Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited


periodicity block


    A term used by Fokker in his lattice theory to describe a whole multi-dimensional region of the lattice that could substitute for another, by means of unison vectors.

    This concept is allied with that of finity, which I developed independently of Fokker. The periodicity blocks quantify the finity of the system.

    Periodicity blocks enclose a certain number of discrete categorical intervals or pitch-classes, and the unison vectors are small enough intervals that pitches within the block can represent or imply pitches outside of it which have different prime/odd-factors, a process which I had named bridging.

    [Paul Erlich comments:

    That's correct, but Fokker generally set the overall prime-limit (either 5 or 7) beforehand and all unison vectors connect pitches (or intervals) within this limit.
    Paul has pointed out that my concept of 'bridges' refers specifically to the kinds of prime-factors and not the numbers of prime-factors, which is what Fokker described with the periodicity block concept.]

    This number of pitches can be calculated by a Matrix determinant, using the prime-factors of the ratios at either ends of the bridges or unison vectors to fill the Matrix.

    [from Joe Monzo, JustMusic: A New Harmony]

    See Erlich's A Gentle Introduction to Fokker Periodicity Blocks.

    also see Kees van Prooijen's periodicity-block webpage.

    * * * * * * * * * * * * * * * * * *

    This is Gene Ward Smith's formula for finding what he calls a "notation" for the ratios enclosed within a JI periodicity-block:

    where M is the matrix composed of a set of i rational vectors {u1/v1, u2/v2,... ui/vi} in which u1/v1 is a step-vector and {u2/v2 ... ui/vi} are commatic or chromatic unison-vector generators of the kernel, and where {h1, h2, ...hi} is the top row of M^-1, and where round() is the function which rounds to the nearest integer,

    for any non-zero a scale can be defined by calculating for 0 <= n < d :

    step[n] = (u1/v1)^round(h1(2)*n/d) * (u2/v2)^round(h2(2)*n/d) * ... (ui/vi)^round(hi(2)*n/d) .

    * * * * * * * * * * * * * * * * * *

    Further notes from Paul Erlich:

    Yahoo tuning-math group, message 2158:

    From: "paulerlich"
    Date: Tue Dec 25, 2001 4:12 am
    Subject: Re: The epimorphic property

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


    > --- In tuning-math@y..., "paulerlich" wrote:
    >
    > > > Does "shape" entail connectedness, or can it be scattered islands
    > > > all over the place?
    > >
    > > The latter. Especially as preimages of ETs, such constructs would be
    > > just fine.
    >
    > Under that definition, PB <==> epimorphic. Are you sure it is the
    > accepted one?

    The only published articles on PBs are Fokker's. Inferring strict definitions from these articles would suggest that a parallelepiped (or N-dimensional equivalent) are the only accepted shape (thus I call these _Fokker_ periodicity blocks, or FPBs), and that if there is an even number of notes, one needs to produce two alternative versions so that symmetry about 1/1 is maintainted.

    [note from Monz: Paul shows how to construct the parallelepiped type of periodicity-block in his A gentle introduction to Fokker periodicity blocks, part 2, and then shows that periodicity-blocks need not be parallelepipeds in his A Gentle Introduction to Fokker Periodicity Blocks: an excursion.]

    * * * * * * * * * * * * * * * * * *

    Onelist Tuning Digest 465
    Message: 6
    Date: Thu, 30 Dec 1999 04:12:36 -0500
    From: "Paul H. Erlich"
    Subject: 11-limit, 31 tones, 9 hexads within 2.7c of just

    ... all JI scales involve a sort-of-arbitrary decision of where to stop. What I've shown is, this decision is not completely arbitrary, it almost always seems to conform to a periodicity-block construction.

    * * * * * * * * * * * * * * * * * *

    Onelist Tuning Digest 340
    Message: 14
    Date: Mon, 4 Oct 1999 18:40:06 -0400
    From: "Paul H. Erlich"
    Subject: RE: Re: Fokker periodicity blocks from the 3-5-7-harmonic lattice

    Let me try graphing the 27-tone periodicity block that I posted on Thu 9/23/99 5:26 PM.

