738 



DIESIS. 



Diesis, is stated to be sesquialtera to the enharmonic diesis, 

 -"V— ' (or i£)=31.350820142 + f+2m, or 31^2-f-3m, and 

 its log. =.9845500,6504. 



DIESIS of Mercator, according to Dr Holder, is r * T d 

 parts of the octave, =2 of his artificial commas, = 

 23.0124496s +f+2m, or 23^2 -f-ffL + 2m, and its 

 log. =.9886403,7752 =.03773584 X VIII, =2.1055714 

 Xc 



DIESIS Chromatic of Dr Calcott, ( S) its ratio is i|j4£ 

 =262 -f 2f-f-2m. See Chromatic Diesis. 



DIESIS Chromatic of Hoyle (^T ) has a ratio 2 \/2 -+-3, 

 = 52.0039312 2 -f-f+4m, or 522 -ff+4im, its com- 

 mon log. =.9744237,3877, its Euler's log. =.084962, 

 and its major comma log. = 4.74070. 



DIESIS Double Enharmonic, (or 2 £) has a ratio 

 5<s 

 4|*|| or 5T? =41.8582012+f+3m,or422-f 4m,and 



its log.=.9794000,S672 ; it is =S-f R, =/-}-€— (5 + 3 

 €=S— 3, =4c— 22, =2t— 4S, 4d— 62, =4S~2t=S 

 —7T, =2 6ths— 4 IIIds=2 VIII— 6 III, by either of 

 which two last equations, it may be tuned on an instru- 

 ment like Mr Liston's organ. 



DIESIS Duodecimal of Aristoxenus, used in form- 

 ing his genera or scales of music, and said to be practi- 

 sed by the Greek musicians, was T * 5 tri of the minor 

 fourth, =8.4895142 +m, or 8 T t-2 +^f+ifm, and its 

 log.=.9958353,7545. Dr Holder informs us, that this 

 interval was so named, because it was thought to be 

 the twelfth part of the major tone ; but the latter ex- 

 ceeds the former by more than f s, or .2001312. It is 

 =.0138346 xVIII=.7719S7xc=8.497376x 2. 



DIESIS Enharmonic, (or £), has the- ratio -£-§4= 

 212 -|-2m. See Enharmonic Diesis. 



DIESIS, Grave, of Listen, and his grave diminished 

 second, (or C), = |§^|=102 -fm, or the Comma Mi- 

 nor, which see. Mr Listen observes, that this interval 

 is near T * 5 th of a major tone, but which is 10.7622262 

 •4-m. 



DIESIS, Greater, of M. Capella, is represented to 

 be T \ (XII— 19c?), or 7.5240382 -fm. 

 . DIESIS, Greater Enharmonic, of Hoyle, (^T) 

 =78.0728649 2 +f+7m, or 782 + llf-f 6|m, and its 

 common log. =.96 16356,0816. 



DIESIS, Greater Enharmonic, of mean tone tem- 

 perament, 2l2+2m (or £). See Diesis of Temper- 

 ed Scales, and our article Enharmonic Diesis. 



DIESIS, Greater, of Quintilian, according to Dr 

 Wallis, had a ratio }*, =25.520192 + f+ 2m, its log. 

 being .9874108,7269. 



DIESIS, Greater, of Rootsey, and which he also 

 calls a quarter of a tone, and also an enharmonic semi- 

 tone, has a ratio £{-,=20.8351562 + 2m, and its log.= 

 .9897808,3482. 



DIESIS, Lesser, of M. Capella, is stated to be T ' 5 th 

 (11th— 17S), or 7.4682752 + m. 



DIESIS, Lesser Chromatic, of Chambers, Good, 

 Holder, &c. (or S), has a ratioi£=362-f f+3m. See 

 Semitone Minor. 



DIESIS, Lesser Enharmonic, of Hoyle, (^T). See 

 Diesis Quadrantalis of Euclid. 



DIESIS, Lesser Enharmonic, of mean tone tempe- 

 rament, which falls between $B and \)C, and between 

 #CE and frF, where the half tones are situate, is =17 

 .8937641 2 -(-2 m, or l7f 2-f f+lJ-m, and its log.= 

 .9912224,3171. See Diesis or Tempered Scales. 



DIESIS, Lesser, of Quintilian, according to Dr 

 Wallis, had. a ratio ^,=24.7983352-f-f+2m, and its 

 log.=.9877655,4358. Mr Holder, in different parts of 

 his works, calls this interval the Accidental Tempera, 

 merit, bearing comma, and quarter of a tone. 



