66. The 3 basic relations of the elementary melodic moves (themes). Chromatic, melodic, harmonic .

Here is a table of the analogy and correspondence of the levels of the musical language and Speaking languages

MUSICAL LANGUAGE	SPEAKING LANGUAGE
Note	Letter  of the alphabet
Interval (3 elementary melodic moves)	Syllables
Melodic moves or themes (5 basic  melodic patterns)	Words that make a simple proposition (subject verb object)
Chords duration may contain many musical themes	Sentences from a point to  a next point , that may contain many simple propositions

A theme melodic move, can easily have three factors that characterize it

1) If it is sad (-) or joyful (+) (we may call it minor or major  melodic move, although its underground chords sometimes , rarely  may be a  major or a minor chord respectively).

2) Its melodic density (see the 4 melodic speeds or densities, chromatic, diatonic, middle harmonic and high harmonic in post 68)

4) Its range as an interval (this is related somehow by inequality to the density as in 2). melodic theme-moves that their range is more than one octave are special in stressing the nature of being sad or joyful. 

These three parameters still do not define the melodic move-theme even if we know its first note. As we see melodic theme-moves are much more complicated than 3 or 4 notes chords! When creating a melody through melodic theme-moves, ideas similar to those that structure a good chord progression may apply.

We may device a symbolism for a melodic theme move based on the above three factors as follows An1Bn2(-) or An1Bn2(+) where An1 is the first note and Bn2 the last note of the move (n1 n2 denote the piano octave of it) and a minus - or plus + sign if its is sad (minor)  or joyful (major) e.g. G5A4(-). In this way we write the dynamics of he melody as a theme-progression ,much like a chord progression. 

What we do in this post is that we highlight and simplify the relations-transitions  of the musical moves (themes) to 3 only , with exactly the same names as the 3 basic Harmonic relations-transitions of chords for obvious reasons. (See post 30, but also post 61 and post 59). Nevertheless, in spite the fact that the name of the relations are the same, the interpretation is different when we are talking about chords compared to when we speak  about elementary melodic moves. There is an intended  similarity but the details are different. 

1) CHROMATIC or Connected-Complementary-sequential melodic moves (Wheel of 2nds)  (this is similar to complementary chords e.g. C->Dm Bdim->C etc)

Definition: Connected-Complementary melodic moves x-y  means that  their union contains a sequence of 3, 4 , 5 or more consecutive notes of the full chromatic 12-notes scale , or  a diatonic scale or in general a used scale at that time. And that this did not exist in each melodic move separately. Each of the x, y may be on notes of different chords, or tones of different   scales. Most often the y is the transpose or shifted morphological transformation  of x by 1-semitone or in general by one  step in a scale. 
Here is an example ofa  melody with themes that are mutaually Connected-Complementary
https://www.youtube.com/watch?v=PBAg10F7CPw
Some times a whole theme T over a chord or over a chord transition, consist of 3 or 4 melodic moves m1,m2,m3,m4  that are in sequence Connected-Complementary and then this theme is transformed in to T2, T3 , T4  so that in sequence these themes they are mutually relative which is the next relation!

As the relation of connected-sequential melodic moves requires one step in the scale we are interested in chord transitions x->y that the chords have notes at a distance of one step of the scale. a) For complementary chords transitions (see post 30) it would be at the roots b) For resolutional chords transitions  x->y , that is, y being a perfect 4th higher than x (5 semitones) such a "diode" it would be the 2nd-middle note   or the 3rd note of x and 1st-root note of y in normal position. Alternatively it could be the 1st root note of x and the 3rd note of y which are one octave apart but can become identical with an extension of x where it repeats its root one octave higher .  If x is a dominant 7nth chord with 4-notes such a "diode" it would also between the 4th note of x and the middle 2nd note of y. c) For relative chords transitions  x->y it would be the case y being y=Rmaj, x=Rmin or vice versa and the notes would be the middle 2nd notes of the chords (in normal position). Therefore this relation of the melodic move requires according to the underlying chord transition appropriate placement of the melodic move relative to the chords to utilize the previous"diodes" we described.

An example of this is the next melody. https://www.youtube.com/watch?v=M4OoR0g1FDo  

And another example is the next https://www.youtube.com/watch?v=Vf0x-kV7wqA

Melodic moves that are complementary create a very peculiar feeling, as the heart wants to contain everything! 

We give here two examples with the well known melodies of a) Resquesdos de L'Alhambra
b) Besame muschos.

