90. ISOMORPHIC LAYOUTS OF NOTES AND GUITAR TUNING BY ALTERNATING 3RDS. THE AMAZING, OPTIMAL DIATONIC GUITAR OR HARMONICA-GUITAR OF 6-STRINGS OR 12-STRINGS , (JARANA) TUNING.

THE AMAZING,  OPTIMAL    DIATONIC GUITAR OR HARMONICA-GUITAR OF  6-STRING OR 12-STRING , TUNING.

See also post 164 and 407.

ABOUT  ISOMPORPHIC 2-DIMENSIONAL LAYOUTS FOR KEYBARDS  STRING INSTRUMENTS  TUNINGS AND SOFTWARE PADS  FOR ARRANGING  THE MUSICAL NOTESAND THEIR IPORTANCE IN IMPROVISING SEE POST 310.

THE TERM ISOMORPHIC REFERS TO THE CHORD-SHAPES THAT REMAIN THE SAME (ARE ISOMORPHIC) WHEN CHANGING THE ROOT NOTE AS LONG AS THE TYPE OF THE CHORD REMAINS THE SAME.

Isomorphic layouts: What they are and why they are awesome for your music

https://www.youtube.com/watch?v=ZczraF3dzU0&t=20s&fbclid=IwAR1bAiGG3HFPE9OciNXGPE5p_GD_L5AJaKXsabiypconKg_HYk_RAz5hR9s

THE MAIN IDEA OF THIS TUNING OF THE 6-STRING GUITAR IS  TO APPLY THE DIATONIC TUNING OF AN HARP ON EVEN OR ODD STRINGS OR AN HARMONICA, VERTICALLY AMONG THE STRINGS, WHILE LETTING THE FRETBOARD ENHANCE IT CHROMATICALLY IN THE USUAL WAY.

 Here is a diatonic rather than chromatic version of the isomorphic layout based on alternating major-minor 3rds:

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

Normally a guitar with the standard tuning is a chromatic instrument e.g.  compared to diatonic wind instruments like a recorder or a diatonic tuned Celtic harp.  But there is a natural harmonic diatonic tuning of the guitar.

An  optimal but unknown tuning for the 6-string guitar when chord-playing is the main target and not so much solo playing is 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 ( default scale Bb) or F2-A2-C3-E3-G3-B3 (default scale F major) or A2-C3-E4-G4-B4-D4 (default scale C major)  or  G2-B2-D3-F#3-A3-C4 (default scale G major) or B2-D3-F#4-A4-C#4-E4 (default scale D major)  or  G2-Bb2-D3-F3-A3-C4 (default scale Bb major)  

Notice also the tuning of alternate minor and major thirds seem to occur for a 5-string Mexican instrument the Jarana huasteca https://en.wikipedia.org/wiki/Jarana_huasteca)

As notes ,

The 1st string , counting from lower to higher defines the Aeolian 6th mode, the 2nd string the Ionian 1st mode, the 3rd string the Phrygian 3rd mode , the 4th string the Myxolydian 5th mode , the 5th string the Locrian 7nth mode, and the 6th string the Dorian 2nd mode.

As triad-chords, 

1) the first 3 strings , starting from the lower to the higher (e.g. in the last tuning, the G2-Bb2-D3) create the Aeolian mode of the default diatonicscale (here Bb). 

2) The next 3 strings (e.g. in the last tuning, the Bb2-D3-F3 ) create the Ionian mode of the default diatonic scale (here Bb). 

3) The next 3 strings (e.g. in the last tuning, the D3-F3-A3 ) create the Prygian  mode of the default diatonic scale (here Bb). 

4) Finally the last 3 strings (e.g. in the last tuning, the F3-A3-C4 ) create the Myxolydian  mode of the default diatonic scale (here Bb). 

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 (as in strumming) but only 3 or 4 strings.

The easiness with which one can improvise melodies within a diatonic scale (all notes within 3  frets and in a very symmetric zig-zag pattern) together with 3-notes chords of the scale (gain all chord patterns within 3-frets) is unsurpassed.

