As the combinations which are to be dealt with are complex, forms of arrangement have been employed which reduce all the phenomena in practice and theory to a few simple types. They were not employed by General Thompson.
There are two forms of symmetrical arrangements. Those which I employ I shall distinguish co-ordinate symmetrical arrangement ; other forms employed by Mr. Poole and Mr. Brown may be called key-relationship symmetrical arrangments. Into these latter I shall not enter, but only remark that those which depend on co-ordinates possess all symmetrical properties in a more extended form than the others ; i.e., they include the key=relationship and more besides.
Mr. Brown's position relations are exactly the same as Mr. Poole's, with omission of the two series of sevenths.
In my co-ordinate symmetrical the arrangements of a number of equal temperament semi-tones, or twelfth parts of an octave, is taken as abscissa, and deviation or departure from the note thus arrived at as ordinate. Thus, the exact pitch of notes can be expressed by reference to co-ordinates in a plane.
To express a series of equal temperament (E.T.) fifths in this manner : the E.T. fifth is 7/12 of an octave. Taking twelve fifths up, we have seven octaves exactly. The result is expressed by abscissae only.
To express a series of perfect fifths in this manner :
The perfect fifth is seven semi-tones and a departure = .01955 E.T.S. = 1/51151 * .'. twelve fifths are seven octaves and a departure = .23460 E.T.S, the Pythagorean comma. The resulting ntoes have ordinates which increase uniformly as we pass along the series of fifths ; and the position arrived at after the twelve fifths has an ordinate which represents the Pythagorean comma when the fifths are perfect, and abscissa 7 x 12 = 84 or seven octaves.
The form of symmetrical arrangement employed in the generalize key-board, is arrived at by arranging a series of notes in the order of the scael as far as the abscissae are concerned, and taking for ordinates the distances proportional to those thus arrived at. That is to say, the ordinate of any note is proportional to its distance from a fixed point in the series of fifths.
The series of fifths is selected for the application of the conditions because it is the most convenient ; but the variations of all the concords in any system are linked together in such a manner that it is indifferent which is taken as independent variable, so to speak ; the results would always be the same.
The generalized key-board, of which the harmonium exhibited offers an example, may be convenienty descibed with reference to abscissae from left to right, and horizontal ordinates on plan from back to front. The vertical ordinates are one-third of the horizontal ones.
In the abscissae, half an inch corresponds to an E.T. semi-tone. Twelve semi-tones make an octave. The octave measures 6 inches in abscissa, and nothing in ordinate.
In the ordinates on plan, 3 inches correspond to the Pythagorean comma or departure of twelve fifths. Thus the difference between the ordinates of two notes on same abscissa, between which one series of twelve fifths lies, is 3 inches.
The ordinates of the intermediate fifths are increased by 1/4 inch at each step upwards in the series of fifths, so that twelve steps upwards in the series correspond to 3 inches.
I have describe the key-board as connected with the system of perfect fifths ; and it is so in this harmonium to all intents and purposes. But it is clear that if each fifth have any departure from E.T. whatever, this may be equally represented by the ordinates in question, as no use has been made of the amount of the departure ; and we can say that a key=board, constructed in the form of a co-ordinate symmetrical arrangement, forms a graphical representation of the interval relations of any set of notes belonging to a regular successions of fifths.
Thirds can always be referred to fifths. In systems such as that of perfect fifth - which we are dealing with here - by means of a theorem brought into notice by Helmholtz ; in other cases, in other ways.
The most important property of key arrangements which form graphical representations of their intervals is, that any combination of intervals has the same form to the finger on whatever notes or in whatever key it may be taken. Thus a common chord always has the same form.
No co-ordinate or key-relationship symmetrical arrangements, such as those of Mr. Poole and Mr. Brown, possess a similar property of more limited extent. In these it is, for instance, possible that a common chord may assume different forms to the finger in cases where the key relationship is differently assumed : not so in co-ordinate arrangements.
I will only allude to one property of the division of the octave into 53 equal intervals, according to which the harmonium exhibited is tuned.
The mode in which the number 53 is arrived at has been explined by me, as part of a general theory. But we can verify its properties independently by noticing, that if we take 31 units for the fifth of the system, then 12 x 31 = 372 and 7 x 52 = 371 ; so that we see directly without formulae, that the departure of twelve fifths = 1/53 of an octave = [] of a semi-tone, which is the error of the fifth of the system. Hence, we may say that the system of fifty-three is sensibly identical with a system of perfect fifths.
In the enharmonic organ recently constructed, I have applied to a generalized key-board of forty-eight notes per octave, Helmholtz's approximately just intonation, and also the mean tone system, which is of historical interest. Each system is brought on to the key-board separately by a draw-stop. In the same way all systems of interests are accessible ; it is this imployment of the key-board that I would at present commend to those who inquire into its utility.
onsidering the facilities that we see about us for manipulating just and approximately just systems, it is difficult to see why mere book knowledge should continue to be alone regarded in the study of this portion of the elements of music. When it is taught, for instance, that certain vibration ratios correspond to certain musical effects, the lesson should be taught experimentally ; as it is , musicians for the most part only know what consonances are from descriptions in books. As illustrations, I may point out that we have, in the harmonium now exhibited, the means of distinguishing three different kinds of minor thirds, whose ratios are 6:5, 32:27, 7:6 ; and these sound quite different to the ear. Again, Pythagoream thirds can be contrasted with exact thirds, the harmonic seventh compared with other forms of minor seventh, and numerous outher theoretical results reduced to practical knowledge.
Of the applications of the various systems, I will only say that in my opinion it is a mistake to apply ordinary music to them indiscriminatly. Just systems especially, which have both thirds and fifths nearly perfect, must be studies and written for before they can be used with advantage. I need hardly say that I think, when this is done, the advantage will be great.