Music theory - Tonality

Modalities and modes

Modality

Modality is a subset of the formal system. E.g. white piano keys form the 7-tone modality cdefgah, which we call natural. This modality corresponds to the number 1387 (ie. binary 010101101011) and distance scheme 122122(2). Only very few songs that we hear every day (if we are not lovers of oriental music ..) deviates from the natural modality.

Natural modality - China, Europe, Arabia, Persia - in China as early as the third millennium BC. Modality is (after the formal system) an another idealization, which reduces the variety of music. The bulk of the existing music was written only in a few selected modalities. Short song or a section of song often covers one modality.

Modality of 12 tone system

In total, the 12-tone system has 352 classes [Janecek], which consists of 2 ¹² = 4096 different tone clusters (chords, modes).

Primitive cultures use two or three-tone modalities. 2(10); 4(8); 12(9); 32(7); 21(9). A similar modalities can also be find in the bird singing: 12(9) thrush, tits, 11(10), 21(9) warblers, 13(8) finch, 53(4) oriole ,…

Modalities of four-tones are known from e.g. Eskimo music. 223 (5). Four-tone cells (from the Greek scales) are called Tetrachords and are the cornerstones of multi-tone modalities, (according to the Greek and medieval theory).

The increasing number of tones increases the number of modalities. There is 66 five-tone modalities (pentatonics) (in 12-tone system) and the same number of seven-tone modalities (heptatonics):

 (0)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)(11)(12)
 ────────────────────────────────────────────────────────
 1  12 66 220  495 792 924 792 495 220  66  12  1  instances (M)
 1   1  6  19   43  66  80  66  43  19   6   1  1  classes (m)

Some of the pentatonic are known under the names. Sometimes there is a whole range of alternative names.

E.g. natural pentatonic is called as Chinese, right, blues, etc. Some names come from Indian music: Megharanji 1131(6), Hindola 2312(4), Kaušika a Méhga 2322(3),…

Like the tetrachords also some pentachords were named (according to the medieval church scales).

Modalities with a semitone or more semitones are called hemitonic, modality without semitones are anhemitonic, modalities with two semitones disemitonic.

Term Diatonics is generally used for modalities changing smallest interval steps (1,2, ...), term Chromatics for modalities with a predominance of the smallest intervals (1), e.g. chromatic hexatonic 11111 (7).

List of named hexatonics and hexachords is relatively short although the number of 6-tonal structures is highest of all (total of 80 modalities). Tones of Guidonian hexachord 22122(3) appears in nightingale singing.

The diatonic include most of the 7-tone modalities. In addition to conventional (natural gypsy, harmonic major a minor,...) - number of interesting modalities (píšťalková, Podhalská ...) is located in folk songs.

An extreme example of the ambiguity of names is "Podhalská folk" heptatonic 222 121 (2), for which there exists many different names: Acoustic, Altered, Mirror, Ascending melodic minor, Lydian-mixolydian, Mixolydian with diminished sexta, Moravian, Hypoharmonic, Plane-altered, Bartok, Jazz, ...

Natural pentatonic 2322(3) and heptatonic 221221(2) are used practically throughout the world.

The following are examples of octatonics

(8-tone structures) a nonatonics

(9-tone structures)

Symmetry of modalities

For each modality modality exists an inverse - e.g. 1312(5) and 2131(5) are two mutually inverse modalities or modality alone is its own inverse - e.g. natural modality 221222(1). Inverted scale to [c d ef g a hc] is according to Helmholtz and Riemann [e d ch a g fe].

Own compositional method based on inverted scales used J.Rut.

Construction of modality

- the introduction of notation according to natural modality - some simplifications and clarification of enrollment; But also misleading terms - enharmonic confusion of tones, differentiation of enlarged seconds and minor thirds, ...

(Relationships between the structures of modalities - eg blues contains in itself harmonious minor, ...)

