RE: Repost: A gentle introduction to Fokker periodicity blocks, part 2

Thanks, Paul!

> -----Original Message-----
> From: Paul H. Erlich [SMTP:PErlich@Acadian-Asset.com]
> Sent: Monday, November 15, 1999 4:15 PM
> To: 'tuning@onelist.com'
> Subject: [tuning] Repost: A gentle introduction to Fokker periodicity
> blocks, part 2
>
> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> You may need to switch to a non-proportional font (such as Courier) to
> view
> this correctly:
>
> ************************************************************
> *A gentle introduction to Fokker periodicity blocks, part 2*
> ************************************************************
>
> Last time we saw how choosing a single "unison vector", or interval too
> small to warrant a distinction in pitch, reduces the infinite resources of
> 3-limit just intonation (or Pythagorean tuning) to a finite scale. The
> number of pitches in this scale turned out to be the number of fifths in
> the
> chain between the two notes defining the unison vector. Our examples were
> a
> pentatonic scale, defined by a unison vector of 5 fifths, a "Pythagorean
> limma" of 90 cents; and a 12-tone scale, defined by a unison vector of 12
> fifths, a "Pythagorean comma" of 23 cents.
>
> Now let's consider 5-limit just intonation. It can be considered an
> infinite
> succession of 3-limit JI systems, separated by just major thirds (5:4
> ratio). Let us put our familiar 3-limit system in the middle, and stack
> its
> transpositions by a 5:4 up, upwards, and by a 5:4 down, downwards:
> . . . . . . .
> . . . . . . .
> . . . . . . .
> | | | | | | |
> | | | | | | |
> ...--50/27---25/18---25/24---25/16---75/64--225/128-675/512-..
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> ...--40/27---10/9-----5/3-----5/4----15/8----45/32--135/128-...
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> ...--32/27---16/9-----4/3-----1/1-----3/2-----9/8----27/16--...
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> ...-256/135--64/45---16/15----8/5-----6/5-----9/5----27/20--...
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> | | | | | | |
> ...1024/675-256/225-128/75---32/25---48/25---36/25---27/25--...
> | | | | | | |
> | | | | | | |
> . . . . . . .
> . . . . . . .
> . . . . . . .
> We could have used 5:3 instead of 5:4, but this way is a bit easier to
> remember: gaining a factor of 3 means moving one place to the right, and
> gaining a factor of 5 means moving one place upwards.
>
> Now say we again start in the middle (1/1) and start adding notes around
> it.
> Unlike the 3-limit case, the direction we move in this lattice may affect
> which small intervals we end up finding. Say we just look at the notes
> adjoining 1/1. The interval between 6/5 and 5/4, as well as that between
> 8/5
> and 5/3, is a chromatic semitone or about 71 cents. This interval is
> always
> produced on this lattice by moving one step to the left and two steps up.
> In
> other words, it corresponds to losing (dividing by) a factor of 3 and
> gaining (multiplying by) two factors of 5. In vector notation, we may
> write
>
> (-1 2) = 71 cents.
>
> Unlike the 3-limit case, a single unison vector in the 5-limit lattice
> does
> not collapse the infinite pitch variety into a finite set. It may be
> considered to collapse it into a single vertical line (since every note is
> equivalent to another note in an adjoining vertical line), or a set of two
> parallel horizontal lines (since every note is equivalent to a note two
> horizontal lines away), or a diagonal band, but in any case the number of
> distinct pitches is still infinite. We need to define one more unison
> vector
> to get a finite set of pitches.
>
> If we add a few more notes related by fifths to those adjoining 1/1, we
> soon
> find pairs of notes with small intervals between them. The interval
> between
> 10/9 and 9/8, as well as that between 16/9 and 9/5, is a syntonic comma or
> about 22 cents. On the lattice, it corresponds to any move of four steps
> to
> the right and one step down:
>
> (4 -1) = 22 cents.
>
> With these two unison vectors, we may now find a finite set of pitches
> which
> is equivalent to the whole infinite lattice. How do we go about this?
