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Tuesday, March 1, 2016

40. Classification of the intervals in the fretboard, and INTERVALS IMPROVISATION.

We must remind here the fundamental philosophy of musical composition and improvisation.
Musical improvisation is not a technical skill that one “learns to do.” It is a natural spontaneous process that occurs first in the imagination. It is often a natural language of the soul, as we have the language of words. But that is why it is understood by people that do not even speak the same language of words

The main goal of musical composition and improvisation is not the output musical piece, but the EXISTENTIAL FUNCTION of the process of creating and listening the musical piece. 
Here is an example of mediative improvisation by Estas Tonne

https://www.youtube.com/watch?v=7gphiFVVtUI

For the ideas discussed in this post see also post 35 about interval-chords

The place of intervals among other entities like notes, melodic themes and chords is the next table. 
We may compare the harmonic method of musical composition, with the way that we shape sentences of meaning in our minds before we choose the exact words to speak  them. We need at first an analogy between the musical language and speaking language.

Here is a table of the analogy and correspondence of the levels of the musical language and Speaking languages



MUSICAL LANGUAGE
SPEAKING LANGUAGE
Note
Letter  of the alphabet
Interval (3 elementary melodic moves)
Syllables
Melodic moves or themes (5 basic  melodic patterns and Dolphin words as in post 101). Chords
Words that make a simple proposition (subject verb object) 


Full melody (propositions) . Chords progressions at a duration that may contain many melodic themes (Phrase).
Propositions . Phrases or sentences from a point to  a next point , that may contain many simple propositions 




This classification must be studied before , the classification of triads, chords and scales in the fretboard.
When playing melodies in the guitar what matters more is the relative position of notes rather, than the absolute pitch of them. All our western musical system is based on uniformity of translating (transposing) melodies in lower or higher pitch, which makes the distances of the notes a priority in what matters.

We may also classify the intervals to 

VERY SMALL  in semitones  1, 2 THEY ARE PLAYED IN THE SAME STRING USUALLY

SMALL              in semitones 3, 4 THEY ARE PLAYED IN TWO SECLUSIVE  STRINGS                                                            USUALLY
MEDIUM           in semitones  5,6 7 THEY ARE PLAYED IN TWO SECLUSIVE  STRINGS                                                            USUALLY                                                                         
BIG                  in semitones 8,9,10,11, 12 THEY ARE PLAYED USUALLY IN STRINGS,                                                                         THAT ARE TWO    STRINGS  APART  E.G.6TH-                                                                       4TH, 5TH-3RD ETC  
VERY BIG          in semitones >12            THEY ARE PLAYED USUALLY IN STRINGS,                                                                           THREE    STRINGS  APART  E.G.6TH-3RD,                                                                             5TH-2ND ETC      

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.


Why we have 12 notes in music?


https://www.youtube.com/watch?v=IT9CPoe5LnM&t=409s

Alternative to western system is the chinese musical system

https://simple.wikipedia.org/wiki/Chinese_musical_system

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.

The remarkable book by Helmholtz On the sensation of tone is online here


https://archive.org/details/onsensationsofto00helmrich



In page 193 of the book of Helmholtz is found also the experimental diagram of how the intervals of just temperament sound in dissonance or not in the human ear. The higher the score on the vertical axis the greater the dissonance. The lower end of  the horizontal x-axis is the note C, and we see also across the horizontal axis the notes e, f, g, a , b    etc.


The smaller a musical interval within an octave the more it signifies sadness, and the more distant, (with maximum a 4th or 5th, as we consider also their inversions) the closer it is to joy.

But the next harmonic score is not about melodies and sadness-joy feelings but more about harmony and stress-serenity feelings.

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      (or 9/8)                      1/(25+24)
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).

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

See also about the Pythagorean tuning

https://en.wikipedia.org/wiki/Pythagorean_tuning

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.

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

Now here one may ask, why then a major triad sounds more harmonious and joyful than a minor triad which sounds more sad? The answer is that the same harmonic score of the minor 3rd interval (3 semitones) sounds less harmonious when it is in lower frequencies, than when it is in higher frequencies. And this explains the different feeling of the major and minor chord, although they have the same harmonic score! 

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

Therefore the harmonic order of the above chords are

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! 

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 other words for the minor triad , the 1st  inversion has the highest harmonic score,while the 2nd inversion has also higher score than the root position!



We may as well calculate the harmonic score of the moves of a melody, and produce an oscillating diagram of it (Melody harmonic score oscillator)


Here is a not irrelevant video on microtonal and polychromatic music

https://www.youtube.com/watch?v=yVKIXCH-5gE



All the intervals,  on the guitar fretboard from the root met in the chords R, Rm, R7, Rmaj7, Rm7, Rdim Raug  in other words the m2=1,M2=2,m3=3,M3=4,P4=5,P5=7,M6=9,m7=10,M7=11.

Minor 2nd interval=1 semitone

Minor 2nd interval starting on the 1st, 4th and 6th strings
Minor 2nd interval starting on the 2nd, 3rd and 5th strings


Major 2nd interval=2 semitones

Major 2nd interval starting on the 1st, 4th and 6th strings

Major 2nd interval starting on the 2nd, 3rd and 5th strings


Minor 3rd intevals= 3 semittones

Minor 3rd interval starting on the 1st, 4th and 6th strings

Minor 3rd interval starting on the 2nd, 3rd and 5th strings


Major 3rd interval= 4 semitones

Major 3rd interval starting on the 1st, 4th and 6th strings

Major 3rd interval starting on the 2nd, 3rd and 5th strings

Perfect 4th interval =5 semitones
Perfect 4th interval starting on the 1st, 4th and 6th strings

Perfect 4th interval starting on the 2nd, 3rd and 5th strings

Pute 5th interval=7 semitones

Perfect 5th interval starting on the 1st, 4th and 6th strings

Perfect 5th interval starting on the 2nd, 3rd and 5th strings

Major 6th interval=9 semitones

Major 6th interval starting on the 1st, 4th and 6th strings

Major 6th interval starting on the 2nd, 3rd and 5th strings

Minor 7th interval=10 semitones

Minor 7th interval starting on the 1st, 4th and 6th strings

Minor 7th interval starting on the 2nd, 3rd and 5th strings

Major 7th interval=11 semitones

Major 7th interval starting on the 1st, 4th and 6th strings

Major 7th interval starting on the 2nd, 3rd and 5th strings



Summarizing with root on the 6th string the M2, M3, P4, P5 ,M6, M7






See also

https://youtu.be/ooPJNYm299k



https://www.youtube.com/watch?v=nmtYaj59slU

http://www.fretjam.com/guitar-intervals-fretboard.html



FOR INTERVAL IMPROVISATION SEE E.G.

https://www.youtube.com/watch?v=binjvIqAN1s

There are of course more types of interval improvisation , that no root is kept always fixed, but the only rule is that at each time an interval sounds rather than only ne  note.


A very useful remark for improvisation of melody within a particular chord is the next.
Suppose we are at a note y1 of the melody which fits the underlying chord with notes x1x2x3 (whatever that may mean), then depending on the particular position of y1 relative to the x1x2x3, a shift by an interval of 3rd, 4th, 5th, and 6th wil lead to a note y2 that will again fit the chord!. This is because the relative positions of the notes x1x2x3 of the chord are intervals of major, minor 3rd and pure 5th, and their complementary intervals relative to the octave are minor or major 6th, and pure 4th 

SINGING INTERVALS:
https://www.youtube.com/watch?v=Sz-U0X7LBRA