THE CHORDS PLAY THE ROLE OF A CHOIR PLAYS IN A BAND
In mathematics and in particular in the area of Universal Algebra, a relational algebra is a set endowed with (possibly partially defined) relations and operations of its elements.
Chord-Relations
1) The relation of the root distance (Chords that are Melodically in series, complementary chords, Chords resolution by intervals of pure 4ths)
2) the relation of common notes (strong and weak relatives) (Chords harmonically in series and in parallel)
3) The group relation or n-tuple relation of n-chords , as belonging in the same scale,n=3,4,5,6,7 etc
Chord-order
4) The harmonic order by the harmonic score (see post 40)
Chord-operations
5) The operation symbolized by V of superposition or composition of chords (3.g. two 3 note chords to a 4-note chord).
The relation algebra of chords is beyond a single tonality. But it is not arbitrary atonality or arbitrary multi-tonality either.
In the next videos one can see how melodic themes of notes but also of chords and mutations of them can be created by keeping invariant an initial pattern of interval shifts of a note or chord and initial pattern of sequence of melodic themes of notes or chords after seeminly random pauses (omittings) of the parts of the fixed pattern
https://www.youtube.com/watch?v=7HPkTMYoJnI
https://www.youtube.com/watch?v=sb3e4Mq6y3s
https://www.youtube.com/watch?v=w0-Ljf5gm4A
https://www.youtube.com/watch?v=Fc16Y1gKUDc
https://www.youtube.com/watch?v=w0-Ljf5gm4A
Also small chord-sub-progressions may be considered as chord-progressions themes, and the whole chord progression may be derived from them by variations (See post 106 about the harmonic seed)
For the correlation of melodies with chords that fit to them, or conversely , melodies that can be improvised over a chord progression the next local concept is very significant: The melodic closure of a chord: This is defined as the closed interval of notes from all the 12-tone (chromatic) scale) with lower end the lowest note of the chord, and highest end the highest note of the chord. The chord is assumed within an octave, and normal positions, 1st inversion, and 2nd inversion have different closures. It holds the next interesting theorem. The melodic closure is also conceivable as transportable in all octaves, and not only where the chord is. If we define randomly a melody within a the closure of a chord in normal position and no other note outside it, with uniform probability of occurrence of any of the notes of the closure, then according to the local condition of fit of a piece of melody with a chord the only chord in normal position that would fit this melody is the one with this as its closure!. Or more generally of we define as probabilities of sounding a note on all the octave an equal value for all notes except at the notes of the chord X where we have as probability the double this value (e.g. sound each note of the octave once but the notes of the chord twice) then any such random melody with this probability structure will have as its fitting underlying chord the chord X.
The diatonic progressions is the sequence (iii->vi->ii->V->I->IV->VII->iii), This progression leads from sadness to joy, from the triad of minor chords to the triad of major chords.
In a diatonic scale, the triad of minor chords (sad triad) is the (iii->vi->ii) where the (iii, vi) and (vi,ii) are consecutive in the cycle of pure 4ths, with standard resolutions (iii7-> vi) , (vi7->ii) and the
(ii, iii) are complementary chords, in other words all of their notes give all the notes of the scale except one.
The triad of joy or triad of major chords is the (V, I, IV) , where the (V, I) and (I,IV) are consecutive in the cycle of pure 4ths, with standard resolutions (V7-> I) , (I7->IV) and the (IV, V) are complementary chords, in other words all of their notes give all the notes of the scale except one.
The bridge between these two triads is the well known jazz progression (ii7, V7, I) , where again the (ii, V) and (V,I) are consecutive in the cycle of pure 4ths, with standard resolutions (ii7-> V) , (V7->I), and the (ii, I) are complementary chords, in other words all of their notes give all the notes of the scale except one.
The relational algebra of chords, relative to a diatonic mode or scale can be visualized very well with the 2-dimensional hexagonal tonality grid for chords geometric representation. (See post 34)
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.
