50. The oriental Harmonic minor (Hijazz) 4-notes sub-scale 1-3-1 and the creation of all possible 7-notes scales by combinations with diatonic tetra-chords . All sub-scales of length a pure 5th

The oriental  tetra-chord which is called also Hijazz,  is 1 semitone-3 semitones-1 semitone 
(1-3-1) in total 5 semitones. Its origin is in ancient Greece and Byzantine empire. Thus if it is to combine it with other tetra-chords of total sum 5 it should be combined with one tone distance to make in total 12 semitones (5+2+5=12).

Now combining it with all possible diatonic tetra-chords that is (1-2-2),  (2-2-1), (2-1-2), (2-2-2) it gives the scales

1) Harmonic minor (called also mode of Hijazz or Byzantine minor) = (1-3-1)-2-(-1-2-2).
It is the inverse 7-notes scale of the first 13 overtones e.g. in a string or natural trumpet C-D-E-F-G-G#-B-C if we interprete the 7nth overtone as B rather than Bb. While if we perceive it as Bb, then it is the scale 2-2-1-2-1-2-2 which is the acoustic overtones scale or melodic minor scale.

OF SCPECIAL INTEREST IS THAT INVERSE OF THE HARMONIC MINOR 2-2-1-2-1-3-1  IS DERIVED FROM THE FIRST 13 OVERTONES OR HARMONIC SERIES OF A STRING OR NATURAL TRUMPET IF  THE 7NTH HARMONIC (E.G.  C-D-E-F-G-G#-B-C )  if we perceive the 7nth overtone as B rather than Bb (in reality it is somewhere in the middle)  AND SO IT  is the inverse 7-notes scale of the first 13 overtones  ON THE OTHER HAND THE MELODIC MINOR OR THE 7-NOTE SCALE 2-2-1-2-1-2-2 E.G. C-D-E-F-G-G#-Bb-C WHICH IS DERIVED FROM THE FIRST 13 OVERTONES HARMONIC SERIES IN A NATURAL TRUMPET IF WE PERCEIVE THE 7NTH HARMONIC AS Bb rather than B.   BOTH SCALES  ARE   CONSTRUCTED FROM THE 4-CHORD 2-2-1 AND THE 5-CHORD.  

A mode (cyclic permutation) of the harmonic minor is called also Blue-scale in American folk music, although sometimes as blue scale is referred to a 6-notes scale (see post 54), and another cyclic permutation of  it the Romani minor scale. Beautiful melodies are composed by alternating within two octaves the diatonic, the harmonic minor and the pentatonic scale. E.g. by translating a simple theme of consecutive 1 semitone 2 semitones by one 4th higher within the diatonic scale and then also inverting it from ascending to descending one 4th higher with the same rhythmic pattern and then doing the same with a simple interval of 3 semitones inside the pentatonic scale creates beautiful melodies.

If we superimpose a harmonic minor e.g. A-harmonic minor, with the diatonic scale e.g. a-minor scale, then we get a 8-notes scale, here in the particular example the scale

a-b-c-d-e-f-g-g#-a which is called n the Gypsy-jazz the BEBOP MAJOR SCALE. It has interval structure  2-1-2-2-1-2-1-1. 

The Bebop dominant 8-notes scale is the 2-2-1-2-2-1-1-1 which is also  a mode of the maximal harmonic 8-notes scale (see post 117)

Chords of the harmonic minor (intervals 2-1-2-2-1-3-1 and from it we may derive the 6-tones scale
1-3-1-2-1-4):



Triads:           min              dim aug         min           maj         maj       dim
Extended
4-notes chord: min/maj7    m7b5 maj7#5 min7 dom7 maj7     dim7

For example for the A harmonic minor(A,B,C,D,E,F,G#)  (intervals 2-1-2-2-1-3-1),the chords are

i	iidim	III	iv	V	VI	VII
Amin	Bdim	Caug	Dmin	Emaj	Fmaj	G#dim
Aminmaj7	Bm7b5	Cmaj7#5	Dmin7	E7	Fmaj7	G#dim7

Typical progression

Typical chord progressions in A harmonic minor
i - iv - V7	Am - Dm - E7
ii - V7 - i	Bm7b5 - E7 - Am

