# Consonance and dissonance

The distinction between consonance (from Latin con 'together' and Latin sonare ' to sound' ) and dissonance (from Latin dis 'apart' ) has been related to the quality of two sounds since ancient times . In occidental teachings of polyphony ( discantus , counterpoint ) it became the basis of syntax . In the 17th century it was expanded to include multiple sounds. The assignment of intervals and chords to one of these categories, the justification for this assignment and the justification for the distinction itself can vary, depending on the music-theoretical tradition or depending on the field of knowledge in which the contrast was also discussed ( physics , physiology , psychology ).

## Ancient and Middle Ages

As συμφωνία ( symphonía ) have been used in Greek music theory since the 4th century BC. The (pure) fourth , fifth and octave , as well as (in some sources) their octave extensions up to the double octave plus fifth are excellent. With regard to the lengths of the strings with which the tones involved u. a. can be generated on the monochord , with the exception of the undecimal ( ) numerical ratios , these intervals correspond to the form (multiple proportion) or (divided proportion): fourth , fifth , octave , duodecime , etc. ${\ displaystyle {\ frac {8} {3}}}$${\ displaystyle {\ frac {an} {n}}}$${\ displaystyle {\ frac {n + 1} {n}}}$${\ displaystyle {\ frac {4} {3}}}$${\ displaystyle {\ frac {3} {2}}}$${\ displaystyle {\ frac {2} {1}}}$ ${\ displaystyle {\ frac {3} {1}}}$

The attribute 'easy to understand' is assigned to these proportions. With regard to the sounds, there is talk of a 'mixture' of different individual tones to form a unit (which is why the harmony is not considered a symphonía ).

The discovery of the relationship between euphony and the comprehensible numerical ratio is ascribed to Pythagoras in the legend of the blacksmith . The Pythagoreans only allow proportions between the numbers of the tetraktys (i.e. the numbers 1, 2, 3 and 4) to be considered symphonía and therefore exclude the decimals . These proportions are for them a symbol of cosmic order (see harmony of the spheres ). In the Pythagorean tuning all intervals are derived from combinations of the symphonoi , which u. a. leads to the fact that the thirds have considerable beats .

Since Aristoxenus and Eukleides, the term διαφωνία ( diaphonía ) is no longer used generally in the sense of discord, but also in a narrower sense for intervals that do not belong to the symphonoi but are still considered to be musically useful (so-called emmelic sounds, e.g. B. the whole tone ).

In the late antiquity imprinted Boethius ( De musica institutione to 500) the Latin term consonantia that was used previously in more general meanings, synonymous of Symphonia (such as the Pythagoreans, he expects the peace and eleventh not do so). The word dissonantia serves him as an opposite term (but he does not give concrete examples of dissonant intervals). Boethius' definitions are handed down to modern times and retain a high degree of authority.

## Counterpoint theory

### Three interval classes

In the 14th century, for the teaching of established counterpoint fundamental distinction of the intervals in

• perfect (or: perfect) consonances: pure prime, pure octave, perfect fifth, and
• imperfect (or: imperfect) consonances: major and minor third, major and minor sixth.

All other intervals, insofar as they are considered musically useful, are dissonances: first the minor and major seconds and sevenths , later also some diminished and excessive intervals such as the diminished fifth and the tritone .

The fourth has a special position:

• As the primary interval (every interval between the bass and another part - in two-part there are therefore only primary intervals) the fourth is usually treated as a dissonance.
• As a secondary interval (any interval between two upper voices), the fourth is the difference between a primary sixth and a primary third or a primary octave and a primary fifth and is treated like an imperfect consonance.

For these interval classes, counterpoint theory formulates different typesetting guidelines:

• Perfect consonances are particularly suitable as opening and, above all, closing sounds. The prohibition of parallels applies to them .
• Dissonances must be able to be shown as syncope dissonance , passing grade , alternating grade or anticipation.

### Pure thirds

In its upgrading of thirds and sixths to consonances, the counterpoint tradition initially confirms a musical custom without justifying it theoretically. Because the established tuning system was initially still the Pythagorean, with its beat-rich thirds and sixths (see above). Already in the early 14th century Walter Odington ( De speculatione musice ) emphasized the proximity of the Pythagorean thirds ( ditone with the proportion 81:64, semiditone with the proportion 32:27) to the pure major (5: 4) and minor (6th) : 5); In musical practice, according to him, the thirds are often adjusted to pure (that is, beat-free) thirds. In the following centuries, Bartolomé Ramos de Pareja (1482), Lodovico Fogliano (1529) and Gioseffo Zarlino (1558) laid the further theoretical basis for pure tuning .

