Music theory - Harmonic kinetics

You're reading an English translation of the original Czech page.

Harmonic stream

Harmonic kinetics deals with harmonic connections and their effect. The sequence of harmonic connections is called harmonic stream.

Hostinský Otakar

Hostinský Otakar , 1847-1910, Czech esthetician, historian, musicologist and music theorist. The founder of modern Czech aesthetics and musicology, author of numerous historical works of music. He emphasized that science should proceed from the simple to the more complex, for example triad consists of 6 pairs of interval. He understands sound color, defended mediant connections, performed comparative analysis of variants of songs. He looked for rules of Czech music declamation (accents of syllables). He studied the theory and history of fine arts and literary theory. He considered program music to be composed art.

(The term dynamics has been already introduced in the music context - in connection with emphasizing of tones and with volume of sound, as opposed to music statics therefore is uses the term kinetics).

Harmonic connection

m_harcon

Harmonic connection arises by immediate succession of two tone clusters.

Tones are carriers of acoustic qualities and characteristics (frequency, amplitude, location in time ...).

We derive other characteristics (continuity, pulse, ...) from tone relationships, while the mean characteristics of individual links within a given time are considered as determining.

Harmonic connections act as the same, if they binding corresponds (by quality and quantity).

More harmonic connections in a row makes a harmonic stream.
Distribution of potentials in the tonality determine certain levels of individual harmonies. Relationships of levels give rise to musical tension, forcing the harmonic current to the next movement.

Contrasts of tension and relaxation can also result from different sonance (entropy) of chords. Harmonic slope (trend) is however determined only by location of chord in the tonality, it is not affected by tension (entropy) within chords.

According to the direction of the continuity we distinguish direct harmonic stream (with a positive continuity to every next chord) and reverse harmonic stream (with a negative continuity).

Harmonic connection is the connection between two chords. Each tone of the first chord is associated with each tone of the second chord.

A long time musical theory used only the links between the essential tones of chords.

m_stream L.Janáček began to consider links of all tones of the first chord to the basic tone of the second chord.

Harmonic phrase C-Dmi⁷-G⁷-C contains of 3 connections.

C-Dmi⁷:        Dmi⁷-G⁷:       G⁷-C:
  c  e  g       f  a  c  d       g  b  d  f
  2 –2 –5 d     0 –4  5  3 f     0 –4  5  2 g
  0 –4  5 c    –3  5  2  0 d    –3  5  2 –1 e
 –3  5  2 a     6  2 –1 –3 b     5  1 –2 –5 c
  5  1 –2 f     2 –2 –5  5 g

Kinetic characteristics of chords (potential functionality, ...) depends also on modality structure.

Trivial connections

Ami7-Ami7:
a c e g
------------
-2 -5 3 0 g
-5 4 0 -3 e
3 0 -4 5 c
0 -3 5 2 a

Trivial harmonic connection is the connection of two identical groupings. Harmonic statics is a subset of harmonic kinetics: isolated chord can be seen as trivial harmonic connection.

Sequential connections

Harmonic connections will usually take place in one particular modality. Sequence is such a succession of certain musical units, where the unit (chord) itself changes, but keeps (if possible) shape. According to relationship to modality we distinguish: [Filip]

translation, i.e. shifts of the structure in modality, e.g. C=>Dmi=>Emi=>F
transposition, i.e. shifts in the system (outside of the last modality), e.g.. C=>C#=>D
mutation (reevaluation to other modality), e.g. C=>Dmi=>Eb.

The list of links

Let us calculate the number of links with a given interval (–5..6) for selected connections:

Connections of identity

0	1	–1	2	–2	3	–3	4	–4	5	–5	6	spoje
3	0	0	0	0	1	1	1	1	1	1	0	CC, AmiAmi
3	0	0	0	0	2	2	0	0	0	0	2	Hmi^5–Hmi^5–

Connections of continuity

Connections with the prevalence of continuity:

0	1	–1	2	–2	3	–3	4	–4	5	–5	6	spoje
1	1	0	1	1	0	1	0	1	3	0	0	CF, GC
1	1	0	1	1	0	1	0	1	3	0	0	EmiAmi, AmiDmi
1	0	0	2	1	0	2	0	0	2	0	1	DmiG
1	0	1	2	0	0	1	0	1	2	0	1	Hmi^5–Emi
1	0	1	2	0	0	1	0	1	2	0	1	FHmi^5–

