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Because 3/2 is not a rational power of two, the angle on the chromatic circle that represents a perfect fifth is not a rational multiple of 2 π {\displaystyle 2\pi } , and similarly other common musical intervals other than the octave do not correspond to rational angles.A tuning system is a collection of tones used to compose and play music. For instance, the equal temperament commonly used for the piano is a tuning system, consisting of 12 tones equally spaced around the chromatic circle. Some other tuning systems do not space their tones equally, but instead generate them by some number of consecutive multiples of a given interval. | https://en.wikipedia.org/wiki/Steinhaus_conjecture |
An example is the Pythagorean tuning, which is constructed in this way from twelve tones, generated as the consecutive multiples of a perfect fifth in the circle of fifths. The irrational angle formed on the chromatic circle by a perfect fifth is close to 7/12 of a circle, and therefore the twelve tones of the Pythagorean tuning are close to, but not the same as, the twelve tones of equal temperament, which could be generated in the same way using an angle of exactly 7/12 of a circle. Instead of being spaced at angles of exactly 1/12 of a circle, as the tones of equal temperament would be, the tones of the Pythagorean tuning are separated by intervals of two different angles, close to but not exactly 1/12 of a circle, representing two different types of semitones. If the Pythagorean tuning system were extended by one more perfect fifth, to a set of 13 tones, then the sequence of intervals between its tones would include a third, much shorter interval, the Pythagorean comma.In this context, the three-gap theorem can be used to describe any tuning system that is generated in this way by consecutive multiples of a single interval. Some of these tuning systems (like equal temperament) may have only one interval separating the closest pairs of tones, and some (like the Pythagorean tuning) may have only two different intervals separating the tones, but the three-gap theorem implies that there are always at most three different intervals separating the tones. | https://en.wikipedia.org/wiki/Steinhaus_conjecture |
In music theory, a natural (♮) is an accidental which cancels previous key signatures or accidentals and represents the unaltered pitch of a note. | https://en.wikipedia.org/wiki/Natural_note |
In music theory, a natural analogue of doubling is the octave (a musical interval caused by doubling the frequency of a tone), and a natural analogue of a cube is dividing the octave into three parts, each the same interval. In this sense, the problem of doubling the cube is solved by the major third in equal temperament. This is a musical interval that is exactly one third of an octave. It multiplies the frequency of a tone by 2 4 / 12 = 2 1 / 3 = 2 3 {\displaystyle 2^{4/12}=2^{1/3}={\sqrt{2}}} , the side length of the Delian cube. | https://en.wikipedia.org/wiki/Duplicating_the_cube |
In music theory, a neutral interval is an interval that is neither a major nor minor, but instead in between. For example, in equal temperament, a major third is 400 cents, a minor third is 300 cents, and a neutral third is 350 cents. A neutral interval inverts to a neutral interval. | https://en.wikipedia.org/wiki/Neutral_second |
For example, the inverse of a neutral third is a neutral sixth. Roughly, neutral intervals are a quarter tone sharp from minor intervals and a quarter tone flat from major intervals. In just intonation, as well as in tunings such as 31-ET, 41-ET, or 72-ET, which more closely approximate just intonation, the intervals are closer together. Neutral second Neutral third Neutral sixth Neutral seventh | https://en.wikipedia.org/wiki/Neutral_second |
In music theory, a ninth chord is a chord that encompasses the interval of a ninth when arranged in close position with the root in the bass. The ninth chord and its inversions exist today, or at least they can exist. The pupil will easily find examples in the literature . | https://en.wikipedia.org/wiki/Ninth_chord |
It is not necessary to set up special laws for its treatment. If one wants to be careful, one will be able to use the laws that pertain to the seventh chords: that is, dissonances resolve by step downward, the root leaps a fourth upward. Heinrich Schenker and also Nikolai Rimsky-Korsakov allowed the substitution of the dominant seventh, leading-tone, and leading tone half-diminished seventh chords, but rejected the concept of a ninth chord on the basis that only that on the fifth scale degree (V9) was admitted and that inversion was not allowed of the ninth chord. | https://en.wikipedia.org/wiki/Ninth_chord |
In music theory, a nondominant seventh chord is both a diatonic chord and a seventh chord, but it does not possess dominant function, and thus it is not a dominant seventh chord. Since the V and viio chords are the dominant function chords, the "major minor seventh" V7 and "half-diminished seventh" viiø7 are the dominant seventh chords. Since the nondominant function chords are I, i, ii, iio, iii, III, IV, iv, vi, and VI, the nondominant seventh chord qualities include the augmented major seventh chord, major seventh chord, minor major seventh chord, minor seventh chord, and major minor seventh chords that do not possess dominant function, such as, in melodic minor, IV7m. To analyze seventh chords indicate the quality of the triad; major: I, minor: ii, half-diminished: viiø, or augmented: III+; and the quality of the seventh; same: 7, or different: 7M or 7m. With chord letters used to indicate the root and chord quality, and add 7, thus a seventh chord on ii in C major (minor minor seventh) would be d7.As with dominant seventh chords, nondominant seventh chords usually progress according to the circle progression, thus III+7M resolves to vi or VI, for example. When possible, as in circle progressions, resolve the seventh of nondominant seventh chords down by step to the third of the following chord. | https://en.wikipedia.org/wiki/Nondominant_seventh_chord |
In music theory, a parameter denotes an element which may be manipulated (composed), separately from the other elements. The term is used particularly for pitch, loudness, duration, and timbre, though theorists or composers have sometimes considered other musical aspects as parameters. The term is particularly used in serial music, where each parameter may follow some specified series. Paul Lansky and George Perle criticized the extension of the word "parameter" to this sense, since it is not closely related to its mathematical sense, but it remains common. The term is also common in music production, as the functions of audio processing units (such as the attack, release, ratio, threshold, and other variables on a compressor) are defined by parameters specific to the type of unit (compressor, equalizer, delay, etc.). | https://en.wikipedia.org/wiki/Parameter |
