Consonance and dissonance

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifths which are "pure" or perfect, with ratio ${\displaystyle 3:2}$. This is chosen because it is the next harmonic of a vibrating string, after the octave (which is the ratio ${\displaystyle 2:1}$), and hence is the next most consonant "pure" interval, and the easiest to tune by ear. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2 (i.e., the untempered perfect fifth), which is ≈ 702 cents wide.

The system dates back to Ancient Mesopotamia;. (See Music of Mesopotamia § Music theory.) It is named, and has been widely misattributed, to Ancient Greeks, notably Pythagoras (sixth century BC) by modern authors of music theory. Ptolemy, and later Boethius, ascribed the division of the tetrachord by only two intervals, called "semitonium" and "tonus" in Latin (256:243 × 9:8 × 9:8), to Eratosthenes. The so-called "Pythagorean tuning" was used by musicians up to the beginning of the 16th century. "The Pythagorean system would appear to be ideal because of the purity of the fifths, but some consider other intervals, particularly the major third, to be so badly out of tune that major chords [may be considered] a dissonance."

In Western music theory, a chord is a group of notes played together for their harmonic consonance or dissonance. The most basic type of chord is a triad, so called because it consists of three distinct notes: the root note along with intervals of a third and a fifth above the root note. Chords with more than three notes include added tone chords, extended chords and tone clusters, which are used in contemporary classical music, jazz, and other genres.

Chords are the building blocks of harmony and form the harmonic foundation of a piece of music. They provide the harmonic support and coloration that accompany melodies and contribute to the overall sound and mood of a musical composition. The factors, or component notes, of a chord are often sounded simultaneously but can instead be sounded consecutively, as in an arpeggio.

In music, tension is the anticipation music creates in a listener's mind for relaxation or release. For example, tension may be produced through reiteration, increase in dynamic level, gradual motion to a higher or lower pitch, or (partial) syncopations between consonance and dissonance.

Experiments in music perception have explored perceived tension in music, and perceived emotional intensity.

Minimal music (also called minimalism) is a form of art music or other compositional practice that employs limited or minimal musical materials. Prominent features of minimalist music include repetitive patterns or pulses, steady drones, consonant harmony, and reiteration of musical phrases or smaller units. It may include features such as phase shifting, resulting in what is termed phase music, or process techniques that follow strict rules, usually described as process music. The approach is marked by a non-narrative, non-teleological, and non-representational approach, and calls attention to the activity of listening by focusing on the internal processes of the music.

The approach originated on the West Coast of the United States in the late 1950s and early 1960s, particularly around the Bay Area, where La Monte Young, Terry Riley and Steve Reich were studying and living at the time. After the three composers moved to the East Coast, their music became associated with the New York Downtown music scene of the mid-1960s, where it was initially viewed as a form of experimental music called the New York Hypnotic School. In the Western art music tradition, the American composers Moondog, La Monte Young, Terry Riley, Steve Reich and Philip Glass are credited with being among the first to develop compositional techniques that exploit a minimal approach. The movement originally involved dozens of composers, although only five (Young, Riley, Reich, Glass, and later John Adams) emerged to become publicly associated with American minimal music; other lesser known pioneers included Dennis Johnson, Terry Jennings, Richard Maxfield, Pauline Oliveros, Phill Niblock, and James Tenney. In Europe, the music of Louis Andriessen, Karel Goeyvaerts, Michael Nyman, Howard Skempton, Éliane Radigue, Gavin Bryars, Steve Martland, Peter Michael Hamel, Henryk Górecki, Arvo Pärt and John Tavener exhibits minimalist traits.

Resolution in Western tonal music theory is the move of a note or chord from dissonance (an unstable sound) to a consonance (a more final or stable sounding one).

Dissonance, resolution, and suspense can be used to create musical interest. Where a melody or chordal pattern is expected to resolve to a certain note or chord, a different but similarly suitable note can be resolved to instead, creating an interesting and unexpected sound. For example, the deceptive cadence.

