- Groupings and chords
- Well-balanced chord
- Chord density
- Doubling of tones
- Ommiting of tones
- Softening of dissonances

*
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.

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):

Chord | Variant | Mark | Used name | Span | Density |
---|---|---|---|---|---|

[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 |

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 change distribution of energy in bands.

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.

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.

1. Increasing of continuity:

h1 h1 h2 h2 -> g#1 -> g#2 -> -> e2 c1 c1 c2 c2

Whole triads would soften dissonance more then pure fifths (K.Janeček).

2. Changing of potentials:

c#2 + -------- + c#2 c2 + h1 + -------- + h1 + c1Collision 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

Harmonic bindings