Harmonic minor scale degrees IV.
The inverse harmonic minor scale
1^{st} degree at the 13th fret
This section was created by Sándor Marosi...
...and Tamás Farkas...
...back in 2006. I would like to thank them for the great help! The basis of their thoughts on the inverse harmonic minor scales was the existence of the sister basic scales described in the Basic scales III. section. The concept of the sister basic scales is based on the following observation: the total number of variations of the basic scale inner structures exceeds the seven basic scales; i.e. more than the seven basic scales can be created using the inner structures of the basic scales. Sándor and Tamás followed this path and applied the concept of the sister scales on the harmonic minor scales. Let's see how this works:
Are there scales, other than the known harmonic minor scale degrees, that use the five kinds of trichord combinations that harmonic minor scales use?
To start with, let's see these trichord combinations:
1
2
3
The following two trichords are typical of the harmonic minor scale and are considered very important trichord combinations:
4
5
Obviously, these two trichord combinations don't just randomly follow each other. The dependencies are demonstrated in the following graph:
The graph shows how the trichord combinations may follow each other (the arrows), and the difference of how many semitones between the trichord combinations the relevant step involves (the numbers next to the arrows; they are called weights on the graph's edges). For example, from trichord combination no. 3 you can either proceed to trichord combination no. 1 or no. 4; both cases involve a difference of two semitones between the original trichord combination and the subsequent one.
And here comes the question: Is there a path of seven units in the graph, with the same starting and ending points, and a total weight of 12?
Obviously, there has to be such a path; and it actually checks with the infinite harmonic minor scale:
31234513123451...
In addition, there is another path that fulfills the above criteria:
31345123134512...
We can easily come to the conclusion that, apart from these two infinite scales, there are no other scales that use those five trichord combinations. The mathematical proof of this suggests that the relevant path can't involve the 54 edge, as that would only be possible if the path was 345451. A 345451 path could only be closed, if there was a 2unit path from trichord combination no. 1 to no. 3. (you have already used 10 units in the 345451 path). Similarly, it is also certain that the 22 edge can't be involved in the path either, since that would result in the 3123122 or 12223131 paths mentioned in Sister basic scales section, or you would not be able to close 3451 path. This implies you have 7 edges left, which you can use for a 7unit path. Thanks to the 13 edge, you have got two options for closing the circuit, as described above.
The 'new' harmonic minor scale we have just discovered is literally the inverse of the original harmonic minor scale. Beside its name being inverse harmonic minor, this scale has several denominations:

Mela Chakravakam,

Raga Ahir Bhairav,

Bindumalini,

Maqam Hicaz,
...all of which sounding very Arabic!
Let's see what the scale notes are, starting the first degree from the F note:
FGA flatB flatBDE flat(F)
And here come the scale degrees in mirror images, brought to you by OSIRE scale software.
1^{st} degree at the 13^{th} fret