225. THE GENEALOGIES OF SCALES AFTER THE ANCIENT GREEK MUSICAL SYSTEM

225. THE  GENEALOGIES OF SCALES AFTER THE ANCIENT GREEK MUSICAL SYSTEM  

In ancient Greece according to Aristoxenos, and Euclid's katatomi kanonos, there were 3 musical gendres , the diatonic , the chromatic and enharmonic which were also tetrachords  (total length an interval of 4th)  and pentachords (total legth an interval of 5th and consisting from an initial or final tone and a tetrachord as before) and by combing a tetrachord with a pentachord they got various   7-notes scales inside an octave . In other words the 7-notes scale was of the form Tetrachord1+2+tetrachord2 , in which case it was called two "divorced" tercahords or Tetrachord1+tetrachord2+2 in which case it was called two adjoint tetrachords.  1) the Diatonic  was balanced feelings (that eventually after more than 1 thousand years resulted to the 7-notes diatonic scale, and a simplified rounding of their tetrcahords to a single one was the 2-2-1 and most probably all permutations of it )  the 2) Chromatic  was sad feelings (that eventually after more than 1 thousand years resulted to the 7-notes harmonic minor and double harmonic minor scales, blues scale etc , And a simplified rounding of their tetrachords to a single one was the 3-1-1 nd most probably all permutations of it  ) and 3) the enharmonic  was happy feelings (used major 3rd intervals and a simplified rounding to a single one was the 4-(1/2)-(1/2)) and most probably all permutations of it )  and has many similarities with the Japanese 5-notes scales like the Akebono with interval structure 4 - 2 - 1 - 4 - 1 e.g. at the mode 2-1-4-1-4  e.g. A B C E F A. (See e.g. the video https://www.youtube.com/watch?v=vv7CO-nVFj8 ) 

THE RESRICTED ENUMERATION-GENERATION OF SCALES  ACCORDING TO EUCLID'S KATATOMH KANONOS OF ANCIENT GREEK MUSICAL SYSTEM

This is based on the next restrictions

1) Smaller intervals in a tetrachord are always at lower frequencies

2) When joining two tetrachords either we put an inteval of one tone in betwenn, or at the end of the 2nd tetrachord

In ancient Greece according to Aristoxenus the tetrachords where 6 and of 3 gendres

They divided the semitone in to 6 equal segments, or the octave in to 72 segments or the pure 4th in to 30 segments.

Then the tetrachords were the next (the integers are Aristoxenus segments, the fractions of a tone by Euclid in Katatomh kanonos))

ENHARMONIC  GENDRE

3-3-24  or semitone/2 -- semitone/2 --2 tones

CHROMATIC GENDRE

Soft 4-4-22  or 1/3 tone 1/3 tone  11/6  of a tone

hemiolion  4.5 -4.5 -21  or 3/8 tone 3/8 tone and  7/4 of a tone


tonal   6-6-18  or 1 semitone 1 semitone 3 semitones

DIATONIC GENDRE

uniform soft  6-9-15  or 1/2 tone 3/4 tone and 5/4 tone

syntono strong  6-12-12  or 1/2 tone , 1 tone 1 tone 

Now two tetrachords make a scale of an octave in two ways 

The disjunctive way  = tetrachord+1tone+tetrachord=5+2+5 semitones=12 semitones

Or the adjoint way = tetrachord+tetrachord+1 tone=5+5+2 semitones=12 semitones

E.g. syntono+1tone+syntono=diatonic scale lydian mode
or syntono+syntono+1tone=diatonic scale Locrian mode

or tonal+tonal+1tone=6-6-18-6-6-18-12 which is a mode ofthe by now known as inverse persian todi theta scale 
The same with the tonal+1tone+tonal.

etc

See also post 227


For reasons of completeness we enumerate here the trichords which are of length an interval of 3rd.

All of them are of the DIATONIC FAMILY

1-2

2-1

2-2 

Because half semitones are not palyable in he piano, and we want to enumarate scales and sbscales of the ancient Greek system playable in the piano, we simplyfy the tetrachords to the next 

ENHARMONIC  GENDRE

1-4  semitones, inteeval of 4th

CHROMATIC GENDRE

1-1-3 semitones

DIATONIC GENDRE

1- 2-2  semitones 

Then we start combining tetrachords to obtain  scales of one octave.

We can combine a gendre with itself (pure gendre) or a gendre with another gendre to obtain an hybrid gendres.

E.g. all 3*3=9 cases are

1) Diatonic-Diatonic

2) Chromatic -chromatic (double chormatic)

3) Enharmonic-enharmonic (double enharmonic)

4)  Diatonic-Chromatic

5) Chromatic-Diatonic

6) Diatonic-enharmonic

7) Enharmonic-diatonic

8) Chromatic-enharmonic

9) Enharmonic-chromatic

To this we may add the possibility that the combination is diazeuksis (one tone in between) , or syzeuksis (a tone at the end) , In principle we get 18 combinations but some of them give the same scale.