(the post has not been written completely yet)

2) MELODIC or Relative melodic moves (Wheel of 3rds) (this is similar to relative  chords, e.g. C->Am , Em->G etc)

Two melodic moves x ,y are defined as relative if y is the shift of x higher or lower in the scale over another chord , or if y is a pitch or rhythmic inversion of x or y is a morphological variation of x (e.g. from wave to scale or spike or vice versa). Sometimes in this relation x->y we may shift from an emotion of joy to an emotion of sadness or vice versa , exactly as in the relative chords that we may move from minor to major chords or vice versa.

3) HARMONIC or Anxiety resolutional melodic moves (Wheel of 4ths) (this is similar to pairs of  chords, that the second resolves in serenity the anxiety first. E.g. G7->C, Bdim7->C etc)

Two melodic moves x ->y are defined as resolutional if y is a move on the notes of a chord (of the underlying chord progression) while  x is a move on notes  of 1 or 2 semitones distance between them and around notes of the chord , thus containing notes outside the chord and x->y is thus creating the emotional effect of resolving of anxiety to serenity and harmony. The order is sometimes reversed and from serenity we shift to anxiety. 

An obvious question is of course that if x->y melodic moves are in relation R(complementary, relative, resolutional) and their underlying chords are C(x)->C(y), would also these chords are in relation R((complementary, relative, resolutional) ? The answer is that in composition-improvisation we very often want it to be so, but in general it may not be so. 

We may find such very clear examples  e.g. in Paganini's caprice 24 (which has 12-broad themes ) https://www.youtube.com/watch?v=98y0Q7nLGWk  or the famous song tico-tico 

https://www.youtube.com/watch?v=B1i0dlegFJQ. 

We need to notice that these three types of relations of melodic moves correspond to the types of chord transitions and to the melodic speeds. 
(See also posts 30,68)

1) Connected-complementary melodic moves are usually covered and fit to complementary chords in chord transitions and to the chromatic/diatonic melodic speed.
2) Relative melodic moves are usually covered and fit to relative chords in chord transitions and to the middle harmonic  melodic speed.
3) Resolution melodic moves are usually covered and fit to successive resolution chords in chord transitions and to the high harmonic  melodic speed.

There is   correspondence to the 3-melodic densities or speeds  of the melodies that fit to such chord transitions of chord progressions. 

1) The complementary chords in a 2-chords transition corresponds to the chromatic/diatonic melodic speed or density. 

2) The relative chords in a 2-chords transition corresponds to the middle harmonic melodic speed or density. 

3) The successive resolutional  chords in a 2-chords transition corresponds to the high harmonic melodic speed or density. 

ANGLES IN FRETBOARD AND MELODIC SPEEDS

1) When playing the melodies on the fretboard in the guitar, the chromatic/diatonic speed is played mainly along the length of a string, so it is the zero angle.

2)  When playing the melodies on the fretboard in the guitar, the middle harmonic  speed is played mainly at an angle which relative to the horizontal is about 45 degrees and moves from the keys of the guitar to the sounding body as the melody descends in pitches! This is is because it consists of intervals of 3 or 4 semitones that in two successive strings is such an angle.

3)  When playing the melodies on the fretboard in the guitar, the high  harmonic  speed is played mainly at an vertical  angle  relative to the horizontal because the strings are tuned at intervals of 5 semitones (and one string in 4 semitones). Also the interval of 7 semitones (5th) when played in descending the pitches makes an angle  larger than vertical or 90 degrees (e.g. 135 degrees) and moves from  the the sounding body of the guitar to the keys of the guitar  as the melody descends in pitches!

PLAYING ONE MELODIC THEME PER ONE STRING:CORRESPONDING THE GEOMETRY OF THE STRINGS TO HARMONIC MODULATIONS OF THE MELODIC THEMES.
This technique of playing a melody,is particular easy and simple when creating by improvisation the melody. Since the string in the guitar and the other portable string instruments are tuned by 5 or 7 semitones apart ( intervals of pure 4th or 5th) then shifting  a melodic theme by a 4th up or down , which is also the distance of chords in the resolutions relations and cycle of 4ths, means that we should play the theme shifted almost in the same place of the fretboard but on string higher or lower. In other words the different strings symbolize the  steps of modulating or shifting the melodic theme along the wheel of 4ths, which is also a very common pattern of harmony. So the different strings are to be used for different modulations of the melodic theme. This simple idea  corresponds, the geometry of the strings on the fretboard to the harmony of modulations of the melodic theme. 