At the same time , the easiness with which one can make diatonic scale modulations, chromatic (1 semitone apart) or by changing a minor to a major chord and vice versa and continuing in a relevant diatonic scale, is unsurpassed again! 

The main advantages are

1) Greater number of major or minor triads per number of frets, thus easier chord playing

2) Less number of chord-shapes, thus easier guitar to learn

3) The shapes of chords require less number of frets 

4)  1st inversion chords require only 1 or 2 frets, thus have easier shapes and are easier to play

5) In the melodies the notes are closer in total thus easier  to find by the fingers

6) Because the chords are in one or 2 only frets, their arpeggios also and neighboring melodies are in fewer frets thus easier to play as finger picking style. 

7) The odd number of strings or the even number of strings are in intervals of 5 (7 semitones) thus as in the tuning of Cello, Octave mandolin , Viola, Violin , mandolin , Irish bouzouki, mandocello etc therefore any  one trained to play solos in the previous instruments can keep his knowledge and play the same solos in such an harmonic tuned guitar (on odds or even strings).
Also the isokratic technique  in the 3-courses instruments tuned so as to contain a 5th like bouzouki, boulgari, tampour , saz  where two string a 5th apart accompany the melody on another string still applies in the harmonic guitar as all odd and even number strings are tuned a 5th apart. Nevertheless compared to the previous instruments (violon , mandolin etc) in the harmonic guitar the 3-notes or 4-notes major or minor chords , or diminished and augmented chords are played radically easier with one or two rarely 3 frets and is  the densest such placement on the fretboard among all tunings of the guitar . At the same time any guitar  jazz player one trained to play 3-notes chords (triads) in the standard guitar and especially on the 4 highest strings can keeps his knowledge and with slight modifications apply it to the harmonically tuned guitar. Furthermore any one playing the panduri (a russian or georgian folk 3-courses instrument coming from ancient Geek panduris, which is tuned in open major or minor 3-notes chord) will pass his knowledge to this harmonic tuned guitar in the upper 3 or middle 3 and lower 3 strings! 

This harmonic tuning by alternating minor-major 3rds, allows, for all  4-notes chords of e.g. the D major scale in   the 3rd octave (c3,d3,e3,f3,g3,a3,b3), Cmaj7->Em7->G7->Bm6->Dm7->Fmaj7->Am7 in 1st normal position across the fretboard, something not possible with the standard tuning of the guitar. In the standard guitar it is possible only by 2nd or 3rd inversion, or by shifting to the 4th octave or 2nd octave. Therefore there are important very natural voicing of the 4-notes chords of the 3rd  octave that we miss with the standard tuning and in the harmonically tuned guitar, it is in a single octave!

THERE ARE 4 VERY SYMMETRIC WAYS THAT THE CHORDS IN A WHEEL BY 4THS CAN BE REPRESENTED AND PLAYED IN THE FRETBOARD WITH THIS  HARMONIC TUNING. 

WE ENLARGE IN THE NEXT WITH MAPS OF THE CHORDS AND THEIR SHAPES IN THE THREE WHEELS, THE ONE BY 4THS, THE ONE BY 3RDS AND THE CHROMATIC.


In post 67 are also described harmonically  tuned 4-course (Greek 4-courses bouzouki, ukulele, mando-lele etc)  or 3-courses (Balalaika, Greek 3-course Bouzouki) instruments that are easier to play but have the same remarkable advantages due to the harmonic tuning.

E.g. for 4-course , the tuning D3D4-F3F4-A4A4-C4C4 abd for 3-courses the F3F4-A4A4-C4C4 or 

G3G4-Bb4Bb4-D4D4. 

THE DIATONIC-GUITAR OR HARMONICA-GUITAR  HAS A DIRECT ADVANTAGE OF APPLYING THE CLASSICAL GROUP OF VARIATIONS OF MELODIC THEMES AS IN POST 279 (TONAL TRANSLATIONS BY 3RDS AND 5THS OR 4THS IN AT LEAST 2/3 OF THE CASES AND CHROMATICALL IN AT MOST 1/3 OF THE CASES). THE MELODIC THEME SUCH TONAL  TRANSLATION BY 3RDS OR 5THS/4THS IS SIMPLY SHIFTING THE MELODIC THEME FROM ONE STRING ON THE SAME FRET  VERTICALLY TO  THE ADJACENT STRING ( A 3RD) OR NEXT TO ADJACENT STRING (5TH OR 4TH).



