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. 

54. The also Western major and minor pentatonic scales and the Chinese pentatonic scale

The western major and minor scales (they are the same scale but different modes or cyclic permutations), historically is known also as the Egyptian /Mongolian scale)

The  1st mode of the pentatonic scale is the major mode 2-2-3-2-3

The Egyptian  mode of the pentatonic scale (2 3 2 3 2 ) which is the 2nd mode of the pemtatonic scale, the Mongolian or standard 1st or major mode (2 2 3 2 3)  , The American Indian Pentatonic scale and the minor Pentatonic (3 2 2 3 2)  which is the 5th mode of the pentatonic scale and it exists in the Shakuhachi flutes and Native American flutes    are different cyclic permutations or modes of the same scale. There is also a  northern American Indian scale (3 2 2 2 3) which is not a mode of the standard pentatonic scale, but a permutation of its intervals.

It seems nevertheless that the Indian music as well as the folk Irish music in the Celtic harp use often also the next mode of the pentatonic 2-3-2-2-3 E.g. in C major scale it would be C-D-F-G-A-C, (because this is easy to tune by levers in a C major scale tuned Celtic harp). We might call it Celtic mode of the pentatonic.

Finally there is also the next mode of the pentatonic : 3-2-3-2-2 , which It does not happen to know if it has a name, but it could be called 2nd minor mode of the pentatonic as it starts with the 3 semitones as the minor mode

The minor mode of the pentatonic can be obtained by starting the scale at  the 5th step
Conversely to obtain the major mode of the pentatonic we must start the scale at the 2nd step.
E.g. The C major pentatonic C-D-E-G-A-C has as minor mode pentatonic the relative scale 
A-C-D-E-G

The Egyptian mode of the pentatonic is if we start the major mode pentatonic scale at the 2nd step.  E.g. from the C-D-E-G-A-C we get the D-E-G-A-C-D. And from the Egyptian mode to obtain the major mode we must start from the 5th step.

The Celtic mode of the pentatonic is if we start the major mode of the pentatonic at the 4th step
E.g. from the  C-D-E-G-A-C we get the G-A-C-D-E-G. While from the Celtic mode of the pentatonic to obtain the major mode we must start from the 3rd step.

Finally the 2nd minor mode of the pentatonic can be obtained from the major mode by starting at the 3rd step. E.g. from the   C-D-E-G-A-C we get the E-G-A-C-D-E. Conversely to obtain the major mode of the pentatonic from the 2nd minor mode we must start at the 4th step.

The way to get the standard (Mongolian) major mode of the pentatonic scale 2-2-3-2-3 from a 7-notes diatonic scale in major mode is to  eliminate the 2nd note of the 1st semitone interval and the 1st note of the 2nd semitone interval . 

The way to get the Celtic mode of the pentatonic scale 2-3-2-2-3 from a 7-notes diatonic scale in major mode is to  eliminate the 1st notes or the 2nd notes of the two semitone intervals.

The way to get the minor mode of the pentatonic scale 3-2-2-3-2 from a 7-notes diatonic scale in major mode is to  eliminate the 1st notes  of the two semitone intervals and shift one step forward.

A c major pentatonic scale can be considered also as the C major chord with 6th and 2nd notes

 C major pentatonic

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

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

1, 9/8 , 5/4 , 3/2 , 7/4. 



Similarly the 7-notes diatonic scale

C---1 harmonic

D---9

E---5

F---11

G---3

A---7

B---15

C---2

Notes of this scale:

C; D; E; G; A; C;

Interval structure of this scale:

2-2-3-2-3

Chords that fit in this scale:

Normal Triads: we may start with the G power chord=(G,D,G) and also the  C     Am  Other Triads: Csus2     Dsus4     Dsus2     Gsus4     Gsus2     Asus4  4 Notes Chords: C6     D7sus4     D7sus2     Am7     A7sus4     C\E     C\G

We notice that the minor pentatonic is actually a mode of the major pentatonic, that is a cyclic permutation of it. 

