21. Composing alternative ethnic scales in western 12-tone equal temperament scale . Flamenco, Gypsy Jazz, Greek Bouzouki, Arabic, Chinese etc scales, The algebra of tetrachords

All three 7-notes scales made only  from tones and semitones


At first a nice video about scale emerge from the harmonics of a single note!

https://www.youtube.com/watch?v=JDFa8TSn6vY

As it is known the diatonic scale (major or minor mode) is made from successive steps of one tone or one semitone. In particular of 5 tones and 2 semitones in the next order (not counting their cyclic permutations that produce the different diatonic modes) 

1) Western Diatonic=2-2-1-2-2-2-1 (e.g. C D E F G A B /C). This scale is  optimal at the next aspect: It has the maximum number of major and minor triad-chords.About playing it it see post 4.

 Notice that the diatonic scale is not directly identical with its reverse order , but it is identical with one of its cyclic permutations (e.g. starting from F ), therefore it is considered inverse-symmetric

But are there more scales (not counting their cyclic permutations that would produce their different  modes) made also from  successive steps of one tone or one semitone? From the mathematical point of view it is easy to find them all.

We are restricting to 7-notes scales


At first we notice that as it is a 7-notes scale it cannot have more than 5 whole notes but neither less than 5 whole notes, as this wold not give in total 12 semitones with 7 addition terms.

Therefore all such scales would be permutations of the pattern 2-2-1-2-2-2-1.

In particular all of them would have 2 semitones!.Now these semitones may be separated by 2 or 3 tones as in the diatonic scale, or by one only semitone or by zero semitones. This gives two other scales.

2) Minor melodic=(1-2-1-2-2-2-2) and all cyclic permutations of it that give its modes (known also as Hindu scale see http://www.scales-chords.com/) , This scale is identical with its inverse order
 and the 
3) Altered natural or second minor melodic or leading whole tone or Arabian scale or Neopolitan major =(1-1-2-2-2-2-2) and all cyclic permutations of it that give its modes. Again this scale is identical with its inverse order.

I found the term altered natural in the next classification list  https://psg.com/~dlamkins/lamkins-guitar/Tools/scales.html and http://www.scales-chords.com/

It seems to me that besides the harmonic minor and Romani double minor, Stephan Grappelli and Django Reinhardt were using these two scales together with 8-notes scales made only from tones and semitones (see 10 of them below) in their improvisational embellishments. 

Nevertheless if we do not restrict to 7-notes scales then of course there are more. 

For example we may also classify all 8-notes scales made only from semitones and tones:

There are at least 10 such! 

1) The 8-notes scale 1-1-1-1-2-2-2-2  and non-cyclic permutations of them like 
2) 2-1-2-1-2-1-2-1  (diminished scale see https://en.wikipedia.org/wiki/Octatonic_scale ) . 
3) Also the 8-tones Spanish or Jewish 1-2-1-1-1-2-2-2
4) and the 8-notes Jewish (Abot) (inverse of 8-notes Spanish-Jewish)1-2-1-2-2-2-1-1

(For Spanish-Jewish music see https://www.youtube.com/watch?v=f5cdoq8on1w&list=PLFEBD04189100E6EC&index=29 )

5) and the Japanese 8-tones Ichikosucho 2-2-1-1-1-2-2-1 (it is self inverse) e.g. (C D E F G  A  A# B C)
6) 1st Alternative of Spanish-Jewish 8-notes 1-1-2-1-1-2-2-2 (again it is self inverse)
7) 2nd  Alternative of Spanish-Jewish 8-notes 1-1-2-2-1-1-2-2 (again self inverse) 
8) 3rd Alternative of Spanish-Jewish 8-notes An extrapolation  of the diatonic scale 2-2-1-2-2-2-1  and of the Harmonic minor scale (2-2-1-3-1-2-1) to an 8 notes scale made by 

2-2-1-2-1-1-2-1  (C D E F G G# A B C) (it is self-inverse) In this scale the chord progression of the Andaluzian Cadenza fits exactly 

9) 4th Alternative of Spanish-Jewish  8-notes (1-2-1-1)-2-(2-1-2)  

10) 5th Alternative of Spanish-Jewish 8-notes    1-2-2-1-1-2-1-2  (inverse of the previous)



