COMMA, in the mathematical study of sound, is applied to two small intervals, which, by comparison and calculation, arise as the difference between the proportions of cer tain other intervals of the diatonic scale. The larger, but seldomer occurring C. is called the C. ditonicum, or the Pythagorean C., being the difference between the true octave, whose ratio is 2 : 1, and the interval which arises when the octave is obtained .by,tuning a progression of twelve perfect fifths, or arithmetically by adding their values together; by which process it is found that the last sound is greater than the true octave in the proportion of 531,441 to 524,288. The smaller C., C. syntonum, or C. of Didymus, is—First, The difference between the large whole tone, the ratio of which is 9 : 8, and the small whole tone 9 : 10, found in the compound of these ratios produced together respectively their antecedents, 9 and 9, and their conscquents, 8 and 10, to be 81 : 80. Second, The difference between the great limma, 27 : 25, and the great half-tone, 15 : 16, which is found by the same process, and then reducing the resulting ratio to its least terms, to be also 81 : 80. Third, The difference between the diesis, 128 : 125, and the diaschisma, 2,048 : 2,025, which by the same process gives 81 : 80; and lastly, the difference between the small limma, 133 : 128, and the small half-tone, 24 : 25, which again gives the proportions 81 : 80. The difference between
theC. ditonicum and the C. syntonum is exactly the schisma; therefore, the aggregate of the diaschisma and schisma, if they be added together, is neither more nor less than the syntonic comma. This C., again added to the diaschisma, makes up the dicsis, and added to the great half-tone, makes up the great limma. It follows therefore that, practically, two enharmonic tones in perfect tune never differ by a syntonic C., and it is wrong to say that d flat is higher than c sharp by a C., while the real difference is that of a diesis, 128 : 125. In the equal-tempered scale, these varieties do not exist. The term syntonic comes from the Greek, and means equal-sounding.
in military matters, is a temporary commander, in place of the real chief; such as a captain-commandant, lieutenant-commandant, etc. In foreign armies, the designation is more frequently applied than in the British, especially to the com manders of garrisons.