ABOUT THE HARMONIC ORDER OF CHORDS BY THE HARMONIC SCORE.
This hexagonal grid,is defined by the following rules
1) Horizontal sequences of notes differ by the interval of perfect 5th (7 semitones) or its inverse perfect 4th (5 semitones) (Chords harmonically in series)
2) Diagonal sequences of notes differ by an interval of major 3rd (4 semitones) o minor 3rd (3 semitones) (Chords harmonically in parallel)
Chords, are represented as triangles or rombuses in this hexagonal grid.
(See e.g. the visualization software for music MAM, http://www.musanim.com/
http://www.musanim.com/player/MAMPlayerUserGuide.pdf )
As an alternative we may define a square grid. The rules are:
1) Horizontlly from left to right the interval is a perfect 4th (5 semitones)
2) Vertically from down to up it is alternating 3rd major and 3rd minor intervals.
In the latter square grid, every 3x3 square are the notes of a diatonic scale. The relation of scales and chords with common notes are directly asilly visible. Roots of alternating major moinor relative chords are on vertical lines.
Also we may define a 24, cycle in alternating major/minor intervals of 3rds , and also alternating major/minor chords , so that consecutive chords are relative chords, and as 3s+4s=7s, it is also a refinement of the cycle of 5ths. In the next we state this cycle in the reverse order, which seems as if a refinement of the cycle of 4ths
(434343434343434343434343)
G C F Bb Eb Ab Db Gb B E A D
Em Am Dm Gm Cm Fm Bbm Ebm Abm Dbm Gbm Bm
We may add two more series based on that mnot-major chords with the same root are relative chords too, so as to havea 2-dimensional grid based of the relation of relatives.
Gm Cm Fm Bbm Ebm Abm Dbm Gbm Bm Em Am Dm
G C F Bb Eb Ab Db Gb B E A D
Em Am Dm Gm Cm Fm Bbm Ebm Abm Dbm Gbm Bm
E A D G C F Bb Eb Ab Db Gb B
It is often very instructive to chart the chords of a song over this 24-cycle of relatives (or double cycle of 4ths) or the above 2-dimensional grid.
Two consecutive chords in the 12-chords sub-cycle of 4ths in this 24-cycle (in red) have only one note in common, the roots differ by an interval of 5th, as they extend beyond one octave, (e.g. the chords G, C extends from C to D of the next octave of the Cmajor scale ) and that is why they are useful in melodies that may extend in one octave. Three consecutive chords in the major 12-chords sub-cycle of 4ths in this 24-cycle (in red), define essentially a tonality, with root the root of the middle chord. But it is obvious that the use of this 24-cycle is beyond one tonality, but it respects the basic relations of chords derived within one tonality. It is therefore a map of systematic and meaningful cycle of modulations.
Most often a song is a connected interval of chords in the cycle of 4ths, with their intermediate relative chords.
See also https://www.youtube.com/watch?v=QRbUgIdy3HY
and also
https://www.youtube.com/watch?v=TRz73-lSKZA
https://www.youtube.com/watch?v=RGe62Rpo7TM
See also a video about the 2-dimensional Terpstra Keyboard
which is an improvement of the Lippens Keyboard
Terpstra Keyboard:
https://www.youtube.com/watch?v=Nb_TQpwam54&t=293s
Lippens Keyboard
https://www.youtube.com/watch?v=k3XveYlhyFs&t=20s
In mathematics and in particular in the area of Universal Algebra, a relational algebra is a set endowed with (possibly partially defined) relations and operations of its elements.
Chord-Relations
1) The relation of the root distance (Chords that are Melodically in series, complementary chords, Chords resolution by intervals of pure 4ths)
2) the relation of common notes (strong and weak relatives) (Chords harmonically in series and in parallel)
3) The group relation or n-tuple relation of n-chords , as belonging in the same scale,n=3,4,5,6,7 etc
Chord-order
4) The harmonic order by the harmonic score (see post 40)
Chord-operations
5) The operation symbolized by V of superposition or composition of chords (3.g. two 3 note chords to a 4-note chord).