More general the chords that fit to the harmonic minor (not taking necessarily the notes in alternating order , that is take one leave one  or 1-3-5  etc) are the next

Notes of this scale:

A; B; C; D; E; F; G#/Ab; A;

Interval structure of this scale:

2-1-2-2-1-3-1

Chords that fit in this scale:

Normal Triads: Caug     Dm     Ddim     E     Eaug     F     Fm     Fdim     G#aug     G#dim     Am     Bdim  Other Triads: Dsus2     Esus4     Asus4     Asus2  4 Notes Chords: Dm6     Dm7     Dm7b5     Dº7     D7sus2     E7     E7#5     E7sus4     F6     Fm6     Fmaj7     Fm(maj7)     Fº7     G#º7     Am(maj7)     Bm7b5     Bº7

The Romanian Kaval 5-holes flutes are in the lydian mode of the E harmonic minor scale!

Again a nice way to play the Harmonic minor on the fretboard with the rule of this shape is 2 notes per string, is the next

5 other ways to play this scale on the guitar fretboard with the rule 2, 3,or 4 notes per string are the next 

2) One that may be called 2nd Harmonic minor , called also mode of Qurdi or Kasigar =(1-3-1)-2-(2-1-2) and

3)  One that may be called 3rd Harmonic minor, called also mode of Shamba=(1-3-1)-2-(-2-2-1).

4) And one than maybe called 4th Harmonic minor. If we take the Shamba in the reverse order we get the Neopolitan scale= (1-3-1)-1-(-2-2-2)

The Neopolitan scale , from D has the next chords (intervals 1-2-2-2-1-3-1)

Notes of this scale:

D; D#/Eb; F; G; A; A#/Bb; C#/Db; D;

Interval structure of this scale:

h W W W h (W+h) h

Chords that fit in this scale:

Normal Triads: C#aug     Dm     D#     Faug     Gm     Gdim     Aaug    A#     A#m  Other Triads: Dsus4     D#sus2     Gsus2     A#sus4  4 Notes Chords: Dm(maj7)     D#maj7     D#7     D#7b5     D#7sus2     F7#5     Gm7     Gm7b5     G7sus2     A7b5     A7#5     A#6     A#m6     A#maj7     A#m(maj7)

5) While if combined by itself it gives what is called Romani (Hungarian or Gypsy or Byzantine  double  minor or Harmonic double minor or mode of the Niavent)=(1-3-1)-2-(-1-3-1).

Here is a relevant video

https://www.youtube.com/watch?v=PRDD83LUL5M

These names are of Arabic origin but are used in Greek folk music with Buzuki  , as they  are played on the western 12-semitones Bach equal temperament scale.

We remark here that the double harmonic minor (1-3-1)-2-(-1-3-1) and by cyclic permuttaion 1-1-3-1-2-1-3 . has in it the tetrad (tetrachord) 1-1-3 which in Aristoxenos studies of Ancient Greek music and its tetrachords it is called the tonal tetrachord of the Chromatic generation.

The chords that fit to  the Romani or Harmonic double minor scale , we take here as an example the D Romani or Hungarian double  minor are the next:

Notes of this scale:

D; E; F; G#/Ab; A; A#/Bb; C#/Db; D;

Interval structure of this scale:

2-1-3-1-1-3-1

Chords that fit in this scale:

Normal Triads: C#     C#m     C#aug     Dm     Ddim     Faug     A     Aaug     A#     A#m     A#dim  Other Triads: Dsus2     Asus4  4 Notes Chords: C#6     C#m6     Dm(maj7)     E7b5     Amaj7     A#maj7     A#7     A#7b5     A#m7     A#m(maj7)     A#m7b5

Two nice positions of the  Romani (Gypsy)  double minor scale are the next. The first is the shape of major7 chord together with its shift by one semitone! The rule of this shape is 2 notes per string.

Other positions are the next 

Two variations of the Byzantine double minor scale are the Persian and inverse Persian scales 
But only the Byzantine (harmonic) double minor contains two tetra-chords 1-3-1 !