In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of the first five consecutive notes in a diatonic scale. The perfect fifth (often abbreviated P5) spans seven semitones, while the diminished fifth spans six and the augmented fifth spans eight semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. | https://en.wikipedia.org/wiki/Just_perfect_fifth |
In a diatonic scale, the dominant note is a perfect fifth above the tonic note. The perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. | https://en.wikipedia.org/wiki/Just_perfect_fifth |
It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of its Greek names, diapente. Its inversion is the perfect fourth. The octave of the fifth is the twelfth. A perfect fifth is at the start of "Twinkle, Twinkle, Little Star"; the pitch of the first "twinkle" is the root note and the pitch of the second "twinkle" is a perfect fifth above it. | https://en.wikipedia.org/wiki/Just_perfect_fifth |
In music theory, a phrase (Greek: φράση) is a unit of musical meter that has a complete musical sense of its own, built from figures, motifs, and cells, and combining to form melodies, periods and larger sections. A phrase is a substantial musical thought, which ends with a musical punctuation called a cadence. Phrases are created in music through an interaction of melody, harmony, and rhythm. Terms such as sentence and verse have been adopted into the vocabulary of music from linguistic syntax. | https://en.wikipedia.org/wiki/Musical_phrase |
Though the analogy between the musical and the linguistic phrase is often made, still the term "is one of the most ambiguous in music....there is no consistency in applying these terms nor can there be...only with melodies of a very simple type, especially those of some dances, can the terms be used with some consistency. "John D. White defines a phrase as "the smallest musical unit that conveys a more or less complete musical thought. Phrases vary in length and are terminated at a point of full or partial repose, which is called a cadence." | https://en.wikipedia.org/wiki/Musical_phrase |
Edward Cone analyses the "typical musical phrase" as consisting of an "initial downbeat, a period of motion, and a point of arrival marked by a cadential downbeat". Charles Burkhart defines a phrase as "Any group of measures (including a group of one, or possibly even a fraction of one) that has some degree of structural completeness. What counts is the sense of completeness we hear in the pitches not the notation on the page. | https://en.wikipedia.org/wiki/Musical_phrase |
To be complete such a group must have an ending of some kind … . Phrases are delineated by the tonal functions of pitch. They are not created by slur or by legato performance ... . A phrase is not pitches only but also has a rhythmic dimension, and further, each phrase in a work contributes to that work's large rhythmic organization." | https://en.wikipedia.org/wiki/Musical_phrase |
In music theory, a pitch simultaneity is more than one pitch or pitch class all of which occur at the same time, or simultaneously: "A set of notes sounded together." Simultaneity is a more specific and more general term than chord: many but not all chords or harmonies are simultaneities, though not all but some simultaneities are chords. For example, arpeggios are chords whose tones are not simultaneous. "The practice of harmony typically involves both simultaneity...and linearity. | https://en.wikipedia.org/wiki/Simultaneity_(music) |
"A simultaneity succession is a series of different groups of pitches or pitch classes, each of which is played at the same time as the other pitches of its group. Thus, a simultaneity succession is a succession of simultaneities. Similarly, simultaneity succession is a more general term than chord progression or harmonic progression: most chord progressions or harmonic progressions are then simultaneity successions, though not all simultaneity successions are harmonic progressions and not all simultaneities are chords. | https://en.wikipedia.org/wiki/Simultaneity_(music) |
In music theory, a predominant chord (also pre-dominant) is any chord which normally resolves to a dominant chord. Examples of predominant chords are the subdominant (IV, iv), supertonic (ii, ii°), Neapolitan sixth and German sixth. Other examples are the secondary dominant (V/V) and secondary leading tone chord. Predominant chords may lead to secondary dominants. | https://en.wikipedia.org/wiki/Predominant_chord |
Predominant chords both expand away from the tonic and lead to the dominant, affirming the dominant's pull to the tonic. Thus they lack the stability of the tonic and the drive towards resolution of the dominant. The predominant harmonic function is part of the fundamental harmonic progression of many classical works. | https://en.wikipedia.org/wiki/Predominant_chord |
The submediant (vi) may be considered a predominant chord or a tonic substitute.The dominant preparation is a chord or series of chords that precedes the dominant chord in a musical composition. Usually, the dominant preparation is derived from a circle of fifths progression. The most common dominant preparation chords are the supertonic, the subdominant, the V7/V, the Neapolitan chord (N6 or ♭II6), and the augmented sixth chords (e.g., Fr+6). | https://en.wikipedia.org/wiki/Predominant_chord |
In sonata form, the dominant preparation is in the development, immediately preceding the recapitulation. Ludwig van Beethoven's sonata-form works generally have extensive dominant preparation — for example, in the first movement of the Sonata Pathétique, the dominant preparation lasts for 29 measures (mm. 169–197). | https://en.wikipedia.org/wiki/Predominant_chord |
In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the context of the common practice period, most or all of the melody and harmony of a musical work is built using the notes of a single scale, which can be conveniently represented on a staff with a standard key signature.Due to the principle of octave equivalence, scales are generally considered to span a single octave, with higher or lower octaves simply repeating the pattern. A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance (or interval) between two successive notes of the scale. | https://en.wikipedia.org/wiki/Music_scale |