DIESIS, Lesser, of Rootsey, and which he also calls 

 a quarter of a tone and an enharmonic semitone, has a 

 ratio||.,=8. 9883632 -f-m, and its log. =.9955908, 8 108. 

 DIESIS, Major, of Lord Brouncker, Holder, &c. 

 (S),.'=|4, = 36z+f+3m, or the Semitone Minor, 

 which see. 



DIESIS Major, of Maxwell, (€) has a ratio §§||, 

 = 102 -f-m or the Comma Minor, which see. 



DIESIS Major, of Quintilian, according to Dr 

 Wallis, had a ratio 4{,=27.102452-|-f-f 2m, or 27.251 

 7062 -j- 2m, and its log. =.9866360,3844. M. Chladni 

 observes, that this is the error ($) of the trumpet 

 fourth. See our article Chromatic French Horn. 



DIESIS Minor, of Maxwell (2), has a ratio f§£ff, 

 = 2, or the Schisma, which see. 



DIESIS Minor, of Quintilian, according to Dr 

 Wallis, had a ratio i-| : ,=26.28796-ff.f2m,and its log. 

 = .9870350,2284. 



DIESIS Quadrantalis of Aristoxenus, was T ' 5 th 

 of the minor fourth, or T ^x4th,=25.326742-ff-j-2m, 

 or 25f 2-j-|f+2fm, and its log.=.987506l,2634. 



DIESIS Quadrantalis of Euclid, was one fourth 

 of the major tone, or ^T,=2^- V /2 V /3,=25.92713532 

 -f-f+2m, or262-f-|f+2|m, and its log.=.9872118, 

 6939. Mr Hoyle calls this the lesser enharmonic diesis. 

 DIESIS Quadruple Enharmonic, (or 4 £) has a 

 94,4. 14.0 62*1 V* 

 ™*° 2, t T£, OT |-,=«M0 8 5 41 , +Sf+7m< or, 

 842 + 8m; its common log.=. 9588001, 7344, its Euler's 

 log. =.1368&1, its schisma log.= 84.062904, and it is 

 =7.636628 X c. 



This interval is an important one, as being the least 

 sum of the tierce temperaments, in any douzeave sys- 

 tem, or 4VIII — 12III=4£, on which account it has a 

 place in our Table, Plate XXX. in Vol. II. Dr Boyce, 

 Clagget, and Holder call it a Note. The following 

 equations will shew some of its relations to other inter- 

 vals, viz. 4£ = 7ct + m,=2S-f 2R,=2/+2€, = 2S+2J, 

 =2ct + 6€,=3/— d,=4t— 8-S,=8c— 42,=8d— 122,= 

 8S— 4t,=P + 2R-f D=2E-t-4€ + 22,=t— 2r— 52,=8 

 T— 8t— 42,=43ds-f-4 4ths— 8111, =4 6th— 8III,= 12 

 6ths— 9VIII, and =4VIII— 12III, by which last and 

 several preceding equations, this interval may be tuned. 

 DIESIS Sesquialtera of Aristoxenus, in his ge- 

 nus chromatic hemiolion, is reckoned 4^ thirtieth parts 

 of the minor fourth, or ^ x 4th (and was supposed to 

 be one and a half enharmonic diesis, whence the name) 

 = 38.0604942 -f. f + 3 m, or 38^2 + |f -f 3 T ^m, 

 and its log.=.9812591,8951. This interval exceeds 1 ft £ 

 by 6.606742 2 + m. 



DIESIS Sesquialtera of Euclid, in his genus 

 chromatic sescuplum, is |ths of a major tone, or |T,= 

 38.96'55332-ff-f 3m, or 392-f-i.f-f- 2 T 7 m, and its log. = 

 .9808178,0408. This interval was supposed equal to 

 l^E; but the latter falls short of the former by 7 

 .6073322 -f-m. 



DIESIS or Tempered Scales, are of two kinds, the 

 greater and the lesser. Dr Robert Smith in his Harmonics, 

 Prop. III. &c. shews, that in every regularly tempered 

 system the octave is made up of five equal tones, and 

 two equal major limmas, or VIII=5T4-2L; the dif- 

 ference between these he calls a minor limma, or T — 

 ~L—l, and the difference between the major and minor 

 limmas, or L — /, he calls the Diesis (D) of all such 

 scales. 



Mr Farey, in a paper in the Philosophical Magazine, 

 vol. xxxvi. p. 40, has given several equations or formu- 

 la for adapting Dr Smith's Interval to his notation, and 

 for rendering the calculations of temperaments more 

 easy and useful than heretofore. For this purpose he 



Diesis. 



