4----15/8
> > | |
> > | |
> > | |
> > | |
> > | |
> > | 7/4----21/16
> > |.'| |.'|
> > 1/1-|---3/2 |
> > | |
> > | |
> > | |
> > | |
> > | |
> > 7/5----21/20---63/40--189/160
> >
> > OK, anyone interested in pursuing this subject further should now have
> the
> > tools to do so. I'll be happy to answer any questions on this subject
> and
> > create further tutorials if necessary. Happy tuning!
>
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