79. THE MELODIC-HARMONIC 7 NOTES, 2 OCTAVES SCALE (WHEEL OF 3RDS). The 7 notes 3 and 4 octaves scales of 4ths and 5ths.

Before this post the reader must study the posts 40 (that classifies intervals), the classification of  2-string triads , and  38 (that classifies 3-string triads) and 35. 

This scale is also the Harmonic tuning of the guitar (see post 1) which is optimal when chord playing is mainly the traget and not soloing so much

OPTIMAL GUITAR TUNING FOR CHORDS PLAYING MAINLY

2.) An more optimal but unkown tuning for the 6-string guitar when chord-playing is the main target and not so much solo playing is and even better by alternating minor and major 3rds. In semitones for the 6 strings   4-3-4-3-4 or 3-4-3-4-3
E.g. Bb2- D3-F3-A3-C4-E4 or F2-A2-C3-E3-G3-B3 or A2-C3-E4-G4-B4-D4
THIS MAY BE CALLED THE HARMONIC TUNING OF THE GUITAR AS IT IS BASED ON THE HARMONIC 2-OCTAVES 7-NOTES SCALE (see also post 79)
The latter is the most natural open tuning. There the same shape for major and minor chords and only 3 of them and in only one or frets compared to the 6 in the standard tuning guitar. If we want also dominant and major 7nth chords we use again only 2 frets. The same with the aug chords Only the dim7 chords require 3 frets. Because  of the symmetry of the tuning among the strings, the relations of relative chords and also chords in the wheel of 4ths is immediate to grasp also geometrically. Of course when we say shape of chords as it is standard in jazz, we do not play all 6-strings but only 3 or 4 strings.
Within 3 frets exist all chords of the  8-notes scale with interval structure 2-2-1-2-2-1-1-1 which is an extension and variation of the melodic double minor 2-2-1-1-1-1-2-2 or (1-1-1)-(2-2)-(2-2-1)
But also all chords of diatonic scale!

All the next chords are superposition of 2-notes relatives chords of types major, minor, diminished, and augmented. 


Discussion of the defnition-creation  and roles of the 8 basic such chords (in root position)
433=majVdim=R7
434=majVmin=Rmaj7
343=minVmaj=Rm7=R'6, where R' the root of  a  relative chord
333=dimVdim=Rdim7
334=dimVmin=Rm6=R'm7b5 , called also half diminsihed where R' the root of  a  relative chord
444=augVaug=Raug
344=minVaug=Rminmaj7
443=augVmaj=Rmaj7#5 

4 of the above chords can be 4-notes chords of the diatonic scale e.g. the R7, Rmaj7, Rm7, Rm7b5. But the Rdim7, Raug, Rminmaj7, Rmaj7#5 cannot be 4-note chords of the diatonic scale. Nevertheless there exist in other scales as we shall see like Harmonic minor, and harmonic double minor Melodic minor (Hindu)and Melodic double minor (Arabic) and other oriental scales as in the scales of post 52.

Some of the above  chords appear as chords of the HARMONIC LONG 12 NOTES 7 OCTAVES SCALE  (see post 42, and 34)  Wheel of 3rds
(successive distance in semitones) (434343434343434343434343) 

Besides the long scale we may define the HARMONIC 2 OCTAVES , 7 NOTES SCALE of 7 notes that spans 2 octaves, and every 3 or 4 successive notes is a chord of the above types 

Here it is (STARTING FROM THE  C3)

etc -C3-E3-G3-B3-D4-F4-A4-C5- etc  (all steps intervals of 3rds)

As a structure of intervals over semitones it is the next sequence

-4-3-4-3-3-4-3-

And the sequence of 3-notes chords , by successive 3 notes in this scale is the chord progression

C->Em->G->Bdim->Dm->F->Am

while as sequence of 4-notes chords is the

Cmaj7->Em7->G7->Bm6->Dm7->Fmaj7->Am7 or 

Cmaj7->G6->G7->Bm7b5=Dm6->F6->Fmaj7->C6

THE MELODIC CORRIDOR 

Constructing therapeutic harps, or similar instruments (e. thump  pianos https://en.wikipedia.org/wiki/Mbira or hang and handpans ) or such keyboards with this scale, has the property that almost what ever we play sounds harmonic as 3 or 4 notes in sequence are well known cords, and overlapping such triads , are relative chords. In fact we may design such an harmonic diatonic  double row keyboard

 as follows 

  -D3-F3-A3-C3- E4-G4-B4-

-C4-E3-G3-B3-D4-F4-A4-C5- 

Similarly this can be a beautiful , practical and harmonic tuning for harmonicas.