Determination of modality

Modality of music phrase arises from the real sounding. It is therefore always somewhat indeterminate: it depends on the number of tones, their intensities, timbre, ... In each modality structure is numerous number of substructures, which are intertwined. If any of the substructures is highlighted at a certain point of harmonic phrases, it causes deviations from the trends of it superior modality.

Modes and scales

Each modality has several modes, i.e ways - in which order to play tones (mode = Latin way). The number of ways is equal to the number of tones in modality, ie the level of class. Mode played sequentially is called scale (scales preserve names of modes). If we play tones starting with tone 'c' we gain C major scale, with the tone 'a' a minor scale.

Earlier naming of the scales is slowly fading. From 7-modes used in the Middle Ages, nowadays only major scale (Ionic) and minor (Aeolian) are usually taught. We use the term 'mode' only in conjunction with older music. 'Scale' is a modality with a fixed ordering of tones.

1387  7 12 010101101011  1 2 2 1 2 2( 2)  

0/ 0,1,3,5,6,8,10  1 2 2 1 2 2( 2)  h,c,d,e,f,g,a  Hypo-phrygian
1/ 10,0,1,3,5,6,8  2 1 2 2 1 2( 2)  a,h,c,d,e,f,g  Aeolian (minor)
2/ 8,10,0,1,3,5,6  2 2 1 2 2 1( 2)  g,a,h,c,d,e,f  Mixo-lydian
3/ 6,8,10,0,1,3,5  2 2 2 1 2 2( 1)  f,g,a,h,c,d,e  Lydian
4/ 5,6,8,10,0,1,3  1 2 2 2 1 2( 2)  e,f,g,a,h,c,d  Phrygian
5/ 3,5,6,8,10,0,1  2 1 2 2 2 1( 2)  d,e,f,g,a,h,c  Doric
6/ 1,3,5,6,8,10,0  2 2 1 2 2 2( 1)  c,d,e,f,g,a,h  Ionic (major)

Naming of modes

Greek (descending) and medieval (church, Christian, Gregorian, ascending) naming

Church: Ending - autentic (basic scale) - plagial (hypo scale)

Modes of limited transpositions

There are 17 modalities that do not have the full number of transpositions. These modalities are usually called Lendvai's or Messiaen's and consists of 76 different modes

(Into system G(2,12) total 17 classes is nested, having total 2∙1+1∙2+2∙3+3∙4+9∙6 = 76 instances. )

System  Distance schema  Example                    Name 
────────────────────────────────────────────────────────────────────────
G(2,1):           
       0 0(0)                                       Silence
    4095 11111…1(1)     c,c#,d,d#,e,…,a#,h          12-sound
G(2,2):       
    1365 22222(2)       c,d,e,f#,g#,a#              Wholetone scale
G(2,3):        
     585 333(3)         c,d#,f#,a                   Diminished tetrad
    1755 1212121(2)     c,c#,d#,e,f#,g,a,a#         Inverse dim. tetrad
G(2,4):        
     273 44(4)          c,e,g#                      Augmented triad
     819 13131(3)       c,c#,e,f,g#,a
    1911 11211211(2)    c,c#,d,e,f,f#,g#,a,a#       Inverse augm. triad
G(2,6):         
      65 6(6)           c,f#                        Tritonus
     195 151(5)         c,c#,f#,g                   .
     325 242(4)         c,d,f#,g#                   ..
     455 11411(4)       c,c#,d,f#,g,g#              Messiaen‘s modes     
     715 12312(3)       c,c#,d#,f#,g,a              of limited transpositions
     845 21321(3)       c,d,d#,f#,g#,a              ...      
     975 1113111(3)     c,c#,d,d#,f#,g,g#,a         ..
    1495 1122112(2)     c,c#,d,e,f#,g,g#,a#         .
    2015 111121111(2)   c,c#,d,d#,e,f#,g,g#,a,a#    Inversion of tritone

Harmonic variety

Assuming also chords and relations arising in the environment of modes (scales), we talk about the tonality (eg. in C major and A minor).