THERE ARE 3 MAIN WAYS THAT TWO TRIAD-CHORDS HAVE HARMONIC RELATION ON THE FRETBOARD OF A  6-STRING SUCH GUITAR

1) By inversting the interval of 5th that each fret has with the  samE fret but two strings higher

2) By and interval of 4th that each fret has 2 frets lower and 2 strings higher (as created by the same interval of 5th as before but one tone lower onthe higher string)

3) By an interval of 4th as created on two succcesive strings that have always an intervalof 3rd and an additional interval of 2nd twoards higher.

89. SIMPLE SYMMETRIC PATTERNS OF PENTATONIC SCALES ON 4-STRING INSTRUMENTS FOR FAST TRILL SOLOING

The pentatonic scale has entirely simple symmetric patterns-shapes on 4-string and 3-string instruments, like 3-string Greek lyra, 4-double strings mandolin, 4-strings ukulele, 3 or 4-double strings Greek bouzouki. And this can be taken in full advantage for fast "dancing" soloing.

The Egyptian scale the Mongolian and the minor Pentatonic are different cyclic permutations or modes of the same scale.

Here is a video with the penantonic scale called "tik" in the Pontian 3-string Lyra.

https://www.youtube.com/watch?v=vUY2OBI_msQ.

For 4-string instruments like Ukulele or Greek Bouzouki (or 4 higher strings of Guitar)

the pattern of the pentatonic scale is symmetric identical  on the 2-strings and then symmetric identical on the other two strings and it is one of the next variations 

dot-empty-empty-dot

dot-empty-empty-dot

dot-empty-dot         

dot-empty-dot

OR

           dot-empty-dot

           dot-empty-dot

dot-empty-empty-dot

dot-empty-empty-dot

          

OR 

   dot- empty-  empty-dot        

    empty -dot-    empty-dot

    dot-     empty-dot-  empty          

    dot-     empty-dot- empty 

Based on this extreme symmetry, the soloing can be very fast by hammering , pull-ups and , taping.

In addition by adding semitones at two of the ends we get diatonic scales, on which we may also solo in fast way.

The patterns for Mandolin, Violin, and in general instruments that two or more strings have interval distance of pure 5th , that is 7 semitones, are even simpler and more symmetric. If all strings have distance 4th or all have distance a pure fifth, then the pattern is symmetric on all 4-strings and not two by two. 

Here is fast soloing on the Greek Pontian lyra on the scale "tik" pentatonic (tuning Β3, Ε4, Α4) 

https://www.youtube.com/watch?v=ldy189LmLDs

Because the  hammering and pull-ups is really fast but  also is for at most 3 notes per string, it may be considered an horizontal harping or trill along a single string, where the trill is not on the same note but on notes at in interval of 2 or 3 semitones. Thus the whole of the soloing is like harping vertically among the strings and trilling horizontally across a string within at most 3-4 frets. This is done mainly close to the upper bridge, so as to use also the open strings.


Here is an example of this type of soloing with mandolin and not Lyra. (Here it may not be within Pentatonic scale only)

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

https://www.youtube.com/watch?v=Y_NRdhL-j6U

https://www.youtube.com/watch?v=qk8xsqKGWFo

https://www.youtube.com/watch?v=Y-tJ1dY2jEE

In older centuries music (that still is practiced as traditional folk music) that the concept of harmony and chords had not been developed yet, the accompanying of melodies was not with chords but with a secondary counter-melody. E.g. if a melody was in a diatonic scale then a secondary counter-melody in the pentatonic sub-scale of the diatonic was accompanying, and actually, intervals of notes that were played (trilled) sequentially in time (like 3rds-6ths  and 4ths or 5ths) were playing the role of chords!
ACCOMPANYING A MELODY WITH INTERVALS INSTEAD OF WITH CHORDS.
Another simple idea is that the countermelody (especially when it is on a simpler scale e.g. a pentatonic ) can be used to accompany the melody not with chords but with intervals from the simpler scale of the countermelody. E.g. we may utilize intervals by 3rds as in the role of minor  diminished or augmented chords and the intervals of 4th or 5th as major or power chords

88. How to play slow soling on any scale across the fretboard, by knowing only chord-shapes and not scales-shapes!

How to play slow soling on any scale across the fretboard, by knowing only chord-shapes and not scales-shapes!

