83. A SCALE OF CHORDS ORIENTED IMPROVISATION METHOD VERSUS SCALE OF NOTES OR CHORDS OF A SCALE.

(see also post 9)

Usually, the improvisation in Jazz, is melody-oriented in the sense, that it usually has a predetermined chord progression and based on that many melodies are improvised. We described in detail the composition and improvisation method that starts with the chord progression at first and then composes the melody in post 9. 

We will develop here a second level of the improvisation where we improvise at first on the chord progression and parallel to it, based on the melodic centers we improvise on melodies. The melodic centers (simplicial sub-melody)  are not too much more complicated than the chord progression.

For this end, we describe again the 3 wheels of chords, the chromatic, the one by intervals of 3rds and the one by intervals of 4ths, and 3 types of chord cycles from them, the short (3-chords), the middle (6-8 chords) and the full cycle (12-15 chords). In the wheel by 3rds there are 2 medium size sub-cycles (See below)    These subcycles of chords, for the chord progressions improvisation, play the roles that scales play for the solo improvisations!
(see also post 17 and post 32)

The most common scale of chords is a connected arc of chords in the wheel by 4ths. That is 3 or 4 or 5 or 6 chords , that each is a pure 4th (or pure 5th) away from the previous and it  is either major or minor chord. Two such successive chords have one end-note in common. This is the equivalent of  the traditional scale, and because the chords may turn from minor to major and vice versa, it is in fact a bundle of different scales with modulations between them. A simple way to improvise over such a "scale" of chords is to play the melody, go up and down according to the demands of the emotions, and then use the chords as areas around the main and more permanent notes of the melody (simplicial sub-melody or centers of the melody) , but putting these centers of the melody as middle notes usually (but not only) of the chords of the chord-scale. To be more certain for the good result we may alternate the melody with the chords, but use only chords of the predetermined scale of chords. (See also post 147 for combining melodic themes with chords)


We enlarge more on it.

A GOOD IMPROVISED OR COMPOSED CHORD PROGRESSION X(1) X(2) X(3),...X(N) IS ONE THAT HAS ONLY THE NEXT 3 CHORD-TRANSITIONS

1) X(I)->X(I+1) IS RESOLVING , THAT SUCCESSIVE IN THE WHEEL OF 4THS
OR  ANY OF THE TWO HAS BEEN SUBSTITUTED WITH RELATIVE CHORD OF IT WITH 2 COMMON NOTES (ALTERNATING MINOR TO MAJOR OR VICE VERSA)
OR
2) X(I)->X(I+1) ARE RELATIVE CHORDS WITH 2 COMMON NOTES, ALTERNATING MINOR TO MAJOR OR VICE VERSA (SUCCESSIVE IN THE WHEEL OF 3RDS)
OR
3) X(I)->X(I+1) HAVE ROOTS IN DISTANCE OF ONE SEMITONE OR ONE TONE (SUCCESSIVE IN THE WHEEL OF 2NDS)

An example of such a chord-progression improvisation is the next progression, that can be played from the 4th neighborhood of the guitar to the first open chords neighborhood

Em->G->Bm->Bb->F->E7->Am->D7->Em->G->D->F#m->F->C->Em->D#->Bb->A7->D

Now we can extend these rules and at the same time simplified them for guitar players. We always assume playing only on the higher 4 strings of the guitar, and the chords are essentially the triads played  only on the 3 higher strings.

A GOOD IMPROVISED OR COMPOSED CHORD PROGRESSION X(1) X(2) X(3),...X(N) IS ONE THAT HAS ONLY THE NEXT 2  CHORD-TRANSITIONS:

1) WHEN THE CHORDS ARE DISTANT IN DIFFERENT OCTAVES, THEN THE CHORD TRANSITION    X(I)->X(I+1)  MUST BE EITHER A) RESOLUTION AS SUCCESSIVE CHORDS IN THE WHEEL OF 4THS B) RELATIVES (ALTERNATING MINOR TO MAJOR AND VICE VERSA) AS SUCCESSIVE CHORDS IN THE WHEEL OF 3RDS. IT IS SUGGESTED THAT THE TRANSITIONS AS SUCCESSIVE CHORDS IN THE WHEEL OF 4THS ARE MUCH MORE THAN THE TRANSITIONS AS RELATIVE CHORDS

2) WHEN THE CHORDS, AND THEIR EXACT VOICING ON THE FRETBOARD, ARE IN THE SAME OCTAVE (AND WE DO NOT REDUCE THEM TO EQUIVALENT IN THE SAME OCTAVE) THEN THE CHORD TRANSITION X(I)->X(I+1) MUST BE SUCH THAT THEIR SHAPES AS PLAYED HAVE AT LEAST ONE FRET IN COMMON (MAYBE 2 OR 3 COMMON FRETS TOO AND MAYBE ONE, OR TWO NOTES IN COMMON). IT IS SUGGESTED THAT THE TRANSITIONS AS SUCCESSIVE CHORDS IN THE WHEEL OF 4THS ARE MUCH MORE THAN THE TRANSITIONS AS RELATIVE CHORDS AND THE REST OF THE TRANSITIONS AS CHROMATIC TRANSITIONS WITH ROOTS ONE OR TWO SEMITONES APART, OR ONE ONLY COMMON FRET ARE MUCH LESS IN NUMBER. ALSO IF JOYFUL SONGS ARE INTENDED THEN AT LEAST 2/3 OF THE CHORDS ARE TO BE MAJOR AND  LESS THAN 1/3 MINOR CHORDS. IN ADDITION THE PATTERN OF CHORD TRANSITIONS AS OF THE 3 BASIC TYPES MUST SOMEHOW REPEAT IN THE CHORD PROGRESSION EVEN WITH DIFFERENT CHORDS.

3) BASED ON THE IMPROVISED CHORD PROGRESSION, THEIR ARPEGGIOS DEFINE AT A SECONDARY ORGANIZATION LEVEL SECONDARY MELODIC AND SOLOING IMPROVISATION AT THE LEVEL OF NOTES NOW AND NOT CHORDS!

NOW WE DO NOT NEED TO PLAY ALL THE CHORDS OF AN IMPROVISED CHORD-PROGRESSION WITH EQUAL SIGNIFICANCE OR TIME DURATION. SOME CHORDS MAY BE GHOST-CHORDS




THE GENERAL PATTERN OF PROGRESSIONS WITH ALTERNATING CHORD-RELATIONS OF 
CHROMATIC-MELODIC ,CHROMATIC-HARMONIC , HARMONIC-MELODIC , HARMONIC-HARMONIC, MELODIC-MELODIC, CHROMATIC-CHROMATIC CHORD-TRANSITIONS.

This is a progressions X1->X2->X3->...->Xn  where the Xi->Xi+1  and Xi+1->Xi+2 is an alternation of chord relation and  transitions of the chromatic-melodic   , chromatic-harmonic,  melodic-harmonic,  chromatic-chromatic, melodic-melodic or harmonic-harmonic relations. 
Such constant alternating patterns of chord relations somehow determine also that the melodic themes (either within a single chord or within a chord transition), are structured and translated or inverted or expanded with similarly alternating intervals of 2nd, 3rd or 4th/5th. 

THE GENERAL PATTERN OF A CHROMATIC DOUBLE SCALE OF CHORDS 

Here is an alternative way to produce not harmonic scales of chords (based on the harmonic relation of chords) but chromatic scales of chords based on the harmonic relation of chords but which still involve the other two chord relations the melodic and the harmonic 

WE START WITH A CHROMATIC CADENZA OR ASCENZA  in semitones 2->2->1  or 1-3-1 or 1-3-1-1-3-1 in harmonic and double harmonic minor scales,   and we paralel chords rooted on such notes X1->X2->X3->X4 with chords 

Y1->Y2->Y3->Y4, such that the relation of Xi with Yi is either in a relation of being  relative chords (melodic relation of chords) or a 4th apart (harmonic relation of chords) 

Of course the less total number of different chords that we may use is better and it sounds more familiar if such chords belong to an harmonic personality (diatonic or harmonic minor or double harmonic minor etc).We may use either minor or major chords. 

TRIPLE ALTERNATION OF CHORD-TRANSITIONS

More generally   and we paralel chords  X1->X2->X3->...->Xn  that are in  one of the relations chromatic, melodic harmonic , with chords X1->X2->X3->...->Xn so that the relation of Xi with Yi is always constantly in one of the 3 basic relations  relative chords (melodic relation of chords) or a 4th apart (harmonic relation of chords) .

When playing the scale as progression X1->Y1->X2->Y2->... it is equivalent with having a triple alternation of chord relation and  transitions of the chromatic-melodic   , chromatic-harmonic,  melodic-harmonic,  chromatic-chromatic, melodic-melodic or harmonic-harmonic relations and a third which is variable. 

Such constant alternating patterns of chord relations somehow determine also that the melodic themes (either within a single chord or within a chord transition), are structured and translated or inverted or expanded with similarly alternating intervals of 2nd, 3rd or 4th/5th. 