Let us look in detail to the scales.

1) Diatonic-Diatonic  gives 

1-2-2-2-1-2-2 (diazeyksis)  or 1-2-2-1-2-2-2 (syzeyksis) which is again the well known 7-notes diatonic scale.

2) Chromatic -chromatic (double chormatic)  gives

1-1-3-2-1-1-3 (diazeuksis) and 1-1-3-1-1-3-2 (syzeyksis) which is aagain the same scale  (but different modes) which is  the parachromatic scale, or the blues scale with a leading tone.  It is called also inverse persian scale and Purvi theta scale. 

3) Enharmonic-enharmonic (double enharmonic) gives 

The pentatonic scale 

1-4-2-1-4 (diazeuksis) and 1-4-1-4-2 (syzeuksis) both are the same pentatonic scale. It  is called Lydian pentatonic and it is also  met in japan and is also called Hirajoshi scale. The japanese santuri, called Koto, is tuned in it. It is met also in India as Raga Amritavarshini, Malashri, Shilangi, and in Ethiopia as Batti Lydian  (See post 227)

See also      https://en.wikipedia.org/wiki/Hiraj%C5%8Dshi_scale

4)  Diatonic-Chromatic  gives 

1-2-2-2-1-1-3 (by diazeyksis) or 1-2-2-1-1-3-2 (by syzeuksis), that are different scales! 

The 1-2-2-2-1-1-3 is called sabach  .

For the sabach see

 https://www.youtube.com/watch?v=TqpI3JpCopw&ab_channel=%CE%9A%CF%89%CE%BD%CF%83%CF%84%CE%B1%CE%BD%CF%84%CE%AF%CE%BD%CE%B1%CE%9A%CF%89%CF%83%CF%84%CE%AD%CE%BB%CE%B7-KonstantinaBouzoukiLessons

It is also met in India under the name Mela Senavati (see post 227) 

The 1-2-2-1-1-3-2  may be called dual of sabach, as it is also obtained as 

Cromatic-Diatonic instead of Diatonic-Chromatic  . 

The Neapolitan minor is  the inverse of the sabach scale . See 

https://en.wikipedia.org/wiki/Neapolitan_scale

The 6) Diatonic-enharmonic gives 

1-2-2-2-1-4 , which is a hexatonic scale , met at least in india under the name Mela Suddha Simantini (see post 227) . It has the 2 chords 4M 3 m . It is also met as in Cretan folk melodies in Lyra.  It is related also (it contains as subsale) to the pentatonic scale Akebono.

The 7) Enharmonic-Diatonic  gives 

the hexatonic scale 1-2-2-1-4-2  , which is met in japan as Honchchi plagal form (see post 227) 

and is also a hexatonic version of the diatonic  Locrian mode.

Finally the 

The 8) Chromatic-enharmonic  gives the  hexatonic 

1-1-3-2-1-4, which is met in India as inverse Raga Tilang.

The 9) Enharmonic-chromatic  gives the hexatonic

1-4-2-1-1-3  , which is met in India as  raga kalakhanthi 

The alternatives diazeuksi or syzeyksi again lead the same scales. 

THE  1ST FREE  GENERATION OF SCALES AND MODES AFTER THE ANCIENT GREEK MUSICAL SYSTEM.

This is based on the next rules

1) Any order of the intervals in semitones in a tetrachord. We allow permutations of  the intervals of the ancient Greek tetrachords in all the 3 gendres

2) Any comination of a tetrachord and a pentachord. 

3) We retain the 1/2 semitone intervals in the 1st desgn but then we simpolify two consecutive of  them to an interval of semitone so as the scale finally si playable in the piano.

In Ancient Greece, Byzantine and middle east, the 7-notes scales were created by combining 4-notes (tetrachords) that have total length an interval of 4th and 5-notes (pentachords) that have total length and interval of 7th. In this way all  such 7-notes scales contain the power-5 chord 1-5-1' (e.g. C3-G3-C4).

The representation of such tetrachords and pentachords can be with the sequence of intervals in semitones. Thus to find all of them is in mathematical terminology to find all partitions of 5 in ti 3 numbers (having sum 5) and all partitions of 7 in to 4 numbers (having sum 7).

The solutions is the next

ALL 6 TETRACHORDS

DIATONIC GENDRE (FAMILY)

(containing only 2 and 1)

2-2-1  syntono major  , Ionian/mixolydian (known also as Rast/Tsargiak/Mahour)

1-2-2  syntono minor  , Phrygian/Locrian (known also as Qurdi)

2-1-2   minor, Dorian/Aeolian (known also as Bousselik/Nichavent/Ousak)

CHROMATIC GENDRE (FAMILY)

(containing 3)

1-3-1   2nd chromatic  (known also as Hijaz)

1-1-3    parachromatic tonal

3-1-1    1st chromatic

ENHARMONIC GENDRE (FAMILY)

1/2-1/2-4
1/2-4-1/2
4-1/2-1/2

 

ALL 20 PENTACHORDS

All pentachords contain a single power-5 chord that power-5 chord which is defined by the end-notes of the pentachord.

