COMMON MEASURES of Musical Intervals. In our article Co :6INIENS on A LE Intervals, we have shewn, that, strictly speaking, there elm be no such thing as a com mon measure or unit, by means of which all other in tervals might be expressed in whole numbers, or in de cimals that terminate, or even that circulate or repeat; Lut that all such numbers expressing intervals (being logarithms of a particular species) will have decimal values indefinitely large, without any law in the conti nuation of the same being discoverable. Before the important invention of Lord Napier, and the construc tion of copious tables of logarithms, even the best ma thematicians had but imperfect views of the relations and values of the prime ratios to which musical inter vals are allied, and hence the many mistakes and in consistencies in some of the best of the ancient writings on music, (see diristorenfanComMON M SURESO where they treat on the minute relations and values of inter vals. Mersennus, for instance, concluded from his cal culations, that 58} major commas made an octave, or 612 2 +12f+53m, instead of 638.979614 2+ 12f+55m, the real value of so many major commas, which error was detected by Nicholas Mercator, who, according to Dr William Holder, (Treatise, 1st edit. p. 79,) in an unpublished manuscript, mentions having calculated by logarithms, that there were more than 55 commas (55c +12/-2-Am) in an octave, and that he had " thence deduced an ingenious invention of finding and applying a least common measure to all harmonic intervals, not precisely perfect, hut very near it." Of the method or process by which Mercator obtain ed his series of artificial commas, answering to the 19 intervals, which Dr Holder has given as an extract from the manuscript, we are quite uninformed ; but to us it seems probable, from the relation, that having calcula ted the number of commas to the nearest whole number in each case, answering to the 19 intervals mentioned, and found them as follows, (see Plate XXX. Vol. II.) c=1, E=2, 1=3, S=4, S=5, S=6, t =8, T=9; 3rd=15, III=18, 4th=23, IV=27, and 5th=28; V= 33, 6th=38, VI=41, and 7E11=46; and VII=51, and V111=56 ; and trying the relations of these by addi tion and subtraction of these commas, he thus found, that all those from e to T inclusive, exactly answered to the known and corresponding values and relations of these intervals, as found by the multiplication of the terms of their ratios, according to the equations given in our articles A PO1031E, COMMA, ; and that again, from the 3rd to the 5th inclusive, the relations of these
.ommas were proper to each other, though each was an unit too much, when compared by the proper addi :Ions and subtractions with the first part of the series ; and, further, that from the Vth to the inclusive, the relations were true with respect to each other, but each was 1 comma more than the last part of the series, and two commas more than calculations from the first part would have given them ; and, lastly, that the Vile' and Vint° v, ere 3 commas too great for comparison with the first part of the series : and thus probably it was, that the in genious Mercator was led to reduce all the numbers to the first part, and so deduced his series of artificial commas, 1, 2, 3, 4. 5, 6, 8, and 9 ; 14, 17, 22, 26, and ; 31. 36, 39, and 44 ; 43 and 53 ; answering to the 19 intervals above mentioned, as Dr Holder has given them.
We are not aware. that this mode of accounting for Mercator's unexpected discovery of the curious proper ties of the above series was ever before published, or that the same attracted the notice of any curious person in these inquiries after Dr Holder, until the year 1807, when :1Ir John Farey sen. having extracted all the inter vals contained in the Overend, and other manuscripts which he had perused, and, for the convenience of fu turc reference, had, with considerable labour, reduced them all into one notation, by the intervals marked Z,f. and in, as shewn in part in Plate V. in vol. xxviii. of the Philosophical Magazine, and in our 30th Plate in Vol. II. Ile then quickly discovered, that the number of in's it. the last column of his Table, answered exactly to Mcr cator's number taxed to the same interval in Dr Ilol der's Treatise ; and nut only sheaved the reason thereof but his table of intervals, so arranged, furnished a far more accurate and extensive set of artificial commas, of under commas 6i2, to the octave, by extracting the number of Z's in the first column.