104. HOW TO CALCULATE THE HARMONIC THE CHROMATIC AND CHORD PROGRESSION SIMPLISTIC SUB-MELODIES OF A CHORD PROGRESSION. THE MELODIC SIMPLICITY SUB-MELODY OF THE CENTERS OF A MELODY

Here we describe a basic technique of the composition method that starts first from the chord progression and then the melody introduced in post 9  (as in jazz improvisation).

A simplicial (or simplistic) sub-melody is a bit more varying than a drone (isocratic)  melody, and is also the  source  of the bass lines. 

1) Chromatic simplicial sub-melody (CSS , minimum distance notes) .  The simplicial submelody is defined by the next rules. 

 When two successive chords of the chord progression have notes that are one semitone distance only, we chose these two notes as notes of the simplicial sub-melody. For reasons of flexibility we allow two notes per chord if necessary. This happens for all cases that the two consecutive chords in a diatonic scale that are at roots distance of an interval of pure 4th (5 semitones) or pure 5th (7 semitones) or if they are mutually complementary chords (with roots of one step of  the scale apart). In general it is a good idea to chose as notes of the simplicial submelody for two successive chords in the chord progression, two notes, one from each chord with the minimum distance in semitones from the notes of the two chords. E.g. if the chords are , the first chord is the C major=(c4,e4,g4) and the 2nd chord is the F major=(f4,a4,c5), then the notes are e4-f4 that is 3rd-1st. If the chords are the first chord is the C major=(c4,e4,g4) and the 2nd chord is the D minor=(d4,f4,a4), then the notes are e4-f3 that is 3rd-3rd. 

 If the two consecutive chords are mutually relative with two common notes, the notes of the simplicial submelody for each chord are either a common note or the note that the other chord does not contain! That is the 1st-5th order notes. E.g. If the first chord is the C major=(c4,e4,g4) and the 2nd chord is the E minor=(e4,g4,b4) then the notes are b4-c5, that is 5th, and higher 1st. But if the chords are  C major=(c4,e4,g4) and the 2nd chord is the A minor=(a3,c4,e4), then the notes are a4-g4  , that is the higher 1st. and the 5th. If the chords are major-minor relatives : C major=(c4,e4,g4) and the 2nd chord is the C minor=(c4,eb4,g4), then the notes are eb4-e4 , that is 3rd-3rd. 

1.3) Chromatic links simplicial submelody (also bass lines) In general we may have the next rule. If X1, X2 are two succesive chords of the chord progression, and we are at X1, a chromatic link or chromatic bridge  is defined by finding two notes a1 in X1, a2 in X2, so that a1-a2 is at the minimum interval distance among all other chord notes of X1 , X2. Then the chromatic link starts with a1, b1,b2....,bn,a2 , and ends with a2 and all the intermediate steps are one semitone distance. 

1.3) Minimal chromatic drone sub-melody (MCD sub-melody).
This simplicial sub-melody is like the chromatic sub-melody, except that we utilize preferably the common notes of the chords, and we require it  
1.3.1) of as few notes as possible and
1..3.2)  of as little distance as possible. 
The rules are the next

Rule 1: We start from the chord and we find a common note with its next chord. If there are two common notes, we look at the next 3rd chord and chose this that is also either a note of the 3rd--next chord or minimal distance of a note of it. We proceed in this way till the last chord of the underlying chord progression. 
It can be proved that if the chord progression are chords of a diatonic scale, then the minimal  chromatic drone melody, can have only some or all of the first 3 notes of the scale (e.g. in a C major mode diatonic scale the c, d, e)  
This is very useful in double flutes or whistles or double reed-winds playing where in the first it is played a minimal chromatic drone sub-melody, and in the 2nd a full melody.

A minimal chromatic drone sub-melody need not be a kind of bass-line! It very well be a kind of very high register or octave simple melodic line. Personally I prefer the latter.

2) Harmonic simplicial sub-melody (HSS, maximum distance notes) . Probably the best method of creating  the simplicial sub-melody which is based on preferring intervals distances of the notes of the simplicial sub-melody (opposite to the previous method) that are large intervals ,namely intervals of 5ths , 4th 6th or 8th.  . The simplicial sub-melody is somehow the centers or oscilaltion boundaries of the final melody and most often it is one note per chord of the chord progression . They may be also the start and end of the melodic themes. Or they can be just centers that the melodic theme must pass from them. It can also be considered as a very simple bass line parallel to the melody. So the rule to choose the simplicial sub-melody is the next

2.1) If we have two successive chords X(1) -> X(2) in the chord progression, and a is the note of the simplicial sub-melody belonging to chord X(1) , and b is the note of the simplicial sub-melody belonging to the chord X(2), then a->b is an interval of maximum distance and the preference in intervals is in the following order of preference 5th, 4th, 8th, 6th. 
For the notes of maximum distance between successive chords we have the next choices : 

If the X(1) -> X(2) are in the relation of resolution (succesive in the wheel by 4ths) e.g. G->C then we have 3 choices for a->b, the g->c, or b->e, or d->g. If the X(1) -> X(2) are in the relation of relative chords (two common notes) e.g. C->Em then we have 2 choices for a->b,

c->g, or e->b. And if the X(1) -> X(2) are in the chromatic or complementary relation of  chords (roots that differ by one step of the scale) e.g. C->Dm, then we have one only best choice of a->b, here the c-> f, and a 2nd best choice the c-> which is an interval of 6th.
The notes of maximum distance would be two notes per chord. The 1st would be the maximum distance from the previous chord and the 2nd the maximum distance from the next chord. We prefer usually to simplify it it in to one only note but either two or one only note  if necessary we shift to the next octave so as to have the rule that two successive notes of two successive chords of the harmonic simplicial sub-melody have always distance large intervals of  5th, 4th, 8th, 6th. 
2.2) After we have defined the simplicial harmonic and the chromatic sub-melody then we may create bridges between its notes by smaller intervals e.g. 3rds or 2nds for a  full melody. The best ways is to start from the first note of the Chromatic Simplicial submelody (CSS) of the chord relevant to the previous chord, pass from the unique note of the Harmonic simplicial submelody (HSS)  of the chord and end at the 2nd note of the chromatic simplicial submelody (CSS) of the chord relevant to the next chord. (See post 109).
The notes of the harmonic submelody of a chord progression may be used to be  somehow the centers or oscilaltion boundaries of a final melody and most often.  They may be also be the start and end of the melodic themes. It depends if we create melodic themes inside the chord and around of a note of it which serves as it center or melodic themes linking two of them  and their successive chords. For the first way , the melodic themes inside the chord and around the note of the harmonic simplicial submelody can be created as in the post 103 using the chord-local 7-notes scale for each one note of the harmonic simplicial submelody.