THE DIATONIC FAMILY
(containing only 2, and 1)

2-2-2-1       Major Lydian    Contains a major chord (known also as Natzi/Zaouil)
2-2-1-2        Major Ionian/Mixolydian    Contains a major chord.(known also as Rast/Mahour)
2-1-2-2       Minor Dorian    Contains a minor chord (known also as Bousselik/Nichavent)
1-2-2-2       Minor Frygian/Aeolian    Contains a minor chord (known also as Qourdi)


THE CHROMATIC FAMILY
(containing  3 )

1-1-3-2
1-1-2-3
1-3-2-1   Contains a major chord and a diminished chord
1-2-3-1   Contains a minor chord and a diminished chord
1-2-1-3   Contains a minor chord and a diminished chord
1-3-1-2   Contains a major chord and a diminished chord (known also as Hijaz)
3-2-1-1    Contains  a diminished chord
2-3-1-1
3-1-2-1   Contains a major chord and a diminished chord
2-1-3-1   Contains a minor chord and a diminished chord (known also as Nevesser/Nikriz)
3-1-1-2   Contains a major chord (known also as Saz-kar)
2-1-1-3   Contains a minor chord (known also as Samba)


THE ENHARMONIAN GENDRE (FAMILY)
(containing  4 )

1-1-1-4   Contains a minor chord
1-1-4-1
1-4-1-1
4-1-1-1   Contains a major chord




By combining a treachord and a pentachord we get a 7-notes scale. We remind the reader that in this book by scale we mean a sequence of notes giving an octave and all cyclic permutations of them are considered the same scale but also 7 different modes of it.

As a rough estimation of an upper bound on the number of the scales of this genealogy we calculate that if x is the number of different such scales then x<=6*20*2*2=480, because we will combine 6 tetrachords with 18 pentachords thus 6*20=120 such combinations, but also combining first 18 pentachords and then 6 tetrachords will give twice as many, thus 6*2*20=240 . And by taking also the inverse order on all notes of them at most  6*20*2*2=480 scales. Of course some of them will turn out to be simply modes (cyclic permutations ) of some previous or next, thus 480 is an upper bound not an exact number.

In this way we can get the next scale

1) The diatonic
2) The harmonic minor
3) The inverse of the harmonic minor
4) The Neopolitan minor scale
5) The inverse of the Neopolitan minor  scale
6) The double harmonic minor
7) The melodic minor
8) The double melodic minor or Arabic scale or Neopolitan major
9) Parachromatic Byzantine scale (1-1-3-2-1-1-3)
10) Inverse parachromatic Byzantine
11) Debussy's 7-tonic (2-1-1-1-1-3-3) .
etc

Modes of the above 1-8 scales include the Byzantines minor and Byzantine double minor , Hungarian minor, Gypsy major and minor, Rast scale, Nikriz scale, Hitzaz scale, Hitzaskar scale , Nihavent scale, Neveser scale, 7-notes Sambach scale, Ousak scale , Qurdi scale , kartzigiar scale etc


Some japanese folk scales

Also arabic scales (projected in the Bach equally temperament  12-notes chromatic scale ) under the names  Shouzinak , Housam, Shegiach , Moustaar  , Persian or todi theta scale, the Enigmatic scale,
Debussy's 7-tonic (2-1-1-1-1-3-3) etc
 etc

See also post 21.

And many more beautiful scales that are not known with any name!

I

OF SCPECIAL INTEREST IS 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.   AND IT IS  CONSTRUCTED FROM THE 4-CHORD 2-2-1 AND THE 5-CHORD 2-1-2-2. IT IS THE MELODIC MINOR OR ACOUSTIC OVERTONES SCALE IF THE 7NTH HARMONIC E.G. IN C-D-E-F-G-G#-B-C  if we interprete the 7nth overtone as B rather than Bb. BUT 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 perceive THE 7NTH HARMONIC AS   B


ALL 7-NOTES SCALES OF THE ENHARMONIC GENDRE BY COMBINING A DIATONIC TETRACHORD WITH AN ENHARMONIC PENTACHORD

2-2-1-1-1-1-4
2-2-1-4-1-1-1
2-2-1-1-4-1-1
2-2-1-1-1-4-1

1-2-2-1-1-1-4
1-2-2-4-1-1-1
1-2-2-1-4-1-1
1-2-2-1-1-4-1

2-1-2-1-1-1-4
2-1-2-4-1-1-1
2-1-2-1-4-1-1
2-1-2-1-1-4-1

ALL 6-NOTES (ENHARMONIC GENDER)  SCALES THAT CONTAIN THE POWER-5 CHORD AND AT LEAST ONE MAJOR 3RD INTERVAL

Many of  the irish melodies belong to such scales!