However, there is no need for scale steps to be equal within any scale and, particularly as demonstrated by microtonal music, there is no limit to how many notes can be injected within any given musical interval. A measure of the width of each scale step provides a method to classify scales. | https://en.wikipedia.org/wiki/Music_scale |
For instance, in a chromatic scale each scale step represents a semitone interval, while a major scale is defined by the interval pattern W–W–H–W–W–W–H, where W stands for whole step (an interval spanning two semitones, e.g. from C to D), and H stands for half-step (e.g. from C to D♭). Based on their interval patterns, scales are put into categories including diatonic, chromatic, major, minor, and others. A specific scale is defined by its characteristic interval pattern and by a special note, known as its first degree (or tonic). | https://en.wikipedia.org/wiki/Music_scale |
The tonic of a scale is the note selected as the beginning of the octave, and therefore as the beginning of the adopted interval pattern. Typically, the name of the scale specifies both its tonic and its interval pattern. For example, C major indicates a major scale with a C tonic. | https://en.wikipedia.org/wiki/Music_scale |
In music theory, a secondary leading-tone chord is a secondary chord that is rooted on a tone that is a leading-tone of (in short, has a strong affinity to resolve to) a tone just 1 semitone from that root (typically 1 semitone above, though can be below). Like the secondary dominant it can be used as tonicization of only one subsequent chord (which will be rooted in the resolution tone), or the music can continue with other chords/notes in the key of that chord's root for a phrase, or even longer to be considered a modulation to that key. This one-semitone-apart resolution of the secondary leading-tone is in contrast to the secondary dominant which resolves through a wider distance of perfect fifth below or perfect fourth above the chord's root (as per the two distances between dominant and tonic). While the root of a secondary leading-tone chord needs to be the leading-tone, the other notes may vary and form with it one of: the triad or one of the diminished sevenths (as in seventh scale degree or leading-tone, not necessarily seventh chord) where the type of the diminished seventh is typically related to the type of tonicized triad: If the tonicized triad is minor, the leading-tone chord is fully diminished seventh chord. | https://en.wikipedia.org/wiki/Secondary_supertonic |
If it is major, the leading-tone chord may be either half-diminished or fully diminished, though fully diminished chords are used more often.Because of their symmetry, secondary leading-tone diminished seventh chords are also useful for modulation; all four notes may be considered the root of any diminished seventh chord. They may resolve to these major or minor diatonic triads: In major keys: ii, iii, IV, V, vi In minor keys: III, iv, V, VIEspecially in four-part writing, the seventh should resolve downwards by step and if possible the lower tritone should resolve appropriately, inwards if a diminished fifth and outwards if an augmented fourth, as the example below shows. | https://en.wikipedia.org/wiki/Secondary_supertonic |
Secondary leading-tone chords were not used until the Baroque period and are found more frequently and less conventionally in the Classical period. They are found even more frequently and freely in the Romantic period, but they began to be used less frequently with the breakdown of conventional harmony. The chord progression viio7/V–V–I is quite common in ragtime music. | https://en.wikipedia.org/wiki/Secondary_supertonic |
In music theory, a semiditone (or Pythagorean minor third) is the interval 32:27 (approximately 294.13 cents). It is the minor third in Pythagorean tuning. The 32:27 Pythagorean minor third arises in the 5-limit justly tuned major scale between the 2nd and 4th degrees (in the C major scale, between D and F). | https://en.wikipedia.org/wiki/Pythagorean_minor_third |
It can be thought of as two octaves minus three justly tuned fifths. It is narrower than a justly tuned minor third by a syntonic comma. Its inversion is a Pythagorean major sixth. | https://en.wikipedia.org/wiki/Pythagorean_minor_third |
In music theory, a theoretical key is a key whose key signature would have at least one double-flat () or double-sharp (). Double-flats and double-sharps are often used as a result of the main key being a specific flat or sharp. For example, the sixth note of C-sharp minor is A while in D-flat minor, C-sharp minor’s enharmonic equivalent, that note becomes B double flat. | https://en.wikipedia.org/wiki/A-sharp_major |
Due to the fact that there is no way to represent the A note by using a single flat key.) Using double flats and double sharps, however, makes the music too complex to read. This is why some musical keys, such as G-sharp major, are not normally used. | https://en.wikipedia.org/wiki/A-sharp_major |
In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, was further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.Conventionally used with a predominant function (resolving to the dominant), the three most common types of augmented sixth chords are usually called the Italian sixth, the French sixth, and the German sixth. | https://en.wikipedia.org/wiki/Augmented_sixth_chord |
In music theory, an eleventh chord is a chord that contains the tertian extension of the eleventh. Typically found in jazz, an eleventh chord also usually includes the seventh and ninth, and elements of the basic triad structure. Variants include the dominant eleventh (C11, C–E–G–B♭–D–F), minor eleventh (Cm11, C–E♭–G–B♭–D–F), and major eleventh chord (Cmaj11, C–E–G–B–D–F). Using an augmented eleventh produces the dominant sharp eleventh (C9♯11, C–E–G–B♭–D–F♯) and major sharp eleventh (Cmaj9♯11, C–E–G–B–D–F♯) chords. | https://en.wikipedia.org/wiki/Eleventh_chord |
A perfect eleventh creates a highly dissonant minor ninth interval with the major third of major and dominant chords. To reduce this dissonance the third is often omitted (such as for example in the dominant eleventh chord that can be heard 52 seconds into the song "Sun King" on The Beatles' Abbey Road album), turning the chord into a suspended ninth chord (e.g. C9sus4, C–G–B♭–D–F), which can be also notated as Gm7/C.Another solution to this dissonance is altering the third or eleventh factor of the chord to turn the problematic minor ninth interval within the chord into a major ninth. A dominant eleventh chord can be altered by lowering the third by a semitone for a minor eleventh chord, or by raising the eleventh by a semitone for a dominant sharp eleventh chord, implying the lydian dominant mode. | https://en.wikipedia.org/wiki/Eleventh_chord |