Here we are taking mainly not for the 6-string guitar but for 4-string instruments that , inherit the tuning from the higher 4-strings of a 6-strong guitar either with the exact frequencies (D,G,B,E) , or only isomorphic-ally (G,C.E,A), (D,F,A,D) etc like Greek 4-double string Bouzouki, Ukulele, baritone ukulele, ukulele-charango etc (For such instrument see  post 67).

Such instrument do not only give exact voicing of the 4-notes chords (no repeating notes) , but this also means that the arpeggios of the major or minor or also with 6th or 7th such chords, are identical with the chord-shape! This has the advantage that we do not need to learn scale-shapes across the fretboard, because as we shall see the scale-shape on all of the fretboard can be obtained as a simple sequence of  (usually only) 3 basic chords of the scale!

Le us take the example of the diatonic scale (e.g. D major) . Let us denote the 7 notes of such a major -mode scale by I, II, III, IV, V, VI, VII, I(=VIII). For the D major-mode scale it wold be 

D4, E4, F#4, G4, A4 , B4 C#5, D5

Now it is known that the chords, with roots on the notes of the scale are also denoted by latin numerals, capital if major and small if minor

 I, ii, iii, IV, V, vi, vii(dim), I(=VIII)

And by substituting the minor chords with their upper or lower major relative chords we get the chord progression

I, IV, I, IV, V, IV, V, I or for the D major-mode on particular the chord progression

D4, G4, D4, G4, A4 , G4 A4, D5  (=1,4,1,4,5,4,5,1)

In the symbols for guitar chords as in post 23 that pin down the place in the fretboard and the shape of the chord, the chord progression is the next:

D->(3E)G->(5A)D->(5D)G->(5E)A->(10A)G->(8D)A->(12D)D

Notice that

D, and G or (3E)G is at the 1st neighborhood o the fretboard (see posts 5, 13)

5A)D->(5D)G->(5E)A are at the 2nd neighborhood

and (10A)G->(8D)A->(12D)D are at the 3d and 4th neighborhood (see posts 5, 13).

When improvising, by listening to the 3 chords D, G, A, the sounds of the notes of the D major scale are created in the subconscious, and then by playing single notes based on he shapes of the above sequence of chords,  the full D major scale is deployed under our fingers!

It is clear that the notes of these chords do cover the scale and in fact contain no more notes than those of the scale (we always talk only for the higher 4 strings of the 6-string guitar and similar 4-string instruments, see post 67) .

So the way to play the scale, in slow soloing, would be to play the shapes of the above chords in that order, but not strumming the guitar, and only playing one note  of the chord shape , and not the rest of the notes of the chord. (The rest might be used in a enrichment of the melody). This chord progression not doubt will give all the notes of the D major-mode  scale.
We are saying slow soloing because obviously , the speed with which we change chord-shapes on the fretboard, is slower than the speed we may play single notes of scale.  But fast soloing is not always the beautiful or required. Slow soloing is more soulful and melodic, giving the opportunity for intermediate chord sounds too.
Notice that this requires that we know all the variations of an open chord-shape as non-open chord shale across the fretboard. But this is also easy and has been discussed together with the concept of 3 basic neighborhoods of the fretboard e.g. in post 3 and post 13.

Now all other 7 modes of the diatonic scale have again as chord progression to play them a cyclic permutation of the above chord progression.

Similarly other scales . e.g. like harmonic minor, or Hungarian minor (=Harmonic double minor) have similar alternating sequence of  usually 3 basic chords that create the scale-shape across the fret-board. So the above technique still applies.