Because the power chord is contained, then there must exist an interval of 4th and an interval of 5th. If the interval of 4th contains a major 3rd then the only solutions are

4-1, 1-4 . Otherwise the interval of major 3rd must be contained inside the interval of 5th, and thus we have the 5-chords of the enharmonic gender


1-1-1-4   Contains a minor chord
1-1-4-1
1-4-1-1
4-1-1-1   Contains a major chord

In total we enumerate at first all such scales with 2 major 3rd intervals

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

If there is only one major 3rd interval (4 semitones) 

Then we combine the 1-4 , 4-1 with any 5-chord in the diatonic or chromatic gender

4) 1-4-2-2-2-1
5) 1-4-2-2-1-2
6) 1-4-2-1-2-2 
7) 1-4-1-2-2-2  
8) 1-4-1-1-3-2
9) 1-4-1-3-2-1 
10) 1-4-1-2-3-1
11)  1-4-1-2-1-3 
12)  1-4-1-3-1-2
13) 1-4-2-3-1-1
14) 1-4-3-1-2-1
15) 1-4-2-1-3-1
16) 1-4-3-1-1-2
17) 1-4-2-1-1-3 

18) 4-1-2-2-1-2
19) 4-1-2-1-2-2  
20) 4-1-1-2-2-2  
21) 4-1-1-1-3-2
22) 4-1-1-2-3-1
23)  4-1-1-2-1-3 
24) 4-1-3-1-1-2
25) 4-1-2-1-1-3 

Among these 25 scales we filter the combinationof diatonic 4-chords and only one major third
thus haveing the 7 diatonic-enharmonic hexatonic scales

I have seydel harmonicas with these dtaitonic-enharmonic hexatonic scales

4) 1-4-2-2-2-1 (HEX ARABIC OR HEX NEAPOLITAN MAJOR 1M =inverse of 20) ) 
5) 1-4-2-2-1-2 (HEX MELODIC MINOR 1M-2d  inverse of 19))
6) 1-4-2-1-2-2 (HEX DIATONICENHARMONIC 1m-2d)
7) 1-4-1-2-2-2  (HEX AKEBONO-CRETAN 1M-7m)
18) 4-1-2-2-1-2 (HEX ENHARMONIC 1M-7d)
19) 4-1-2-1-2-2  (HEX HMIN ENHARM 1m-7d inverse of 5) )
20) 4-1-1-2-2-2  (HEX ARABIC OR HEX NEAPOLITAN MAJOR   1m)


Among these 25 scales we filter those that contain the 4-3, or 3-4 which is triad chord as with more harmonic sound

Major chord 4-3:

14) 1-4-3-1-2-1
16) 1-4-3-1-1-2
By cyclic permutation we may re-write them as 

14) 4-3-1-2-1-1
16) 4-3-1-1-2-1

Minor chord 3-4:
23)  4-1-1-2-1-3 
25) 4-1-2-1-1-3 

By cyclic permutation we may re-write them as 

23) 3-4-1-1-2-1
25) 3-4-1-2-1-1 


ALL 5-NOTES  SCALES THAT CONTAIN THE POWER-5 CHORD AND AT LEAST ONE MAJOR 3RD (4 SEMITONES)

This already mean that we know tow notes of the scale . The other 3 

1) All in the lower interval of 5th . Thus a 5-chord as above and then an interval of  4th or 5 semitones. 

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

2) All in the upper interval of 4th . Thus a 4-chord as above as as upper 3 notes and as lower notes the  3-4, 4-3 E.g.

5) 4-3-2-2-1 Known in this blog as the inverse maximal harmonic 5-tonic scale (see post 117  and 204 )
6) 4-3-1-2-2
7) 4-3-2-1-2
8) 4-3-1-3-1 
9) 4-3-1-1-3
10) 4-3-3-1-1

and

11) 3-4-2-2-1
6) 3-4-1-2-2
7) 3-4-2-1-2
8) 3-4-1-3-1 
9) 3-4-1-1-3
10) 3-4-3-1-1

3) 2 notes in the lower interval of 5th 

3-2-2, 2-2-3, 4-2-1, 4-1-2 , 2-1-4, 1-2-4, 3-3-1, 1-3-3, 3-1-3, 1-1-5, 5-1-1, 1-5-1