Also the Egyptian scale is a mode of this scale with semitone patterns 

2-3-2-3-2 (e.g. D, E, G, A, C, D)

If we include only minor major and R5 chords the chords of the pentatonic would be

C, D5, G5, Am 

Here is a video of the use of pentatonic scale as soloing around any chord!
(actually, the pentatonic scale can be considered the arpeggio of the chord with 2nd and 6th notes too or 6th and added 9th C6add9 )


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

 C minor pentatonic

Notes of this scale:

C; D#/Eb; F; G; A#/Bb; C;

Interval structure of this scale:

3-2-2-3-2

Chords that fit in this scale:

Normal Triads: Cm     D#  Other Triads: Csus4     D#sus2     Fsus4     Fsus2     A#sus4     A#sus2  4 Notes Chords: Cm7     C7sus4     D#6     F7sus4     F7sus2     Csus4\F     Csus4\G     Cm\D#     Cm\G     Cm\Eb

Again if we include only minor major and R5 chords the chords of the minor pentatonic starting from C would be

Cm, D#, F5, A#5 

Some standard ways to play the pentatonic scale , where the rule here is 2 notes per string, are the next :

 A minor pentatonic

Notes of this scale:

A C D E G

Interval structure of this scale:

(W+h) W W (W+h) W (W: Whole tone, h: half tone)

Scale structure:

1 b3 4 5 b7

Chords that fit in this scale:

Normal Triads: C    Am Other Triads: Csus2    Dsus4    Dsus2    Gsus4    Gsus2    Asus4 4 Notes Chords: C6    D7sus4    D7sus2    Am7    A7sus4    C\E    C\G 5 and 6 Note Chords: C6/9    D9sus4    D9sus2    Am11    C6\G    C6/9\E    C6/9\G

Scales Equivalent to A minor pentatonic :

C major pentatonic;

Scales wich notes are within A minor pentatonic:

B minor pentatonic

Notes of this scale:

B D E F# A

Interval structure of this scale:

(W+h) W W (W+h) W (W: Whole tone, h: half tone)

Scale structure:

1 b3 4 5 b7

Chords that fit in this scale:

Normal Triads: D    Bm Other Triads: Dsus2    Esus4    Esus2    Asus4    Asus2    Bsus4 4 Notes Chords: D6    E7sus4    E7sus2    Bm7    B7sus4 5 and 6 Note Chords: D6/9    E9sus4    E9sus2    Bm11

Scales Equivalent to B minor pentatonic : D major pentatonic;

D minor pentatonic

Notes of this scale:

D F G A C

Interval structure of this scale:

(W+h) W W (W+h) W (W: Whole tone, h: half tone)

Scale structure:

1 b3 4 5 b7

Chords that fit in this scale:

Normal Triads: Dm    F Other Triads: Csus4    Csus2    Dsus4    Fsus2    Gsus4    Gsus2 4 Notes Chords: Dm7    D7sus4    F6    G7sus4    G7sus2    Csus4\F    Csus4\G 5 and 6 Note Chords: Dm11    F6/9    G9sus4    G9sus2

Scales Equivalent to D minor pentatonic :

F major pentatonic;

And the same patterns are for the minor pentatonic as it is a cyclic permutation (mode) of the major pentatonic. Among them the pattern 4 seems attractive due to its simplicity 

BUT it is interesting to try a non-cyclic permutation of the western pentatonic scale like 

2-2-2-3-3   !! This is called dominant pentatonic=2-2-3-3-2

E.G.  C-D-E-F#-A-C

It is also called Northern America Indian Scale

http://www.flutopedia.com/scale_Northern.htm

We may try to find all 5-notes scales containing semitones, tones and 3-tones. 

It seems that there are exist only these two!

1) Western/Mongolic pentatonic 2-2-3-2-3 (self-inverse)

2) Dominant western pentatonic 2-2-2-3-3 (self-inverse)



Notice also that an extrapolation of the western pentatonic  are the 7-notes and 8-notes scales that contain twice the tetra-chord 1-3-1, 1-3-1 separately 

2-(1-1)-3-(1-1)-3  (Romani or Harmonic double minor )

(1-1)-(1-1)-3-(1-1)-3

This also shows a way of chord progressions, where one starts with the diatonic scale, 2-2-1-2-2-2-1, then interpolates (simplifies) to the pentatonic 2-2-3-2-3 then extrapolates to the harmonic double minor 2-1-1-3-1-1-3 and then backwards again. 






