57. Extrapolating major, minor, diminished and augmented triad chords to 4,5,6,7,8 notes scales in improvisation and musical composition

Extrapolating major, minor, diminished and augmented triad chords to 4,5,6,7,8 notes scales  in improvisation and musical composition. This is a very important concept, in other words an enveloping local  sub-scale for each chord which allows when soloing to play notes outside the chord in at most 1/3 of the dime duration, while at 2/3 of the duration notes of the chord.

One of the most well know technique is the extrapolation of minor or major chord to a pentatonic (minor or major) with the same root.

Other well known extrapolations is to take the arpeggio of chord extensions to 4 or 5 notes chords like C->C6, C-C6add9 etc

Discussion of examples


A1) 4-scale 1-3-1
Minor Chord 3-4-5  -----> 3-4-(1-3-1) (tetra-chord 1-3-1)
Major Chord 4-3-5 ------> 4-3-(1-3-1) (tetra-chord 1-3-1)
Augmented Chord 4-4-4  -------> 4-4-(3-1)-----> 4-3-1-3-1 (tetra-chord 1-3-1)

A2) Pentatonic 
Minor chord 3-4-5 ---->     3-4-(3-2) ------> 3-(2-2)-(3-2) pentatonic western
Major Chord 4-3-5 ------>  4-3-(2-3)-------> (2-2)-3-(2-3) western pentatonic
Major Chord 4-3-5 ------>  4-3-(3-2)-------> (2-2)-3-(3-2) dominant pentatonic
Minor chord 3-4-5 ---->     3-4-(2-3) ------> 3-(2-2)-(2-3) dominant pentatonic


Here is a nice video about soul and blue scales  as extrapolation of chords

https://www.youtube.com/watch?v=mjttaiOq-8Q 


And more about blues arpeggio extrapolation of minor and major chords

In fact, a blue arpeggio of a 7nth chord with interval structure 

3-1-3-4-1  (called here blue pentatonic scale of a chord)

may be considered a pentatonic scale corresponding exclusively to that chord!


https://www.youtube.com/watch?v=y-gV5RGJbLo


A3) Diatonic scale
Major Chord 4-3-5----->  (2-2)-(1-2)-(2-2-1) diatonic
Major chord 4-3-5----->   (2-2)-(2-1)-(2-2-1) diatonic
Minor Chord 3-4-5----->  (2-1)-(2-2)-(2-1-2) diatonic
Minor Chord 3-4-5------>(1-2)-(2-2)-(1-2-2) diatonic

Diminished chord 3-3-6 ------>  3-3-3-3 dim7

A4) Melodic minor (Hindu), characteristic 4-subscale 1-2-1
Major Chord 4-3-5------> (2-2)-(1-2)-(1-2-2)  Hindu or melodic minor
Minor Chord 3-4-5------> (2-1)-(2-2)-(2-2-1)  Hindu or melodic minor

(Melodic minor=6,7,1,2,3,4#,5#,6'=(in semitones) (2-1)-(2-2)-(2-2-1))

A5) Melodic double minor (Arabic or leading wholetone or neopolitan major scale see also post 21) characteristic 3-subscale 1-1
Minor Chord 3-4-5------->(1-2)-(2-2)-(2-2-1) Arabic or melodic double minor
Major Chord 4-3-5------->(2-2)-(1-1-1)-(1-2-2) Arabic or melodic double minor

A6) Harmonic Minor ,characteristic 4-subscale 1-3-1
Minor Chord 3-4-5----->  (2-1)- (2-2)-(1-3-1) harmonic minor
Major Chord 4-3-5----->  (3-1)-(2-1)-(2-2-1) harmonic minor

A7) Harmonic double Minor, characteristic two  4-subscales (1-3-1)-(1-3-1)
Major Chord 4-3-5------>  (1-3)-(1-2)-(1-3-1) harmonic double minor
Minor Chord 3-4-5------>  (1-2)-(1-3)-(1-1-3) harmonic double minor

A good concept when creating melody after the choice of a chord (see harmonic method of composition post 9) , is to determine also the extrapolation scale of the chord before the choice of the melodic pattern. Then choose a butterflying pattern, and finally the melody. In other words, determine at first the triple (chord, extrapolated scale, butterflying pattern). The notes of the simplicial sub-melody (see post 9 about the concept of simplicial sub-melody) are  centers and are notes of the chord, while the rest of the melody (the part that sounds parallel to the particular chord) is a butterflying within the extrapolated scale around the centers of the simplicial melody. 