Connections of impulse

Connections with the prevalence of impulse:

1	–1	2	–2	3	–3	4	–4	5	–5	6	spoje
2	0	1	2	0	0	0	1	2	0	1	EmiF
1	1	1	2	0	0	1	0	1	2	0	CHmi^5– , Hmi ^5–Ami
0	1	3	1	0	1	0	0	1	1	1	FG
0	1	3	1	0	1	0	0	1	1	1	DmiEmi
1	0	2	2	0	1	0	0	2	1	0	CDmi,GAmi

Volek Jaroslav

Volek Jaroslav , 1927-1989, Czech musicologist, music theorist and esthetician. He dealt with philosophy and music theory in general terms. He has defined so called responsible binding consisting of melody in older music, and harmony in music later. He supported the Janáček's theory of harmonic connections. He advocated integration of mediants into system of harmonic functions.

Mediants

“Mediant” connections of major and minor triads.
The number of common tones is in the first column. E.g. [c,e,g] and [a,c,e] has 2 common tones {c,e}.

Number of linkages for the interval (–5..6):

0	1	–1	2	–2	3	–3	4	–4	5	–5	6	spoj
2	0	0	0	1	2	0	1	1	0	2	0	Ami-C
2	1	0	0	0	1	1	0	2	2	0	0	Emi-C
1	0	1	0	1	3	0	0	1	0	1	1	Ami-Cmi
1	1	1	0	0	1	0	1	3	1	0	0	Emi-Cmi
1	0	1	0	1	3	0	0	1	0	1	1	A-C
1	1	1	0	0	1	0	1	3	1	0	0	E-C
0	0	2	1	1	2	0	0	1	0	0	2	A-Cmi
0	1	2	0	0	1	0	2	2	0	1	0	E-Cmi

Mediant connections

with two common tones (Ami-C,..,C-Emi) have highest continuity.
with one common tone - chords have the same genus (Ami-Cmi,..,C-E)
without common tones - so called chromatic mediants (A-Cmi,..,Cmi-E), have highest impulse.

Characteristics

The degree of connection between two chords is determined by the total value of the characteristics (continuity or impulse).

If the connection of two chords (given by vectors u, v) has the characteristic of f (given by matrix F) then f(u,v) = u∙F∙v = ∑∑F(h_k,l)/(m∙n), where h_k,l is interval structure, k=0..m–1, l=0..n–1, m,n are dimensions of u,v.

Intervals c e g --------- –3 5 2 a 5 1 –2 f 0 –4 5 c	Continuity: c e g ---------- 0 4 0 a 4 0 0 f 0 2 4 c	Impulse: c e g ---------- 1 0 3 a 0 12 3 f 0 0 0 c
	c(C-F)=(3∙4+2)/9=14/9=+1.56	i(C-F)=(12+2∙3+1)/9= 19/9=2.11

E.g. continuity (impulse) of harmonic connection [C,F] is product: of vector of the chord C i.e.: u=[1,0,0,0,1,0,0,1,0,0,0,0],

matrix of continuity (impulse) and vector of the chord F: v = [1,0,0,0,0,1,0,0,0,1,0,0]

Selected connections of the natural 7-tonal modality in 12-tone system (connections of consonant triads) have the following values continuity and impulse:

Selected connections

Harmonic connection	∑Connection	∑Impulse	øConnection	øImpulse
EmiAmi, GC, CF	3c₁+c₂	i₁+2i₂+i₃+i₄	+2.33	3.06
EmiC, AmiF	2c₁+2c₂	i₁+2i₃+2i₄	+2.00	1.89
EmiF	2c₁+c₂	2i₁+3i₂+i₄	+1.67	4.89
Hmi^5-Emi, FHmi^5-	2c₁+c₂	i₁+2i₂+i₃+i₄	+1.67	3.06
DmiG	2c₁	3i₂+2i₃	+1.33	2.61
CAmi, FDmi, GEmi	2c₁	i₁+2i₃+2i₄	+1.33	1.89
Hmi^5-C, AmiHmi^5-	c₁+c₂	2i ₁+3i₂+i₄	+1.00	4.89
Hmi^5-G, DmiHmi^5-	c₁+c₂	i₂+3i₃+i₄	+1.00	1.44
GAmi, CDmi	c₁	i₁+4i₂+i₃	+0.67	4.44
DmiEmi, FG	0	i₁+4i₂+i₃	0.00	4.44
AmiAmi, CC	0	2i₃+2i₄	0.00	0.56
Hmi^5- Hmi^5-	0	4i₃	0.00	0.89
AmiEmi, CG, FC	-3c₁-c₂	i₁+2i₂+i₃+i₄	-2.33	3.06

Comparison of connections

Two harmonic connections are equivalent (similar) if all the individual links are the same (similar). E.g. [Dmi,Hmi^5–] has the same bindings as [Hmi^5–,G]; [G,C] is equivalent with [Emi,Ami].