and one in the upper interval of 4th

2-3, 3-2, 1-4 , 4-1 Thus

11) 3-2-2-4-1
12) 3-2-2-1-4
13) 2-2-3-1-4
14) 2-2-3-4-1 Known also as Greek pentatonic 2-2-3-4-1 (known also as Raga Chitthakarshini)
15) 4-2-1-2-3
16) 4-2-1-3-2
17) 4-2-1-1-4
18) 4-2-1-4-1 known also as the Japanese  Hirajoshi 5-notes scale and In 5-notes scale and Iwato 5-notes scale and as Akebono 5-notes scale
19)  4-1-2-3-2
20) 4-1-2-1-4
21) 4-1-2-4-1
22) 2-1-4-2-3 known also as the Japanese  Insen 5-notes scale 
23) 1-2-4-2-3
24) 1-2-4-3-2
25) 1-2-4-4-1
26) 3-3-1-1-4
27) 3-3-1-4-1
28) 1-3-3-4-1
29) 3-1-3-1-4
30) 3-1-3-4-1
31) 1-1-5-1-4
32) 1-1-5-4-1
33) 5-1-1-1-4
34) 5-1-1-4-1
35) 1-5-1-4-1
36) 1-5-1-1-4

4) 1 note in the lower interval of 5th and 2 notes in the upper interval of 4th

thus 1-6, 6-1 and 1-1-1-2, 1-1-2-1, 1-2-1-1, 2-1-1-1, but none contains 4 , thus no new 5-notes scales.


2-1-4-1-4

THE 2ND FREE GENERATION  OF SCALES AND MODES AFTER THE ANCIENT GREEK MUSICAL SYSTEM

This is based on the same rules as with the 1st free enumaration except 

1) We do not need to obtain an octave scale, only by combining a tetrachord (interval of 4th) and a pentachord (interval of 5th) .

2) We stil keep the 3 geentres as

a) Pure diatonic gendre= Only intervals of 2nds

b) Pure Chromatic gendre=only intervals of 2nd and mior 3rd

c) Pure Enharmonic gendre=only intervals of 2nds and major thirds.

ALL THE 7 NOTE SCALES MADE ONLY FROM TONES AND SEMITONES

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.  Stephan Grappelli soloing is utilizing the diatonic speed , with almost no (middle harmonic speed)  waving but approximation of continuous movements at diatonic speed together with occasional spike jump intervals  at high harmonic speed or even higher jumps. 

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)


We may compare them with the 

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

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

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. 

The chords of  the 8-notes Spanish-Jewish are the next when starting from C

Notes of this scale:

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

Interval structure of this scale:

h W h h h W W W

Chords that fit in this scale:

Normal Triads: C#     C#m     Caug     Cdim     D#m     Eaug     Fm     F#     G#     G#aug     A#m     A#dim  Other Triads: C#sus4     C#sus2     D#sus4     D#sus2     Fsus4     F#sus2     G#sus4     G#sus2     A#sus4     A#sus2  4 Notes Chords: C7b5     C7#5     C#6     C#m6     C#maj7     C#m(maj7)     Cm7b5     D#m6     D#m7     D#7sus4    D#7sus2     Fm7     Fm(maj7)     F7sus4     F#6     F#maj7     F#7     F#7b5     F#7sus2     G#6     G#7     G#7#5     G#7sus4     G#7sus2     A#m7     A#m7b5     A#7sus4     A#7sus2

Also 9- notes scales like  2-2-1-2-1-1-1-1-1 or 2-1-1-2-1-1-2-1-1 and non-cyclic permutations of them. 
E.g. The 9-tones Moorish Phrygian =1-2-1-1-2-1-2-1-1

or  9-tones scales like 2-1-1-1-2-2-1-1-1 (C, D, D# ,E F, G A A# B C)

or  9-tones scales like 2-1-2-1-1-1-2-1-1 which is an extrapolation from the Harmonic double minor 2-1-3-1-1-3-1

We may compare these scales also with the Chinese 5-notes scales 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
Formula: Quadra-step, Whole, Half, Quadra-step, Half
See e.g.
http://www.pianoscales.org/chinese.html


The chords that fit to the the melodic minor scale are the next: we take as example the D melodic minor

Notes of this scale:

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

Interval structure of this scale:

2-1-2-2-2-2-1

Chords that fit in this scale:

Normal Triads: C#aug     C#dim     Dm     Em     Faug     G     A     Aaug     Bdim  Other Triads: Dsus4     Dsus2     Esus4     Gsus2     Asus4     Asus2  4 Notes Chords: C#7b5     C#7#5     C#m7b5     Dm6     Dm(maj7)     Em6     Em7     E7sus4     G6     G7     G7b5     G7sus2     A7     A7#5     A7sus4     A7sus2     Bm7b5

A way to play this melodic minor or Hindu scale on the 4 highest strings, with the rule of 2 notes per string is the next

The chords that fit the second melodic minor scale or also called leading whole tone scale are the next (we take the example of the C leading whole tone (or second melodic minor ) scale