> Fokker's solution was to draw a parallelogram having the two unison
> vectors
> as sides. In this case, two opposite sides of the parallelogram would
> slope
> one step to the left and two steps up, while the other two sides would
> slope
> four steps to the right and one step down. This parallelogram would be
> wide
> enough along each of the unison vectors to contain one and only pitch from
> each equivalence class. Moreover, the parallelogram shape would tile the
> plane, so that every pitch would fall into one and only one parallelogram.
> Let us illustrate this for our example:
> . . . . . . .
> . . . . . . .
> . . . . . . .
> | | | | | | *
> * | * | | | |* | |
> ...-50/27*---25/18---25/24---25/16---75/64--225/128-675/512-..
> | * | | | | * | |
> | | * | | | | |
> | *| * | | | * | |
> | | | * | | | |
> | * | * | | * | |
> | | | | * | | |
> | |* | | * | * | |
> ...--40/27---10/9-----5/3-----5/4----15/8*---45/32--135/128-...
> | | * | | | * | |
> | | | | | | * |
> | | * | | | *| * |
> | | | | | | | *
> * | | * | | | * |
> | * | | | | | |
> | * | * | | | |* |
> ...--32/27---16/9*----4/3-----1/1-----3/2-----9/8----27/16--...
> | | * | | | | * |
> | | | * | | | |
> | | *| * | | | * |
> | | | | * | | |
> | | * | * | | * |
> | | | | | * | |
> | | |* | | * | * |
> ...-256/135--64/45---16/15----8/5-----6/5-----9/5*---27/20--...
> | | | * | | | * |
> | | | | | | | *
> * | | | * | | | *|
> | * | | | | | |
> | * | | * | | | *
> | | * | | | | |
> | | * | * | | | |*
> ..1024/675-256/225-128/75*---32/25---48/25---36/25---27/25--...
> | | | * | | | | *
> | | | | * | | |
> . . . . . . .
> . . . . . . .
> . . . . . . .
> Using our two unison vectors, we have divided the plane into identical
> parallelograms, each of which has seven lattice points (notes) inside of
> it.
> The parallelogram that is completely visible in the diagram above
> surrounds
> the ratios that most JI enthusiasts associate with the major scale. Every
> other parallegram has exactly the same configuration of notes, and each of
> its notes is equivalent, through trasposition by one or more unison
> vectors,
> to one and only one note of the central major scale. Thus we may want to
> say
> that the major scale is a periodicity block in the 5-limit lattice.
>
> Unlike the 3-limit case, though, moving the boundaries of the periodicity
> block does not necessarily lead to the same scale. For example, we could
> just as easily have drawn our parallelograms like this:
> . . . . . . .
> . . . . . . .
> . . . . . . .
> | | | | * | |
> | * | | | |* | |
> ...-50/27----25/18---25/24---25/16---75/64--*225/128-675/512-..
> | * | | | | * |* |
> | | | | | | * |
> | *| | | | * | |*
> * | | | | | | |
> |* * | | | * | |
> | * | | | | | |
> | |* | | | * | |
> ...--40/27---10/9---*-5/3-----5/4----15/8----45/32--135/128-...
> | | * |* | | * | |
> | | | * | | | |
> | | * | |* | *| |
> | | | | * | | |
> | | * | | |* * |
> | | | | | * | |
> | | * | | | |* |
> ...--32/27---16/9-----4/3-----1/1-----3/2-----9/8---*27/16--...
> | | * | | | | * |*
> * | | | | | | |
> |* | *| | | | * |
> | * | | | | | |
> | |* * | | | * |
> | | * | | | | |
> | | |* | | | * |
> ...-256/135--64/45---16/15--*-8/5-----6/5-----9/5----27/20--...
> | | | * |* | | * |
> | | | | * | | |
> | | | * | |* | *|
> | | | | | * | |
> | | | * | | |* *
> | | | | | | * |
> | | | * | | | |*
> ...1*24/675-256/225-128/75---32/25---48/25---36/25---27/25--...
> |* | | * | | | | *
> | * | | | | | |
> . . . . . . .