As its upper extensions (7th, 9th, 11th) constitute a triad, a dominant eleventh chord with the third and fifth omitted can be notated as a compound chord with a bass note. So C–B♭–D–F is written as B♭/C, emphasizing the ambiguous dominant/subdominant character of this voicing. In the common practice period, the root, 7th, 9th, and 11th are the most common factors present in the V11 chord, with the 3rd and 5th typically omitted. The eleventh is usually retained as a common tone when the chord resolves to I or i. | https://en.wikipedia.org/wiki/Eleventh_chord |
In music theory, an enharmonic scale is "an gradual progression by quarter tones" or any " scale proceeding by quarter tones". The enharmonic scale uses dieses (divisions) nonexistent on most keyboards, since modern standard keyboards have only half-tone dieses. More broadly, an enharmonic scale is a scale in which (using standard notation) there is no exact equivalence between a sharpened note and the flattened note it is enharmonically related to, such as in the quarter tone scale. As an example, F♯ and G♭ are equivalent in a chromatic scale (the same sound is spelled differently), but they are different sounds in an enharmonic scale. | https://en.wikipedia.org/wiki/Enharmonic_scale |
See: musical tuning. Musical keyboards which distinguish between enharmonic notes are called by some modern scholars enharmonic keyboards. (The enharmonic genus, a tetrachord with roots in early Greek music, is only loosely related to enharmonic scales.) | https://en.wikipedia.org/wiki/Enharmonic_scale |
Consider a scale constructed through Pythagorean tuning. A Pythagorean scale can be constructed "upwards" by wrapping a chain of perfect fifths around an octave, but it can also be constructed "downwards" by wrapping a chain of perfect fourths around the same octave. By juxtaposing these two slightly different scales, it is possible to create an enharmonic scale. The following Pythagorean scale is enharmonic: In the above scale the following pairs of notes are said to be enharmonic: C♯ and D♭ D♯ and E♭ F♯ and G♭ G♯ and A♭ A♯ and B♭In this example, natural notes are sharpened by multiplying its frequency ratio by 256:243 (called a limma), and a natural note is flattened by multiplying its ratio by 243:256. A pair of enharmonic notes are separated by a Pythagorean comma, which is equal to 531441:524288 (about 23.46 cents). | https://en.wikipedia.org/wiki/Enharmonic_scale |
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.In Western music, intervals are most commonly differences between notes of a diatonic scale. Intervals between successive notes of a scale are also known as scale steps. The smallest of these intervals is a semitone. | https://en.wikipedia.org/wiki/Minor_interval |
Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. | https://en.wikipedia.org/wiki/Minor_interval |
Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C♯ and D♭. Intervals can be arbitrarily small, and even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic frequencies. | https://en.wikipedia.org/wiki/Minor_interval |
For example, any two notes an octave apart have a frequency ratio of 2:1. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio. | https://en.wikipedia.org/wiki/Minor_interval |
In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.). Examples include the minor third or perfect fifth. These names identify not only the difference in semitones between the upper and lower notes but also how the interval is spelled. The importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭. | https://en.wikipedia.org/wiki/Minor_interval |
In music theory, an inversion is a type of change to intervals, chords, voices (in counterpoint), and melodies. In each of these cases, "inversion" has a distinct but related meaning. The concept of inversion also plays an important role in musical set theory. | https://en.wikipedia.org/wiki/Inversion_(music) |
In music theory, chord substitution is the technique of using a chord in place of another in a progression of chords, or a chord progression. Much of the European classical repertoire and the vast majority of blues, jazz and rock music songs are based on chord progressions. "A chord substitution occurs when a chord is replaced by another that is made to function like the original. | https://en.wikipedia.org/wiki/Chord_substitution |
Usually substituted chords possess two pitches in common with the triad that they are replacing. "A chord progression may be repeated to form a song or tune. Composers, songwriters and arrangers have developed a number of ways to add variety to a repeated chord progression. There are many ways to add variety to music, including changing the dynamics (loudness and softness). | https://en.wikipedia.org/wiki/Chord_substitution |
In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism. In interval complementation a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is also known as its inverse or inversion. | https://en.wikipedia.org/wiki/Complement_(music) |
Note that the octave and the unison are each other's complements and that the tritone is its own complement (though the latter is "re-spelt" as either an augmented fourth or a diminished fifth, depending on the context). In the aggregate complementation of twelve-tone music and serialism the complement of one set of notes from the chromatic scale contains all the other notes of the scale. For example, A-B-C-D-E-F-G is complemented by B♭-C♯-E♭-F♯-A♭. Note that musical set theory broadens the definition of both senses somewhat. | https://en.wikipedia.org/wiki/Complement_(music) |
In music theory, contrapuntal motion is the general movement of two melodic lines with respect to each other. In traditional four-part harmony, it is important that lines maintain their independence, an effect which can be achieved by the judicious use of the four types of contrapuntal motion: parallel motion, similar motion, contrary motion, and oblique motion. | https://en.wikipedia.org/wiki/Contrapuntal_motion |
In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows (ordered sets). A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence ..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence." | https://en.wikipedia.org/wiki/Equivalent_set |
