Chord arrangement

Groupings and chords

Each grouping represents a set of possible chords.

E.g. the grouping {h,d,f} represents the chords {h1,d2,f2}, ...,{d0,f1,h1,d2,f2,d3}, and so on.

A grouping having n tones have n (soprane) positions, n! shapes; i.e. (n-1)! basic shapes and n rotations of every basic shape.

E.g. the grouping {ab} has 2 shapes (ab), (ba); basic shape [ab].
The grouping {abc} has 6 shapes (abc), (bca), (cab), (acb), (bac), (cba); 2 basic shapes [abc] and [acb].

Grouping of n tones could be written as a chord with r tones r>n. Number of such chords is equal to the number of permutations with repetition.

Well-balanced chord

Chord is well-balanced if energy of all tones is in given bounds. Theorist recomend, that chord should be to some degree "homogennous". E.g. intensity of one tone should not be extremely high, intervals of tones should not be too different, ... In the following paragraphs we will observe density of chords, doubling and ommiting of tones and softening of dissonances.

Chord density

The interval between lowest tone and highest tone of chord is named chord span.
Density of chord is defined: r = k / s,
where k is system order and s chord span.

E.g. grouping {c,e,g} has 6 shapes (2 main shapes):
ChordVariantMarkUsed nameSpanDensity
[c1,e1,g1] 012 5 chord of fifth 7 1.71
[c2,e1,g1] 120 8 chord of sixth 8 1.50
[c2,e2,g1] 201 3 of sixth+fourth 9 1.33
[c2,e1,g2] 102 5 - 15 0.80
[c1,e2,g1] 210 3 - 16 0.75
[c3,e2,g1] 102 8 - 17 0.71
Narrow span (when s<=7),extended span (s>7; s<15) and wide span (s>=15) are distinguished.
Density of a chord influences entropy and stability of the chord.
Adding of other tones into the same span of a chord is called thickening of the chord.
Density often changes symmetricaly (bass voice | upper voices).

Doubling of tones

If two or more tones of a chord put energy into the same formal band, we call this phenomenon "doubling".
Doubling change distribution of energy in bands.
E.g. the tone {e} in the chord {e1,c2,e2,g2,e3} disturbes formal root {c} of grouping {c,e,g}.

The rules of classical harmony do not recommend doubling of sensitive tones {b,f} of natural modality {c,d,e,f,g,a,b}:
E.g. in the chord {b,d,f} only doubling of tone d was allowed.
Doubling of tones was tolerated in melodic sequences.
(Tonality and its sensitive tones and harmonic functions have a second-rate significance in sequences).

Permissions for doubling do not depend on structure of the chord, but on (formal) potentials of tones.
More energy is the bands of sensitive tones changes natural structure of the modality.

Ommiting of tones

Some connection of two groupings requires ommiting of a tone. E.g.:
      D7     G              Dmi7   G
      --------              --------
      c   -  h              c   -  h
      a   ?                 a   ?
      f#  -  g              f   ?  g
      d   =  d              d   =  d
Tone a can be omitted.    Tone f or a can be omitted.
Bindings having higher impulse (e.g. f#-g) are preffered.

Softening of dissonances

Optimization of real sound of the chord.

1. Increasing of continuity:

 h1        h1        h2      h2
    ->    g#1           ->   g#2
    ->                  ->   e2
 c1        c1        c2      c2
It is an acoustic phenomenon of partitoning of beats.
Whole triads would soften dissonance more then pure fifths (K.Janeček).

2. Changing of potentials:

   c#2 + -------- + c#2
    c2 +
    h1 + -------- + h1
                  + c1
Collision of halftones - removed with help of octave transposition. Tone c1 has a higher potential than tone c2, so the chord [c1,h1,c#2] is better ordered (has lower entropy).

3. Decreasing of density:

                  + e1
   g0 + --------- + g0
   e0 +
   c0 + --------- + c0

Some musical styles intentionally ignore this recommendation (e.g. see Beethoven's bass parts).
Harmonic bindings