Iacobus Leodiensis [Iacobus de Montibus, Iacobus de Oudenaerde]

BIBLIOGRAPHY

H.L.F. Helmholtz: Die Lehre von den Tonempfindungen (Brunswick, 1863, 4/1877; Eng. trans., 1875, 2/1885/R, as On the Sensations of Tone)

A. Daniélou: Tableau comparatif des intervalles musicaux (Paris, 1958)

H.-P. Reinecke: Cents Frequenz Periode: Umrechnungstabellen für musikalische Akustik und Musikethnologie (Berlin, 1970) [also Eng. text]

F. Lieberman: ‘Working with Cents: a Survey’, EthM, xv (1971), 236–412

R. Plomp: Aspects of Tone Sensation (London, 1976)

E.M. Burns and W.D. Ward: ‘Intervals, Scales and Tuning’, The Psychology of Music, ed. D. Deutsch (New York, 1982, 2/1999), 241–69

MARK LINDLEY/MURRAY CAMPBELL, CLIVE GREATED

Interval class.

An understanding of interval whereby the order of the notes is left unspecified. If the two notes, each of which exemplifies a Pitch class, are ordered (for example, if one comes after the other), the ‘ordered pitch-class interval’ between them is obtained by subtracting the first pitch-class number from the second, modulo 12: this ordered pitch-class interval is 10 from E to C (1 minus 3) but 2 from C to E (3 minus 1). Where the two notes are considered as a pitch-class Set, without order, the 12 possible ordered pitch-class intervals reduce to seven interval classes, usually numbered from 0 to 6, so that each class is denoted by the smallest ordered pitch-class interval that belongs to it. Thus the set {C, E} exemplifies interval class 2.

JOHN ROEDER

Intonation (i).

In plainchant, the initial melodic phrase usually sung solo by the priest or cantor before the other voices enter. In particular, the term is used for the phrases ‘Gloria in excelsis Deo’ and ‘Credo in unum Deum’ at the beginnings of the second and third sections of the Ordinary of the Mass, which are sung by the priest, and to the notes preceding the reciting note in the Gregorian tones. When the latter are used for the psalms, the intonations are sung only for the first verse (which is sung by the cantor alone, the choir joining in after the colon). Subsequent verses omit the intonation and start on the reciting note. For the Magnificat and Benedictus, the intonations are slightly more elaborate and are used in each verse. Nowadays the intonation is often first played quietly on the organ to give the correct pitch to the singer. SeeInflection, and Psalm, §II.

GUY OLDHAM

Intonation (ii).

The treatment of musical pitch in performance. It is usually thought of as the acoustical and artistic accuracy of pitch in singing or playing, but it has an indispensable role in musical expression through the deliberate inflection of pitch to shade and colour melody, to create excitement or tension, or as a means of characterizing a particular repertory or style of performance. In contrasting the intonation of the violin virtuosos Joachim and Sarasate (quite different, as one may hear on recordings), Bernard Shaw (1893) could have been describing the intonational art of much of the world’s music: ‘[T]he modes in which we express ourselves musically …, though in theory series of sounds bearing a fixed pitch relation to one another, are in practice tempered by every musician just as the proportions of the human figure are tempered by a sculptor’. The sounding of a given series of musical intervals is subject to the laws of acoustics, but undergoes also an elusive, more or less subtle shaping (Shaw’s ‘tempering’) not unlike the shaping of precise rhythmic values by tempo rubato.

The perception of intonation is influenced by a number of often inseparable factors besides the most obvious one, the relative pitch of a musical note in context. These include ambient acoustics, loudness and tempo; timbre plays a particularly critical role (see Sethares), but perhaps the strongest influence is the ear’s expectation, which is created by prevailing, culturally determined intonational norms.

The intonational basis of Western art music has undergone two drastic structural shifts since the time of the earliest polyphonic music, the intervals of which were constructed from the pure 5ths (ratio 3:2) and octaves (2:1) of Pythagorean intonation. The admission of the pure major 3rd (5:4), in use in the British Isles by the 13th century, was momentous, and the sensuousness of its sonority should certainly be considered a humanistic attribute (medieval music theory permitted only musical intervals generated from the trinity of the first three integers, proscribing five, which generates the major 3rd, in part because in ancient numerologies – particularly that of Plato – five was the first ‘human’ number). The 5:4 major 3rd opened the way, through a transitional period (roughly the 14th century), for the triadic sonorities of Just intonation and its keyboard surrogate, Mean-tone temperament, which together dominated intonational theory and practice until the mid-19th century. The next shift, which began in the early 18th century, was from mean-tone intervals through various irregular keyboard temperaments to 12-tone equal temperament (see Temperaments, §§6–8); the change was due largely to the increasing dominance of keyboard instruments with their limited number of fixed pitches. In the 20th century intonation was dominated by the conservatory standard of Pythagorean-shaded 12-equal, but after about 1950 became increasingly inclusive and reformist, under the influence of non-Western music and, especially, of the early-music movement, which brought about a return to just and mean-tone intonation, along with a growing understanding that no one intonational basis fits all Western music.

