Iacobus Leodiensis [Iacobus de Montibus, Iacobus de Oudenaerde]

Interval.

The distance between two pitches. The term ‘harmonic interval’ (as opposed to ‘melodic interval’) indicates that they are thought of as being heard simultaneously. Intervals are traditionally labelled according to the number of steps they embrace in a diatonic scale, counted inclusively: thus from C up to D or down to B is a 2nd, another step up to E or down to A makes a 3rd, etc. These names are applied in non-diatonic contexts so that an interval embracing five degrees of a pentatonic scale is still called an octave (from Lat. octavus: ‘eighth’) and an interval in 12-note music embracing six degrees of the chromatic scale is called a 4th. Qualifying adjectives as shown in ex.1 lend precision to this terminology but the terms ‘major’ and ‘minor’ (or ‘greater’ and ‘lesser’) are also sometimes used to distinguish slightly different forms of semitone, whole tone etc. Some music employs intervals not included in ex.1 because they are not represented faithfully by standard Western notation; 3rds intermediate in size between major and minor, for instance, are characteristic of scales used in some varieties of European folk music, in the art music of several Islamic countries etc.; melodically such an interval is likely to consist of a whole tone plus a step intermediate in size between a whole tone and semitone.

Apart from their musical context many intervals can be identified by their correspondence to a simple ratio of sound wavelengths or frequencies. (Before the development of wave theory in the 18th century, monochord string lengths provided, in effect, a means of discovering wavelength ratios.) An octave is identified by the ratio 2:1, meaning that the frequency of the upper note is twice that of the lower note. Notes an octave apart tend to sound so identical that they are said to belong to the same ‘pitch class’: they bear the same name (e.g. C, B, G); two intervals which together make up an octave are said to be ‘inversions’ of each other (for example, the minor 3rd is the inversion of the major 6th); and intervals an octave larger than those in ex.1 are considered their ‘compounds’.

The uniquely important role played by the octave interval can be partly explained by recalling that a single musical note usually contains a harmonically related set of frequency components. The note A sung by a bass, for example, will contain components with frequencies 110, 220, 330, 440, 550, 660 Hz etc. If the singer leaps to a, an octave higher, the new note will have components 220, 440, 660 Hz etc.; the frequency of each component will have doubled. It can be seen that the jump of an octave has in a sense introduced nothing fundamentally new: each component of the new note was already present in the original note.

When two notes an octave apart are sounded simultaneously, the harmonic components of the upper note coincide with the even harmonic components of the lower note, and the two sounds can blend to give the perception of a single pitch. If the frequency ratio is not exactly 2:1, the coincidence will be imperfect, giving rise to the amplitude fluctuations described as Beats. For example, if the octave A–a is ‘stretched’ to a ratio of 2·01:1, keeping the frequency of the first component of A at 110 Hz, the frequency of the first component of a will become 221·1 Hz. This will beat against the second component of A at 220 Hz, giving an audible amplitude fluctuation with a frequency of 1·1 Hz (just over one beat per second). The elimination of beats is used by organ tuners to establish true octaves.

Although medieval theorists emphasized the perfection of small whole number ratios, it is not in fact the case that musical octaves always correspond to an exact 2:1 ratio. Well-tuned pianos usually exhibit a degree of octave stretching, arising at least in part from the fact that the frequency components of a piano note are not strictly harmonic (see Inharmonicity). It is also well established that when asked to judge octave intervals in a melodic rather than a harmonic context, musicians tend to favour a frequency ratio slightly greater than 2:1.

The 19th-century acoustician Helmholtz explained the relative dissonance of musical intervals in terms of the extent of the beating between the two corresponding sets of harmonic frequency components when the notes are heard simultaneously. If there are many coincident frequency components the beating is reduced and the interval sounds relatively smooth. Thus a perfect fifth, with a frequency ratio of 3:2, is a smooth interval with low beating since the third harmonic of the lower note coincides with the second harmonic of the upper; the perfect fourth, with a frequency ratio of 4:3, is also relatively smooth since the fourth harmonic of the lower note coincides with the third harmonic of the upper note.

While the ratios 3:2 and 4:3 have been regarded almost unanimously as the ideal paradigms of a perfect 5th and 4th, there is no single ideal for major or minor 2nds, which in Western music derive their identity melodically far more than harmonically. Medieval theorists who favoured exclusively 9:8 (3:2 ÷ 4:3) for the whole tone were obliged to uphold 81:64 (9:8 x 9:8) and 27:16 as the ratios of a proper major 3rd (‘ditonus’) and 6th, and it was only in the 16th century that the simpler 5:4 and 5:3 ratios became the standard European theoretical ideal. The change corresponded, albeit belatedly, to an earlier change in the practical status of 3rds and 6ths as consonant intervals (see Harmony and Consonance, §1). Elaborate codifications of intervals have been developed on the basis of ratios involving numbers even larger than 81 or prime numbers larger than 5, a vulnerable aspect of such theories often being their relation to practice. The ‘blue 7th’ of jazz, however, sometimes thought of as representing the ratio 7:4, is indeed characteristically produced on the cornet or trumpet by overblowing to the seventh natural note. And no doubt the harmonic distinction between 9:8 and 10:9 may be featured in certain kinds of music, such as in India, in which the scale is set out against the background of a drone.