GHOST-CHORDS PROGRESSION METHOD OF IMPROVISATION OVER A SINGLE CHORD:

MOST OF THE TEACHERS OF IMPROVISATION SUGGEST  USING THE ARPEGGIO OF THE UNDERLYING CHORD, EITHER AS PURE TRIAD OF NOTES OR AS EXTENSION TO 4 OR 5 NOTES SUCH CHORD. BUT THERE IS ANOTHER INTERESTING TECHNIQUE THAT INVOLVES GHOST-CHORDS (NAMELY THAT ARE NOT REALLY HEARD). E.G. OF WE ARE TO IMPROVISE SAY ON C MAJOR CHORD, THEN IT IS NOT ENOUGH TO USE ITS ARPEGGIO, BUT DO THE NEXT: CONSIDER C IN THE CHORD PROGRESSION OF THE SONG, AND TAKE TWO OTHER CHORDS OF THE SONG PREFERABLY IN THE WHEEL OF 4THS,  THE 2 NEIGHBORHOOD CHORDS (EITHER AS MAJORS OR MINORS) HERE E.G. LET US TAKE THE MAJORS G->C->F ASSUMING THEY WHERE IN THE SONG. IF THERE IS NOT SONG YET WE JUST TAKE . IN THE WHEEL OF 4THS,  THE 2 NEIGHBORHOOD CHORDST (that define here the C major-mode diatonic  scale) . THEN  TAKE THE ARPEGGIOS OF THESE THREE CHORDS AND PLAY THEM IN RHYTHMIC , FAST AND RATHER RANDOM PERMUTATION  WAY, AS IF A VERY FAST CHANGE OF CHORDS IS MADE IN THE THREE  G->C->F, SO FAST THAT G, F CHORDS ARE RATHER TRANSIENT WHILE WE REMAIN MOST OF THE TIME ON C. THE SEQUENCE OF THE CHORDS THROUGH THEIR ARPEGGIOS DEFINE ALSO A SOLOING. THE RESULT WILL BE AN IMPROVISATION ON ALMOST A WHOLE 7-NOTES SCALE, WITH UNDERLYING SINGLE CHORD THE C.IN ADDITION THE SOLOING TAKES IN CONSIDERATION   AT LEAST TWO  OTHER CHORDS OF THE SONG. IF THERE IS MELODY IN THE SONG WE MAY CONSIDER MIMICKING THE MELODY WITH WAVINGS AND "DANCING AROUND THE NOTES OF IT, IN NOTES THAT EXIST IN THE CHORDS OF THE MELODY. OR WE MAY APPLY DIFFERENT TRANSFORMATIONS IN THE MELODIC THEMES THAN THE TRANSFORMATIONS THAT EXIST IN THE MELODY. THE RESULT WILL BE A DIALOGUE BETWEEN THE MELODY AND THE SOLOING


The diatonic progressions is the sequence (iii->vi->ii->V->I->IV->VII->iii), This progression leads from sadness to joy, from the triad of minor chords to the triad of major chords. In the symbolism of the 24-cycle of  chords the diatonic scale is the arc of the next chords   (x-2, X-1,x-1,X,x,X+1,x+1)=(vii,V,iii,I,vi,IV,ii). (see post 34). 

There is also the inverse or descending  diatonic progression which is the (I->V->ii->vi->iii->VII->IV->I)

In a diatonic scale, the triad of minor chords (sad triad) is the (iii->vi->ii) where the (iii, vi) and (vi,ii) are consecutive in the cycle of pure 4ths, with standard resolutions (iii7-> vi) , (vi7->ii) and the 
(ii, iii) are complementary chords, in other words all of their notes give all the notes of the scale except one. 

The triad of joy or triad of major chords  is the (V, I, IV) , where the (V, I) and (I,IV) are consecutive in the cycle of pure 4ths, with standard resolutions (V7-> I) , (I7->IV) and the  (IV, V) are complementary chords, in other words all of their notes give all the notes of the scale except one. 

The bridge between these two triads is the well known jazz progression (ii7, V7, I) , where again  the (ii, V) and (V,I) are consecutive in the cycle of pure 4ths, with standard resolutions (ii7-> V) , (V7->I), and  the  (ii, I) are complementary chords, in other words all of their notes give all the notes of the scale except one. 



Alternative closures 
The diatonic progression closes also to a cycle by utilizing the triad progressions 
(IV->IV#7->VII7->iii) or (IV7->VIIb->vi). 
Or IV->V7->I or IV->ii7->V7->I
Or IV->IVdim7->G7->I   (see e.g, Bach prelude and Fugue C major, BWV 846)
Or IV->IV#dim7->VII7->iii


HERE IS THE 24-CHORDS CYCLE IN THE REVERSE ORDER BY 5TH RATHER THAN BY 4TH WHICH IS THE ACTUAL 

WHEEL OF 4THS
1) The short 3-chords sub-cycle of the wheel of 4ths is a set of chords X1, X2, X3 where the previous chords are successive in the wheel of 4ths but they may be either minor or major and alternating also. Thus there are 2^3=8 types of such small sub-cycles. Essentially they define a diatonic scale or a mode of it

2) The medium size sub-cycle is a sequence of 6-chords X1, X2, X3, X4 X5, X6,  which they are again successive chords in the wheel of 4ths and again they may be either major or minor or alternating in any combination (e.g. two successive minor then one major etc).For this to be a sub-cycle, the X1, and X6 must differ in their root notes only a semitone.  For the choices of major or minor there are 2^6 such types of sub-cycles. We may also add the possibility that they are dominant 7tnth or major 7nth chords, or chords with 6th etc.

3) Similarly the 4 or 5 successive chords in the wheel of 4ths  X1, X2, X3, X4 X5 maybe considered closing if X5 and X1 ar relative chords e.g.

Bm->Em->Am->Dm->G, as G and Bm are relative chords

But also the 4-chords sequence is also

Em->Am->Dm->G, as G and Em are relative chords.

1) The minimal 3-chords cycle.
This is 3 successive chords in the wheel of 4ths

e.g. G->C->F->G
or Em->Am->Dm->Em.

Slight enlargement to this is the  Small 4 chords cycles of relatives mutation (by // we denote the relatives mutation)

Examples:  A->D->G//Em    or D->G->C//Am    or   G->C->F//Dm

2) Medium  6 chords cycles of chromatic mutation (by // we denote the chromatic mutation)

Examples:  A->D->G//F#->Bm->E    or A->D->G//F#->B7->Em

or  A->D->G->C->F->Bb//A

D->G->C//B7-> Em->A7  or D->G->C//B7-> E->Am

G->C->F//E7->Am->D  or G->C->F//E7->A->Dm

o

(Notice here that if we would restrict to a diatonic scale the cycle G->C->F//E7->Am->D  or G->C->F//E7->A->Dm   would be

G->C->F//Em7->Am->Dm  .

This cycle can be extended to an 8-cycle in the following way:

The 6-cycle  A->D->G->C->F->Bb//A  can be extended to the 8-cycle

A->D->G->C->F->Bb->D#->Edim7->A

or the A->D->G//F#->Bm->E  will become A->D->G->C->C#dim7->F#->Bm->E

With double chromatic mutation we have the progression 7-chords cycles

A->D->G//C#->F#->Bm->E  or D7->G7->C7->F7->Bb//E7->A7, which also can be alternating in minor major: Dm7->G7->Cm->F7-<Bbm//E7->Am7  or D7->Gm7->C7->Fm7-<Bb//E7->Am7 etc.

See also the double Andaluzian cadenza above and the standard Jazz 7-chords progression

So the  suggested cycles already contain a modulation that combines e.g. two diatonic scales o a diatonic and a harmonic minor etc, and is necessary so as to have 2/3 or more major chords in the chord progression. So the rule is: The 2/3 rule of major chords in the chord progression of 6 chords necessarily   involves modulations, and cannot be conducted within a single diatonic scale!!!

C->F->Bb//A7->Dm->G or C->F->Bb//A7->D->Gm

etc.

Notice that an alternating even only or odd only sequence of chords in such 7 chords cycles with double chromatic mutation gives the Andaluzian cadenza and Jazz 7-chords progression.

Example of  beautiful chord progressions that one can obtain with the above 6 or 7 chords cycles are the next with the next rules

1) All chords are from the above 7-cycle cycle with chromatic mutation and are with 7th chords

2) Any two successive chords are either successive in the above cycle ( that is successive in the wheel of 4th too or are in chromatic 1 semitone relation) or are relative chords, or inverses in order in the above relations

3) All successive chords alternating are minor then major or major then minor.

4) There is a starting and ending pair of chords which is successive in the wheel of 4ths and are both major chords.

An example of a chord progression with the above rules is the next

C7 F7 Bbm A7 Dm7 D7 Gm7 C7 Am7 D7 Gm7 C7 Am7 Bbmaj7 A7 Dm7 D7 Gm7 C7 Am7 F7 Dm7 A7 Am7 D7 Gm7 C7 F7.