We neglect the other factors.
E.g. differences between actual (not formal) links, or differences caused by different energy of bands,...

Particular connections

Particular harmonic connection includes a subset of bindings of a given harmonic connection. Studying these subsets may be important for distinction of musical styles.

Connections of quints

The table of connections of quints is sorted according to the value of impulse:

Connection	∑Continuity	∑Impulse	øContinuity	øImpulse	Quints
cg-cg	0	0	0.00	0.00	Identical
cg-cf, cg-dg	±2c₁	i₂	±3.00	1.63	Live
cg-a#d#, cg-ae	±c₂	i₂₊2i₃	±0.75	2.63	Soft
cg-d#g#, cg-he	±2c₂	i₁₊i₃	±1.50	3.50	Sharp
cg-a#f, cg-da	±c₁	2i₂₊i₃	±1.50	3.75	Soft Mozartian
cg-c#g#, cg-hf#	±c₂	2i₁	±0.75	6.00	Sharp Mozartian
cg-c#f#	0	2i₁	0.00	6.00	Neutral

Potential of chord

Javorskij Boleslav Leopoldovič

Javorskij Boleslav Leopoldovič [], 1877-1942, Russian music theorist, creator of dynamic musical concept, fundamentally opposed to the concept of Riemann. Rejected the concept of musical forms such as architectural design, regarded it as a phenomenon primarily procedural, psychological. Works with the formation and release of tension, proposed a theory of gradual rhythm. He greatly influenced the Soviet musicology.

Potential chord is the sum of the potentials (F-potentials) all participating bands.

U(X) = ∑(U(X_i)/L, where L is level (number of tones) of the chord, iε(0,L-1)

This is a key point for understanding of the functional theory of harmony submitted by Riemann. Each chord tone contributes to the overall potential of the chord. Each tone is a carrier functionality.

U(C) = U(Ami)	6.33
U(Emi)= U(F)	4.33
U(Dmi)= U(G)	3.67
U( Hmi^5-)	1.33

The following F-potentials, U, correspond to the selected chords of the natural modality:

E.g. for the chord C:
U(C)=(c,e,g) =(U_c+ U_e+ U_g)/3= (6+6+7)/3 = 6.33

Formal potential (F-potential) of a chord is given by the sum of the formal potentials of the participating bands.

We call tonicity of chord the potential reduced by the entropy of sounding.

Contrast of potentials

The distribution of F-potential in the modality determines certain levels of particular groupings. The difference between the the potential levels results in tension.

Contrast of potentials is defined:
F(i,j)= U(j) – U(i);
where U(k) is potential of chord k.

Energy of bindings

i	j	i+j	i∙j	(i+j)/(i∙j)
1	1	2	1	2.00
1	2	3	2	1.50
1	3	4	3	1.33
1	4	5	4	1.25
2	2	4	4	1.00
2	3	5	6	0.83
2	4	6	8	0.75
3	3	6	9	0.66
3	4	7	12	0.58
4	4	8	16	0.50

Energy of bindings depends on the energy of sounding tones..
If the connection has multiple linkages, then the energy pertaining to a given linkage is lower.

Table contains ratios (i+j)/(i∙j) for i, j <5.

E.g. connection of two triads has 3+3 = 6 portions of energy and 3∙3= 9 bindings.
Connection of two tetrads has 4+4 = 8 portions of energy and 4∙4= 16 bindings.

Functionality

Factor	Connection	Chord
Connection	Fluency g → c	Consonance g c
Impulse	Impulsity h -- c	Dissonance c h
Continuity + impulz	Functionality g → c h -- c	Entropy c h g

Impressiveness of the connection

Continuity makes the sound of the connection flowing. Impulse adds effect to the connection. Let us suppose that there is such a characteristic (impressiveness of the connection) which depends on both (continuity and impulse).

