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:
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
Harmonic connection |
Cont. |
Imp. |
Harmonic connection |
Cont. |
Imp. |
EmiAmi, GC, CF |
+1.56 |
2.11 |
Bmi5-C, AmiBmi5- |
+0.67 |
3.67 |
EmiC, AmiF |
+1.33 |
1.56 |
GAmi, CDmi |
+0.44 |
2.78 |
EmiF |
+1.11 |
3.67 |
DmiEmi, FG |
0.00 |
2.78 |
Bmi5-Emi, FBmi5- |
+1.11 |
2.11 |
AmiAmi, CC |
0.00 |
0.22 |
DmiG |
+0.89 |
1.22 |
EmiC, AmiF |
-1.33 |
1.56 |
CAmi, FDmi |
+0.89 |
0.56 |
AmiEmi, CG, FC |
-1.56 |
2.11 |
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:
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.