2

1864.]

on Instruments with Fixed Tones.

99

the major scale of D are played as G BD, D F#†4, and A‡CE, instead
of +4 C+E. The dominant chord is therefore too flat by a comma, and
in passing from the chord of A to that of D, as in the ordinary cadence,
the note A has to be changed into †A. If A is the highest or lowest note
in the chords, the effect is decidedly bad. The flatness of the "leading
note" C, in place of C#, although only a comma in extent, is felt as
annoying in the succession ‡c#, d. The result is such that it would not be
worth while to invent new instruments with such a defect in common scales.
On the same instrument I have the three chords of the major scale of A
tuned as DFA, A‡CE, E‡G# B, in which the subdominant chord is
now a comma too sharp. As the subdominant is a much less important
chord than the dominant, the effect is better, but trouble arises from having
occasionally to alter the tonic note A itself. Even the dissonance of the
dominant seventh, when played as E G# Bd is perceptibly harsher than
the correct EG B‡d (both forms lie on the instrument), although the
added seventh d now forms a true minor third with the fifth B, whereas
the correct note ‡d forms a dissonant Pythagorean minor third with the
same note B. When, however, the first E is omitted, the chord of the di-
minished fifth G# B‡d is not so pleasant as G# Bd. Again, on the same
instrument, instead of having DFA, as the synonymous major of
‡dfa in the scale of a minor, I have only D F#†4, which is a comma too
sharp. The rarity of the chord, however, renders the bad effect of less
importance. Again, I am obliged to modulate from D major to ‡d minor
instead of d minor. Even here the error of a comma is perceptible.
general result, therefore, is that commatic substitution, even within the same
melody or succession of chords, is inadmissible in just intonation.

The

Professor Helmholtz (op. cit. pp. 433 & 484) has suggested what may be termed schismatic substitution, or the use of one note for another which only differs from it by a schisma, the eleventh part of a comma. Having one concertina tuned to equal temperament, and another to just intervals, the equation ||g=g has enabled me to test this suggestion by practice. I find that in slow chords, the altered fifth cg, the altered major third gb, and the altered minor third eg are all decidedly, though only slightly, dissonant. In rapid chords the effect would be necessarily much less perceptible. Such chords as CEG, e gb are far superior either to > the Pythagorean C+E G, teg tb (of which I can produce the counterparts F+AC, dfta), or the still worse tempered chords C||E||G, ||e||g ||b. If we modified Professor Helmholtz's suggestion, and, where practicable, used only entire chords which are too flat or too sharp by a schisma, so that the schismatic errors would only occur in harmonies where a note was prolonged from a chord to which it belonged into another for which it was too sharp or too flat by a schisma, then there could be no objection whatever to schismatic substitution, which would be quite inappreciable in melody. Now schismatic substitution will materially reduce the number of different tones required. By referring to Table IV. it will be seen that all the

VOL. XIII.

I

MICHIGAN

tones in Table V., lines 1 to 8, throughout all the columns are exactly one schisma flatter than the corresponding tones in lines 10 to 17. Hence we only require the tones in lines 5 to 13 in order to reproduce the whole Table, with the help of schismatic substitution. It is, however, more convenient to use columns III., IV., lines 14 to 17, in place of columns I. and II., lines 5 to 8; and columns VII., VIII., lines 1 to 4, in place of columns IX. and X., lines 10 to 13. In this case only 48 tones will be required. If the schismatic substitution of tf, b, c for e#, g, to were allowed, which would introduce three schismatic errors of no great importance, the number of tones would be reduced to 45, which is the lowest possible number of tones by which a complete scale can be played. All these tones are enumerated in Table III.

There are several ways of realizing such a scale in whole or in part*. The following appears to be the most feasible, as it would render the mere mechanism of playing a perfect scale on an organ or harmonium easier than that of playing the tempered scale on the same instruments.