The relation algebra of chords is beyond a single tonality. But it is not arbitrary atonality or arbitrary multi-tonality either.
In the next videos one can see how melodic themes of notes but also of chords and mutations of them can be created by keeping invariant an initial pattern of interval shifts of a note or chord and initial pattern of sequence of melodic themes of notes or chords after seeminly random pauses (omittings) of the parts of the fixed pattern
https://www.youtube.com/watch?v=7HPkTMYoJnI
https://www.youtube.com/watch?v=sb3e4Mq6y3s
https://www.youtube.com/watch?v=w0-Ljf5gm4A
https://www.youtube.com/watch?v=Fc16Y1gKUDc
https://www.youtube.com/watch?v=w0-Ljf5gm4A
Also small chord-sub-progressions may be considered as chord-progressions themes, and the whole chord progression may be derived from them by variations (See post 106 about the harmonic seed)
For the correlation of melodies with chords that fit to them, or conversely , melodies that can be improvised over a chord progression the next local concept is very significant: The melodic closure of a chord: This is defined as the closed interval of notes from all the 12-tone (chromatic) scale) with lower end the lowest note of the chord, and highest end the highest note of the chord. The chord is assumed within an octave, and normal positions, 1st inversion, and 2nd inversion have different closures. It holds the next interesting theorem. The melodic closure is also conceivable as transportable in all octaves, and not only where the chord is. If we define randomly a melody within a the closure of a chord in normal position and no other note outside it, with uniform probability of occurrence of any of the notes of the closure, then according to the local condition of fit of a piece of melody with a chord the only chord in normal position that would fit this melody is the one with this as its closure!. Or more generally of we define as probabilities of sounding a note on all the octave an equal value for all notes except at the notes of the chord X where we have as probability the double this value (e.g. sound each note of the octave once but the notes of the chord twice) then any such random melody with this probability structure will have as its fitting underlying chord the chord X.
The diatonic progressions is the sequence (iii->vi->ii->V->I->IV->VII->iii), This progression leads from sadness to joy, from the triad of minor chords to the triad of major chords.
In a diatonic scale, the triad of minor chords (sad triad) is the (iii->vi->ii) where the (iii, vi) and (vi,ii) are consecutive in the cycle of pure 4ths, with standard resolutions (iii7-> vi) , (vi7->ii) and the
(ii, iii) are complementary chords, in other words all of their notes give all the notes of the scale except one.
The triad of joy or triad of major chords is the (V, I, IV) , where the (V, I) and (I,IV) are consecutive in the cycle of pure 4ths, with standard resolutions (V7-> I) , (I7->IV) and the (IV, V) are complementary chords, in other words all of their notes give all the notes of the scale except one.
The bridge between these two triads is the well known jazz progression (ii7, V7, I) , where again the (ii, V) and (V,I) are consecutive in the cycle of pure 4ths, with standard resolutions (ii7-> V) , (V7->I), and the (ii, I) are complementary chords, in other words all of their notes give all the notes of the scale except one.
The relational algebra of chords, relative to a diatonic mode or scale can be visualized very well with the 2-dimensional hexagonal tonality grid for chords geometric representation. (See post 34)
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.
ABOUT THE HARMONIC ORDER OF CHORDS BY THE HARMONIC SCORE.
After the Helmholtz studies (see post 24 ) how good or not good an intervals of notes sounds depends on how many common harmonics they have. The Helmholtz diagram defines also the harmonic hierarchy of the intervals, where the best sounding is of course the unison (0 semitones) distance then the 2nd best is that of the octave (12 semitones) and the 3rd best that of the 5th (7 semitones). These two intervals 0-7-12 define also the R5 chord. The next table gives the harmonic order of the intervals.