Barbershop vocal harmony is a style of a cappella close harmony, or unaccompanied vocal music, characterized by consonant four-part chords for every melody note in a primarily homorhythmic texture. Each of the four parts has its own role: generally, the lead sings the melody, the tenor harmonizes above the melody, the bass sings the lowest harmonizing notes, and the baritone completes the chord, usually below the lead. The melody is not usually sung by the tenor or baritone, except for an infrequent note or two to avoid awkward voice leading, in tags or codas, or when some appropriate embellishment can be created. One characteristic feature of barbershop harmony is the use of what is known as "snakes" and "swipes". This is when a chord is altered by a change in one or more non-melodic voices. Occasional passages may be sung by fewer than four voice parts.

Barbershop music is generally performed by either a barbershop quartet, a group of four typically male singers with one on each vocal part, or a barbershop chorus, which closely resembles a choir with the notable exception of the genre of music.

A suspended chord (or sus chord) is a musical chord in which the (major or minor) third is omitted and replaced with a perfect fourth or a major second. The lack of a minor or a major third in the chord creates an open sound, while the dissonance between the fourth and fifth or second and root creates tension. When using popular-music symbols, they are indicated by the symbols "sus4" and "sus2". For example, the suspended fourth and second chords built on C (C–E–G), written as C and C, have pitches C–F-G and C–D-G, respectively.

Suspended fourth and second chords can be represented by the integer notation {0, 5, 7} and {0, 2, 7}, respectively.

In music, extended chords are certain chords (built from thirds) or triads with notes extended, or added, beyond the seventh. Ninth, eleventh, and thirteenth chords are extended chords. The thirteenth is the farthest extension diatonically possible as, by that point, all seven tonal degrees are represented within the chord (the next extension, the fifteenth, is the same as the root of the chord). In practice however, extended chords do not typically use all the chord members; when it is not altered, the fifth is often omitted, as are notes between the seventh and the highest note (i.e., the ninth is often omitted in an eleventh chord; the ninth and eleventh are usually omitted in a thirteenth chord), unless they are altered to give a special texture.

Chords extended beyond the seventh are rarely seen in the Baroque era, and are used more frequently in the Classical era. The Romantic era saw greatly increased use of extended harmony. Extended harmony prior to the 20th century usually has dominant function – as V, V, and V, or V/V, V/ii etc.

In music, a factor or chord factor is a member or component of a chord. These are named root, third, fifth, sixth, seventh, ninth (compound 2nd), eleventh (compound 4th), thirteenth (compound 6th), and so on, for their generic interval above the root.In harmony, the consonance and dissonance of a chord factor and a nonchord tone are distinguished, respectively.

Chord factors are taken into consideration in voicing and voice leading. A chord contains exactly as many factors as it contains unique pitch names (octaves don't matter), while a voicing can have any number of voices that draw from and represent some or all the factors of a chord in various octaves. Thus, a chord with three unique pitch names always has three factors, even if some of those pitches are doubled or omitted in a particular voicing. For example, the figure to the right shows a four-note voicing of a C Major triad, which has three chord factors. The "root" chord factor (pitch name "C"), is represented twice in the voicing by voices 1 and 4 in different octaves. The chord factor called the "fifth" (pitch name "G") is represented in voice 2 (shown in red).

A semitone, also called a minor second, half step, or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.It is defined as the interval between two adjacent notes in a 12-tone scale (or half of a whole step), visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C♯; the interval between them is a semitone.

In a 12-note approximately equally divided scale, any interval can be defined in terms of an appropriate number of semitones (e.g. a whole tone or major second is 2 semitones wide, a major third 4 semitones, and a perfect fifth 7 semitones).

In jazz, a bar-line shift is a technique in which, during improvisation, one plays the chord from the measure before, as an anticipation of a chord, or after the given chord, as a delay, either intentionally or as an "accident."

Bar-line shifts may be caused by a novice having lost their place in the chord progression, but is most often attributable to: "(1)...harmonic generalization, as in the case of playing a II to V7 (+5, +9) progression [II-V-I turnaround] as only a V7 (+5, +9); or (2) the player wanted to play the previous chord (though it has already transpired), but was either pausing momentarily (as in taking a breath), and decides to adopt the 'better later than never' attitude." An example of a "very intentional" bar-line shift may be found on Cannonball Adderley's solo on "So What," "in which he deliberately enters and exits the bridge early, causing considerable tension, since the chord of the A section (D-) is one-half step lower than the chord of the bridge (E♭-)." Other notable performers to use this technique include Charlie Parker, Coleman Hawkins, and Pat Metheny.