Traditionally, octave equivalency is assumed, while inversional, permutational, and transpositional equivalency may or may not be considered (sequences and modulations are techniques of the common practice period which are based on transpositional equivalency; similarity within difference; unity within variety/variety within unity). A definition of equivalence between two twelve-tone series that Schuijer describes as informal despite its air of mathematical precision, and that shows its writer considered equivalence and equality as synonymous: Two sets , P and P′ will be considered equivalent if and only if, for any pi,j of the first set and p′i′,j′ of the second set, for all is and js , if i=i′, then j=j′. (= denotes numeral equality in the ordinary sense). | https://en.wikipedia.org/wiki/Equivalent_set |
Forte (1963, p. 76) similarly uses equivalent to mean identical, "considering two subsets as equivalent when they consisted of the same elements. In such a case, mathematical set theory speaks of the 'equality,' not the 'equivalence,' of sets." | https://en.wikipedia.org/wiki/Equivalent_set |
However, equality may be considered identical (equivalent in all ways) and thus contrasted with equivalence and similarity (equivalent in one or more ways but not all). For example, the C major scale, G major scale, and the major scale in all keys, are not identical but share transpositional equivalence in that the size of the intervals between scale steps is identical while pitches are not (C major has F♮ while G major has F♯). The major third and the minor sixth are not identical but share inversional equivalence (an inverted M3 is a m6, an inverted m6 is a M3). A melody with the notes G A B C is not identical to a melody with the notes C B A G, but they share retrograde equivalence. | https://en.wikipedia.org/wiki/Equivalent_set |
In music theory, fake books and lead sheets aimed towards jazz and popular music, many tunes and songs are written in a key, and as such for all chords, a letter name and symbols are given for all triads (e.g., C, G7, Dm, etc.). In some fake books and lead sheets, all triads may be represented by upper case numerals, followed by a symbol to indicate if it is not a major chord (e.g. "m" for minor or "ø" for half-diminished or "7" for a seventh chord). An upper case numeral that is not followed by a symbol is understood as a major chord. The use of Roman numerals enables the rhythm section performers to play the song in any key requested by the bandleader or lead singer. | https://en.wikipedia.org/wiki/Roman_numeral_analysis |
The accompaniment performers translate the Roman numerals to the specific chords that would be used in a given key. In the key of E major, the diatonic chords are: Emaj7 becomes Imaj7 (also I∆7, or simply I) F♯m7 becomes IIm7 (also II−7, IImin7, IIm, or II−) G♯m7 becomes IIIm7 (also III−7, IIImin7, IIIm, or III−) Amaj7 becomes IVmaj7 (also IV∆7, or simply IV) B7 becomes V7 (or simply V; often V9 or V13 in a jazz context) C♯m7 becomes VIm7 (also VI−7, VImin7, VIm, or VI−) D♯ø7 becomes VIIø7 (also VIIm7b5, VII-7b5, or VIIø)In popular music and rock music, "borrowing" of chords from the parallel minor of a major key is commonly done. As such, in these genres, in the key of E major, chords such as D major (or ♭VII), G major (♭III) and C major (♭VI) are commonly used. | https://en.wikipedia.org/wiki/Roman_numeral_analysis |
These chords are all borrowed from the key of E minor. Similarly, in minor keys, chords from the parallel major may also be "borrowed". For example, in E minor, the diatonic chord built on the fourth scale degree is IVm, or A minor. | https://en.wikipedia.org/wiki/Roman_numeral_analysis |
However, in practice, many songs in E minor will use IV (A major), which is borrowed from the key of E major. Borrowing from the parallel major in a minor key, however, is much less common. Using the V7 or V chord (V dominant 7, or V major) is typical of most jazz and pop music regardless of whether the key is major or minor. Though the V chord is not diatonic to a minor scale, using it in a minor key is not usually considered "borrowing," given its prevalence in these styles. | https://en.wikipedia.org/wiki/Roman_numeral_analysis |
In music theory, harmonic rhythm, also known as harmonic tempo, is the rate at which the chords change (or progress) in a musical composition, in relation to the rate of notes. Thus a passage in common time with a stream of sixteenth notes and chord changes every measure has a slow harmonic rhythm and a fast surface or "musical" rhythm (16 notes per chord change), while a piece with a trickle of half notes and chord changes twice a measure has a fast harmonic rhythm and a slow surface rhythm (1 note per chord change). Harmonic rhythm may be described as strong or weak. | https://en.wikipedia.org/wiki/Harmonic_rhythm |
According to William Russo harmonic rhythm is, "the duration of each different chord...in a succession of chords." According to Joseph Swain (2002 p. 4) harmonic rhythm, "is simply that perception of rhythm that depends on changes in aspects of harmony." | https://en.wikipedia.org/wiki/Harmonic_rhythm |
According to Walter Piston (1944), "the rhythmic life contributed to music by means of the underlying changes of harmony. The pattern of the harmonic rhythm of a given piece of music, derived by noting the root changes as they occur, reveals important and distinctive features affecting the style and texture. "Strong harmonic rhythm is characterized by strong root progressions and emphasis of root positions, weak contrapuntal bass motion, strong rhythmic placement in the measure (especially downbeat), and relatively longer duration. | https://en.wikipedia.org/wiki/Harmonic_rhythm |
"The 'fastness' or 'slowness' of harmonic rhythm is not absolute, but relative," and thus analysts compare the overall pace of harmonic rhythm from one piece to another, or the amount of variation of harmonic rhythm within a piece. For example, a key stylistic difference between Baroque music and Classical-period music is that the latter exhibits much more variety of harmonic rhythm, even though the harmony itself is less complex. For example, the first prelude (BWV 846) from J. S. Bach's The Well-Tempered Clavier, illustrates a steady harmonic rhythm of one chord change per measure, although the melodic rhythm is much faster. | https://en.wikipedia.org/wiki/Harmonic_rhythm |
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name. | https://en.wikipedia.org/wiki/11-limit_interval |
In music theory, passus duriusculus is a Latin term which refers to chromatic line, often a bassline, whether descending or ascending. A line cliché is any chromatic line that moves against a stationary chord. There are many different types of line clichés—most often in the root, fifth or seventh—but there are two named line clichés. The major line cliché moves from the fifth of the chord to the sixth, then back to the fifth. | https://en.wikipedia.org/wiki/Chromatic_note |