In the Helmholtz just intonation or temperament of intervals (see e.g. https://en.wikipedia.org/wiki/Just_intonation and http://www.phy.mtu.edu/~suits/scales.html ), the next small number rational ratios of frequencies correspond to the next musical intervals.
Interval Bach scale semitones Helmholtz scale rational ratio Harmonic score
Unison (Bach scale 0 semitones) 1 1/1
Octave (12 semitones) 2 1/2
5th perfect (7 semitones) 3/2 1/(3+2)
4th pure (5 semitones) 4/3 1/(4+3)
6th major (9 semitones) 5/3 1/(5+3)
3rd major (4 semitones) 5/4 1/(5+4)
3rd minor (3 semitones) 6/5 1/(6+5)
7th minor (9 semitones) 9/5 1/(9+5)
2nd major (2 semitones) 9/8 1/(9+8)
7th major (10 semitones) 15/18 1/(15+18)
6th minor (8 semitones) 25/16 1/(25+16)
2nd minor (1 semitone) 25/24 (9/8) 1/(25+24) or 1/(8+9)
5th diminished (6 semitones) 45/32 1/(45+32)
The smaller the numerator and denominator of the rational ratio of the pitches (in the Helmholts scale) the larger the number of harmonics that are common, when two notes sound with this ratio and thus the better and more harmonic , the interval sounds. In the Bach scale this is a bit ruined but not too much (see also post 24, and if the reader understands Greek also post 25, in the .pdf manuscript page 53).
Therefore the order of the intervals in the above table is also the order of how good their harmony is when the interval sounds is isolation of anything else.
This harmonic clasification of intervals goes back to ancient Pythagorean ideas, but also to modern discoveries of the nature of senses. For example the flavors and smells are good or bad according to if the "chord" of ultra high frequency sonic high frequencies of its molecules, is harmonic or not . About this see for example the next video.
http://www.ted.com/talks/luca_turin_on_the_science_of_scent
This harmonic clasification of intervals goes back to ancient Pythagorean ideas, but also to modern discoveries of the nature of senses. For example the flavors and smells are good or bad according to if the "chord" of ultra high frequency sonic high frequencies of its molecules, is harmonic or not . About this see for example the next video.
http://www.ted.com/talks/luca_turin_on_the_science_of_scent
Notice that the intervals of perfect 4th and 6th sound better than the intervals of 3rd, and the interval of 7th minor better than the interval of 2nd major. The worst of all is the interval of 6 semitone (4th augmented or 5th diminsihed).
Utilizing the harmonic scores in the above table we can define also harmonic scores of the various chords, by adding the scores of all of their intrevals in all possible ways they shape, divided by the number of these shaped intervals. Let us calculate for example the harmonic scores of the major, minor triads , the R5 chord and the some more chords.
The major triad has intervals M3 m3, P5 , so its harmonic score H(R) is
H(R)=1/3*([1/(5+6)]+[1/(4+5)]+[1/(3+2)])=([1/11]+[1/9]+[1/5])/3=0.134
The harmonic score ofthe minor is the same:
H(Rm)=1/3*([1/(5+6)]+[1/(4+5)]+[1/(3+2)])=([1/11]+[1/9]+[1/5])/3=0.134
The harmonic score of the R5 chord is the highest:
H(R5)=([1/(3+2)]+[1/(4+3)]+[1/2])/3=0.28
The harmonic score of the R7 is the average of the harmonic scores of it intervals that is