11) The 8-notes scale 2-1-1-3-1-2-1-1 (might be called blues arpeggio of triple maj-min-maj chord) which is derived from blues arpeggios of the triple chord maj-min-maj e.g.
from c e g b d (which includes the c-major c e g, the e-minor e g b and the g major g b d) 
by extrapolation the middle note of each chord by 1-semitone making it from major minor and vice versa (in other words 3 blues arpeggios of the 3 chords) we get the 8-notes scale
c, d, eb, e, g, g# ,bb,b, c

12) Similarly the 8-notes scale 2-1-1-2-1-3-1-1 (might be called blues arpeggio of triple min-maj-min chord) which is derived from blues arpeggios of the triple chord min-maj-min e.g.
from a, c , e, g  (which includes the a-minor a c e, the c-major c-e-g and the e minor e g b) 
by extrapolation the middle note of each chord by 1-semitone making it from major minor and vice versa (in other words 3 blues arpeggios of the 3 chords) we get the 8-notes scale
a ,b,c.c#,eb, e, g, g# a


We may compare them with the 

1-3-1-1-1-1-3-1

and the 10-notes scale 1-1-1-1-1-1-2-1-2-1

We may also notice the interesting 8-notes scale

2-2-1-1-2-2-1-1 which is very symmetric as it re[etas the pattern 2-2-1-1

A variation of this scale so as to include the chromatic pattern 1-3-1, is the next 8-note scale

2-2-1-1-1-1-3-1. 

SEE ALSO THE HARMONIC SCALES DERIVED FROM THE HARMONICS OF A SINGLE TONE IN POST 81.
All the above scales may be considered extrapolations to intervals of only 1,2 semitones, of arpeggios of 3-notes chords of the type major,minor, diminished and augmented. 

Finally we mention the COMPREHENSIVE 9-NOTES SCALE, (or CHINESE CHROMATIC DIATONIC 9-NOTES SCALE)  which is the minimal extension of the diatonic scale, that contains also the harmonic minor scale (1st Byzantine scale)  but also the Harmonic double minor (or 2nd Byzantine scale, or Hungarian or Gypsy minor scale)
We only need to add the D# and G# (2nd and 5th) notes at 7-notes diatonic scale

Thus with root C it is the 

C  D, D# , E, F, G, G#, A, B C

with interval structure (at a different mode from C)

2-1-1-1-2-1-1-2-1.

(Notice the successive symmetry of increasing the number of successive semitones after a tone 2).

The 7-notes sub-scale C  D, E, F,  G#, A, B C is the E harmonic minor while the 

7-notes sub-scale C D#, E, F, G#, A, B  C# with intervals structure 3-1 -1-3-1-2-1-
which is the E double harmonic minor scale.

I have created also a caval Romanian flue with 6 front and 2 thump holes with a permutation of the previous 9-note comprehensive diatonic scale, that is

A3-B3-C4-D4-D4#-E4-F4-G4-G4#-A4

and interval structure 2-1-2-1-1-1-2-1-1

which can play the natural diatonic A minor (A B D D E F G A) , but also the E4 harmonic minor (A B C D E F G# a) and E4 double harmonic minor (A B C D# E F G# A). 

A similar  scale is used in some 7+1 holes XIAO flutes. E.g. 

D D#,  E, F#, G, A, A#, B C# with intervals structure 2-1-1-1-2-1-1-2-1

which is cyclic permutation from the previous thus the same scale. 

With root C it would be

C-D-D#-E-F-G-G#-A-B-C



A slightly alternative such 9-tones scale with pattern of intervals

1-1-2-1-2-2-1-1-1 or in notes e.g. starting from Bb is the 

Bb-B-C-D-D#-F-G-G#-A-Bb which is met in traditional Thailand flutes and the inverse of it by chromatic Quenas.

In more detail chromatic Quenas use t 8+1 holes giving the 9-notes scale 


2-1-1-1-1-1-2-2-1 e,g.

G-A-Bb-B-C-C#-D-E-F#-G

See also the maximal harmonic 9-notes scale in post 117. Still there is an alternative 9-tones scale with pattern of intervals is 

2-1-1-1--2-1-1-1-2   (which is an expansion by semitones at 3-semitones intervals of the mode of the pentatonic 2-3-2-3-2 known as Egyptian 5-tonic scale ) and is also the maximal harmonic 9-notes scale (which  is the 9-notes scale with the maximum number of major or minor triads see post 117) 

This scale is  sometimes is used in folk bass Armenian duduk-like winds

e.g. Bb2-C3-C#3-D3-D#3-F3-F#3-G3-G#3-Bb3

and it happens that I have one such wind.