In the 4-double strings instruments (see post 67) the melodic themes modulations are as follows
1) Assume that we are playing a melodic theme on the highest pair of strings .
2) We utilize the 2nd lower pair of strings to shift the melodic theme a) one pure 4th (or 5 semitones) lower b) one pure 5th (or 7 semitones) lower c) to play the second voice which is a third lower.
3) We utilize the 3rd lower pair of strings to shift the melodic theme, 3 semitones  tome higher but also one octave lower from that (interval of 6th or 9 semitones=+3-12=-9)
4) We utilize the 4th lower pair of strings to shift the melodic theme, one   tome lower but also one octave lower from that (or 14 semitones=-2-12=-14)
5) Of course when shifting the melodic theme among the strings we change it also in the 
absolute distances among its notes, or its rhythm, or from ascending to descending and vice-versa etc.

We remind some observation is composing melodies

Summarizing in  simplistic way the correspondence of melodic pitch dynamics and the 4-basic emotions in music (joy, sadness, anxiety, serenity) we have 

1) Up pitch moves correspond to joy

2) Down pitch moves to sadness

3) Small pitch intervals of 1 or 2 semitones (chromatic or interval of 2nd) correspond to anxiety

4) Large pitch intervals (e.g. 4th, 5th octave etc) correspond to harmony and serenity. 

More instructive remarks in creating the final melody based on the chords are the next.

1) In the part of the chord progression with minor chords, utilize descending melodic moves so that sadness from melody and sadness from harmony fit. Similarly ascending melodic moves for  major chords.

2) In the sad melody parts of the melody (and minor chords) utilize rhythmic patterns that start with faster notes and end with slower notes, and the reverse for the happy part (and major chords).

3) In a triad or 7 nth 4-notes chord the most characteristic notes are the middle 2nd note (in 1-3-5 interval notation  is the 3) and the 7 nth (if it exists). So for the anxiety part of the melodic moves we may utilize 1-semitone trills around these two notes, or waving with 1 or 2 semitones steps and notes outside the chord in the interval of minor 3rd (3 semitones) of the chord. Alternatively instead of trill or small amplitude waves we may utilize chromatic monotone scaling by steps of 1 semitone , or scaling with steps by intervals of 2nd of the scale,  that go from these previous notes of the chord to the same such notes in the next octave. But always make sure that the notes of the chord sound in the average longer, than the notes of these anxiety transition moves with notes outside the chord. 

4) Alternate up (happy) and down (sad) pitch moves , or chromatic moves (anxiety), with harmonic (on chord notes) moves (serenity-harmony).

5) Utilize at least 2 octaves, or even 3 for the melodic moves repeating the notes of the underlying chord on the next octaves , so there is sufficient space for melodic moves, to express with sufficiency the emotions.

6) For the duality of emotions anxiety-serenity, it may be utilized also harmonic waves or monotone scaling over 2 octaves at least,  on the notes of the chord, but also chromatic trill wave over the notes of this wave or scaling (modulated wave on wave or move) and then return to the pure harmonic wave or scaling on the notes of the chord.

7) A chromatic wave by 1-semitones steps or all notes of the scale (steps by intervals of 2nd) that goes up and down at least 2 octaves, corresponds to a chord sub-progression of the song , of our choice that utilizes almost all the chords of the scale!

The next software called palette is a good attempt to help create melodies , but it is lacking the full concept of basic 3 relations of melodic moves (motives), although it utilizes what it calls motives, development (transformations) 

http://www.palette-mct.com/manual_eng/table_of_contents.html

DEFAULT MELODIES  FOR A CHORD PROGRESSION.
Given a  chord progression it is direct how to create a melody that fits the chords, with the following rules
1) During  each chord, the entry note of the simplicial submelody , is the middle note of the chord.
2) During  each chord, the exit note of the simplicial submelody (two notes per chord here), for major chords (including 7nth chords and extensions) is the upper note of the chord, for minor, diminished and augmented chords it is the lower note of the chord.
3) During the chord the melody follows an harmonic theme in one or more octaves span, in other words from notes of the chords, and is walking the chord by a spike, straight scaling or waving (these are parameters for the composer or improviser to choose) from middle and down to up (joy) if the chord is major, or from middle and upper to down (sadness) if it is minor, diminished or augmented. If the chord is simply major or minor we may enhance its harmony by extending it with its upper and lower relatives thus  by an interval of 3rd at the highest note and up , or at the lowest note and lower (in normal position). In other words making it a chord with 6th and/or 7nth.
4) At chord transitions x->y , the melody utilizes a dense melodic move (anxiety), with steps from 1 or 2 semitones, and within a scale (including the chromatic 12-notes scale) from the exit note of x of to the entry note of y , of the simplicial submelody.
5) The harmonic move   lasts longer than the transitional dense melodic move , as the latter  takes less than 30% of the duration of x, and y.