When playing such chord progressions we may move slowly all the way  up and then all the way down in th fretboard.

3) Full 12 chords cycle: A->D->G->C->F->Bb->D#(=Eb)->G#(=Ab)->C#(=Db)->F#->B->E

where at most 1/3 that is at most 4 chords can be minor chords.

WHEEL OF 3RDS

SOME EXAMPLES OF THE SMALL AND MEDIUM SUB-CYCLES

The next sequence of intervals  (3333444)=(C,D#,F# ,A,C,E,G#,C) defines a   closed cycle of 4 relative chords:   Cdim7, Am ,Eaug, G#7=Ab7 They sound  resolving in the next sequence Cdim7 Eaug  Ab7  Am



The next sequence of intervals  (3334434)=(C,D#,F#,A,C#,F,G#,C) defines a   closed cycle of 7 relative chords: Cdim7, D#dim, F#m, Aaug, C#, Fm, G#




The 7 (8)-chords diatonic closed cycle of relative chords is the  (4343343)=(CEGBDFAC) (with a slight repetition of G and G7):

C, Em,G, Bdim ,Dm, F,Am or

C, Em,G, Bm, G7 ,Dm, F,Am or

C, Em,G, Bm, D ,Dm, F,Am

and there are more such 8-chords cycles if we play with changing minor to major

e.g. C, Em,G, Bm, D, F#m , A, Am etc
or 

C, Em,G, Gm , D, Dm F , Am


The 24-chords chromatic closed cycle of relative chords (434343434343434343434343):

G        C         F         Bb          Eb        Ab         Db          Gb          B           E           A         D             
     Em   Am     Dm        Gm       Cm       Fm        Bbm       Ebm     Abm     Dbm     Gbm   Bm  


We may add two more series based on that mnot-major chords with the same root are relative chords too, so as to havea 2-dimensional grid based of the relation of relatives.


Gm    Cm       Fm        Bbm       Ebm     Abm     Dbm     Gbm   Bm     Em       Am     Dm  
G        C         F         Bb          Eb        Ab         Db          Gb          B           E           A         D             
     Em   Am     Dm        Gm       Cm       Fm        Bbm       Ebm     Abm     Dbm     Gbm   Bm  
     E        A         D           G        C              F         Bb            Eb        Ab         Db         Gb      B  

This grid which is also a table as below shows clearly the chords that are  harmonically in series(interval of 4th or 5th) and chords that are harmonically in parallel (intervals of 3rds or 6ths). But it does not show of course the relations of chords that are melodically in series

Gm

Cm

Fm

Bbm

Ebm

Abm

Dbm

G

C

F

Bb

Eb

Ab

Db

Em

Am

Dm

Gm

Cm

Fm

E

A

D

G

C

F

	Gbm		Bm		Em		Am		Dm
	Gb		B		E		A		D
Bbm		Ebm		Abm		Dbm		Gbm		Bm
Bb		Eb		Ab		Db		Gb		B

It is often very instructive to chart the chords of  a  song over this 24-cycle of relatives (or double cycle of 4ths) or the above 2-dimensional grid.

Most often a song is a sequence of connected intervals or arcs of chords in the cycle of 12ths shifting by relatives to a corresponding similar connected arc in the parallel cycle of 4th in the overall cycle of fifths. We call this concept a harmonic multi-tonality. Simple  tonality is simply 3 -successive major chords in the 12-cycle of 4ths. 

Conversely any connected sequence of arcs of chords of this 24-cycle of chords (defining harmonic multi-tonality), is the chord progression  of a nice song with nice sounding modulations. Normally in harmonic multi-tonality  we are keeping the qualities major-minor as in the 24-cycle but a more free approach allows altering them , from minor to major and vice versa or to more complicated qualities like 7th, 6ths etc.  The same for chord progression for improvisation (see post 11) . To the rule of harmonic multi-tonality in the 24-cycle, we may allow as transition to a next chord, a shift by one semitone or tone of the root of the current chord. (see also post 30)

See also https://www.youtube.com/watch?v=TRz73-lSKZA


TO ACCOMMODATE A SCALE OF CHORDS ORIENTED IMPROVISATION METHOD, A SPECIAL TUNING FOR THE 6-STRING GUITAR CAN BE GIVEN THAT REFLECTS THE WHEEL OF 3RDS.

OPTIMAL GUITAR TUNING FOR CHORDS PLAYING MAINLY

1.) An more optimal but unknown tuning for the 6-string guitar when chord-playing is the main target and not so much solo playing is and even better by alternating minor and major 3rds. In semitones for the 6 strings   4-3-4-3-4 or 3-4-3-4-3
E.g. Bb2- D3-F3-A3-C4-E4 or F2-A2-C3-E3-G3-B3 or A2-C3-E4-G4-B4-D4
THIS MAY BE CALLED THE HARMONIC TUNING OF THE GUITAR AS IT IS BASED ON THE HARMONIC 2-OCTAVES 7-NOTES SCALE (see also post 79)
The latter is the most natural open tuning. There the same shape for major and minor chords and only 3 of them and in only one or frets compared to the 6 in the standard tuning guitar. If we want also dominant and major 7nth chords we use again only 2 frets. The same with the aug chords Only the dim7 chords require 3 frets. Because  of the symmetry of the tuning among the strings, the relations of relative chords and also chords in the wheel of 4ths is immediate to grasp also geometrically. Of course when we say shape of chords as it is standard in jazz, we do not play all 6-strings but only 3 or 4 strings.
Within 3 frets exist all chords of the  8-notes scale with interval structure 2-2-1-2-2-1-1-1 which is an extension and variation of the melodic double minor 2-2-1-1-1-1-2-2 or (1-1-1)-(2-2)-(2-2-1)

But also all chords of diatonic scale!

The easiness with which one can improvise melodies within a diatonic scale (all notes within 3  frets and in a very symmetric zig-zag pattern) together with 3-notes chords of the scale (gain all chord patterns within 3-frets) is unsuprassed.
At the same time , the easiness with which one can me diatonic scale modulations, chromatic (1 semitone apart) or by changing a minor to a major chord and vice versa and continuing in a relevant diatonic scale is unsurpassed again! 

This harmonic tuning by alternating minor-major 3rds, allows, for all  4-notes chords of e.g. the D major scale in   the 3rd octave (c3,d3,e3,f3,g3,a3,b3), Cmaj7->Em7->G7->Bm6->Dm7->Fmaj7->Am7 in 1st normal position across the fretboard, something not possible with the standard tuning of the guitar. In the standard guitar it is possible only by 2nd or 3rd inversion, or by shifting to the 4th octave or 2nd octave. Therefore there are important very natural voicing of the 4-notes chords of the 3rd  octave that we miss with the standard tuning and it is in a single octave!



Other types of chromatic wheel of chords are the personalities defined by various musicologists.

(See post 36 )

 According to musicologists of Jazz and Ethnic music (as appearing e.g. in music arrangement software like the music machines of Microsoft), the next progressions of chords (that cover the steps of the chromatic 12-tone scale) correspond to the next harmonic personalities.

This concept also can be considered as chromatic wheels of chords or chord-scales as described in post 83.

These tables give also a system to create chord progressions that cover (but not necessarily belong) any scale

But it is also a system to adopt a song from its original chord progression to another (as we may change a song from a minor mode to a major mode and vice-versa) 

HOW IT APPLIES TO SONGS

The way that the table below can be used on an existing song with melody written say on a known particular diatonic scale and mode(at least in its musical score)   is the next.

Take the known chord progression of the song, and substitute, each chord of it with root R with the corresponding chord of root R  in the table below. This applies to all possible Roots R in the 12-notes chromatic scale, and in the 7 tone scale of the song too. The result is that there is a change in the "personality" of the sound of harmony of the song. By changing the chord-type we may need to change some of the notes of the melody to fit in the new type chord as it was flitting to the old type of chord. This is in accordance to our philosophy which in general defines the harmony of a song not through a single scale but throughout chord progression (see post 49) 

For the definition of the chords see e.g. www.all-guitar-chords.com, or the internet.

M symbolizes  major, m symbolizes  minor, dim=diminshed, aug=augmented.

For example C63=CEA, C65=CGA, C42=CDFG, C43=CEFG, C7#9=CEGBbD#, Cm7b5=CEbGbBb, Cm11=CEbGBbDF, Cdim9=CD#F#A#D, C9#11=DF#CEBb , Cm67=CEbGAB

C2=CDEG,

One way to utilize this table for a song already written in one of the scales major, minor, harmonic minor, or the modes Mixolydian, Phrygian, Dorian ,  so as to give it the coresponding jazz personality, is to alter a major or minor chord of the song and at any root, with the atributes below (dim, ,7, 63, 65, 9, 11, 7#9 etc) and making it thus sound with a different personality. This concept is based on tonality but it is actually beyond one only scale and tonality, it substitutes the concept of modulation, and is based rather on the concept of covering sets of chords of all posible steps of any melody. 

