1equal Length, all correſponding with one an
other at Right Angles. The neareſt to this is
the Seſquialtera, and the Seſquitertian alſo may
be reckoned among the ſhorter Areas. Theſe
three Proportions therefore, which we may alſo
call ſimple, are proper for the ſmaller Plat
forms. There are likewiſe three others, which
are proper for middling Platforms: The beſt
of all is the Double, and the next beſt is that
which is formed of the Seſquialtera doubled,
which is produced as follows: Having ſet
down the leaſt Number of the Area, as, for
Inſtance, four, lengthen it to the firſt Seſqui
altera, which will make ſix, and then add the
Seſquialtera of this ſix, which will produce
nine. Thus the Length will exceed the Breadth
in a double Proportion, and one Tone more.
4 0000Seſquialtera6 0000009 000000000Seſquialtera doubledother at Right Angles. The neareſt to this is
the Seſquialtera, and the Seſquitertian alſo may
be reckoned among the ſhorter Areas. Theſe
three Proportions therefore, which we may alſo
call ſimple, are proper for the ſmaller Plat
forms. There are likewiſe three others, which
are proper for middling Platforms: The beſt
of all is the Double, and the next beſt is that
which is formed of the Seſquialtera doubled,
which is produced as follows: Having ſet
down the leaſt Number of the Area, as, for
Inſtance, four, lengthen it to the firſt Seſqui
altera, which will make ſix, and then add the
Seſquialtera of this ſix, which will produce
nine. Thus the Length will exceed the Breadth
in a double Proportion, and one Tone more.
FOR moderate Platforms alſo, we may uſe
that Proportion which ariſes from the Seſqui
tertian doubled in the ſame Manner as the for
mer; wherein the Length and Breadth will
be as nine and ſixteen.
9 000000000Seſquitertia12 00000000000016 0000000000000000Seſquitertia doubledthat Proportion which ariſes from the Seſqui
tertian doubled in the ſame Manner as the for
mer; wherein the Length and Breadth will
be as nine and ſixteen.
HERE the longer Line contains the ſhorter
twice, excluding one Tone of that ſhorter
Line. In the longeſt Areas we either add the
Duple to the Seſquialtera, which will produce
the Triple; or add the Seſquitertia to the
Duple, which will make the Proportion as three
to eight; or laſtly make the Lines correſpond
to each other in a Quadruple Proportion. We
have now ſpoke of the ſhorter Platforms,
wherein the Numbers anſwer to each other
equally, as two to three, or three to four, and
of the Middling, wherein they correſpond as
two to four, or as four to nine, or as nine to
ſixteen: And laſtly of the longeſt, wherein
the Numbers anſwer in a Triple or Quadruple
Proportion, or as three to eight. We may
join together or compound all the three Lines
of any Body whatſoever, by Means of theſe ſe
veral Number, which are either innate with
Harmony itſelf, or produced from other
Proportions in a certain and regular Me
thod. We find in Harmony thoſe Num
bers from whoſe mutual Relations we may
form their ſeveral Proporions, as in the Duple,
the Triple and the Quadruple. For In
ſtance, the Duple is formed of the ſimple Seſ
quialtera, with the Addition of the Seſquitertia,
in the following Method. Let the leaſt Num
ber of the Duple be two; the Seſquialtera of
this is three, and the Seſquitertia of this Num
ber three is four, which is juſt the Double of
two before-mentioned.
00000The Seſquialtera0000The Seſquitertia or Dupletwice, excluding one Tone of that ſhorter
Line. In the longeſt Areas we either add the
Duple to the Seſquialtera, which will produce
the Triple; or add the Seſquitertia to the
Duple, which will make the Proportion as three
to eight; or laſtly make the Lines correſpond
to each other in a Quadruple Proportion. We
have now ſpoke of the ſhorter Platforms,
wherein the Numbers anſwer to each other
equally, as two to three, or three to four, and
of the Middling, wherein they correſpond as
two to four, or as four to nine, or as nine to
ſixteen: And laſtly of the longeſt, wherein
the Numbers anſwer in a Triple or Quadruple
Proportion, or as three to eight. We may
join together or compound all the three Lines
of any Body whatſoever, by Means of theſe ſe
veral Number, which are either innate with
Harmony itſelf, or produced from other
Proportions in a certain and regular Me
thod. We find in Harmony thoſe Num
bers from whoſe mutual Relations we may
form their ſeveral Proporions, as in the Duple,
the Triple and the Quadruple. For In
ſtance, the Duple is formed of the ſimple Seſ
quialtera, with the Addition of the Seſquitertia,
in the following Method. Let the leaſt Num
ber of the Duple be two; the Seſquialtera of
this is three, and the Seſquitertia of this Num
ber three is four, which is juſt the Double of
two before-mentioned.
OR elſe the ſame is done in the following
Manner: Let the ſmaller Number be, for In
ſtance, three; I add one to make it a Seſqui
tertia, and it becomes four, to which adding a
Seſquialtera, it makes it ſix, which, compared
to three, is juſt in a double Proportion.
000The Duple0000Seſquitertia000000SeſquialteraManner: Let the ſmaller Number be, for In
ſtance, three; I add one to make it a Seſqui
tertia, and it becomes four, to which adding a
Seſquialtera, it makes it ſix, which, compared
to three, is juſt in a double Proportion.
THE Triple is likewiſe made of the Duple,
and of the Seſquialtera joined together: For
Inſtance, let the ſmaller Number here be two;
this being doubled, makes four; to which
adding a Seſquialtera, it becomes ſix, which is
the Triple of two.
00The Triple0000doubled000000Seſquialteraand of the Seſquialtera joined together: For
Inſtance, let the ſmaller Number here be two;
this being doubled, makes four; to which
adding a Seſquialtera, it becomes ſix, which is
the Triple of two.
OR the ſame Thing is done as follows;
placing the ſame Number of two for the
ſmaller Number, take the Seſquialtera, and
you will have three, which being doubled,
gives ſix, and ſo we ſhall have the Triple of
two.
00The Triple000Seſquialtera000000doubledplacing the ſame Number of two for the
ſmaller Number, take the Seſquialtera, and
you will have three, which being doubled,
gives ſix, and ſo we ſhall have the Triple of
two.
BY Means of the ſame Extenſions we may
produce the Quadruple, by compounding one
Duple with another, ſince it is indeed nothing
more than the Duple doubled, which is alſo
called Diſdiapaſon, and is performed as follows:
Let the ſmaller Number here, for Inſtance, be
two; double this, and it makes the Diapaſon,
that is to ſay four, which is the Duple of two,
and doubling this four, it makes the Diſdiapa
ſon, which is as eight to two.
produce the Quadruple, by compounding one
Duple with another, ſince it is indeed nothing
more than the Duple doubled, which is alſo
called Diſdiapaſon, and is performed as follows:
Let the ſmaller Number here, for Inſtance, be
two; double this, and it makes the Diapaſon,
that is to ſay four, which is the Duple of two,
and doubling this four, it makes the Diſdiapa
ſon, which is as eight to two.
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