74 Mr. Farey’s Letter on musical Intervals, &c. 
Yn the work last quoted, vol. XIII, p. 41, the 59 notes 
of Mr. Liston’s Scale will be found expressed in the nota- 
tion that has now been described; from which I have ex- 
tracted, and placed in Table I. the 12 notes of his original 
Scale already mentioned herein. 
The seven last lines of the above ‘Table have been added, 
in order to include the concords of minor Sixth, the minor 
Tone, the major Semitone and the Comma; and the Sehis- 
ma, the lesser Fraction, and the most Minute. This Table 
can hardly need further explanation : 1 will therefore pro- 
ceed to some further remarks. 
The expressions in col. 3 of the above Table, accurately 
express the values of the Intervals in cols. 4 and 2: and 
such is the peculiar and natural connection, between the 
rates of increase upwards in the ‘Table, for each of the terms 
=, fand m, that whatever result or truth appears conjointly 
from the three terms, after any Pepe of adding or sub- 
tracting Intervals in any manner, has been performed, the 
same result or truth appears also, from each of ats terms 
separately : there being here, no carrying or borrowing (in 
whole numbers, at least) from one column to another, as in 
common Arithmetic. ‘Thus, independently of the other 
two columns = and m, the f column, in every result of ope- 
rations performed with these expressions, as giving a rough 
value in artificial Semitones, or 12th parts of the octave : 
every like result in the m column will give a considerably 
more accurate value, in the artificial Commas of Nicholas 
Mercator, or 53d parts of the octave, very nearly : and 
every result of like operations in the © column, will give 
rigidly accurate results, in all such Euharmonic or untem- 
pered calculations as are alluded to above, and will approx- 
imate to the truth, abundantly sufficient for nearly every 
practical purpose of Harmonics, being extremely near to the 
612th parts of the octave, and to the 1ith parts of the 
major comma. 
When Tempered Systems are to be calculated, fractional 
parts of the major comma, expressed in Schismas, may be 
joimed with these artificial commas without at all disturbing 
the consistency of their results: thus, if the fLsotonie scale 
of equal Semiiones were required to be calculated, the flat 
ao of the Vth 1s known to be extremely near 
to =,th of a major comma, which is =, and the true Vth 