The set of chords of a personality, is indeed based on tonality, but only at the level of the chords. Not at the level ofthe melody. The melody need not be within a diatonic scale. The fact that personality chords are defined even for steps outside a diatonic scale makes it clear.

Another way to utilize it, is to compose songs regardless of scale, but utilizing the chords of a prticular personality , and as the chords of a personality have roots all the 12 notes, one has all possible roots for chords of the song  over a melody with all possible notes. This concept is based on tonality but it is actually beyond one only scale and tonality,  it substitutes the concept of modulation and is based rather on the concept of covering sets of chords of all posible steps of any melody, creating so composite chord-melodies (see also post 11).

HOW IT APPLIES TO SONGS

The way that the table below can be used on an existing song with melody written say on a known particular diatonic scale and mode(at least in its musical score)   is the next.

Take the known chord progression of the song, and substitute, each chord of it with root R with the corresponding chord of root R  in the table below. This applies to all possible Roots R in the 12-notes chromatic scale, and in the 7 tone scale of the song too (covering the case of chords outside the 7-notes scale in the case of multiple modulations to different scalles during the song). The result is that there is a change in the "personality" of the sound of harmony of the song. 
ALTERNATIVELY, the table below can be used as a guide of the kind of chord we would choose in the maximal correspondence of each note of a melody with a chord with the root on it before we simplify the chord progression to one with fewer chords.
Still, ALTERNATIVELY we may use the corresponding chord types as in the table when the underlying note of the simplicial sub-melody (see post 65 )is the one in the corresponding position in the full 12-notes scale.

Steps of the chromatic scale/Harmonic Personality

1

1#

2

2#

3

4

4#

5

5#

6

6#

7

Righteous(Major)

M(ajor)

Dim7

m(inor)

Dim7

m

M

Dim7

7

Dim7

m

4,2

Dim7

Honest (major)

M

Dim7

m

Dim7

m

M

Dim7

7

Dim7

m

M

dim

Hopeful (Mixolydian)

M

M

m

M

dim

M

m

m

M

m

M

dim

Serious (Phrygian)

m

M

6,3

M

6,3

m

6,3

dim

M

6,5

m

6,5

Upbeat(Mixolydian)

M

M

m

M

m

M

M

M

M

m

M

dim

Searching(Dorian)

m

M

m

M

dim

M

dim

m

M

dim

M

dim

Lonely (major)

7

Dim7

m7

Dim7

7

7

Dim7

7

7

7

7

7

Funky(Mixolydan)

7#9

7

m7

9

m7

9

7

7

7

m7

6,3

7

Sad(minor)

m7

M

6,3

M7

6,3

m7

6,3

m7

M7

6,5

M

6,5

Romantic(major)

M

6,3

m7

6,3

m7

M7

6,3

9s

6,3

m7

M

m7b5

Boogie(major)

7

Dim7

7

Dim7

7

7

Dim7

7

Dim7

7

7

7

Noble(major)

M

6,3

m

6,3

6,3

M

6,3

M

4,3

6,3

4,2

6,3

Bittersweet(Harmonic minor)

m

6,5

m7b5

M

Dim7

m

Dim7

7

M

m7b5

M

dim7

Adventurous(H-minor)

m

M

6,3

M

6,3

m

6,3

m

M

6,5

M

6,5

Striving(minor)

M7

M

6,3

M7

6,3

m7

6,3

m7

M7

6,5

M

6,5

Sophisticated(major)

6,9

Dim7

m11

Dim9

m7

M9#11

Dim7

13

Dim7

m11

9#11

Dim9

Miserable(mixolydian)

13

Dim7

m11

Dim9

m11

13

Dim7

13

Dim7

m11

9#11

Dim9

Complex(major)

2

6,3

m11

Dim7

m6,7

2M7

6,3

9s

2M7

6,3

2

6,3

An alternative way of course would be to correspond personalities only to the 7 modes, and then besides the 3-notes chords with roots one every note of the 7-tone mode, add all the other notes to make 12-notes of the chromatic scale, and extend, the triads to 4-notes chords making sure the added note (that would be of the melody and possibly outside the original chord) is always the highest. In this way 5 more composite chords (for chord-melodies) are in general added to the mode which is now the chromatic 12-tone scale. 

82. CHORD-INDEPENDENT AND UNACCOMPANIED MELODIES: THE TWO NOTES STATISTICAL HARMONY. The chromatic , melodic and harmonic melody notes transitions (speeds)

So far in post 9, we have developed the chord-progression first and afterwards  the melodic themes, system of musical composition, which is mainly the pattern that takes place in improvisation , especially in jazz. 

The main reason for this, is of course,  because chord progressions is a simpler pattern than the melody. But what if the melodic themes is a simpler pattern than the chords progression? 

We must remark here that there melodies that either

a) No single chord is best for accompany it all 
b) or any chord can accompany it.
c) Too many chords fast changing (ghost chords) are appropriate to accompany it.

d)  A small number of chords but only very fast changing (ghost chords) are appropriate to accompany it.

THE STATISTICAL TWO-NOTES  HARMONY

What if the melodic theme is as here the simplicial melodic themes  which is simple a melodic interval of two notes (actually melodic vectors as orientation does matter) which is simpler than the 3 notes of  chord, and we start composing the melody with such melodic intervals? From this point of view there are only 3 types of such simplicial melodic themes: 1) Of intervals of 2nd and inverses (Chromatic transitions)  2) of intervals of 3rd and inverses (melodic transitions)  3) of intervals of 5th or 8ths and inversions (Harmonic transitions) .

These three types of transitions of notes , namely chromatic, melodic and harmonic are like the three basic relations of chords transitions (see post 30). So improvisation of a melody which is sounding alone without chords underlying it, but probably only some rhythmic percussion can be conducted solely by choosing each time how much melody, harmony or chrome we want!  

See also post 68 about the 3 melodic densities or speeds (chromatic, melodic, harmonic)

In the context of 2-notes harmony or Interval harmony of melodies the closest counterpart of a chord is an interval of 5th or its  inverae an interval of 4th. Oscillations between tow notes at an interval of 5th or 4th is the counterpart of the guitar harping within a chord.



Of course we can afterwards add 3-note or 4-notes chords and study such an harmony of the melody. But the visible and undisputed harmony of the melody is the two notes harmony, or the statistics and sequence of intervals of the melody.
If such a two-notes harmony was tried to be covert it to a 3-notes chords harmony, not only the extension would not be unique, but also it may be that a close fitted such extension might give a so much fast changing chord progression, that would rather be ghost-chords progression (see post 87 ).

For example let us assume that we improvise a melody that we ascend with such steps that would suggest chords at the odd steps of the diatonic scale (I, II, V) and the descend with steps that suggest the even chords (II, IV, VI). The change from odd to even and vice versa is done with intervals of 2nds, 7nths and 4ths. 

Usually the melody would give so fast changing chords that would be classified as ghost chords (see post 87 ). 

It is important of course to have here the simplest possible relation of the simplicial melodic themes with the chords , which is one note per chord or the melodic theme is the transition bridge between successive chords. 

One of the best methods to choose relation of the notes of the simplicial submelody (one note per chord) and the chord progression, is so  that the intervals between the notes are minimized (1 semitone) or maximized (interval of 5th or octave and inverses). For successive chords in the wheel of 4th both extremes are feasible. For chords that are  relatives, intervals of 4ths are possible, while for chords with roots one tone apart, intervals of 5th are also possible. For chords with roots one semitone apart, obviously the minimum 1 semitone is feasible.  Of course a meaningful (in respect to joy and sadness , anxiety and serenity) repetitive pattern must occur both in the simplicial melodic themes and in the chord progression.  

The notes of the simplicial melodic themes (one for each chord at its simplest version) need not justify the chord progression. The justification will be with other embellishment notes that make the full melody as contrasted to the simplicial sub-melody. The initial justification of the chord progression is the feelings that it creates parallel to the simplicial sub-melody. The chord progression magnifies the attention and feelings of parts of the simplicial sub-melody and shrinks the feelings and roles of other parts of it. Of course both flows that of the simplicial melodic themes and of the chords progression must be harmonically compatible. In other words each note of he simplicial sub-melody must be also a note of the underlying chord. 

As the wheel of 4ths is the basic tool to design chord progressions, so scales is the basic rule to design melodic vectors (simplicial melodic themes).

As we wrote in the post 40, the intervals of  5th/4ths have higher harmonic score than the intervals of 3rd which in their turn have higher harmonic score than the intervals of 2nd.

So many beautiful melodies have this distribution of  the percentage   of  intervals in them. In other words % of 5ths/4ths> % of 3rds>% % 2nds.

Some of the melodies of the music od Incas, Andes etc, but also of all over the world composers have this property.

We should notice also that although the diatonic 7-notes scale is closed to intervals of 2nd, 3rds and 5ths or 4ths (but not both) the standard pentatonic scale is  closed  to intervals by 5th and by 4ths .

We say that a scale is closed to  intervals by nth, if and only if starting from any note of it if we shift higher or lower by an interval by nth, we are again in a note of the scale.