On a board of manuals similar to that now in use for the organ, introduce two additional red manuals (of the same shape as the black, but with a serrated front edge to be recognizable by blind and colour-blind performers, as in some cases on General Perronet Thompson's organ) in the two gaps between B and C, and between E and F, so as to make 14 manuals in all. Let there be 16 stops worked as pedals with the foot, as in Mr. Poole's Euharmonic Organ (loc. cit. p. 209). Let one of these stops give the equally tempered tones to the manuals, so that any piece could be played in the tempered scale, and thus compared with the same piece when played with just intervals. Let the 15 other stops give the tones required for the 15 keys C to +C, as shown in Table II., and be numbered 7b, 6b...1b, natural, 1#....7#. When any pedal is put down, let the seven white manuals give the seven tones of the primary major scale of the corresponding key, and the seven coloured manuals give seven out of the nine other tones required to complete the key, omitting the acute fourth (which would be found in the key of the dominant) and the grave seventh (which would be found in the key of the subdominant). To the right of each white manual let there be its conjugate coloured manual, of such a value that, if the seven tones of the major scale be indicated by the numbers 1 to 7, the tones corresponding to the manuals in any key may be

Table II. shows the tones associated with the manuals in each stop; capital letters indicate white manuals, small letters black, and small

*Singers and performers on bowed instruments and trombones can produce any scale whatever. Other instruments are more limited in range and would require special treatment, similar to the "crooks" of the horns and the various clarinets.

capitals red*. By this arrangement the fingering of every key would be the same. The performer would disregard the signature except as naming the pedal, and play as if the signature were natural. Table V. would inform him whether the accidentals belonged to the key, its dominant, or any other key; and if they indicated another key, he would change the pedal. It would be convenient to mark where a new pedal had to be used; but no change would be required in the established notation §.

Mr. Poole's organ, which suggested the above arrangement, has 11 stops, from 5 to 5#, and only 12 manuals, which appear to be associated with the following tones on each stop:

The two manuals whose notes are put in parentheses are inadequately described. Mr. Poole's scale does not include the synonymous minor chords, which he plays by commatic substitution.

Another method of realizing such a scale is by additional manuals and additional boards of manuals. Thus three boards of manuals, each with 23 manuals, containing the tones in Table V. cols. III. to VIII., lines 4

On examining Table II. it will be found that 10 different tones lie on each pair of manuals, so that there are only 70 different tones. The two missing tones are, necessarily, t†f (the acute fourth of the key of †C#), and ‡‡ (the grave seventh of the key of Cb); and to this extent the scheme is defective. It would probably be more convenient to the instrument-maker to use all the 70 tones in this arrangement than to take the inferior number 45 due to schismatic substitution. A full-sized harmonium at present employs from 48 to 60 vibrators to the octave, so that the mechanical difficulties to be overcome in introducing 70 are comparatively slight. By omitting the two very unusual keys of C and †C#, the 8 tones denoted by t‡db, 1Fb, ‡‡ƒ, abb and †D, gx, +B, ttb in Table II. would be saved, and the number of vibrators required would be reduced to 62, nearly the same as that actually in use. As each new key introduces 4 additional tones, and the key of Chas 14 tones, the number of vibrators required for any extent of scale is readily calculated. Thus for the 11 keys from 5 flats to 5 sharps, or D, Aɔ, Eɔ, Bɔ, F, C, G, D, A, E, B, which is Mr. Poole's range, and is sufficiently extensive for almost all purposes, only 4× 10+14=54 vibrators to the octave would be required, distributed over 11 stops (exclusive of the tempered notes); and such a number of vibrators and stops is in common use. § If in Table V. we reject the marks †,‡, consider 16 4=27 C, 64 E=81 C, 2187 2048 2187'

128 B = 243 C, 2048'

leaving the value of the other letters un

changed, the Table will represent the Pythagorean relations expressed by the usual notation (which is quite unsuited to the equally tempered scale). The chords thus formed were too dissonant for the Greek or Arabic ear to endure, although Drobisch and Naumann (loc. cit. ad finem) desire this system to be acknowledged as "the sole, really sufficient acoustical foundation for the theory of music" (als einzige, wahrhaft genügende akustische Grundlage der theoretisch-mukalischen Lehre).

to S, 7 to 11, and 10 to 14 respectively would be nearly complete. The manuals might be similar to those on General T. Perronet Thompson's Enharmonic Organ, which has 3 boards, with 20, 23 and 22 manuals respectively, and contains the chords in Table V. cols. III., lines 6 to 11; IV. 6 to 12; V., VI., VII., 5 to 12; VIII. and IX., 6 to 12 (four chords belonging to col. IX., lines 6 to 9, are not in the Table, but can be readily supplied, as well as the additional lines 0, -1, named below).