Melodic and harmonic considerations pull intonation in opposite directions: in melody, towards the brightness of Pythagorean tuning, with its small diatonic semitones, high sharps and low flats; and in harmony, towards the just tuning of vertical sonorities, giving wide diatonic semitones, low sharps and high flats in melody. Zarlino (1558, p.163) was one of the few theorists to write about the practical intonation of his time: ‘Voices … seek the perfection of intervals … [and] can tune intervals higher or lower as desired and through this bring to perfection any composition’. Singers accustomed to tuning pure harmonic triads usually make the quite small adjustments that are necessary to reconcile the conflicting intonational demands of 3rd, 5th and octave. These are not mechanical nor even necessarily rational adjustments, but require intonational flexibility (lucidly described by Lloyd, Grove5); the intervals of a Palestrina motet sung in tune cannot be captured in a table of interval ratios. Roger North's observation of 1726 still rings true:

if the sounding part [of music] had bin left to the Voice, which conformes to all truth of accords, whereof the ear is judge, there never had bin any suspicion of such majors, minors, dieses, commas, and I know not what imaginary devisions of tones, as some clumsye mechanick devices called Instruments have given occasion to speculate.

These devices were nevertheless called upon to replicate the intonational beauty of vocal music, as witnessed by the various experiments with vastly expanded keyboards, such as Vicentino's 36-note-per-octave arcicembalo of ?1555 (see Enharmonic keyboard). Even with only 12 to 14 keys per octave keyboard instruments were able to reproduce in mean-tone much of the colour and sonority of just tuning, if not the modulatory range of, say, the vocal works of Marenzio or of Gesualdo, which carry the implications of the intervals of the pure triad to the very limits of the ear's comprehension. Plucked fretted strings (lute, vihuela, guitar, etc.) have always been able to simulate the sonorities of mean-tone intervals quite well, having a certain degree of intonational latitude even with non-adjustable frets. Guitarists regularly make slight tuning adjustments in order to improve the intonation of one key or another.

By 1700 the influence of keyboard instruments was such that their fixed pitches began to replace the voice and the monochord as intonational arbiter. Traditionalists maintained with Quantz (1752: XVII.vi.20) that it was the melodic instruments that gave the intervals ‘in their true ratios’ and recommended that tempered keyboard instruments defer in performance in order to avoid clashes. Others, pre-eminently Marpurg and Sorge, insisted that all intonation should conform to that of the keyboard. Theoretical treatises and methods for instruments and voice show, nonetheless, that just/mean-tone continued through 18th century as the standard of intonation; the main evidence is the persistence of the traditional distinction between the diatonic and chromatic semitones (see Chesnut, 1977; Barbieri, 1991; Haynes, 1991). In contrast to this established practice, a return to Pythagorean intonation began toward 1800, introduced first by virtuoso violinists for the sake of a more brilliant sound.

The rise of musical amateurism among the middle class in the 18th and 19th centuries also had considerable influence on intonational practice. Some violin tutors, including those of Geminiani (1751) and Spohr (1832), suggested that amateur players need not distinguish between the diatonic and chromatic semitones. The traditional modes were consolidated into the simplified system of tonality, with its 24 major and minor keys. The motive force in tonality and its linear formal strategies was above all the dynamic effect of the dominant triad. The edgy, unstable 3rds and 6ths of 12-equal impart energy to the dominant triad through the ‘leading note’ (a tendentious term inappropriate for the sub-semitone in general), which does not lead so much as it is pushed by its irritable harmonic relation to the fifth degree of the scale. What is intonationally useful for the dominant function is, however, a liability for the major tonic triad, in which a semblance of repose is called for. The harshness of bare triads can be mitigated considerably by the addition of 6ths, 7ths, 9ths and so on, creating sonorities that became familiar in 20th-century music, for example in popular and jazz piano styles. The intonational defects of 12-equal can also be concealed by changes of timbre, adjustments of dynamic levels or the use of vibrato. Though known since antiquity, 12-equal became a musical reality only with the commercial success of the metal-frame piano, and with the institution of universal standardized measurement necessitated by 19th-century industrial technology. Many 19th- and 20th-century musicians and theorists, from Helmholtz and Stanford to Hindemith and Kodály, rejected 12-equal as musically deficient, criticizing its inflexibility and lack of intervallic variety, as well as the ‘clouding’ effect of its clashing harmonics and spurious difference-tones; just intervals were the intonational basis of John Curwen’s widely successful Tonic Sol-fa system for teaching choral singing. As absolute values, however, the intervals of 12-equal gained a pre-eminent position in 20th-century music theory; Schoenberg wrote that he ‘always requested tempered intonation’, and defined the semitone as precisely half a tone, ‘without any relationship to harmonic questions’.