All 7-notes scales made from semitones, tones and at least one  3-semitone.

Many such scales are used in the Greek folk music with Buzuki, and have their origin in ancient Greece, Byzantine empire, and Arabic music. Many of them have Arabic names although they are played on the 12-semitone Bach equal temperament scale.

For the names of some of these scales see http://www.scales-chords.com/

Therefore its is worth finding them all. We have already found all of them that they do not contain a 3-semitone (see post 51, the diatonic, the melodic minor and the second melodic minor) . So let us find all that contain at least one 3-semitone We already know so far 

1) the Harmonic minor= (1-3-1)-2-(-1-2-2),   and 

2)the Romani (or Hungarian /Ukrainian/Flamenco) double minor (modes also of Niavent and Hijaskar) also called Byzantine or Harmonic double minor scale (and by some also called abydos Egyptin scale)=(1-3-1)-2-(-1-3-1) (notice that the inverse order of it is identical with it). 

The Romanian kaval flutes with 5 holes play the 6-notes scale 2-1-3-1-1-4   E.G. A4-B4-C5-D#5-E5-F5-A5 and by adding a 6th thump hole to play the G5# it becomes a Harmonic double minor scale A4-B4-C5-D#5-E5-F5-G5#-A5  or 2-1-3-1-1-3-1  which is a mode of the E5 harmonic double minor scale.

3) A slight alternation of it is the Persian scale or todi theta scale=(1-3-1-1-2-3-1) 

4)  Inverse Persian scale or Purvi Theta scale (3-1-1-3-2-1-1) or Byzantine parachromatic scale 
The last 3   contain two  3-semitones, but only the Harmonic double minor two tetrachords 1-3-1 ! And the other two  scales of post 50 are 
This scale (as the harmonic double minor or Byzantine double minor too) too  is directly derivable from the pentatonic scale e.g., the Egyptian mode of it 2-3-2-3-2 gives the 
 2-1-1-3-1-1-3 This scale can be easily played in the Shakuhachi minor pentatonic flutes. 



5) the 2nd Harmonic minor or  Kurdi or Kassigar=(1-3-1)-2-(-2-1-2) (which is the inverse order of the Harmonic minor),
6) and the 3rd Harmonic minor or Shamba (which is the inverse of the Neopolitan scale below) =(1-3-1)-2-(-2-2-1) 
all of them containing the oriental tetra-chord 1-3-1.

We may take the inverse order of the Shamba which is called the Neopolitan scale

7) The 4th Harmonic minor or Neopolitan scale (different from  the major and the minor Neopolitan)= (1-3-1)-1-(-2-2-2)

Notice that the Neopolitan scale is made by the inverses of 2 ancient Greek tetrachords the 1-1-3 and the 1-2-2 withe and in between tone 2 (disjunction of tetrachords or divorced tetrachords) . The  1-1-3 was called in ancient Greece the tonal tetrachor of the Chromatic Generation and the 1-2-2 the syntono tetrachord of the Diatonic generation.

Now are there more? Certainly there are! Many of them are modern versions in the 12-semitones scale realizations of ancient Byzantine 7-notes scales or "sounds" (ηχοι). If there are two 3-semitones as in the Romani double minor, we may have a permutation of it , which is not a  cyclic permutation (mode of the Romani double minor) which are the next 

8) (3-1-3-2-1-1-1)
9) Second Harmonic or  Romani   double minor=(3-1-3-1-2-1-1) or gypsy hexatonic or Mela Gayakapriya, Raga Kalakanti (see post  227 )
10) Third  Harmonic or Romani double minor=(3-1-3-1-1-2-1) inverse of Mela Ganamurti, Raga Ganasamavarali (see post  227)
11) (3-1-3-1-1-1-2)
12) (3-1-2-3-1-1-1)




The next scales do not sound too much as minor scales as the 3-3 , or 3-2-3 is not sad and are mainly extrapolations of the western or Chinese/Mongolian  pentatonic !