Here is a video showing a practical way to figure out which scales are define by a chord progression. Strictly speaking there is an algorithm that could be programmed in a software program that you give the chord progression and it defines the smallest set of scales that contain these chords. 

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


The easiest algorithm to find the smallest set of scales that have the chords of a chord progression is based on the rule that represents the chords of a diatonic scale on the 24 cycle, which is the next (see post 34)
The chords of a diatonic scale in this 24-cycle are easily defined as the chords of an arc of 3 consecutive major chords together with their 4 relative minors. The root of the diatonic scale is the middle major chord. In the symbolism above it is the next arc of 7 chords  (x-2, X-1,x-1,X,x,X+1,x+1)=(vii,V,iii,I,vi,IV,ii).

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. 

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!

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. 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 characteristic of at least  4-notes sub-scales (tetra-chords) of the above extrapolation scales are the next


Diatonic
2-2-1, 
1-2-2
2-2-2, 
2-1-2
Melodic minor  
1-2-1
Melodic double minor
2-1-1, 1-1-2
Harmonic minor 
1-3-1
Harmonic double minor
(1-3-1)-(1-3-1) , 


And if we restrict to only 4-notes sub-scales (tetra-chords) , having inverse such scales not different, then we are left with a small number of 10 of such characteristic tetra-chords
They are also all such tetra-chords containing intervals of 1,2,3, and where inverses and cyclic permutations of them do not count as different 

Diatonic

2-2-1, (major, natural minor rast, ussak)

2-2-2, (major, augmented)

Melodic minor, double minor (shabach)

1-2-1

Harmonic minor (Hijazz,Huzam)

1-3-1

Harmonic double minor

1-2-3,
 Diminished 

3-3-3 , (diminished 7nth)

3-3-1, 

Pentatonic

3-3-2, 
2-2-3

Chromatic

1-1-1

We shall also show how these extrapolations of the chords can be realized on the guitar fretboard.

For example The  pentatonic scale as extrapolation of major and minor chord in of the shapes D, E, A at the 3-highest strings is the next 

                                                                   D major shape

E major shape

A major shape 

E minor shape

D minor shape

A minor shape 

See also the next video on how to relate chords with their arpeggios and relevant scale.

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

56. The psychology of intervals progressions in scales , melodies and chords

The psychology of the harmony intervals when the notes are sounding simultaneously or very close in time goes back to the ancient Greek Pythagoras and also the studies of the physicist Helmholtz is his studies and remarkable book "The theory of sound" see post 24. Here we will only make some remarks about not the harmony of the intervals but of the melodic dynamics, and only as long as the topic is not covered in the post 18. 

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. 
5) 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. (see post 78)
6) Ascending with larger steps that those of descending indicates favor of joy. E.g. Ascend with intervals of 2nd and major 3rd and descend with 2nds and minor 3rds. 
7) Accelerating ascending indicates more joy, while decelerating ascending less joy. The converse with descending. E.g. Ascend by first intervals of 2nd then 3rd and finally 4th or 5th while descend at first with intervals of 5th or 4th, then 3rds and finally 2nds. 
8) Melodies that move with steps of minor 2nds and minor 3rds mainly may be called minor feeling melodies, while melodies that move with steps of major 2nds and major 3rds mainly may be called major-feeling melodies.