Connection	Continuity	Impulse	Sum
Emi -F ,	1.67	4.89	4.78
Emi –Ami, G -C	2.33	3.06	4.56
Hmi^5-C , Ami-Hmi^5-	1.00	4.89	4.33
G -Ami,	0.67	4.44	3.22
Hmi^5-Emi, F -Hmi^5-	1.67	3.06	3.22
Emi -C , Ami-F	2.00	1.89	2.89
Dmi –Emi, F -G	0.00	4.44	2.78
C -Ami	1.33	1.89	1.44

Connections in the table are sorted by the sum of continuity and impulse.

Harmonic density

The density of functions - the number of harmony in the block, density of changes of polarity functions (or rate of polarity of function - T extreme polarity)

- dominant density - maximum - tonic density - minimum

Harmonic tendencies

Each tone has a certain trend in the context of a modality. This tendency depends on the structure of the modality, particularly on the potential tones and impulse bindings.

Filip Miroslav

Filip Miroslav , 1932-1973, Slovak musicologist and educator engaged in music theory and acoustics. In an essay on the evolution of classical harmony he extended term of sensitive tone to routing (directional) tones and routing (directional) chords.

Tendency of impulse

Impulse tendency is analogous to gravitational attraction.

Tones connected by impulse binding are called tied tones. The opposite of tied tones are free tones.

The following table compares three modalities:

                      Free tones    Tied tones
Natural pentatonic    c,d,e,g,a     ---
Natural diatonic      d,g,a         e,f,h,c
Full chromatic        ---           c,c#,d,...h

Diatonics are most variable modality, containing both free and bound tones.

Natural diatonic has bound tones e-f and h-c, see table.

       |  c   d   e   f   g   a   h  | ∑(i)
     --+-------------------------------------
     c |  0   3   0   0   0   1  12  | 16
     d |  3   0   3   1   0   0   1  |  8
     e |  0   3   0  12   1   0   0  | 16
     f |  0   1  12   0   3   0   0  | 16
     g |  0   0   1   3   0   3   0  |  7
     a |  1   0   0   0   3   0   3  |  7
     h | 12   1   0   0   0   3   0  | 16

Impulse connection - so hard that grips both harmony together and separates them from the neighboring harmonic currents. E.g. C-Hmi5 - Dmi-Ami (before Dmi - breath, "pause", blank)..

Main and sensitive tones

Let us consider two tied notes. If the potentials of these tones differs, then the tone with a higher potential is called the main tone and the tone with a lower potential sensitive tone.
E.g. in natural modality are the main tones c, e, and sensitive tones h,f:

Tone	c	d	e	f	g	a	h
Impulse ∑(i)	16	8	16	16	7	7	16
potential U	6	4	6	0	7	7	0

Tone h has a low potential and binding h-c high pulse. Transition h-c terminates the tone h. After the extinction of the tone h, tone c excels (M.Filip).

Sensitive tones are 'focused' ie. played closer to the main tones (maximization of impulse). Tendence of tone has always a certain direction. We distinguish between upward and downward trend. E.g. in natural modality, the tone h is increasing, while the tone f decreasing.

According to the classical theory of harmony is the most stable tone c. Tones e, g are also referred to as stable because they bind to the tone c. Tones a, d, f are (in this order) considered unstable and expands downward a=>g , dàc, f=>e. The most labile of the tones h is lead upwards hàc. All chromatic tones are unstable and tend to be lead the nearest diatonic tone. But it is expected that e.g. tone ab is different from tone g#, because they lead: > ab=>g, g#=>a.

Modality structure

Marks ß a=> show a tendency of tones:

	Natural	Podhalska	Harmonic minor	Gypsy
Schema	221222(1)	221212(2)	212213(1)	213113(1)
Modality	cdefgahc	cdef gg#a#c	ahcdef g#a	ahc d#ef g#a
∑	5 4 5 3 5 5 3 5	6 3 2 4 4 2 3 6	5 2 4 2 6 3 2 5	3 3 4 3 6 3 4 3
Sensitive	eß f ,h =>c	e=> f , g ßg#	h=> c , e ß f ,g# =>a	d#=> e, e ßf,…

Fifth skeleton

m_kernel Fifth skeleton consists of at least two contiguous quint, e.g. {f,c,g}.
Fifth skeleton helps to establish contrast in modalities, particularly in the case of semitone cores.
Semitone core ìs group of 2 semitones next to each other, for example. {h, c, c#}. E.g. In Bardos' modality {g, h, c, c#, f}, (structure 1142(4)), c tone has great potential and tones h, c# have the least potential.

