1.  1C   to 1G                                  segment  2/3    for 1G
2.  1G    " 1D Multiply  2/3      by 4/3, gives segment  8/9      " 1D
3.  1D    " 1A    "      8/9      "  2/3   "     "      16/27     " 1A
4.  1A    " 1E    "     16/27     "  4/3   "     "      64/81     " 1E
5.  1E    " 1B    "     64/81     "  2/3   "     "     128/243    " 1B
6.  1B    " 1F[#] "    128/243    "  4/3   "     "     512/729    " 1F[#]
7.  1F[#] " 1C[#] "    512/729    "  4/3   "     "    2048/2187   " 1C[#]
8.  1C[#] " 1G[#] "   2048/2187   "  2/3   "     "    4096/6561   " 1G[#]
9.  1G[#] " 1D[#] "   4096/6561   "  4/3   "     "   16384/19683  " 1D[#]
10. 1D[#] " 1A[#] "  16384/19683  "  2/3   "     "   32768/59049  " 1A[#]
11. 1A[#] " 1F    "  32768/59049  "  4/3   "     "  131072/177147 " 1F
12. 1F    " 2C    " 131072/177147 "  2/3   "     "  262144/531441 " 2C

Now, this last fraction should be equivalent to 1/2, when reduced to its lowest terms, if it is destined to produce a true octave; but, using this numerator, 262144, a half would be expressed by 262144/524288, the segment producing the true octave; so the fraction 262144/531441, which represents the segment for 2C, obtained by the circle of fifths, being evidently less than 1/2, this segment will yield a tone somewhat sharper than the true octave.  The two denominators are taken in this case to show the ratio of the variance; so the octave obtained from the circle of fifths is sharper than the true octave in the ratio expressed by 531441 to 524288, which ratio is called the ditonic comma.  This comma is equal to one-fifth of a half-step.

We are to conclude, then, that if octaves are to remain perfect, and we desire to establish an equal temperament, the above-named difference is best disposed of by dividing it into twelve equal parts and depressing each of the fifths one-twelfth part of the ditonic comma; thereby dispersing the dissonance so that it will allow perfect octaves, and yet, but slightly impair the consonance of the fifths.

We believe the foregoing propositions will demonstrate the facts stated therein, to the student’s satisfaction, and that he should now have a pretty thorough knowledge of the mathematics of the temperament.  That the equal temperament is the only practical temperament, is confidently affirmed by Mr. W.S.B.  Woolhouse, an eminent authority on musical mathematics, who says:—­

“It is very misleading to suppose that the necessity of temperament applies only to instruments which have fixed tones.  Singers and performers on perfect instruments must all temper their intervals, or they could not keep in tune with each other, or even with themselves; and on arriving at the same notes by different routes, would be continually finding a want of agreement.  The scale of equal temperament obviates all such inconveniences, and continues to be universally accepted with unqualified satisfaction by the most eminent vocalists; and equally so by the most renowned and accomplished performers on stringed instruments, although these instruments are capable of an indefinite variety of intonation.  The high development of modern instrumental music would not have been possible, and could not have been acquired, without the manifold advantages of the tempered intonation by equal semitones, and it has, in consequence, long become the established basis of tuning.”