From the rule of local fitness of a melody to a  chord  progression , such a default melody will fit the chord progression.

(the post has not been written completely yet)

65. The centers of a melody and the simplicial sub-melody. The simplicial submelody as the channel of the melody

The concept of the centers of a melody within a time interval , can be defined more precisely in  mathematical-statistical way as follows: Divide all notes of the melody in to equal smaller ones (e.g. by the smallest duration note on the melody), and then create s a statistical histogram with statistical probabilities of how often the particular note and pitch occurs in the melody. The highest 3 peaks of this histogram define the top 3 centers of the melody within the particular time interval (it can be a note that sounds non-continuously but  repeatedly within the melody). This may be used to find a chord (according to the local criterion only see post 27) to much the melody in thsi time interval. If we utilize a moving time interval (e.g. of one or few measures) we may define the centers for all the melody. (See in post 27 also the scientific papers  http://research.microsoft.com/en-us/um/redmond/projects/songsmith/ and  http://research.microsoft.com/en-us/um/people/dan/mysong/ )

If the melody has also an underlying chord progression then there is an alternative way to find the centers of the melody as in the post 118. 

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. 

If the melody consists of a finite set of repeating themes, and each theme consists of one or more of the 4 basic melodic moves (see post 59 ), them the centers are most often the tops and bottoms of the 4 basic melodic moves but also the notes of the underlying chord. So they constitute what could be said as supper or resistance levels for the pitch to go up or down while waving. 

In a triad or 4-notes with 7 nth chord , the most characteristic notes are the middle 2nd note (in 1-3-5 interval notation  is the 3) and the 7 nth (if it exists) as the define their character as minor-major and 7nth or extended in general. Therefore these two notes have higher probability to be the notes of the simplicial submelody. In general in a chord transition X-->Y , the notes of simplical sub-melody can be one starting   note from  X and and ending note in Y. 
The themes of a melody consist of a plot or sequence of the 4 basic moves (see post 59)  which by itself says an emotional story without the help of the harmony. If we have (as here we assume we do) an underlying chord progression, then utilizing almost all the notes of the chords and one theme for each of the 3-harmonic-types of chord transitions , we may define the set of themes of the melody in easy way. Alternatively we may define a theme for each type of emotion, sad, joy, anxiety or serenity, or a theme for each type of chord respectively minor (sad) major (happy), 7nth or diminished or augmented (anxiety) and r5 (serenity.) The chord progression serves as a way to transform and make variations of the themes. The notes of the simplicial submelody may be the centers of the melody that sound longer and are usually the tops and bottoms of the 4 basic melodic moves that create the themes of the melody but also the notes of the underlying chord. 

The next video in jazz improvisation music shows that in fact any of the 12 notes of the chromatic scale and not only the 3 or 4 notes of the chord can be chosen as the note of the simplicial sub-melody during a chord!


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


Another more practical way of course as we have mentioned in other posts (see e,g. post 72) is defining the simplicial sub-melody through the stating and ending points of the melodic themes.


1) Chromatic simplicial sub-melody. A second a but more sophisticated way is to do exactly the same except that the 1st voice is defined not by the highest note in the chords but through the simplicial submelody.  The simplicual submelody is defined by the next rules. 
1.1) 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. And alternatively for a 2nd voice we may take the 2 notes in the chords respectively with the maximum distance  between them for maximum action of waving movements! This in general may lead to two notes per chord in the chord progression, the second note is reserved for the 2nd voice etc.  The more correct rule to find the simplicial submelody  is as few notes per chord as possible that give the basic feeling of the melody. 
1.2) 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! 

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 ling or chromatic bridge  is defined by finding two notes a1 in X1, a2 in X2, so taht a1-a2 is at the minimum interval distance among all other chord notes. Then the chromatic link starts with a1, b1,b2....,bn,a2 , and with a2 and all the intermediate steps are one semitone distance. 