Modality	Potential of tones	Core
Bardosova ( 327)	h(-3) c(2) c#(-3) f(3) g(3)	h(-3) c(2) c#(-3)
Blues ( 663)	g(4) g#(-5) a(2) c(8) d(4) f(3)	g(4) g#(-5) a(2)
Combined (1367)	h(-2) c(2) c#(-2) d#(-1) f(4) g(4) h(-1)	h(-2) c(2) c#(-2)

Fifth skeleton supporting the halftone kernel is part of e.g.. Bardos', blues, combined, gypsy and other modalities.

Tendency of continuity

Tones {h, f} have the lowest potential because of their links to other tones include only one fifth. Let use calculate the two sums of continuity.

   |  c   d   e   f   g   a   h  | ∑c    | ∑|c|
 ------------------------------------------------
 c |  0   0  –2  +4  –4   0   0  | –2    |  10
 d |  0   0   0   0  +4  –4   0  |  0    |   8
 e | +2   0   0   0   0  +4  –4  | +2    |  10
 f | –4   0   0   0   0  –2   0  | –6    |   6
 g | +4  –4   0   0   0   0  –2  | –2    |  10
 a |  0  +4  –4  +2   0   0   0  | +2    |  10
 h |  0   0  +4   0  +2   0   0  | +6    |   6

The first sum, ∑c, gives extreme value of the tones h (+6) and f(–6). Signs of values differs - we say that h has a dominant tendency and the tone f a subdominant tendency.

Tones with higher values of the sum Σ|c| have a higher potential, i.e. greater stability.
These tones are the pillars of modality, they produces the main skeleton of modality.

Sensitive tones h f leads to tones of modality skeleton c,e.

Chinese modalities

Basic mode	Derived modalities
2232(3)	22322(1); 22212(3); 222122(1)
2323(2)	23231(1); 23113(2); 231131(1)
3232(2)	32321(1);
2322(3)	23222(1); 23112(3); 231122(1)
3223(2)	32231(1); 32113(2);

A special case of modality structures based on fifth skeleton and sensitive tones are Chinese modalities. These are based on the five Chinese pentatonic modes, 2232 (3). Semitones were added the fifth year and (or) an octave from below.

Effect of tone color on melody

m_itimbr The spectrum of overtones tends to affect the tendency of tone. Usually we assume that the chromatic step has no continuity, only impulse.
Now consider the main overtones. We see a strong link of continuity (c-f).
For example, we consider the halftone ( h - c ) and wholetone ( d - c ) processes in pure form in two-way. In fact, aliquots decide their tendency.

Bindings arise arise hf#d# à cge, resp. df#a=>ceg.

The only interval that remains practically unaffected by the color spectrum is triton (see, e.g., binding between c#a#f# and egc).

Triton (symmetric interval) has neither continuity nor pulse and even is not affected by timbre.

Root in the context

Tones sounding prior or after the given tone can support or disturb rootness of the tone. Prof. Risinger shows how musical context can change root (and consonance amount) of a given chord. Particular attention is given to quart:

    f 70% + 10% = 80% 
    c 30% - 10% = 20%

Let us consider, for example interfal of quarts c- f, where deeper tone c will be played very weakly - it gets only 30% of the total energy (meanwhile F 70% of energy). Suppose now that the binding continuity c=>f move 10% of the total energy and thus causes, c it that c will have 20% and f 80% of energy. Then the f becomes the root and quart interval sounds consonant.

    f 30% + 10% = 40%
    c 70% - 10% = 60%

Otherwise, if c is greater than g (which may occur for example in duo played by pulling bow with equal force on both strings) we get a different energy distribution. Interval of quart sounds dissonant (and a higher tone requires leading ...)

Change of modality

Modulation

Prokofjev Sergej Sergejevič

Prokofjev Sergej Sergejevič [], 1891-1953, Russian composer and pianist, master of vertical polytonality, i.e. insertion of different keys to each other.

Modulation is a change of the existing modality into another modality. In this sense it is very similar to harmonic connection.