We may make a  classification  also of 4-notes scales  made from intervals of semitone 1, tone 2, 3-semitones 3 and 2-tones 4. Such 4-scales are essentially chords arpeggios of 4-notes chords  (see post 28) There are 7 of them

1) 4-3-3-2
2) 4-3-4-1
3) 3-4-3-2
4) 3-3-3-3
5) 3-3-4-2
6) 3-4-4-1
7) 4-4-3-1


In addition if in these arpeggios of the 4-notes chords, by doubling the lowest note one octave higher make 5-notes pentatonic scales. And if the  intervals of 4 are extrapolated to 2-2 or 1-3, 3-1, and intervals of 3 to 1-2 , 2-1, then are shaped 6-notes  scales,in soloing.


A celebrated example of 6-notes scale derived from the major pentatonic scale C; D; E; G; A; C;  (interval structure 2-2-3-2-3 )is the 
MAJOR BLUES SCALE  (ALSO CALLED COUNTRY SCALE) C; D; Eb;E; G; A; C  with interval structure 2-1-1-3-2-3.  

The major blues scale e.g. D major Blues scale might be supposed to be used for improvisations of 12-bars blues on  the D major scale, as it can also be also derived from the major D pentatonic by adding   a  blue note 3b which will make it 6-notes scale. E.g. the blues major D scale D, E, F, Gb ,A, B , D, whould play over the chords Dmajor D-F#-A,  (D-Gb-A),  A major A-C#-E, or A major with 6th A-C#-E-Gb and G major G-B-D. Obviously some notes  would creare a dissonance, but it may be considered part of the chromaticity of  the Blues major scale improvisation. Similarly the D major blues could be used to improvise over the C major and G major 12-bars blues.


Notice that the 3-notes 2-1-1 in the major blues scale exist also in  the double minor scale or Arabic scale. Also notice that the blue note here the  D#=Eb  is also the 2nd "blue" note of the double harmonic A minor scale which has 2 blue notes the d# =3b=2# and the g#=5#  . This suggest also the 7-notes scale C D# E F G A B C with interval structure 3-1-1-2-2-2-1 which is nothing else than the Neapolitan scale with harmonic personality 1M 2#m 3m 4M 5M 6m 7M 1M! 

Furthermore we may notice that the 1-1-3 tetrachord of the blues scale is the chromatic tetrachord of Ancient Greek music, that exists also in the parachromatic Byzantine scale (see post 235 ) which was played in the Emperor's palace , but not in the church (In the church was played the harmonic minor and double harmonic minor scale).

There are two other ways to extend the major blues scale 2-1-1-3-2-3 to a 7-notes scale and come from breaking the 3nd 3 to 2-1 or 1-2, giving

2-1-1-3-2-1-2 or  2-1-1-3-2-2-1 . Both sound a bit softer than the Byzantine parachromatic (see post 235 ) and have been classified in post 21. It can be conceived as combination of 3 notes of the  7-notes melodic double minor scale with interval structure (2-1-1-2-2-2-2) , with the major pentatonic scale to give   the 6-notes  2-1-1-3-2-3 ). Other refinement of the pentatonic scale to a 6-tone scale is the next: 2-2-3-1-1-3 !  E.g.
 C-D-E-G-G#-A-C This scale like the standard major blue scale is a minimal chromatic variation of the pentatonic scale.It can be called DOUBLE MINOR BLUES SCALE




On the other hand if we add to the major pentatonic scale  C D E G A C the 1st blue note of the harmonic minor that is the g#, it becomes the next 6-notes scale

C D E G G# A C with interval structure 2-2-3-1-1-3 .

If on the other hand we add the G#, but substitute the G with F it becomes the 

C D E F G# A C with interval structure 2-2-1-3-1-3 which is an Erik Satie-like scale as it is a slight variation of the Erik  Satie 6-notes scale (see also post 21, the Erik Satie scale has structure 2-1-3-1-2-3  or e.g.  D E F G# A B D ) 

Similarly, the MINOR BLUES SCALE is derived from the minor pentatonic scale
C; D#/Eb; F; G; A#/Bb; C;,(interval structure 3-2-2-3-2  ) to give the 6-notes scale C; Eb; F; F# G; Bb; C; with interval structure  3-2-1-1-3-2, which is cyclic permutation of the major blue scale. (notice that the 3-notes 2-1-1 exist also in the double minor scale or Arabic scale. That is why it can be conceived as combination of 3 notes of the  7-notes melodic double minor scale with interval structure (2-1-1-2-2-2-2) , with the minor pentatonic scale to give the  the 6-notes   3-2-1-1-3-2).`

