Harmony

The harmonious major triad is composed of three tones. Their frequency ratio corresponds approximately 6:5:4. In real performances, however, the third is often larger than 5:4. The ratio 5:4 corresponds to an interval of 386 cents, but an equally tempered major third is 400 cents and a Pythagorean third with a ratio of 81:64 is 408 cents. Measurements of frequencies in good performances confirm that the size of the major third varies across this range and can even lie outside it without sounding out of tune. Thus, there is no simple connection between frequency ratios and harmonic function.

In music, harmony is the use of simultaneous pitches (tones, notes), or chords.^[1] The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them.^[2] Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic line, or the "horizontal" aspect.^[3] Counterpoint, which refers to the interweaving of melodic lines, and polyphony, which refers to the relationship of separate independent voices, are thus sometimes distinguished from harmony.

Definitions, origin of term, and history of use

The term harmony derives from the Greek ἁρμονία (harmonía), meaning "joint, agreement, concord",^[4] from the verb ἁρμόζω (harmozo), "to fit together, to join".^[5] The term was often used for the whole field of music, while "music" referred to the arts in general.

In Ancient Greece, the term defined the combination of contrasted elements: a higher and lower note.^[6] Nevertheless, it is unclear whether the simultaneous sounding of notes was part of ancient Greek musical practice; "harmonía" may have merely provided a system of classification of the relationships between different pitches. In the Middle Ages the term was used to describe two pitches sounding in combination, and in the Renaissance the concept was expanded to denote three pitches sounding together.^[6]

It was not until the publication of Rameau's 'Traité de l'harmonie' (Treatise on Harmony) in 1722 that any text discussing musical practice made use of the term in the title, though that work is not the earliest record of theoretical discussion of the topic. The underlying principle behind these texts is that harmony sanctions harmoniousness (sounds that 'please') by conforming to certain pre-established compositional principles.^[7]

Current dictionary definitions, while attempting to give concise descriptions, often highlight the ambiguity of the term in modern use. Ambiguities tend to arise from either aesthetic considerations (for example the view that only "pleasing" concords may be harmonious) or from the point of view of musical texture (distinguishing between "harmonic" (simultaneously sounding pitches) and "contrapuntal" (successively sounding tones).^[7] In the words of Arnold Whitall:

While the entire history of music theory appears to depend on just such a distinction between harmony and counterpoint, it is no less evident that developments in the nature of musical composition down the centuries have presumed the interdependence—at times amounting to integration, at other times a source of sustained tension—between the vertical and horizontal dimensions of musical space.

—^[7]

The view that modern tonal harmony in Western music began in about 1600 is commonplace in music theory. This is usually accounted for by the 'replacement' of horizontal (of contrapuntal) writing, common in the music of the Renaissance, with a new emphasis on the 'vertical' element of composed music. Modern theorists, however, tend to see this as an unsatisfactory generalisation. As Carl Dahlhaus puts it:

It was not that counterpoint was supplanted by harmony (Bach’s tonal counterpoint is surely no less polyphonic than Palestrina’s modal writing) but that an older type both of counterpoint and of vertical technique was succeeded by a newer type. And harmony comprises not only the (‘vertical’) structure of chords but also their (‘horizontal’) movement. Like music as a whole, harmony is a process.

—^[8][9]

Descriptions and definitions of harmony and harmonic practice may show bias towards European (or Western) musical traditions. For example, South Asian art music (Hindustani and Carnatic music) is frequently cited as placing little emphasis on what is perceived in western practice as conventional 'harmony'; the underlying 'harmonic' foundation for most South Asian music is the drone, a held open fifth (or fourth) that does not alter in pitch throughout the course of a composition.^[10] Pitch simultaneity in particular is rarely a major consideration. Nevertheless many other considerations of pitch are relevant to the music, its theory and its structure, such as the complex system of Rāgas, which combines both melodic and modal considerations and codifications within it.^[11] So although intricate combinations of pitches sounding simultaneously in Indian classical music do occur they are rarely studied as teleological harmonic or contrapuntal progressions, which is the case with notated Western music. This contrasting emphasis (with regard to Indian music in particular) manifests itself to some extent in the different methods of performance adopted: in Indian Music improvisation takes a major role in the structural framework of a piece,^[12] whereas in Western Music improvisation has been uncommon since the end of the 19th century,^[13]. Where it does occur in Western music (or has in the past), the improvisation will either embellish pre-notated music or, if not, draw from musical models that have previously been established in notated compositions, and therefore employ familiar harmonic schemes.^[14]

There is no doubt, nevertheless, that the emphasis on the precomposed in European art music and the written theory surrounding it shows considerable cultural bias. The Grove Dictionary of Music and Musicians (Oxford University Press) identifies this quite clearly:

In Western culture the musics that are most dependent on improvisation, such as jazz, have traditionally been regarded as inferior to art music, in which pre-composition is considered paramount. The conception of musics that live in oral traditions as something composed with the use of improvisatory techniques separates them from the higher-standing works that use notation.