Notes of this scale:

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

Interval structure of this scale:

2-2-2-2-2-1-1

Chords that fit in this scale:

Normal Triads: Caug     Daug     E     Eaug     F#aug     G#aug     G#dim     A#aug     Bm  Other Triads: Esus2     Bsus4  4 Notes Chords: C7b5     C7#5     D7b5     D7#5     E7     E7b5     E7#5     E7sus2     F#7b5     F#7#5     G#7b5    G#7#5     G#m7b5     A#7b5     A#7#5     Bm6     Bm(maj7)

A way to play this previous scale sometimes cal led Arabic scale on the 4 highest strings, with the rule of 2 notes per string is the next

The next are 5 ways to play on the guitar fretboard the melodic minor scale

On the higher 4 strings of the guitar (or on a ukulele or on a 4-string Buzuki) the next is the pattern on the fretbaord of the second melodic or leading whole tones scale (although without the notes names that would not be accurate on the strings as the ukulele is tuned a bit higher than the guitar)

We may compare these scales also with the Chinese 5-notes scales 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
Formula: Quadra-step, Whole, Half, Quadra-step, Half
See e.g.
http://www.pianoscales.org/chinese.html

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, 

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 Neapolitan scale (different from  the major and the minor Neopolitan)= (1-3-1)-1-(-2-2-2)

Notice that the Neapolitan 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)  Anasazi flute scale or  

Mela Vanaspati, Raga Bhanumati

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) Inverse Anasazi flute scale or inverse 

Mela Vanaspati, Raga Bhanumati

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

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.

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 with interval structure

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

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

And the 6-tone scale of the Hang instrument (raised Celtic minor )(see https://en.wikipedia.org/wiki/Hang_(instrument) )

2-1-2-2-2-3 

This 6-tone Hang-scale can be derived from the dominant pentatonic 

3-3-2-2-2 by splitting the 3 o 2-1, or can be derived also

from the 7-tone diatonic scale 2-1-2-2-2-1-2 by merging 1-2 to 3. 

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

The pentatonic scale is the arpeggio of a major chord with 6th and 9th (or 2nd) which is the usual pentatonic scale

2-2-3-2-3.

Here  also we may mention the blue arpeggio (arpeggio of both minor and major chord)  of a  chord with 6th, which is a pentatonic scale of the next structure

3-1-3-2-3

which can be considered as the harmonic minor that we subtracted the 4th and 7nth note!



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 (raised Celtic Minor ) 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. 

If we ask to find all scales that use steps of only 2 or 3 semitones, then they must be 5-tonic or 6-notes and the next  are all the possible cases

5-tonic
3-3-2-2-2 dominant pentatonic
2-3-2-3-2  pentatonic

6-notes
2-2-2-2-2-2  (Whole tone)

224. THE 7 BASIC IMPROVISATIONAL EXERCISES ON A WIND

The 6 basic improvisational exercises on a whistle or  transverse flute or quena (rim blown flutes) or winds with reeds are the next 6


1) Correct covering of holes and scale fingering

2) Overtones awareness and octaves crossing

3) Ornamentations (Cuts or hammers, double cuts, slides (rolling or bends), vibratos, etc)

4) 2,3,4,5 etc Translations (tails or Alankaras)

5) Rotations (or permutations, including arpeggios of chords in 2 octaves and power chords too in 2 octaves and arpeggios or permutations or rotations of vector intervals of 3rd (trichords), 4th (tetrachords) and 5ths (pentachords).

6) Inversions (in pitch or rhythm of small musical themes.

7) Improvisations restricted in a pentachord (as in post 225) and over 3 chords.

We enlarge more on them.  But the next 18+7 videos are very instructive too



A) 18 Videos by Phil (Kerry whistles) 

https://www.youtube.com/watch?v=SQq2dbN5Xsk&list=PLiYGJiKPn5B3RN_xjtQ6ZktWFAg4HWv7Z&index=18


B) 7 Videos by Das (flute maker)

https://www.youtube.com/watch?v=7Ox55nMD9RE

223. MINORIZATION-MAJORIZATION AS A UNIQUE HARMONIC TRANSFORMATION (NEGATIVE--POSITIVE HARMONY) MODULATIONS VERSUS TRANSPOSITIONS OF MELODIES AND HARMONIES

Here it is described why and how the transformation of a song from a major mode diatonic scale to a minor mode diatonic scale (and vice versa) is not just an ordinary change of mode among the 7 possible modes of the diatonic scale (e.g. from the ionina to the aeolian) but a  unique harmonic and melodic transformation.

In the nxt video this transformation from a  major mode to the minor mode (but also as scale modulation) is described under the term negative harmonics.

