64. Symbolism of the melodic themes and their harmonic transformations structure in a melody. The algebra of melodic moves-themes. Melodic maths by Max Martin

THE KEY-WORD HERE IN THE 4TH GENERATION DIGITAL MUSIC FOR THE MUSICAL-THEORETIC IDEAS OF THIS   POST (AS FAR AS MORDEN SOFTWARE FOR MUSIC MAKING IS ) IS MELODY-SEQUENCERS 

THE TERM  SEQUENCER MEANS HERE A LOOP OR RHYTHMIC CYCLE OF   A  MELODIC THEME THAT IS VARIATED INTERACTIVELY BY THE USER  IN A MELODIC SEQUENCER.

THERE MANY GOOD SOFTWARE PROGRAMS FOR THIS COMPOSITION AND IMPROVISATION LIKE FUGUE MACHINE, YAMAHA MOBILE SEQUENCER, THUMPJAM ETC. ALAO ARPIO  AND ARPEGGIONOME FOR GENERAL ARPEGGIOS ALTERNATED WITH MELODIC IMPROVISATIONS


MELODIC MATHS BY MAX MARTIN AND SYSTEMS OF CREATING MELODIC THEMES AND MUTATING MELODIC THEMES AND RHYTHMS

In the next videos one can see how melodic themes of notes  (but also of chords) and mutations of them plus repetitive combinations of them, can be created by keeping invariant an  initial germ-pattern or  melodic-seed of  interval shifts and pause (GERM PATTERN)  of a note (or chord) or  of  initial pattern of sequence of melodic themes of notes or chords after  seeminly random pauses (omittings) of the parts of the fixed pattern. 

Melodic themes of notes can be considered and created also as repettitive combinations of a small set of interval-steps (pitch transformations) in a scale plus a pause wchich may be called MELODIC GERM . A melodic germ  as basic invariant can give many melodic themes with an internal affinity which can  be considered a system of muttations of melodic themes

Comparing the melody with a speaking language suggests the next correspondence

Let us correspond to each vowel a number of steps inteval shift insidea scale

E.g.

empty space=pause
A=0 step
E= 1 steps
I= 2 steps
O=3 steps
OU=4 steps


Then the content of vowels of any phrase can be translated as a GERM-PATTERN for creating melodic themes as muttaions of this germ-pattern  (and latter also repettitive combinations of them)

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


https://www.youtube.com/watch?v=sb3e4Mq6y3s


https://www.youtube.com/watch?v=w0-Ljf5gm4A


https://www.youtube.com/watch?v=Fc16Y1gKUDc



https://www.youtube.com/watch?v=w0-Ljf5gm4A

This post should be read together with post 106, about the Melodic Seed of a song, where starting from a small number of melodic themes that are independent (abstract algebraic dependence system or closure system) as far as variation transformations is concerned, from which the full melody of the song is derived by an algebra of transforming variations.

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 and  themes (5 basic  melodic patterns, 4 basic transformations of melodic themes.)	Words (corresponding to the simple melodic moves) that make a simple proposition (subject verb object, that correspond to the melodic themes)
Chords duration may contain many musical themes	Sentences from a point to  a next point , that may contain many simple propositions

In this post we introduce special symbolism for the basic themes, their rhythm and their transformations structure in  melodies (Time and pitch inversions, time and pitch translations 1st 2nd 3rd 4th voices versions  of them inside or outside a scale , morphological type transformation, and harmonic relations) . This is very useful in musical composition and improvisation as introduces simplicity in the complexity of a melody. It is so important in improvising melodies as is important to write the chords in a  chord progression. The technique of this symbolism utilizes the more simple , mathematical exact and compact way of musical writing as presented  in the post 7.
As we have defined them in this book, the 3-elementary pitch moves are like the 3-elementary particles of electron , neutron and proton, the 4-basic melodic moves are like types of atoms, the melodic themes are like molecules, and a melody is like a material mixture of molecules. The symbolism we describe is not only like the chemical symbolism of long organic molecules. The only reason we introduce this symbolism is for purposes of simplifying composition and playing and  so as to have simplified perception of the structure of the melody in a stratified-structure and the 3 basic  harmonic transformations  of its themes (see post 30) . In this way we can easily visualize how the harmony of the chord progression of the song contributes in the structure of the melody.
The general shape of such structure is a follows. If a1, a2, a3, a4, are the themes of the melody, and f1,f2,f3,f4,f5 some of the transformations of the theme, then such a symbolism would have a shape for the melody as follows a1a2a4f1(a1)f2(a1)f3(a1)f4(a2)a3a1f3(a4) etc.........Nevertheless each of the  ai i=1,2,3,4 can be further decomposed in the way a theme is created from the 4 basic melodic moves  itch moves, andeach fi i=1,2,3,4 can be decomposed in the 5 basic theme transformations. In this way writing contrapuntal fugue, like those of J.S. Bach  becomes a lot easier.