    
    
    The matrix of unison vectors is
    
         2    -3     1
         3    -1    -3
        -4     3    -2
    
    and the coordinates are
    
              -3           2          -2         329
              -2           1          -2         645
              -2           1          -1         414
              -2           2          -3          62
              -1           0          -2         960
              -1           0          -1         729
              -1           1          -3         378
              -1           1          -2         147
               0          -1          -1        1045
               0          -1           0         814
               0           0          -3         694
               0           0          -2         462
               0           0          -1         231
               0           0           0           0
               0           1          -4         111
               1          -2           0        1129
               1          -1          -3        1009
               1          -1          -2         778
               1          -1          -1         547
               1           0          -3         195
               2          -2          -2        1094
               2          -2          -1         862
               2          -1          -3         511
               2          -1          -2         280
               3          -3          -1        1178
               3          -2          -2         596
               4          -3          -2         911
    
    
    I think I'll go back to the usual orientation for this one:
    
    
         5
        /:\
       / : \
      /  7  \
     /,'   `.\
    1---------3 
    
    
    And we're off:
    
    
    329
      \`.
       \  62
        \ :   414
         \: ,'   `.
         645-------147
           \`.   ,'/:\             0
            \ 378 / : \         ,'/
             \ : /.729-------231 /
              \:/,'**\`.\  ,'/:\/
              960-------462 / :/\
                 `. : ,' : / 814 \
                   694--\:/,'195`.\
                     \  1045-/:\--547
                      \    `/ : \' : `.
                       \   / 778-\-----280
                        \ /.'  \`.\@@,'/:\
                        1009------511 / : \
                                 \ : / 862 \
                                  \:/,'   `.\
                                  1094------596
                                             :\
                                             : \
                                            1178\
                                               `.\
                                                 911
    
                                               
    **111 goes here, but there was not enough room to write it.
    @@1129 goes here, but there was not enough room to write it.
    
    

    This one is amazingly rich in 7-limit triads and tetrads, so much so that it's straining my meager ASCII symbology to show all the connections.

    * * * * * * * * * * * * * * * * * *

    Onelist Tuning Digest 340
    Message: 16
    Date: Mon, 4 Oct 1999 22:00:20 -0400
    From: "Paul H. Erlich"
    Subject: JI pentatonic, "diatonic", and decatonic scales as periodicity bl ocks

    These three basic scales, identified in my paper as the melodic bases for 3-, 5-, and 7-limit harmony, respectively, have JI representations that come out as Fokker periodicity blocks when the typical "chromatic" interval implied by the scales is used as a unison vector.

    Interestingly, all these periodicity blocks are of the "most natural" type catalogued by Kees van Prooijen (he skipped the 3-limit ones but they're trivial -- apparantly not enough so to satisfy Carl [Lumma]?)

    In the 3-limit pentatonic case, modulating the scale by a single ratio of 3 simply moves one note by 256:243. Using this as the unison vector (only one is needed since octave-equivalent 3-limit space is one-dimensional), we get the following periodicity block:

    
    
    ratios for major
    
    1/1-------3/2-------9/8------27/16-----81/64
    
    ratios for minor
    
    32/27-----16/9-------4/3-------1/1-------3/2
    
    

    Interestingly, as I was writing this, Carl posted something about 1D periodicity blocks, to which this may relate.

    In the 5-limit diatonic case, 81:80 is already assumed as a unison vector, and the chromatic interval by which one note moves when modulating by a ratio of 3 is 25:24. Using these as the unison vectors, the resulting periodicity block is:

    
    
    ratios for major
         
         5/3-------5/4------15/8
         / \       / \       / \
        /   \     /   \     /   \
       /     \   /     \   /     \
      /       \ /       \ /       \
    4/3-------1/1-------3/2-------9/8
    
    or
    
    10/9-------5/3-------5/4------15/8
       \       / \       / \       / 
        \     /   \     /   \     /   
         \   /     \   /     \   /     
          \ /       \ /       \ /       
          4/3-------1/1-------3/2
    
    or
    
    100/81-----50/27
        \       / \
         \     /   \
          \   /     \
           \ /       \     
          40/27-----10/9-------5/3
                       \       / \
                        \     /   \
                         \   /     \
                          \ /       \
                          4/3--------1/1
    
