ments. If the performers tried to be true to these, the gist of the story, as it were, would be lost in the telling. The performance, therefore, may be psychologically true and yet mathematically false. There is a natural tendency to raise notes under the influence of extremely excited, passionate phrases and to depress them in sinking to a state of repose. But, irrespective of such changes, errors must certainly be made by singers and violinists of the nature of those now to be defined, however they may be glossed over and hidden.
Mr. Ellis illustrates his system of "Duodenes" by the first line of the hymn "My country, 'tis of thee," which is better known to German and Anglo-Saxon peoples as "God save the Queen." It will be employed here, for it is not only familiar and very simple, but usually has equally simple harmonies that may be conveniently illustrated with low numbers and a few fractions.
Whatever numbers may by chosen for convenience, the proper proportions of the melodic notes are 15 : 16 : 18. Those of the chords in columns 1, 4, 5, 6, are in the ratio of 4, 5, 6, and those in columns 2, 3 in the ratio of 10 : 12 : 15. It is understood that the doubling or halving any of these chordal numbers will merely represent the note in another octave, above or below, it will not change its character so as to affect the investigation.
3. When the melodic proportions are true, the chords are untrue.
| 1 | 2 | 3 | 4 |
| C 48 | C 48 | D 54 | B 45 |
| G 36 | A 40 | A 40 | G 36 |
| E 30 | E 30 | F 3225 | D 27 |
| C 24 | A 20 | D 27 | G 18 |
At the third note of the melody the chord is out of proportion, for the interval D : A, 27 : 40, should be D : A, 27 : 4012 to be in the required ratio of 2 : 3 or 10 : 15.
4. When the chords are true, the melody is temporarily out of proportion.
| 1 | 2 | 3 | 4 | 5 | 6 |
| C 128 | C 12935 | D 144 | B 120 | C 128 | D 144 |
| G 96 | A’ 108 | A’ 108 | G 96 | G 96 | G 96 |
| E 80 | E’ 81 | F 86 35 | D 72 | E 80 | B 120 |
| C 64 | A’ 54 | D 72 | G 48 | C 64 | G 48 |