M3, m3, P5, m3, m7, m5
H(M3)=[1/(4+5)]=0.111, H(m3)=([1/(5+6)]=0.090, H(P5)=[1/(3+2)]=0.2, H(m7)=[1/(9+5)]=0.071
H(m5)=[1/(32+45)]=0.012
H(R7)=1/6*([1/(5+6)]+[1/(4+5)]+[1/(3+2)]+[1/(6+5)]+[1/(9+5)]+[1/(45+32)])=0.095
The harmonic score of the Rmaj7 is better than that of R7:
It is the average of the harmonic scores of its intervals M3, m3, P5, M3, P5, M7, and H(M7)=1/(15+18)=0.030
H(Rmaj7)=1/6*([1/(5+6)]+[1/(4+5)]+[1/(3+2)]+[1/(15+18)]+[1/(4+5)]+[1/(3+2)])=0.123
The harmonic score of the Rm7 is less than that of Rmaj7 but greater than tha of R7, and it is the average of the harmonic scores of its intervals M3, m3, P5, m3, P5 , m7,
H(Rm7)=1/6*([1/(5+6)]+[1/(4+5)]+[1/(3+2)]+[1/(3+2)]+[1/(6+5)]+[1/(18+15)])=0.120
The harmonic score of the Rm7b5 is the average of the harmonic scores of its intervals m3, m3, m3,m5, m5, m7,
H(Rm7b5)=1/6*([1/(5+6)]+[1/(6+5)]+[1/(6+5)]+[1/(45+32)]+[1/(9+5)]+[1/(45+32)])=0.06
While the harmonic score of the Rdim7 is slightly better than that of Rm7b5 and it is the average of the harmonic scores of its intervals m3, m3, M2,m5, P4, m6,
H(Rdim7)=1/6*([1/(5+6)]+[1/(6+5)]+[1/(24+25)]+[1/(45+32)]+[1/(16+25)]+[1/(4+3)])=0.063
The harmonic score of the Raug is better than that of Rdim7 ,Rm7b5 and is the average of the harmonic scores of its intervals M3, M3, m6,
H(Raug)=([1/(4+5)]+[1/(4+5)]+[1/(25+16)])/3=0.082
We notice also here that the more notes, a chord has the less would be its harmonic score, compared to chords with few only notes
H(Raug)=([1/(4+5)]+[1/(4+5)]+[1/(25+16)])/3=0.082
We notice also here that the more notes, a chord has the less would be its harmonic score, compared to chords with few only notes
Therefore the harmonic order of the above chords are
R5>R, Rm>Rmaj7> Rm7>R7>Raug>Rm7b5>Rdim7
R5>R, Rm>Rmaj7> Rm7>R7>Raug>Rm7b5>Rdim7
We must remark here that these harmonic scores refer to the root position of the chord. The invesrions will have different harmonic score!
In othe words for the minor triad , the 1st invesrion has the highest harmonic score,while the 2nd inversion has also higher score than the root position!
The harmonic score of the invesrions of the major and minor triads are the next
H(R, 1st invesion 3-5-1)=1/3*(H(m3)+H(P4)+H(P5))=1/3*([1/(5+6)]+[1/(4+3)]+[1/(3+2)])=0.144
H(R, 2nd invesion 5-1-3)=1/3*(H(M3)+H(P4)+H(P5))=1/3*([1/(4+5)]+[1/(4+3)]+[1/(3+2)])=0.151
In othe words for the major triad , the 2nd invesrion has the highest harmonic score,while the 1st inversion has also higher score than the root position!
H(Rm, 1st invesion 3-5-1)=1/3*(H(M3)+H(P4)+H(P5))=1/3*([1/(5+6)]+[1/(4+3)]+[1/(3+2)])=0.151
H(Rm, 2nd invesion 5-1-3)=1/3*(H(m3)+H(P4)+H(P5))=1/3*([1/(4+5)]+[1/(4+3)]+[1/(3+2)])=0.144
In othe words for the minor triad , the 1st invesrion has the highest harmonic score,while the 2nd inversion has also higher score than the root position!
This hexagonal grid,is defined by the following rules
1) Horizontal sequences of notes differ by the interval of perfect 5th (7 semitones) or its inverse perfect 4th (5 semitones) (Chords harmonically in series)
2) Diagonal sequences of notes differ by an interval of major 3rd (4 semitones) o minor 3rd (3 semitones) (Chords harmonically in parallel)
Chords, are represented as triangles or rombuses in this hexagonal grid.