We may compare these 6-tone scales with the raised Celtic minor scale
2-3-2-2-1-2, which is used in the hang drums. e.g. C-D-F-G-A-Bb-C

It is a mode of the maximal harmonic 6-notes scale (see post 117)


Here is a nice video about soul and blue scales

https://www.youtube.com/watch?v=mjttaiOq-8Q
Here is more about the C minor blue scale

In fact, a blue arpeggio of a 7nth chord with the 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


Other such derived 6-notes scales of blue arpeggios over 2 chords of the triple min-maj-min chord is the  3-1-2-1-3-2 
And from  blue arpeggios over 2 chords of the triple maj-min-maj chord is the  3-1-3-1-3-1  

Musical Scale Info: C minor blues

Notes of this scale:

C; D#/Eb; F; F#/Gb; G; A#/Bb; C;

Interval structure of this scale:

(W+h) W h h (W+h) W  or 3-2-1-1-3-2

Chords that fit in this scale:

Normal Triads: Cm     Cdim     D#     D#m  Other Triads: Csus4     D#sus2     Fsus4     Fsus2     A#sus4     A#sus2  4 Notes Chords: Cm7     Cm7b5     C7sus4     D#6     D#m6     F7sus4     F7sus2     Csus4\F     Csus4\G     Cm\D#    Cm\G     Cm\Eb  5 and 6 Note Chords: Cm11     D#6/9     F9sus4     F9sus2     Cm7\D#     C7sus4\G     Cm7\Eb

Scales Equivalent to C blues :

;

Scales whose notes are within C blues:

D#/Eb major pentatonic; C minor pentatonic;

Still another interesting 6-notes scale which has less minor chords and is an alternative extension of the major pentatonic which could be called half-pentatonic is this 

2-1-2-3-2-2  (half diatonic +half pentatonic)

Also still alternative 7-notes extension of the pentatonic and the major  Blue too, is the 

1-1-1-1-3-2-3  (Chromatic+pentatonic, which is the no 13,  7-notes scale of post 52)

Or the 7-notes extension of the major blue 

2-1-1-3-1-1-3  (which is the no 4  7-notes ,inverse Persian scale or Purvi Theta scale of post 54)



Or classify all 4-notes scales that contain steps by 2,3, 5 semitones, where again cyclic permutations and inverses do not count as different. There are only two of them! 

1) 3-2-5-2   e.g. B-D-E-A-B 
which is also a chord arpeggio  of Asus2sus4 or  E7sus4 or D6add9(no 3rd)
2)  3-5-2-2  e.g. B-D-G-A-B
which is also a chord arpeggio of Bm7#5  or Gadd9

Such scales as the two above , sometimes are used by Bobby  McFerrin

See e.g. https://www.youtube.com/watch?v=BmXsA-siNvo

It is easy to realize that both  are interpolations of the diatonic scale

3-2-5-2   -->  (1-2)-2-(1-2-2)-2

3-5-2-2 -->  (1-2)-(2-1-2)-2-2


In general, any 4-note chord, by doubling  the lowest note one octave higher makes  5-notes pentatonic scale. E.g. the chord R7b5 makes the 4-notes scale (arpeggio) 
4-2-4-2 . 

We may also try to find all 6-notes scales containing 3 3-semitones. There seem to be 4 of them 

1) 3-3-3-1-1-1 (self-inverse)

2) 3-3-1-3-1-1  and its inverse

3) 1-1-3-1-3-3

4) 3-1-3-1-3-1 (self-inverse)

The next 6-tone scale is an interpolation of the diatonic, and is used in Jazz soloing

1-2-2-2-1-4

Or compare them with other 4-notes scales like 3-3-3-3 and


6-tonic scales can be derived from the Harmonic minor with the known interval  structure   
2-2-1-3-1-2-1

the 

2-2-1-3-1-3 which is by adding to the major chord the 2nd, the 4th, the 5# and the 6th

or also

2-3-3-1-3.

