EVIL. See THEOLOGY.
EULER's LOGARITHMS, for musical calculations, are a series of artificial numbers invented by this author, for expressing musical intervals in their relations decimally to the octave, as unity, forming a set of BINARY Loga rithms, which see. The writer has known a learned stu dent of Euler's famous work on music, wherein these logarithms are used, stopped for want of being able to deduce common logarithms and lengths of strings from Euler's numbers affixed to the intervals ; not consi dering, that one constant ratio must obtain between the corresponding logarithms in any two tables of different kinds, and another constant ratio between their recipro cals respectively.
In the present case, the reciprocal logs. (or those which increase with the numbers or intervals) of Euler's, and the common log. of the octave, (or ratio are, 1 and .30103 respectively, nearly : we have only, there fore, to multiply any of Euler's log. by .30103, to obtain nearly its recip. common log. If, for example, the me dals semitone, the first of Euler's notes above F, were given, we have .076814 x .30103= .0231233, whose re cip. .9768767 is the common log. near enough for most experimental purposes, and the number answering to this last is .94815, the length of string to the unison 1.
In like manner, if Farcy's artificial commas were want ed from Euler's logs, since both of these are of the na ture of reciprocal logs. the octave being expressed by 612 and 1 in them respectively, we have only to multi ply Euler's logs. by 612, to obtain nearly their artificial commas. The last example will stand thus : .076814 X 612=47.010168, or 47, as in the schisma column (z ) of Plate XXX. Vol. II.; and it may be remarked hereon, that whenever the product approaches a whole number, either above or below it, that such wnole number is, in general, the proper number of artificial commas sought.
EULER's SCALE of musical intervals. It is proba ble, we think, that 111. Euler made the first, though an unsuccessful attempt, at the grand harmonic improve ment, (which Mr Liston has lately effected, by his scale of perfect harmony, and his EUHARMONIC Organ, which sec), by endeavouring, on a very limited scale, to avoid tempered or imperfect concords, by means of more notes introduced in the octave than the 12 in common use.
M. Euler extended these to 24, eight of which new notes were only a major comma higher than his notes Dt:;, E, Fes, A, B12, and B ; and four others of them a minor cordina lower than his notes D, F, 6, and C: at the same time, that three notes of his original scale fer from Liston's, viz. and B bp, in having the24 128 5 9 ratios — and instead of — —, and —, which 25' 75' 225 135 6 16 are Liston's notes respectively.
In order to facilitate the labours of those, who, like the gentleman alluded to in the last article, may be de sirous of trying, either by calculation or experiment, the effect and extent of this scale in producing perfect har monics, we have been at the trouble of reducing M. Eu ler's vingtquatreave scale, from the octave F to f, in which he has published it, to that of C to c, in which all musical scales are given in our work ; and we have added in the columns of the following Table, the lengths of strings, and number of artificial commas, answering to each of Euler's notes ; and in the last column but one, we have set down the notes on Liston's organ, (see the Phil. Mag vol. xxxix. p. 419, or the Monthly Mag. vol. xxxvi. p. 217,) whereon the several chords in this sys tem might be tried, and whence these notes might be transferred to a new instrument, if it were thought worth any one's while to make such a one, having a pe dal for bringing on the eight acute notes, and another for supplying the four that are minor comma flats, in the place of their respective original notes, as often as they are wanted in performance, in this scale proposed by M. Euler. The last column shews the numeral value of the notes, major and minor, above C.
The want of minor consonances above C, are very ob vious in the above systems, which most probably M. Eu ler never tried any more than the equal temperament, although his name has been often enrolled among the very numemus theoretic recommenders of the IsoToxic system. (f)