Modulation is any change of modality, related modalities are those which have a similar distribution of formal potentials. Effect of modulation depends primarily on the impulse. The greatest impulse is generally achieved by moving the whole modality by semitone.

There are two basic types of modulation:

Translation of modality, ie. change of modality with conservation of the structure. Deformation of modality, ie. change of modality structure Without changing of tonic With changing of tonic

Effect of modulation depends on the duration of the transition to the new modality. Chord with tones, which are common to both modalities, represents a kind of theoretical transition between modalities. But not for all of modulations there is such a (suitable) chord (of the tonality).

Imagine a formation of modality projected in the coordinate system. Let the vertical axis represents the potential of the formation and a horizontal axis continuity to the tonic. Every chord and even every modality get its coordinates. At the beginning of the system is the tonic.

Similarity of modalities

Similarity of modalities is given by characteristics of their mutual bindings.
It depends on the number of common tones, continuity, impulse, potentials of tones.

E.g. let us compare two pairs of modalities:
1/ [cdefgab]-[c#defg#ab] and 2/ [cdefgab]-[cd#ef#gab]
Modalities have two distinct tones 1/ {c-c#, g-g#}, 2/{d-d#,f-f#}.
In case 1/ basic tonic tones {c,g} changes; modalities are therefore less similar.

Let v(A,B) is sum of values (eg. of continuity, impulse,...) for all bindigs from the modality A to the modality B. Then similarity r(A,B) of modalities A and B is:

r(A,B) = |v(A,B) – v(B,A)|

Oscillation

Oscillation of modality is repetitive deformation of modality.

Oscillation often affects only a few selected tones. Other notes remain fixed. (Fixed tones form the skeleton for all used modalities. They could therefore have a high potential in any of the modalities).

Mozart Wolfgang Amadeus

Mozart Wolfgang Amadeus [mócárt], 1756-1791, Austrian composer, one of the greatest musical geniuses of all time.

In the following example from Mozart's opera Don Giovanni is a modality changes with every chord.

Harmony	Modality	Scheme	p
X:	a,h,c,d,e,f#,g#	#10#111	4
Ami:	a,h,c,d,e,f,g	#10#100	2
Dmi6-:	a,a#,c,d,eb,f,g	#00#011	2
G, Ami:	a,h,c,d,e,f,g	#10#100	2
A:	a,h,c#,d,e,f,g	#11#100	3
Dmi:	a,h,c#,d,e,f,g	#11#100	3
	a,a#,c,d,e,f,g	#00#100	1
A#, F7:	a,a#,c,d,eb,f,g	#00#000	0
Ami,Esus4,E:	a,h,c,d,e,f#,g#	#10#111	4
Ami	...

Solid tones are here {a,d} (marked #), other tones oscillate (marked 0,1), p - number of ones.

Schema of oscillation: (a),(a#,h),(c,c#),(d),(eb,e),(f,f#),(g,g#)

Oscillation modality change potential distribution. This creates new trends of tones, changes the sensitive tones and harmonic functions,...
Eg. ascending melodic minor scale consists of modality {a,h,c,d,e,f#,g#} with major dominant and subdominant, D={e,g#,h}, S = {d,f#,a}.

The same descending scale forms a natural modality {a,h,c,d,e,f,g} with minor dominants S={d,f,a}, D={e,g,h}.

Indeterminacy of modality

During analyzis it is (at all times) necessary to identify existing modality and to see emerging sessions in its context. Modality is determined only by those bands that get their energy from tones.

Considerations on fixed modalities and groupings are, however, idealizations. These formations are in reality created dynamically by polyphony.
Extra-tonal tones are tolerated especially if their energy soon extinguished due to the reactivity bands. That is precisely the case of Neapolitan chord Db; energy in bands C#, G# are canceled by entering of tones D, G of the dominant chord G.

The exact definition of the modality is formed through gradual sounding of individual tones, by compilation of the whole from fractional compositional structures.

Mozart - Alla Turca March - 10 tones appears in the first few bars (all except F# C#), but not all have the same weight. Changing substructures of the basic modality is associated with the courses of the contributing melodic lines. The closest linkage is in monophonic songs, these (folk songs etc) are therfore the best material for research of melodic tendencies. Melodic tendencies. E.g. Let the phrase with notes of {c, d, e, f, g, a, h} is divided into two separate sections with tones {c, e, f, g, a} and {f, a, h, c, d}.
Independent combining of two modalities is not the same thing as the unification.