> . . . . . . .
> . . . . . . .
> giving the JI _minor_ scale as the periodicity block. This scale is not
> quite the sixth mode of the major scale; one note needs to be transposed
> by
> a syntonic comma to make one scale a mode of the other. Or another
> possibility is to center the parallelogram on 1/1, giving:
> . . . . . . .
> . . . . . . .
> . . . . . . .
> |* | | | * | * | |
> | | | | | * |
> ...-50/27*---25/18---25/24---25/16---*75/64-225/128*675/512-..
> | | | | | | *
> * | * | | | *| | |
> * | | | | | |
> | * | | | * | |
> | * | | | | |
> | * | * | | |* | |
> | | * | | | |
> ...--40/27---*10/9----5/3--*--5/4----15/8*---45/32--135/128-...
> | | | * | | |
> | *| | | * | * | |
> | | | | * | |
> | * | | | * | |
> | | | | | * |
> | |* | | | * | * |
> | | | | | | *
> ...*-32/27---16/9*----4/3-----1/1-----3/2----*9/8----27/16--...
> * | | | | | |
> | * | * | | | *| |
> | * | | | | |
> | | * | | | * |
> | | * | | | |
> | | * | * | | |* |
> | | | * | | |
> ...-256/135--64/45---*16/15---8/5--*--6/5-----9/5*---27/20--...
> | | | | * | |
> | | *| | | * | * |
> | | | | | * |
> | | * | | | * |
> | | | | | | *
> | | |* | | | * |
> * 256| | | | | |
> ..1024/675-*--/225-128/75*---32/25---48/25---36/25---*27/25-...
> | * | | | | |
> | | * | * | | | *|
> . . . . . . .
> . . . . . . .
> . . . . . . .
> sort of a JI "dorian" scale. In general, then, we can only say that the
> two
> unison vectors we have chosen define a periodicity block that is some sort
> of diatonic scale, but we can't be totally precise as to its JI
> construction, as any of the notes may be transposed by a unison vector and
> the important properties of the periodicity block will be maintained.
>
> Mathematically, the unison vectors define 7 equivalence classes in the JI
> lattice. No matter where we put the paralellograms, each one will have
> exactly 7 lattice points inside it, equivalent (through unison vectors) to
> the 7 lattice points inside every other parallelogram. This is true
> because
> the area of each parallelogram is exactly 7 (if you consider each step in
> both directions of the lattice to be of length 1) and it can be proved
> that
> a parallelogram of area N whose edges are defined with integer vectors
> always contains exactly N lattice points no matter where you put it. (If
> you
> put an edge right on a lattice point, you can consider it to be 1/2 inside
> and 1/2 outside the parallelogram, and if you put a corner right on a
> lattice point, you can consider it to be 1/4 inside, or a fraction
> determined by the angle of that corner divided by 360 degrees -- either
> way,
> you will always end up counting exactly N lattice points inside the
> parallelogram.)
>
> Is there a way to calculate the area of the parallelogram from the vector
> representation of the unison vectors? Yes, there is! First, put the unison
> vectors together into a matrix (the order doesn't matter):
>
> ( 4 -1)
> ( )
> (-1 2)
>
> (those are supposed to be big parentheses around the matrix)
>
> Now, calculate the determinant of the matrix:
>
> | 4 -1|
> | | = 7.
> |-1 2|
>
> In case you didn't know, the formula for the determinant of a
> 2-by-2-matrix
> is:
>
> |a b|
> | | = a*d - b*c
> |c d|
>
> If you like geometry, you can convince yourself that this is indeed the
> formula for the area of a parallelogram whose sides are defined by vectors
> (a b) and (c d). Don't worry if the determinant comes out negative; you
> can
> throw out the minus sign for these purposes.
>
> Let's try another one, since most of us would consider a chromatic
> semitone
> a large enough interval to distinguish on our JI instruments. Proceeding
> to
> add notes related by a major third to those adjacent to 1/1, we find that
> 25/16 is close to 8/5, and 5/4 is close to 32/25, the difference in each
> case being a lesser diesis, or about 41 cents. As a unison vector, this is
> written:
>
> (0 3) = 41 cents.
