The musical intervals are not merely referred to by their mathematical fractions. Here we'll derive their more common names, and by doing that, fill in the final notes of the 8 note, octave scale, of two types: Hard Melody and Soft Melody.
If you have some familiarity with music, then by this point, you've probably realized that the intervals we keep referring to as 4/5, 2/3, and so on, also go by other names. We're going to derive those here.
Take a look at the sets of harmonic divisions. Here I have them with the melodic intervals marked in:
What you'll notice is that while in both sets the 3/4 interval is the third note, the same interval, 3/4 also exists between the second and the fifth notes in the series. Here, the 3/4 interval contains within it three melodic intervals: 9/10, 8/9, and 15/16. For this reason, it is named, using the Greek term, “diatesseron,” or, “over four,” because the three intervals fall in between four notes. Today, we simply call it a “fourth”.
If you've already gone back to look at the original 3/4 division, you should be raising some objections. Three-fourths is the interval we called a fourth, and so should span three intervals; so why would the fourth here, whether we look at either the hard and the soft system, span only two? Wouldn't that make it a third?
We'll come back to that in a moment. For now, let's finish naming the rest of the intervals. You can see here that 2/3 is one major tone more than the fourth, and for that reason, it's called a “diapente,” or, “fifth”.
The 5/8 and 3/5 divisions are one semitone and one minor tone more than the fifth, and get the names hard and soft “diahex,” what we today call the major or minor “sixth”.
And finally, 5/6 and 4/5 are one minor tone and one semitone less than the fourth, earning the names hard and soft “third”.
We'll use these names to refer to the intervals from now on. Now back to the fourth at the bottom of the system. We saw before that rather than spanning three intervals, as the other fourths both did, it only spans two: either the soft third, 5/6 and 9/10, or the hard third 4/5 and 15/16. Looking back to the middle of the system, there's an interval of 4/5 here, which is clearly broken into the two smaller intervals: 8/9 and 9/10. We also find a soft third, broken into 8/9 and 15/16.
This means that the thirds, which share the 8/9 interval, can be further divided into melodic intervals.
In order to complete the fourth at the bottom of the system, I'm going to create a new division of the string, this time not a harmonic division, but one which is a lawful derivative from them, and which maintains the characteristic of the 3/4 interval as a “fourth”.
For the same reason, we need to add another division at the top of the system, to complete the fourth which lies between the fifth and the octave, or the 2/3 division and the half string. We'll make this one a major tone, or 8/9 lower than the octave—although in music, a semitone is commonly used here. We'll hear that in a moment.
The just-completed system now includes seven intervals between eight tones, hence the name “octave”; or “diapason”, “over all”.
Think back to how this investigation of sound began; we first looked at sound as a continuum. Now, through a process of physical experimentation and reasoning, we've coaxed that continuum into revealing to us its inherent, underlying structure. From that, we've constructed a system which allows us to move through the domain of sound by a series of discrete steps, with two different variants—each physically derived from a principle of harmony.
However, music is not contained within just one octave; the question is, can we extend the system, generating multiple octaves, by repeating the structure of one octave that's just been generated? If you think so, you might be surprised...
Book III, Chapter Five
Book III, Chapter Six
Book III, Chapter Seven