Now that the creator had harmony in mind with the creation of the heavens, the question comes up: does harmony exist only in the individual and pairwise relations of planets, or do they all in fact agree, as if within a single structure?
This is the subject of Book V Chapter V.
Here we will discover that the heavens express the same harmony that is found in the musical scales of man, both of the hard and soft type (as learned in Book III), no matter where you start or in which direction you move. We will ignore all discrepancies less than a semi-tone ( for these will be dealt with later), and will bring all the planets into one octave, and see what we come up with.
The notes that are set out in the position of an octave were found in Book III and set out below in the table. These numbers, Kepler says, are to be understood as refering to pairs of strings, with the “motions” of the strings being in an inverse proportion to lengths, as will be elaborated more below.
The symbol in the chart | h | represents “B natural” and the note |b | represents "B flat".
With this refreshing look at the proportions, let's see what we can do with the planets.
As you can see in the table below, the actual proportions of all the planets (taken from daily apparent arc lengths) leaps over many octaves. For example, take the motion of Saturn at aphelion (107") and the motions of Earth at aphelion (3423") this leap is over 5 octaves!
3423/107 = ~32
Now 32 is 2^5 which is 5 Octaves.
In order to better grasp the single structure, we must put them into the same octave with each other by "continuous halving", bringing them into tune with Saturn. This can be done because "intervals of one or more octaves are identical by opposition" (see Book III Chapter 1, Proposition 1). Now we can take the motions and put them into the “same octave” with each other by continuous divsions by 2. The numbers in the table representing daily arc lengths, have been converted from minutes and seconds, and then brought down in the final column. The pluses and minuses indicate a small discrepency that arises from dividing by 2 many times.
Notice, that Saturn's perihelion is at 107". Kepler uses this in the construction of the scales because it fits with Earths aphelion, and the difference of one second is not detectable by astronomy, and the difference in music is of the proportion of 106:107, less than a comma.
Now, if the heavens are contained "as within a single structure" we should be able to assemble them into scales which "have an affinity with the rest."
So we have the planets already brought into the same octave, next we can see, that by applying the same proportions that we used in Book III to find the hard and soft scales, we can assemble the Solar System into something remarkably similar (see table below).
You can see that the planets come very close to the harmonies, within commas, or intervals less that a diesis, that are not detectable by the ear, and whose importance will be discussed later in the book.
Keep in mind that we are dealing with motion, not distance, so that we are using the inverse proportions to determine note placement.
For example, in the chart below, the diatessaron is represented by Mars at perihelion as the note |c |. If you were cutting a string, and you wanted to find that note, you would take 3/4ths of the string, but the sound, frequency or motion of the note increases by the proportion 4/3rds, the inverse. So, by taking Saturns perihelion and multiplying by 4/3, you get the motion of Mars at perihelion.
107 x 4/3 = 142 2/3 (or 143-)
Now we have something to work with. The solar system is shown to express the scales constructed below. Notice that the note |a| is missing from the hard scale, which was not represented by the harmonic divisions in Book III Chapter III, either.
First the Hard type:
The funny looking X's next to the |c | and |f | are Keplers characters representing a "sharp".
Next all the notes for the soft scale are represented, except for the note |f |, by most of the planets at aphelion and perihelion, especially the ones left out of the previous scale: that is, Venus at aphelion, and Mars at perihelion. Let's begin the soft scale with Saturn's perihelion as the note |g |.
There you have Harmony in the Heavens, a remarkable thing, having the same affinity to the hard and soft scales produced by mankind. The only difference in these scales is that they are transposed, starting on different notes; whereas when we produced the scales previously (Book III) we proceeded to create a scale from the same note: one hard, the other soft.
In the following diagram, the motions of the heavens are shown as transposed scales (notice the clefs have shifted), where the |b| natural (represented by the note |h | in the book) in the hard scale (red note top diagram), becomes the |g | in the soft (red note bottom diagram). Take note, that the fractions on this diagram are portions of the total "length of string", taking 2160 as the total length (or tonic).
Next you have the progression in the heavens through the harmonic divisions. The fractions in the diagram below reflect the proportions between neighboring notes, not to the whole (or tonic).
So the proportions in the heavens are maintained with the proportions of the hard and soft scales; for 2160/1800 is equal to 6/5, as is also true in the former diagram of the heavens, the ratio is 1728/1440 which is also equal to 6/5. Test out the rest of the proportions to see if they hold true!
You will now therefore wonder no more at the embellishment of the most excellent order of the sounds or steps in the musical system or scale by men, since you see that all they were doing in this respect is emulating God the Creator, and as it were acting out a particular scenario for the ordering of the heavenly motions.
Kepler gives another method of grasping the double musical scale in the heavens at the end of Book V Chapter V.