58. All 10 different , 4-notes sub-scales (or tetra-chords) containing steps of 1,2 or 3 semitones. All 8-scales of length a pure 5th.

There is  small number of  exactly 10 characteristic tetra-chords (=4-notes sub-scales)  containing intervals of 1,2,3, semitones and where inverses and cyclic permutations of them do not count as different  (e.g. the 2-2-1, the cyclic permutation of it as 2-1-2, and the inverse of the first as 1-2-2 are considered the same tetra-chord)

Diatonic

2-2-1, (major, natural minor, Ancient Greek syntono,  Rast, Ussak, Kurdi)

2-2-2, (major, augmented)

Melodic minor or Hindu,  Melodic double minor or Arabic, (Shabach)

1-2-1

Harmonic minor (Hijazz,Huzam, Ancient Greek chromatic)

1-3-1

Harmonic double minor (Piraeus, Niavent)

1-2-3,
 Diminished 

3-3-3 , (diminished 7nth)

3-3-1, 

Pentatonic

3-3-2, 
2-2-3

Chromatic 

1-1-1

We should be also familiar with the ways we can play them in 1 , 2 or 3 strings.



If we are interested for 5-chords that have length exactly a pure 5th (7 semitones) so that it is within a classical 3 chord major or minor then the only cases are the next 3 (always here cyclic permutations and inversions are counted as the same 5-chord)

1) 2-2-2-1  (diatonic)

2) 1-2-1-3  (Harmonic minor)

3) 2-3-1-1  (Piraeus, Hungarian)




If on the other hand we set the restrictions
a) 4-notes sub-scales (tetra-hords)
b) Made from steps of 1,2,3 or 4 semitones
c) Make total sum 5 (interval of pure 4th) or 7  (interval of pure 5th) then all the possible combinations where inverses and cyclic permutations of them do not count as different  are the next 5 only, 

Diatonic

2-2-1, (major, natural minor, Ancient Greek syntono,  Rast, Ussak, Kurdi)

Harmonic minor (Hijazz,Huzam, Ancient Greek chromatic)
1-3-1

Diminished 

3-3-1,

Pentatonic

2-2-3

Chinese
1-2-4


THE CLASSIFICATION OF ALL SUB-SCALES OF TOTAL LENGTH AN INTERVAL OF PURE 5TH (7-SEMITONES).

We make here this classification of all 1 note, 2 notes, 3 notes, 4 notes, 5 notes, 6 notes and 7 notes sub-scales of total length a pure interval of 5th (7-semitones) . As usually we use the convention that cyclic permutations of a scale are modes and not a different scale. The numbers denote semitones.
The classification is the next 8-scales
1) 1 note scale: 7 That is an interval of pure 5th. This is actually not counting as a                                     scale
2) 2-notes scales 4-3 , Intervals of 3rd ( 3 or 4 semitones)
3) 3-notes scales a) 2-3-2  It occurs as sub-scale in  the standard pentatone  scale
                              b) 1-3-1-2 It occurs in the Harmonic minor scale, and has oriental                                       sound
                               c) 1-1-3-2 
4) 4-notes scales  2-2-1-2  It occurs in the diatonic scale.
5) 6-tones scales a) 1-1-1-2-2  Small chromatic sound
                             b) 2-1-2-1-1    Hebrew scales sound
6) 7-notes scale    1-1-1-1-1-1-1-1 Full chromatic scale sound.