13)  (3-2-3-1-1-1-1) (with a very peculiar sound)
14) (3-2-1-3-1-1-1) 
15)   (3-3-2-1-1-1-1) (with a very peculiar sound)
16)   (3-3-1-2-1-1-1)
17)    (3-3-1-1-2-1-1)
18)  (3-3-1-1-1-2-1)
19)   (3-3-1-1-1-1-2) (this is the inverse order of scale 15)

While if it has only one 3-semitone we may take non-cyclic permutations of the 3) and 4) like the next fr which  ma nor aware of names, but maybe there are in modern Arabic music (At first we make all possible combinations of different intervals around 3, and then all possible non-cyclic permutations of the rest of the intervals) 

20) (1-3-2)-1-(-2-1-2)
21) (1-3-2)-1-(-1-2-2)
22) (1-3-2)-2-(-1-1-2)
23) Called Enigmatic (1-3-2)-2-(-2-1-1)
24) (1-3-2)-2-(-1-2-1)
25)  (1-3-2)-1-(-2-2-1). This scale is  by combining  the ancient Greek tonal tetrachord of the Chromatic generation 1-1-3 in a disjunctive way -2- ith the syntono tetrachord 1-2-2 of the Diatonic generation.

26) (2-3-1)-1-(-2-1-2)
27) (2-3-1)-1-(-1-2-2)
28) (2-3-1)-2-(-1-1-2)
29) (2-3-1)-2-(-2-1-1)
30) Called Hungarian major scale (2-3-1)-2-(-1-2-1)
31)  (2-3-1)-1-(-2-2-1) This scale is
refered in this video https://www.youtube.com/watch?v=mjttaiOq-8Q
as the 7-notes soul scale and is refred as the major pentatonic scale with added flat 3rd and flat 7nth!
32) (2-3-2)-1-(-2-1-1) 
33) (2-3-2)-1-(-1-1-2)
34) (2-3-2)-1-(-1-2-1)
35) (2-3-2)-2-(-1-1-1)

It is clear that these scales can be ordered according to how many usual chords they define (like major, minor diminished, augmented). The more non-weird chords the higher in the list. In other words they are not all of them the same good from the point of view of harmony of the chords they define, although they may seem very similar from the melodic point of view.


There are also the 6 notes Hirajoshi scale

http://www.flutopedia.com/scale_Hirajoshi_Extended.htm

and the Miyako Bushi extended scale

http://www.flutopedia.com/scale_MiyakoBushi_Extended.htm

ALL 4-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 , 1-1-3,  or two 3's 

1) 1-3-1-7

2) 1-3-7-1

3) 1-3-3-5

4) 1-3-5-3


ALL 5 CHROMATIC 5-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 , 1-1-3,  or two 3's 

1) 1-3-1-3-4 (a 6-notes sub-scale of the double harmonic minor)

2) 1-3-1-4-3 

3) 1-3-3-1-4

4) 1-1-3-3-4  

5) 1-1-3-4-3  


ALL 6-NOTES SCALES THAT CONTAIN THE PATTERN IN SEMITONES 1-3-1 .

1) 2-1-3-1-2-3 (the Erik Satie 6-note scale)

2) 1-3-1-3-2-2 

3) 1-3-1-3-1-3 (a Messiaen  scale) 

4) 1-3-1-3-3-1

5) 1-3-1-4-2-1

6)  1-3-1-1-4-2

7) 1-3-1-2-4-1

8) 1-3-1-5-1-1

9) 1-3-1-1-5-1


NOTICE THAT THE FAMOUS BLUES 6-NOTES SCALE CONTAIN IS NOT THE 1-3-1 BUT THE 1-1-3:

3-2-1-1-3-2

Some of the  4 and 5-notes sub-scales (tetra-chords and penta-chords) of the above 7-notes scales have known Arabic names

Tetra-chords (all the next are diatonic tetra-chords)

Rast     2-2-1
Ussak  1-2-2
Kurdi    2-1-2

The next is from the second melodic minor or leading whole tone scale

Shamba    2-1-1 

The next contain a 3-semitone

Niavent     2-1-3
Hijazz       1-3-1
Huzam      3-1-1
Piraeus     1-3-2

And the next are 5-notes sub-scales (penta-chords)  that are essentially diatonic

Rast     2-2-1-2
Ussak  1-2-2-2
Kurdi    2-1-2-1 (this is from the melodic minor)
Minor    2-1-2-2