—^[15]

Yet the evolution of harmonic practice and language itself, in Western art music, is and was facilitated by this process of prior composition (which permitted the study and analysis by theorists and composers alike of individual pre-constructed works in which pitches (and to some extent rhythms) remained unchanged regardless of the nature of the performance).^[16]

Historical rules

Some traditions of music performance, composition, and theory have specific rules of harmony. These rules are often held to be based on natural properties such as Pythagorean tuning's law whole number ratios ("harmoniousness" being inherent in the ratios either perceptually or in themselves) or harmonics and resonances ("harmoniousness" being inherent in the quality of sound), with the allowable pitches and harmonies gaining their beauty or simplicity from their closeness to those properties. While Pythagorean ratios can provide a rough approximation of perceptual harmonicity, they cannot account for cultural factors.

Early Western religious music often features parallel perfect intervals; these intervals would preserve the clarity of the original plainsong. These works were created and performed in cathedrals, and made use of the resonant modes of their respective cathedrals to create harmonies. As polyphony developed, however, the use of parallel intervals was slowly replaced by the English style of consonance that used thirds and sixths. The English style was considered to have a sweeter sound, and was better suited to polyphony in that it offered greater linear flexibility in part-writing. Early music also forbade usage of the tritone, as its dissonance was associated with the devil, and composers often went to considerable lengths, via musica ficta, to avoid using it. In the newer triadic harmonic system, however, the tritone became permissible, as the standardization of functional dissonance made its use in dominant chords desirable.

Although most harmony comes about as a result of two or more notes being sounded simultaneously, it is possible to strongly imply harmony with only one melodic line through the use of arpeggios or hocket. Many pieces from the baroque period for solo string instruments, such as Bach's Sonatas and partitas for solo violin, convey subtle harmony through inference rather than full chordal structures; see below:

Example of implied harmonies in J.S. Bach's Cello Suite no. 1 in G, BWV 1007, bar 1.

Types

Carl Dahlhaus (1990) distinguishes between coordinate and subordinate harmony. Subordinate harmony is the hierarchical tonality or tonal harmony well known today, while coordinate harmony is the older Medieval and Renaissance tonalité ancienne, "the term is meant to signify that sonorities are linked one after the other without giving rise to the impression of a goal-directed development. A first chord forms a 'progression' with a second chord, and a second with a third. But the former chord progression is independent of the later one and vice versa." Coordinate harmony follows direct (adjacent) relationships rather than indirect as in subordinate. Interval cycles create symmetrical harmonies, which have been extensively used by the composers Alban Berg, George Perle, Arnold Schoenberg, Béla Bartók, and Edgard Varèse's Density 21.5.

Other types of harmony are based upon the intervals used in constructing the chords used in that harmony. Most chords used in western music are based on "tertian" harmony, or chords built with the interval of thirds. In the chord C Major7, C-E is a major third; E-G is a minor third; and G to B is a major third. Other types of harmony consist of quartal harmony and quintal harmony.

Intervals

An interval is the relationship between two separate musical pitches. For example, in the melody "Twinkle Twinkle Little Star", the first two notes (the first "twinkle") and the second two notes (the second "twinkle") are at the interval of one fifth. What this means is that if the first two notes were the pitch "C", the second two notes would be the pitch "G"—four scale notes, or seven chromatic notes (a perfect fifth), above it.

The following are common intervals:

Therefore, the combination of notes with their specific intervals —a chord— creates harmony. For example, in a C chord, there are three notes: C, E, and G. The note "C" is the root, with the notes "E" and "G" providing harmony, and in a G7 (G dominant 7th) chord, the root G with each subsequent note (in this case B, D and F) provide the harmony.