OVERTONES-UNDERTONES AND HARMONICS-SUBHARMONICS 

When we  utilize the undertones or subharmonics the effect of minor sad chord apprears . In other words if a is a fundamental frequency the undertones are the 1/2a  ,1/3 a, 1/4a , 1/5a  etc

In a string of length l giving frequency a the undertones  will be produced by multiplying  the length of  the string from l, to 2l , 3l 4l 5l etc.

Similarly a fretboard of n equal length l of frets will produce the n undertones  of  mini-string of length l (but not oft he whole string of n frets)

WHAT IS VERY INTERESTING IS THAT THE INITIAL MAJOR CHORD IN OVERTONES HAS A CORRESPONDING MINOR CHORD OF UNDERTONES!

This is also significant in understanding the sad emotion correlated with the minor chord as it is by contraction and lowering of a fundamental frequency compared to expansion and raising of fundamental frequency by overtones which gives the major chord. 

HARMONIC POLES AND HARMONIC WAVING IN THE CHORD PROGRESSION AND STRUCTURE OF THEMES IN THE MELODY BASED ON THE  EMOTION OF THE HARMONY OF THE INTERVALS:

The emotional parallel of the harmony of intervals, suggests that the chord progression and harmonic structure of the themes of the melody, waves (Harmonic waves) between the emotion of stress,intervals of 1 or 2 semitones to the pole of harmonic serenity ,intervals of 5 (4th) , 7 (5th) or 12 (octave) semitones.

We remind that in the post 30 we described the three relations of chord transitions corresponding to the intervals of 2nd, 3rd and 4th or 5th. 

1) Resolution relation of chords

MELODIC MEANING : When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 5 (4th) or 7 semitones (5th) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).

2) Relative chords

MELODIC MEANING: When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 3 (minor 3rd) or 4 semitones (major 3rd) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).


MELODIC MEANING: When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 3 (minor 3rd) or 4 semitones (major 3rd) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).


3) Complementary transition of chords


MELODIC MEANING: When such a pair of chords accompanies a melody then taking the themes of this melody either the theme is inside the chords is or is a bridge theme relating the two chords. This means that either the interval of 1 (minor 2nd) or 2 semitones (major 2nd) appears as a shift or translation of the theme (chord-theme) or exists inside the theme (bridge-theme).

Much of the remarkable art of scale butterflying improvisation (see post 55) and in general of melodies and their pitch dynamic patterns (see post  18 ) is that  it can create the emotions of sadness, joy etc not through harmonic simultaneous sounding of notes, and common harmonics, but through progressions of intervals. 

As we have remarked small interval like 1 semitone is closer to sadness than longer intervals like 1 tone etc.

But what is more effective is the acceleration or deceleration in the movement of the pitch.

Ascending pitch is of course correlated to joy and descanting to sadness. But the acceleration amplifies it.

So an ascending sequence of tones (measured in semitones)  2-2-2 will become even more joyful if it accelerates. That is like this 2-3-5. And a descending sequence of notes will become even more sad if it accelerates 2-2-2 becoming 2-3-5. An ascending sequence of notes will become more sad if it decelerates.

So an ascending 3-2-1 is more sad than the ascending 2-2-2. And when descending the deceleration reduces sadness. So the descending 2-2-2 is more sad than the descending 3-2-1.

See also post 59 for the basic melodic moves spike, scaling, waving, isokratic waving.

1) up/down spike (=large interval of 5th or larger in one or very few steps, extreme emotional effect, excitement , emotional intensity)

2) up/down waving (also called in this blog butterflying, emotional effect of playing either sad of with joy, emotional complication and ambiguity)

3) up/down scaling (direct ascending or descending of notes in a monotone order without waving, emotional effect straight usually with simplicity, emotional clarity)

4) Iso-kratic waving (=horizontal waving with repeating same note, peculiar emotional effect of internal symmetry , and emotional stability )

From these 4 patterns the 1) and 3) are simple and with emotional clarity. The 2) and 4) are with emotional complication and ambiguity.

Another classification would make them 5!