1.4) 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  
1.4.1) of as few notes as possible and
1.4.2)  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)  


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. 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 . 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 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. 

3) Default simplicial sub-melody.  This is simply the melody created by the roots of the chords of the chord progression.

MELODIC THEMES TRANSFORMATIONS AND SIMPLICIAL SUBMELODY

We have mentioned in this post that the simplicial submelody is usually  the starting or ending notes of simple melodic themes, that can be external bridges of the chord transitions (of density diatonic or middle harmonic etc). Therefore here we apply the 3 basic transformations and starting from a single melodic theme ending to the first note of the simplicial submelody we translate or invert or vary rhythmically thsi theme, and make it end (or start) on the next note of the simplicial submelody. The transformed melodic themes derived in this way cover most often two chords or a chord transition or chord relation


Still another important remark is that we may have HIGHER ORDER SIMPLICIAL SUBMELODIES. In other words except the 1st simplification ofthe melody, which is the 1st order simplicial submelody, we may have the 2nd order simplicial submelody, the 3rd order simplicial submelody, each one simpler that its previous. A path from the complexity to simplicity. One of them should correspond of course to the complexity of the chord-progresion, that is have one note for each chord of the chord progression. 

The starting and ending notes of the melodic themes may define a simplicial submelody, while the centers of the melody a higher order simplicial submelody.

Summarizing in  simplistic way the correspondence of melodic pitch dynamics and the 4-basic emotions in music (joy, sadness, anxiety, serenity) we have 

1) Up pitch moves correspond to joy

2) Down pitch moves to sadness

3) Small pitch intervals of 1 or 2 semitones (chromatic or interval of 2nd) correspond to anxiety

4) Large pitch intervals (e.g. 4th, 5th octave etc) correspond to harmony and serenity. 

The 3 elementary melodic themes, as we mentioned earlier (e.g. in posts  66 and 69 ) , are  the ascending melodic interval of two notes, the isokratic melodic interval of two equal notes and the descending melodic interval of two notes.

The 3 basic transformations of them are

1) the translation (either with intervals of 2nd , (or diatonic density) or intervals of 3rd (or middle harmonic density) or  of intervals of 4th or 5th (or high harmonic density))

2) The inversion where the ascending move becomes descending.

3) Rhythm transformation (which may vary)

The 5 basic melodic moves (see e.g. post 69)  , being more complicated have more types of transformations, as derived by the writing in a pentagram :

1) Translation

2) Inversion relative to a point

3) Reflection relative to an horizontal line

4) Reflection relative to a vertical line.

5) Rhythm transformation

to the above five we may add the

6) Acceleration (e.g. from the diatonic speed or density to the middle harmonic speed or density) or Deceleration (vice-versa).

Bach has often used the above 6 transformations in his fugue.

More complicated  ways to transform a theme are at least the next 5 and combinations of them (see also post 41)

1) Translate it in different pitches (within a scale or not changing possibly the pitch distances )

2) Translate in time (repeat it)

3) Invert it in time or change its rhythm (if at the begging is slower and at the end faster it will be now the reverse etc)

4) Invert it or distort it in pitch (Create 1st 2nd 3rd or 4th voice versions, utilizing the chord progression as rules of transformation of the theme, or if it is ascending now it will be descending etc)

5) Change it as morphology  (from a non-waving ascending it may become waving ascending or isocratic). We prefer spikes and scaling as the main morphological types, while the waving and isocratic as intermediate bridges. 

Often melodic bridges from a chord to the next, may start with harmonic speed or density covering the first chord A and then decelerate to diatonic speed or density when reaching to the next chord B

More instructive remarks in creating the final melody based on the chords are the next.

1) In the part of the chord progression with minor chords, utilize descending melodic moves so that sadness from melody and sadness from harmony fit

2) In the sad melody parts of the melody (and minor chords) utilize rhythmic patterns that start with faster notes and end with slower notes, and the reverse for the happy part (and major chords).

