1that the ſame Numbers, by means of which
the Agreement of Sounds affects our Ears with
Delight, are the very ſame which pleaſe our
Eyes and our Mind. We ſhall therefore bor
row all our Rules for the finiſhing our Pro
portions, from the Muſicians, who are the
greateſt Maſters of this Sort of Numbers, and
ſrom thoſe particular Things wherein Nature
ſhews herſelf moſt excellent and compleat:
Not that I ſhall look any further into theſe
Matters than is neceſſary for the Purpoſe of the
Architect. We ſhall not therefore pretend to
ſay any thing of Modulation, or the particular
Rules of any Inſtrument; but only ſpeak of
thoſe Points which are immediately to our Sub
ject, which are theſe. We have already ob
ſerved, that Harmony is an Agreement of ſeve
ral Tones, delightful to the Ears. Of Tones,
ſome are deep, ſome more acute. The deeper
Tones proceed from a longer String; and the
more acute, from a ſhorter: And from the mu
tual Connection of theſe Tones ariſes all the
Variety of Harmony. This Harmony the An
cients gathered from interchangeable Concords
of the Tones, by means of certain determinate
Numbers; the Names of which Concords are
as follows: Diapente, or the Fifth, which is
alſo called Seſquialtera: Diateſſaron, or the
Fourth, called alſo, Seſquitertia: Diapaſon, or
the Eighth, alſo called the double Tone; Dia
paſon Diapente, the twelfth or triple Tone, and
Diſdiapaſon, the fifteenth or Quadruple. To
theſe was added the Tonus, which was alſo
called the Seſquioctave. Theſe ſeveral Con
cords, compared with the Strings themſelves,
bore the following Proportions. The Seſqui
altera was ſo called, becauſe the String which
produced it bore the ſame Proportion to that
to which it is compared, as one and an half
does to one; which was the Meaning of the
Word Seſqui, among the Ancients. In the Seſ
quialtera therefore the longer String muſt be
allowed three, and the ſhorter, two.
3 000Seſquialtera.2 00the Agreement of Sounds affects our Ears with
Delight, are the very ſame which pleaſe our
Eyes and our Mind. We ſhall therefore bor
row all our Rules for the finiſhing our Pro
portions, from the Muſicians, who are the
greateſt Maſters of this Sort of Numbers, and
ſrom thoſe particular Things wherein Nature
ſhews herſelf moſt excellent and compleat:
Not that I ſhall look any further into theſe
Matters than is neceſſary for the Purpoſe of the
Architect. We ſhall not therefore pretend to
ſay any thing of Modulation, or the particular
Rules of any Inſtrument; but only ſpeak of
thoſe Points which are immediately to our Sub
ject, which are theſe. We have already ob
ſerved, that Harmony is an Agreement of ſeve
ral Tones, delightful to the Ears. Of Tones,
ſome are deep, ſome more acute. The deeper
Tones proceed from a longer String; and the
more acute, from a ſhorter: And from the mu
tual Connection of theſe Tones ariſes all the
Variety of Harmony. This Harmony the An
cients gathered from interchangeable Concords
of the Tones, by means of certain determinate
Numbers; the Names of which Concords are
as follows: Diapente, or the Fifth, which is
alſo called Seſquialtera: Diateſſaron, or the
Fourth, called alſo, Seſquitertia: Diapaſon, or
the Eighth, alſo called the double Tone; Dia
paſon Diapente, the twelfth or triple Tone, and
Diſdiapaſon, the fifteenth or Quadruple. To
theſe was added the Tonus, which was alſo
called the Seſquioctave. Theſe ſeveral Con
cords, compared with the Strings themſelves,
bore the following Proportions. The Seſqui
altera was ſo called, becauſe the String which
produced it bore the ſame Proportion to that
to which it is compared, as one and an half
does to one; which was the Meaning of the
Word Seſqui, among the Ancients. In the Seſ
quialtera therefore the longer String muſt be
allowed three, and the ſhorter, two.
THE Seſquitertia is where the longer String
contains the ſhorter one and one third more:
The longer therefore muſt be as four, and the
ſhorter as three.
4 0000Seſquitertia3 000contains the ſhorter one and one third more:
The longer therefore muſt be as four, and the
ſhorter as three.
BUT in that Concord which was called Dia
paſon, the Numbers anſwer to one another in
a double Proportion, as two to one, or the
Whole to the Halſ: And in the Triple, they
anſwer as three to one, or as the Whole to one
third of itſelf.
2 00300Diapaſon, or doubleTriple1 01 0paſon, the Numbers anſwer to one another in
a double Proportion, as two to one, or the
Whole to the Halſ: And in the Triple, they
anſwer as three to one, or as the Whole to one
third of itſelf.
IN the Quadruple the Proportions are as
four to one, or as the Whole to its fourth Part.
4 0000Quadruple1 0four to one, or as the Whole to its fourth Part.
LASTLY, all theſe muſical Numbers are as
follows: One, two, three, four, and the Tone
before-mentioned, wherein the long String
compared to the ſhorter, exceeds it one eighth
Part of that ſhorter String.
1. 2. 3. 4.8 00000000ToneMuſical Numbers9 00000000,0follows: One, two, three, four, and the Tone
before-mentioned, wherein the long String
compared to the ſhorter, exceeds it one eighth
Part of that ſhorter String.
OF all theſe Numbers the Architects made
very convenient Uſe, taking them ſometimes
two by two, as in planning out their Squares
and open Areas, wherein only two Proporti
ons were to be conſidered, namely, Length
and Breadth; and ſometimes taking them three
by three, as in publick Halls, Council-cham
bers, and the like; wherein as the Length was
to bear a Proportion to the Breadth, ſo they
made the Height in a certain harmonious Pro
portion to them both.
very convenient Uſe, taking them ſometimes
two by two, as in planning out their Squares
and open Areas, wherein only two Proporti
ons were to be conſidered, namely, Length
and Breadth; and ſometimes taking them three
by three, as in publick Halls, Council-cham
bers, and the like; wherein as the Length was
to bear a Proportion to the Breadth, ſo they
made the Height in a certain harmonious Pro
portion to them both.
CHAP. VI.
Of the Proportions of Numbers in the Meaſuring of Areas, and the Rules for
ſome other Proportions drawn neither from natural Bodies, nor from Harmony.
ſome other Proportions drawn neither from natural Bodies, nor from Harmony.
Of theſe Proportions we are now to treat
more particularly, and firſt we ſhall ſay
ſomething of thoſe Areas where only two are
uſed. Of Areas, ſome are ſhort, ſome long,
and ſome between both. The ſhorteſt of all
is the perfect Square, every Side whereof is of
more particularly, and firſt we ſhall ſay
ſomething of thoſe Areas where only two are
uſed. Of Areas, ſome are ſhort, ſome long,
and ſome between both. The ſhorteſt of all
is the perfect Square, every Side whereof is of
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