## Harmony

As early as 1610 Johannes Lippius transferred the pair of terms consonance-dissonance to triads . In the course of the 18th century, a new understanding of consonance and dissonance gradually established itself, based on the category of the triad, and which increasingly competed with the contrapuntal teaching tradition. The writings of Jean-Philippe Rameau had a particular influence .

In the Traité de l'harmonie , Rameau traces all dissonant chords back to the layering of more than two thirds. Consonant are the accord parfait ( major and minor triad ) and its inversions :

"Pour se rendre les choses plus familieres, l'on peut regarder à present les Tierces comme l'unique objet de tous les accords: En effet, pour former l'accord parfait , il faut ajoûter une Tierce à l'autre, & pour former tous les accords dissonans , il faut ajoûter trois ou quatre Tierces les unes aux autres; [...]. »

“To make things a little more convenient, we can consider the thirds as the single element of all chords: in fact, to make the accord parfait, one has to add one third to another, and for all dissonant chords one has to add three or four thirds [ ...]. "

- Jean-Philippe Rameau : Traité de l'harmonie. Paris 1722, p. 33.

In the late 19th century, Bernhard Ziehn wrote :

“The words“ consonance ”and“ dissonance ”have nothing in common with the terms“ good sound ”and“ discordant ”(or even“ bad sound ”) . These designations are only to be regarded as generic names for accords and intervals. Consonances are the major and minor triads, as well as the intervals that occur in such a triad; namely major and minor third, major and minor sixth, perfect fifth, fourth and octave [...]. All other accords, as well as all remaining intervals, are dissonances. "

- Bernhard Ziehn : Harmony and modulation theory . Berlin 1888, p. 4.

In the Traité , Rameau derives the major triad from the arithmetic division of the fifth (proportion 4: 5: 8), whereby it refers to the length of the strings in the spirit of the monochord tradition. It was only in Génération harmonique (1737) that he revised this concept under the influence of findings from physics, and now derived the major triad from the overtone series (fundamental, 3rd and 5th partial).

### "Essential" dissonance and a tone foreign to the chord

The sevenths in seventh chords called Johann Kirnberger as "essential dissonance", "because they are not placed at the site of a consonance, which it soft again, but a place for themselves claim". All other dissonant chords, on the other hand, contained "accidental dissonances", "which can be viewed as suspensions [...] which for a short time take the place of the consonant ones, and longer [!] Duration of the root with which they dissonate in their next consonances skip ". A seventh in a seventh chord does not represent a chord tone, but is one itself. In this way Kirnberger marks the departure from the contrapuntal concept of syncope dissonance in favor of the terms chord dissonance and lead.

• The sevenths under a) are therefore leads, ie “tones not related to harmony”, which represent a tone of a triad inversion (the sixth of a sixth chord).
• The seventh under b) are not a lead, but part of seventh chords.

In harmony, 'foreign to chords' means 'dissonant' at the same time, while on the other hand there are dissonances that are considered part of a chord.

### Dissonance treatment

The seventh of the dominant seventh chord , the diminished fifth and the seventh of chords derived therefrom (diminished and half diminished seventh chords of the seventh degree) as well as the ninth in the dominant seventh chord are usually resolved by a second step down, like the 'old' syncope dissonance. However, these chords can also be in an unstressed time position and can occur freely (so they do not need any particular type of preparation).

Other “essential” seventh and ninths are still treated as syncopated dissonance.

Other types of dissonance treatment that became a matter of course in the 18th century include: a .:

### Functional theory

Even in his early writings, Hugo Riemann justified the consonance of the major and minor triad in a dualistic sense with the fact that their fundamental, third and fifth tones can be interpreted as the 2nd, 3rd and 5th overtone or undertone of a tone. Later, however, he emphasizes that acoustically consonant sounds can be musically dissonant and that the psychological instance of the "sound concept" is fundamental for the distinction between consonance and dissonance.

Against this background, Riemann coined the term pseudo- consonance : In Riemann's theory of functions , only the triads of the tonic , subdominant and dominant count as consonances; the triads of the other degrees are dissonances. The expression Scheinkonsonanz have Rudolf Louis and Ludwig Thuille later by the term view dissonance replaced.

## Emancipation of dissonance

In the early 20th century, musicians such as Arnold Schönberg and Ferruccio Busoni rejected the traditional technical compositional distinction between consonance and dissonance and at best allowed it to be regarded as a gradual distinction (instead of a dichotomy ).