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 are 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. 

Summarizing in  a 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. 

Moving in the melody from intervals of 1,2 semitones to 7 (5th) , 12(octave) is moving from anxiety and stress to joy and serenity.

6) Ascending with larger steps that those of descending indicates favor of joy

7) Accelerating ascending indicates more joy, while decelerating ascending less joy. The converse with descending.

 A single melodic theme has a simple emotional meaning and this is a simple interplay or move inside the duality of emotions (positive-negative emotions).

A melodic  theme-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(-)(x) or An1Bn2(+)(x) 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) and (x)=1,2,3,4 denotes the dominating density of it is chromatic x=1, if it is diatonic x=2, if it is middle harmonic x=3 and high harmonic x=4  (see post 68)  e.g. G5A4(-)(2) . In this way we write the dynamics of he melody as a theme-progression ,much like a chord progression. 

The 4 basic transformations of them are

(we should remark that such transformations may be interpreted not only in one pitch dimension of the scale, but in modern digital musical instruments with 2-dimesional scales layouts (like Musix (see post 12 and post 312) e.g. horizontally by 2nds andvertically by 3rds) we may have 2-dimensional interpretaion of the transformations or melodic themes variations.  In 2-dimensional interprtation except of these 4 we may also have rotations! )

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)). Translations with intervals of 3rd, may applied without changing the underlying chord, or changing it to a relative chord. Translations with intervals of 4th and 5ths, occur when the underlying chords are in resolution-relation that is successive chords on the wheel of 4ths. Translation by one semitone or chromatic translation may happen in the cases where the underlying chords are in resolution relation (successive chords on the wheel of 4ths) and the first is a dominant 7th chord, or when the underlying chords also have roots at  distance of one semitone. 

2) The melodic density change , density contraction or expansion (see post 68 and 78) . Often it is neither expansion neither contraction but rotation in the sense of stationary waving like an harping in a chord.

3) The inversion where the ascending pitch move becomes descending.

4) Rhythm transformation (which may vary)

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 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!

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)

(This post has not been written completely  yet)

63. How to create a melody and 3 more parallel melodies from a chord progression that fits to them. The 66% or 50% in the chord harmonic method

In this post we utilize the way to create a melody , from a chord progression (as in the harmonic method of composition in post 9 ) and then we extend it to create 3 more parallel melodies (in total 4) that fit to the chord progression. In order to create 3 parallel melodies it is best of the chord progression consists from 3-notes chords , while in  order to create 4 parallel melodies it is best of the chord progression consists from 4-notes chords. The key here is the simplicial sub-melody that will be the centers of the melody . We must utilize also  and the types of melody morphology we want to use (as in posts 18,19). Again the guiding principle are the intended plot of emotions, and the relation of the harmony of the chord progression and melody morphology to the emotions, of sadness, joy, serenity anxiety etc. If J. S. Bach was living right now, he would have given us excellent hints and discussions, but he is not living and we have to  figure it out ourselves.

See for Bach fugue and 4 voice harmony e.g. https://www.youtube.com/watch?v=dFDx-L7PcrY

https://www.youtube.com/watch?v=ddbxFi3-UO4

https://www.youtube.com/watch?v=ikYa01tuJPs



IN MY APPROACH IN THIS BOOK I FAVOR MIXTURE OF AN IN ADVANCED COMPOSED MUSIC PIECE AND  A LATER IMPROVISATION OVER IT, RATHER THAN A 100% PRIMA-VISTA IMPROVISATION. THE REASON IS OBVIOUS. THERE ARE ADVANTAGES OF MUSICAL COMPOSITION THAT WILL TAKE MORE TIME THAN THE DURATION OF THE MUSICAL PIECE OVER A DIRECT IMPROVISATIONAL CREATION OF IT AS WE LISTEN TO IT. THE FORMER GIVES US THE OPPORTUNITY OF A BETTER QUALITY MUSICAL CREATION AND A BETTER BALANCE OF THE PREVIOUS TRIANGLE OF MUSICAL MENTAL IMAGES, SOUND FEELINGS AND FINGER ACTIONS WHEN WE IMPROVISE LATER ON THE ALREADY COMPOSED MUSICAL PIECE.

