| 1 | 2 | 3 | 4 | 5 | 6 |
| C 252 | C 252 | D 288 | B 240 | C 256 | D 288 |
| G 189 | F-sharp 180 | F-sharp 180 | G 192 | G 192 | G 192 |
| E 157 12 | D 144 | C 126 | D 144 | E 160 | B 120 |
| C 126 | A 108 | D 144 | G 96 | C 128 | G 96 |
After exploring the whole known field of harmony, and calculating the elevations and depressions consequent on using more elaborate chords, it is asserted that the exact pitch could not be regained.
The formulated results need not be stated here; it is sufficient to give the conclusions to which they point. But assuming that the composer could succeed in so planning his chords that the second half of this melody would so correct the eccentricities of the key-note in the first half that at its completion the composition would be rounded off at the true pitch, it is easy to see that if the first strain were repeated, and the second left unrepeated, or any such ordinary change made, all his elaborate calculations would be of no avail.
Mr. Ellis, who proposes a system with 117 notes within the octave, is thus shown that an infinite number of notes is required, for there is no synonymity in any system when the key-note moves. At each change of pitch the whole series is changed. Mr. Bosanquet, with 53 notes to the octave, offers to provide musicians with materials for 84 scales; and thus we are more reminded of the musical formulæ of the ancient Hindoos—their 16,000 keys—than informed how the above simple melody may be correctly rendered.
It is somewhat amusing to find Mr. Ellis seriously proposing to employ three harmoniums, the three players having to touch the notes that happen to fall to their respective instruments, not only because, as shown above, the music would still be out of tune, but because no performer would play a complete melody by himself, but a note here, another there, unconnectedly. For, however neatly this might be managed, expression or artistic rendering would be unattainable. Yet it is remarked, "The performers would merely require a little drilling and practice together."
Logarithms may be piled and compiled to define scales, but it is not so easy to reconcile the conflicting principles that appear in actual composition. The musician baffles the mathematician, who fails to follow him in his operations, as proved by the hitherto unnoticed discrepancies between melodic and harmonic proportions herein demonstrated. Although the composer's notation is not an exact statement, the performers do not experience practical difficulties: the intention is
