The Ancient Library

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in such a form as to serve for an introduction to the original sources. Hence it necessarily consists in a great measure of technical details, which, how­ever, can present no difficulty to persons acquainted with the first elements of the modern theory ; and nothing has been said in the way of deduction, except in one or two cases where the interest of the subject and the apparent probability of the conclusions seemed to permit it.

The term 'Ap/AoviK-f] was used by the Greek writers to denote what is now called the Science of Music ; povariKT] having, as has been already re­marked, a much wider- signification. 'A/tytoi/i/cT?

Kal Trp.aKriK^) tt)s rov


Kal $La(TT7]{J.d>Tb>v, ttomv to^lv (Euclid. Int. Harm. p. 1.)

The following sevenfold division of the subject, which is adopted by the author just quoted, as well as by others, will be partly adhered to in the pre­sent article : — I. Of Sounds (irepl tyQoyyow). II. Of Intervals (irep), S/ao-r^^Tcyi/). III. Of Genera (irepl -yeVco;/). IV. Of Systems (Trep.l crv(TTr]/j,d-. tuv). V. Of Modes (Trepl t<*i/wz/).* VI. Of Transition (irepl jueragoA^s). VII. Of Composi­tion (Trep\ jueTujTTOuas).

A sound is said to be musical when it has a de-^ terminate pitch (ratm). When two sounds differ in pitch, one is said to be more acute (o£us), the other more grave ( fiapvs) : or, in common language, one is called higher and the other lower. The term ^/j./j.^X-fjs applied to a sound either signifies simply, that it is capable of being used in a melody ; or relatively, that it is capable of being used in the same melody ^ith some other sound or system of sounds ; the latter is its most common meaning.

An Interval is, the difference or rather distance between two sounds of different pitch. When we compare the intervals between two pairs of sounds, we judge them in certain cases to be similar, or equal. If the more acute sound of one of them be then raised, that interval is said to become greater than the other. It is this property of intervals (their being comparable in respect of magnitude) which enables us to classify tkem, and enumerate their several kinds.

Intervals are either consonant- (crv/Atpaiva) or dis­sonant (§iaL$(av&), according as the two sounds may or may not be heard at the same time without offending the ear. (EucL p. 8.) Strictly speaking it is impossible to define the limit between the two classes, and this seems to be acknowledged by the later writers, who distinguish various degrees of consonance and dissonance. Originally, the only intervals reckoned consonant were the Octave or eighth (Sioi ira<ru>v\ the Fifth (Sia Trez/re or 5Y b^ei&v), the Fourth (Sia Tecrffdptw or ffvXKa^T]), and any interval produced by adding an octave to one of these. But all intervals less than the fourth, or intermediate between any two of those just enumerated (as the sixth, tenth, &c.), were con­sidered as dissonant. The principal intervals, less

* Ttfvos is used in several different senses. First it signifies degree of tension, and so pitch, whence its application to denote mode, the modes being scales which differed in pitch : and then it is taken for result of tension ; whence its meaning as the name of an interval, tone, because a tone is the in­terval through which the voice is most naturally raised at one effort. (See Aristid. p. 22 ; Eucl. 19.)



than the fourth, employed in Greek music were the double tone \§irovov}, nearly equal to the modern major third ; the tone and half (rpi^iro-viov}, nearly the same as the minor third ; the tone (r6vos\ equal to the modern major tone ; the half tone (^iroviov") and the quarter tone (Slecrts). (Eucl, pi 8.) Other writers speak of o^uo^owa or unison, avTKpcwia or the consonance of the octave, and Trapatyow.ia or the consonance of the fourth and fifth. See Arist. Probl. xix. 39, and Gaudentius, p. 1 1. The latter author considers irapa<j><avia to be intermediate between consonance and dissonance, and mentions the tritone or sharp fourth as an ex­ample of it.

If two strings, perfectly similar except in length, and stretched by equal tensions, be made to vi­brate, the number of vibrations performed in a given time by each is inversely proportional to its length ;. and the interval between the sounds produced is found to depend only on the ratio of the lengths, i. e. of the numbers of vibrations. Thus if the ratio be 1 the interval is an octave, ~

f -f



M a fifth, „ a fourth, « a major tone.

The discovery of these ratios is attributed, pro­bably with truth, to Pythagoras. But the accounts of the experiments by which he established them (see Nicomachus, p. 10) are plainly false, since they contradict the known fact that when similar and equal strings are stretched by different tensions, the number of vibrations are as the square roots of the tensions. (See Whewell's Dynamics, part ii. p. 331, ed. 1834.)

The tovos or tone was defined to be the dif­ference between the fourth and fifth ; so that the corresponding ratio would be determined either by experiment, or by simply dividing f by f.

It is remarkable that each of the four ratios enumerated above is superparticular *; i. e. the two terms of each differ from one another by unity. And all tjje intervals employed in the modern theory are either such as correspond to superpar­ticular ratios, or are produced from such by com­pounding them with the octave. Thus the ratio corresponding to the

major third minor third minor tone

is *

" f " 7% major semitone „

It seems therefore extraordinary, that analogy should not have led at once to the discovery at least of the major and minor third, as soon as the connection between intervals and ratios had been observed. However no such discovery was then made, or if made it was neglected ; and this affords a,t once an explanation, of the fact that intervals less than the fourth were reckoned dissonant: for the fi'novov, or double major tone, is greater than the true consonant major third (which consists of a major and minor tone) by an interval expressed by the ratio •§ ^ ; a. difference quite sufficient to de-

* Euclid seems to consider no intervals conso­nant except su.qh as correspond to superparticular (eTTijAopios) or multiple (TroAAaTrAatnVj/) ratios ; the latter being such as -y, f, f, &c. On thi3 theory the octave and fourth (-f) would be dis­sonant, but the octave and fifth (^) consonant. (See Eucl. Sect. Can. p. 24.)

3 d 3

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