H. W. Poole on Perfect Haney etc. 19 
natural scale as its third. If this is desirable, I have provided 
a finger-key at the back me of each black key, giving the minor 
third to the key-note, to which this black key gives the major 
third.- For example, at the end of the key e, is Eb, minor third 
to C. This key is on a vertical wire with a circular ib 
above the black key; and the whole key can be rem 
found to be in the way when not wanted. The sound, if intro- 
duced, = form a series VI ($12), and will ses ‘but eight 
tra or, ipes, whatever the number of signatures. The 
key-note of the ninth signature above, is but 0:09 "of comma 
sharp for a minor third; but applying the correction ($80) 
needed for other notes, these minor thirds are made cePene ¥ 
cally perfect. Thus the key-notes (see enharmonic table, p. 1 
C to Dy, become the series Dbb to Eb (series VI, $12), minor 
thirds respectively above the key-notes Bpb to 
y transposing a scale by fifths, above or below, we first 
obtain a pitch widely sees and afterward approach _ 
from which we started. For example, beginning with C w 
have G far removed, but in the bast transposition comes D, iale 
one diatonic interval—the tone—above Proceeding in the 
same pee we shall ascend until at the seventh ste 
rge chromatic semitone above C. Attwelve fc 
By i is sages above C by what is called the comma of ‘ap fie 
caer or 1:09252 comma. This is the “circle” of the equal 
temperament, where the tuner divides this excess among the 12 
fifths: making tolerable fifths, but intolerable substitutes for 
thirds. With twelve other notes for thirds, fair intervals could 
be had. At each successive twelve transpositions we rise by 
same comma, until at fifty-three sharps the key-note is closer 
than ever to the starting | sate ep 0°16813 of comma sharp. 
If this small interval is divided among the 53 fifths, leaving 
each flat 000816 of comma, the circle will meet, and the octave 
will be divided into 53 equal intervals.* These are nearly exact 
for all the combinations of the triple diatonic scale, in these 53 
ree The fifths are as stated. e major thirds are 0065 of 
near the limit where they draw into tune 
. agai! In equal temperament they are oe 0°63582. 
37. The following table, page 21, serves to show: 1. this equal 
* This is most readily done by its logarithm, viz., bey "i 88943 ,04142 ; 
SS a of the oct 
pace ooresionstg Mr. Briggs, in Giactvetiog hin logutiinen, in otder Jona 
first small ee: began wi th the number 10, and its logarithm 1, and extract 
root of the ast number, and bisected its ts logari till he 
at the 58d sad Bath roo ts and their annexed logarithms as agers below : ee 
nd that th e decimal inthe natural numbers are to each other in the ratio of the 
logarithms, or as tk a 
Logarithms. 
100000, nol preempt ener 0-00000,00000, 00000, 11102,23024,62515,65404 
be 54 | 1-00000,00000,00000, 12781, 91493, 20032,35 000000, 700000;00000,05851" 11512,31257,82702 — 