In order to create the melody over a chord progression we may proceed in a simple way as follows.

1)We compose a simplicial sub-melodies one for each part of the song ,  with one note per chord, over the chord progression preferably at a chromatic sequence ascending and descending . 

2) We create moves or waves or cycles for each note of the simplicial sub-melody by sequencing during the chord with two types of notes a fast (usually outside the chord) and a slow of double duration on the notes of the chord again ascending or descending with smaller waves. I this way more than 50% of the time of the duration of the melody will be notes of the chord. 

3) We arrange a continuous sound instrument to play the simplicial sub-melody only and a discrete sound (guitar mandolin etc) to play the full waves melody.

1) An obvious and simple way of course would be to arrange the chords in normal position of the chord progression in sequence with their timing , and conceive each note of the chord (1st 2nd 3rd or also 4th according to the pitch order ) as the note of one of the (1st 2nd 3rd or also 4th) voices or melodies. As these note change from one chord to the next (or remain the same of the successive chords are relatives with common notes) we define faster in time melodic moves (scaling, or waving, or spike or isokratic) that bridge as transient notes a note to its new position in the next chord. (see e.g. https://www.youtube.com/watch?v=tCuxVS3CI3U or https://www.youtube.com/watch?v=LlvUepMa31o  Or see e.g. how one melody could be created parallel to chords where in this example we eliminate the chords and leave only the one melody https://www.youtube.com/watch?v=JhLhsbza1Ic or https://www.youtube.com/watch?v=tCuxVS3CI3U or https://www.youtube.com/watch?v=tCuxVS3CI3U  )  In this way we have 3 (or 4) voices or melodies. The freedom here is the pattern that will move one note to its next. We may even break the duration of each  note of each chord to many times repeating same note and then create fast waving melodies from all these small notes that both make an trill harping of he chord (extrapolated refinement) and the same with the bridge to the next chord.Different variations of doing it create different voices but alraedy harmonized due to the initial chord progression. (See e.g. https://www.youtube.com/watch?v=uoqFH-i7jYY  and https://www.youtube.com/watch?v=T2aR9eq1fzQ or https://www.youtube.com/watch?v=tbWqPnRbq3M&index=1&list=RDtbWqPnRbq3M )  We may chose such patterns that repeat from one voice to another with delay giving the effect of what is known as fugue contrapuntal . In this simple method it is not used the concept of simplicial submelody. We have also some additional degrees of freedom in choosing the chords as inversions not in their normal position, and  also not all in the same octave. Notice that we did not use here the concept of scale. Maybe the chord progression consists of chords that are not all in the same scale. This can be easily checked by placing these chords on the 24-cycle of chords , and see if they are within an arc of 4 major and 3 minor chords.But also if they are not we can find in this way the smallest number of diatonic scales that contains the chords of the chord progression. If a single scale is defined in this way, we obviously prefer to define the bridge melodic moves from one note of the chord to its corresponding next, with steps only from this diatonic scale.  It is not very difficult to modify slightly the 4 voices so most often either only one sounds at each note , or only 2  sound , in an interval of the used scale or move with notes very fast, in which case the ear does not hold harmony, and rarely when all three or four have notes that sound simultaneously , called  the centers of the melody where notes sound more time, then it is always a 3-notes chord or 4-notes chord of the scale. If an initial chord progression is  defined in advanced this is not as easy to design  for the 3 or 4 voices-melodies but becomes easy if we allow fo rthe chord-progression repetitions of the chords and probably extension with more chords (chords butterflying). Of course it may be easier to create at first one melody, then vary it without changing its centers to different melodies with different morphology, and then put them together according to their centers and the chord progression , possibly modifying them slightly. Another simpler idea is to take only the basic theme of the melody, and create many simultaneous  harmonic variations of it (extrapolated refinement) that when played simultaneously create at the centers different chords of the scale and of the chord progression. Do it for all the chords in the progression and label all the pieces according to the chord. Then combine in sequence the theme and its variations according their harmony and the order of the chords in the chord progression, and pass the theme to another voice when the variations of the first voice sound, and continue with variation pieces according the following chords. And the same with all the other voices. When Bach was creating his fugue the music composition based on chords was not popular or well understood, but nowadays we can create easier fugue contrapuntal with chord progressions.  If one is sophisticated he even create the same theme in different time scales (like  a fractal) and thus embellish harmonically  the larger theme with itself in shorter time scale.

