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.
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