Pentatonic scales can  be derived  also from the harmonic minor with the known interval  structure   2-2-1-3-1-2-1

Then we may derive the two pentatonic scales

4-1-3-1-3 

or the 

2-2-4-1-3 

From which we may derive the next two 4-notes scales

2-2-4-4

and 

5-3-1-3



Or the Chinese-Japanese  5-notes scale    that have 2-tones steps , in other words interval structure 
Interval from the root : 1, 3, 4#, 5, 7
Intervals in steps: 4 - 2 - 1 - 4 - 1  (the tetra-chord 1-4-1 is used instead of the oriental 
1-3-1) It is called also the AKEBONO SCALE  (some say that it is also the indian scale Raga Katyayani ) (Usually as A B C E F A) It can be obtained e.g. by the major pentatonic scale C D E G A C by making flat the 3rd E, and the 5th A (or 6th A in the diatonic scale). Some call it the double minor pentatonic.

(listen to it here https://www.youtube.com/watch?v=gsBbWXD1NNI  or 
here https://www.youtube.com/watch?v=aSQg-ff5exE
https://www.youtube.com/watch?v=GcSDv7Bfabw
)
Formula: Quadra-step, Whole, Half, Quadra-step, Half
See e.g.
http://www.pianoscales.org/chinese.html
C: C, E, F#, G, B, C  (we notice that it is a sub-scale of the F-major 7-notes scale)


For example if we take the mode of the diatonic scale starting from F

F G A B C D E F , a Chinese scale  will be shaped by keeping  the semitones B C,, E, F, but eliminating the notes G, D , Thus it will be

F, A, B, C, E, 

This scale in an inverse mode can also be obtained from the 

C double harmonic minor  (2nd minor Byzantine) scale  

C Db E F G Ab B C by eliminating the notes with flats, that is the Db and Ab obtainingt hsu the  pentatonic scale

C E F G B C with interval structure 4-1-2-4-1

In the Shakuhachi flutes in D4 pentatonic minor or D4 Native American flute this scale can be payed by substituting the 1st and 4th hole with its 2/3 open version

An inverse mode of the AKEBONO scale called also Pelog scale 

In other words both the western and Chinese pentatonic scales can be shaped by eliminating two notes (interpolating) of the diatonic 7-notes scale

We notice that there is a complementarity  with the western pentatonic , in the sense that the western pentatonic skips the 1-semitone intervals of the 7-note diatonic while the chinese , keeps them and skips 2  in between notes.

`
The inverse order of the Chinese pentatonic is called Pelog scale in other words the 

Pelog scale=(1-2-4-1-4)

It is interesting also to try non-cyclic permutations of the Chinese pentatonic scale like

4-2-4-1-1

and  the inverse 1-1-4-2-4 

There is also the japanese pentatonic scale Insen  with interval structure

1-4-2-3-2


All 5-notes scales containing the intervals 1,2,4 are the next (not including their cyclic permutations, that is their modes)

1) 4-1-4-2-1
2) 1-2-4-1-4
3) 4-2-4-1-1
4) 1-1-4-2-4
5)  4-4-1-1-2
6) 2-1-1-4-4
7) 4-4-1-2-1

Here are all 5-notes scales containing 4, 1, 3:

1) 1-3-1-4-3
2) 3-1-3-4-1 (blue pentatonic arpeggio of major 7th chord)
3) 1-3-4-2-2
4) 1-4-3-2-2
5) 3-1-4-2-2
6)  3-4-1-2-2 (this scale is the join arpeggio of two major chhords that are succesive in the wheel by 4ths e.g. G-C, see also post 92)
7) 4-3-1-2-2
8) 4-1-3-2-2  
9) 4-1-2-3-2 
10) 1-4-2-3-2 (mode of the Japanese Insen scale)
11) 1-3-1-3-4

The next pentatonic does not contain the 4 but the 5 (interval of 4th)

10) 1-3-1-2-5 E.g.  (E F G# A B E)

I have found improvisation solos in Gypsy Jazz climbing an octave by the steps of the previous scales (1-1-4-2-4)! (e.g Bossa Dorado song) which is of course an interpolation to 5-notes scale of the Arabic 7-notes scale (1-1-2-2-2-2-2).


In jazz often 5-notes (pentatonic) scales are shaped by extrapolating the 4-notes arpeggios of a 4-notes chords (as those in post 28). E.g. an interval of 3 is extrapolated to 2-1, and an interval of 4 to 2-2 or 3-1.
Or combinations of the tetra-chords 1-3-1 and 1-4-1 like 6-notes scales 

1-3-1-4-1-2

The 4-notes sub-scale or tetra-chord
1-1-4 
was called in ancient Greece Enharmonio (actually instead of 1 semitone it was only 1/4 of the semitone)