In the Hypolydian, the semitones occur between the fourth and fifth, and seventh and eighth:
The Dorian (E), Phrygian (commencing on F♯ with the fourth sharped), and the Lydian (A♭ major scale) modes we have already explained. In the Mixolydian, the semitones occur between the first and second, and fourth and fifth degrees:
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 Æolian 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].
The fifteen strings of the kithara were tuned according to this scale, and the A, recurring three times in it, acquired something of the importance of a tonic or key note. As yet, however, this scale allowed of no transposition of a mode to another pitch; in order to accomplish this the second tetrachord was used as the first of another similar system. Thus, considering the second tetrachord, E F G A, as first of the new scale, it would be followed by A B♭ C D, and the two disjunct tetrachords would be formed. Followed by the two upper conjunct tetrachords, and the proslambanómenos added, our system on a new pitch would be complete. This procedure has come down almost unchanged to our times; for we have but two modes, major and minor, which are used on every pitch, constituting various keys. These Greek modes are the basis on which all our modern ideas of tonality rest; for our major mode is simply the Greek Lydian, and our minor mode the Æolian.