(See e.g. the visualization software for music MAM, http://www.musanim.com/
http://www.musanim.com/player/MAMPlayerUserGuide.pdf )
For hexagonal representation of tonality effects see also the schismatic temperament
See also post 99 about the hexagonal Tersptra keyboards
AND POST 310 ABOUT ISOMORPHIC 2-DIMENSIONAL LAYOUTS OF NOTES
1) Horizontlly from left to right the interval is a perfect 4th (5 semitones)
2) Vertically from down to up it is alternating 3rd major and 3rd minor intervals.
In the latter square grid, every 3x3 square are the notes of a diatonic scale. The relation of scales and chords with common notes are directly asilly visible. Roots of alternating major moinor relative chords are on vertical lines.
Also we may define a 24, cycle in alternating major/minor intervals of 3rds , and also alternating major/minor chords , so that consecutive chords are relative chords, and as 3s+4s=7s, it is also a refinement of the cycle of 5ths. In the next we state this cycle in the reverse order, which seems as if a refinement of the cycle of 4ths
(434343434343434343434343)
G C F Bb Eb Ab Db Gb B E A D
Em Am Dm Gm Cm Fm Bbm Ebm Abm Dbm Gbm Bm
We may add two more series based on that mnot-major chords with the same root are relative chords too, so as to havea 2-dimensional grid based of the relation of relatives.
Gm Cm Fm Bbm Ebm Abm Dbm Gbm Bm Em Am Dm
G C F Bb Eb Ab Db Gb B E A D
Em Am Dm Gm Cm Fm Bbm Ebm Abm Dbm Gbm Bm
E A D G C F Bb Eb Ab Db Gb B
This grid which is also a table as below shows clearly the chords that are harmonically in series(interval of 4th or 5th) and chords that are harmonically in parallel (intervals of 3rds or 6ths). But it does not show of course the relations of chords that are melodically in series
Gm
|
Cm
|
Fm
|
Bbm
|
Ebm
|
Abm
|
Dbm
|
||||||
G
|
C
|
F
|
Bb
|
Eb
|
Ab
|
Db
|
||||||
Em
|
Am
|
Dm
|
Gm
|
Cm
|
Fm
|
|||||||
E
|
A
|
D
|
G
|
C
|
F
|
Gbm
|
Bm
|
Em
|
Am
|
Dm
|
||||||
Gb
|
B
|
E
|
A
|
D
|
||||||
Bbm
|
Ebm
|
Abm
|
Dbm
|
Gbm
|
Bm
|
|||||
Bb
|
Eb
|
Ab
|
Db
|
Gb
|
B
|
It is often very instructive to chart the chords of a song over this 24-cycle of relatives (or double cycle of 4ths) or the above 2-dimensional grid.
Two consecutive chords in the 12-chords sub-cycle of 4ths in this 24-cycle (in red) have only one note in common, the roots differ by an interval of 5th, as they extend beyond one octave, (e.g. the chords G, C extends from C to D of the next octave of the Cmajor scale ) and that is why they are useful in melodies that may extend in one octave. Three consecutive chords in the major 12-chords sub-cycle of 4ths in this 24-cycle (in red), define essentially a tonality, with root the root of the middle chord. But it is obvious that the use of this 24-cycle is beyond one tonality, but it respects the basic relations of chords derived within one tonality. It is therefore a map of systematic and meaningful cycle of modulations.
Most often a song is a connected interval of chords in the cycle of 4ths, with their intermediate relative chords.
See also https://www.youtube.com/watch?v=QRbUgIdy3HY
and also
https://www.youtube.com/watch?v=TRz73-lSKZA
https://www.youtube.com/watch?v=RGe62Rpo7TM
See also a video about the 2-dimensional Terpstra Keyboard
which is an improvement of the Lippens Keyboard
Terpstra Keyboard:
https://www.youtube.com/watch?v=Nb_TQpwam54&t=293s
Lippens Keyboard
https://www.youtube.com/watch?v=k3XveYlhyFs&t=20s