2) 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. 
2.1) 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 submelody. This happens for all cases that the two consecutive chords 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. 
2.2) If the two consecutive chords are mutually relative with two common notes, the notes of the simplicial submelody for each chord are the notes that the other chord does not contain! 
The previous rules of minimum distance notes and disjoint notes of relative chords for two consecutive chords of the chord progression, determine at least one simplicial submelody for each chord progression! Then we extend as in 1) the simplicial submelody to the full melody where the original simplicial submelody are centers of the full melody. This means notes that sound more time than the other notes. The rules of the simplicial sub-melody give a  more passionate melody with conflicts and resolutions according to the chord progression. After we defined fine the notes of the simplicial sub-melody, then we  create the full melody by composing bridges between its notes , with other sizes of intervals. 


3) Harmonic simplicial sub-melody. Finally a third and probably best method is based on creating first the simplicial sub-melody in a different way which is based on preffering 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 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. 

A very useful remark for improvisation of melody within a particular chord is the next.
Suppose we are at a note y1 of the melody which fits the underlying chord with notes x1x2x3 (whatever that may mean), then depending on the particular position of y1 relative to the x1x2x3, a shift by an interval of 3rd, 4th, 5th, and 6th wil lead to a note y2 that will again fit the chord!. This is because the relative positions of the notes x1x2x3 of the chord are intervals of major, minor 3rd and pure 5th, and their complementary intervals relative to the octave are minor or major 6th, and pure 4th 
We may connect the previous remark with the next

LATIN JAZZ AND HARMONIC BUTTERFLYING 

This butterflying is very often utilizing intervals of 3rds (3 or 4 semitones) and 4ths (5 semitones) thus it is ascending or descending chords (chord-scales or chord-arpeggios  , that is why it is called harmonic butterflying) and it is thus chord-harping too, but it involves also intervals of 2nd (1 or 2 semitones) which correspond to chord transitions. A hidden simplicity or invariant in this  butterflying is obviously the underlying chord.  This butterflying maybe of  waving type of melodic move but the amplitudes of the waves may be intervals of 3rds (3 or 4 semitones) and 4ths (5 semitones), instead of intervals of 1 or 2 semitones as in eastern folk music butterflying. And it can be of course of non-waving and monotone scaling type of melodic move . Obviously this butterflying prefers changing strings tuned by 4ths, rather than moving along a single string as in the Greek Bouzouki butterflying.

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
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!

Here simple such 3-voices music by Guillaume Dufay (1397-1474)

https://www.youtube.com/watch?v=8wOEBuhhsLQ


Bach fugue and 4 voice harmony

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

We summarize the basic concerns in the melodic improvisation and composition of a voice or melody (similar to the syntax of phrase with subject verb and object etc).

1) Always use a finite set of melody motives , themes or moves. A theme may consists of the basic 4 melodic moves.  (The theme entity for melodies is so important as the chord in harmony. A theme can  be inside a chord or over a chord transition. Conversely any of the 3 types of chord transitions may define a theme for the melody, thus a finite set of themes for a chord progression)
2) Transform these melodic themes or moves  which will be the invariant of their transformations. The simpler the themes the easier the transformations.
3) Close it by returning to the initial theme. 
The ways to transform a theme are at least the next 5 and combinations of them
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 iso-kratic) . We prefer spikes and scaling as  the main  morphological types, while the waving and isokratic as intermediate bridges. 

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)

Here is also a quite standard way to create a melody from a chord progression

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.

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 aver 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) Itis 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. forthe 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 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) THE 2/3-CHORD HARMONIC METHOD 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. 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 traditional 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.