Tonality is a certain complex of relationships in the formation, which we call modality.

The actions of harmony phenomena can be most easily determined in the tonal context, [3]. The tonality is always made of a certain modality, [13].

This modality has k = 12 transpositions, allowing 12 different musical designs (scale or key of C major, C# major, D major, ... to B major).

The set of all groupings in a tonality is the harmonic variety. Every grouping has its specific properties depending on its position within the tonality.

Comparison of tonalities

Let us compare potentials of chords of two minor keys: natural (aeolian minor) and harmonical.
The first key prevale in the older music, the second in the newer music.

Tonics of both modalities have the same potential, but contrast of dominant and tonic is more marked in the harmonic modality:
Natural: U(D)-U(T)= U(Emi)-U(Ami) = 4.33-6.33 = -2.00
Harmonic: U(D)-U(T)= U(E)-U(Ami) = 1.33-6.33 = -5.00

Characteristics of bindings depends on the octave positions of tones, but we assume that this dependence is secondary (ie. that it is possible neglected it in the first approximation).

Harmonic functions

The basic trinity of functions

For the harmonization of many songs just three chords are needed: "functions" - called tonic, dominant and subdominant. Musicians feels this functions, regardless of the instrument on which to play. We can change the intensity and some octave tones - function still maintain the same effect.

(see formal system).

Harmonic function are formal chords from harmonic variety with some characteristics becoming (in comparison with other groupings) of extreme values:

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).

Function Lydian and Phrygian

The Phrygian function (F), is the grouping with the maximum impulse towards the tonic from above. [10],[3].

The Lydian function (L), is the grouping with the maximum impulse towards the tonic from below. [10],[3].

If appears e.g. at the end of C-major modality instead of a sequence Dmi:G:C the sequence Db:G:C (ie. Neapolitan sixth chord Db),[5], we can not say that we are still in C-major.
The original modality just modulated into modality [c, c#, d, e, f, g, g#, h].

Types of functions

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

• Potential functions
• Directional functions
• Other 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.

Bands of the 12-tone systems affected by natural modality (c,d,e,f,g,a,h) have the following F-potentials:

Grouping with the highest F-potential:

Tones g,a,c,e,... see the table above.

Intervals (g,a), (c,g), (a,e), (c,e), (a,c), (e,g)

Triad (a,c,e) and (c,e,g)

Tetrad (c,e,g,a)

A e.g. (c,g) is "better tonic" than (g,a) because is consonant (has a small entropy of sound).

Anti-tonic (A) is organized grouping of harmonic variety with minimal F-potential..

Directional functions

• Functions of continuity
• Functions of impulse
• Other directional functions

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).

Similarly, we determine values of the continuity towards the tonic for some selected connections. In this case, by counting of continuities of the individual bindings, see the table. Note the extreme values of continuity for the dominant (+2.33) and subdominant (-2.33) towards both tonics.

Now we enumerate the values of continuity and impulse for some selected connections.

Selected harmonic connections

Note the extreme values of dominant (+2.33,+1.11) and subdominant (–2.33) 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 (tonic C):

- relaxation - the difference of sonance of two consecutive chords ( amplifies the effect of the impulse- L.Janáček)

- chord - measuring - kind of instances, which occurs more frequently: development of harmonic current from one point and continuation.

only after return to this point (hierarchy, fractals ...), e.g. Bach C : C …C | D7 … D7 G7 … G7 | C …C |

- relations between the chords at a higher level (not close each other).

Examples

Estimated functions of some heptatonics:

Tonic of medieval modes

At the time of formation - basic modes - insensitivity for potentials; after a time a sensitive tone considered and integrated into all modes - with the exception of the Phrygian.

"Importance" of the chord is proportional to the heaviness of the beat (stretchable harmony x passable harmony)

Natural tonality - Harmonic functions of church modes. According to polarity of functions: (modulated = deviating, unstable (labile) = a rickety)