Interval

Basic Information

The name of the particular interval corresponds with a number of steps of a scale between its elements. Intervals from unison to an octave (simple intervals), as well as their size in semitones, are shown in Table 1.

Table 1 Basic intervals

Intervals larger than an octave are called compounds (compound 2nd would be an octave and 2nd, compound 3rd an octave and 3rd, and so on). However, the first four of them have their own names: 9th, 10th, 11th, and 12th.

Unison, 4th, 5th, and octave are perfect intervals. Others (2nd, 3rd, 6th, and 7th) are imperfect and called minor or major, depending on the number of semitones. The above-mentioned division is associated with the notion of consonance: perfect intervals were always considered as consonants, while the perception of the others has changed with time.

Musical intervals may also be diminished or augmented, that is, reduced or increased by a semitone (doubly diminished and doubly augmented intervals are also possible, although [Page 634]rather rare). What seems particularly significant is augmented 4th or diminished 5th, a strong dissonance called triton because of the three whole tones it embraces. As diabolus in musica (devil in music) it was banned in medieval church music.

Further modifications of intervals include inversion and enharmonic exchange. Inversion consists in reversing the position of the interval elements. As a consequence of this operation, an initial interval together with its inversion always give an octave (e.g., 5th is the inversion of 4th). Moreover, minor intervals become major ones and vice versa; diminished become augmented and vice versa (e.g., minor 3rd and major 6th). Although some intervals may sound the same, in tonal music their notation, names, and meaning are different (for example, augmented 2nd C-D# and minor 3rd C-E♭). Such equivalences with different tonal implications are called enharmonic intervals.

Every interval can be expressed by frequency ratio, that is, the quotient of frequencies of two tones of the interval, where frequency of the tone is defined as the number of its vibrations per second. However, frequency ratios for the same intervals can be different depending on the tuning system: Pythagorean, just, and equal temperament (currently the most popular). For example, in case of just intonation, based on the natural harmonics, these ratios are quite simple (natural frequency ratio is shown in Table 1). According to the well-known rule, the simpler the ratio is, the more pleasant the interval will be.

History

It is believed that Pythagoras (6th century B.C.E.) had obtained an octave, a 5th, and a 4th by dividing the string of a monochord in the ratio of 2:1, 3:2, and 4:3, respectively. The Pythagorean explanation of musical phenomena in terms of numerical relationships provided the basis for the theory of the harmony of the spheres, according to which celestial bodies produced intervals, mostly consonant, inaudible to the human ear. Medieval music was still based on this theory, although it allowed some sharp dissonances. In the baroque period, intervals were important for musical rhetoric (e.g., melodic minor 6th upward could create a figure called exclamatio, symbolizing exclamation or pleading). Since the evolution of the tonal system in the 17th century, intervals have been perceived in the context of major-minor tonality. In post-tonal music theory harmonic references have been loosened. Therefore, traditional interval names are no longer necessary; intervals are labeled according to the number of semitones they contain: unison is 0, minor 2nd is 1, etc.

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