If we know only one voice (eg. the tune of folk songs), a classification is difficult.
E.g. tune {c, d, e, f, g} may be of the modality {c, d, e, f, g}. It also could be sung as duo - together with tones {a, b, c, d, e}; then it could belong to modality {a, h, c, d, e, f, g}.
Or it could be played with the chord accompaniment, eg. C, C#, Fmi. This would correspond to modality {c, c#, d, e, f, g, g#}.

Possible harmonization from above (eg. The pentatonic can be harmonized by diatonic rather than by tritonic)) ((((tritonic) pentatonic) diatonic) chromatic)

The conclusion of the phrase

The convincing conclusion is a measure of the relaxation. The connection is convincing conclusion if it respects this scheme:

                 extreme
        small    ═══════════►  high
       potential continuity  potential

Deceptive closing

Deceptive closing is connection leading to chord, that has an extreme potential, but not with the extreme continuity; e.g. G-Ami.
The listener hears stable harmony, but this chord appears in such a manner that it can not be accepted as final end.

Harmony returns to T in the ends of verses. If a verse is repeated, then first conclusion is misleading connection.

Czech anthem:

    C,G7,C,C7 - Kde domov můj, kde domov můj, voda
    F,C,F,C - hučí po lučinách, bory šumí po skalinách
    G7,C,G7,C - v sadě skví se jara květ, zemský ráj to na pohled
    E,Ami - a to je ta krásná země
    F,C,G7,Ami - země česká, domov můj
    F#dim,C,G7,C - země česká domov můj

Harmonic sequence

Baroque Sequence

Harmonic connection	Continuity	Impulse
EmiAmi, GC, CF	+2.33	3.06
Hmi^5-Emi, Fhmi^5-	+1.67	3.06
DmiG	+1.33	2.61

The following type of sequence is often used in the baroque music: C:F:Hmi^5-:Emi:Ami:Dmi:G:C. [2]. High positive values (1.56, 1.11, 0.89) of continuity are in all the particular connections of these consonant triads in natural modality.

The average continuity of sequence is equal to the average continuity of individual connections:

c(s) = [c(C,F) +c(F,Hmi^5-) +c(Hmi^5-,Emi) +c(Emi,Ami) +c(Ami,Dmi) +c(Dmi,G) +c(G,C)]/7
c(s) = (2.33+1.67+1.67+2.33+2.33+1.33+2.33)/ = +14/7= 2.00

From Mozart's music

The beginning of Mozart's overture to opera Don Giovanni:

Continuity (red arrows) is more concentrated in lower position of voices (bassoon, viola, lower positions of violins). Onsets of connection are apparent in the voices of the bassoon and violin with four-bar period. Progress in impulse intervals (black arrows) is entrusted primarily to parts of the second violins.

Jazz Sequences

Jazz sequences [11] are sequences with a predominance of impulse values this is particularly true for jazz chromatic sequence: (e.g. E⁷:D⁷:C⁷:H⁷)

The normal jazz sequence (e.g. Ami⁷:D⁷:Gmi⁷:C⁷) resembles the baroque one as to the succession of the continuity.

Unusual connections

Every musical style has fixed limits for the values of certain characteristics. Connections that are outside of those boundaries are considered unusual or are directly prohibited by stylistic rules. Restrictions need not be subject to a whole harmonic connection only, but also each of its subset; eg. the prohibition of parallel fifths and octaves in the classical harmony. In contrast to that some of the rules forces partial phenomena to obey the laws of the whole. E.g. rules for the strict leading of the voices in classical harmony permit only such a motion of voices, that the overall change in heights of all voices is minimal.

Allowable connections

Each harmonic connection is allowed. But every connection is perceived in relation to the music system, modality, tonality, ...
Not all combinations of tensions are natural and sounds good.
E.g. various rules allows or prohibits certain types of fifths in harmonic connections.
Janáček wrote about a misstatement of cg-a#f. Other theorists refer to cg-da (soft Mozart fifths) as incorrect inverse connection (in some contexts).

Strict connection

Classical strict harmonic connection must comply with the following rules:

Advance by second - All upper voices should lead to the nearest tones (countermovement with the bass).

The process of quart or a fifth - each of upper voices should proceed no more than third (without changing of voices)

The process of minor third - each of the upper voices should do no more than fourth (in countermovement with the bass).