There are also the 


3) CHORD-PROGRESSION SIMPLICIAL SUB-MELODIES (CPSS) 
 This is defined in the most easy way as consisting from one note per chord of the chord progression and always at the same degree (1st or 3rd or 5th, or 6th, or 7nth or 9nth or 2nd etc) 
Here is relevant video that by extrapolating  this simplicial sub-melody , we get an improvisational melody, an idea of Jerry Bergonzy

https://www.youtube.com/watch?v=2X-WsnWCAaA&t=21s


4) The chord-middle note simplicial sub-melody (CMNSS) This is one of the most simple tupes and most characteristic sub-melodies for the chord progression. The reason is that the middle note characterises a chord of it is major or minor, and thus this sub-melody involves notes that sometimes are the critical notes of modulations e.g. from the natural minor o the harmonic minor or double harmonic minor
IN THE NEXT WE DESCRIBE HOW TO CALCULATE THE SIMPLICIAL SUB-MELODY OF THE MELODIC CENTERS OF A MELODY

The simplicial sub-melody of the melodic centers is defined by the melodic centers of the melody (see also post 65 about the centers ) .

HOW TO FIND THE MELODIC CENTERS OF A MELODY:

The way to do it is the next

1) We partition the melody , to time intervals or connected pieces of it defined by the property that each one of then  has a single underlying chord, and the piece of the melody is maximal with this property

2) Then for each such time interval or piece of the melody, we define as its center, the note of the melody with the maximal time duration. There is one such note for each instance of a chord in the chord progression. The sequence of these notes is the simplicial sub-melody if the melodic centers of the initial melody. 

(This post has not been written completely yet)

103. HOW TO CREATE MELODIES FROM A CHORD PROGRESSION 2/2: THE CHORD-LOCAL 7-NOTES SCALES OF A CHORD PROGRESSION AND THE MELODIC THEMES VARIATIONS THAT THEY DEFINE.

(This post has not been written completely yet)

This post should be read after reading post 92 and post 96.

As we shall see in the next, the Chord-local 7-notes scale should not be confused with the 7-notes arpeggio-scale of a chord , which requires 2-octaves.

As in general the current music is not restricted to the harmony and melody of only scale but of many scales diatonic or not (multi-tonal music) , we will describe the basic process, based only on the chord progression and not on a particular scale. We have wrote in the past that the chord progression (and also wheels of chords or scales of chords) is a substitute of the old concept of mono-tonal scale harmony.

So let as assume that we start with a cord progression CP=(X(1), X(2) ,...,X(n)) .

Then we will define a 7-notes scale S(i) for each chord X(i) , that it will be called Chord-local scale of the chord X(i).

We assume for simplicity that all chords X(i) are 3-note chords, with notes a(i),b(i) c(i).

Now we define also the set of all such notes a(i),b(i) c(i) for all chords  X(i), as the note universe of the chord progression CP and we symbolize it with S(CP) . 

For each chord X(i) ={ a(i),b(i) c(i)} we need to define intermediate notes y1(i), y2(i), y3(i) y4(i) so that y1(i)<a(i)<y2(i)<b(i)<y3(i)<c(i)<y4(i). In this way we may have notes to create 2nds 4ths, 6ths and 7nths extensions of the chord.

To discover such notes y1(i), y2(i), y3(i) y4(i), we use at first the previous chord X(i-1)  , and next chord X(i+1) in the chord progression CP, and if they are not enough then from the set of notes S(CP) defined as above. If still the notes are not enough to define the y1(i), y2(i), y3(i) y4(i), then we choose ourselves notes that are most natural to do so.

Having defined the set of notes S(i)={y1(i)<a(i)<y2(i)<b(i)<y3(i)<c(i)<y4(i)} for the chord X(i), we notice that we already have a 3+4 notes or a 7-notes scale, which we denote by S(i). WE cal it chord-local 7 -notes scale or Chord-neighborhood 7-notes scale.

If the chord progression is e.g. all the chords of a diatonic scale, then for all chords the scales S(i) are the same diatonic scale (at different modes) and we have a mono-tonal harmony. But in general we may have more than one scale therefore modulations. The scales S(i) may be diatonic scales but maybe also other type of scales like Harmonic minor, or even scales without a particular known name.

Now once we have a chord-local scale for each chord of the chord progression, then a melodic theme created as in the post 92, will give a piece of melody that fits the chord X(i). It was called this this post the Chord-courtyard melody (sub-melody here), denoted here by M(X(i)) or simply M(i). In total the duration of the sounding of the notes of the melody that belong also to the chord should be longer (preferable more than 2/3 of the total time) compared to the duration of the sounding of the notes ouside the chord and inside the chord-local 7-notes scale.

In addition all the basic procedures of variation of this melodic theme, like inner scale translation (see post 100) or translation-modulation of the theme from scale S(i) to the scale S(i+1) etc are definable. Also pitch inversions inside these scales, rhythmic inversions and dilation (contractions or expansions) or homemorphic variations ,etc as in the post 96, that we repeat here as a summary of the possible variations techniques.
Now as we move from  chord to chord in the chord progression , then such 7-notes chord-local scales are defined, also chord-court pieces of the melody M(i) , that we apply variations to them as the scales change, to create finally the complete melody M(CP) of the chord-progression CP.


