Things are not always what then seem. The sun appears to revolve round the earth; and the Ptolemaic system of Astronomy founded on that appearance was believed in for more than fourteen hundred years. Things are not what they seem in Music any more than in Astronomy. The first note of the octave scale, the root of the tonic chord, appears to be the one on which the other notes of the scale are founded; and this note, adopted by the ancients as the root of the scale, has been carried own to the very last edition of the Encyclopedia.
    Since the unit of the ratios ascending is really the root of the subdominant and not the root of the tonic; when the root of the tonic is made the generating root of the ratios ascending, as a matter of course, notes will be produced foreign to the scale intended. And as one error leads to another, so these notes and numbers really foreign to the scale are suppose to be different from what they are.
     The first six of the ascending notes in the generation of the scale appear to be in arithmetical series of one, two, three, four, five, six. They certainly are in that order, but not as an arithmetical series. It is only because these six numbers contain nothing but what belongs to the proper geometrical genesis of the scale that they do not lead to an error. The arithmetical series being geometrically right up to six, made it appear that consecutive whole numbers from one onwards was the order of the ascending notes; and all the numbers from one to sixteen have been used in the genesis of the scale.
     The difference between the arithmetical series and the geometrical series in the production of the scale does not tell in the way producing wrong notes till the number seven is reached. D, represented by 7, is made to follow 6, whereas this should have been 6 3/4, geometrically derived. The musical scale