87. WHY MOST SUGGESTION ABOUT SOLOING PARALLEL TO CHORDS ARE INADEQUATE . THE ROLE OF THE MELODY AND GHOST-CHORDS

Most of the suggestions about soloing parallel to chords are of the type:

1) Play the arpeggio or chord-tone of the chord

2) Play the pentatonic scale, minor or major, with the same root

3) Play a mode or scale that the song is in it

etc

And although applying the above will not sound ugly when soloing, still all of the above are inadequate for good licks and multiplicative (meaning dense and chatty)  soloing by an instrument in a song! Of course we are taking about songs that do have a melody and a chord progression!
The reason is the next: A song has a singable melody and chords and when soloing, the soloing must not only fit the chord progressions but also resemble the melody that the singer sings especially at the pattern of repeating and transforming simple melodic themes!

Now the melody has simple themes that repeat, ascend or descend and expand or contact. So the soloing must refect the simple theme repetition and transform them in more complicated ways. That is why all the 1),2), 3) are not really adequate.

Here is an example of the fitting of the melody that the singers sings and the instrumental multiplicative soloing

https://www.youtube.com/watch?v=RYXYoUb13nM

and one more from ethnic music from Andes

https://www.youtube.com/watch?v=EW5-w-Dr34w

See also the post 71 about

Dialogue of a simple human singable melody and dense chatty-birdy instrumental multiplied melody

In other posts of this book, we have enlarged on the structure of the melody from simple themes that somehow repeat and the simplicial sub melody. E.g. the soloing must have also the same simplicial sub melody.

GHOST-CHORDS PROGRESSION METHOD OF IMPROVISATION OVER A SINGLE CHORD:

MOST OF THE TEACHERS OF IMPROVISATION SUGGEST  USING THE ARPEGGIO OF THE UNDERLYING CHORD, EITHER AS PURE TRIAD OF NOTES OR AS EXTENSION TO 4 OR 5 NOTES SUCH CHORD. BUT THERE IS ANOTHER INTERESTING TECHNIQUE THAT INVOLVES GHOST-CHORDS (NAMELY THAT ARE NOT REALLY HEARD). E.G. OF WE ARE TO IMPROVISE SAY ON C MAJOR CHORD, THEN IT IS NOT ENOUGH TO USE ITS ARPEGGIO, BUT DO THE NEXT: CONSIDER C IN THE CHORD PROGRESSION OF THE SONG, AND TAKE TWO OTHER CHORDS OF THE SONG PREFERABLY IN THE WHEEL OF 4THS,  THE 2 NEIGHBORHOOD CHORDS (EITHER AS MAJORS OR MINORS) HERE E.G. LET US TAKE THE MAJORS G->C->F ASSUMING THEY WHERE IN THE SONG. IF THERE IS NOT SONG YET WE JUST TAKE . IN THE WHEEL OF 4THS,  THE 2 NEIGHBORHOOD CHORDST (that define here the C major-mode diatonic  scale) . THEN  TAKE THE ARPEGGIOS OF THESE THREE CHORDS AND PLAY THEM IN RHYTHMIC , FAST AND RATHER RANDOM PERMUTATION  WAY, AS IF A VERY FAST CHANGE OF CHORDS IS MADE IN THE THREE  G->C->F, SO FAST THAT G, F CHORDS ARE RATHER TRANSIENT WHILE WE REMAIN MOST OF THE TIME ON C. THE SEQUENCE OF THE CHORDS THROUGH THEIR ARPEGGIOS DEFINE ALSO A SOLOING. THE RESULT WILL BE AN IMPROVISATION ON ALMOST A WHOLE 7-NOTES SCALE, WITH UNDERLYING SINGLE CHORD THE C.IN ADDITION THE SOLOING TAKES IN CONSIDERATION   AT LEAST TWO  OTHER CHORDS OF THE SONG. IF THERE IS MELODY IN THE SONG WE MAY CONSIDER MIMICKING THE MELODY WITH WAVINGS AND "DANCING AROUND THE NOTES OF IT, IN NOTES THAT EXIST IN THE CHORDS OF THE MELODY. OR WE MAY APPLY DIFFERENT TRANSFORMATIONS IN THE MELODIC THEMES THAN THE TRANSFORMATIONS THAT EXIST IN THE MELODY. THE RESULT WILL BE A DIALOGUE BETWEEN THE MELODY AND THE SOLOING E.g. The full ghost chord progression may be D G D G D A D, while the chords really played is only D. Of course, the time duration of D in the chord progression of the ghost chords D G D G D A D is longer than any of the other chords