Here is an example of a simple but beautiful melody almost directly generated by  the next chord progression (here by the chord-transitions in pairs of the chord progression)

1) (Am E7 Am) 4 times
2) {GmA7-Dm
G7-C
F-E
E7-Am} 2 times
and then 1),2) 2 times over all
It is the Andes song Cuerdo de la Plata (Silver string).

The way it is composed among other ideas, is that we shift the melodic pattern which is usually at the chromatic and diatonic speed (see post 68) and a simple scaling or wave, till it has as much common notes as possible with the 2-chords desired transition chords-scale (possibly modifying a bit the pattern) and we proceed in this way for all chord-transitions. As the chord-scales are at the middle and high harmonic speed, the total melodic pattern will be in all 4 speeds! At chord-transitions we have the chance to be at chromatic or diatonic speed in the melodic pattern and still mostly within the notes of the chords! In this example it is given equal time to each note in the melodic moves . The requirement here is therefore that while we ascend or descend at diatonic speed (=within a scale) at least 2/3 or even 4/5 of the notes are notes of the underlying chord. This may achieved by interpolating  repeating notes that of course are chosen to belong to the underlying chord. We may start e.g. with an oscillation of the minor 3rd interval of the chord then the major 3rd and end with the pure 5th. But the oscillations can involve also notes outside the chord as long as by repeating notes of the chord, we keep the time  ratios of sounding of them as above. The selection of such moves is easy. E.g. the move from a root of a chord to its 3rd note a 5th away, is a move that belongs to the chord. And if the chord is an R7 it already has 4 from the 7 notes of a diatonic scale. There are moves that belong both to the two chords of a chord transition. E.g. the straight move e,a,e,b,e,c.c,d,e,e belongs both to the Am and E7. Of course we prefer moves that as morphology of ups and downs of pitch they repeat. We may continuously descend the pattern or ascend it or do both, by shifting the chords also among the octaves. The choices of such moves with the previous requirements that we can make are pretty many for the same chord progression. Here the moves start at a note of a chord an end at a note of its transition next chord. Usually start at the middle 2nd note of the chord and end also at the middle 2nd note of the next chord, as they ar the most characteristic of the chords. But also this may happen with the roots or the 4th note of  7th chord. If the chord progression with its chord transitions is beautiful, and the chosen melodic moves have a measure of repetition of morphological pitch moves , then the melody so created will be bound to be beautiful too! 
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  (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 post has not been written yet completely)

62. The role of the relative pitch order of the roots of two chords (position) versus the role of their invesrion voicing

If we define as voicing of a chord as the particular combination, or inversion type of the chord together with possible repetitions of notes (especially in the open chords that have 6 notes, instead of 3 or 4), then the position of a chord is defined by the specification of  what octave  the root of the cord is found and for the guitar fretboard what is its exact fretboard shape and position (see post 23 for a special symbolism for this). 

In particular , the relative positions of two chords e.g. V7-> I (in the roman numbering of a diatonic scale) or X7->X+1 in the symbolism of the wheel of 24 chords (see post 34), is also defined by which root is higher, that of X7 or X+1? Normally for such a dominant->root resolution it is usually intended that the root of X7 is lower that the root of X+1 but in the same octave.  The sound feeling of the succession of such two chords is entirely different if the root of X7 is higher than that of X+1 or vice versa. E.g. The G7->C in particular  G->C as open chords of the guitar have such position that the upper root of G (In the high E-string) is higher that the upper root of the Chord C (on the B-string). While for the D7->G, in particular  D->G , the upper root of D (on the B-string) ,is lower that the upper root of G (on the E-string). 

This distinction becomes even more important if there is parallel melody, which requires that the chord follows it, and that some of the notes of the melody are identical with some of the notes of the chords in  pitch too and not only in name. Then with such a requirement in the melody-chord relation, the melody itself defines if the root of X7 will be higher or lower that the root of X+1, which also may define the type of inversion , shape and position of the chord on the guitar fretboard. 

In the next we shall present some examples, and we shall discuss the role of the relative position of two chords of a chord progression on the guitar fretboard.

Here is a relevant video

https://www.youtube.com/watch?v=hpQfUePH820

(The post has not been written yet completely)