3) In a triad or 7 nth 4-notes chord the most characteristic notes are the middle 2nd note (in 1-3-5 interval notation  is the 3) and the 7 nth (if it exists). So for the anxiety part of the melodic moves we may utilize 1-semitone trills around these two notes, or waving with 1 or 2 semitones steps and notes outside the chord in the interval of minor 3rd (3 semitones) of the chord. Alternatively instead of trill or small amplitude waves we may utilize chromatic monotone scaling by steps of 1 semitone , or scaling with steps by intervals of 2nd of the scale,  that go from these previous notes of the chord to the same such notes in the next octave. But always make sure that the notes of the chord sound in the average twice longer, than the notes of these anxiety transition moves with notes outside the chord. If we intent for a super simple simplicial sub-melody, then the common notes of  maximal subsequences of the chords of the chord progression may be a candidate. Conversely the non-common notes could serve as simplical submelody notes.

4) Alternate up (happy) and down (sad) pitch moves , or chromatic moves (anxiety), with harmonic (on chord notes) moves (serenity-harmony).

5) Utilize at least 2 octaves, or even 3 for the melodic moves repeating the notes of the underlying chord on the next octaves , so there is sufficient space for melodic moves, to express with sufficiency the emotions.

6) For the duality of emotions anxiety-serenity, it may be utilized also harmonic waves or monotone scaling over 2 octaves at least,  on the notes of the chord, but also chromatic trill wave over the notes of this wave or scaling (modulated wave on wave or move) and then return to the pure harmonic wave or scaling on the notes of the chord.

7) A chromatic wave by 1-semitones steps or all notes of the scale (steps by intervals of 2nd) that goes up and down at least 2 octaves, corresponds to a chord sub-progression of the song , of our choice that utilizes almost all the chords of the scale!

We give an example of the simplicial sub-melody with the next melody of folk music with Cuatro https://www.youtube.com/watch?v=ob50UXyr1JE

After the chord progression and simplicial submelody we chose, 

THE DEFINITION OF MELODIC BRIDGES THAN LINK TWO SUCCESSIVE CHORDS BETWEEN THEM AND START AND END AT THE NOTES OF  THE SIMPLICIAL SUBMELODY.

1) WHICH CHORD-TRANSITIONS (PAIRS OF CHORDS) WILL HAVE A MELODIC BRIDGE! (Usually the chord-trasnitions that are in resolutional relation, or resolutional-like relation)

2) THEN WHICH BRIDGES WILL BE ISOMORPHIC IN PITCH AND RHYTHMIC DYNAMIC SHAPE AND WHICH DIFFERENT, DEFINING THEREFORE A PARTITIONING IN THE BRIDGES.

3) THEN IF IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THIS PARTITIONING, THE BRIDGES ARE  EVENTUALLY ASCENDING OR DESCENDING (This besides the emotional significance, determines also where to play the chord in one of the 3 neighborhoods of the fretboard)

4) FINALLY  HOW IN EACH EQUIVALENCE CLASS OF  ISOMORPHIC MELODIC BRIDGES IN THE PARTITIONING, THE COMPLICATED PITCH DYNAMIC SHAPE  OR WAVING AND RHYTHM WILL BE AS A REPETITION  OF SUCH PATTERNS OF PREVIOUS ISOMORPHIC MELODIC BRIDGES, OR VARIATION OF  SUCH PATTERNAS S SO NOT TO BE TOO BORING. (This pitch dynamic shape has again a significant emotional meaning)


5) THE JUSTIFICATION OF THE CHORD PROGRESSION USUALLY IS NOT DONE BY THE CHOICE OF THE MELODIC BRIDGES (THAT IS GIVEN THE MELODIC BRIDGES MAYBE A SIMPLER CHORD PROGRESSION MAY COVER THEM HARMONICALLY). BUT AN INTERMEDIATE HARPING OR STRUMMING OF EACH CHORD WILL ENHANCE  THE MELODY OF THE BRIDGES SO THAT ONLY THIS CHORD PROGRESSION IS JUSTIFIED!

DEFAULT MELODIES  FOR A CHORD PROGRESSION.
Given a  chord progression it is direct how to create a melody that fits the chords, with the following rules
1) During  each chord, the entry note of the simplicial submelody , is the middle note of the chord.
2) During  each chord, the exit note of the simplicial submelody (two notes per chord here), for major chords (including 7nth chords and extensions) is the upper note of the chord, for minor, diminished and augmented chords it is the lower note of the chord.
3) During the chord the melody follows an harmonic theme in one or more octaves span, in other words from notes of the chords, and is walking the chord by a spike, straight scaling or waving (these are parameters for the composer or improviser to choose) from middle and down to up (joy) if the chord is major, or from middle and upper to down (sadness) if it is minor, diminished or augmented. Alternatively  any descending , ascending or waving sequence of notes at diatonic speed such that the odd or even number of them is exactly the notes of the chord (extended probably by 7nth or 6th) and these motes sound e.g. 3 times more than the notes of the rest of the scaling is a melody that fits the particular chord! Irish melodies do it often. If the chord is simply major or minor we may enhance its harmony by extending it with its upper and lower relatives thus  by an interval of 3rd at the highest note and up , or at the lowest note and lower (in normal position). In other words making it a chord with 6th and/or 7nth. 