The blue is blowing and the red is draw. Then with the key we may have the sharps too thus in total

-D3#-F3#-A3#-C3#- E4#-G4#-B4#-

 -D3-  F3- A3-   C3-  E4-  G4-   B4-

-C4-  E3-  G3-  B3  -D4  -F4-  A4-  C5- 
-C4#-E3#-G3#-B3#-D4#-F4#-A4#-C5#- 

OR 

-D3#-F3#-A3#-C3#- E4#-G4#-B4#-

 -D3-  F3- A3-   C3-  E4-  G4-   B4-

-C4#-E3#-G3#-B3#-D4#-F4#-A4#-C5#- 

-C4-  E3-  G3-  B3  -D4  -F4-  A4-  C5- 

This type of tuning can be applied also as setting to 4 strings  frets of  the fretboard of the artiphon the midi-controller, and give it sound of adouble diatonic harp or flute etc.

(for he artiphon see here https://artiphon.com/  )


THIS KIND OF KEYBOARD OR TUNING OF INSTRUMENTS (FLUTES, HARMONICA, ARTIPHON , KEYBOARD ETC)  ALLOWS FOR MELODIES IMPROVISATION IN A FAST  WAY EASILY WITHOUT COMPLICATED FINGERINGS OBSTRUCTIONS AS LONG AS THE MELODY IS A DIATONIC SCALE. IT IS A BIT BETTER THAN PIANO KEYBOARD AS IT FOLLOWS THE SHAPING OF CHORDS BY INTERVALS OF 3RDS. IT IS EVEN AEASIRT THAN SINGING OR WHISTLING FROM THE HARMONY POINT OF VIEW!


The melodic corridor can be also be spotted  on the fretboard of the standard tuning 6-string guitar. It will also define an unusual shape  of the diatonic scale with 2 only notes per string. More on that on the post 94.

Similar to this harmonic scale is the next , with interval pattern 

-4-3-4-4-3-3-3-

or C3-E3-G3-B3-D#4-F#4-A4-C5

And the sequence of 3-notes chords , by successive 3 notes in this scale is the chord progression

C->Em->Gaug->Baug->D#->F#dim->Adim

Which when projected within one octave it is the 

C3-D3#-E3-F3#-G3-A3-B3-C4

which in interval structure it is the

-3-1-2-1-2-2-1

which is nothing else than the Harmonic minor (here we consider a sequence of notes and circular permutation of them the same scale)     

The last remaining permutation of harmonic 2 octaves scale , is the


-4-4-4-3-3-3-3 - or  3-3-3-3-4-4-4- 

e.g. 

C3-E3-G3#-C4-D#4-F#4-A4-C5  or as interval structure within one octave


C3-D#3-E3-F#3-G#3-A3-C4   

3-1-2-2-1-3- 

which is a 6-notes scale which is very close the 7-notes scale

3-1-2-2-1-2-1 

 which is the  2nd Harmonic minor or  Kurdi or Kassigar (see post 72).

Similar to the above 7 notes 2 octaves scale by 3rds are the 7 notes scales by 4ths and 4-octaves scale by 5ths.

In other words the next 7 notes scales

(We may agree to start from the 2nd octave)

7-notes 3 octaves scale by 4ths:

B2-E3-A3-D4-G4-C5-F5-B5

7-notes 4 octaves scale by 5ths:

F2-C3-G3-D4-A4-E5-B5-F6

See also the symmetric scales post 95 and the hexagonal pattern (Terpstra, Keyboards) of harmonies and melodies of the diatomic scales in post 99. 