1) Straight scaling up or down (including spikes) in one or more of the melodic speeds (straight sadness or joy). Here the notes of the simplicial submelody are the starting and ending notes.

2) Ascending or descending waving (complex sadness or joy). Here the notes of the simplicial submelody are the starting and ending notes.

3) Flat equilibrium waving (serenity and equilibrium emotion).Here the notes of the simplicial submelody are the upper level and lower level ofthe flat channel.

4) Flat diminishing waving (serenity and diminishing emotions). Here the notes of the simplicial submelody are the starting upper or lower level and h ending note of the diminishing channel

5) Flat expanding waving resolving up or down  (serenity emotions exploding to either sadness or joy). Here the notes of the simplicial submelody are the starting note and the ending note at the upper or lower level of the expanding channel.

MORE PITCH DYNAMICS AND THEIR PSYCHOLOGY

We may create more complex pitch dynamic patterns than these basic with recognizable psychological meaning. Eg. a melodic theme that has a spike up but then falls back to the same pitch level corresponds ton an emotion of "complaining" or "crying" or angry protest that turns to  sadness"

In general when the melody is ascending through repetitive descending melodic themes (resisting ascension), or is descending through ascending repetitive melodic themes (resisting descending) , the emotion (either joyful or sad) is more dark emotion compared to than when the  the melody is ascending through repetitive ascending melodic themes (irresistible ascension) , or is descending through descending repetitive melodic themes (irresistible descending) , the emotion (either joyful or sad) in which latter case it is an emotion more straight and transparent.

Natural major and minor explained with the psychology of the progression of intervals

This is somehow the reason why the mode of the diatonic scale 2-1-2-2-1-2 is called natural minor while the cyclic permutation of it 2-2-1-2-2-2 is called natural major . The former has twice deceleration in ascending 2-1 at the beginning , which is preemptive in the perception with a bias to the deceleration thus sadness,  and 2-1 towards the end, while the latter starts with non-deceleration which is again preemptive in the perception towards balance rather than sadness and has only one deceleration 2-1 in the middle!

This explains also why the tetra-chord 1-3-1 of the Harmonic minor and Romani double minor sounds so sad.
They accelerate at the beginning 1-3 as ascending but then suddenly  they decelerate fast  by 3-1.
The same with the melodic minor 1-2-1.

Also the same principle explains why a minor chord sounds sad, while a major sounds happy. The human harmonic  perception starts its analysis of the chord from its highest pitch, that in the normal position is called also dominant note , not without a reason! So for example the chord C major c-e-g , is analyzed by the brain, starting from g, then e, then c, and here the intervals are in a sequence which accelerates, from 3 semitones (minor 3rd) to 4 semitones (major 3rd) thus it sounds with joy. While the A minor a-c-e, is analyzed by the brain from e then c then a, and its a deceleration , from major 3rd interval of 4 semitones to a minor 3rd interval of 3 semitones that decelerating, therefore it induces sadness.
Actually there is another factor , probably more robust that explains the different feeling of the sounds of a major and minor chord.
One may ask,  a major triad sounds more harmonious and joyful than a minor triad which sounds more sad? The answer is that the same harmonic score (see post 40 ) of the minor 3rd interval (3 semitones) sounds less harmonious when it is in lower frequencies, than when it is in higher frequencies. And this explains the different feeling of the major and minor chord, although they have the same harmonic score! 

Another more simple explanation is the harmonics of the root. For a major triad we need only 5-harmonics to derive all the 3 roots, while for the minor triad, we need 19 harmonics to derive the roots!

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

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

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

C---1 harmonic

D---9

E---5

F---11

G---3

A---27

B---15

C---2

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

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

Another aspect of the psychology of the intervals progression in melodies is of course the pitch dynamics waving pattern (butterflying) which is described in detail in post 18.
The psychology of such waving patterns have also been studied in the psychology of the speech of married couples by the research group of the professor Gottman  https://www.gottman.com/   http://www.amazon.com/The-Mathematics-Marriage-Nonlinear-Bradford/dp/0262572303 .

