MUST C. 
an o(Slave towards it. a. Some accidental fharps or flats 
are allowed in the inverfion, to preferve the original pro- 
greflion more ftriftly than could otherwife be done. And 
the end of an inverfion may be altered, if a conclufion 
on the fucceeding harmony require it. 3. Every interval 
may be ufed as an accidental one. And thofe effential 
ones which cannot be ul'cd without certain limitations, 
may be covered by a diatonic or chromatic gradual pro- 
greflion to and from them. 
Double Counterpoint of the Twelfth or Fifth. —Two parts 
calculated to be inverted by carrying either of them a 
twelfth towards the other, are called a counterpoint of 
the twelfth. The particular rules of this counterpoint 
depend on that change of intervals which the inverfion 
produces, as exprefled in the following table; 
Intervals before the inverfion x 2, 345 678910x112 
Their inverfion in the twelfth 12 11 10 98 7 65 4 32 1 
Here it appears, that, by the inverfion, the unifon becomes 
a. twelfth, the fecond an eleventh, and fo forth. Hence 
it follows : Firft, that all the intervals, except the fourth 
and the fixth, can be treated in the fame manner as in 
t limple counterpoint; but that the fourth and fixth, when 
inverted, become difcords ; and therefore ought only to 
be introduced by a gradual progrefiion in one of the parts, 
Secondly, that confecutive thirds are good, becaufe they 
become thirds again. See Ex. 6. where, at a, the fubjedt 
is in the lower, and a counterpoint to it in the higher, 
part; at b is one inverfion of the fame, by carrying the 
higher part a twelfth towards the lower ; and at c the 
other inverfion, by carrying the lower part a fifth towards 
the higher, and the higher part an oftave towards it ; 
at d and e are farther examples of this kind of inverfion. 
When this counterpoint is inverted in the o(Stave, ei¬ 
ther before or after its inverfion in the twelfth, it becomes 
a counterpoint of the eleventh, which may be inverted 
in both in the fame manner as that of the twelfth. 
Double Counterpoint of the Tenth or Third. —Two parts 
calculated to be inverted by carrying either of them a 
tenth towards the other, are called a counterpoint of the 
tenth. The particular rules of this counterpoint alfo 
depend on that change of intervals which the inverfion 
produces, as exprefled in the following table : 
Intervals before the inverfion 1 23456789 10 
Their inverfion in the tenth 10 98765432 1 
Here it appears, that, by the inverfion, the unifon be¬ 
comes a tenth, the fecond a ninth, and fo forth. From 
which it follows : Firft, that ail the intervals, except the 
fourth, may be treated the fame as in the Ample counter¬ 
point. Secondly, that the equal motion muft be avoided 
throughout. 
Ex. 7. at a, is the former fubjeft with a counterpoint 
to it; and at b and c, its two inverfions in the tenth ; d 
and e are other examples of the fame kind of inverfion. 
When this counterpoint is inverted in the oftave, like 
the former one, it produces a counterpoint of the thir¬ 
teenth. 
Double Counterpoint of the Fourteenth or Seventh. —Two 
parts calculated to be inverted by carrying either of them, 
a fourteenth towards the other, are called a counterpoint 
of the fourteenth. The particular rules of this counter¬ 
point again depend on that change or intervals which the 
inverfion produces, viz. 
Intervals before the inverfion 1 2 3 4 567891011121314 
Inverfion in the fourteenth 14 13 12 1 1 10 9 8 7 6 5 4 3 2 1 
Here it appears, that, by the inverfion, the unifon becomes 
a fourteenth, the fecond a thirteenth, and fo forth. From 
■which it follows : Firft, that ali the intervals, except the 
fixth and the oftave, may be treated as in the Ample coun¬ 
terpoint. Secondly, that equal motion muft be avoided 
throughout. 
F.x. 8. at «, is the former fubjeft, with a counterpoint 
to it; and at b, c, its inverfions in the fourteenth. 
When this counterpoint is inverted in the oftave, 
Vol. XVI. No. 1 j 16. 
337 
like the two former, it produces a counterpoint of the 
ninth. 
