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Intervals may occur two ways:
An '' Interval Class '' is an interval measured by the shortest distance possible between its two Pitch Class es. FREQUENCY RATIOS In Just Intonation intervals are commonly labelled according to the Ratio of Frequencies of the two pitches. Important intervals are those using the lowest integers, such as 1/1, 2/1, 3/2, etc. This system is frequently used to describe intervals in non-Western music. This method is also often used in theoretical explanations of equal-tempered intervals used in European tonal music which explain their use through their approximation of just intervals. INTERVAL NUMBER AND QUALITY In Diatonic or tonal theory intervals are labelled according to their Diatonic Function and according to the number of members or Degrees they span in a Diatonic Scale . The ''interval number'' of a note from a given Tonic note is the number of Staff Position s enclosed within the interval, as shown at right. Intervals larger than an octave are called ''compound'' intervals; for example, a tenth is known as a compound third. Intervals larger than a thirteenth are rarely spoken of, since going above this by stacking thirds would result in a double octave (but see 8va for use of ''15ma''). The name of any interval is further qualified using the terms Perfect , Major , Minor , Augmented , and Diminished . This is called its ''interval quality''.
It is possible to have doubly-diminished and doubly-augmented intervals, but these are quite rare. Shorthand notation Intervals are often abbreviated with a P for perfect, '''m''' for minor, '''M''' for major, '''d''' for diminished, '''A''' for augmented, followed by the diatonic interval number. The indication M and P are often omitted. The octave is P8, and a unison is usually referred to simply as "a unison" but can be labeled P1. The Tritone , an augmented fourth or diminished fifth is often '''π''' or '''TT'''. Examples:
For use in describing Chords , the sign + is used for augmented and '''−''' for diminished. Furthermore the 3 for the third is often omitted, and for the seventh, the plain form stands for the minor interval, while the major is indicate by '''maj'''. So for example:
Enharmonic intervals Two intervals are considered to be Enharmonic if they both contain the same Pitch es spelled in different ways; that is, if the notes in the two intervals are enharmonic with one another. Enharmonic intervals contain the same number of Semitone s. For example C#-D#, a Major Second , and C#-Eb, a Diminished Third , are enharmonic. Steps and skips Linear (melodic) intervals may be described as ''steps'' or ''skips'' in a diatonic context. Steps are linear intervals between consecutive , an interval comprising two Semitone s, is still considered a skip. The words ''conjunct'' and ''disjunct'' refer to melodies composed of steps and skips, respectively. PITCH CLASS INTERVALS Post-tonal or Atonal theory, originally developed for equal tempered European classical music written using the Twelve Tone Technique or Serialism , Integer Notation is often used, most prominently in Musical Set Theory . In this system intervals are named according to the number of half steps, from 0 to 11, the largest interval class being 6. Ordered and unordered pitch and pitch class intervals In atonal or Musical Set Theory there are numerous types of intervals, the first being Ordered Pitch Interval , the distance between two pitches upward or downward. For instance, the interval from C to G upward is 7, but the interval from G to C downward is −7. One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, somewhat similar to the interval of tonal theory. The interval between pitch classes may be measured with ordered and unordered pitch class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. For unordered pitch class interval see Interval Class . GENERIC AND SPECIFIC INTERVALS In Diatonic Set Theory , Specific and Generic Interval s are distinguished. Specific intervals are the interval class or number of semitones between scale degrees or collection members, and generic intervals are the number of scale steps between notes of a collection or scale. CENTS The standard system for comparing intervals of different sizes is with Cents . This is a Logarithmic Scale in which the octave is divided into 1200 equal parts. In Equal Temperament , each Semitone is exactly 100 cents. The value in cents for the interval ''f''1 to ''f''2 is 1200×log2(''f''2/''f''1). COMPARISON OF DIFFERENT INTERVAL NAMING SYSTEMS It is possible to construct just intervals which are closer to the equal-tempered equivalents, but most of the ones listed above have been used historically in equivalent contexts. In particular the tritone (augmented fourth or diminished fifth), could have other ratios; 17:12 (603 cents) is fairly common. The 7:4 interval (the used in Jazz . The diatonic intervals, as well, have other Enharmonic equivalents, such as Augmented Second for Minor Third . CONSONANT AND DISSONANT INTERVALS Consonance And Dissonance are relative terms referring to the stability, or state of repose, of particular musical effects. Dissonant intervals would be those which cause tension and desire to be ''resolved'' to consonant intervals. These terms are relative to the usage of different compositional styles.
All of the above analyses refer to vertical (simultaneous) intervals. INVERSION An interval may be '' Inverted '', by raising the lower pitch an octave, or lowering the upper pitch an octave (though it is less usual to speak of inverting unisons or octaves). For example, the fourth between a lower C and a higher F may be inverted to make a fifth, with a lower F and a higher C. Here are the ways to identify interval inversions:
:A full example: E flat below and C natural above make a ''major sixth''. By the two rules just given, C natural below and E flat above must make a ''minor third''.
Interval roots Although intervals are usually designated in relation to their lower note, David Cope and Hindemith both suggest the concept of interval root. To determine an interval's root, one locates its nearest approximation in the harmonic series. The root of a perfect fourth, then, is its ''top'' note because it is an octave of the fundamental in the hypothetical harmonic series. The bottom note of every odd diatonically numbered intervals are the roots, as are the tops of all even numbered intervals. The root of a collection of intervals or a chord is thus determined by the interval root of its strongest interval. As to its usefulness, Cope provides the example of the final tonic chord of some popular music being traditionally analyzable as a "submediant six-five chord" ( Added Sixth Chord s by popular terminology), or a first inversion seventh chord (possibly the dominant of the mediant V/iii). According the interval root of the strongest interval of the chord (in first inversion, CEGA), the perfect fifth (C-G), is the bottom C, the tonic. INTERVAL CYCLES Interval Cycle s, "unfold a single recurrent interval in a series that closes with a return to the initial pitch class", and are notated by George Perle using the letter "C", for cycle, with an interval class integer to distinguish the interval. Thus the diminished seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0-11 to indicate the lowest pitch class in the cycle. (Perle 1990, p.21) OTHER INTERVALS There are also a number of intervals not found in the chromatic scale or labeled with a diatonic function which have names of their own. Many of these intervals describe small discrepancies between notes tuned according to the tuning systems used. Most of the following intervals may be described as Microtone s.
See List Of Musical Intervals for more. See Musical Interval Mnemonics for popular musical fragments that feature common intervals SOURCES
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