According to the best evidence (in the works of Ptolemy, "Harmonics," second book, and Aristides), these were approximately the actual pitch of the modes as compared one to another. And now the difficulty was to weld all these modes together into one scale, so that all should be represented and yet not be complicated by what we should call accidentals. This was accomplished in the following manner, by simple mathematical means: We remember that the Dorian, which was the most greatly favoured mode in Greece, was divided into two tetrachords of exactly the same proportions, namely, semitone, tone, tone. By taking the lowest note of the Mixolydian, B, and forming a Dorian tetrachord on it, B C D E were acquired. Adding to this another Dorian tetrachord, E F G A (commencing on the last note of the first), and repeating the same series of tetrachords an octave higher, we have in all four Dorian tetrachords, two of which overlap the others. The two middle tetrachords, constituting the original Dorian mode, were called _disjunct_, the two outer ones which overlap the middle ones were called _conjunct_ or _synemmenon_ tetrachords. If we consider this new scale from octave to octave, commencing with the lowest note, that is to say from B to B, we find that it coincides exactly with the Mixolydian mode; therefore this was called the Mixolydian octave. The octave in this scale from the second note, C to C, coincides exactly with the Lydian mode, and was called the Lydian octave; from the third note, D, up to its octave gives the Phrygian; from the fourth note, E, the Dorian; from the fifth, F, the Hypolydian; from the sixth, G, the Hypophrygian; and from the seventh, A, the Aeolian or Hypodorian octave. Add one note to the lower end of this universal Greek scale, as it was called, and we see that the whole tonal system was included within two octaves. To each of the notes comprising it was given a name partly derived from its position in the tetrachords, and partly from the fingering employed in lyre playing, as shown in the diagram on page 87.