173 



TETRACHORD. 



TETRAMETHYLAMMONIUM. 



171 



likely to aid an unbiassed reader in approaching, should it please him 

 so to do, the mass of different accounts which have been given. 



All parties seem agreed that the Greek scale, which at first consisted 

 of only two or three leading consonances, was gradually enlarged until 

 it comprehended two octaves, or fifteen notes. It is generally stated 

 that this scale, when it was what we now call diatonic (a word which 

 means the same with us as with the Greeks), was minor in its character, 

 so that in fact it would be represented by 



A B c D E F o A'B'C'D'E' F'o 1 .* 1 . 



It is also known that the Greeks were early in possession of the mode 

 of dividing a string so as to produce their several notes ; and that, by 

 the time of Ptolemy at least, they took the rapidity of the vibra- 

 tions (on which they 'knew the pitch to depend) to be inversely as the 

 lengths of the strings. 



Their scales were numerous : three were considered classical, if we 

 may use the word, and were called enharmonic, chromatic, and dia- 

 tonic ; the two first words not having the same meaning as with us. 

 The remaining scales had names of locality attached to them, Lydian, 

 Dorian, &c. The distinction between these lay in the different modes 

 of dividing the octave, as seems to be now generally agreed, though 

 there have been those who have thought that these terms, Lydian, 

 4c., were the names, not of scales, but of single notes. 



Of enharmonic, chromatic, and diatonic scales, Ptolemy lays down 

 fifteen from his predecessors, and eight from himself. In each of them 

 is an octave, and all of them agree in two particulars : first, each has 

 the fourth and fifth of the fundamental note perfect ; secondly, each 

 has the tetrachord made by the fundamental note and its fourth 

 divided in precisely the same manner as that of the fifth and the 

 octave. That is, if we call the notes of this octave 



CPQFOR8C 1 ; 



then c F ia a fourth, and c o a fifth, always ; and the interval c p, p Q, 

 Q F are severally equal to the intervals o B, B 8, 8 c'. Thus it appears 

 that the fourth was to the Greeks what the octave ia to us, the unit, 

 as it were, of the scale, in the subdivision of which consisted the 

 differences of their systems. We now give a tetrachord from each of 

 these twenty-three scales, assigning the intervals first by the ratios of 

 the vibrations, next by the number of mean semitones they contain, as 

 in the article SCALE. We prefix the Latin rendering of Ptolemy's 

 appellatives from Wallis. 



And first as to enharmonic scales, which are mentioned first, and 

 seem to have been ancient, and regarded with high approbation. 



It seems then that the enharmonic system would allow only of the 

 following notes in an octave 



CJSPFOBQC 1 ; 



where p means a note about half way between E and f, and Q one half 

 way between B and o 1 . An odd scale truly for a.modern musician to 

 look at ; but, it may be, not incapable of pleasing effects to ears not 

 accustomed to music in parts. 



The chromatic scales come next in order, as follows : 



To make something as like as we can to these scales, we should 

 write down in modern music 



F o 



n; c 1 



The diatonic scales, Ptolemy allown, ore more agreeable to the ear, 



and his specimens are as follows : we shall now write the scale with 

 .he usual letters throughout. 



These scales have all so far the diatonic character that they divide 

 the tetrachord into two larger intervals followed by a smaller one : the 

 scale of Didymus would have been exactly the modern untempered 

 diatonic scale, if he had inverted the order of the two larger intervals 

 in his second tetrachord. As to the other modes, the Dorians, &c., 

 there is much confusion in Ptolemy respecting them, arising from the 

 corruptness of the text, which Wallis has endeavoured to remedy. 

 According to him, there are divisions of the octave, somewhat more 

 fantastic than those which precede. In more recent times the idea has 

 been started of their being simply different keys, or rather answering 

 to different variations of the diatonic scale, by using intermediate 

 semitones instead of some of the notes : it would be difficult, we 

 think, to produce authority enough for this conjecture. 



If it were true, as supposed, that the two octaves of the Greek 

 scale, beginning, say with A, were minor, it would follow that Ptolemy, 

 in his diatonic scales, exhibited the octave from o to c 1 , as we have 

 supposed. Accordingly, the principal mode of exhibiting the forma- 

 tion of the octave from two tetrachords and a tone would be the one 

 we have taken, namely, 



(c D E F) (o A B c 1 ) 

 But it is frequently supposed that it was the following : 



{>() ABC) 

 or the following 



A {B c D (E} p a A) 



On this point we shall only say that there never was, we believe, so 

 strong a union of the three characters of scholar, mathematician, anil 

 musician, as was seen in Dr. Smith, the author of the Harmonics. He 

 had studied the Greek scale attentively, and to him the first of these 

 methods was a matter of course. " The Greek musicians " (' Har- 

 monics,' 1749, p. 45), "after dividing an octave into two-fourths, with 

 the diazouctic or major tone in the middle between them, and admitting 

 many primes to the composition of musical ratios, subdivided the 

 fourth into three intervals of various magnitudes placed in various 

 orders, by which they distinguish their kinds of tetrachords." 



\Vu do not, we confess, though admitting that it is exceedingly hard, 

 and probably impossible, to reconcile the Greek writers with them- 

 selves and each other, find that sort of difficulty which Dr. Buruey 

 owned to, when he said that he neither understood those writers 

 himself, nor had met with any one who did. He was a musician, and 

 was looking out for an intelligible mode of arriving at and distributing 

 the most agreeable concords, with a strong predetermination to arrive 

 at musical truth or nothing. But the Greek writers were arithmeti- 

 cians, with as strong a determination to find natural foundations in 

 integer numbers : they did not ask how to find sounds which would 

 best suit the ear, but how to discover triplets of fractions which 

 multiplied together should produce four-thirds of a unit. Pleased with 

 the simplicity of the ratios which give the fourth, fifth, and octave, 

 their efforts at musical improvement were confined to the attempt at 

 discovering magic numbers to fill up the intervals. It was not until 

 one of these philosophers had laboured at his abacus, and tasked his 

 metaphysics to find <i priori confirmation of some question in arith- 

 metic, that he strung his monochord and tried how his scale sounded : 

 it would have been hard indeed if his ear had refused to sympathise 

 with his brain. In all probability the musicians, whose object was 

 simply to please, laughed at the arithmeticians, as Tycho Brane" did at 

 Kepler, when the latter had discovered reason for the distances of the 

 planets in the properties of solid bodies : they had motive enough, and, 

 beyond all question, reason more than enough. 



TETRAGON (properly a four-angled figure), a term usually ap- 

 plied to the square only, when used, which it seldom is. [REGULAR 

 FIGURES.] 



TETRAHEDRON (a solid of four faces), a term usually applied to 

 the regular tetrahedron. [REGULAR FinuiiES.] 



TETRAMETHYLAMMONIUM. [ORGANIC BABES, Methylamine.] 



* This is also Ptolemy's Ditonici Diatonicn, 