Another characteristic of the happy and joyful melodies is to define two notes (or interval) for the simplicial sub-melody for each chord so that in over all the melody is maximally harmonic (see post 40) and we  may use almost exclusively  the maximum large intervals (within a scale) that exist in the chords of the song. And this would be intervals of 8th, 6th (for triad-chords) , 5th and 4th. In other words we use almost exclusively the maximum harmonic melodic  speed that the chords allow (see post 68). 

This idea of maximum harmonic speed in melodies is also an idea that can give pretty directly improvisation melodies over a chord progression! This is good for happy melodies. It directly defines improvisational beautiful melodies from the chord progression,  because the maximum intervals of a chord are unique or very few for each chord! In fact a single large such interval from each chord can define the melodic-rhythmic pattern for each chord! 

The standard preference is to use 

a1) For  a major chord x1-x2-x3, the 1st x1-3rd x2   notes interval of pure 5th (7 semitones), or the 1st nx1-2nd x2 notes interval of major 3rd (4 semitones)

a2) For  a minor chord x1-x2-x3, the 1st x1-3rd x2   notes interval of pure 5th (7 semitones), or the 1st x1-2nd x2 notes interval of minor 3rd (3 semitones)

a3) For  a dominant 7th and major 7th chord x1-x2-x3-x4, the 1st x1-3rd x2   notes interval of pure 5th (7 semitones), or  the 1st x1-4th x4   notes interval of minor 7th (8 semitones), or of  major 7th (9 semitones). 
An interesting case of simplicial submelody is  the first choice  always  (interval of 5th or 4th). 

Or we may allow this interval of 4th or 5h of each chord sound 2/3 of the time of the chord sounding and 1/3 of the time the other middle x2 note for minor or major  , or 7th note of the 7th chords.
 Still another case is the minimal harmonic simplicial submelody (but always with notes of the chords) where we take always the 2nd choice (the x1-x2 interval of 3rd, or x1-x4 interval of 7th) where this sounds 2/3 of the time and 1/3 of the time the 3rd note of the chord. This simplicial submelody gives emphasis to the character of each chord, that is being minor , major or 7th etc. 

But another more maximal  harmonic  method is based on the next rules

b1) For each chord the simplicial submelody consists of at least two notes one entry and one exit (that may though coincide)

b2) Complementary chords (e.g. Cmajor, Dminor) can transition with intervals of 5 or 7 semitones (e.g. exit note of Cmajor is the c, and entry note of Dminor is the f).

b3) Successive chords in the cycle of 4ths or 5ths, and relative chords have common notes, this the exit note of the first chord and the entry note of the 2nd chord are identical.

b4) If the entry note of the a chord and its exit  note is an interval of minor 3rd  (3 semitones) we may add two more notes during the chord which is twice the 3rd note of the chord, but at one octave distance, and convert the minor 3rd interval to major 3rd (4 semitones) which has higher harmonic score (see post 40). E.g. G7-->C-->E7 , entry of C=g3, exit of C=e2, so we add c2, c3, and the simplicial submelody goes like this g3-c2-c3-e2, duringthe chord C. We converted the minor 3rd interval g-e, to a major 3rd c-e. 

b5) It is prefered that intervals of 1,2,3,4 semitones are converted to their complemntary of 11,10,9,8 semitones, by changing octave.

The so derived simplicial submelody singles less melody than the chord progression itself!

E.g. for the Chord progression Am->F->G7->C->G7->C->G7->C->E7->Am, the sumblicial submelody with these rules would be a3-a2a2-f2f2-g3g3-g3g3-g3g3-g3g3-g3g3-c2c3e2e2-e3e3-a3.
This simplicial submelody can be the centers of  full melody over this chord progression

4) At chord transitions x->y , the melody utilizes a dense melodic move (anxiety), with steps from 1 or 2 semitones, and within a scale (including the chromatic 12-notes scale) from the exit note of x of to the entry note of y , of the simplicial submelody.