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 must emphasize a basic philosophy  here that when playing the musical instrument, we do not only produce musical sounds but also independently emotions in us. And I do not mean that the musical sounds produce the emotions I mean that our self is producing the emotions parallel and in fitness with the musical sounds

55. Scale-butterflying improvisation versus scale of chords improvisation

This post is to be read after posts 18-19 and 11, and also after post 83.

The term "versus" in the title is just a way of speaking, since we already know that scale means also a set of chords and progressions of chords. But the converse is not true. In other words scale (tonality) is a special type of chord progressions. Nevertheless such types of chord progressions (chords of a  scale) are handled easier together with melody improvisation, by just remembering the cyclic interval structure of the scale rather than the chords of the scale. On the other hand chord-progressions have the advantage that may involve really many different scales that someone does not need to know by heart (see e.g. posts 51,52). For each chord one needs to know only the chord-scale or arpeggio, and create his own scale with notes of the chord-scale that last 2/3 of the time and any other note outside the chord scale that lasts 1/3 of the total time. This would give plenty many different scales, for different improvisations on the same chord-progression

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.

Scale butterflying improvisation. Here the chord progression is not strictly determined in advance , but the main invariant is a scale. Then butterflying is applied inside the scale, with skew-ascending or descending channels, or horizontal channels around a chord-area of the scale, or spikes. This is most often the improvisation by Bouzouki in Greek folk music. A guitarist following the musician who improvises may try to find and sound in the background the appropriate chords bu this is quite unpredictable, as the main invariant of the improvisation is not a chord-progression but a scale, and the butterflying pattern (which is the main concern of the instrument improviser) Scale-butterflying can become so rich and abundant in complexity and variations while at the same time  with very simple and strong invariants like a) the scale b) the butterflying patterns that by itself can substitute the concept of melody and chord progressions in the background. E.g. Very complex and rich scale-butterflying can be conducted with one only chord in the background. 

Experienced jazz guitarists claim that , as in usual chord progression playing, the left hand holds for some time a chord, in scale-butterflying  the improviser holds for sufficient long  time  in his imagination a whole scale along  all the fretboard , and butterfly in it. Then they may change the scale (modulation) and hold another scale on the whole of the fretboard for sufficient long time, butterflying again in it 

In the scale butterflying , if it is from intervasl of 1 or 2 semitones , it seems easier to conduct it by moving the left hand along the length of one or two strings rather than moving vertically to the strings among the 6-strings.   In the guitar fretboard, since at the 9th and 10th fret the pitch is identical with that of two strings higher, the movement along a single string can be repeated usually three times on three strings where at the end of the one string we go one with the other end of the next string (e.g. 1st 3rd 5th or 2nd, 4th, 6th) covering 2 or 3 octaves.

See e.g. https://www.youtube.com/watch?v=vvCwj8NWclc

The smaller a musical interval within an octave the more it signifies sadness, and the more distant, (with maximum a 4th or 5th, as we consider also their inversions) the closer it is to joy.

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. 

I believe the best idea is combining  the scale-butterflying improvisation with the chord progression improvisation. We must be utilizing the chord progression as rules of transformation of the theme.

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. 

Some instructive remarks in the composition of the melody based on the chord progression

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 main parameters of a "butterflying" or "oscillation"  or "waving" embellishments  of melodies, that go from one part of the scale or one octave to  a next , are at least the next 8. 

1) Number of ascending notes (up move)
2) Number of descending notes (down move)
3) Retrace order (which defines the speed of  ascending or descending of the waving in the scale. e.g. 1234-2345-3456 etc or 1234-3456-5678 etc)
4) Repeating notes pattern or trill pattern  (e.g. if in ascending or descending move one is repeated twice or more times or two notes alternate as trill).
5) The pattern of two or more notes sound simultaneously among the other that sound repeatedly  
6) Rhythmic pattern (if the faster part is at the end or at the beginning of the up or down move)
7) The interval pattern of the up or down move (if it is 2nd or 3rd or 4th and higher  intervals). Or the scale in which  the butterflying is conducted. 
8) The 2-level butterflying pattern or larger  pattern of up and down moves e.g. (up , up down) (up , up down) , (up down down) ,(up , up down), (up , up down) , (up down down) (common in the Greek folk Buzuki layer and composer Zambetas ) etc . And the larger pattern of repeating and alternating whole butterfly patterns. 

