69 
arrangement of the notes in the diatonic scale, and to suggest 
grounds for believing that no other division would be equally 
convenient. 
Let us resume Fig. 1. It will be seen that the musical 
circle representing the diatonic scale is divided into twelve equal 
portions, and that these twelve equal portions are divided into 
two groups, one consisting of four portions, or two whole tones, 
the other consisting of six portions, or three whole tones, by 
the two shaded portions corresponding to the semitones. Hence 
it is obvious that the shifting of the semitones so as to take 
one from the larger division of three tones and to add it to 
the smaller division of two tones, will leave a musical circle 
divided exactly as before; that is, there will still be two great 
divisions of two tones and three tones respectively, separated 
by semitones. The arrangement reproduces itself. 
Not only is this the case, but it is easy to see that no 
other arrangement of the semitones would produce the same 
result. Suppose for instance we Fig. IV. 
have the circle divided as in Fig. 
Iv., that is, into two groups of one 
tone and four tones respectively, 
separated by the two semitones; in 
other words, regarding C as the 
tonic, suppose that we have a flat 
third. Then it is manifest that by 
no shifting of the semitones can this 
arrangement be made to reproduce 
itself, In fact the problem of making such a self-reproducing 
scale is merely that of dividing 5 into two parts, such that if 
unity be taken from one and added to the other the two parts 
shall be the same as before. It is manifest that the division 
into 2 and 38 is the only solution. 
The division of the circle represented in Fig. Iv. is some- 
what interesting from the fact that it is the actual division 
