in eadem consonantia neque in duarum consonantiarum successione misceri possunt, error etiam ab acutissimo auditu percipi non poterit." It will appear in the sequel that these assertions, when tested by experiments on instruments with fixed tones, are all incorrect.

The musical scale has formed the subject of many recent investigations* ; but I have been unable to find a complete account of the necessary conditions to be fulfilled by a perfect scale, the least number of fixed tones required, and the practical means of producing them uncurtailed without inconvenience to the performer, although instruments which produce a limited number of just tones have been practically used by Gen. Perronet Thompson, Mr. Poole, Prof. Helmholtz, Prof. Wheatstone, myself, and others. This is therefore the subject of the present paper.

The following notation is employed. I have introduced it for the purpose of supplying a want which has been greatly felt by all writers on the theory of music. It is founded on the principle of retaining the whole of the usual notation unaltered, but restricting its signification so as to prevent ambiguity, and introducing the smallest possible number of additional signs to express the required shades of sound with mathematical accuracy, selecting such signs as are convenient for the printer, and harmonize with the ordinary notation of accidentals on the staff.

A letter, as C, called a note, will represent both a certain tone and its pitch, defined to be the number of double vibrations in one second, to which the tone is due. The letters D, E, F, G, A, B represent other

tones and pitches, so that

8D=9C, 4E = 5 C, 3 F = 4 C, 2 G = 3 C, 3 A = 5 C, 8B = 15 C.

and similarly for other letters. The pitch of c is that of the "tenor or middle c," usually written on the leger line between the treble and bass staves; and the other letters are noted on the staff as usual in the scale of C major.

*Gen. T. Perronet Thompson, F.R.S., Instructions to my daughter for playing on the Enharmonic Guitar, 1829; Just Intonation, 6th ed. 1862. H. W. Poole, On a perfect musical Intonation, Silliman's American Journal of Science, 2nd ser. vol. ix. pp. 68 and 199. W. S. B. Woolhouse, Essay on Musical Intervals, 1835. Prof. A. De Morgan, Cambridge Philosophical Transactions, vol. x. p. 129. M. Hauptmann, Die Natur der Harmonik, Leipzig, 1853. M. W. Drobisch, Ahhandlungen der Fürstlich Jablonowskischen Gesellschaft, 1846; Poggendorff's Annalen, vol. xc. C. E. Naumann, Ueber die verschiedenen Bestimmungen der Tonverhältnisse, Leipzig, 1858. Prof. H. Helmholtz, Lehre von den Tonempfindungen, Braunschweig, 1863. To this last writer we owe the first satisfactory theory of consonance and dissonance.

The following symbols always represent the fractions, and are called by the names written against them:

The name and pitch of the tones represented by any such notes as

and the ratio of their pitches to the corresponding notes in the scale of C major is therefore precisely indicated. In ordinary musical notation on the staff, it is only necessary to prefix the signs †,‡, 7, ¶, b, to those already in use. These symbols suffice for writing any tone whose index is the product 2. 3". 5o. 7 (see Tables I. and III.). For equally tempered tones, when it is necessary to distinguish them, the sign || is prefixed to the usual names, and read "equal." Since

llg: c = 27: 1 = 0·998871384584 × 3,

and

bg: c

=

bg, and hence represent the

by

we may without sensible error consider g equally tempered scale

C, c, d, e, le, If, fg, |, ||a, 86, ||8

||ab, lla, lub, 116

c, ¶3, 2d, ¶3¿, b'ie, ¶f, Lotƒt, bg, ¶^a?, b3ta, ¶2, stb.

In calculating relative pitches or intervals, and in all questions of tem

perament, it is most convenient to use ordinary logarithms to five places, because the actual pitches, and the length of the monochord (which is the reciprocal of the relative pitch), can be thus most easily found. In Table I. the principal intervals are given as fractions, logarithms, and degrees. If we call 0.00568 one degree, then 53 degrees=0·30104=log 2-0·00001, and 31 degrees=0·17608=log -0.00001. If we moreover represent the addition and subtraction of 0.00035 (or one-sixteenth of a degree) by an acute or grave accent respectively, then 17' degrees=0·09691=log 5, and 1' degree=0·00533-log-0.00007. Two numbers of degrees which differ by a single accent of the same kind, as 17′, 17′′ represent notes whose real interval is a schisma (thus e has 17' degrees; and dx, =¶e, has 17" degrees), having a difference of logarithm=0.00049 or 0' degrees +0.00014. By observing this, degrees may be very conveniently used for all calculation of intervals between tones of pitches represented by 2m. 3". 5o. Table IV. contains a list of tones which differ from each other by a schisma, and will be useful hereafter.

The conditions of a perfect musical scale are not discovered by taking all the tones which can be expressed by one of Euler's "exponents," nor by forming all the tones which are consonant with a certain tone, and then all the tones consonant with these, as Drobisch has done. Such processes produce many useless, and omit many necessary tones. music depends on the relations of harmonies, and not necessary to find what consonant chords of three tones are most closely connected *.