(This post has not been written completely yet)

78. HARMONIC POLES AND HARMONIC WAVING IN THE CHORD PROGRESSION AND STRUCTURE OF THEMES IN THE MELODY

HARMONIC POLES AND HARMONIC WAVING IN THE CHORD PROGRESSION AND STRUCTURE OF THEMES IN THE MELODY (RULES SO AS TO PLAY ALMOST ANYTHING AND STILL SOUND HARMONIOUS AND BEAUTIFUL) :

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 (CHROMATIC CHANGES) to the pole of harmonic serenity ,intervals of 5 (4th) , 7 (5th) or 12 (octave) semitones (HARMONIC CHANGES).

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).



This harmonic waving layer is more macroscopic than the emotional and melodic meaning of single melodic theme. A single melodic theme has a simple emotional meaning and this is a simple interplay or move inside the duality of emotions (positive-negative emotions).

We remind the reader that 
1) Ascending with larger steps that those of descending indicates favor of joy

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

Based on the idea of the three relations of the chords (see post 30) , we may compose beautiful chord progressions. Two general rules that create chord progressions and therefore overlying melodic themes, in such a way that almost anything we play sounds harmonic and beautiful are the next:

A11 . 1st general rule for harmonic chord progressions:  Progressions by arcs in the 12-chord cycle by intervals of 4th, with substitutions with relative chords

This cycle defines by every connected arc of it a chord progression , where a chord may be substituted  with its same root relative major or minor chord , or its lower or upper minor relative chord. Of course  as they are an arc of the above 12-cycle they are successive chords or in the harmonic relation of resolution.

E.g. B7->Em->Am->D7->G->Bm-> etc 

Or B7->Em->Am->D7->G->C->(Am orA7)->D7 etc

E.g. The well known song of Frank Sinatra "Fly me to the moon" is using this technique in its sequence of chords 

https://www.youtube.com/watch?v=U8zvLvfUzCQ

Another example is the song of Nat King Cole L.O.V.E.

(main arc is the (Em or E7)->A7->D7->G(or Bm or Gm7) ->E7 etc with backwards retraces by one chord)

https://www.youtube.com/watch?v=cZ4TfRt1NK0&list=PLCD6F28C13A0A9AC2&index=12

 A12 . 2nd general rule for harmonic chord progressions:  Two arcs  in the 12-chord cycle by intervals of 4th (substituting any of the chords with its minor if it is major or vice versa) that have distance at the closest ends either 1 , or 2 or 3 or 4 semitones!

E.g. D7->G ,(1 semitone apart)  Db7->Fm 

or D7->G, (2 semitones apart) E7->Am 

or D7->G , (3 semitones apart) B7-> Em 

or Am->D7->G, (1 semitone apart) F#7->B7->Em.

77. LONG-SHORT , TWO-NOTES, IN 3-or-4 BEATS , MELODIC THEME PATTERNS TO COMPOSE MELODIES

In the traditional Irish melodies that utilize 2-3 only major chords, while the melodic moves are 4-5 or 6-7 , but also in the traditional Greek music of the Aegean Islands, the starting and ending point of the melodic move is during the duration of a single chord and are notes of the chord! But still the rule 2/3 -1/3 for notes internal and external to the chord still holds, and the starting and ending notes of the melodic move may define the simplicial submelody. The same of course with waltz songs. 

In the harmonic method of composition (see post 9) we conversely start with the chord progression, and its chord transitions, we select the starting and ending points of the melodic moves, and then  the morphological type of the melodic move , their length , their  rhythm , harmonic speeds etc. 