James Tenney (August 10, 1934 – August 24, 2006) was an American composer, music theorist, and pianist. He made significant early musical contributions to plunderphonics, sound synthesis, algorithmic composition, process music, spectral music, and microtonal tuning systems including extended just intonation. His theoretical writings variously concern musical form, texture, timbre, consonance and dissonance, and harmonic perception.

In classical music from Western culture, a diminished seventh (play) is an interval produced by narrowing a minor seventh by a chromatic semitone, and its inversion is the augmented second. For instance, the interval from A to G is a minor seventh, ten semitones wide, and both the intervals from A♯ to G, and from A to G♭ are diminished sevenths, spanning nine semitones. Being diminished, it is considered a dissonant interval.

The diminished seventh is used quite readily in the minor key, where it is present in the harmonic minor scale between the seventh scale step and the sixth scale step in the octave above.

A seventh chord is a chord consisting of a triad plus a note forming an interval of a seventh above the chord's root. When not otherwise specified, a "seventh chord" usually means a dominant seventh chord: a major triad together with a minor seventh. However, a variety of sevenths may be added to a variety of triads, resulting in many different types of seventh chords.

In its earliest usage, the seventh was introduced solely as an embellishing or nonchord tone. The seventh destabilized the triad, and allowed the composer to emphasize movement in a given direction. As time passed and the collective ear of the western world became more accustomed to dissonance, the seventh was allowed to become a part of the chord itself, and in some modern music, jazz in particular, nearly every chord is a seventh chord. Additionally, the general acceptance of equal temperament during the 19th century reduced the dissonance of some earlier forms of sevenths.

Johannes Brahms (/brɑːmz/; German: [joˈhanəs ˈbʁaːms] ; 7 May 1833 – 3 April 1897) was a German composer, virtuoso pianist, and conductor of the mid-Romantic period. His music is noted for its rhythmic vitality and freer treatment of dissonance, often set within studied yet expressive contrapuntal textures. He adapted the traditional structures and techniques of a wide historical range of earlier composers. His œuvre includes four symphonies, four concertos, a Requiem, much chamber music, and hundreds of folk-song arrangements and Lieder, among other works for symphony orchestra, piano, organ, and choir.

Born to a musical family in Hamburg, Brahms began composing and concertizing locally in his youth. He toured Central Europe as a pianist in his adulthood, premiering many of his own works and meeting Franz Liszt in Weimar. Brahms worked with Ede Reményi and Joseph Joachim, seeking Robert Schumann's approval through the latter. He gained both Robert and Clara Schumann's strong support and guidance. Brahms stayed with Clara in Düsseldorf, becoming devoted to her amid Robert's insanity and institutionalization. The two remained close, lifelong friends after Robert's death. Brahms never married, perhaps in an effort to focus on his work as a musician and scholar. He was a self-conscious, sometimes severely self-critical composer.

In music, a pedal point (also pedal note, organ point, pedal tone, or pedal) is a sustained tone, typically in the bass, during which at least one foreign (i.e. dissonant) harmony is sounded in the other parts. A pedal point sometimes functions as a "non-chord tone", placing it in the categories alongside suspensions, retardations, and passing tones. However, the pedal point is unique among non-chord tones, "in that it begins on a consonance, sustains (or repeats) through another chord as a dissonance until the harmony", not the non-chord tone, "resolves back to a consonance".

Pedal points "have a strong tonal effect, 'pulling' the harmony back to its root". Pedal points can also build drama or intensity and expectation. When a pedal point occurs in a voice other than the bass, it is usually referred to as an inverted pedal point (see inversion). Pedal points are usually on either the tonic or the dominant (fifth note of the scale) tones. The pedal tone is considered a chord tone in the original harmony, then a nonchord tone during the intervening dissonant harmonies, and then a chord tone again when the harmony resolves. A dissonant pedal point may go against all harmonies present during its duration, being almost more like an added tone than a nonchord tone, or pedal points may serve as atonal pitch centers.