5) As more general alternative to the above rules 1)-4) , we may define melodic moves not for each chord but for each chord-transition, and preferably for the        X7-->x+1 type of transitions (see the symbolism of post 34) e.g. E7-->Am.
Then the chord X7 has only one note x1 for simplicial submelody  the starting note of the melodic move, and the end note x2 of the melodic move is the next simlicial submelody note and one note of the chord x+1 not common with the chord X7. If the latter note x2 is not the root of x+1, it is created a tension that has to be resolved later where x2 would be the root of x+1. The tension is highest if the x2 is the 3rd note, middle of it is the 2nd note and resolving if it is the root note. In between the x1 and x2, the rule is that at least 2/3 of the notes belong to the underlying chord, and this can be achieved by repeating notes of the underlying chord if necessary. The move x1->x2 may involve  each of the chords X7, x+1 , twice in two octaves each instead of once in one octave only, which may create very impressive melodic effects. This gives an even better opportunity to use in the melodic move, intervals of 8th, 4th and 5th (high harmonic speed, see post 68) , that have higher harmonic score than the other intervals (see post 40). The at most 1/3 of the total duration of the move x1->x2 ,of notes that play with underlying the 1st chord but may be outside the starting chord, might be unusually at chromatic and diatonic speed (see post 68), and sometimes might belong to the next chord or even to none of the two chords. The chromatic or diatonic speed applies usually when approaching the ending note of the melodic move.  The melodic moves x1-->x2 can be called chord-transition melodic moves and must have an element of repetition in length and rhythm. In the transitional Irish melodies that utilize 2-3 only major chords, while the melodic moves are 4-5 or 6-7 , but also in the traditional Greek music of the Aegean Islands, the starting and ending point of the melodic move is during the duration of a single chord and are notes of the chord! But still the rule 2/3 -1/3 for notes internal and external to the chord still holds, and the starting and ending notes of the melodic move may define the simplicial submelody.
6) The harmonic move   lasts longer than the transitional dense (chromatic or diatonic harmonic speed) melodic move , as the latter  takes less than 30% of the duration of x, and y.

7) From the rule of local fitness of a melody to a  chord  progression , such a default melody will fit the chord progression.

In the example below the chord progression is Am E7 Am E7 Am E7 Am E7 Am A7 Dm G7 C F E7 Am and the centers of the melody are correspondingly for each of the above chords the  E E E E E  B A B A A F G E F D A . The melody-moves consist of 10 notes ,the first 9 belong to the first chord and the last 10th to the next. All the moves are on the chord transitions of the form X->(x+1) in the symbolism of the cycle of 24 chords (see post  34). E.g. E7->Am, or Am->E7, or A7->Dm, or G7->C. An exceptions is the transition F->E7.  The notes that belong to the chord for each of these moves are 6 from the 9, that is 2/3 of the notes. They achieve it ,as we said , by repeating notes of the chord. And even in the transition F->E7 the notes hat do not belong to the chord F, while F sounds , do belong to the next chord E7 and so they prepare the ear for the next chord. The melody has all the 4 harmonic speeds (see post 68).  They start (ignoring the repeating notes) from the root A of Am and end to the root E of E7,they go back and forth, then from the root A of Am go to the dominant B of E7 and back to the root A of Am. Then they repeat. Then from the root A of Am which is also of A7, they go to the middle note F of Dm. Then from the root G of G7 to the middle E of C. Then from the root of F to the chord F to the 4th note (7th) D of E7, and close back to the root A of Am.  Starting from the root of X7 and ending in the middle (2nd note)  or dominant (3rd note) of (x+1), (e.g. starting at a of A7 and ending at f of Dm) creates a tension, which resolves at the end of the cycle of 16 moves by ending at the root of minor chord (x+1) (here at a of Am).

Here is the result.

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

The simplicial submelody can be viewed also as the channel submelody.  With this we mean that the simplicial submelody defines a channel in the pitch-time diagram, where the melody waves. But the shape of the move of the channel is defined by the simplicial submelody.  In the harmonic method of composition, what  is of interest is how to derive a full melody, from the simplicial submelody. And the idea to conceive the simplicial submelody as defining the channel where the full melody waves and being probably the staring and ending notes of the themes (here usually waves of the channel), is the key to do so. We just sing a waving with small waves that start and end at the notes of the simplicial submelody. The waving may be at 2/3 inside the chord and 1/3 outside it. (See also post 72)

(This post has not been written completely yet)