The phrase “ emancipation of dissonance” was first used by Rudolf Louis, albeit in a negative sense. Arnold Schönberg , however, applied it positively:

“Apart from those who still find enough today with a few tonal triads [...] most living composers have drawn certain conclusions from the work of Wagner, Strauss, Mahler, Reger, Debussy, Puccini etc. in terms of harmony , the result of which is the emancipation of dissonance. "

- Arnold Schönberg : Mind or Knowledge? (1926)

This tendency is noticeable z. For example, because, from a traditional point of view, dissonant intervals are carried out in mixtures in parallel or used in closing sounds, or that sounds can no longer be clearly traced back to layers of thirds with notes that are not chordal.

## Mathematical and physical reasons

Since the 17th century ( Marin Mersenne , Galileo Galilei ) no longer abstract numerical ratios, but ratios of tone frequencies were considered as the basis of the degree of consonance. In the tentamen novae theoriae musicae (1739) Leonhard Euler proposed a mathematical formula for determining the " gradus suavitatis " ( degree of loveliness) of intervals and chords.

The phenomenon of overtones has Joseph Sauveur recorded in 1701 first physically closer ( Principes d'acoustique et de musique, ou système général of the interval, the son ). Rameau, Riemann et al. a. an explanation for the consonance of the major triad; but the consonance of the minor triad could not be derived from it in a satisfactory way.

Guerino Mazzola developed a mathematical theory of counterpoint, where the ban on parallel fifths and the dissonant fourth result from mathematical structures . This theory embeds the Fux theory as one of a total of six counterpoint worlds . This theory has also been extended to microtonal contexts.

## Physiological justifications

### Beat theory

Hermann von Helmholtz tried to explain the historically grown distinction between consonance and dissonance on the basis of the criterion of roughness , i.e. the number and intensity of beats . Accordingly, two overtones of both tones coincide at consonant intervals. With this approach, the distinction between consonance and dissonance is gradual and also depends on whether the intervals sound in a higher or lower register and whether the sounds are rich or poor in overtones. Helmholtz 'approach was continued by Heinrich Husmann under the name "Coincidence Theory".

### Fusion theory

Carl Stumpf , on the other hand, used the criterion of "fusion" as a basis: "the relationship between two [...] sensory contents according to which they do not form a mere sum but a whole". The degree of fusion determines the degree of consonance.

Since then, the degree of roughness on the one hand and fusion on the other has often been summarized in the term sonance ( character ) in music psychology .

## Psychoacoustic Research

The hypothesis that consonance is determined by the frequency ratio is disproved by experiments with musically educated and uneducated test persons who were exposed to dichotically offered intervals of sinus tones . Further factors such as beat are therefore a necessary condition.

The assumption that in addition to sonance, acculturation is also a determining factor in the consonance-dissonance distinctions developed in music history, a. researched using cognitive modeling .

## Sources and literature (chronological)

Wiktionary: Dissonance  - explanations of meanings, word origins, synonyms, translations
Wiktionary: Consonance  - explanations of meanings, word origins, synonyms, translations

## Individual evidence

1. Since the 11th century, successive intervals have also been temporarily called consonantia ; see Beiche 2001, p. 11 f.
2. Beiche 2005, p. 3 f.
3. Reckow 1971.
4. Book 1, chap. 3-8, 16, 28, 31; Book 4, chap. 1; Book 5, chap. 7, 11.
5. See Beiche 2001, pp. 8–11.
6. Sachs 1974.
7. Zarlino 1558, chap. 30th
8. One exception contains e.g. B. the cadenza doppia .
9. Lippius 1610, f. B2: "Trias Musica ex Tribus Sonis & Dyadibus Radicalibus distinctis constituta est: ex consonis, consonans: dissonans ex dissonis."
10. See Christensen 1993.
11. Kirnberger 1774, p. 30
12. Kirnberger 1774, p. 28
13. Riemann 1873, p. 31.
14. Riemann 1880, p. 62 f.
15. Riemann 1893, p. 77; see also Seidel 1966, p 58: "With the far more theoretical than practical requested realized Scheinkonsonanz the minor chords in major or major chords in a minor key is now [in the Simplified harmony made (1893)] definitely serious."
16. Schönberg 1922, p. 17f. Busoni 1922, p. 179.
17. Louis 1893, pp. 55, 80.
18. Helmholtz 1863, chap. 10-12.
19. Helmholtz 1863, p. 283.
20. Husmann 1953.
21. Stumpf 1890, p. 128.
22. Wellek 1963.
23. Eberlein 1993, p. 480.
24. Eberlein 1993, p. 483.