Although unrecognized in mainstream music theory, intervals derived from the 7th and 11th harmonics are quite commonly encountered in Western music (see Harmonics, Table 1). It is not unusual for a minor 3rd (most often from the third to the fifth degree of the major scale) to be divided into two roughly equal three-quarter tones, which closely approximate the intervals between the 10th, 11th and 12th harmonics. This division occurs in the Highland bagpipe scale (Bagpipe, §3 (i)), in the vocal and Hardanger fiddle tradition of Norway, in the so-called Alphorn-fa note in some yodelling styles (see Yodel, §3) in some Irish traditional music practice, and in the music of parts of southeastern Europe (see also Blue note). The 7th harmonic (7/4) occurs as a considerably flattened minor 7th added to a major triad in a cappella barbershop-style close harmony and jazz-gospel vocal ensembles. The intonational characteristics of particular performance styles – including those whose music is notated – are always learned and transmitted aurally. The slight mistunings of unisons, whether intended or unavoidable, is often conisdered a musical asset, an enrichment or enlivening of the sound. This added vibrancy, which is a calculated effect in a few organ stops (piffaro, unda maris, for example), occurs naturally in a section of stringed instruments or voices.

BIBLIOGRAPHY

Grove5 (‘Just Intonation’; Ll.S. Lloyd)

H. von Helmholtz: Die Lehre von den Tonempfindung (Brunswick, 1863, 6/1913/R: Eng. trans., 1875, as On the Sensations of Tone, 2/1885/R) [Eng. trans. includes addl notes and an appx]

W. Pole: The Philosophy of Music (London, 1879, rev. 6/1924 by H. Hartridge), 76–88, 128–59

C.V. Stanford: Musical Composition (London, 1911/R), 13–18

O. Ševčík: School of Intonation on an Harmonic Basis for Violin: a Short Treatise for Students, op.11 (New York, 1922) [in Eng., Fr., Ger.]

G.B. Shaw: Music in London 1890–94 (London, 1932/R), ii, 276–7

P. Hindemith: A Composer's World (Cambridge, MA, 1952), 81–92

Z. Kodály: Let us Sing Correctly (London, 1952) [Eng. trans. of Énekeljünk tisztán (1941); 107 intonation exercises]

J. Rufer: Das Werk Arnold Schönbergs (Kassel, 1959, 2/1974); Eng. trans. as The Works of Arnold Schoenberg (London, 1962), 143

J. Wilson, ed.: Roger North on Music (London, 1959), 204–5

M. Kolinski: ‘Gestalt Hearing of Musical Intervals’, The Commonwealth of Music, in Honor of Curt Sachs, ed. G. Reese and R. Brandel (New York, 1965), 368–74

M. Vogel: Die Lehre von den Tonbeziehungen (Bonn, 1975; Eng. trans., 1993, as On the Relations of Tone)

A.H. Benade: Fundamentals of Musical Acoustics (New York, 1976/R), 274–7, 296–300

J.H. Chesnut: ‘Mozart's Teaching of Intonation’, JAMS, xxx (1977), 254–71

Ll.S. Lloyd and H. Boyle: Intervals, Scales and Temperaments (London, 2/1978)

J.C. Leuba: A Study of Musical Intonation (Seattle, 1980)

J. Herlinger: ‘Fractional Divisions of the Whole Tone’, Music Theory Spectrum, iii (1981), 74–83

T.H. Podnos: Intonation for Strings, Winds, and Singers (Metuchen, NJ, 1981)

R. Stowell: Violin Technique and Performance Practice in the Late Eighteenth and Early Nineteenth Centuries (Cambridge, 1985), 245–56

D. Leedy: ‘A Question of Intonation’, Journal of the Conductors' Guild, viii (1987), 107–20

P. Barbieri: ‘Violin Intonation: a Historical Survey’, EMc, xix (1991), 69–88

B. Haynes: ‘Beyond Temperament: Non-Keyboard Intonation in the 17th and 18th Centuries’, EMc, xix (1991), 356–81

R. Covey-Crump: ‘Pythagoras at the Forge: Tuning in Early Music’, Companion to Medieval and Renaissance Music, ed. T. Knighton and D. Fallows (London, 1992), 317–26

B. Gratzki: Die reine Intonation im Chorgesang (Bonn, 1993)

M. Lindley and R. Turner-Smith: Mathematical Models of Musical Scales: a New Approach (Bonn, 1993), 134–242

M. Perlman: ‘American Gamelan in the Garden on Eden: Intonation in a Cross-Cultural Encounter’, MQ, lxxviii (1994), 510–55

B. Suchoff, ed.: Béla Bartók Studies in Ethnomusicology (Lincoln, NE, 1997)

W.A. Sethares: Tuning, Timbre, Spectrum, Scale (London, 1998)

DOUGLAS LEEDY (with BRUCE HAYNES)