We repeat some of the discussion of the posts 92 and 96 here. First from, the post 92

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, or micro-rhythmic melodic theme, called MELODIC WORD (a concept from poetry) 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).
For example, we may compose the melody from 3-notes micro-themes, the first and last inside the chords and the middle possible outside the chords.

PITCH OSCILLATIONS AND THE MELODIC MICRO-RHYTHMIC-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. but also of 8th E.g.one of the most common such dancing pattern (waltz) 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. It corresponds also to the basic harping of a 3-notes chord.  It is also a micro-rhythmic pattern that repeat either inside or outside the chord. In this way by going up and down the diatonic scale,   this very micro-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 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 or 8ths. There is also acceleration and deceleration as the melodic theme starts and ends.

If one wants to use the calculated Harmonic simlicial sub-melody of the chord-progression as in post 104, then inner translation and oscillations that would be inside the chord-local 7-notes scale of the chord and aroundthe note of the harmonic simplicial submelody , then they are  modulated when moving at the next chord-local 7-notes scale of the next chord and around the next note of the next chord of the harmonic simplicial sub-melody . Similar remarks apply for the chromatic simplicial sub-melody (see post 104) that usually has two notes per chord.

We remind the calculation of the harmonic simplicial submelody 

1) Harmonic simplicial sub-melody. Probably the best method of creating  the simplicial sub-melody which is based on preferring intervals distances of the notes of the simplicial sub-melody (opposite to the previous method) that are large intervals ,namely intervals of 5ths , 4th 6th or 8th.  . The simplicial sub-melody is somehow the centers or oscilaltion boundaries of the final melody and most often it is one note per chord of the chord progression . They may be also the start and end of the melodic themes. It can also be considered as a very simple bass line parallel to the melody. So the rule to choose the simplicial sub-melody is the next

1.1) If we have two successive chords X(1) -> X(2) in the chord progression, and a is the note of the simplicial sub-melody belonging to chord X(1) , and b is the not of the simplicial sub-melody belonging to the chord X(2), then a->b is an interval in the following order of preference 5th, 4th, 8th, 6th. 

If the X(1) -> X(2) are in the relation of resolution (succesive in the wheel by 4ths) e.g. G->C then we have 3 choices for a->b, the g->c, or b->e, or d->g. If the X(1) -> X(2) are in the relation of relative chords (two common notes) e.g. C->Em then we have 2 choices for a->b,

c->g, or e->b. And if the X(1) -> X(2) are in the chromatic or complementary relation of  chords (roots that differ by one step of the scale) e.g. C->Dm, then we have one only choice or a->b, here the c->f. 
1.2) After we have defined the simplicial sub-melody then we may create bridges between its notes by smaller intervals e.g. 3rds or 2nds for a  full melody. 
The notes of the harmonic submelody of a chord progression may be used to be  somehow the centers or oscilaltion boundaries of a final melody and most often.  They may be also be the start and end of the melodic themes. It depends if we create melodic themes inside the chord and around of a note of it which serves as it center or melodic themes linking two of them  and their successive chords. For the first way , the melodic themes inside the chord and around the note of the harmonic simplicial submelody can be created as in the post 103 using the chord-local 7-notes scale for each one note of the harmonic simplicial submelody.

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, pitch inversion, dilation and rhythmic-inversion  transformations. 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). A simple and common way to crate such an oscillations is to take for example a simple chord harping-waving that conatins also with the previous rules less than 50% of the time also notes outside the chord , and then half of this simple theme translate it one octave higher, and so oscillate between the two octaves. The interval of 3rd will become 6th , the interval of 5th, a 4th and an interval of 2nd , will become 7nth. See e.g. the folk Irish melody Kerry Polka below

RULE 4 OF ORDER-TOPOLOGICAL  STRUCTURE BALANCE
The melody if it ascend then it descends and vice versa. The imbalance of this rather slight to indicate joy or sadness respectively. (For the Affine or ordert-opolpgical  structure of a melody see post 97)

RULE 5 OF PITCH SCALE-COMPLETENESS
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-COURTYARD MELODY OR SIMPLER THE CHATTY COURTYARD 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 )

The order-topological theme (see post 101 about the Dolphin Language) may be a musical word or micro-rhythmic melodic theme as in post 92, or concatenations of them creating an order-topological shape (see post 101) so that in total the total  time duration of notes of it that are notes also of the chord is longer and preferably >=2/3 of the total time compared to the total duration of the notes of it that are outside the chord and its arpeggio-scale  (e.g. inside the chord-local 7-notes scale).  

Chopin  uses an beautiful technique (but also a technique  in Greek folk melodies of Rebetika) where , the notes of the melody are most often  pairs of simultaneous notes (harmonic intervals) of the arpeggio-scale, but also the notes outside the arpeggio scale are again pairs of simultaneous notes (in harmonic intervals of 3rds, 4ths, 5ths, 6ths, 8ths etc) that are borrowed from the next or previous (or in general any other) chord of the chord progression (or of the underlying 7-notes scale if there is one). In this way even the chromatic outside the chord parts of teh melody have harmony!

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 all , some or none of the previous rules to some or of the chords.

Here is the way to create melodies with at least 2/3 of the intervals that  are the larger intervals of 3rds , 5ths/4ths or 8ths. 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) 

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.

Usually the pattern of the melody e.g. in Celtic folk music is with underlying chords two successive in the wheel by 4ths, that is e.g. D7->G (actually the requirement is to cover the diatonic scale so it could also be D->A, D->Bm etc) . E.g. there is an ascending  excitation movement to the next octave, maybe also one more fifth higher (may be called upwards melodic movement) , during the D7, while there is descending waving return to G (maybe called downwards melodic movement) , which goes quite low so that finally the melody closes with waving ascending return to D from where it started. In general the repeated waving of the melody is large within an interval of 8th , or  large-medium within an interval of 5th or medium within an interval of 3rd.