Assuming the starting chord is the tonic, the simplest form of the major line cliché forms a I–I+–vi–I+ progression. The minor line cliché moves down from the root to the major seventh, to the minor seventh, and can continue until the fifth.From the late 16th century onward, chromaticism has come to symbolize intense emotional expression in music. Pierre Boulez (1986, p. | https://en.wikipedia.org/wiki/Chromatic_note |
254) speaks of a long established "dualism" in Western European harmonic language: "the diatonic on the one hand and the chromatic on the other as in the time of Monteverdi and Gesualdo whose madrigals provide many examples and employ virtually the same symbolism. The chromatic symbolizing darkness doubt and grief and the diatonic light, affirmation and joy—this imagery has hardly changed for three centuries." When an interviewer asked Igor Stravinsky (1959, p. | https://en.wikipedia.org/wiki/Chromatic_note |
243) if he really believed in an innate connection between "pathos" and chromaticism, the composer replied "Of course not; the association is entirely due to convention." Nevertheless the convention is a powerful one and the emotional associations evoked by chromaticism have endured and indeed strengthened over the years. To quote Cooke (1959, p. 54) "Ever since about 1850—since doubts have been cast, in intellectual circles, on the possibility, or even the desirability, of basing one's life on the concept of personal happiness—chromaticism has brought more and more painful tensions into our art-music, and finally eroded the major system and with it the whole system of tonality. "Examples of descending chromatic melodic lines that would seem to convey highly charged feeling can be found in: | https://en.wikipedia.org/wiki/Chromatic_note |
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships under consideration, the models may be multidimensional. Models of pitch space are often graphs, groups, lattices, or geometrical figures such as helixes. | https://en.wikipedia.org/wiki/Pitch_space |
Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class spaces, which represent relationships between pitch classes. (Some of these models are discussed in the entry on modulatory space, though readers should be advised that the term "modulatory space" is not a standard music-theoretical term.) Chordal spaces model relationships between chords. | https://en.wikipedia.org/wiki/Pitch_space |
In music theory, pitch-class space is the circular space representing all the notes (pitch classes) in a musical octave. In this space, there is no distinction between tones that are separated by an integral number of octaves. For example, C4, C5, and C6, though different pitches, are represented by the same point in pitch class space. Since pitch-class space is a circle, we return to our starting point by taking a series of steps in the same direction: beginning with C, we can move "upward" in pitch-class space, through the pitch classes C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, and B, returning finally to C. By contrast, pitch space is a linear space: the more steps we take in a single direction, the further we get from our starting point. | https://en.wikipedia.org/wiki/Pitch_class_space |
In music theory, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of post-tonal thinking. Twelve-tone technique orders the twelve notes of the chromatic scale, forming a row or series and providing a unifying basis for a composition's melody, harmony, structural progressions, and variations. | https://en.wikipedia.org/wiki/Music_Theory |
Other types of serialism also work with sets, collections of objects, but not necessarily with fixed-order series, and extend the technique to other musical dimensions (often called "parameters"), such as duration, dynamics, and timbre. The idea of serialism is also applied in various ways in the visual arts, design, and architecture"Integral serialism" or "total serialism" is the use of series for aspects such as duration, dynamics, and register as well as pitch. Other terms, used especially in Europe to distinguish post-World War II serial music from twelve-tone music and its American extensions, are "general serialism" and "multiple serialism".Musical set theory provides concepts for categorizing musical objects and describing their relationships. | https://en.wikipedia.org/wiki/Music_Theory |
Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte (1973), drawing on the work in twelve-tone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally tempered tuning system, and to some extent more generally than that.One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well. | https://en.wikipedia.org/wiki/Music_Theory |
In music theory, the bass note of a chord or sonority is the lowest note played or notated. If there are multiple voices it is the note played or notated in the lowest voice (the note furthest in the bass.) Three situations are possible: The bass note is the root or fundamental of the chord. The chord is in root position. | https://en.wikipedia.org/wiki/Bass_note |
One of the other pitches of the chord is in the bass. This makes it an inverted chord The bass note is not one of the notes in the chord. Such a bass note is an additional note, coloring the chord above it. | https://en.wikipedia.org/wiki/Bass_note |
The name of such a chord is also notated as a slash chord.In pre-tonal theory (Early music), root notes were not considered and thus the bass was the most defining note of a sonority. See: thoroughbass. In pandiatonic chords the bass often does not determine the chord, as is always the case with a nonharmonic bass. | https://en.wikipedia.org/wiki/Bass_note |
In music theory, the chromatic hexachord is the hexachord consisting of a consecutive six-note segment of the chromatic scale. It is the first hexachord as ordered by Forte number, and its complement is the chromatic hexachord at the tritone. For example, zero through five and six through eleven. On C: C, C♯, D, D♯, E, Fand F♯, G, G♯, A, A♯, B.This is the first of the six hexachords identified by Milton Babbitt as all-combinatorial source sets, a "source set" being "a set considered only in terms of the content of its hexachords, and whose combinatorial characteristics are independent of the ordering imposed on this content" (Babbitt 1955, 57). | https://en.wikipedia.org/wiki/Chromatic_hexachord |