>
> If we take this and the syntonic comma as our two unison vectors, we may
> draw periodicity blocks like so:
> . . . . . . .
> . . . . . . .
> . . . . . . .
> | * | * | | | | * |
> | * | | | | |
> ...--50/27---25/18-*-25/24---25/16---75/64--225/128*675/512-..
> | | * | | | |
> | | * | * | | | * |
> | | | * | | |
> | | * | | * | | * |
> | | | | * | |
> | | * | | | * | * |
> | | | | | * |
> ...--40/27---10/9--*--5/3-----5/4----15/8----45/32-*135/128-...
> | | | | | | *
> | | * | | | | * | *
> | | | | | | |
> | | * | | | | * |
> | | | | | | |
> | | * | | | | * |
> | | | | | | |
> ...--32/27---16/9--*--4/3-----1/1-----3/2-----9/8--*-27/16--...
> | | | | | | |
> | | * | | | | * |
> | | | | | | |
> * | | * | | | | * |
> * | | | | | |
> | * | * | | | | * |
> | * | | | | |
> ...-256/135--64/45-*-16/15----8/5-----6/5-----9/5--*-27/20--...
> | | * | | | |
> | | * | * | | | * |
> | | | * | | |
> | | * | | * | | * |
> | | | | * | |
> | | * | | | * | * |
> | | | | | * |
> ...1024/675-256/225*128/75---32/25---48/25---36/25-*-27/25--...
> | | | | | | *
> | | * | | | | * | *
> . . . . . . .
> . . . . . . .
> . . . . . . .
> This periodicity block has 12 notes in it, and corresponds to one of the
> proposals for a 12-tone JI system (was it Ramos?) Another famous system
> defined from the same unison vectors, but only a mode of the other when
> two
> notes are transposed by a lesser diesis, is shown here:
>
> . . . . . . .
> . . . . . . .
> . . . . . . .
> * | * | | | | * | |
> * | | | | | |
> | * | | | | * | |
> | * | | | | |
> | * | * | | | * | |
> | | * | | | |
> ...--50/27-*-25/18---25/24-*-25/16---75/64-*225/128-675/512-..
> | | | * | | |
> | * | | | * | * | |
> | | | | * | |
> | * | | | | * | |
> | | | | | * |
> | * | | | | * | * |
> | | | | | | *
> ...--40/27-*-10/9-----5/3-----5/4----15/8--*-45/32--135/128*...
> | | | | | | |
> | * | | | | * | |
> | | | | | | |
> | * | | | | * | |
> | | | | | | |
> | * | | | | * | |
> | | | | | | |
> ...--32/27-*-16/9-----4/3-----1/1-----3/2--*--9/8----27/16--...
> | | | | | | |
> * | * | | | | * | |
> * | | | | | |
> | * | | | | * | |
> | * | | | | |
> | * | * | | | * | |
> | | * | | | |
> ...-256/135*-64/45---16/15-*--8/5-----6/5--*--9/5----27/20--...
> | | | * | | |
> | * | | | * | * | |
> | | | | * | |
> | * | | | | * | |
> | | | | | * |
> | * | | | | * | * |
> | | | | | | *
> ...1024/675*256/225-128/75---32/25---48/25-*-36/25---27/25-*...
> | | | | | | |
> | * | | | | * | |
> . . . . . . .
> . . . . . . .
> . . . . . . .
> Can we verify that these parallelograms always have area 12, and so always
> define a 12-tone system, from the numbers alone? Yes:
>
> |4 -1|
> | | = 4*3 - (-1)*0 = 12-0 = 12
> |0 3|
>
> I know this one was much more difficult than the first one, and I'm sure a
> lot of things could have been explained better. So questions, please, and
> next time we'll consider some 3-D (7-limit) examples, and we'll have to
> get
> a little more abstract since it's hard to show the configuration of soild
> figures in a 3-D lattice using ASCII text!
>
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