As it is understood the number of combinations of  these parameters and creation of different ways of butterflying is very big. 

From the point of view of the pitch-shape of the butterflying or waving we may have the next classifications

HORIZONTAL

1) Standing horizontal channel of constant width  (Balance)

2) Standing horizontal channel of diminishing  width (Contraction, expiration with little sadness)

3) Standing horizontal channel of increasing  width  (Expansion. initiation with joy)

In 2) and 3)  one of the sides of the channel maybe horizontal which is an isocratic pattern

SKEW 

4) Ascending  waving in channel of constant width  (Joy)

5) Descending waving in channel of constant width  (Sadness)

6) Ascending  sequence   (Joy)

7) Descending sequence   (Sadness)

ALMOST VERTICAL 

6) Up spike (extreme joy)

7) Down spike (extreme sadness)

We may simplify the morphology of the melody moves to the next 4 (see also post 59)

1) up/down spike (=large interval of 5th or larger in one or very few steps, extreme emotional effect, excitement , emotional intensity)

2) up/down waving (also called in this blog butterflying, emotional effect of playing either sad of with joy, emotional complication and ambiguity)

3) up/down scaling or monotone ascending/descending  (direct ascending or descending of notes in a monotone order without waving, emotional effect straight usually with simplicity, emotional clarity)

4) Iso-kratic waving (=horizontal waving with repeating same note, peculiar emotional effect of internal symmetry , and emotional stability )

From these 4 patterns the 1) and 3) are simple and with emotional clarity. The 2) and 4) are with emotional complication and ambiguity.

From the 4 basic patterns only the up/down scaling is in the context of calm and long term stable emotionalism. The other patterns are in the context of  complicated and fast changing emotionalism.

Of course if we want more elementary classification of the pitch moves (like not analyzing substances to their chemical type but resort t the 3 elementary particles of electron , neutron and proton), then there are only 3-types  a) the (non-waving) up move of pitch (upward vector) , b) the (non-waving) down move of pitch (downward vector) c) and the sustained sounding of a note (horizontal vector)

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. 

We summarize the basic concerns in the melodic improvisation 

(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. 

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 

LATIN JAZZ AND HARMONIC BUTTERFLYING 

This butterflying is very often utilizing intervals of 3rds (3 or 4 semitones) and 4ths (5 semitones) and their complementary (6th,  8 and 9 semitones and 5th, 7 semitones when changing octave too),   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. If we move by an interval of 3rd lower to the root
 (inside the scale) of a 3-note chord of the scale, we get a 4-note chord with 6th, which extends the initial 3-notes chord by its lower relative, and we get sounds as in Romani (Gypsy) Jazz.





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.

There is  small number of  exactly 12 characteristic tetra-chords (=4-notes sub-scales)  containing intervals of 1,2,3, semitones and where inverses and cyclic permutations of them do not count as different 

Diatonic

2-2-1, (major)

2-2-2, (major, augmented)

2-1-2 (natural minor)

Melodic minor  

1-2-1

Melodic double minor

2-1-1,

Harmonic minor 

1-3-1

Harmonic double minor

1-2-3,

 Diminished 

3-3-3 , (diminished 7nth)

3-3-1, 

Pentatonic

3-3-2, 
2-2-3

Chromatic

1-1-1

We should be also familiar with the ways we can play them in 1 , 2 or 3 strings.

There is also the CHORD BUTTERFLYING, which is to start with a chord progression, and wave some parts of the progression. This is very common to the songs.