Nevertheless , other proportions of  percentages of 5ths/4ths/8ths, of 3rds and of 2nd are known to give characteristic types of melodies among the different cultures.

Other observed profiles of percentages are


%2nds> %3rds+%4ths/5ths/8ths 
(e.g. the 2nds double more than the rest of the intervals, ratio 3:1 ) :
Oriental and Arabic Music,  GypsyJazz, and Jazz Stephan Grappelli soloing

%3rds+%4ths/5ths/8ths>% 2nds : or %  2nds<=50%.:
(e.g. the 2nds less than half compared to the rest of the intervals,ratio 3:1 )
 Music of Incas, and countries of the Andes. Celtic music Ancient Egyptian  music

For independent from chord progression free improvised melodies a simple way to create melodies with number of 2nds intervals <=50% is to alternate intervals of 2nds with intervals of 3rds, 4ths, 5th,s and 6ths. 

We must remark here that there are melodies that either

a) No single chord is best for accompany it all 
b) or any chord can accompany it.
c) Too many chords fast changing (ghost chords) are appropriate to accompany it.
d)  A small number of chords but only very fast changing (ghost chords) are appropriate to accompany it.

1) Such melodies are e.g. fast sequential successive sounding by 2nds for 2 or 3 octaves
2) But very close to is also melodies with alternating 3rds or 6ths and 2nds in a diatonic scale

MELODIES WITH 100%  INTERVALS OF 2NDS
Such melodies give the feeling of coherence in the melodic moves.
We must notice that even if we use, say 100%, intervals of 2nds in a scale, we may still create harmony. E.g. in such an extreme case of melody, the harmony is created by persisting and passing  again and again from notes of an interval of the scale. E.g. if such an interval of successive notes of the scale is x(i), x(i+1), x(i+2) , or  x(i), x(i+1), x(i+2), x(i+3), x(i+4) , and the time spend on each of these notes is in the average equal, then the harmony and underlying chord that is created is the  x(i),  x(i+2),  x(i+4) because these are more than the   x(i+1),  x(i+3), and thus the melody spends more time in the chord x(i),  x(i+2),  x(i+4) among other chords for the time that is in the interval  x(i), x(i+1), x(i+2), x(i+3), x(i+4).  
SPENDING MORE TIME ON SPECIFIC NOTES:
Similarly if we take again the extreme and simplistic example of melodies with 100% intervals of 2nds, and we want for a time interval to have as harmony and underlying accompanying a chord  X=(y1,y2,y3), then its about sufficient to spent about equal time to all other notes except in one or more of these  notes y1,y2,y3, that we spent much more time (making theme temporary centers of the melody)  creating thus the effect of this underlying chord harmony. Of course if we did not have the restriction of 100% intervals by 2nds, another method would be instead of spending more time to the notes of the chord , spending almost no time at all to notes not to the chord.


On the other hand if %3rds+%4ths/5ths/8ths>=2*(% 2nds)  or  number of 2nds intervals <=33%  then essentially the melody is sequence of chords linked by intervals of 2nds.


The way to create melodies with at least 2/3 of the intervals to by the larger intervals of 3rds , 5ths/4ths or 8ths, is to apply harping in a chord say with 6 or 8 steps on notes, where it is added only one intermediate note in the chord (e.g. 7nh, 6th, 4th or 2nd) and so that the created intervals of 2nd are only 2 in the 6 or 8 intervals. Then we shift to a relative chord an interval of  3rd away or to a resolution transition which is a chord in an interval  5th or 4th away , or we even shift to a chord a 2nd away in which case we do not use any additional note, and we continue so.  So finally %3rds+%4ths/5ths/8ths>=2*(% 2nds) .

Or a way to create melodies with number of 2nds intervals <=33% is
alternate for every one interval of 2nds two intervals of 3rds, 4ths, 5th,s and 6ths. Which of course usually mean of course in a diatonic scale that we arpeggio on chords successively one after the other in any order. But as we have said for god chord progressions, it is better to arpeggio i minor or major successive chords on the wheel by 4ths and 3rds


A way to take short notes of such beautiful melodies is to write the chord progression, and then one note with small letters above or below the chord denoting which neighboring note (by interval of 2nd usually)  is the extension of the chord in the melody.

%4ths/5ths/8ths/6th>%3rds>% 2nds :
(e.g. the 2nds +3rds less than half compared to the rest of the intervals,ratio 3:1,  )
The way to create such melodies with at least 2/3 of the intervals to by the larger intervals of 5ths/4ths or 8ths, compared to 3rds ,  and 2nds is to apply the same technique as before, but when harping inside the chord we use the intervals of 4th and 5th and 8th of the  normal position and   2  inversions, instead of the 3rds in  the normal position! In this way in the fast soloing or harping on the notes of the the chord has more intervals of 4th, 5th and 8th than of 3rds!

Nevertheless the chord progression over which this technique produces fast melodies may contain very fast chord changes, and may not be identical with the actual chord progression that the instruments play as background to the melody. This is the concept of "ghost chords" in the melody as described in the post 87. E.g. The full ghost chord progression may be D G D G D A D. While the chords really played is only D. 

String instruments, that the melody is played mainly along a string (e.g. Greek Bouzouki, sazi etc) the scales that are favored for melodies improvisation are ones with many 2nds and in particular scales with many occurrences of semitone intervals of 2nds (oriental type)

Diatonic instruments like Irish whistles, and flutes, diatonic panflutes etc are best for melodic improvisations with melodies with many intervals of 2nds. Intervals of 2nds provide the feeling of pitch continuation in the melody (chromaticity) but not of harmony. 

The Zampona pan-flute and the melodic corridor arrangements of the notes of a diatonic scale (see post 94 ) are best for melodies with high harmonic profile that is  %  2nds<=50%. or 
%  2nds<=33%.


THE "CHORD-AND-TRANSITION MELODIC THEME"  METHOD OF UNACCOMPANIED MELODIC IMPROVISATION.

Studying some of the musical pieces of Bach for solo instruments (e.g. 

BWV 1013 - Partita in A Minor for Solo Flute  https://www.youtube.com/watch?v=Datoqxx-biw) we may inspired for the next method of unaccompanied improvised soloing:

The melody is like a man wandering-walking in a town, who spends much time at squares and garden of the town that his is exploring. The squares and garden  are the chords, played in the solo , preferably a single chord spanned in two octaves, At this time of the soloing the emphasis is on the harmony. But when continuing till the next "garden or square"   then the emphasis is on the melodic themes and their affine structure dynamics (see post 97 and 101). How much time is spend in "gardens" ( chords) or "street-walking" (transition melodic themes) is a matter of choice of the improviser. If the melody is unaccompanied, then at the same time the chord progressions is improvised in this way. 

We remind the reader what we wrote in post 82, about creating melodies by melodic micro-themes. The reason we repeat the discussion here is that although this method initially was devised to compose melodies when already a chord progression is given, some of its aspects  can apply also as a method to create chord-independent melodies without having in advance a chord progression, but creating the melody with sufficient hidden harmony in it. Here is what we wrote

Here we concentrate one only simple organization structure which the closest corresponded in the poetic language and lyrics is the word. So we introduce a concept of micro-melodic theme, called

MUSICAL WORD that we may agree to symbolize say by w. It consists of a very small number of beats higher than 2 e.g.  3 or 4, and we may symbolize it with 0,s and 1,s , which means that at this beat if no sound is heard it is zero, while if a sound is heard it is 1. E.g. (0101) or (011) etc Now we divide the word in its LONG PART , that symbolize by L(w) , and SHORT PART . that we symbolize by S(w) and so that in time duration, or beats it holds that L(w)/S(w)>=2 (e.g. L(w)/S(w)=3 etc).

PITCH OSCILLATIONS AND THE MELODIC MICRO-THEME
The musical-words or melodic micro-themes need not be by intervals of 2nds! They can be by intervals of 3rds and 5ths or 4ths! Actually as we shall see in the RULE OF OSCILLATION below its ends may be the required oscillation which most often is an interval of 5th or 4th. E.g.on of the most common such dancing pattern is the (1,1,1), where 2 of the 1's is the long part and 1 is the short part. It may start so  that these 3, 1's are the notes of the underlying chord a kind of harping) , but then it dances away so that only two of the 1's are eventually notes of the underlying chord. The number 3 here most often in dancing comes from the 3-like steps of the running horse. In this way by going up and down the diatonic scale,   this very rhythmic structure of the melodic micro-theme, by odd and even steps creates chords and diatonic harmony. Of course the chord changes may be fast , so actually we are talking about ghost-chords! (see post 87 about ghost chords ). 
When playing or improvising  such melodies, with the vibraphone (metallophone) , the 2 , 3 or 4  mallets, correspond to this oscillating melodic micro-theme.

Such melodic micro-themes when improvised rhythmically (especially within a diatonic scale) , with statistically higher percentage of intervals of 3rds (4ths, 5ths also) compared to intervals of 2nds , will create chord-independent melodies with good harmony in them! 