Furthermore, the rule can be extended to the optional part    of the rule which is that we are at least 1/3  of the time (preferably more than 2/3 of the time) at intervals of 3rds in the 2-octave 7-notes scale by thirds, which is always chords, or higher intervals of 4ts and 5ths and the rest of the time with intervals of 2nds. If the chords are mainly in the resolution relation (4ths) or relatives (3rds) the faster the changes of the chords relative to the duration of the musical-words, that may be with intervals by 2nds, the more the higher intervals of 3rds, 4ths, 5ths are in the total melody. The shifting a musical-word or micro-theme which is based, say, in intervals by 3rds inside the underlying chord X(i), is already a translation of the theme by intervals of 3rds, 4ths or 5ths. And at the transition of the chords X(i)->X(i+1), we may consider that the musical-word micro-theme translates also by the interval of the roots of the chords (although this is not absolutely necessary always). Therefore if the chord transitions X(i)->X(i+1) are mainly in the relation of resolution (intervals by 4ths or 5ths) or relative chords (interval of 3rd) then transitioning in the next chord again translated the micro theme by intervals by 3rds 4th or 5ths. Therefore in total, we may have at least more than half of the successive intervals of the melody by intervals of 3rds , 4th, 5ths or 6ths. 
This works even better if for every resolution pair X(i)->X(i+1) we involve as parallel mirror of it its relative pair Y(i)->Y(i+1) where Y(i) relative chord to X(i) and Y(i+1) relative chord to X(i+1). (e.g. to the resolution pair Am->Dm the relative pair is the C->F In the language of intervals for the simplicial sub-melody, this means that we may descend with an interval of 4th (5 semitones) and ascend by a lower relative intervals of 4th again E.g. f4->c4-> e3->a3 ). 
When we solo around say a major chord e.g. C , that we may consider as root chord of a major diatonic scale , the out of chords notes are the 7th, 2nd, 4th, and 6th (b, d, f, a) . But the 2nd, 4th, 6th are the notes of the minor chord ii (Dm)  , which is the lower distant relative chord of the IV (F). Thus it also belong to the V6 (F6)  . While the 7nth (b) is in the V (G) or in the same chord C7. Also the 6th, may be considered as belonging to the I6 (C6). Therefore the sequence C7->F6 , or the G->C->F6, which is in the wheel by 4ths, covers such soloing. Different soloing is a permutation of such triads or pairs. We may also consider that it is covered in the wheel by 3rds, as the ascending sequence of 5 chords  with 3 minors 2 majors (minor oriented) Em->C->Am->F->Dm  or the 5 chords sequence with 2 minors and 3 majors (major oriented)  G->Em->C->Am->F. The latter consideration in the wheel by 3rds seems more natural. Therefore soloing around a chord like C,=(c,e,g) as interval of 7 notes b-c-d-e-f-g-a, is covered by an arc of 5 successive chords in the wheel by 3rds , and the soloing can be patterned by permutations of these chords, as fast-ghost chord progression (see post 87 ) while in reality we may play only 2 major or 3 major chords only.  The same method as we may continue further left or right in the wheel by 3rds defines also the modulations that lead us away from the initial diatonic scale.

For example, 
1) if X(i)->X(i+1) are two chords successive in the wheel by 4ths e.g. G->C, then the chord-pair sub-scale od join-arpeggio of the two successive chords is the pentatonic  scale (B,C,D,E,G) with interval structure 1-2-2-3-4.
2) if X(i)->X(i+1) are two chords successive in the wheel by 3rds e.g. C->Em then the chord-pair sub-scale of join-arpeggio of the two successive chords is the 4-notes scale (B,C,E,G) with interval structure 1-4-3-4. If it is the pair C->Am, then the chord-pair sub-scale of join-arpeggio of the two successive chords is the well known and standard  5-notes major pentatonic scale (C-D-E-G-A) with interval structure 2-2-3-2-3 

3) if  X(i)->X(i+1) are two chords successive in the wheel by 2nds e.g. Dm->Em then the chord-pair sub-scale of join-arpeggio of the two successive chords is the 6-notes scale (B,D,E,F,G,A)
with interval structure 3-2-1-2-2-2. Or if it is the pair F->G then it is the 6-notes scale (F,G,A,B,C,D) with interval structure 2-2-2-1-2-2. On the other hand if it the pair E->Am then it is a pentatonic  scale  (C,E,G#,A,B) with an interval structure 4-4-1-2-1. While if it is the pair Am->G it is the 6-notes scale (A,B,C,D,E,G). And if the G is with dominant seventh G7, so Am->G7, then it is all the 7-notes diatonic scale (A,B,C,D,E,F,G)! If it is the power chord Gpower, so Am->Gpower, then the chord-pair sub-scale of join-arpeggio of the two successive chords is the minor pentatonic scale (A, C, D, E, G)! 
The same if we have the chord progression 
Am->Gpower->C, again the chord-triad sub-scale of join-arpeggio of the three successive chords is the minor pentatonic scale (A, C, D, E, G)! Some beautiful folk songs have this chord progression, and melody in the corresponding pentatonic scale as above.
In the same way, the chord progression G->Am->C  would as join-arpeggio scale the 6-notes scale C-D-E-G-A-B, with internal structure (2-2-3-2-2-1)
Or the progression C-E7->Am the join arpeggio the 7-notes scale C,D,E,G,G#,A,B with interval structure 2-2-3-1-1-2-1.
And of course the join-arpeggio of the chords progression C-F-G or Em-Am-Dm is all the diatonic scale.


W e may strengthen the harmony of the melody by the following observations
THE BEAUTIFUL PROPORTIONS MELODY:  % of intervals of 5ths/4ths> % of intervals of 3rds>% % of intervals of 2nds.

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! 

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 of 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 :
(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

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) 

%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!

