Leading of voices








Strict harmonic connection

Classical strict harmonic connection have to satisfy the following rules:

These rules have been respected also in other styles, mostly in central voices of polyphony.

Optimal link-up

Optimal link-up is such a leading of voices, in which the total value of bindings impulse in particular linear voice bindings is maximum:
max imp(h(i,j))

This task is similar to minimization of interval sizes in matrix of connection:
min |h(i,j)|


E.g. matrix of the harmonic connection [e,g,c]->[g,b,d]:
Intervals:                    Absolute values:
    e   g   c |                  e   g   c |
  ------------+---             ------------+---
   -2  -5  +2 | d                2   5   2 | d
   -5  +4  -1 | b                5   4   1 | b
   +3   0  -5 | g                3   0   5 | g

These possible connections exists: (Do you remember the combinatorial problem of placing queens to the chessboard?)

  a/ * . .  b/ . . *  c/ * . . d/ . . * e/  . * .  f/ . * .
     . * .     . * .     . . *    * . .     . . *     * . .
     . . *     * . .     . * .    . * .     * . .     . . *

Sum of values corresponding to the marks (*):
a b c d e f
11 9 3 7 9 15

The sum is minimal in case of the variant c/:

       e  ->  d
       c  ->  b
       g  ->  g


Other example; the connection [c,f,a]->[d,g,b]:
    c   f   a |
  ------------+--
    1   6   2 | b
    5   2   2 | g
    2   3   5 | d

Sum of values corresponding to the marks (*):
a b c d e f
8 6 6 10 10 16

The sum is minimal in case of the variant b/ and c/.

For more precise result we need the matrix of impulse values:

    c   f   a |
  ------------+--
   12   0   3 | b
    0   3   3 | g
    3   1   0 | d

Sum of impulse values corresponding to the marks (*):
a b c d e f
15 9 16 4 6 0
The sum is maximal in case of the variant c/.

       c  ->  b
       a  ->  g
       f  ->  d

This connection is "correct" (with regard to classical rules):

       c  ->  b
       a  ->  g
       f  ->  d
      -----------
       f  ->  g

Other examples:
Harmonic bindings