These rules have been and are respected also in other styles, mostly in middle voices of polyphony.

Unusual connections of classical music

Natural modality

Consonance with the highest continuity to the tonic (dominants) were rarely associated with non-tonic tones. Connections started on the dominant sorted according to the values of continuity:

Connection	Continuity	Impulse	Note	Continuity	Impulse	Difference
GC	3c₁+c₂	i₁+2i₂+i₃+i₄	to tonic	+2.33	3.06	-0.73
GEmi	2c₁	i₂+2i₃		+1.33	1.89	-0.56
GAmi	c₁	i₁+4i₂+i₃	deceptive connection	+0.67	4.44	-3.77
GF		i₁+4i₂+i₃	unusual	0.00	4.44	-4.44
GHmi^5-	-c₁-c₂	i₂+3i₃+i₄	complex G7	-1.00	1.44	-2.44
GDmi	-2c₁	3i₂+2i₃	unusual	-1.33	2.61	-3.94

Harmonic minor modality

Connections started on the dominant sorted according to the values of continuity:

Connection	Continuity	Impulse	Note	Continuity	Impulse	Difference
EAmi	2c₁+c₂	2i₁+i₂+3i₄	k tónice	+1.67	3.56	-1.89
EF	c₁	3i₁+i₂+i₃+2i₄		+0.67	5.06	-4.39
EDmi	c₁	2i₁+2i₂+2i₃	unusual	+0.67	4.56	-3.89
EC⁵⁺	c₁	i₁		+0.67	1.33	-0.67
EHmi^5-	-c₁	i₁+i₂+3i₃	unusual	-0.67	2.72	-3.39
EG#^5-	-c₁-c₂	i₂+3i₃+i₄	Complex E⁷	-1.00	1.44	-2.44

Other unusual connections

Other unusual connections: AmiC,DmiF,EmiG ( natural modality), FAmi, Hmi5-Dmi (harmonic minor modality)

Harmonic connection	Note	Continuity	Impulse
Hmi^5-Dmi	complex Hmi^7/5-	-1.00	1.44
AmiC (DmiF, EmiG)	complex Ami⁷	-1.33	1.89
FAmi (CEmi)	complex Fmaj⁷	-2.00	1.89

Harmonic skeleton of melody

m_skelet

Harmonic skeleton

Harmonic power is total of (time integrated) energy in zones.
Harmonic skeleton of melody is a set of tones which contribute mostly to the harmonic power.
Essential tone is every tone of the given chord. If another tone appears, while the chord is still sounding, we call it non-essential tone.
Harmonic skeleton of melody usually consists of essential tones. Stability of the melody depends on number and location (light x heavy beats) of the non-essential tones.
E.g. let us have two chords {F,G}. The melody {a,h,d,c} has a bit different stability than melody {h,a,c,d}. In the first melody are the essential tones {a,d} on heavy beats, in the second one on light beats.

Harmonic cover

Harmonic cover is a measure of harmonic stability of melody.
We define: c = a/d,
where d is duration of given phrase (usually duration of a chord} and a is duration of essential tones only.
E.g.:

Harmony	Melody g,h,c,f	Harmonic cover
Dmi	x x x 1	1/4 = 0.25
Ami	x x 1 x	1/4 = 0.25

C	1 x 1 x	2/4 = 0.50
Hmi-	x 1 x 1
Emi(G)	1 1 x x
F	x x 1 1

C-F	1 x 1 1	3/4 = 0.75
G-C	1 1 1 x	3/4 = 0.75

G-F	1 1 1 1	4/4 = 1.00

Every musical phrase has certain bounds for the values of harmonic cover. The less harmonic cover is ussualy allowed only in case of higher mobility of the melody.
E.g. the broken chords have c=1.00, melodic ornaments approx. c=0.5. Melody having c<0.5 appears rarely (e.g. blue tones).

Blue tones

Blue tones are a special case of non-essential tones. The melody uses other modality than the harmony.
E.g. while the chords respect the natural modality, the tune uses tones of blues modality.

The blue tones are often half tone below the essential tones.

Chord	Essential tones	Blue tones
C7	c,e,g,b	d#,f#,a#,(g#)
Dmi7	d,f,a,c	g#,(a#)
G7	g,b,d,f	a#
Ami	a,c,e	d#,(f),(g)