The next contain a 3-semitone

Shamba   2-1-1-3
Nikriz       2-1-3-1
Hijazz      1-3-1-2
Huzam     3-1-1-2


See also https://www.youtube.com/watch?v=B6xddWJFmt8 
For Arabic names of many of the previous scales see

http://www.maqamworld.com/

And 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 10 of such characteristic tetra-chords
They are also all such tetra-chords containing intervals of 1,2,3, and where inverses and cyclic permutations of them do not count as different . Obviously all of the above scales are compositions of two of them, with possibly an extra interval between them

Diatonic

2-2-1, (major, natural minor rast, ussak)

2-2-2, (major, augmented)

Melodic minor, double minor (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

Ir is obvious also that by extrapolating the steps 3-semitones to 2+1 we get 8-notes and 9-notes scales that  are made only from steps of 1 and 2 semitones. 

For a list of scales with their chords see http://www.scales-chords.com/

We may compare these scales with the 6-notes minor blue scale derived from the minor pentatonic scale , with interval structure

3-2-1-1-3-2   (see e.g. http://www.jazzguitar.be/minor-blues-scale.html ) which is analysis of the western pentatonic below.

(for the 6-notes major blues scale derived from the major pentatonic scale see post 54) 

The 6-tone scale 1-3-3-1-3-1, the inverse which is 1-3-1-3-3-1 and the 
   1-3-1-3-1-3

Or the 5-tone Egyptian / Mongolian mode  of the western pentatonic 2-3-2-3-2

The 6-tone Prometheus  2-2-2-3-1-2

The 6-tones inverse Prometheus 2-1-3-2-2-2

 or the 6-tone 2-3-2-2-1-2 which is extrapolation of the 5-tonic

We may compare these scales with the Western pentatonic scale that has  also two 3-semitones, that is of  interval step structure 2-2-3-2-3  (also known as Egyptian/Mongolian).

Or compare them with the oriental 6-note scale  1-3-1-3-1-3

Or compare them with other 4-notes scales like 3-3-3-3 

the 3-5-3-1 and  2-1-4-5  or 3-2-2-5 or 4-2-1-5, and  2-2-3-5 that extrapolations to 4-notes scales of  3-notes major or minor chords arpeggios-scales

Or the 8-notes Algerian=2-1-2-1-1-1-3-1

Or the 8-notes extrapolation of the  Romani double minor

1-3-1-1-1-1-3-1

Or the Chinese 5-notes scale    that have 2-tones steps , in other words interval structure 
Interval fro the root : 1, 3, #4, 5, 7
Intervals in steps: 4 - 2 - 1 - 4 - 1  (the tetra-chord 1-4-1 is used instead of the oriental 
1-3-1)
Formula: Quadra-step, Whole, Half, Quadra-step, Half  4 - 2 - 1 - 4 - 1
See e.g.
http://www.pianoscales.org/chinese.html

C: C, E, F#, G, B, C  (we notice that it is a sub-scale of the F-major 7-notes scale)


For example if we take the mode of the diatonic scale starting from F

F G A B C D E F , a Chinese scale  will be shaped by keeping  the semitones B C,, E, F, but eliminating the notes G, D , Thus it will be

F, A, B, C, E, / F


Somehow all the above scales may be considered extrapolations of arpeggios of 3-notes chords of the type major,minor, diminished and augmented. 

The oriental (Hijazz) tetra-chord and the creation of all possible 7-notes scales by combinations with diatonic tetra-chords . 


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),


A mode (cyclic permutation) of the harmonic minor is called  also Blue-scale in American folk music, and another cyclic permutation of  it the Romani minor scale

Chords of the harmonic minor (intervals 2-1-2-2-1-3-1):



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

2) One that may be called 2nd Harmonic minor , called also mode of Kurdi 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).

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.

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 variations of the Byzantine double minor scale are the Persian and inverse Persian scales 
But only the Byzantine (harmonic) double minor containes 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

We may compare these scales with the melodic minor (e.g. A melodic minor) and its chords

The chord chart shows the traid and 4 note extended chords belonging to the key of A Melodic Minor.

• A Melodic Minor scale notes: A B C D E F# G#

Triads:	min	min	aug	maj	maj	dim	dim
Extended:	min/maj7	min7	maj7#5	dom7	dom7	m7b5	m7b5

i	ii	III	IV	V	vi	vii
Amin	Bmin	Caug	Dmaj	Emaj	F#dim	G#dim
Aminmaj7	Bm7	Cmaj7#5	D7	E7	F#m7b5	G#m7b5

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.