We've now seen how a complete system of a unit octave can be constructed from the seven possible harmonic ways to divide the octave, and the melodic intervals derived from them. However, since the human species, which sings, consists of both man and woman, it seems that our musical system may require more than just one octave (as is confirmed in the solar system).
Taking the musical system of the octave for soft melody seen the in diagram below, let's see what happens when we continue the same series of intervals, beginning from our octave ”g” (or taking ½ the string as the new “whole” string), to create two octaves of soft melody (Click Here:
). Sounds pretty good, doesn't it?
With the repetitions of the same series of intervals, each new note is exactly one octave above (or half the string length of) the scale we generated before. But let's examine things a little more closely here, because when we repeat the same intervals from the lower octave into the higher, we find something that we haven't encountered before. For example, in the soft scale, we have a Major tone, or 8/9, as the interval at the top of the scale ("F" to "g"), and immediately another interval of 8/9 at the bottom of the next octave ("g" to "a"). This creates two intervals of 8/9 in a row, something which didn't occur within our single octave (see diagram).
Those two Major tones make the interval of the major third created between the “F” to the “a” of the next octave 64/81 (8/9 times 8/9). But the major third from G to the B of the lower octave was 4/5, or 64/80 (8/9 times 9/10). That's a difference of 80/81, the interval to which Kepler gives the name “comma”. This same difference occurs between the “fourth” from “F” to “b-flat” (since it includes the oversize third of "F" to "a", plus one semitone ), and the fourth of the lower octave, “G” to “C”, making the interval between the “F” and “b-flat” an “oversize fourth”. So if “F” to “b-flat” is an oversize fourth, that would make the fifth which completes the octave “F” to “f” an undersized fifth (see diagram). In other words, since the fourth oversteps its bounds, the fifth must give way in order to maintain the perfection of the octave "F" to "f".
So, while we've built the system of the octave based upon the seven possible harmonic divisions, by repeating the same intervals for a second octave, there arises in the process another octave (from “F” to “f”) which is divided in a non-harmonic way. (If we examine the thing, we find that the same is the case for the octaves from “E” to “e”, and “D” to “d”, and so on.) Now, one could say (quite rightly) that the difference between the “perfect” fourth from “G” to “C” and the “oversize” fourth from “F” to “b-flat” is imperceptible, and is something to be brushed over. However, we, being concerned not just with practicality, but with principle, can't just smooth over such a discrepancy. Why would it be the case that our success in discovering how to fit together the system of harmonic proportions only seems to hold within the initial octave, and seems to break down as it is extended upwards? There must be some remedy for this.
Aha! The difference between our Major tone (8/9) and our minor tone (9/10) is exactly 80/81, so by substituting a minor tone between “g” and “a” for the Major tone which is there, then both the “a” and the “b-flat” can be lowered by a comma, and the minor tone between "b-flat" and "c" becomes a major tone (which enlarges the undersized diapente or fifth). This shrinks our two oversized intervals by exactly a comma (80/81), and the octave “F” to “f” is now divided harmonically by the new, flatter “b-flat”!
The same “quick fix” remedies the same problem which occurred in the octave from “E” to “e”, previously divided non-harmonically by the “a” and the “b”, and also from “D” to “d”. Thank goodness for that minor tone, or else our system would get more and more unharmonic the further we went up!
Hold on a second. If I've lowered both the “a” and the “b-flat” by 80/81, won't I then have increased the size of the intervals which these notes make with the “g” above? By adjusting to try to right one relationship, I merely transferred the problem to a neighboring interval, and I still end up destroying the very harmonic divisions I started with.
If it is going to be possible to construct a perfectly harmonic system, it almost seems as if some of the tones (such as the “a”) would have to be able to vary in pitch in order to simultaneously meet the harmonic demands from both octaves! More to come on that in Book V...