In the larger context of thirty-five source hexachords catalogued by Donald Martino, it is designated "Type A" (Martino 1961, 229–30). Applying the circle of fifths transformation to the chromatic hexachord produces the diatonic hexachord (Babbitt 1987, 93). As with the diatonic scale, the chromatic hexachord is, "hierarchical in interval makeup," and may also be produced by, or contains, 3-1, 3-2, 3-3, 3-6, and 3-7 (Friedmann 1990, 111). | https://en.wikipedia.org/wiki/Chromatic_hexachord |
Serial compositions including Karlheinz Stockhausen's Kreuzspiel and Klavierstück I feature the chromatic hexachord in permuted orderings, as do certain pieces composed by Milton Babbitt, Alban Berg, Ernst Krenek, Luigi Nono, Karlheinz Stockhausen, Igor Stravinsky, and Anton Webern in various fixed-order derivations (twelve-tone rows and arrays). Babbitt's Second Quartet and Reflections for piano and tape feature the hexachord (Babbitt 1987, 93). | https://en.wikipedia.org/wiki/Chromatic_hexachord |
The retrograde-symmetrical all-interval series employed by Luigi Nono for the first time in Canti per tredeci in 1955, also used in his Il canto sospeso and nearly all subsequent works up to Composizione per orchestra n. 2: Diario polacco ’58 in 1959, is built from two chromatic hexachords (Nielinger 2006, 97–98). Stefan Wolpe's Suite in Hexachord (1936) begins with a chromatic hexachord on G, introducing the complementary hexachord in the final movement, while Elliott Carter calls his own piece, "Inner Song" for solo oboe—the second movement of Trilogy for oboe and harp (1992)—"some thoughts about Wolpe's hexachord" (Schiff 1998, 146). | https://en.wikipedia.org/wiki/Chromatic_hexachord |
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of diminished sixth to be treated as a fifth). If C is chosen as a starting point, the sequence is: C, G, D, A, E, B (=C♭), F♯ (=G♭), C♯ (=D♭), A♭, E♭, B♭, F. Continuing the pattern from F returns the sequence to its starting point of C. This order places the most closely related key signatures adjacent to one another. It is usually illustrated in the form of a circle. | https://en.wikipedia.org/wiki/Circle_of_fifths |
In music theory, the concept of root is the idea that a chord can be represented and named by one of its notes. It is linked to harmonic thinking—the idea that vertical aggregates of notes can form a single unit, a chord. It is in this sense that one speaks of a "C chord" or a "chord on C"—a chord built from C and of which the note (or pitch) C is the root. When a chord is referred to in Classical music or popular music without a reference to what type of chord it is (either major or minor, in most cases), it is assumed a major triad, which for C contains the notes C, E and G. The root need not be the bass note, the lowest note of the chord: the concept of root is linked to that of the inversion of chords, which is derived from the notion of invertible counterpoint. | https://en.wikipedia.org/wiki/Root_note |
In this concept, chords can be inverted while still retaining their root. In tertian harmonic theory, wherein chords can be considered stacks of third intervals (e.g. in common practice tonality), the root of a chord is the note on which the subsequent thirds are stacked. For instance, the root of a triad such as C Major is C, independently of the vertical order in which the three notes (C, E and G) are presented. | https://en.wikipedia.org/wiki/Root_note |
A triad can be in three possible positions, a "root position" with the root in the bass (i.e., with the root as the lowest note, thus C, E, G or C, G, E, from lowest to highest notes), a first inversion, e.g. E, C, G or E, G, C (i.e., with the note which is a third interval above the root, E, as the lowest note) and a second inversion, e.g. G, C, E or G, E, C, in which the note that is a fifth interval above the root (G ) is the lowest note. Regardless of whether a chord is in root position or in an inversion, the root remains the same in all three cases. Four-note seventh chords have four possible positions. That is, the chord can be played with the root as the bass note, the note a third above the root as the bass note (first inversion), the note a fifth above the root as the bass note (second inversion), or the note a seventh above the root as the bass note (third inversion). Five-note ninth chords know five positions, etc., but the root position always is that of the stack of thirds, and the root is the lowest note of this stack (see also Factor (chord)). | https://en.wikipedia.org/wiki/Root_note |
In music theory, the dominant seventh flat five chord is a seventh chord composed of a root note, together with a major third, a diminished fifth, and a minor seventh above the root (1, ♮3, ♭5 and ♭7). For example, the dominant seventh flat five chord built on C, commonly written as C7♭5, is composed of the pitches C–E–G♭–B♭: It can be represented by the integer notation {0, 4, 6, 10}. This chord is enharmonically equivalent to its own second inversion. That is, it has the same notes as the dominant seventh flat five chord a tritone away (although they may be spelled differently), so for instance, F♯7♭5 and C7♭5 are enharmonically equivalent. | https://en.wikipedia.org/wiki/Dominant_seventh_flat_five_chord |
Because of this property, it readily functions as a pivot chord. It is also frequently encountered in tritone substitutions. | https://en.wikipedia.org/wiki/Dominant_seventh_flat_five_chord |
In this sense, there are only six "unique" dominant seventh flat five chords. In diatonic harmony, the dominant seventh flat five chord does not naturally occur on any scale degree (as does, for example, the dominant seventh chord on the fifth scale degree of the major scale e.g. C7 in F major). In classical harmony, the chord is rarely seen spelled as a seventh chord and is instead most commonly found as the enharmonically equivalent French sixth chord. In jazz harmony, the dominant seventh flat five may be considered an altered chord, created by lowering the fifth of a dominant seventh chord, and may use the whole-tone scale, as may the augmented minor seventh chord, or the Lydian ♭7 mode, as well as most of the modes of the Neapolitan major scale, such as the major Locrian scale, the leading whole-tone scale, and the Lydian minor scale. | https://en.wikipedia.org/wiki/Dominant_seventh_flat_five_chord |
In music theory, the dominant triad is a major chord, symbolized by the Roman numeral "V" in the major scale. In the natural minor scale, the triad is a minor chord, denoted by "v". However, in a minor key, the seventh scale degree is often raised by a half step (♭ to ♮), creating a major chord. | https://en.wikipedia.org/wiki/Dominant_function |