LONG-SHORT TWO-STEP PATTERN
A very simple trick to create beautiful melodies like those of Irish songs is to use a two-step rhythmic and melodic scheme of one short-duration note and one long duration (like in ancient Greek language that vowels are divided into two categories long and short duration). The long duration is, say two (or three) times longer, and the long duration is a note belonging to the chord while the short duration it does not! As the timing of this elementary melodic theme pattern, is 3-beats, some times it is 3 notes but two of them the same note and we may call it also triad-pattern. Usually, the two-step pattern is of diatonic density that is the interval of the two notes is a tine or semitone. But sometimes it may be 3 or 4 semitones, that is, of middle harmonic density. One of the goals of the melody is e.g. to walk down or up an octave or an interval of pure fifth or fourth that is to go from simplicial sub-melody center to another. In this way, both internal and external bridges among the chords can be created. We compose small ripples of this walking up or down (bridges) by the two-step pattern so that the long note is always a note of the underlying chord. Thus given a chord progression we may easily compose such beautiful melodies! If there are lyrics and the lyrics e.g. are in the Greek language we immediately derive an appropriate rippling of the melody.
The transformations of this short-long 2-notes melodic theme patter may be
1) Horizontal, time-order Inversion, where the long is first then the short note
2) Vertical, pitch-order inversion, where if the short is lower note and the long is higher now the long is lower and the short is higher pitch.
3) Elongation, where if the pitch distance of the short and long is say 1 or 2 semitones it may become 3, 4, 5, or 5 semitones etc
4) Half-repetition, where the long or the short not  is repeated in the next pair of short-long. 
5) Unification, where both the short and long notes become of the same pitch and are now a single note.
6) And of course pitch translation , where we shift the pair short-long inside a scale or chord.

Such melodies can be fast e.g. 4 times the 2-notes pattern per measure, or 2 times per chord strumming.
Since the 2-notes pattern is also found in the rhythm of the dance Waltz they can be danced as waltz. That is why sometimes such 2-notes short-long notes created melodies , we may call them drunken waltzes . 

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.

When composing melodies through bridges (see post 72), the bridges themselves are not sufficient to justify the choice of the chords, and we have to walk through the notes of each chord like arpeggios and between the bridges so as to have a full melody that justifies the particular choice of the chords. But when composing melodies after a chord progression through the 2-step patterns (internal-long, external-short notes for the chord) we walk the octave at the area of each chord, therefore we use all the notes of the chord and the melody immediately justifies the choice of the chords.

Here is an example of such a melody
(Frost is all over Irish melody)

http://ungaretti.racine.ra.it/ireland/music/frost.mid

(More traditional and beautiful Irish tunes in midi files here
http://ungaretti.racine.ra.it/ireland/music/irshmenu.htm )

In this way of composing chord-transition melodic moves, the starting and ending points are of paramount importance. Generally speaking, they are not identical with the centers of the melody, as they do not last in general longer than the other notes. They can be used though to define the simplicial submelody.    


IN MORE DETAIL

Here we concentrate one only simple organization structure which the closest corresponded in the poetic language and lyrics is the word. So we introduce a concept of micro-melodic theme, called

MUSICAL WORD that we may agree to symbolize say by w. It consists of a very small number of beats higher than 2 e.g.  3 or 4, and we may symbolize it with 0,s and 1,s , which means that at this beat if no sound is heard it is zero, while if a sound is heard it is 1. E.g. (0101) or (011) etc Now we divide the word in its LONG PART , that symbolize by L(w) , and SHORT PART . that we symbolize by S(w) and so that in time duration, or beats it holds that L(w)/S(w)>=2 (e.g. L(w)/S(w)=3 etc).

The musical-words or melodic micro-themes need not be by intervals of 2nds! They can be by intervals of 3rds and 5ths or 4ths! 

Such musical words may be ascending, descending or waving. Ascending as excitation may be small (intervals of 2nd) low middle (intervals of 3rds) or high middle (interval of 5th or 4th) or high (intervals of  8th or higher) Of course, as they are combined, they definitely create the effect of waving. BUT the waving is not the very standard by intervals by 2nds but a richer one, that involves many intervals of 3rds and even 5ths, and 8ths. The simplicial sub-melody of such melodies are movements mainly with intervals by 3rds and 5ths. There is also acceleration and deceleration as the melodic theme starts and ends.