Euler's "genus cujus exponens est 2". 37. 52," as developed in his Tentamen, p. 161, must be considered as adapted for an instrument with two boards of ordinary manuals, such as some harmoniums are now constructed. His "soni primarii" would occupy the lower, and his "soni secundarii" the upper board. If to these we add their schismatic equivalents, inclosed in brackets, and distinguish white and black manuals by capital and small letters as in Table II., Euler's scheme will appear as follows, where the notation interprets his arithmetical expressions of pitch ("soni"), and not his notes ("signa sonora "), which are too vague.

EULER'S DOUBLE SCHEME.

Schism. Equival... [‡C, tab, "Soni Secundarii " B, c,

Upper Board.

‡D, , F, IF, gh, IG, d, B2, 12, Cx, d‡, †E, E‡, †ƒ‡, Fx, g‡, †A, ta Lower Board.

"Soni Primarii " C, Ic, D, ‡d, E, F, f G, A, Schism. Equival... [‡Db, ‡‡ɑb, Eɔɔ, ‡Ã, ‡Fɔ, ‡Gɔɔ, ‡gh, Ahh, ‡ah, ‡B?”, ¡l

Although it is evident from his notation that Euler regarded schismatic equivalents as identities, he has not especially alluded to them. The above scheme would contain Table V. col. V., lines 0 to 14, and the major third †† in 15 (with the schismatic error of ¶¶¶‡DF for B‡D F), col. VI. 1 to 15; VII. 9 to 24; VIII. 10 to 24; IX. 18 to 24; III. -1 to 5; IV. 0 to 6. It would be therefore nearly complete in major scales, but would have only ‡d, a, e, b, f, c, g minor, and their comparatively useless schismatic equivalents. It would have no single complete key, and would therefore require many commatic substitutions in modulation, and the use of the Pythagorean major third in the major chords of the comparatively common minor scales of If, ‡c, Ig. If only the "soni primarii" of the lower board are used the substitutions become very harsh, as for example D F, D F 4 for B D F, DF†A.

Euler's "soni primarii" may be compared with Rameau's scale *, which was as follows,

C, tc, ID, te, E, F, tf, G, g, 4, †, B,

* Traité de l'Harmonie, 1721. The values of the tones are determined from his arithmetical expression of the intervals.

and therefore only contained the following perfect harmonies, and two perfect scales, A major and a minor :

Prof. Helmholtz has tuned an harmonium with two boards of manuals, somewhat in Euler's manner, as follows:

:

HELMHOLTZ'S DOUBLE SCHEME.

Upper Board.

Schism. equiv. [C, db, ID, b, F, F, F, G, α, A, b, c] Tones tuned.. B, tc, Cx, d‡, †E, †E‡, †ƒ‡, FX, g, GX, †a‡, †B.

Lower Board.

Tones tuned.. C, c, D, td, E, E, F, G, A, a, B Schism. equiv. [Dbb, ‡ab, Ebb, ‡eb, ‡Fb, ‡F, ‡gh, Abb, tab, ubb, ‡bb, ‡C].

This scheme has nearly the same extent and the same defects as Euler's. The concertina, invented by Prof. Wheatstone, F.R.S., has 14 manuals to the octave, which were originally tuned thus, as an extension of Euler's 12-tone scheme.

C, tc, D, td, E, te, F, f, G, ‡g, A, tab, B, †Ùɔ.

It possessed the perfect major and minor scales of C and E. The harshness of the chords +B DF, D F A, for B D F, DF†4 has, however, led to the abandonment of this scheme, and to the introduction of a tempered scale. I have taken advantage of the 14 manuals to contrive 4 different methods of tuning, so that 4 concertinas would play in all the common major and minor scales. Two of these I have in use, and find them effective and very useful for experimental purposes. The following gives the arrangement of the manuals in each, together with the scales possessed by each instrument, major in capitals, and minor in small letters. Where commatic substitution makes the dominant chord too flat in major scales, parentheses () are used; where it makes the subdominant chord too sharp, brackets [] are used. Minor scales in brackets have only the subdominant tone too sharp.

The major chord G BD and the tone C being common to all four instruments, determine their relative pitch. The method of tuning these and all justly intoned or teleon* instruments is very simple. C being tuned to any standard pitch, the fifths above and below it are tuned perfect. To any convenient tone thus formed, as Citself, form the major thirds above,

* A convenient name, formed from réλeov diáornμa, a persect interval.