In the musical scale, there are twelve pitches. Each pitch is referred to as a "degree" of the scale. The names A, B, C, D, E, F, and G are insignificant. The intervals, however, are not. Here is an example:

As can be seen, no note always corresponds to a certain degree of the scale. The "root", or 1st-degree note, can be any of the 12 notes of the scale. All the other notes fall into place. So, when C is the root note, the fourth degree is F. But when D is the root note, the fourth degree is G. So while the note names are intransigent, the intervals are not. In layman's terms: a "fourth" (four-step interval) is always a fourth, no matter what the root note is. The great power of this fact is that any song can be played or sung in any key—it will be the same song, as long as the intervals are kept the same, thus transposing the harmony into the corresponding key.

When the intervals surpass the Octave (12 semitones), these intervals are named as "Extended intervals", which include particularly the 9th, 11th, and 13th Intervals, widely used in Jazz and Blues Music.

Extended Intervals are formed and named as following:

• 2nd Interval + Octave = "Ninth" Interval / 9th
• 4th Interval + Octave = "Eleventh" Interval / 11th
• 6th Interval + Octave = "Thirteenth" Interval / 13th

Apart from this categorization, intervals can also be divided into consonant and dissonant. As explained in the following paragraphs, consonant intervals produce a sensation of relaxation and dissonant intervals a sensation of tension.

The consonant intervals are considered to be the Unison, Octave, Fifth, Fourth and Major and Minor Third. The Third is considered Imperfect while the former are considered Perfect. In classical music the fourth may be considered to be dissonant when its function is contrapuntal.

All the other intervals, such as the 7th, 9th, 11th, and 13th are considered Dissonant and require resolution (of the produced tension) and usually preparation (depending on the music style used).

Chords and tension

In the Western tradition, harmony is manipulated using chords, which are combinations of pitch classes. In tertian or tertial harmony, so named after the interval of a third, the members of chords are found and named by stacking intervals of major and minor thirds, starting with the "root", then the "third" above the root, and the "fifth" above the root (which is a third above the third), etc. (Note that chord members are named after their interval against the root, not by their numerical inclusion in the building of the chord.) Traditionally, a chord must have at least three members to be called a chord, although 2-member dyads are sometimes treated as chords, especially in rock (see power chords). A chord with three members is called a triad because it has three members, not because it is necessarily built in thirds (see Quartal and quintal harmony for chords built with other intervals). Depending on the widths of the intervals being stacked, different qualities of chords are formed. In popular and jazz harmony, chords are named by their root plus various terms and characters indicating their qualities. To keep the nomenclature as simple as possible, some defaults are accepted (not tabulated here). For example, the chord members C, E, and G, form a C Major triad, called by default simply a "C" chord. In an "A♭" chord (pronounced A-flat), the members are A♭, C, and E♭.

In many types of music, notably baroque and jazz, chords are often augmented with "tensions". A tension is an additional chord member that creates a relatively dissonant interval in relation to one or more of the other chord members. Following the tertian practice of building chords by stacking thirds, the simplest first tension is added to a triad by stacking on top of the existing root, third, and fifth, another third above the fifth, giving a new, potentially dissonant member the interval of a seventh away from the root and therefore called the "seventh" of the chord, and producing a four-note chord, called a "seventh chord". Depending on the widths of the individual thirds stacked to build the chord, the interval between the root and the seventh of the chord may be major, minor, or diminished. (The interval of an augmented seventh reproduces the root, and is therefore left out of the chordal nomenclature.) The nomenclature allows that, by default, "C7" indicates a chord with a root, third, fifth, and seventh spelled C, E, G, and B♭. Other types of seventh chords must be named more explicitly, such as "C Major 7" (spelled C, E, G, B), "C augmented 7" (here the word augmented applies to the fifth, not the seventh, spelled C, E, G#, Bb), etc. (For a more complete exposition of nomenclature see Chord (music).)

Continuing to stack thirds on top of a seventh chord brings in the "extended tensions" or "upper tensions" (those more than an octave above the root when stacked in thirds), the ninths, elevenths, and thirteenths, and creates the chords named after them. (Note that except for dyads and triads, tertian chord types are named for the widest interval in use in the stack, not for the number of chord members, thus a ninth chord has five members, not nine.) Extensions beyond the thirteenth reproduce existing chord members and are (usually) left out of the nomenclature. Complex harmonies based on extended chords are found in abundance in jazz, modern orchestral works, film music, etc.