Do you know beautiful songs written in a Aeolian or minor mode of a diatonic scale? Then try the positive harmony, in other words transform them melodically and harmonically to a major mode and make them happier.

https://www.youtube.com/watch?v=qHH8siNm3ts&t=168s


TRANSCRIPTIONS:
We must discriminate between the concept of transposition transcription  of the  melody and harmony of a song from that of modulation transcription .  A transposition transcription preserves the intervals among the notes of the melody and types of chords and it is played on higher or lower pitch. But a modulation transcription does not preserves the intervals among the notes of the melody neither the types of the underlying chords as it is "translating within the scale" and not within the 12-notes chromatic scale, the melody and its chords starting from a different note of the scale (different mode). Both though require that all the melody (or song) is written in a single scale (or mode) and the transcription is another scale or mode of equal number of notes. 

222. THE 7 MODES OF A DIATONIC SCALE AS TRIADS OF CHORDS

We cannot stress much the importance of triads of chords (but also of notes intervals and scales) in improvisation.

A good application is to realize that as a diatonic scale is characterized as chords on the wheel by 4ths by its 3 chords on the steps 1M, 4M , 5M  (harmonic triplet of chords) ,  OR 7d 1M 2m (chromatic tripler of chords) the same can apply for the modes of the diatonic scale.

The chords of a mode of a diatonic scale at the steps 1,4,5 (or 7,1, 2) of it characterize its mood.

For example for the mixolydian mode of the diatonic scale from C ,(G, A, B , C, D, E, F, G) the chords at the steps 1,4 ,5 are the G, C, Dm) thus the 1, 2, 5 or I, ii, V of the original ionian and are the base of blues jazz.


Based on this perception, the 7 modes of the diatonic scale have the following chord triads:

(that is chords on their 1st 4th and 5th step, and we symbolize by m=minor M=major d=diminsihed.
Based on this perception, the 7 modes of the diatonic scale have the following chord triads:

(that is chords on their 1st 4th and 5th step, and we symbolize by m=minor M=major d=diminsihed.



1. IONIAN  :  HARMONIC TRIPLET (1M, 4M, 5M)
2. DORIAN  : HARMONIC TRIPLET (2m, 5M, 6m)
3. PHRYGIAN : HARMONIC TRIPLET (3m, 6m, 7d)
4. LYDIAN: HARMONIC TRIPLET  (4M, 7d,1M)
5. MIXOLYDIAN: HARMONIC TRIPLET  (5M, 1M,2m)
6. AEOLIAN: HARMONIC TRIPLET (6m, 2m, 3m)
7. LOCRIAN: HARMONIC TRIPLET  (7d, 3m, 4M)

We may clasify them according the the major and minor chords

A.ALL 3 MAJOR CHORDS (ALL HAPPY) 

1. IONIAN  :  HARMONIC TRIPLET (1M, 4M, 5M)

B. 2 MAJORS AND ONE MINOR OR DIMINISHED CHORD (ALMOST HAPPY, ALMOST MAJOR)

4. LYDIAN: HARMONIC TRIPLET  (4M, 7d,1M)

5. MIXOLYDIAN: HARMONIC TRIPLET  (5M, 1M,2m)

C.  2 MINORS (OR DIMINISHED) AND ONE MAJOR  (LITTLE SAD,  ALMOST MINOR )

2. DORIAN  : HARMONIC TRIPLET (2m, 5M, 6m)

7. LOCRIAN: HARMONIC TRIPLET  (7d, 3m, 4M)

D. THREE MINORS OR DIMINISHED (ALL SAD) (ALL MINOR) 

3. PHRYGIAN : HARMONIC TRIPLET (3m, 6m, 7d)

6. AEOLIAN: HARMONIC TRIPLET (6m, 2m, 3m)

We may compare the basic harmonic triplets with the basic chromatic triplets (see also post 322)

1. IONIAN  :  CHROMATIC TRIPLET (7d, 1M, 2m)
2. DORIAN  :  CHROMATIC TRIPLET (1M, 2m, 3m )
3. PHRYGIAN : CHROMATIC TRIPLET(2m,3m, 4M)
4. LYDIAN:  CHROMATIC TRIPLET(3m, 4M, 5M)
5. MIXOLYDIAN: CHROMATIC TRIPLET (4M, 5M, 6m)
6. AEOLIAN:  CHROMATIC TRIPLET (5M, 6m, 7d)
7. LOCRIAN:  CHROMATIC TRIPLET (6m, 7d, 1M)



If we want an ordering of how much "minor" and "sad" or "anxious" is a mode and how much "major" and "happy" or "serene" is a mode based on the ordering d<m<M then we get the next partial ordering (non-linear ordering)

PHRYGIAN<AEOLIAN<DORIAN<MIXOLYDIAN<IONIAN

and

LOCRIAN<DORIAN

LYDIAN<MIXOLYDIAN

But the Locrian and Lydian are not comparable with all the others so as to put all of them in a linear order.