When we solo around say a major chord e.g. C , that we may consider as root chord of a major diatonic scale , the out of chords notes are the 7th, 2nd, 4th, and 6th (b, d, f, a) . 
We may also consider that it is covered in the wheel by 3rds, as the ascending sequence of 5 chords  with 3 minors 2 majors Em->C->Am->F->Dm  or the 5 chords sequence with 2 minors and 3 majors G->Em->C->Am->F. The latter consideration in the wheel by 3rds seems more natural. Therefore soloing around a chord like C,=(c,e,g) as interval of 7 notes b-c-d-e-f-g-a, is covered by  an arc of 5 successive chords in the wheel by 3rds , and the soloing can be patterned by permutations of these chords, as fast-ghost chord progression  while in reality we may play only 2 major or 3 major chords only. The same method as we may continue further left or right in the wheel by 3rds defines also the modulations that lead us away from the initial diatonic scale.

The next video is an example from folk Cretan (Greece) music:
https://www.youtube.com/watch?v=Fq_srURnI5o&feature=youtu.be

or the next with Irish soloing

https://www.youtube.com/watch?v=S3d-aFpwia0



In the next example of the famous Erotokritos song of Creta (Greece), the simple chords of the song are only two ,Em, D. But the Ghost-chords progression so as to improvise a dialogue soloing with the melody is the progression

Em, D, G, Bm, G, D7, Em (Rhythm, 0,0,0,1, 0,0,0,1 0,0,0,1 0,1,0 empty)  
D7, G, D, Em, G, F#dim, Bm, G, D7, Em (Rhythm, 0,0,0,1, 0,0,0,1 0,0,0,1 0,1,0 empty)
So placing the ghost chords on the rhythm is as the next 


D    Em                      D    G
0,0,0,1, 0,0,0,1 0,0,0,1 0,1,0 empty
        Bm        G         D7   Em
0,0,0,1, 0,0,0,1 0,0,0,1 0,1,0 empty

D7    G    D   Em       G
1,0,0,1, 0,1,0,1 0,0,0,1 0,1,0 empty

        F#dim  Bm G   D7  Em
0,0,0,1 0,0,0,1 0,1,0,1 0,1,0 empty


The rhythm is that of the 15 syllables poetry. 

The simplified chord accompaniment is with two chords only 



        Em                               
0,0,0,1, 0,0,0,1 0,0,0,1 0,1,0 empty
         D7                           Em
0,0,0,1, 0,0,0,1 0,0,0,1 0,1,0 empty

Em                                  D
1,0,0,1, 0,1,0,1 0,0,0,1 0,1,0 empty

                                      Em

0,0,0,1 0,0,0,1 0,1,0,1 0,1,0 empty

https://www.youtube.com/watch?v=UwDomxyoFgU&t=8s

86.1. Chromatic music techniques and scale-completeness music techniques . The topology of melodies and chord progressions. The concept of melodic closure and interval-melodies.

The term closure is borrowed from mathematics, where in e.g. topology , the closure Cl(A) of a set A  is all the points of A plus all points in contact with points of of A.

In music the closure Cl(M) of melody M is all the notes of the melody, as notes of the 12-notes chromatic scale. Notice that here we project all notes to a single octave. 

A Melody M  is chromatic-complete  if its closure Cl(M) is all the 12 notes of the chromatic scale.
A Melody M  is scale-complete  if its closure Cl(M) is all the  notes of the underlying scale, if there is one (e.g. diatonic 7-notes scale).

The same definitions can apply to the set of notes of the chords of a chord progression. We may define the Closure Cl(P) of a chord progression P

Usually the closure of a melody is only a subset of the 12-notes of the chromatic scale.

A melody or chord progression  M is called coherent or compact or interval-melody , interval-chord-progression respectively if its closure Cl(M)  is all the notes between the end notes of a interval. In symbols Cl(M)=[x1,x2]  E.g. if this interval is an interval of 5th that is [c,g]={c,c# d,d#, e, f, f# g}.

