the instrument and his voice, when he alters the pitch slightly.… When we require a delicate use of the muscles of any part of the human body, as, in this case, of the larynx, there must be some sure means of ascertaining whether success has been attained. Now the presence or absence of beats gives such a means of detecting success or failure when a voice is accompanied by sustained chords in just intonation. But tempered chords which produce beats of their own, are necessarily quite unsuited for such a purpose.'[1]
For performance in just intonation the three quartets of voices, strings, and trombones have a pre-eminent value; but as it requires great practice and skill to control the endless variations of pitch they supply, we are obliged to have some fixed and reliable standard by which they can at first be guided. We must be certain of obtaining with ease and accuracy any note we desire, and of sustaining it for any length of time. Hence we come back once more to keyed instruments, which do not present this difficulty of execution and uncertainty of intonation. The only question is how to construct such instruments with an adequate number of notes, if all the intervals are to be in perfect tune. Theoretically it is necessary that every note on the keyboard should be furnished with its Fifth, Major Third, and Harmonic Seventh, upwards and downwards. There should be Fifths to the Fifths, Thirds to the Thirds, and Sevenths to the Sevenths, almost to an unlimited extent. Practically these conditions cannot be fully carried out, and all instruments hitherto constructed in just intonation have been provided with material for the simpler modulations only. One of the best-known historical examples is General Perronet Thompson's organ, now in the collection of instruments in the South Kensington Museum. In each Octave this organ has forty sounds, which may be divided into five series, the sounds of each series proceeding by perfect Fifths, and being related to those of the next series by perfect Major Thirds. The interval of the Harmonic Seventh is not given. With a regular and consistent form of keyboard it would have been more successful than it was, but the idea of arranging the keys symmetrically had not then been developed. The first application of this idea was made by an American, Mr. H. W. Poole, of South Danvers, Massachusetts. His invention is described and illustrated in 'Silliman's Journal' for July, 1867. The principle of it is that keys standing in a similar position with regard to each other shall always produce the same musical interval, provided it occurs in the same relation of tonality. But if this relation of tonality alters, the same interval will take a different form on the keyboard. There are five series of notes, each proceeding by perfect Fifths:—(1) the keynotes; (2) the Major Thirds to the keynotes; (3) the Thirds to the Thirds; (4) the Harmonic Sevenths to the keynotes; (5) the Sevenths to the Thirds. The Major Thirds below the keynotes, which are so often required in modern music, as for instance in the theme of Beethoven's Andante in F, are not given. So that the range of modulation, though extensive, is insufficient for general purposes.[2]
Owing to the limited number of notes which keyed instruments can furnish, the attempt to provide perfect intervals in all keys is regarded by Helmholtz as impracticable. He therefore proposes a system of temperament which approaches just intonation so closely as to be indistinguishable from it in ordinary performance. This system is founded on the following facts:—We saw that in equal temperament the Fifth is too flat for exact consonance, and the Major Third much too sharp. Also that the interval got by four Fifths up (D—A—E—B—F♯) is identified with the Major Third (D—F♯).[3] Now if we raise the Fifths, and tune them perfectly, the interval D—F♯ becomes unbearable, being sharper than the equal temperament Third. But in a downward series of just Fifths the pitch becomes at each step lower than in equal temperament, and when we reach G♭, which is eight Fifths below D, we find that it is very nearly identical with the just Major Third of D, thus—
The best way of applying this fact is to tune a series of eight notes by just Fifths—say D♭, A♭, E♭, B♭, F, C, G, D; then a similar series forming just Major Thirds with these; whence it will result that the last note of the latter series (F♯) will form an almost exact Fifth with the first note of the former series (D♭).[4]
In applying the ordinary musical notation to systems of temperament of this class, a difficulty arises; for the Major Third being got by eight Fifths downward, would strictly have to be written D—G♭. As this is both inconvenient and contrary to musical usage, the Major Third may still be written D—F♯, but to distinguish this F♯ from the note got by four Fifths up, the following convention may be used. The symbols G♭ and F♯ are taken to mean exactly the same thing, namely the note which is eight Fifths below D. We assume G♭—D♭—A♭—E♭—B♭—F—C—G—D—A—E—B as a normal or standard series of Fifths. The Fifth of B is written indifferently /Gb or /F♯, the acute mark (/) serving to show that the note we mean belongs to the upward, and not to the downward series. The Fifth of /F♯ is written /C♯, and so on till we arrive at /B, the Fifth of which is written //F♯. In like manner, proceeding along a downward series, the
- ↑ 'Sensations of Tone,' pp. 505–510.
- ↑ The keyboard invented by Mr. Colin Brown of Glasgow, is similar in principle to Mr. Poole's, except that it does not give the two series of Harmonic Sevenths. See Bosanquet, 'Temperament.'
- ↑ In general when a series of Fifths is compared with a Major Third, the number of Octaves (by which we must ascend or descend in order to bring the notes into the same part of the scale) is not expressed, but can be easily supplied by the reader.
- ↑ The error, which is called a 'Skhisma,' is about the fifty-first part of a Semitone. This system, therefore, differs so slightly from just intonation, that we shall henceforward treat them as practically identical.