Here is a musicological old qualitative description of it, but the names are not the standard but an older version of them (e. mode V is referred in this table as Lydian , while nowadays is named Mixolydian). To realize about which mode is referenced we must look at the 2nd column with the latin numerals.

Based on this concept it is possible to define tunings  of an harmonica (see post 274 ) with these types of triads of chords so as to improvise with the corresponding harmony on each of the 7 modes!

221. UNLOCKING THE MELODIC AND HARMONIC IMPROVISATION BY UTILIZING TRIADS OF NOTES , INTERVALS , CHORDS AND 3-LEVELS STRUCTURE OF A MELODY

See also posts 217, 216, 214, 208 etc

We enlarge more on it below. It is a generalization of the concept of scale, as a diatonic scale can also be defined by a triad of major (or minor) chords consecutive on the wheel by 4ths.

The triads is the golden middle ground between the extreme simplicity of the one  note ,interval or chord and the more complex combinations of 5 or more notes , intervals or chords.

The simplicity of triad of factors for the melodic action is  mainly the next three factors

1) The triad of polarity (qualitative) of a melodic move (ascending, descending, rotating stationarily or isocratic)

2) The  quantitative  triad of the character of the move :Chromatic, Melodic ,Harmonic

3) The rhythmic pattern of repetitions (e.g. A1 A2 A3 B1)

We remind the reader that in posts 203 and 214 we described a 3-levels structure of melodies which we refresh here:

THE MELODIC IMPROVISATION IS A SEQUENCE OF "ROTATIONS". EACH "ROTATION" HAS THE NEXT 3-LEVEL  RHYTHMIC OR MORPHOLOGICAL STRUCTURE

THE 4 PARTS A1 A2 A3 BI STRUCTURE 

This structure is rhythmic morphological and comes as a pattern from the 15-syllable poetry.

2 lines in a 15-syllables poetry are divided to 4 parts A1 A2 A3 B1, each of the Ai having 4 syllables and the B1 3 syllables. Similarly here in the melody the A1 A2 A3 , B1 are of 2 measures.  Each of the A1 A2 A3 B1 contain at least one melodic theme. If we put the restriction that both the melodic theme and its variations are consisting from intervals that are 2nds in less that 1/3 (=33%) of the cases, then we get an "Irish" but also "south american", melodic/harmonic sound in the melody. The melodic theme of A1 is translated or inverted to that of the A2 and that of the A2 to that of A2. Finally the melodic theme of A3 is mutated to that of B1. The simplicial sub-melody has one note in each measure therefore two notes in each of the A1 A2 A3 B. These two notes signify a MELODIC MOVEMENT (VECTOR) which is the basis of an emotion.  If we superimpose the melody at the first two levels, the detailed improvisational notes of the melodic theme (1st level) and the melodic move or vector of the simplicial submelody (2nd level) we get a new musical entity and concept that we may call WAVING MELODIC VECTOR. The wavings are as in the 7 melodic themes patterns asin post 231. Another classification of them is that they are of 3 classes a) Chromatic (of length an interval of 2nd) b) Melodic (of length an interval of 3rd) c) Harmonic (of length an interval of 5th). The choice in the improvisation of one of these 3 classes is similar to the choice in a chord progression if the chord-transition will me chromatic , melodic or harmonic. Very often the wavings are of diminishing amplitude as we reach the melodic center-end note of it. This is very convenient to have in mind when we improvise because we think that we will move from a persisting note a (melodic center) to a persisting note b (melodic center ) in a scale but with improvisational waving way that we may change each time at willAll the melodic movements of the A1 A2 A3 B make a DOLPHIN WORD (as defined also in post 101, 114 ). Each part A (thus melodic move too) has an underlying chord. The chords of two parts like A1 A2 may be the same or different chords. The melodic move can be also a melodic triad (alternating major minor vector interval of 3rd as in post 208) . The Dolphin word of the "rotation" may be a closed polygon (e.g, triangle) of waving melodic vectors , so that the length of the melodic vector indicates also the length-interval of the waving melodic vector, which may also be the distance as musical interval of the middle notes of the underlying chords, of the melodic move.
This melodic polygone allows for simple and very concise, simple and visually beautiful  methods of writing the improvisation with not less information than the necessary but no more than a minimum too, so that each time we see the written paper and play the melody a different melody will emerge (mainly at the first layer/level of it) while still it will be "the same" song. 

When playing again such an improvisation the simplicial sub melody is repeated (thus the "rotation" A1 A2 A3 B or Dolphin word of it), but the other notes inside the measures that are embellishments may change. All together the "rotations" or Dolphin words make the total melodic phrases of the song. Therefore we have here at least a 3-level structures of the melody a) notes b) melodic moves c) "rotations" or Dolphin words,and in time it is 1, 2^3=8 (2 measures)  and 2^5=32 (A1A2A3B parts).