Similarly we define it relative to a scale rather than all the 12-notes of the full scale.

A melody or chord progression  M is called scale-coherent or scale-compact or scale-interval-melody , interval-chord-progression respectively if its closure Cl(M)  is all the notes of the underlying scale  between the end notes of a interval. In symbols Cl(M)=[x1,x2]  E.g. if this interval is an interval of 5th like C-G and we are in the c-diatonic scale then it  is [c,g]={c, d, e, f,  g}.

We may also define that the length L([x1,x2]) of the maximal interval [x1,x2] in the closure Cl(M), so that all notes in the interval are contained in the closure  , L([x1,x2])= x2-x1 in semitones, may be called chromaticity completeness measure of the melody or chord progression.  A full melody has chromaticity completeness measure equal to 12.

We may also define that the length L([x1,x2]) of the maximal interval [x1,x2] in the closure Cl(M) relative to the underlying scale , so that all notes in the interval and in the scale are contained in the closure  , L([x1,x2])= x2-x1 in intervals of 2nd, and it  may be called scale-completeness measure of the melody or chord progression.  A full melody has scale-completeness measure equal to the number of notes in the scale.

The same applies if we substitute melodies with modes of scales.

E.g. a modulation from the major mode to the Locrian mode would involve 5 notes changed by flat, therefore a melody with closure the major mode, which is transposed to the Locrian mode will have in total chromatic measure 12, that is a full melody. If the modulation would be from the major mode to the minor mode, then since 3 notes will change by flat, the union of the two models will be c, d d# e f g g# a a# b c,  in total a 10-notes scvale with the chromaticity completness  measure of 5 semitones.

An example  of a masters of chromaticity , is the Italian composer Nino Rota (music in the films if Fellini) https://www.youtube.com/watch?v=m9FPo4eiBCg&t=3273s

In general if max(Cl(M)) and min(Cl(M)) are the maximum and minimums notes of the Closure Cl(M) in the 12-notes chromatic scale, then the interval  [min(Cl(M)), max(Cl(M))] is called the chromatic range of the melody or chord progression. This is not to be confused with the actual range of minimum and maximum frequency range of the melody or chord progression.

Similarly if max(Cl(M)) and min(Cl(M)) are the maximum and minimums notes of the Closure Cl(M) in the underlying  scale S, then the interval  [min(Cl(M)), max(Cl(M))] from notes only in the scale S  is called the scale-range of the melody or chord progression. This is not to be confused with the actual range of minimum and maximum frequency range of the melody or chord progression.

Starting the composition of a chord progression and melody , from an interval, that may be the chromatic range, is a good beginning, especially if the melody or chord progression is an interval-melody.

The next Greek folk song (I do not want you any more= Δεν σε Θελω πια) has melodic closure an interval of 6th e.g. {c#, d, d#, e, f, f#, g, g#, a, a#} =[c#, a#]

https://www.youtube.com/watch?v=nLnIimHe1HI

We enlarge more about closures of melodies.

If the closure Cl(M) of a melody is all the notes  of a  scale C (e.g. a diatonic scale) , then obviously the melody is in this scale.

Exercise: 1) Find the closure of the Recuerdos de l Alhambra

https://www.youtube.com/watch?v=lIINjG6DDhc

2)  Find the closure of the next song

https://www.youtube.com/watch?v=G7a-oRHMcxI

3) Prove that the closure of the next chord progression is all the 12 notes of the chromatic scale. That is , it is a full chord progression.

G-> F#7->B7->Em(OR E7) ->C->B7->E7->Am(OR A7) ->F->E7->A7->Dm (Or D7).

4) Verify that the well known melody of harry Potter films, has chromatic degree 11, as it utilizes all notes except F#.

https://www.youtube.com/watch?v=kLQ_ykifs0A

85. Chords that are played sequentially with 3-shapes in 2-octaves along the fretboard neighborhoods , for fuller chord voicing

This technique was mentioned already in various posts like post 3 , post  13 etc. But we devote to it a new separate post.

84. 2-string scales for 2-voices soloing