and so on. Let a regular system of fifths start from c. If they are positive, then at each step the pitch rises further from E. T. It can only return to c by sharpening an E. T. note.

     Suppose that b is sharpened one E. T. semitone, so as to become c; then the return may be effected

at the first; b in 5 fifths,

at the second b in 17 fifths,

at the third b in 29 fifths; and so on.

Thus we obtain the primary positive systems. Secondary positive systems may be got by sharpening bb 2 semitones ; and so on.

     If the fifths are negative, the return may be effected by depressing c# a semitone in 7, 19, 31... fifths ; we thus obtain the primary nega�tive systems ; or by depressing d two semitones, by which we get the secondary negative systems ; and so on.

     An instructive illustration may be made as follows; it requires too large dimensions for convenient reproduction here: -

     Set off on the axis of abscissa the equal temperament series in order of fifths, as above, taking about 10 complete periods. If the distances of the single terms are made 1 centimeter, this will take 1.20 m in length, starting from the origin on the left.

     Select a unit for the E.T. semitone of departure, say 1 decimeter.
     Rule a series of lines parallel to the axis of abscissa, at distances representing integral numbers of E.T. semitones, both above and below.

     Rule, parallel to the axis of ordinates, straight lines through the points representing the E.T. notes.

     Enter on the intersections the names of the E T. notes they represent. Thus the notes on the positive ordinate of c are c c# d..., and so on, each pair separated by 1 decimeter, and the notes on the negative ordinate of c are c-b-bb....

     If we then join the c on the left hand of the axis of abscissa to all the other c's on the figure, except, of course, those on the axis, we obtain a complete graphic representation of all the systems whose orders are included. The rth order is represented by lines drawn to the c's in the rth line above, the - rth by the lines drawn to the c's in the rth line below. This illustration brings specially into prominence the singularity of multiple systems, as all the multiples of any system lie on the same straight line with it, and the representation fails to give all the notes of such systems.

Multiple Systems.

     Multiple systems are such that the number of divisions in the octave (kn) in any such system is a multiple (k) of the number of divisions (n) of some other system.

     Multiple systems have not been as yet practically applied.

     These systems are not strictly regular; for though their fifths are all equal, yet they do not form one continuous series, but several. They are strictly cyclical, i. e. they divide the octave into n equal intervals.

     Theorem v. A multiple system, kn, may be regarded as being of order kr, where n is a system of order r.

     For, n being a system of order r, r + 7n is a multiple of 12; [therefore] also k(r + 7n) is a multiple of 12, which is the condition that the system kn be of order kr.

     This is useful in the investigation of systems of the higher orders.

     If n is a multiple of 12, the system is a multiple of the E. T., and of order zero.

     In the illustration described under Th. iv. the notes of a multiple system (kn) are the same as those of system n, until the latter is com�plete. The rest of the representation consists simply of the same notes repeated over and over again. To obtain the rest of the notes we should have to change the starting-point.

     On the whole, we may regard the system kn as consisting of k different systems n, having starting-points distant from each other by 1/k of the unit of the system n.

     It follows immediately that the system kn is of the krth order ; for in every unit of the system n there are k units of system kn ; and so in r units of system n then are kr units of system kn.

     Any system, when n is not a prime, can be regarded as a multiple system.

     Thus the system of 59 is of the 7th order; 118 consequently a multiple system of the 14th order, in which point of view it is of no interest; but, casting out the 12 from the order, it may be also regarded as an independent system of the 2nd order, in which point of view it is of considerable interest.

Formation of Major Thirds in Positive and Negative Systems.

     The departure of the perfect third is -.13686. Hence negative systems (where the fifth is 7 - δ) form their thirds in accordance with the ordinary notation of music. For if we take 4 negative fifths up, we have a third with negative departure (-4δ) which can approximately represent the departure of the perfect third. Thus c# is either the third to a, or four fifths up from a, in accordance with the usage of musicians.

     Positive systems form their thirds by 8 fifths down; for their fifths are of the form (7 + δ), and 8 fifths down give the negative departure (-8δ). Thus the third of a should be db, which is inconsistent with musical usage. Hence positive systems require a separate notation. Helmholtz proposed a notation for this purpose, which, however, is unsuitable for use with written music. The following notation is here adopted for positive systems in general; it is not intended to be limited to any one system, like Helmholtz's. In fact it may, on occasions, be used even for negative systems.

Notation for Positive Regular Systems.

     The notes are arranged in series, each containing 12 fifths, from f# up to b. These may be called duodenes, adopting a term introduced by Mr. Ellis. The duodene

f# - c# - g# - d# - a# - f - c - g - d - a - e - b,

which contains the standard c, is called the unmarked duodene. No distinction is made in these series between such notes as c# and db. These signs refer only to the E.T. note from which the note in question is derived; the place in the series of fifths is determined by the notation. Continuing the series to the right, each note of the next 12 fifths is affected with the mark / (mark of elevation), drawn upwards in the direction of writing. These notes join on to the unmarked duodene as follows: -

e - b - /f# - /c# - /g# . . . ,

and so on.

     Thus /c is 12 fifths to the right of c, and the interval /c - c is the departure of 12 fifths. The next duodene to the right is affected with the mark //, which joins on to the last as before: -

/e - /b - //f# ....,

and so on.

     Proceeding in the same way, we have notes affected with such marks as ///, ////.

     Return to the unmarked duodene, and let it be continued to the left; the notes in the next duodene on the left are affected with the mark \, (mark of depression), drawn downwards in the direction of writing. The junction with the unmarked duodene will be

\c - \g - \d - \a - \e - \b - f# - c# ....

     The next junction on the left will be

\\e - \\b - \f# . . ;

and, proceeding in the same way, we have notes affected with such marks as \\\, \\\\.

     Thus c - e\ is a major third determined by eight fifths down in the whole series; and \e will have the departure (- 8δ) from the E.T. note e derived from c.

Notation applicable to all Regular Systems, Negative as well as
Positive.

     As this notation simply consists of a determination of position in a continuous series of fifths, it may be applied to all regular systems, positive or negative ; but, it is not commonly needed for negative systems, it is not generally applied to them.

Formation of Harmonic Seventh in Positive and Negative Systems.

     The harmonic seventh is the interval whose ratio is 7:4. It affords a smooth combination, free from beats.

     The departure of the harmonic seventh from the note which gives the E. T. minor seventh is -.31174 (Rule I.).

Concords of regular and regular cyclical systems....398.

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