Such techniques come also from the syllabic poetry measure and also dancing measure.
E.g. in Homer's Iliad and Odyssey alternate lines of 9 and 8 syllables where the syllables in a each line are grouped by the dactylic hexameter (one long two shorts syllables or 1011 or -UU where - =10 and U=1) This is also the rhythmic tempo of the ancient but also modern Greek dance of syrtos (and laso Cretan Sousta). Thus poetic measure, musical rhythm and dancing rhythm all coincide.  It is  not a coincidence that in the modern syrtos called kalamatianos, the dancer moves 7 steps right and 2 steps left and so one, and at the same time the full measure of a poetic line of Homer's poems are 9 syllables.

E.g. we may descend with a chord say Am and its relative C (out of chords would be notes of G), and ascend with its chromatic-complementary thee G7 (out of chord notes would be those of Am or C ) etc. In other words, we ascend with even or odd notes and descend conversely. Here although we may utilize only 3 chords (Am, C, G) the alternating-changing may be fast covering practically all waving and melodies of the pentatonic or diatonic scale. The scale-completion of the melody (see post 86)  , may be at the next octave rather than in the same octave!
The rhythmic repetition 3 times then the 4th is different is more common than 2 times repeated then 2 times a different. The total range of waving say of the first 3 repetitions may be of size a 5th, while the 4th measure a range of size an 8th, or vice versa.

Let us also assume that the chord progression that underlines the melody is the X(1), X(2) ,...X(n).

As we wrote in previous posts, the melody consists by a progression of melodic themes, that are transformed, by the 4 main transformations or translation, inversion, rotation including dilation and rhythmic transformation. This is indeed happening in to the melodic micro-themes or melodic or musical words during the part of the melody that sounds during say the chord X(i) i=1,2...n, BUT we impose here a very important structure which is the key to the beautiful folk melodies, and makes them compatible with the chord progression that underlines, the melody. And this rule is a 

RULE1 OF TRANSIENT AND CHORD NOTES. Obligatory part: In simple words, each musical-word w , that has underlined chord X(i) has the notes of its long part L(w) , to be notes of the chord X(i), (which includes extended forms of X(i) like X(i)maj7 or X(i)7 or X(i)add9 or X(i)sus4 ) while , the notes of its short part S(w) to be transient and belonging to the notes of the neighboring chord that is X(i-1) or  X(i+1), (which includes extended forms of X(i+1) like X(i+1)maj7 or X(i+1)7 or X(i+1)add9 or or X(i+1)sus4) or and more rarely to the rest of the chords of the chord progression. And if so if it contains a note from a non-adjacent chord Y(j) of the progression, then usually somewhere in the progression there is a transition X(i)->Y(j) or Y(j)->X(i) . We keep the transient notes sound at most 1/3 of the time only and the notes of the chord at least 2/3 of the time, because of the rule of long and short parts of the musical word or micro-theme. No mentioning of any scale is necessary in this definition (as usually there are more than one) but only of the chord progression, which is compatible with our enhanced concept of modern harmony. Nevertheless the chord progression over which this technique produces fast melodies may contain very fast chord changes, and may not be identical with the actual chord progression that the instruments play as background to the melody. This is the concept of "ghost chords" in the melody as described in the post 87. E.g. The full ghost-chord progression may be D G D G D A D. While the chords really played is only D. 