Such oscillating musical words may be ascending, descending or waving. Ascending as excitation may be small (intervals of 2nd) low middle (intervals of 3rds) or high middle (interval of 5th or 4th) or high (intervals of  8th or higher) Of course, as they are combined, they definitely create the effect of waving. BUT the waving is not the very standard by intervals by 2nds but a richer one, that involves many intervals of 3rds and even 5ths, and 8ths. The simplicial sub-melody of such melodies are movements mainly with intervals by 3rds and 5ths. There is also acceleration and deceleration as the melodic theme starts and ends.

E.g. we may descend with a chord say Am and its relative C (out of chords would be notes of G), and ascend with its chromatic-complementary thee G7 (out of chord notes would be those of Am or C ) etc. In other words, we ascend with even or odd notes and descend conversely. Here although we may utilize only 3 chords (Am, C, G) the alternating-changing may be fast covering practically all waving and melodies of the pentatonic or diatonic scale. The scale-completion of the melody (see post 86)  , may be at the next octave rather than in the same octave!
The rhythmic repetition 3 times then the 4th is different is more common than 2 times repeated then 2 times a different. The total range of waving say of the first 3 repetitions may be of size a 5th, while the 4th measure a range of size an 8th, or vice versa.

Let us also assume that the chord progression that underlines the melody is the X(1), X(2) ,...X(n).

As we wrote in previous posts, the melody consists by a progression of melodic themes, that are transformed, by the 4 main transformations or translation, inversion, dilation and rhythmic transformation. This is indeed happening in to the melodic micro-themes or melodic or musical words during the part of the melody that sounds during say the chord X(i) i=1,2...n, BUT we impose here a very important structure which is the key to the beautiful folk melodies, and makes them compatible with the chord progression that underlines, the melody. And this rule is a 

RULE1 OF TRANSIENT AND CHORD NOTES. Obligatory part: In simple words, each musical-word w , that has underlined chord X(i) has the notes of its long part L(w) , to be notes of the chord X(i), (which includes extended forms of X(i) like X(i)maj7 or X(i)7 or X(i)add9 or X(i)sus4 ) while , the notes of its short part S(w) to be transient and belonging to the notes of the neighboring chord that is X(i-1) or  X(i+1), (which includes extended forms of X(i+1) like X(i+1)maj7 or X(i+1)7 or X(i+1)add9 or or X(i+1)sus4) or and more rarely to the rest of the chords of the chord progression. And if so if it contains a note from a non-adjacent chord Y(j) of the progression, then usually somewhere in the progression there is a transition X(i)->Y(j) or Y(j)->X(i) . We keep the transient notes sound at most 1/3 of the time only and the notes of the chord at least 2/3 of the time, because of the rule of long and short parts of the musical word or micro-theme. No mentioning of any scale is necessary in this definition (as usually there are more than one) but only of the chord progression, which is compatible with our enhanced concept of modern harmony. Nevertheless the chord progression over which this technique produces fast melodies may contain very fast chord changes, and may not be identical with the actual chord progression that the instruments play as background to the melody. This is the concept of "ghost chords" in the melody as described in the post 87. E.g. The full ghost-chord progression may be D G D G D A D. While the chords really played is only D. 

RULE2 An alternative rule is that a musical-word w , that has underlined chord X(i) has the notes of its long part L(w) , to be notes of the chord X(i), (which includes extended forms of X(i) like X(i)maj7 or X(i)7 or X(i)add9 or X(i)sus4 ) while , the notes of its short part S(w) to be transient and is one only intermediate not between the notes of the  chord X(i) (usually a 2nd away from the notes of X(i) and preferably but not obligatory this additional note to be a note of the other chords of the progression, again preferably and if possible of the previous or next chord, rarely on  of other chords. And if so, if it contains a note from a non-adjacent chord Y(j) of the progression, then usually somewhere in the progression there is a transition X(i)->Y(j) or Y(j)->X(i) .In this way we keep the transient notes sound at most 1/3 of the time only and the notes of the chord at least 2/3 of the time, in addition to the rule of long and short parts of the musical word or micro-theme. Even if we did not have the structure of micro-themes as musical-words with long and short notes , and we are playing in a random way the three notes of the chord plus one transient, in equal time in the average, we are still in the harmony of this chord, because of the proportion 3:1. And this would still hold if we used 2 transient notes in which case we would have the time proportion 3:2.  But in addition to this rule if we want also the intervals of 3rds, 4ths, 5th and 8th to be more than 2/3 of all the intervals the way is to apply harping in a chord say with 6 or 8 steps on notes, where it is added only one intermediate note in the chord (e.g. 7nh, 6th, 4th or 2nd) and so that the created intervals of 2nd are only 2 in the 6 or 8 intervals. Then we shift to a relative chord an interval of  3rd away or to a resolution transition which is a chord in an interval  5th or 4th away , or we even shift to a chord a 2nd away in which case we do not use any additional note, and we continue so.  So finally %3rds+%4ths/5ths/8ths>=2*(% 2nds) . Again the chord progression over which this technique produces fast melodies may contain very fast chord changes, and may not be identical with the actual chord progression that the instruments play as background to the melody. This is the concept of "ghost chords" in the melody as described in the post 87. E.g. The full ghost chord progression may be D G D G D A D. While the chords really played is only D. 

THEREFORE EVERY CHORD PLAYS THE ROLE OF A MINI CENTRAL SUB-SCALE AROUND WHICH THE MELODY DANCES FOR A WHILE ALTHOUGH IT  IS STEPPING ON OTHER NOTES TOO BUT NOT FOR LONG, THAT ARE MAINLY THE NOTES OF THE NEXT CHORD-SUB-SCALE. 

RULE 3 OF OSCILLATION OR BALANCE
THE COURT-MELODY USUALLY  OSCILLATES INSIDE AN INTERVAL OF 5TH OR 8TH. AND IT MAY BE OF THE NOTES OF THE HARMONIC SIMPLICIAL SUBMELODY (oscillating link or bridge of chords) OR THE ROOR-DOMINANT OF THE CHORD, OR MIDDLE 3RD AND 6TH OR 7NTH OFTHE CHORD (internal bridge of a chord).

RULE 4 OF AFFINE STRUCTURE BALANCE
The melody if ir ascend then it descends and vice versa. The imblanace of thsi rather slight to indicate joy or sadness respectively. (For the Affine structure of a melody see post 97)

RULE 5 OF PITCH SCALE-COMPLENTESS
THE MELODY IS DESIRD TO USE AS EVENTUALLY MANY AS POSSIBLE OF ALL THE NOTES OF AN INTERVAL EITHER OF THE 12-TONES CHROMATI SCALE OR OF A 7 NOTES DIATONIC SCALE.


WE MAY CALL SUCH A CHATTY FAST MELODY THE CHORD-COURT MELODY OR SIMPLER THE CHATTY COURT MELODY OF THE CHORD PROGRESSION.
IT IS IMPORTANT TO REALIZE THAT THE COURT-CHATT MELODY MAY USE OSCILLATIONS BETWEEN THE NOTES OF THE HARMONIC SIMPLICIAL SUBMELODY THAT ARE MAILY INTERVALS OF 4TH, 5TH AND 8TH.  (SEE POST 9, 65, 72 )

GENERAL REMARKS ABOUT MELODY-CHORD CORRELATION 

0) When a melody is created without reference to any chord-progression (see e.g. post 82 about INDEPENDENT MELODIES ), then an statistical profile with high percentages of intervals of 5ths, 4ths, and 3rds compared to 2nds is sufficient to make it an beautiful harmonic melody. But if there is already a chord progression, and we improvise with a melody on it, 

1) then during the time interval that a chord is sounding, we may want to have notes of the melody that include at least one note of the chord and in overall the time that notes of the melody that belong to the chord ,sound, is longer that the total time that the rest of the notes not in the chord is sounding during the chord. This is a quite strong rule. 

2) A weaker rule is simply the requirement that the notes of the melody during the sounding of the chord, contain  notes of the sounding chord, and probably that compared to their neighboring notes, the notes in the melody of the chord, sound longer during the sounding of the underlying chord.

3) If we abolish even this rule then we have an independent melody parallel to an independent chord progression, which is entirely acceptable in Jazz. In an independent melody, from the chord progression, we feel the harmony of the chord progression, but we apply statistically to all , some or none of the previous rules to some or of the chords.

We enlarge below more and we give an example.

81. HOW TO DERIVE NEW SCALES FROM THE HARMONICS (OVERTONES) OF A SINGLE TONE. THE HARMONIC PENTATONIC, HARMONIC 7-TONE AND HARMONIC 8-TONE SCALES.

HOW TO DERIVE NEW SCALES FROM THE HARMONICS OF A SINGLE TONE. THE HARMONIC PENTATONIC, HARMONIC 7-TONE AND HARMONIC 8-TONE SCALES.