Typically, in the classical Common practice period a dissonant chord (chord with tension) will "resolve" to a consonant chord. Harmonization usually sounds pleasant to the ear when there is a balance between the consonant and dissonant sounds. In simple words, that occurs when there is a balance between "tense" and "relaxed" moments. For this reason, usually tension is 'prepared' and then 'resolved'.^[17]

Preparing tension means to place a series of consonant chords that lead smoothly to the dissonant chord. In this way the composer ensures introducing tension smoothly, without disturbing the listener. Once the piece reaches its sub-climax, the listener needs a moment of relaxation to clear up the tension, which is obtained by playing a consonant chord that resolves the tension of the previous chords. The clearing of this tension usually sounds pleasant to the listener.^[17]

Perception of harmony

Harmony is based on consonance, a concept whose definition has changed various times during the history of Western music. In a psychological approach, consonance is a continuous variable. Consonance can vary across a wide range. A chord may sound consonant for various reasons.

One is lack of perceptual roughness. Roughness happens when partials (frequency components) lie within a critical bandwidth, which is a measure of the ear's ability to separate different frequencies. Critical bandwidth lies between 2 and 3 semitones at high frequencies and becomes larger at lower frequencies. The roughness of two simultaneous harmonic complex tones depends on the amplitudes of the harmonics and the interval between the tones. The roughest interval in the chromatic scale is the minor second and its inversion the major seventh. For typical spectral envelopes in the central range, the second roughest interval is the major second and minor seventh, followed by the tritone, the minor third (major sixth), the major third (minor sixth) and the perfect fourth (fifth).

The second reason is perceptual fusion. A chord fuses in perception if its overall spectrum is similar to a harmonic series. According to this definition a major triad fuses better than a minor triad and a major-minor seventh chord fuses better than a major-major seventh or minor-minor seventh. These differences may not be readily apparent in tempered contexts but can explain why major triads are generally more prevalent than minor triads and major-minor sevenths generally more prevalent than other sevenths (in spite of the dissonance of the tritone interval) in mainstream tonal music. Of course these comparisons depend on style.

The third reason is familiarity. Chords that have often been heard in musical contexts tend to sound more consonant. This principle explains the gradual historical increase in harmonic complexity of Western music. For example, around 1600 unprepared seventh chords gradually became familiar and were therefore gradually perceived as more consonant.

Western music is based on major and minor triads. The reason why these chords are so central is that they are consonant in terms of both fusion and lack of roughness. they fuse because they include the perfect fourth/fifth interval. They lack roughness because they lack major and minor second intervals. No other combination of three tones in the chromatic scale satisfies these criteria.

Consonance and dissonance in balance

As Frank Zappa explained it,

"The creation and destruction of harmonic and 'statistical' tensions is essential to the maintenance of compositional drama. Any composition (or improvisation) which remains consistent and 'regular' throughout is, for me, equivalent to watching a movie with only 'good guys' in it, or eating cottage cheese."q:Frank Zappa

In other words, a composer cannot ensure a listener's liking by using exclusively consonant sounds. However, an excess of tension may disturb the listener. The balance between the two is essential.

Contemporary music has evolved in the way that tension is less often prepared and less structured than in Baroque or Classical periods, thus producing new styles such as Jazz and Blues, where tension is not usually prepared.

See also

• Barbershop music
• Consonance and dissonance
• Chord (music)
• Chord sequence
• Chromatic chord
• Chromatic mediant
• Counterpoint
• Harmonic series
• Homophony (music)
• List of musical terminology
• Mathematics of musical scales
• Musica universalis
• Peter Westergaard's tonal theory
• Prolongation
• Physics of music
• Tonality
• Unified field
• Voice leading

References

Footnotes

Notations

• Dahlhaus, Carl. Gjerdingen, Robert O. trans. (1990). Studies in the Origin of Harmonic Tonality, p. 141. Princeton University Press. ISBN 0-691-09135-8.
• van der Merwe, Peter (1989). Origins of the Popular Style: The Antecedents of Twentieth-Century Popular Music. Oxford: Clarendon Press. ISBN 0-19-316121-4.
• Nettles, Barrie & Graf, Richard (1997). The Chord Scale Theory and Jazz Harmony. Advance Music, ISBN 3-89221-056-X

External links

• Chord Geometry - Graphical Analysis of Harmony Tool
• Interactive Lessons about harmonizing melodies and scales using different Musical Styles
• Harmonic Progressions with demos and how to harmonize a melody
• General Principles of Harmony by Alan Belkin
• Morphogenesis of chords and scales Chords and scales classification
• A Beginner's Guide to Modal Harmony
• LucyTuning
• Posts about traditional and jazz harmony
• Sonantometry as Natural Harmony Algebra
• Chord Harmonization Tool Online tool that automates chord progression analysis to identify underlying scales and related chords

Further reading

• Prout, Ebenezer, Harmony, its Theory and Practice (1889, revised 1901)

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