2. 9th-century theory.
The concordant intervals were an essential element in early polyphony. This is clear from the way in which polyphonic treatment was at first frequently discussed under headings such as ‘De symphoniis’. The technique of polyphony was based on two practices, both of them probably very old: that of parallel singing in concordant intervals, and that of the use of a drone. At any rate, regular instruction in the form of a systematic course of teaching had become a necessity at just about the time when performance in pure and unornamented parallel motion, and performance using a drone, were on the decline. This decline took the form of a combination of the two practices. In it the new elements of polyphony since the 9th century had their origin – and not only in the decline itself, but also in the way in which musical theorists presented and accounted for it. For only now could the vox organalis be thought of as an increasingly independent moving ‘voice’; only now did alternative ways of singing polyphonically become thinkable and feasible; only now could different ways of shaping a counterpoint be tried out in practice and formulated as theory, and hence the ‘history’ of polyphony really begin. And it was a history at whose beginning there was very little by way of prescriptive theoretical writing. Nevertheless, by about 1100 a measure of freedom had been achieved in the fashioning of counterpoint. Music theory could do little more than give a general idea of this, in the form of contrived examples; and in turn, from the 13th century onwards there are specific references to individual compositions which the reader is expected to have in his mind as models. These, apart from elementary rules of part-writing, take the place of the examples and now serve to stimulate or to corroborate what has been said.
In the organum of the early Middle Ages the vox organalis generally lies beneath the vox principalis (ex.1; see also Score, fig.1). The latter, as the melody pre-existing in its own right, placed as it is in the prominent register, is still heard as the primary voice. Organum theory begins with performance in either parallel 5ths or parallel 4ths. Free interchange between these two intervals was not expressly permitted until about 1100. In even the earliest sources the 4th was favoured. This must have been partly because it was the concordant interval which would most perceptibly have the effect of ‘sounding apart’, and partly because the two voices were then closer together and could most easily converge on to a unison. The Daseian notation used in Musica enchiriadis was, as will be seen later, obviously designed with organum at the 4th in mind.
In the discussion of organum in Musica enchiriadis a crucial role is played by the tritone. It is the tritone’s inconsonantia or absonia which makes deviation from strict parallel movement necessary. It is that also which brings about a change in texture to one in which the vox organalis often clings to a particular note and produces a drone-like effect. Since the writer of Musica enchiriadis was clearly very concerned with such deviations, he found that a system of disjunct tetrachords offered the means whereby the occurrence of tritone intervals in organum at the 4th – and the need for avoidance – could by comparison with the normal octave system be doubled. This system was built of identical tetrachords grouping themselves around the tetrachord of the finals D, E, F and G (ex.2: note the brackets under the letter names; the four degrees of the main tetrachord appear with their Greek number names archoos, deuteros, tritos and tetrardos). Adjacent tetrachords were always separated by a whole tone – hence they were disjunct. Because of this they cut across the octave structure of the normal tonal system; the pattern of intervals repeats at the 5th rather than at the octave. Thus it is possible to sing in parallel 5ths in this system without disturbance. A critical factor was, however, that when singing in 4ths a tritone now occurred on every fourth degree of the scale, between the tritos of one tetrachord and deuteros of the next tetrachord up: B–e, f–b, c'–f', g'–c'' (see ex.2, the brackets above the letter names). By contrast, the tritone in the octave system arises only between B and e, and between f and b. Each of the four degrees of the tetrachord has its own sign, and this sign is modified with its reappearance in each higher tetrachord by being reversed, inverted and reverted respectively – the sign for the tritos being slightly modified in shape also. The practice of changing a sign by turning it round is reminiscent of Greek instrumental notation and is possibly directly influenced by that. The signs themselves were named Daseia signs after their basic sign, which was the Greek for the prosodia daseia (cf Schmid, pp.5–7).
If the teaching of Musica enchiriadis is followed, organum takes shape as a result of the joint operation of the tetrachord system and the law prohibiting the tritone. In order to avoid each tritone the vox organalis must constantly avoid the sonus tritus. To cope with this there is one prime rule: the vox organalis must not, in the course of a phrase, descend beyond the sonus tetrardus; nor, where the vox principalis begins a melodic ascent, must the vox organalis approach the tetrardus from below (cf Schmid, p.49: ‘ut in quolibet tetrachordo in qualibet particula nec infra tetrardum sonum descendat positione nec inchoatione levetur obstante triti soni inconsonantia, qui tetrardo est subsecundus’). The vox organalis is only allowed to ‘stray’ into the register of a neighbouring tetrachord if the vox principalis changes its register correspondingly (ed. Schmid, pp.51–2; cf ex.1: from the syllable -lis onwards the vox organalis shifts into the next tetrachord down). In practice each sonus tetrardus (c, g, d', a') functions as a lower limit of pitch to which the vox organalis clings like a drone. The vox organalis quits this limit in the upward direction only if the vox principalis itself moves beyond a 4th above it, or if the vox organalis moves to a unison with the vox principalis at the end of a section (the latter, as the pre-existing and hence unchangeable melody, thus drawing the vox organalis towards it, as it were). The second of these situations arises in ex.1 at the syllables pi-is; the first does not occur until after the dip down into the tetrachord below at the syllable -lis, the vox organalis therefore having to cling to the same tetrardus for the first nine syllables.
Put negatively, this way of shaping the end of a section, which applies analogously also to the beginning of a section (e.g. the unison opening of ex.1), suggests that a section was supposed never to end on a non-consonant interval. (Without the convergence of voices (convenire) ex.1 would have ended on a 3rd.) This is indeed the argument of Musica enchiriadis: it is because the vox organalis must not go below the sonus tetrardus, and because at the same time a section must not end on a 4th, that a unison is selected (ed. Schmid, p.50: ‘[vox organalis] subtus eundem [tetrardum] non valet positione progredi et ob hoc in finalitate positionum a voce principali occupetur, ut ambae in unum conveniant’). Admittedly examples do also occur in Musica enchiriadis in which the vox organalis does go below the sonus tetrardus and ends on a 4th. But such cases were probably counted as transitional – in accordance with the methodical way of setting out organum theory – though not in fact forbidden (ed. Schmid, pp.36ff). Put positively, this way of ending sections manifests a very strong desire to point up by musical means the structure and phrasing of chants that are to be sung polyphonically. The vox organalis is in truth no more than an ‘ornament’, but it is a great deal more than a mere ‘doubling’ of the chant. It creates sonorous tension by ‘singing apart’, heightens this tension by changing the vertical intervals, and dissolves the tension at each of the caesuras as it establishes a point of rest in the sonority.