SUCH SCALES OCCUR IN THE OVERTONE FLUTES

The pentatonic scale is supposed to be obtained by the first 9 harmonics when reduced to the first octave with frequencies based on a fundamental,  given by the harmonic order (numerator) and power of 2 which is the reduction in the first octave

C, D, E, G, A, C'

1, 9/8 , 5/4 , 3/2 , 7/4. 



Similarly the 7-notes diatonic scale

C---1 harmonic

D---9

E---5

F---11

G---3

A---7

B---15

C---2

To derive the 7 notes of the diatonic scale in major mode requires more harmonics (of a single note) that one may imagine.

See e.g. https://en.wikipedia.org/wiki/Harmonic_series_(music)

Thus the 7-notes  of the diatonic scale in major mode, require 27 harmonics. And the correspondence is the next. The notes are of course lowered to be within one octave, while the harmonics in many higher.

C---1 harmonic

D---9

E---5

F---11

G---3

A---27 (or 7) 

B---15

C---2

So a major scale like C-E-G, requires only the first 5 harmonics

While a minor scale e.g. C-Eb-G requires 19 harmonics as Eb is obtained as the 19th harmonic.

In the Pythagorean method, we derive the 7-notes diatonic scale, by repeating 7 times, the 3rd harmonic of the previous harmonic (thus 3^6=729 harmonics of the deepest tone which is the F here)

So the correspondence in harmonics would be, starting from F this time

F---1 harmonic
C---3 harmonic

D---3^3=27

E---3^5=243

G---3^2=9

A---3^4=81

B---3^6=729

F---2

If the base note is C then

C---1 harmonic
G---3 harmonic

A---3^3=27

B---3^5=243

D---3^2=9

E---3^4=81

F---3^6=729

C---2

With this Pythagorean method, therefore all frequencies of the scale are simple ratios with numerator powers of 3 and denominator powers 2. The harmonics by 3, 3^7=2187, are close for the first time to harmonics by 2, for 2^11=2048. So after 11+1=12 octaves by harmonics as powers of 2  (+1 because we started lower than C) and after 7 intervals by 5th (harmonics 3^7) the two harmonics differ by an amount very close to the discrimination threshold by the human ear, which is called the Pythagorean comma. More formally the  Pythagorean comma, denoted by pc can be defined as the difference pc= log(3/2)/log(2)-7/12=0.001629167..... and it is an irrational number.

Here an analytic table from (https://en.wikipedia.org/wiki/Pythagorean_tuning )

Note	C		D		E		F		G		A		B		C
Ratio	​1⁄1		​9⁄8		​81⁄64		​4⁄3		​3⁄2		​27⁄16		​243⁄128		​2⁄1
Step	—	​9⁄8		​9⁄8		​256⁄243		​9⁄8		​9⁄8		​9⁄8		​256⁄243		—

In full the Pythagorean enharmonic scale is (from https://en.wikipedia.org/wiki/Enharmonic_scale )

The following Pythagorean scale is enharmonic:

Note	Ratio	Decimal	Cents	Difference (cents)
C	1:1	1	0
D♭	256:243	1.05350	90.225	23.460
C♯	2187:2048	1.06787	113.685
D	9:8	1.125	203.910
E♭	32:27	1.18519	294.135	23.460
D♯	19683:16384	1.20135	317.595
E	81:64	1.26563	407.820
F	4:3	1.33333	498.045
G♭	1024:729	1.40466	588.270	23.460
F♯	729:512	1.42383	611.730
G	3:2	1.5	701.955
A♭	128:81	1.58025	792.180	23.460
G♯	6561:4096	1.60181	815.640
A	27:16	1.6875	905.865
B♭	16:9	1.77778	996.090	23.460
A♯	59049:32768	1.80203	1019.550
B	243:128	1.89844	1109.775
C′	2:1	2	1200

Notice that all the ratios of the 7-notes of the enharmonic Pythagorean diatonic scale are quotients powers that have base 2 or 3

C = 1

D= (3^2)/(2^3)=9/8

E=(3^4)/(2^3)=81/64

F=(2^2)/3=4/3

G=3/2

A=(3^3)/(2^4)=27/16

B=(3^5)/(2^6)=243/128 

This approach is very relevant t the Chinese musical system where all ratios are quotients of powers with base 2 or 3 (thus derived from the 3rd and 2 harmonic and their harmonics)

The ancient Chinese musical system depends on very ancient mathematics used to determine sound frequencies. The easiest way to explain it is to work through a real example.

Suppose that somebody wanted to make a musical instrument that could play any song in the ancient Chinese system. Here are the instructions:

Make a wooden box 105 cm long and 60 cm wide. Put guides for the strings near each end of the box, and fix it so that these two guides are 99 cm apart. Multiply 99 cm by 2/3, which is 66 cm. Place a fret all the way across the box on the 66 cm line.
Multiply 66 cm by 4/3, which is 88 cm. Place a fret along the 88 cm line.
Multiply 88 cm by 2/3, which is 58.66...6 cm. Place a fret along this line.
Multiply 58.66...6 cm by 4/3, which is 78.22...2 cm. Place a fret along this line.
Multiply 78.22...2 cm by 2/3, which is 54.148148...148 cm. Place a fret along this line.
Multiply 54.148148...148 cm by 4/3, which is 69.531 cm. Place a fret along this line.
Multiply 69.531 cm by 2/3, which is 46.354 -- and which is too short, so double it to get 92.708 cm. Place a fret along this line.
Multiply 92.708 cm by 4/3...
Multiply the previous answer by 2/3...
Keep going until you have put down eleven frets.

Counting the frequency on the open string and the frequencies on the fretted strings, for each string there will be 12 defined frequencies.

Tune the bottom string to some basic frequency. Tune the next string to the frequency of the bottom string at the first fret. Tune the third string to the bottom string's second fret. Keep going until you have tuned all twelve strings.

When you pluck these strings at all the fretted and unfretted positions, you will get 144 frequencies. Some of them will be duplicates, but not as many as you might think because this system is not like the Equal tempered system now used for almost all Western music.

Out of each twelve frequencies on a single string, you can make many selections of either five frequencies (for the pentatonic scales) or seven frequencies (for the heptatonic scales).

Since the sequence of 5ths is say the F->C->G->D->A->E->B , it is interesting to define successively larger and larger scales that contain many intervals of 5th in this sequence.

1) One interval of 5th C-G
2) 3 notes C,F,G (it is not a  major chord!) Interval structure 5-2-5
3) 4 notes C, D, F, G Interval structure 2-3-2-5
4) 5 notes C,D, F, G, A  Interval structure 2-3-2-2-3 which is the inverse of the well known minor pentatonic scale.
5) 6 notes C, D, E, F, G, A Interval structure 2-2-1-2-2-3. This may be called the Pyrhagorean 6-notes harmonic scale 

6)The scale F1-C2-G2-D3-A3-E4-B4-F5  or short wheel of intervals of 5.  This scale as sequence of intervals of 5 is e.g. F1-C2-G2-D3-A3-E4-B4-F5 has a span of 4 octaves! From F1 to F5 while the last interval B4-F5 is a not of 7 semitones but of 6.

Similarly if we would use intervals of 4 in the reverse order it would be

7) The scale B1-E2-A2-D3-G3-C4-F4-B4 or short wheel of intervals of 4.  This scale as sequence of intervals of 4 is e.g. F1-C2-G2-D3-A3-E4-B4-F5 has a span of 3 octaves! From B1 to B4 while the last interval F4-B4 is a not of 5 semitones but of 6.

Going back to the order in which the simplest harmonics derive the 12-tone chromatic scale, we may put, the intervals, chords, and scales with the maximum number of simplest harmonics, in the next order

1) A SINGLE TONE C ( ALL SIMPLE HARMONICS )

2) AN INTERVAL OF OCTAVE  C(N)-C(N+1) 2ND HARMONIC)

3) POWER CHORD C-G-C (2ND 3RD HARMONIC)

4) MAJOR TRIAD CHORD C-E-G-C (WITHIN THE FIRST 5 HARMONICS)

5) MAJOR TRIAD SUSPENDED 2 OR ADDED 9TH Cadd9 or Csus2 or Em7#5=Em7+
(WITHIN THE FIRST 9 HARMONICS. HERE A COINCIDENCE OF THE NUMBER 9)
THEY CAN BE CONSIDERED ALSO AS A 4-NOTES HARMONIC SCALE C-D-E-G-C
interval structure 2-2-3-5, which is a very common overtones scale of the overtones flutes! 

6) THE HARMONIC PENTATONIC (AN UNNOTICED SO FAR PENTATONIC SCALE!)

C-D-E-F-G-C (SEMITONE STRUCTURE 2-2-1-2-5)
(WITHIN THE FIRST 11 HARMONICS) which is again sometimes  overtones scale of the overtones flutes!  This scale also is directly playable in the Ney-flute which  has hole spanning a 5th of scale rather than an 8th C-D-(Eb)-E-F-(F#)-G

Nevertheless by combining acoustics of open-open and open-closed pipe with the first 9 harmonics we may get the classical mode of the major pentatonic scale. See  this  post below about overtone flutes.

