GEN GEN
Generum Synopfis fecuildum Ariftoxenum.
Genus Enharmonium
G
enus Chromaticum
Genus Diatonicum.
3°
Molle !
Hemiolium
Tonicum.
Molle. 1 Intenfum.
Nete.
Nete.
Nete.
Nete.
Nete.
Nete.
"■ 30
H
-
-
22
21
i8
■5
J2 —
— 24
18
2+
Paranete.
— 18
t
Lichanus.
ii
—
Paranete.
-
-,
—
Lichanus.
_
Paranete
Paranete
9 Trite.
12 z
Trite. ~~
-'..
Lichanus. 6
Trite.
9
Paranete
Paranete
, — 9
Lichanus. 4i
Trite
Lichanus
4 Trite.
I 1
6
Lichanus
3
Trite.
Parypate 6
Hypate.
Parypate. 6
Hypate.
Parypate. '— 6 1 Hypate.
- a
Parypate
4e Hypate.
Parypate.
4 Hypate.
- 3 1
■J
Parypate.
3
Hypate.
The gradual approach of the paranete, in the different fpe- ] cies to the nete, (hews why this chord was called lichanus. The length of this fcheme, or the fide divided into thirty parts, isTuppofed to be the difference of any chord, and its fourth. Ptolemy takes thefedivifions in an afccnding fourth ; that is, the longed of the two chords, or the whole of one chord, is terminated at hypate ; and the termination of the fhorter chord, or of that part of the long chord, founding a fourth to the whole, is at nete. So that we muft imagine ninety parts added upwards to this fcheme beyond nete ; and then, for inftance, the lengths of the chord , or parts of a chord reprefenting the diatonicum intenfum of Ariftoxenus, according to Ptolemy, will be, 120, 1 14, 102 and 90. Ptalem. Haimorr. pai. 92. Edit. Wallh in Oper. Tom. 3. But we muft obferve, that the ninety parts might be added downwards in the foregoing fcheme, from hypate ; and then we fhould have a defcending fourth expreffed by the follow- ing numbers, 90, 96, 108 and 120, which is mod per- • feffly in tuiic, and gives us the femitone major, the tone major, and the tone minor. For #J = -H is the proportion of lengths of chords exprefling the femitone major ; T „ 6 j = I is in like manner a tone major ; and 4li = T o is the ex- predion of the tone minor. On the other hand, Ptolemy's numbers are much out of tune, and fuch as a muficiaii like Ariftoxenus dividing a ftring by his ear could not fall upon. We do not know that any commentator of the antients ever made this obfervation ; but if we purfue it through all the fpecies, we (hall find a very fenfible difference in the ex- preffions of the proportions of the various intervals found according to Ptolemy's manner, and thofe found in the way here mentioned, and exemplified in the diatonicum intenfum. Ariftoxenus and his followers determined the feveral fpecies, or divifions of the fourth by intervals, making the whole diatefl'aron equal to two tones and an half. But the Pytha- goreans fitting afide this method, took another, whereby they fought for three ratios, which added together would produce
the ratio of 4 to 3, being the expreflion of the fourth or diateffaron. And as this might be varioufly done, fo they conftituted various differences of genera. Ptolem. L. I. c. d. Edit. Wallis Oper. Tom. 3.
But they did not admit all ratios indifcriminately. They chiefly regarded fuch as were multiple and fuperparticular j and did not admit of the fupcrpartienr, but through necefhty, and much lefs of incommenfurablc expreflions. WalUs Append, ad. Ptolem. Harm. pag. 166. Oper. Tom. 3. Ptolemy b gives us feveral divifions of the tetrachord, partly his own and partly of others, which we ihall here place in one fhort view from Dr. Wallis's Appendix to Ptolemy's har- monics c .—[ b Ibid. Lib. 1. c. 13, 14, 15 and 16. = Pag. 166. Oper. Tom. 3.]
1 jS
X
4-5- X Yi x T
•M- * V° * t
Enharmonic
Chromatic
Diatonic
Enharmonic
Chromatic
Diatonic
Enharmonic
Chromatic
Diatonic
of Archytas,
\
Vof D
f Enttofthenes
idytnus,
+ 5
i? x « * ! = t it x-4i *.t = t- U x ¥ »'=!
m *i x ■■ * = * m ■* ■« x -
a x ii
v° = t
Enharmonic Croma molle
intenfum
Diatonum molle
. tonicum
— ditonicum -intenfum
rof Ptolemy.
"9 — j
x 's = f jequabile J.
GEOME-