RULE2 An alternative rule is that a musical-word w , that has underlined chord X(i) has the notes of its long part L(w) , to be notes of the chord X(i), (which includes extended forms of X(i) like X(i)maj7 or X(i)7 or X(i)add9 or X(i)sus4 ) while , the notes of its short part S(w) to be transient and is one only intermediate not between the notes of the  chord X(i) (usually a 2nd away from the notes of X(i) and preferably but not obligatory this additional note to be a note of the other chords of the progression, again preferably and if possible of the previous or next chord, rarely on  of other chords. And if so, if it contains a note from a non-adjacent chord Y(j) of the progression, then usually somewhere in the progression there is a transition X(i)->Y(j) or Y(j)->X(i) .In this way we keep the transient notes sound at most 1/3 of the time only and the notes of the chord at least 2/3 of the time, in addition to the rule of long and short parts of the musical word or micro-theme. Even if we did not have the structure of micro-themes as musical-words with long and short notes , and we are playing in a random way the three notes of the chord plus one transient, in equal time in the average, we are still in the harmony of this chord, because of the proportion 3:1. And this would still hold if we used 2 transient notes in which case we would have the time proportion 3:2.  But in addition to this rule if we want also the intervals of 3rds, 4ths, 5th and 8th to be more than 2/3 of all the intervals the way is to apply harping in a chord say with 6 or 8 steps on notes, where it is added only one intermediate note in the chord (e.g. 7nh, 6th, 4th or 2nd) and so that the created intervals of 2nd are only 2 in the 6 or 8 intervals. Then we shift to a relative chord an interval of  3rd away or to a resolution transition which is a chord in an interval  5th or 4th away , or we even shift to a chord a 2nd away in which case we do not use any additional note, and we continue so.  So finally %3rds+%4ths/5ths/8ths>=2*(% 2nds) . Again the chord progression over which this technique produces fast melodies may contain very fast chord changes, and may not be identical with the actual chord progression that the instruments play as background to the melody. This is the concept of "ghost chords" in the melody as described in the post 87. E.g. The full ghost chord progression may be D G D G D A D. While the chords really played is only D. 

76. The 4 basic transformations of the elementary melodic themes: Translation , Inversion, expansion (contraction), rotation (mutation) , . All of them both in pitch or rhythm . Loops and arpeggiators

(This post has not yet being completely written)

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

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

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

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

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

1) The translation (either with intervals of 2nd , (or diatonic density) or intervals of 3rd (or middle harmonic density) or  of intervals of 4th or 5th (or high harmonic density)). Translations with intervals of 3rd, may applied without changing the underlying chord, or changing it to a relative chord. Translations with intervals of 4th and 5ths, occur when the underlying chords are in resolution-relation that is successive chords on the wheel of 4ths. Translation by one semitone or chromatic translation may happen in the cases where the underlying chords are in resolution relation (successive chords on the wheel of 4ths) and the first is a dominant 7th chord, or when the underlying chords also have roots at  distance of one semitone. 

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

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

Translations and inversion may both relative to pitch or relative to rhythm.

4) Mutation:  this practically means that we give up with the particular (order topological) shape of the melodic theme and shift to a a new waving or move or melodic theme.



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

1) Translation
2) Inversion relative to a point
3) Reflection relative to an horizontal line
4) Reflection relative to a vertical line.
5) Rhythm transformation
to the above five we may add the
6) Acceleration (e.g. from the diatonic speed or density to the middle harmonic speed or density) or Deceleration (vice-versa).

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

More complicated  ways to transform a theme are at least the next 5 and combinations of them (see also post 41)
1) Translate it in different pitches (within a scale or not changing possibly the pitch distances )
2) Translate in time (repeat it)
3) Invert it in time or change its rhythm (if at the begging is slower and at the end faster it will be now the reverse etc)
4) Invert it or distort it in pitch (Create 1st 2nd 3rd or 4th voice versions, utilizing the chord progression as rules of transformation of the theme, or if it is ascending now it will be descending etc)

5) Change it as morphology  (from a non-waving ascending it may become waving ascending or isocratic). We prefer spikes and scaling as the main morphological types, while the waving and isocratic as intermediate bridges. 
6) Increase or decrease the size of steps while ascending or descending (pitch dilation)

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

After the chord progression and simplicial submelody we chose, 

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

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

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

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

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


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

MELODIC THEMES TRANSFORMATIONS AND SIMPLICIAL SUBMELODY

We have mentioned in post 72 that the simplicial submelody is usually  the starting or ending notes of simple melodic themes, can be external bridges (see post 72) of the chord transitions (of density diatonic or middle harmonic etc). Therefore here we apply the 3 basic transformations and starting from a single melodic theme ending to the first note of the simplicial submelody we translate or invert or vary rhythmically this theme, and make it end (or start) on the next note of the simplicial submelody.  The transformed melodic themes derived in this way cover most often two chords or a chord transition or chord relation

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

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

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

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

4) Its melodic density acceleration or deceleration 

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

We may device a symbolism for a melodic theme move based on the above three factors as follows An1Bn2(-)(x) or An1Bn2(+)(x) where An1 is the first note and Bn2 the last note of the move (n1 n2 denote the piano octave of it) and a minus - or plus + sign if its is sad (minor)  or joyful (major) and (x)=1,2,3,4 denotes the dominating density of it is chromatic x=1, if it is diatonic x=2, if it is middle harmonic x=3 and high harmonic x=4  (see post 68)  e.g. G5A4(-)(2) . In this way we write the dynamics of he melody as a theme-progression ,much like a chord progression. 