Double Counterpoint of the Eleventh, Thirteenth, and 
Ninth .—It has already been ftiovvn how this counterpoint 
ariles from the three former ones by the Ample procefs of 
an inverfion in the odtave ; viz. the counterpoint of the 
eleventh or fourth, from that of the twelfth or fifth ; the 
counterpoint of the thirteenth or fixth, from that of the 
tenth or third ; and the counterpoint of the ninth or fe¬ 
cond, from that of the fourteenth or feventh. When¬ 
ever, therefore, counterpoints of the eleventh, thirteenth, 
and ninth, are introduced, they are mere derivatives of the 
former ones in the Came manner as the fourth, fixth, and fe¬ 
cond, are derivative intervals of the fifth, third, and feventh. 
Ex. 9. at a, is the original counterpoint, which is in¬ 
vertible in the twelfth, tenth, and fourteenth, and coni'e- 
quently alio in'their inverfions in the oftave. At b, each 
part is fet one degree farther from the other, which is the 
lame as letting one of the parts a third farther from the 
other; and confequently, alfo, the fame as if the higher 
part of it had been the lower, and the two parts inverted 
in the tenth. At c , the higher part of a is carried a third 
higher, and the lower part one degree loyvef, which is the 
fame as if one of the parts had been carried a fourth from' 
the other; and confequently, the fame as if the parts of 
a had been firft inverted in the twelfth, and then a dou¬ 
ble oftave back again. At d, lee the original higher 
part a fet a fourth higher, and the lower two, degrees 
lower, which is the fame as letting one of the parts a fixth 
farther from the other; and confequently, alfo the fame 
as if the counterpoint at a had been firft inverted in the 
tenth, and then in the double oftave back again. Ate, 
the original part is difpofed as if one of them had been let 
a feventh farther from the other, which is the fame as if 
the original higher part had been the lower, and then an 
inverfion had taken place in the fourteenth. 
According to the rules of a mulicalmode, a fubjeft.and 
its counterpoint in any interval may take place on every 
degree of the diatonic fcale; and this produces the differ¬ 
ent fpecies of every lbrt of double counterpoint, as well as 
of its different inverfions, in a fimilar manner as the diffe- 
ent fpecies of intervals and chords arife from their taking 
place on the other degree of the fcale. 
2. Counterpoint may coniilt of more than two inverti¬ 
ble parts; and others may alfo be added as mere ac¬ 
companiments. When it conlifts of three invertible parts, 
it is called triple counterpoint; and four invertible parts 
make a quadruple counterpoint. Thefe are the moil; ufe- 
ful counterpoints ; for quintuple, fextuple, &c. &c. are 
only ufed in canons of fo many real parts. The inverfions 
which a triple and quadruple counterpoint is calculated 
for are—Firft, all pofiible ones in the oftave ; and, fe- 
condly, thofe in other intervals. A triple or quadruple 
counterpoint may be formed in two different ways; viz. 
firft by doubling one or both parts of a double counter¬ 
point in thirds; and, fecondly, by giving every part its, 
own melody or progrefiion. 
The counterpoints whofe parts may be doubled in 
thirds, are all thofe invertible in the tenth ; either with¬ 
out or with being ailb invertible in the twelfth or four¬ 
teenth. And the principle according to which they may 
thus be doubled is ; that the confecutive tenths or thirds, 
which the inverfion produces to the original counter¬ 
point, have fo good an effeft, as to admit being added to 
the original part or parts. But it makes a material differ¬ 
ence whether the original parts lhall be doubled in thirds 
only before their inverfion in the twelfth, tenth, and four¬ 
teenth, or both before and after thole inverfions; as will 
appear in the following particulars. 
If a counterpoint be doubled in thirds only before 
its inverfion in the twelfth, tenth r and fourteenth, it mult 
be obferved : Firft, that in any counterpoint which is in¬ 
vertible in the tenth, one part may be doubled ; viz. the 
higher part in thirds below, or the lower part in thirds 
above. Secondly, that, if the faid thirds Ilia 11 be let to 
4 R both 
