The Differences of the Genera

Let us now set ourselves to consider the origin and nature of the differences of the genera. Our attention must be directed to the smallest of the concords, that of which the compass is usually occupied by four notes—whence its ancient name. [Now since in such an interval the notes my be arranged in many different orders, what order are we to choose for consideration? One in which the fixed notes and the notes that change with the variation in genus are equal in number. An example of the order required will be found in the interval between the Mese and the Hypate: here, which the two intermediate notes vary, the two extremes are left unchanged by genus-variation.] Let this then be granted. Further, while there are several groups of notes which fill this scheme of the Fourth, each distinguished by its own special nomenclature, there is one which, as being more familiar than any other to the student of music, may be selected as that wherein we shall consider how variation of genus makes its appearance. It consists of the Mese, Lichanus, Parhypate, and Hypate.

That variation of genus aries through the raising and lowering of the movable notes is obvious; but the locus of the variation of these notes requires discussion. The locus of the variation of the Lichanus is a tone, for this note is never nearer the Mese than the interval of a tone, and never further from it than the interval of two tones. There lesser of these extreme intervals is recognized as legitimate by those who have grasped the principle of the Diatonic Genus, and those who have not yet mastered it can be led by particular instances to the same admission. The greater of these extreme intervals, on the other hand, finds no such universal acceptance; but the reason for this must be postponed to the sequel. That there is a style of composition which demands a Lichanus at a distance of two tones from the Mese, and that far from being contemptible it is perhaps the noblest of all styles—this is a truth which is indeed far from patent to most musical students of to-day, though it would become so if they were led to the apprehension of it by the aid of concrete examples. But to any one who possesses an adequate acquaintance with the first and second styles of ancient music, it is an indisputable truth. Theorists who are only familiar with the style of composition now in vogue naturally exclude the two-tone Lichanus, the prevailing tendency being to the use of the high Lichani. The ground of this fashion lies in the perpetual striving after sweetness, attested by the fact that time and attention are mostly devoted to chromatic music, and that when the enharmonic is introduced, it is approcimated to the chromatic, while the ethical character of the music suffers a corresponding deflection. Without carrying this line of thought any further, we shall assume the locus of the Lichanus to be a tone, and that of the Parhypate to be the smallest diesis, as the latter note is never nearer to the Hypate that a diesis, and never further from it than a semitone. For the loci do not overlap; their point of contact serves as a limit to both of them. The point of pitch upon which the Parhypate in its ascent meets the Lichanus in its descent supplies a boundary to the loci, the lower locus being that of the Parhypate, the higher that of the Lichanus.

Having thus determined the total loci of the Lichanus and Parhypate, we shall now proceed to ascertain the loci as qualified by genus and shade. The proper method of investigating whether the Fourth can be expressed in terms of any lower intervals, or whether it is incommensurable with them all, is given in my chapter on 'Intervals ascertained by the principle of Concord.' Here we shall assume that its apparent value is correct, and that it consists of two and a half tones. Again, we shall apply the term Pycnum to the combination of two intervals, the sum of which is less that the complement that makes up the Fourth. Let us now, starting from the lower of the two fixed notes, take the least Pycnum: it will consists of the two least enharmonic diesis; while a second Pycnum, taken from the same note, will consists of two of the least chromatic dieses. This gives the two lowest Lichani of two genera—the enharmonic and the chromatic; the enharmonic Lichani being in general, as we saw, the lowest, the chromatic coming next, and the diatonic being the highest. Again, let a third Pycnum be taken, still from the same note; then a fourth, which is equal to a tone; then fifthly, from the same note, let there be taken a scale consisting of a tone and a quarter; then a sixth scale consisting of a tone and a half. We have already metioned the Lichani bounding the first and second Pycna; that bounding the third is chromatic, and the special chroma to which it belongs is called the Hemiolic. The Lichanus bound the fourth Pycnum is also chromatic, and the special class to which it belongs is called the Tonic Chromatic. The fifth scale is too great for a Pycnum, for here the sum of the intervals between the Hypate and Parhypate and between the Parhypate and the Lichanus is equal to the interval between the Lichanus and the Mese. The Lichanus bounding this scale is the lowest diatonic. The sixth scale we assumed is bounded by the highest diatonic Lichanus. Thus the lowest chromatic Lichanus is one-sixth of a tone higher than the lowest enharmonic; since the chromatic diesis is greater than the enharmonic by one-twelfth of a tone—the third of a quantity being one-twelfth greater than the fourth—and similarly the two chromatic diesis exceed the two enharmonic by double that quantity, namely one-sixth—an interval smaller than the smallest admitted in melody. Such intervals are not melodic elements, or in other words cannot take an independent place in a scale. Again, the lowest diatonic Lichanus is seven-twelfths of a tone higher than the lowest chromatic; for from the former to the Lichanus of the hemiolic chroma is half a tone; from this Lichanus to the enharmonic is a diesis; from the enharmonic Lichanus to the lowest chromatic is one-sixth of a tone; while from the lowest chromatic to that of the hemiolic chroma is one-twelfth of a tone. But as a quarter consists of three-twelfths, it is clear that there is the interval just mentioned between the lowest diatonic and the lowest chromatic Lichanus. The highest diatonic Lichanus is higher than the lowest diatonic by a diesis. These considerations show the locus of each of the Lichani. Every Lichanus below the chromatic is enharmonic, every Lichanus below the diatonic is chromatic down to the lowest chromatic, and every Lichanus lower than the highest diatonic is diatonic down to the lowest diatonic. For we must regard the Lichani as infinite in number. Let the voice become stationary at any point in the locus of the Lichanus here demonstrated, and the result is a Lichanus. In the locus of the Lichanus there is no empty space—no space incapable of admitting a Lichanus. The point we are discussing is one of no little importance. Other musicians only dispute as to the position of the Lichanus—whether, for instance, the Lichanus in the enharmonic species is two tones remove from the Mese or holds a higher position, thus assuming but one enharmonic Lichanus; we, on the other hand, not only assert that there is a plurality of Lichani in each class, but even declare that their number is infinite.

Passing from the Lichani we find but two loci for the Parhypate, one common to the diatonic and chromatic genus and one peculiar to the enharmonic. For two of the genera have the Parhypate in common. Every Parhypate lower than the lowest chromatic is enharmonic; every other down to the point of limitation is chromatic and diatonic. As regards the intervals, while that between the Hypate and Parhypate is either equal to or less than that between the Parhypate and the Lichanus, the latter my be less than, equal to, or greater than that between the Lichanus and the Mese, the reason being that the two genera have the Parhypate in common. We can have a melodious tetrachord with the lowest chromatic Parhypate and the highest diatonic Lichanus. Enough has now been said to show how great is the locus of the Parhypate both in respect of its subdivisions and when regarded as a whole.