Persian scale or todi theta scale=(1-3-1-1-2-3-1) 
E.g. starting from C

Notes of this scale:

C; C#/Db; E; F; F#/Gb; G#/Ab; B; C;

Interval structure of this scale:

h (W+h) h h W (W+h) h

Chords that fit in this scale:

Normal Triads: C#     C#m     Caug     E     Eaug     Fm     Fdim     G#aug  Other Triads: C#sus4     Esus2     F#sus4     F#sus2     Bsus4     Bsus2  4 Notes Chords: C#maj7     C#7     C#m7     C#m(maj7)     C#7sus4     E6     Fm(maj7)     F#7sus4     F#7sus2     G#7#5

Inverse Persian scale or Purvi Theta scale= (3-1-1-3-2-1-1) 

E.g. starting from C

Notes of this scale:

C; C#/Db; E; F#/Gb; G; G#/Ab; B; C;

Interval structure of this scale:

h (W+h) W h h (W+h) h

Chords that fit in this scale:

Normal Triads: C     C#m     Caug     C#dim     E     Em     Eaug     G#aug  Other Triads: C#sus4     Esus2     F#sus4     F#sus2     Bsus4     Bsus2  4 Notes Chords: Cmaj7     C#m7     C#m(maj7)     C#m7b5     C#7sus4     E6     Em6     F#7sus4     F#7sus2     G#7#5     C\E     C\G

All the above scales may be considered extrapolations to intervals of only 1,2 ,3 semitones, of arpeggios of 3-notes chords of the type major,minor, diminished and augmented.



If in the above 5 scales of 1st 2nd 3rd and 4th  Harmonic minor and Harmonic double minor scales we extrapolate the intervals of 3-semitones as ascending to  2+1 semitones, we get the next 8-tones scales made only from steps of 2 and 1 semitones

1) From the 1st Harmonic minor 2-1-2-2-1-3-1 the 8-notes scale  2-1-2-2-1-2-1-1 which is a cyclic permutation or mode of what is called in post 51, 3rd alternative of Spanish-Jewish 8-notes scale that contains in its chords the Andaluzian Cadenza.

2) From the 2nd Harmonic minor (1-3-1)-2-(2-1-2)  the 8-notes scale  (1-2-1-1)-2-(2-1-2)
which is a cyclic permutation or mode of what is called in post 51, 4th alternative of Spanish-Jewish 8-notes scale 
3) From the 3rd Harmonic minor (1-3-1)-2-(-2-2-1).  the 8-notes scale  (1-2-1-1)-2-(2-2-1)
which is a cyclic permutation or mode of what is called in post 51, 1st alternative of Spanish-Jewish 8-notes scale 
4) From the 4th Harmonic minor  1-2-2-2-1-3-1 the 8-notes scale  1-2-2-2-1-2-1-1
which is a cyclic permutation or mode of what is called in post 51,  Spanish-Jewish 8-notes scale 
5) From the Harmonic double minor 2-1-3-1-1-3-1 the 9-notes scale 2-1-2-1-1-1-2-1-1

If we restrict to only 4-notes sub-scales (tetra-chords) , having inverse such scales not different, then we are left with a small number of 8 of such characteristic tetra-chords met in corresponding 7-notes scales.

Diatonic

2-2-1, (major)

2-2-2, (major)

2-1-2 (natural minor)

Melodic minor  

1-2-1

Melodic double minor

2-1-1,

Harmonic minor 

1-3-1

Harmonic double minor

1-1-3

1-2-3,

Nevertheless if cyclic permutations of them are not considered different then 2-2-1=2-1-2. 1-2-1=2-1-1, 1-3-1=-1-1-3 and so they are only 5 different 

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 

Diatonic

2-2-1, (major, natural minor Rast, Ussak)

2-2-2, (major, augmented)

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

1-2-1

Harmonic minor (Hijazz,Huzam)

1-3-1

Harmonic double minor

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.

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 7-scales
1) 1 note scale: 7 That is an interval of pure 5th
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 It occurs in the Harmonic minor scale, and has oriental sound
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) 2-1-2-1-2    Hebrew scales sound

6) 7-notes scale    1-1-1-1-1-1-1-1 Full chromatic scale sound.