168 
Proceedings of the Boyal Society of Edinburgh. [Sess. 
The character of the first three pairs is more definite than that of the 
last three, and their designation is sufficiently clear. For the sake of 
simplicity we use the same terms to describe the last three pairs, only 
adopting the prefix “ ultra.” It will be observed that these latter 
co-functionaries are, mathematically, the exterior notes of the scale, 
removed one-fifth upward or downward from the other six. (See 
Diagram II.) 
Let us now consider these co-functionaries a little more closely. 
Dominators C G. — These two notes, forming the central interval, are the 
basis of measurement for all others in the scale, C dominating the major, 
and G dominating the minor, system. This would seem to explain the 
peculiar pre-eminence of the dominant in the minor key. 
Mediants E Eb. — These notes, so familiar as major and minor thirds of 
the common chord, are essentially major thirds (root measures) upward and 
downward from their respective bases, C and G. The two mediants are 
consonant with both dominators. They are distinctive also as relatives, 
E being dominant of the relative minor, and Eb tonic of the relative 
major, key. 
Leaders B Ab. — Here again we are familiar enough with the major 
functionary (B), and need only to observe how the same character of leader 
is natural to Ab in the minor key ; j ust as the upward measured seventh 
leads to its dominator, C, so the downward measured seventh leads to its 
dominator, G. 
Relatives of the key : A being tonic of the relative minor , and Bb 
dominant of the relative major. — These ultra-mediants A and Bb are re- 
markable as creating a unique interval in the chromatic scale ; an interval 
neither diatonic nor chromatic, but essentially harmonic in character and 
in fact. This interval is not diatonic, because the two notes A and Bb do 
not belong to any single diatonic scale; * it is nok' chromatic, because they 
constitute different degrees in the chromatic scale and have not the same 
letter name : it is harmonic, because it constitutes the centre of the 
harmonic scale of C, based on Db (Diagrams IV and V). This unique 
semitone between A and Bb has ratio 27 : 25, and is the largest in the 
chromatic scale. 
Ultra-leaders Fjf and Db. — These are the chromatic extremes (the flattest 
and sharpest notes in the key); theif* character as ultra-leaders is very 
distinctive, F$ leading up to the tonic of the new sharp key, and Db leading 
* In the scales of F and Bb major, D and G minor, either the A or the Bb is a different 
note from the A and Bb in scale of C. (See below, Diagram VII.) 
