The teaching in Musica enchiriadis is characterized by the search for a thoroughly ‘automatic’ process in polyphonic performance. Thanks to the particular nature of the tetrachord system, organum arose more or less of its own accord so long as certain rules and constraints were consistently observed (the only entirely optional factors were the doubling or tripling of both voices at the octave, and their reinforcement by instruments: cf Schmid, pp.38–40). The performers were not directly answerable for the musical effect; nor would they find in this treatise the necessary aesthetic grounding. It was thus possible to perform a vox organalis extempore at any time and to any chant for which it might be desired, after agreement on only a very small number of technical points. Not only this, however, but also the greatest possible uniformity was automatically guaranteed for the polyphonic end product. This very disregard of aesthetic considerations, and of evaluation, saved polyphonic practice from having to think in terms of alternatives, improvements or refinements. The teaching in Musica enchiriadis must surely be conceived in so cryptically codified a fashion precisely because the uniformity of liturgical chant had to be preserved even in polyphonic performance – that wholesale uniformity which had been a prime goal of all reforms in church music ever since Pépin and Charlemagne. This suggestion is supported by the extraordinarily wide distribution of the treatise, for it survives today in more than 40 manuscripts.
However, the future lay in a type of polyphonic teaching that emerged for the first time in the Cologne organum treatise of about 900. This type of teaching gradually began to spread in influence from the time of Guido of Arezzo’s Micrologus (early 11th century). This is a method of teaching that, while firmly based on Musica enchiriadis in its subject matter, differs strikingly in its manner of presentation and argument. Its starting-point was still a set of definitive rules, but now in addition it allowed for ‘exceptions’. In matters of detail it was content to lay down guidelines. It relied on the singers’ experience and judgment; by means of aesthetic argument it strove to analyse current practices, to experiment and develop new methods. The theory of polyphony thus became something like an introduction to the subject, describing possible ways of creating polyphony, and putting forward rules but never expecting blind observance.
The abstract tetrachord system was abandoned along with Daseian notation. The Cologne organum treatise no longer used Daseian notation; Guido criticized the ‘moderns’ openly for having introduced these innovations very carelessly, and at the same time disregarded the early theorists’ recognition of the octave as the only interval that makes perfect consonance (‘perfecte consonat’) rather than the 5th (CSM, iv, pp.112–3). The pitching of the vox organalis a 4th lower than the vox principalis, and also its parallel movement, were retained in essence, as was the principle of pitch limits that the vox organalis must not overstep. Indeed, these were now positively reasserted. The vox organalis no longer had to converge to unison (‘in unum convenire’) with the vox principalis as a matter of necessity just because an artificially produced tritone stood in the way. Rather the contrary: so that the two melody lines ‘can come together in a suitable manner’ at the end of the line – and this is the crux of the matter – the pitch limits should be obeyed merely as rules of thumb (‘ut in finalitatibus vox ad vocem apte convenire possit …, organum inferius descendere non possit’: Cologne organum treatise, ed. Waeltner, p.54).
The determination of the pitch limits was governed by the tonality of the chant. However, their deployment now became transparently clear on aesthetic grounds also. According to Guido, the whole tone and major 3rd (together with the 4th and unison) were the favoured sonorities. The minor 3rd on the other hand was no more than tolerated, the semitone not accepted at all. For this reason the bottom notes of the hexachords, C, F and G, with the particular pattern of intervals that surrounds them, turn out to be the ideal pitch limits, because all the favoured sonorities could be sounded above them (‘Aptissime vero, qui saepissime suaviusque id faciunt, ut tetrardus et tritus in .C. et .F. et .G. Haec enim tono et ditono et diatessaron obsequuntur’: CSM, iv, p.202). The tritone that arises in Guido’s system, between F and B natural, was avoided quite pragmatically by shifting the vox organalis on to G to produce a major 3rd (CSM, iv, p.206).
Guido went significantly beyond Musica enchiriadis in his refining of the way in which cadences were formed. He also for the first time allowed brief crossing of parts. The vox organalis was no longer simply ‘occupied’ at the close by the chant (cf Schmid, p.50) but could now ‘come to meet’ it in what was called the occursus (literally ‘meeting’). Guido viewed the two voices as approaching each other by step, so that they could converge on to unison as far as possible e vicino (‘from nearby’: CSM, iv, p.204). He demonstrated this by means of two examples, significantly presented together as alternative and equally acceptable possibilities (exx.3 and 4). Following traditional practice, the vox organalis in ex.3 (CSM, iv, p.211) clings to the lower pitch limit right through to the penultimate note on the grounds that its distance from the chant is less than a 4th. This is called occursus simplex. Following the new practice, the interval of the major 3rd c–e in ex.4 is passed over by step in the vox organalis via a penultimate d. This is called occursus per intermissas [voces].
When convergence by step in this way is not possible, it is preferable for a phrase to close on a 4th rather than converge on to a unison by leap in the vox organalis (CSM, iv, p.204). This does not apply, however, when the phrase concerned is the last of the whole piece. The progression towards a close may be further refined by a kind of cadential extension of the chant, this being also optional. Against the last note of the chant there occur two notes in the vox organalis, the first being a 2nd below – thus in effect prolonging the penultimate note – and the second note providing a resolution on to the unison (CSM, iv, p.205: ‘Item cum occursus fit tono, diutinus fit tenor finis, ut ei et partim subsequatur et partim concinatur’) (ex.5, based on exx.3 and 4).
The rule that the vox organalis must lie always beneath the chant was also first modified by Guido: if the chant went only briefly below the lower pitch limit (Guido demonstrated it with a limit of f) then the organum voice could remain unchanged. This was called organum suspensum (CSM, iv, pp.205, 212) (ex.6, at the asterisk).