D I A 



730 



D I A 



Diaschisma 



Diatonic. 



tion, (£££ i" this case), whose — power is to be 



sought in another vulgar fraction ; the above ratio be- 

 ing true within less than the smallest known interval, 

 or m. Its value, in Farey's notation, is 22.92516952 + 

 f + 2 ra, or 23 2 + \ f + 2m ; its common log. = 

 .9886813,5414. 



DIASCHISMA of Dr Busby, is an interval, the 

 half of the minor semitone, or ± x d ; whose approxi- 

 mate ratio, found as above, is |-J, which differs only 

 about I m from the true interval. It is =1 8.0708993s 

 _|-2 m, or 182 + £ f + l|m. and its log. .991 1356,1652. 



DIASCHISMA of Euler, is an interval whose ra- 

 tio is i%$r$, = 102m, or the Comma Minor, which 

 see. 



DIASCHISMA of S. Rootsey, is an interval not 

 diatonic, but intended as an approximation to the true 

 «1 mentioned above, whose ratio he states to be ^\, 

 bs 12.0483985 2 + m ; its common log. being 

 .9940911,4039. 



DIASCHISMA Triple, is a compound or multiple 

 interval, which is mentioned here, from its resulting 

 also, as simple or prime intervals usually do, from the 

 subtraction of simple intervals, viz. 3ct = d" — f, =3 

 L— <p, =S + 42— /, =P+42— I =S— c— f, =S — 



2 s7 

 £ — f, &c. Its ratio is — j which is expressed in large 



144111 &c. 

 numbers, of which the first figures are r . n . n .' c — = 



150094, &c. 



35.850339 2 + f + 3 m, or 36 2 + 3 m. Its log. is 

 .9823445,8294. 



DIASPASIS, a genus of plants of the class Pentan- 

 dria, and order Monogynia. See R. Brown's Prodrom. 

 Plant. Nov. Holl. et Ins. Van Dietnen. p. 586, and Bo- 

 tany, p. 175. 



DIATESSARON, in Music, is an interval, sometimes 

 called a tetrachord, whose ratio is |, = 2542 + 5f+ 

 22m, or the Fourth Minor, which see. 



DIATESSARON of Holder, an interval, improperly 

 so called, whose ratio is T 5 ^, =1478 2 + 29f+128m, or 

 the Eighteenth Minor, which see. 



DIATONIC Elements, inMusic,ai*e the Major Tone 

 (T), the Minor Tone (t), and the Major Semitone (S). 

 By the combination of which three intervals, all others 

 which occur in the music now in use, called the diato- 

 nic, may be derived. Intervals thus expressed, as Mr 

 Liston has done throughout his valuable " Essay on 

 Perfect Intonation," are said to be expressed in, or com- 

 puted by a notation of diatonic elements; but Mr Liston 

 follows Dr Robert Smith, in denominating the lesser 

 interval a Hemitone, and marking it H, instead of S, 

 which is used in our Table, Plate XXX. Vol. II. and 

 in our several musical articles. 



Mr Maxwell, in his " Essay upon Tune," calculates 

 by these elements ; but he calls them by the names 

 Greater Tone, Lesser Tone, and Semitone, and marks 

 them G, L, and S, instead of T, t, and S, as we do. 

 Owing to the constant occurrence of negative signs, in 

 the minuter parts of the calculations, unless that an oc- 

 tave, or3G + 2L + 2S, is added to every interval, as 

 Mr Maxwell sometimes does to avoid them ; and ow- 

 ing also to the want of any apparent value in the three 

 terms collectively, this notation often fails to convey 

 readily a most important piece of information, viz. which 

 of two intervals, expressed in it, are the largest ? As, 

 for instance, whether is the c or $B, of Maxwell, p. 194, 

 the most acute, or the largest intervals above the bass C? 

 the former being 3 2 2, and the latter 4 2 in his Ta* 



ble, or the c and B% of Liston, which are expressed by Diatonl 

 the same numbers : whereas, had Farey's notation been cum * 

 used, and these two notes been expressed by 612 2 + s "^"~ 1 

 12 f+53 m, and 602 2+ 12f+52 m, it would at once 

 have appeared that the former is the largest, and that 

 the difference of them is 10 2 +m, instead of the ambi- 

 guous difference 2 S — G, as Maxwell has it. 



For all purposes of perfect harmony, or diatonic 

 calculations, his artificial commas, or the first or largest 

 only of his elements, 2, might be used, as shewn in 

 the Philosophical Magazine, vol. xxxix. p. 419, and by 

 which the whole of the calculations, necessary for un- 

 derstanding and proving Mr Liston's system of perfect 

 harmony, is reduced to the adding or subtracting of 

 numbers, which rarely exceed three figures, and which 

 surely need not deter any practical musician from the 

 attempt, however slight his knowledge of arithmetic. 



By some, the prime digits, 2, 3, and 5, have also been 

 called Diatonic elements, because they are, in every in- 

 stance, composed of these numbers, and no other prime 

 digits, except 1, which does not affect ratios, or the 

 multiplications or divisions by which they are com- 

 pounded. See Musical Primes. (^) 



DIATONICUM, Diatonum, in the music of the 

 Greeks, was distinguished among their genera, accord- 

 ing to Euclid, Eratosthenes, Ptolemy, &c. by a tetra- 

 chord, ascending according to the following numerical 

 ratios, viz. $£| x £ X f- =|> which, in our notation, (see 

 Plate XXX. Vol. II.) is as follows, viz. 



T=1042 + 2f+9m 

 T=104 2 + 2f+9m 

 L= 46 2+ f+4m 



4th=2542 + 5f+22m 



DIATONICUM Equabile, was a genus of Ptolemy, 

 which, according to Dr Wallis, was thus composed, viz. 

 4 t X 44 X i 9 o = h whence we have, 



■£s = 93.000000 2 + 2 f + 8 m 

 44= 84.4013672 + 2f+ 7 m 

 44= 76.598633 2+ f+ 7 m 



4th = 254.000000 2 + 5 f + 22 m 



DIATONICUM Intensum, or Syntonum, this most 

 important of the Greek genera, according to Didymus, 

 Euclid, and many other writers, had a tetrachord thus 

 composed, viz. 4| X to X f-=£> or, in our notation, 



T=1042 + 2f+ 9m 

 t= 93 2 + 2f+ 8m 

 S= 57 2+ f+ 5m 



4th = 254 2 + 5 f + 22m 



Or, accordi g to Ptolemy, thus, 



t= 932 + 2f+ 8m 

 T=104 2 + 2f+ 9m 

 S = 572 + f + 5m 



4th = 254 2 + 5 f + 22 m 



Of all the numerous scales of musical intervals which 

 the Greek musicians used, or their own or subsequent 

 theoretical writers have pretended that they did, the 

 two last only, called the Diatonic, (see Diatonic Ele- 

 ments,} are now in use, since our chromatic scales differ 

 essentially in their construction and use, especially on 

 tempered instruments, from any that are found in the 

 ancient musical writers. 