Another characteristic of such beautiful melodies with the "right harmonic proportions" is that the exhibit the effect of acceleration/deceleration in the movement exactly as the physical bodies. In other words, they start with slow speed (intervals of 2nds), accelerate (intervals of 3rds and then intervals of 5ths/4ths) and finally decelerate when reaching to the right center-note (from intervals of 5ths/4th to intervals of 3rds and then to intervals of 2nds), Of course there many shortcuts where intermediate level of melodic-speed or melodic-density (see post 68 ) are omitted.

The melody understands the chord sequentially rather than simultaneously, and therefore the chord is mainly two poles of notes roots and dominant that are 7 semitones or an intervals of 5th apart. So the melody waves between these two poles, utilizing the middle note but also another intermediate not in the chord, which creates also a few intervals of 2nd. This is normally the high-middle excitation in the waving. For high excitation we jump to intervals at an octave or higher.

ALTHOUGH THE DIATONIC SCALE REQUIRES MANY HARMONICS TO BE DEFINED, (within the first 27 harmonics see post 81) 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. EVEN WIT THIS RESTRICTION BY MAKING SUCH MELODIES AS ABOVE AROUND INTERLEAVS BY 5TH, AND MOVING UP AND DOWN CREATES BEAUTIFUL MELODIES

What ever it is improvised with the previous rules , and also follows a balance between repetition (3 times) and resolution (4th time) will result in to simple joyful and beautiful melodies. 

We remind also the concept of harmonic simplicial sub-melody of the full melody.(posts 9,63,65,72 


 Harmonic simplicial sub-melody. Probably best method of creating first the simplicial sub-melody is based on preferring intervals distances of the notes of the simplicial sub-melody (opposite to the previous method) that are large intervals ,namely intervals of 5ths , 4th 6th or 8th.  . The simplicial sub-melody is somehow the centers of the final melody and most often it is one note per chord of the chord progression . It can also be considered as a very simple bass line parallel to the melody. So the rule to choose the simplicial sub-melody is the next
3.1) If we have two successive chords X(1) -> X(2) in the chord progression, and a is the note of the simplicial sub-melody belonging to chord X(1) , and b is the not of the simplicial sub-melody belonging to the chord X(2), then a->b is an interval in the following order of preference 5th, 4th, 8th, 6th. 
If the X(1) -> X(2) are in a diatonic scale and in  the relation of resolution (succesive in the wheel by 4ths) e.g. G->C then we have 3 choices for a->b, the g->c, or b->e, or d->g. If the X(1) -> X(2) are in the relation of relative chords (two common notes) e.g. C->Em then we have 2 choices for a->b, c->g, or e->b. And if the X(1) -> X(2) are in the chromatic or complementary relation of  chords (roots that differ by one step of the scale) e.g. C->Dm, then we have one only choice or a->b, here the c->f. After we have defined the simplicial sub-melody then we create bridges between its notes by smaller intervals e.g. 3rds or 2nds. 

And then the discussion from the post 96

0) An affine-topological pattern  of the melody which is independent of a realization in  a mode or in a scale (see post 97)

1) Reflection to a horizontal axis (time)

2) Reflection to a  vertical axis (pitch)

3) Point symmetry to a time point

4) Pitch translation

5) Recursive pitch waving or oscillations ,  ascending or descending.

6) Cyclic or balanced behavior   in ascending-descending (standing oscillations).

7) Dilation on the size of intervals (waved changing of the 3 melodic densities or speeds). Usually the melody starts with low melodic speeds or densities , accelerates to higher speeds or densities and then decelerates again to lower speeds or densities, as is also the motion of bodies in dancing.

8) Statistical types of symmetries.

9) Furthermore, the melodic themes may be organized at small time level by the micro-rhythm of the "melodic words" e.g. 3:1 or 2:1 time duration ratio of the long-short notes, the long inside the underlying chord and the short possibly outside the chord. The melodic word is a basic micro-theme of 

the melody. The interval of the long-short notes is a basic step-interval of the melody and it is avoided to me an interval of 2nd , instead an interval of 3rd, 4ths/5th, 6th , 7th or 8th (see post  92 ). The next basic interval in the melody, is the pitch distance among two successive melodic words, which is usually  zero, an interval of 3rd, 4th, 5th etc.

10) or at a larger time scale, by the relevant poetic measure (11-syllables poetry, 15-syllables poetry, 17-syllables poetry) that determine the pattern of repetitions in the melodic themes E.g. 3 repetitions at 4th measure resolution-change or 4 repetitions and at he 5th resolution-change .

11) We may determine a statistical profile of statistical frequency of intervals in the melody such that the highest statistical  frequency of intervals of the melody are mainly the next intervals in the next preference order 5th, 4th, 8th, 6th, 3rd, 2nd. A happy melody tends to avoid sad and dissonant intervals and use instead happy harmonic intervals

12) As the micro-themes (melodic "words") develop over notes ascending and descending over even or odd number steps of the diatonic scale (as in such a way that chords are shaped) the total results, as intended,  is to use eventually all the notes of he diatonic scale, so that the melody has high scale-completeness measure (see post 86 about chromatic music ). This principles somehow determines the preferred chord progressions (E.g. I, IV, V7) .

13) Although we may focus in such an organized symmetry of the melody during a single underlying chord, the true harmony of the fast melody may use "ghost chords" around this single chord (see post 87 about ghost chords ).  

 E.g. if the chord progression is I, IV, V7 used where IV and V7 are ghost chords, then substituting IV with ii or vi and V7 with vii or iii, we get at least 9 more combinations and variations for the ghost-harmony of the melody , that essentially only the chord I is sounding. E.g. (I,ii,vii), (I,ii,V), (I,vi,V), (I,iv,vii) ,(I,ii, iii) ,(I,vi,iii), (I,vi,V), (I,IV,vii), (I,IV,iii).

14) A fast melody should balance properly repetition and  innovation during its development

It is obvious that a simple guitar harping is not a sufficient concept to grasp the required above high organization of the melody even during a single chord. The guitar has only 6-strings while to lay-out the previous organization structures may require many notes and the chord considered at two octaves rather than one only octave.

