Harmonic functions


Types of functions

Harmonic function is a grouping from the harmonic variety having an extreme characteristic.

According to the extreme property we distinguish the following types of harmonic functions:

Some theorists distinguish centrifugal and centripetal functions (K.Risinger,...), e.g. D->T->S, D is centripetal S centrifugal. Others claim that only centripetal functions exist (Kresanek,...), e.g. D->T<-S, both D and S are centripetal.
Both groups are right.
The first group speak about directional functions of continuity, the second about potential functions.
Dominant (D) is a special case - it is potential and also directional function.

Potential functions

Tonic (T) is the best-ordered grouping from the harmonic variety with the maximum formal potential (F-potential). ("Best-ordered" means that tonic should have a small entropy of sounding, i.e. should be consonant.)

Tonicity (T) of grouping (g) is F-potential (U) reduced by the entropy of sounding (H):
T(g) = U(g)-H(g)

The tonic is the grouping with maximum tonicity.

E.g. Bands of 12-tone system influenced by natural modality (c,d,e,f,g,a,b) have the following F-potentials:
gacedfbf#a#g#d#c#
7.07.06.06.04.00.00.0-2.0-2.0-4.0-4.0-4.0

Groupings having greatest F-potential:

And e.g. (c,g) is a "better tonic" than (g,a) because it is consonant (its entropy of sounding is low).

Antitonic function (A) is the best-ordered grouping from the harmonic variety with the minimum formal potential (F-potential).

Functions of continuity

Value of continuity of a harmonic connection is a sum of the particular continuities of all bindings (divided by number of bindings).

Continuity towards the tonic.

Natural modality (the tonic C):
CDmiEmiFGAmiHmi5-
0.00 -0.44+1.33-1.56+1.56-0.89+0.67

Harmonic minor modality (the tonic Ami):

AmiHmiCDmiEFG#mi5-
0.00 -0.67+0.44-1.56+1.11-1.330.00

Note the extreme values for the dominants (+1.56,+1.11) and subdominants (-1.56) towards the tonic.

Dominant (D) is the grouping with the maximum positive continuity towards the tonic.
Subdominant (S) is the grouping with the maximum negative continuity towards the tonic (i.e. the maximum positive continuity in the direction from the tonic).

The local dominant of a given chord is the grouping with the maximum positive continuity towards the chord.
The local subdominant of a given chord is the grouping with the maximum negative continuity towards the chord.

If the local dominant (subdominant) belongs to an another tonality, it is called extratonal dominant (subdominant).

Functions of impulse

Value of impulse of a harmonic connection is a sum of the particular impulses of all bindings (divided by number of bindings).

The Phrygian function (F), is the grouping with the maximum impulse towards the tonic from above.
The Lydian function (L) , is the grouping with the maximum impulse towards the tonic from below.

Other directional functions

Impression on the tonic
Total energy on bindings to the tonic: E=|continuity|+impulse.

Other functions

Similarity to the tonic.
Number of common tones with the tonic.

Natural modality (the tonic C):

CDmiEmiFGAmiHmi5-
3.000.002.001.001.002.000.00

Cogency of closing

Cogency of closing is meassure of relaxation. A connection makes a cogent closing if it respects this scheme. 
                      extreme
        small     ----------------->  big
       potential     continuity       potential

The fallacy (deceptive closing) is the connection leading to the chord having extreme potential, but not with extreme continuity; e.g. G-Ami.
Listener hear a stable chord but this chord appears such a way, that he cannot receive it as a definitive end.

Example of harmonic functions

Estimated functions of some heptatonics:

Modality name Tones Dominant Tonic Subdominant
Natural (minor) gahcdef Emi Ami Dmi
Natural (major) g#ahc#def# E A D
Harmonic major g#ahc#def E A Dmi6
Harmonic minor g#ahcdef E Ami Dmi
Gypsy g#ahcd#ef G#mi C5+ F
Whistle g#ahcdd#f G# F Dmi
Altered g#a#hc#def E7 C# A#mi

Functions of natural modality

Harmonic functions of Christian modes.

Name Example S D T Status
phrygian efgahcde Ami Hmi5- Emi labile
    Ami Dmi E modulated, labile
aeolian ahcdefga Dmi Emi Ami stable
    Dmi E Ami modulated
ionic cdefgahc F G C stable
lydian fgahcdef Hmi5- C F labile
    B C F modulated
dorian defgahcd G Ami Dmi labile
    G A Dmi modulated
mixolydic gahcdefg C Dmi G labile
    C D G modulated

According to polarity of functions:

Functions Name (Status)
S -D -T ionian (stable), lydian, mixolydian (modulated)
Smi-D -T lydian (labile)
S -D -Tmi dorian (modulated)
S -Dmi-T mixolydian (labile)
S -Dmi-Tmi dorian (labile)
Smi-Dmi-T phrygian (modulated,labile)
Smi-D -Tmi aeolian (modulated)
Smi-Dmi-Tmi aeolian (stable)

Harmonic bindings