    ratios for minor
    
         1/1-------3/2-------9/8
         / \       / \       / \
        /   \     /   \     /   \
       /     \   /     \   /     \
      /       \ /       \ /       \
    8/5-------6/5-------9/5------27/20
    
    or
    
     4/3-------1/1-------3/2-------9/8
       \       / \       / \       / 
        \     /   \     /   \     /   
         \   /     \   /     \   /     
          \ /       \ /       \ /       
          8/5-------6/5-------9/5
    
    or
    
     40/27-----10/9
        \       / \
         \     /   \
          \   /     \
           \ /       \     
          16/9-------4/3-------1/1
                       \       / \
                        \     /   \
                         \   /     \
                          \ /       \
                          8/5--------6/5
    
    
    

    In the 7-limit decatonic case, 64:63 and 50:49 are already assumed as unison vectors. If you are unfamiliar with decatonic scales, see my paper. When modulating by a 3-limit ratio, two notes move by a 48:49. Using these as the unison vectors, the resulting periodicity block is:

    
    
    ratios for symmetrical major
    
          5/4------15/8                      7/4------21/16
        ,'/:\`.   ,'/:\`.                  ,'/:\`.   ,'/:\`.
    10/7-/-:-\15/14/-:-\45/28   or      1/1-/-:-\-3/2-/-:-\-9/8
      : / 7/4------21/16\ :              : /49/40----147/80\ :
      :/,'   `.\:/,'   `.\:              :/,'   `.\:/,'   `.\:
     1/1-------3/2-------9/8            7/5------21/20-----63/40
    
    or
    
    16/9-------4/3-------1/1           80/63-----40/21-----10/7
      :\`.   ,'/:\`.   ,'/:              :\`.   ,'/:\`.   ,'/:
      : \32/21/-:-\-8/7 / :              : 160/147-:-\80/49/ :
    56/45-----28/15------7/5    or     16/9-------4/3-------1/1
        `.\:/,'   `.\:/,'                  `.\:/,'   `.\:/,'
         16/15------8/5                     32/21------8/7
    
    or
    
                    5/4                                7/4
                  .'/:\`.                            .'/:\`.
    40/21-----10/7-/-:-\15/14           4/3-------1/1-/---\-3/2
      :\`.   ,'/: / 7/4 \ :              :\`.   ,'/: /49/40\ :
      : \80/49/ :/,'   `.\:     or       : \ 8/7 / :/.'   `.\:
     4/3-------1/1-------3/2           28/15------7/5------21/20
        `.\:/,'                            `.\:/,'
          8/7                                8/5
    
    ratios for symmetrical minor
    
          4/3-------1/1                     40/21-----10/7 
        ,'/:\`.   ,'/:\`.                  ,'/:\`.   ,'/:\`.
    32/21/-:-\-8/7-/-:-\12/7    or   160/147/-:-\80/49/-:-\60/49
      : /28/15------7/5 \ :              : / 4/3-----/-1/1 \ :
      :/,'   `.\:/,'   `.\:              :/,'   `.\:/,'   `.\:
    16/15------8/5-------6/5           32/21------8/7------12/7 
    
    or
    
    40/21-----10/7------15/14           4/3-------1/1-------3/2
      :\`.   ,'/:\`.   ,'/:              :\`.   ,'/:\`.   ,'/:
      : \80/49/-:-\60/49/ :              : \ 8/7-/-:-\12/7 / :
     4/3-------1/1-------3/2    or     28/15------7/5------21/20
        `.\:/,'   `.\:/,'                  `.\:/,'   `.\:/,'
          8/7------12/7                      8/5-------6/5
    
    or
    
                   21/16                              15/8
                  .'/:\`.                            .'/:\`.
     1/1-------3/2-/-:-\-9/8           10/7------15/14/---\45/28
      :\`.   ,'/: 147/80\ :              :\`.   ,'/: /21/16\ :
      : \12/7 / :/,'   `.\:     or       : \60/49/ :/.'   `.\:
     7/5------21/20-----63/40           1/1-------3/2-------9/8
        `.\:/,'                            `.\:/,'
          6/5                               12/7
    
    

    I don't think the pentachordal decatonic scales can be thought of as periodicity blocks, but I could be wrong . . .


Updated: 2001.12.25


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