102. THE METHOD OF UNACCOMPANIED IMPROVISATION BY ALTERNATION OF CHORDS VARIATIONS WITH SHORT MELODIC THEMES VARIATIONS

THE METHOD OF UNACCOMPANIED  IMPROVISATION BY ALTERNATION OF CHORDS VARIATIONS WITH SHORT  MELODIC THEMES VARIATIONS,  

See also post 295

IMITATION ASSIMILATION, INNOVATION.


THIS METHOD BECOMES EASY, FOR 3 OR MORE STINGS INSTRUMENTS (TENOR GUITAR, UKULELE, BOUZOUKI, OUD , SAZ, MANDOLIN, VIOLIN , CELTIC DIATONIC HARP  ETC) WHEN A SINGLE PRIVILEGED DIATONIC (OR CHROMATIC SCALE) IS MARKED ON THE FRETBOARD. ALSO ONOTHER DIATONIC INSTRUMENTS LIKE FLUTE, PANFLUTE, CHALUMEAU CLARINETT SAXOPHONE HARMONICA ETC. THE MELODY IS IMPROVISATIONAL GOING UP OR DOWN AND THE ALTERNATED CHORDS ARE CHORDS AS TRIADS OF MARKED NOTES THUS OF THE DIATONIC SCALE AND ON THE LOCAL 3-STRINGS THAT INCLUDE THE MELODY STRING. IF IT IS A DIATONIC WIND INSTRUMENT THEN THE CHORD IS SUBSTITUTED WITH ITS ARPEGGIO . OF COURSE IT WORKS BEST FOR DIATONIC SCALES BECAUSE EVERY SUCH TRIAD IS A USUAL CHORD MINOR MAJOR OR DIMISHED. FOR CHROMATIC SCALES LIKE HARMONIC MINOR AND DOUBLE HARMONIC MINOR THE CHORDS ARE LESS COMMON. 

IT IS IMPORTANT TO REALIZE THAT THERE IS NO STRICT REQUIREMENT THAT THE CHORD JUST BEFOR OR AFTER THE INTERMEDIATE MELODIC IMPROVISATION FITS HARMONICALLY AS ACCOMPANYING CHORD OF THE MELODY. BUT IT CAN OF COURSE. THUS THIS CONCEPT OF IMPROVISATION WHICH IS MAINLY FR A SINGLE INSTRUMENT IS A WIDER CONCEPT THAT THE USUAL CHORD ACCOMPANIED IMPROVISATIONS

We continue the discussion of the post 82, about chord-independent and unaccompanied melodies  composition and 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 be inspired for the next method of unaccompanied improvised soloing:

IMITATION ASSIMILATION, INNOVATION.

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 spent 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. 

When spending time within a "garden" or "square" in other words within a chord the best idea is to have the chord in 4-notes form e.g. like a with 7nth or with 6th, and in the current octave or in the next. Then start the melody at a note of the chord and end it again at a note of a chord in this or the next octave. For example we may compose the melody from 3-notes micro-themes, the first and last inside the  chords and the middle possible outside the chords.  Since the chord has 4-notes and the scale  7 notes the passing or transient notes are only 3, less than the 4 of the chord, therefore, any such melodic theme fits harmonically to this chord.

Here is an example :

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

To be more specific: This simple and beautiful  method of improvisation is to play in the harp or piano  with the left hand a simple random melody in lower octaves , (It applies to all scales and also to  the Byzantine or harmonic minor and harmonic double minor scales) and then with the right hand at higher octaves , a triad chord with always middle note the last note of the simple melody of the left hand (or root or dominant note but always the same rule during the improvisation)  . For guitarist that have fingernails at the right hand and no finger nails at he lefts hand playing a simple low pitch melody and then such an alternation of chord with the right hand is easier and more convenient (See also the Colombian harpist Edmar Castandeda). Of course this improvisation can be done also with a (diatonic or harmonic simple minor  or harmonic double minor) wind instrument, and instead of chord we play the arpeggio of the chord in single or in two octaves.

An interesting elaboration is that at the melodic theme we give rhythm that creates harmony. By this we mean that we apply a rhythm with long-short syllables in the musical word as in post  77 and 92. E.g. rhythms like 1/2, 1/4 1/4, or 1/3, 1/3 1/3. The long duration notes usually will fall on the even or odd steps of the scale thus will create more often chords.

Then also a good choice of the next alternating chord is that the chord has end-note or root note the same end-note of the previous melodic theme, and as first note or last note so that the chord spans at the side lower or higher of the least note of the melodic theme, where  the melodic theme spans more notes. 
Still a more elaboration in the improvisation is that when  it is the turn to play the chord we play actually 2 chords that are relative chords. E.g. if the melody starts from lower pitches and ends with C, we not only play the chord Am =A-C-E , but after that the lower-relative chord F=F-A-D. If on the other wand the melody ended at C again but descending from higher pitches , then we play again the chord Am= A-C-E but then we play the upper-relative chord   C=C-E-G. After the two chord we play again another transition melody , and so one. 
In a keyboard or harp , at least according to my preferences, it is more beautiful, to play the melodic themes at lower pitches e.g. in the 3rd octave, and the chords at higher pitches of the 5th or 6th octave. Chords are not comprehended well by the human perception in the 1st or 2nd octave, while at the higher octaves are vivid and clear. Also ifthe alternation of melodic themes and harmony is a type of dialogue, it is more natural that the harmonic answer which is more perfect answer is given at the higher frequencies, while the wondering questions at the lower frequencies. 

We must remark here also the next: We argue that any note of the diatonic scale fits harmonically with any major of minor chord of the same diatonic scale because the number of harmonic  mutual intervals created (that is intervals of 3rds, 6ths, 45ths, or 4ths) is greater than the number of chromatic intervals created (that is intervals of 2nds or 7nths).