The conventional scientific unit of measure for intervals, devised by A.J. Ellis in about 1880, is derived by dividing the 2:1 octave into a theoretical microscale of 1200 cents (100 cents = an equal-tempered semitone). Other such units are the millioctave (1/1000 of an octave) and the savart (named after Félix Savart, 1791–1841, but virtually identical with the ‘eptameride’ – 1/301 of an octave – proposed by Joseph Sauveur in 1701). The number of savarts in an interval of known frequency ratio is equal to 1000 times the logarithm (to the base 10) of the ratio; the number of cents is equal to the logarithm of the ratio multiplied by 1200/log2, or 3986·3. One savart is thus approximately 4 cents.

A tendency to divide the octave, 5th or major 3rd into equal quantities can be discerned in many kinds of music. The slendro and pelog scales of the Indonesian gamelan sometimes approximate to a division of the octave into five or seven equal parts respectively. The ‘neutral 3rds’ alluded to above divide the 5th virtually in half. The mean-tone temperaments of Renaissance and Baroque keyboard music divided the major 3rd into two equal whole tones, while equal temperament divides the octave into equal semitones. To calculate the frequency ratios of such intervals (if they are to be precisely equal) involves roots of prime numbers and hence irrational numbers.

Intervals too small to be used melodically are encountered by theoreticians, experimenters and tuners of instruments with fixed pitches (such as keyboard instruments). The terms ‘diesis’ and ‘subsemitone’ have been used for the difference between enharmonic pairs (such as D–E) in any mean-tone temperament, an interval which, like the diesis used melodically in the ancient Greek enharmonic genre, is distinctly smaller than a semitone. Finer intervals of discrepancy are called ‘commas’ (one traditional rule of thumb being that a comma is one ninth of a whole tone) or ‘schisma’ (the most important schisma amounting theoretically to 1·95 cents, i.e. about one hundredth of a whole tone). Many Hindu theorists have considered the octave to be divided into 22 śruti usually deemed not to be of uniform size but all smaller than a semitone (see India, §III, 1(ii)(a)).

Table 1 gives the size in cents of intervals smaller than a tritone, according to various systems of intonation that have been proposed for Western music since the Middle Ages. In each column the various forms of the interval in question are listed in order of their size, decreasing for 4ths, minor 3rds and diatonic semitones and increasing for major 3rds, whole tones and chromatic semitones. Where a certain form of semitone has been used freely by musicians both diatonically and chromatically, it is listed here as a diatonic semitone. The intervals of each scheme make a single row across the table, except in three cases: just intonation, Pythagorean intonation, and the division of the octave into 53 equal parts. Intervals of just intonation are identified by the ratios 4:3, 5:4, 6:5, 9:8, 10:9, 16:15 and 25:24. In addition to the most characteristic intervals of Pythagorean intonation, the use of the Pythagorean apotomē as a diatonic semitone in some 15th-century keyboard music enabled major and minor 3rds impure by merely a schisma to be employed as well; thus the major 3rd labelled ‘pure–schisma’ occurred in Pythagorean intonation as the compound of eight pure 5ths and 4ths (e.g. E–G derived from the chain E–A–D–G–C–F–B–E–A; see Pythagorean intonation). The 53-part octave division contains very close approximations to justly intoned as well as Pythagorean intervals. In the table, equal temperament is labelled ‘12-part division’; the system most commonly known as mean-tone temperament is labelled ‘1/4-comma temperament’.

TABLE 1

Systems

4ths

Major 3rds

Minor 3rds

Whole tones

Diatonic semitones

Chromatic semitones

pure +

317.60

schisma

14/53

316.98

8/53

181.13

octave

octave

pure 10:9

182.40

19-part

8/19

505.26

6/19

378.95

5/19

315.79

3/19

189.47

2/19

126.32

1/19

63.16

division

octave

octave

octave

octave

octave

octave

pure 6.5

315.64

2/7-comma

pure +

504.19

pure –

383.24

pure –

312.57

10.9 +

191:62

16:15 +

120.95

25:24

70.67

temperament

2/7 comma

1/7 comma

1/7 comma

3/7 comma

3/7 comma

pure –

384.36

schisma