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The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them.
In music and in music theory, "seventh interval" refers to the following musical intervals: major seventh, minor seventh, augmented seventh, or; diminished seventh. It also refers to inverted second intervals.
The major seventh interval is considered one of the most dissonant intervals after its inversion the minor second. For this reason, its melodic use is infrequent in classical music. However, in the genial Gavotte from J.S. Bach’s Partita in E major for solo violin, a major seventh features both as a chord (bar 1) and as a melodic interval ...
The dramatic clout of the interval was further exploited by operatic composers during the nineteenth century. Robert Donington (1963, p. 175) heard the dark, atmospheric Prelude to Wagner ’s opera Siegfried as "a kind of elemental brooding…Its material is an uneasy sequence of thirds low in the bass, and separated by a diminished seventh ...
For example, the interval from A 3 to G 4 is a minor seventh, as the note G 4 lies ten semitones above A 3, and there are seven staff positions from A 3 to G 4. Diminished and augmented sevenths span the same number of staff positions, but consist of a different number of semitones (nine and twelve, respectively).
A few steps of the bisection method applied over the starting range [a 1;b 1].The bigger red dot is the root of the function. In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs.
The major seventh chord, sometimes also called a Delta chord, can be written as maj 7, M 7, Δ, ⑦, etc. The "7" does not have to be superscripted, but if it is, then any alterations, added tones, or omissions are usually also superscripted. For example, the major seventh chord built on C, commonly written as Cmaj 7, has pitches C–E–G–B:
The harmonic seventh arises from the harmonic series as the interval between the fourth harmonic (second octave of the fundamental) and the seventh harmonic; in that octave, harmonics 4, 5, 6, and 7 constitute the four notes (in order) of a purely consonant major chord (root position) with an added minor seventh (or augmented sixth, depending ...