Of course this does not mean that a note fits the same well with any of the chords of the same diatonic scale. Not at all!

This is mild statement to that claim of jazz that any note of the 12-notes chromatic scale fits in improvisation with any major or minor chords as long as the type and number of created intervals chromatic  or harmonic dissonant or consonant is desirable as musical and expression effect.

For example an harmonic profile difference may exist in the part of the melody that is wandering in the "Garden" of a chord  (or  in more technical terminology in the chord-local 7-notes scale see post 103) in which case we may have percentage of intervals by 3rds or 6ths +4ths 5ths>=66%  and when it is "street walking" between the "gardens" in which case we may have percentage of intervals by 2nds >=50% 

An example of this method is e.g.  in the major F pentatonic scale in the harp (when it is tuned in C major) where we alternate the root chord F major with any combination of notes and then also the D minor again with any combination of notes.

This method of alternation of chord and transition melodies" becomes even more clear if we take    as simplistic and extreme example   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 PER NOTE 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 per note to all other notes except in one or more of these  notes y1,y2,y3, that we spent much more time per note (making theme temporary centers of the melody)  creating thus the effect of this underlying chord harmony. So for the method of "chord-and-transition melodies" where all the melodies is 100% from intervals by 2nds, simply when we wander in the "garden" of a chord, we spend more time per note on the notes of  the chord. While when we move as transition from one chor to another we spend equal time per note to all the non-chord notes. 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.

Especially the intermediate time that the improviser needs for his next melodic theme idea, is always good to be spent harping on a chord, around the last note. Obviously the result will be much more recognizable if we move always inside a scale. e.g. a diatonic scale, therefore being at a note there are 3 primary choices of a chord around it, and the rest of the choices are more distant and secondary.

Of course the simplest transition melodic theme is the strait scale of intervals of 2nds. But for more beautiful and elaborate unaccompanied melodies , a more complicated affine structure of melody is required (see post 97, 101)

Conversely of course if one has in his mind a chord progression then this may guide him  in the unaccompanied melody improvisation. But the chords duration and time is variable when improvising in an unaccompanied melody.

As a chord progression has an simplicial sub-melody (harmonic or Chromatic) or as default simplicial sub-melody we may take the roots of the chords, we may compose the chord progression after composing a simplicial sub-melody of the roots of its chords, that   are also the centers of the full melody. (See post 114 about higher-order syntax of the Dolphin language as method of composition of melodies)

Thus the "squares"now of random walk are not chords but single notes, that are chose by improvisation at first and then we "walk" in waves and randomly , possibly by repeating patterns in moving till the next "square" which is the next note of the simplicial sub-melody. So both the full melody and simplicial sub-melody are improvised at the same time but the simplicial sub-melody is an easier simpler improvisation which is done at first, while the afterwards improvisation of the full melody fill the details. Like in a journey that we may chose easily the "town stations" but the in between details of walking inside around the "town stations" are decided when being there and not before. 

For the structure of the bridging melodic themes we may remind what we wrote in the post 96

In an chord-progression independent melody improvisation no chord progression is determined from the beginning. Nevertheless chords are emerging in the melody as it is improvised in the current methodology of this post. Once such a chord is highlighted  my the melodic movement , then of course a chord-local 7-notes scale is definable for this chord as in post 103. And melodic themes variations may occur inside this chord- local 7-notes scale as in post 96, before we move to a next chord of the independent melody. We may notice though that the melodic moves between such chords of the melody need not highlight any chord at all, so the above variations inside the chord-local 7-notes scales apply only during chord-time of the melodic improvisation. But the variations as in post 96 apply also during moves between chords.

We have already mentioned types of symmetry for the melodic themes that are

0) An affine pattern of the melody which is independent of a realization in  a mode or in a scale (see post 97)

1) Reflection to a horizontal axis (time)

2) Reflection to a  vertical axis (pitch)

3) Point symmetry to a time point

4) Pitch translation

5) Recursive pitch waving ascending or descending.

6) Cyclic or balanced behavior   in ascending-descending.

7) Dilation on the size of intervals (waved changing of the 3 melodic densities or speeds). Usually the melody starts with low melodic speeds or densities , accelerates to higher speeds or densities and then decelerates again to lower speeds or densities, as is also the motion of bodies in dancing.

8) Statistical types of symmetries.

9) Furthermore, the melodic themes may be organized at small time level by the micro-rhythm of the "melodic words" e.g. 3:1 or 2:1 time duration ratio of the long-short notes, the long inside the underlying chord and the short possibly outside the chord. The melodic word is a basic micro-theme of 

the melody. The interval of the long-short notes is a basic step-interval of the melody and it is avoided to me an interval of 2nd , instead an interval of 3rd, 4ths/5th, 6th , 7th or 8th (see post  92 ). The next basic interval in the melody, is the pitch distance among two successive melodic words, which is usually  zero, an interval of 3rd, 4th, 5th etc.

10) or at a larger time scale, by the relevant poetic measure (11-syllables poetry, 15-syllables poetry, 17-syllables poetry) that determine the pattern of repetitions in the melodic themes E.g. 3 repetitions at 4th measure resolution-change or 4 repetitions and at he 5th resolution-change .

11) We may determine a statistical profile of statistical frequency of intervals in the melody such that the highest statistical  frequency of intervals of the melody are mainly the next intervals in the next preference order 5th, 4th, 8th, 6th, 3rd, 2nd. A happy melody tends to avoid sad and dissonant intervals and use instead happy harmonic intervals

12) As the micro-themes (melodic "words") develop over notes ascending and descending over even or odd number steps of the diatonic scale (as in such a way that chords are shaped) the total results, as intended,  is to use eventually all the notes of he diatonic scale, so that the melody has high scale-completeness measure (see post 86 about chromatic music ). This principles somehow determines the preferred chord progressions (E.g. I, IV, V7) .

13) Although we may focus in such an organized symmetry of the melody during a single underlying chord, the true harmony of the fast melody may use "ghost chords" around this single chord (see post 87 about ghost chords ).  