75. The correlation of the accumulative pitch shift of melodic themes with the pitch shift of their chords in the 12-cycle of chords by intervals of 4th

(The post has not been written completely yet)

Most easy to sing songs use the triads chords of the diatonic scale, either the major triad , (V, I, IV) or the relatives minor triad   (iii, vi, ii). Now these triads are successive chords in the 12-cycle of chords by intervals of pure 4ths (5 semitones) (See e.g. post 32, qnd 34 ).

But as we have said the best fit of melody and chords is one that there is a correlation between the "emotional story" of the themes ofthe melody and the "emotional story" of the parallel chord progression (see e.g. post 27).

The simplest such correlation is that, when utilizing e.g. the triad (V, I, IV) for the song, the theme that is parallel to a  chord transition e.g. I->IV is ascending in total accumulation  in pitch (easily seen in the simplicial sub-melody), while while going to V (either from IV, or from I) is descending. In other words also the transition resolutions  V->I, I->IV are ascending or descending, while the transitions that are inverse of the resolution,  I->V, IV->I are descending or ascending respectively, but always with a fixed correlation during the song. Thus cycling in the triad is essentially alternating going up and going down in the total accumulated pitch of the melodic themes which also creates a feeling of balance and easy to sing concept. The major triad is also partially or totally substituted by the minor triad (iii, vi, ii), but always with the same pitch correlation rule, before we return back to the major triad again. We are always considering the golden rule: At least  2/3 of the notes (as duration) as notes of the chord and at most 1/3 of the notes (as duration) as notes outside the chord. And at least 2/3 of the chords (as duration) being major chords, and at most 1/3 of the chords (as duration) as minor chords.
This correlation seems to be the strongest among the "pitch emotional story" of the melodic themes and the "pitch emotional story" of the chord progression. By utilizing the chord transition relation of relative chords, and substituting major chords on the 12-cycle of chords by  4ths, with one of their relative minor chords , the correlation is inherited in the minor chords, while now there appear transitions of complementary chords, (e.g. I->IV->I , may become I->ii->I). But still the pitch-shift  correlation is that of the original chords in the 12-cycle rather than that of complementary chords.

A natural way to define the pitch dynamics of the melodic themes, and  the "emotional story" of the melodic themes is directly from the speaking language, when we shape phrases and speak them. There is a natural shaping also of the pitch dynamics that comes from the meaning of the words, or not exactly from the meaning of the words but from the intended emotional background of our speaking. It can be soothing agitating, reassuring, irritating, angry, calming , waving, falling in itch ,or increasing in pitch etc. The common way of composition starts from the words, and their speaking pitch dynamics, it goes to the melody, melodic themes  pitch dynamics and then to the harmony. But in this book we prefer the harmonic method. The harmonic method does not necessarily restrict the composer, to define the pitch dynamics of the melodic themes from the speaking language dimension, as any such pitch dynamic can be derived also from pre-defined chord progressions, if we chose properly the inversions and scales of the chords of the chord-progression.

By utilizing this method of relative chords , where we include the 3rd relative relation (chords of the same root one major the other minor e.g. A->Am) we may have scale modulations where no # or b shift occurs in the chords of a single scale, For such modulations ,it is still possible to reduce virtually the chords only to the major triad of a single diatonic scale , and check the correlation of of pitch shifts, of the melodic themes and chord transitions. This can be done with more complicated scale modulations , where locally in time we are still in a single diatonic scale (definable easily by an arc, in the 12-cycle of chords by 4ths)

To play and observe these correlations of melodic and chord shifts on the guitar see also the post 13 about the 3 neighborhoods of the guitar fret-board