Harmonic Bindings (brief abstract)

We present a musically theoretical model of relations among the tones in the harmonic music stream. Model unifies the Janeček's theory of imaginary tones[Janeček,1965] with the two Risinger's principles of functional relations [Risinger,1969].

Harmonic system, modality, tonality

Harmonic system is a relation on a set of tones. The regular (harmonic) system has the frequencies of tones ordered according to a geometric progression. The formal system (F-system) identifies tones with the 2:1 frequencies (octave identity). Each tone in a F-system represents the corresponding equivalence class of the regular system. Modality is a subset of F-system tones. Tonality is a modality with some restrictions on possible groupings. The harmonic variety is a set of all groupings in the tonality.

Energy zones, harmonic bindings

Assume that the energy zone is a carrier of the energy of a given F- system tone. Harmonic binding is a carrier of the energy pertaining to the given musical interval. We assume that two basic processes exist:

• The reactivity among the adjacent zones. The potential energy of the zones changes into the binding energy; the measure of change is called impulse of binding .
• The resounding of zones whose frequencies are in ratios of small integers (approximately). One zone gains part of the potential energy from the other zone; the measure of the transferred potential energy is called continuity of binding .

Continuity and impulse

To obtain some more precise (numerical) values of continuity and impulse we need understand the very substance of the interactions described above. In relevant literature there are known various amounts about roots (= the tone having the zone equipped with maximal energy), [Risinger,1969], and consonance , [Janecek,1965], of the selected chords. The consonance of the grouping depends on the distribution of energy among zones. The maximal dissonance (entropy) appears when the portions of energy in zones are balanced. The highest impulses are associated with the following intervals: the semitone, whole tone and minor third, whereas the highest continuity with the perfect fifth (resonance 3:2) and major third (5:4). The continuity does depend on the acting direction: the descending fifth, similarly the descending major third, have a positive continuity.

Harmonic connections

The measure of the link-up between groupings is the total value of the continuity/ impulses in the bindings. The values for some selected connections (in natural modality) are shown in Table 1. We distinguish between direct harmonic stream (with the positive continuity to every next grouping) and reverse harmonic stream (with the negative continuity).

Table 1: Some selected connections

For example, some musical styles have their marked harmonic progressions. The following succession can be often found in baroque music: $C: F: Bmi5-: Emi: Ami: Dmi: G: C$. We see that there are practically the maximal positive values (1.56, 1.11, 0.89) in all partial connections (see Table 1).

Potential levels

By modality, only finite zones have their energy directly from the sounding tones. If energy input in these zones is balanced, we derive some characteristics from the modality structure itself. Formal potential (F-potential) of a zone is the sum of individual binding influences going from other zones to the given zone. We call tonicity of the grouping the F-potential reduced by the entropy of the sounding. The distribution of the F-potentials in the tonality determines certain levels of particular groupings. The transition from one level to another one results in a tension. For example, the following F-potentials correspond to the adequate groupings of the natural modality:

p( C)  = p(Ami)= 6.33;

p( Emi)= p(F)  = 4.33;

p( Dmi)= p(G)  = 3.67;

p( Bmi5-)      = 1.33.

Harmonic functions

Harmonic functions are groupings from a harmonic variety with extreme properties. They are defined as follows:

• Tonic function is the grouping with the maximum tonicity.
• Dominant function is the grouping with the maximum positive continuity towards the tonic.
• Subdominant function is the grouping with the maximum negative continuity, i.e. the maximum positive continuity in the direction from the tonic.
• Phrygian function, is the grouping with the maximum impulse towards the tonic from above.
• Lydian function, is the grouping with the maximum impulse towards the tonic from below, [10], [3].

Bibliography

Janecek,1965: Janeček Karel: Základy moderní harmonie (Fundamentals of Modern Harmony; in Czech), Prague 1965.

Risinger,1969: Risinger Karel: Hierarchie hudebních celků (Hierarchy of Musical Units; in Czech), Prague 1969.

Music theory