These chords may also appear as seventh chords: typically as a dominant seventh chord, but occasionally in minor as a minor seventh chord v7 with passing function: As defined by the 19th century musicologist Joseph Fétis, the dominante was a seventh chord over the first note of a descending perfect fifth in the basse fondamentale or root progression, the common practice period dominant seventh he named the dominante tonique.Dominant chords are important to cadential progressions. In the strongest cadence, the authentic cadence (example shown below), the dominant chord is followed by the tonic chord. A cadence that ends with a dominant chord is called a half cadence or an "imperfect cadence". | https://en.wikipedia.org/wiki/Dominant_function |
In music theory, the double-diminished triad is an archaic concept and term referring to a triad, or three note chord, which, already being minor, has its root raised a semitone, making it "doubly diminished". However, this may be used as the derivation of the augmented sixth chord. For example, F–A♭–C is a minor triad, so F♯–A♭–C is a doubly diminished triad. | https://en.wikipedia.org/wiki/Augmented_sixth_chord |
This is enharmonically equivalent to G♭–A♭–C, an incomplete dominant seventh A♭ 7, missing its fifth), which is a tritone substitute that resolves to G. Its inversion, A♭–C–F♯, is the Italian sixth chord that resolves to G. Classical harmonic theory would notate the tritone substitute as an augmented sixth chord on ♭2. The augmented sixth chord can either be (i) an It+6 enharmonically equivalent to a dominant seventh chord (with a missing fifth); (ii) a Ger+6 equivalent to a dominant seventh chord with (with a fifth); or (iii) a Fr+6 equivalent to the Lydian dominant (with a missing fifth), all of which serve in a classical context as a substitute for the secondary dominant of V. All variants of augmented sixth chords are closely related to the applied dominant V7 of ♭II. Both Italian and German variants are enharmonically identical to dominant seventh chords. | https://en.wikipedia.org/wiki/Augmented_sixth_chord |
For example, in the key of C, the German sixth chord could be reinterpreted as the applied dominant of D♭. Simon Sechter explains the chord of the French sixth chord as being a chromatically altered version of a seventh chord on the second degree of the scale, . The German sixth is explained as a chromatically altered ninth chord on the same root but with the root omitted. The tendency of the interval of the augmented sixth to resolve outwards is therefore explained by the fact that the A♭, being a dissonant note, a diminished fifth above the root (D), and flatted, must fall, whilst the F♯ – being chromatically raised – must rise. | https://en.wikipedia.org/wiki/Augmented_sixth_chord |
In music theory, the flamenco mode (also Major-Phrygian) is a harmonized mode or scale abstracted from its use in flamenco music. In other words, it is the collection of pitches in ascending order accompanied by chords representing the pitches and chords used together in flamenco songs and pieces. The key signature is the same as that of the Phrygian mode (on E: no accidentals; on C: four flats), with the raised third and seventh being written in as necessary with accidentals. Its modal/tonal characteristics are prominent in the Andalusian cadence. | https://en.wikipedia.org/wiki/Flamenco_mode |
There are three fundamental elements which can help define whether or not something really is flameco: A flamenco mode -or musical tonality-; the compás -rhythm- and the performer...who should be a Flamenco! ...For example, if a composer writes a song using a flamenco key- usually called a mode- won't be writing a flamenco piece. Flamenco, which is a harmonic system of false relations constitutes, according to Manuel de Falla, "one of the marvels of natural art." | https://en.wikipedia.org/wiki/Flamenco_mode |
Only the flamenco guitar, de Falla noted, "can flexibly adjust itself to the ornate melodic embellishments of the flamenco mode." The exact chords depend on the song form (palo) and guitar chord positions since chord voicings in flamenco often include nontriadic pitches, especially open strings. It is characteristic that III, ♭II, and I appear as dissonant chords with a minimum of four tones (for example seventh chords or mixed third chord). | https://en.wikipedia.org/wiki/Flamenco_mode |
Since the tetrachord beginning on the tonic may ascend or descend with either G-sharp or natural (Phrygian tetrachord), the mixed-thirds clash between the major third degree (G♯) in the melody and the minor third degree (G♮) in the accompanying harmony occurs frequently and is characteristic of the flamenco esthetic, as with the blues scale on a major chord. This tetrachord may be copied in the second, producing a D♯ and allowing an augmented sixth chord on the second degree: B7♭5/F.Lou Harrison composed a "Sonata in Ishartum" (1974 or 1977), which has been arranged by Tolgahan Çoğulu (2001), part of his Suite. In early scholarship regarding a Babylonian cuneiform inscription tuning tablet from the eighteenth century BC, "Ishartum" was equated with the modern Phrygian, but now it isconsidered equivalent to the Ionian mode/major scale. Çoğulu's arrangement, at least, is the white note mode on E in Pythagorean tuning, as follows (): F-, C-, G-, D-, A, E, B (F♯+, C♯+, G♯+), or E (1/1), F- (256/243), G- (32/27), A (4/3), B (3/2), C- (128/81), D (16/9), E (2/1), with G♯+ being 81/64. | https://en.wikipedia.org/wiki/Flamenco_mode |
In music theory, the half-diminished seventh chord (also known as a half-diminished chord or a minor seventh flat five chord) is a seventh chord composed of a root note, together with a minor third, a diminished fifth, and a minor seventh (1, ♭3, ♭5, ♭7). For example, the half-diminished seventh chord built on B, commonly written as Bm7(♭5), or Bø7, has pitches B-D-F-A: It can be represented by the integer notation {0, 3, 6, 10}. The half-diminished seventh chord exists in root position and in three inversions. | https://en.wikipedia.org/wiki/Half_diminished_chord |
The first inversion shares identity with a chord on the minor sixth:In diatonic harmony, the half-diminished seventh chord occurs naturally on the seventh scale degree of any major scale (for example, Bø7 in C major) and is thus a leading-tone seventh chord in the major mode. Similarly, the chord also occurs on the second degree of any natural minor scale (e.g., Dø7 in C minor). It has been described as a "considerable instability". | https://en.wikipedia.org/wiki/Half_diminished_chord |
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