Since modern on scales, it is

Three tones whose pitches are as 4: 5: 6, or 10: 12: 15 form a major or minor consonant chord respectively. The same names are used when any one or more of the pitches is multiplied or divided by a power of 2, notwithstanding the dissonant effect in some cases. Thus, C: E:G= 4: 5: 6 is a major, and c: te: g=10: 12: 15 is a minor chord, and the same names are applied to e: g2: c1-5: 2×6: 22 × 4, and G: †eb: c2= 152x 122 x 10, although these chords are really dissonant (Helmholtz, ib. p. 333-4). I shall consequently use a group of capitals, as CEG, to represent a major chord, and a group of small letters, as c teg, to represent a minor chord, irrespective of the octaves. The three notes in this order, being the first, third and fifth of the major or minor scale commencing with the first, are called the first, third and fifth of the chords respectively. Both chords contain a fifth, a major and a minor third. If the interval of the fifth is contained by the same tones in a major and minor chord, as

*There are consonant chords of four tones, such as gb dzf, and these are insisted on by Poole (loc. cit.); but, though they are quite consonant and agreeable, and much pleasanter than the dissonant chords by which they are replaced, such as gb d2 f2, they do not form a part of modern music, for reasons clearly laid down by Helmholtz (op. cit. p. 295). Dissonant chords must always arise from the union of tones belonging to two consonant chords, or from the inversions of consonant chords; and therefore their tones are determined with those of the others.

CEG, cte g, or A‡CE, ac e, the chords are here termed synonymous. If the interval of the major third is contained by the same tones, as CE G, ace; or +E GB, cte g, they are termed relative. If two chords, major or minor, have the fifth tone of the one the same as the first tone of the other, as FAC, CEG; ftab c, e te g; ftab e, CEG; FAC, cte g, they are here termed dominative. If a chain of three such dominative chords be formed (as FAC, CEG, GBD, or ƒta? c, e te g, g to d, the minor and major chords being interchanged at pleasure), the first is called the 'subdominant, the second the tonic, and the third the dominant. Three such chords contain seven tones, and if such octaves of these tones are taken that all seven tones may lie within the compass of one octave they form a scale, of which 24 varieties can be formed by varying the major and minor chords, and beginning with the first of any one of the three chords. These scales include all the old ecclesiastical modes and several others. If all three chords are major and the scale begins on the first of the tonic chord, the result is the major scale, C, D, E, F, G, A, B, c. If all three chords are minor and the scale begins on the first of the tonic chord, the result is the minor descending scale, ca, tb, tab, g, ƒ, teb, d, c. If the first and second are minor, and the third major, or if the first and third are major and the second minor, we have the two usual ascending minor scales, c, d, ten, f, g, taɔ, b, c2, or c, d, te, f, g, a, b, c2. Three major chords may therefore be considered to represent a major scale, but. both major and minor chords are necessary for the various minor scales. If to each of three dominative major chords we form the relative and synonymous minor chords, the synonymous and relative majors of these, and the relative minor of this synonymous major, we shall have a group of 9 major and 9 minor chords, which I shall call the key of the first of the tonic chord. Thus the following is the

These chords contain 16 tones, which, when reduced to the compass of the same octave, form the complex scale c, ‡c#, ‡d, d, teb, e, ƒ, (†ƒ), ‡ƒ‡, g, tg, tab, a, †, (b), c2, of which the acute fourth (†), and the grave seventh (16), have been enclosed in parentheses, as being of rare occurrence. From this complex scale 54 scales of 7 tones each may be formed, similar to the 24 scales already named. A selection of 12 tones, such as c, ‡c‡, d, te, e, f, ‡ƒ‡, g, tab, a, tlo, b, c forms the so-called chromatic scale, which, however, has no proper existence except in equal temperament.

Now proceed to form a series of seven dominative major chords, as

EIG Bb, B IDF, FAC, CEG, GBD, DF‡†4, †4 C‡†E, and form the five related chords of each as before. The result will be five keys, as those of Bb, F, C, G, D, such that the primary major scales of each will have either two major chords, or one major chord in common with the original primary major scale. I call these five keys the postdominant, subdominant, tonic, dominant, and superdominant keys, and the whole group of 21 major and 21 minor chords, with the 30 tones which they contain, I term the system of the first tone of the tonic chord of the original primary major scale, which tone may be called the tonic of the system.

A piece of music is written in a certain system, determined by the compass or quality of tone of the instruments or voices which have to perform it, and rarely exceeds that system*. It is only in the system that the true relation of the tones of a piece of music, the course and intention of the modulation, and the return to the original key or scale can be appreciated. I have not yet found these relations fully expressed in any theoretical work on music; but their full expression was necessary to the solution of the problem here proposed.

It will be found practically that only 11 systems are used in music. These are, in dominative order, the systems of D, A, E, B, F, C, G, D, †A, †E, †B, which contain the 11 keys of the same name, together with the 4 keys of Cb, G, and +F#, +C. In Table V., columns III. to VIII., the whole of the major and minor chords of these 15 keys are exhibited in dominative order §. This Table, therefore, furnishes the tones which must be contained in a perfect musical scale of fixed tones, or the conditions of the problem.

On examination it will be found that these six columns contain 72 different notes. Hence the extent of a perfect scale is fixed at 72 tones to the octave. It is therefore six times as extensive as the equally tempered scale. Some means of reducing this unwieldy extent is required. The most obvious is that proposed by Euler, in the passage already quoted, namely, the use of certain tones for others which differ from them by a comma or diaschisma. Such substitution within the same chord creates intolerable dissonance. But in melody and in successions of chords it might seem feasible. I have had a concertina tuned, so that the three chords of

* The use of the equally tempered scale has much diminished the feeling for the relations of the system, by confounding tones originally distinct, and has thus led to the confusion of the corresponding notes. Thus such a note as

will have to be read as 19#, g, ty‡; ‡aɔ, aɔ or taɔ, according to the requirements of the system, for all six tones are represented by one on the equally tempered scale.

The Table of Key-relationships (Tonartenverwandtschaften) in Gottfried Weber's Theorie der Tonsetzkunst (3rd ed. 1830, vol. ii. p. 86), may be formed from Table V., by suppressing the signs †, ‡, supposing all the notes to represent tempered tones, contracting the names of the chords to their first notes, and extending the Table indefinitely in all directions.