7) THE HARMONIC 6-TONES SCALE (not to be confused with the Pythagoream 6-notes harmonic scale above)

C-D-E-F-G-Ab-C  (SEMITONE STRUCTURE 2-2-1-2-1-4)
(WITHIN THE FIRST 13  HARMONICS).


8) THE MELODIC MINOR  7-TONES SCALE (Not to be confused with the harmonic minor or major scale!)

C-D-E-F-G-Ab-Bb-C  (SEMITONE STRUCTURE 2-2-1-2-1-2-2 NOTICE THAT IT IS SYMMETRIC RELATIVE TO THE CENTRAL TONE INTERVAL OF 2 SEMITONES ON F-G. THIS SCALE IS KNOWN ALSO AS HINDU SCALE )

(WITHIN THE FIRST 14 HARMONICS).

NOTICE THAT COMPARED TO THE DIATONIC 7-NOTES SCALE, IT IS DERIVED WITHIN THE FIRST 14 HARMONICS WHILE THE 7-NOTES DIATONIC IS DERIVED WITHIN THE FIRST 27 HARMONICS!

9) THE HARMONIC 8-TONES SCALE 

C-D-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 2-2-1-2-1-2-1-1 )

(WITHIN THE FIRST 15 HARMONICS).

(NOTICE THAT BY ELIMINATING THE Bb, WE RESULT TO THE

7-NOTES  1ST BYZANTINE SCALE OR HARMONIC  MINOR SCALE

WITH AMAZING SOUND

C-D-E-F-G-Ab-B-C  (SEMITONE STRUCTURE 2-2-1-2-1-3-1 ) AGAIN WITHIN THE 15 HARMONICS!

10) THE HARMONIC 9-TONES SCALE 


C-Db-D-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 1-1-2-1-2-1-2-1-1 )

(WITHIN THE FIRST 17 HARMONICS).

(NOTICE THAT BY ELIMINATING THE D, WE RESULT TO A

SECOND HARMONIC 8-NOTES HARMONIC SCALE

WITH AMAZING SOUND

C-Db-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 1-3-1-2-1-2-1-1 ) AGAIN WITHIN THE 17 HARMONICS!

AND BY ELIMINATING THE Bb IN THIS SCALE WE GET THE REMARKABLE

C-Db-E-F-G-Ab-B-C  AGAIN WITHIN THE 17 HARMONICS,  WITH SEMITONE STRUCTURE 1-3-1-2-1-3-1  WHICH IS NOTHING ELSE THAN THE 2ND BYZANTINE SCALE OR HARMONIC DOUBLE MINOR OR HUNGARIAN MINOR OR GYPSY MINOR SCALE!



11) THE HARMONIC 10-TONES SCALE 

C-Db-D-Eb-E-F-G-Ab-Bb-B-C  (SEMITONE STRUCTURE 1-1-1-1-1-2-1-2-1-1 )

NOTICE THE BLUE-NOTE Eb-E, THAT ALLOWS BOTH C MAJOR AND C MINOR CHORD.

(WITHIN THE FIRST 19 HARMONICS).

12) THE DIATONIC 7-TONES SCALE 

C-D-E-F-G-A-B-C  (SEMITONE STRUCTURE 2-2-1-2-2-2-1 )


If on the other hand we have an overtone flute based on C4, by alternating open-close-open-close etc and blowing each time more we get the 9 notes sequence of notes spanned on 3 octaves (4th, 5th, 6th) and up to the first 5 harmonic-overtones  by the open-pen acoustics and first 4 harmonic-overtones of the open-closed acoustics, in total 1st 2nd 3rd, 4th, 5th, 7th, 9th harmonics 

C4(open 1st basic harmonic)-G4 a bit sharped(closed 3rd harmonic)-C5(open 2nd harmonic )-E5(closed 5th harmonic)-G5(open 3rd harmonic)-A5 a bit sharped(closed 7nth harmonic )-C6(open 4th harmonic )-D6 a bit sharped (closed 9th harmonic)-E6(open 5th harmonic)

with intervals structure in semitones 

7-5-4-3-2-3-2-2

Main Chords C major C6
               Aminor    etc

which when reduced to a single octave it is the 6-notes scale by overtones

C-D-E-G-A-C  with interval structure  2-2-3-2-3

which is nothing else than the standard (Mongolian)  mode of the (major C) pentatonic scale



(WITHIN THE FIRST 27 HARMONICS).

ALTHOUGH THE DIATONIC SCALE REQUIRES MANY HARMONICS TO BE DEFINED, IT CAN BE PROVED THAT IT HAS THE LARGEST NUMBER OF MAJOR AND MINOR TRIADS COMPARED TO THE OTHER SCALES.

NEVERTHELESS THE STANDARD PENTATONIC SCALE IS THE MAXIMAL SUB-SCALE OF THE DIATONIC WHICH IS CLOSED TO INTERVALS BY 5TH (7 SEMITONES) iN OTHER WORDS STARTING FROM A NOTE OF THE SCALE BY GOING UP OR DOWN A 5THS (7 SEMITONES) WE ARE AGAIN BACK TO A NOTE OF THE SCALE. THE DIATONIC IS NOT CLOSED. IT IS CLOSED ONLY IF WE TOLERATE EITHER AN INTERVAL OF  5TH OR OF  4TH



Here is how it is made willow flutes without holes, playing on scale of...harmonics

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

80. The pitch translation homomorphism between melodic themes and underlying chords.

The pitch translation homomorphism between melodic themes and underlying chords.

When the melody is composed from little pieces called melodic themes M1, M2, M3 etc and each one of them or a small number of them (e.g. M1, M2)  , have the same underlying chord C1, then we have a particular simple and interesting relation between the chords C1, C2 , C3 and the melodic themes (M1, M2), (M3, M4) ,...etc. This is not the case when the melodic themes start at one chord and end to the next, that we usually call in the book, as "external melodic Bridges" . We are in the case of "internal melodic Bridges". This relation is based on the pitch translations of the melodic themes and of the chords. Actually this is also a scheme of composition of melodies based on small melodic themes (see post 9), when the chord progression is given or pre-determined.

So let us say that the melodic themes (M1, M2), have underlying chord C1. Then as we have said there are only 3 possible chord-transition relations in a chord progression (see post 30 ): C2 will be either in resolution relation with C1 which means that the rood of C2 is a 4th lower or higher relative to the root of C1, or the root of C2 is an interval of  3rd  away from the root of C1 , or and interval  of 2nd away from the root of C1. Let us symbolize by tr4(), tr3(), tr2() , where tr() is from the word translation, of these three pitch shifts. Then we may also translate the melodic themes similarly

tr4(M1), tr4(M2), or tr3(M1), tr3(M2), or tr2(M1), tr2(M2), Then automatically the new translated melodic themes will have as appropriate underlying chord the C2. Actually in the case of intervals of 3rd or 5th, the melodic themes tr3(M1), tr3(M2)  or tr5(M1), tr5(M2) may as well as appropriate underlying chord the C1 again as the 3rd and 5th of the chord is a pitch translation that leads to a note again inside the chord. This is the reason we called this relation homomorphism and not isomorphic. In mathematics , and correspondence H is call homomorphism relative to some relations R, if the the objects H(x1), H(x2) are in relation R , if the objects x1, x2 are in relation R. Here H(M1)=C1 and H(tr(M2))=C2 and C2=trn(C1) that is are in distance of interval of n (=2,3,4 etc) if tr(M1) and M1 are in distance of interval of n. It may happen that H(x1)=H(x2). But if when x1 is different from x2 then always also H(x1) is different from H(x2) we say that H is an isomorphism. Here because it may happen that H(M1)=C1, and H(tr(M1))=C1 again the correspondence of melodic themes and chord is not 1-1, that is H is an homomorphism not an isomorphism in general. 

By continuing in this way translating in pitch the initial melodic themes M1, M2 according to the interval shifts of the roots of the chord progression an remaining inside a diatonic scale , we compose a melody (or simplicial sub-melody too, see post 9). Of course in order for the melody not to be too monotonous we may vary also the melodic themes from ascending to descending etc.

Other translations of the melodic themes can be during the same underlying chord, and are obviously of an interval of 3rd.

Now even when we are at external melodic bridges e.g. M1 which starts at underlying chord C1 and ends in underlying chord C2, even then this homomorphism is of use! The way to make it work is to take the range of the melodic theme (usually starting and ending note as simplicial submelody) equal as interval to the interval of the roots of the underlying chords C1, C2. 

Chord progressions that two successive chords  are always either 1) an interval of  4th , that is successive n the wheel of 4ths 2) Relative chords where major turns to minor and vice-versa, thus roots-distance  an interval of 3rd 3) Chromatic relation , in other words the roots differ by a semitone
are best chord progressions for parallel translations of melodic themes by intervals of octave, 4th-5th, 3rd and semitone. 

Here is a video of jazz improvisation which uses this idea. The chord progression id C, F, G, and the of pitch translations of the initial melodic themes are always intervals of 4th or 5th.  

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