 E.g. if the chord progression is I, IV, V7 used where IV and V7 are ghost chords, then substituting IV with ii or vi and V7 with vii or iii, we get at least 9 more combinations and variations for the ghost-harmony of the melody , that essentially only the chord I is sounding. E.g. (I,ii,vii), (I,ii,V), (I,vi,V), (I,iv,vii) ,(I,ii, iii) ,(I,vi,iii), (I,vi,V), (I,IV,vii), (I,IV,iii).

14) A fast melody should balance properly repetition and  innovation during its development

It is obvious that a simple guitar harping is not a sufficient concept to grasp the required above high organization of the melody even during a single chord. The guitar has only 6-strings while to lay-out the previous organization structures may require many notes and the chord considered at two octaves rather than one only octave.

A video about melody writing

https://www.youtube.com/watch?v=rl-V2IsUprQ


HERE ARE MORE IDEAS

MY PERSONAL APPROACH:

My personal approach to similar type of improvisation is as follows

HARDWARE  IDEAS

1) INSTRUMENT I utilize a 4-strings instrument instead of 3-stringsm although sometimes a 3-strings too. Of course I may use a 6-string instrument.

2) STARTING SCALE I chose a basic scale, usually the natural defoult scale of the instruent which is marked on its fretboard.

3) The TUNINGS are preferably overtones cuatro tunings thus cavaco cuatro or Swedish troll fiddle tuning. But I can also use the venezuala cuatro or ukulele (guitar tuning of 4 highest strings)

4) CHROMATIC TONALITY
Although I chose a diatonic scale, that is from 7 notes I involve also some or all of the 5 blue notes that result to all 12-notes of the musical universe. This is usually done in chords by convrting major to minor and vice versa E.g. a stanard chromatic extension ofthe chords of the diatonic scale is 3M instead of 3m and 7M instead of 7d. But also the minors  6m can become 6M when resolving to 2m and also the 3m to 3M when resolving to 6m, and the 2m to 2M7 when resolving to 5M. In addition we may suround each major chord with two dimisnsihed chords an interval of 2nd away, that resolve to them, in the same way that the 7dimished resolves to 1M. I have a composed collection of more than 800 chord progressions in tonality , chromatic tonality and general multitonality in which I can resort to improvise with melodic bridges metween the chords as in 10) and 11) below.

For the next steps I will enlarge more in the subsequents paragraphs

CHORD-BASED IDEAS

5) I improvise on major-minor lternations of HARMONIC CYCLES
with bridging the chords small melodic themes too.

6)  I improvise on major-minor lternations of CHROMATIC  CYCLES (like Andaluzian cycle)
with bridging the chords small melodic themes too.

7) I improvise on ISOMORPHIC CYCLES OF CHORDS (X1-X2-X3etc) with (Y1-Y2-Y3 etc) Most often of chromatic sequences that are melodic or harmonic isomorphic . I bridge them with small melodic themes too.

8) 2-VOICES AMBIGUITY REVEALED
I ascend or descend  a mode of the scale with 2-voices simulteneous notes intervals of 3rds. The ambiguity of  them is that they can be part of a minor or dimished chord  (sad feeling) or a major chord (happy feeling).  So 1st I play the 2-voice interval andthen I reveal it as major ot minor chord depending on the mood and to if I am ascending or descending.

MELODIC THEMES BASE IDEAS

IMITATION ASSIMILATION, INNOVATION.

9) SMALL MELODIC THEMES VARIATIONS
The idea here is to start with short and rythmic melodic themes and apply the 3 basic variations , like translation (chromatic, melodic, harmonic) inversions and mutations. The structure of the melodic themes may mimic the chords accompanying a melody. Instead of chords here we may have an isokratic a drone  or intermittently repeating melodic theme , which may be an "arpeggio" of the closure of a chord, and over this we add sequentially a higher octave melodic theme that is varied and evolves.

MIXED CHORD AND MELODIC THEMES IDEAS

10) "WALKING THROUGH PARTS OF  A  TOWN", FLUID-CHORD-TRIADS or 2-VOICE ACCOMPANYING OF MELODY.

This idea is very old idea  in ancient folk music before the era of classical music when complicated melodies where accompanied by power chords that are essentially 2-voices chords. It is an idea also similar to the next below at 11) except that instead of alternating standard 3-voices chords with melodic themes, we  are struming or arpeggiating on a single triad which is not standard major or minor and so that it is constantly changing at each note of the melodic themes so that the 2 lower notes are rather stable and are used as 2-voice chord which accomnies the 3rd note which is changing faster and is usually the higher.

11) "TRAVELING AMONG AND WITHIN TOWNS"

This is a generic type of improvisation where I improvise "randomnly"  with melodic themes that are translated , inverted and mutated mainly within the diatonic scale, but at their standing notes (melodic centers) I apply arpegiation or struming of the close local major or minor/diminishd chord which fits to the melody at that time. The choices are only 2-3 at each time. Here the metaphor for the melodic themes is traveling and the metaphor for chords is "town"

12) OVERTONES VERTICAL GATES
This applies especially well in overtones tunings and of course on the full 6-string 1st overtones tuning. I improvise horizontally with melodies and at the standing note a cross-over vertically at the local overtones "gate" which is of course an extended overtones arpeggio of a major or minor chord.



OF COURSE IN THE 4TH GENERATION DIGITAL  MUSIC  MAKING THAT KNOW  IS BEEN BORN, THE ABOVE METHOD OF ALTERNATION OF CHORDS (ARPEGGIATED) AND MELODY CAN BE VERY WELL PLAYED WITH SOFTWARE THAT CREATES MUSIC CALLED ARPEGGIONOME (for iOs ) OR ARPIO (for Android) by Alexander Randon. (https://www.linkedin.com/in/alexandernaut/) Furthermore withother less automated softare like the ThumbJam we can still make such improvisations ina very efficient ,easy and beatifull way.