2 parallels: Herb Pomeroy - Yuzef Kon.
All those who studied at Berklee, are familiar with the course, referred to as Line Writing; who founded the Herbie Pomeroy in the late 60's. One of the interesting moments in this technique is to calculate the level of dissonant voicings, that is actually the development of the Hindemith's theory.
But who knows, that in 1970 in Russia, published an article by musicologist Yuzef Kon "A property of the vertical in the atonal music of A. Schoenberg and A. Berg," where he suggested - quite apart from Pomeroy - his method calculate the degree of dissonant chords, very reminiscent of the approach J. Pomeroy (Pomeroy actually learned the power of Line Writing from someone else, but I did not know his name). This article has never been translated into other languages, so that the West is practically unknown.
This approach relates to Kon atonal harmonies, while Line Writing Pomeroy refers to the arrangement on the basis of modality;however, Kohn calculations are made much more detail.
The system works fantastically, especially in the absence of tonal, modal and functional relationships between chords. They can simply invent!
Y. Kon modified the interval series in order of increasing dissonant by Hindemith.
Each interval is assigned a number, denoting the degree of its dissonant, compactness and saturation. Unison (0) and octave (1) occupy a special position as consisting of the same sounds, then there is a gap in the numbering, and lasts from 3 to 13.
enharmonic intervals is not taken into account.
This numbering is suitable only for simple intervals .interval wider than an octave requires correction, which is denoted by subtracting the rate of the simple interval of the corresponding octave – i.e. 1.Each subsequent expansion octave requires the following amendments: more than 2 octaves - minus 0,5; more than 3 octaves - minus 0.25, more than 4 octaves - minus 0,125.
This factor-correction is already characterized, in addition to dissonant, and the location - a narrow or wide.
Practical applications with examples from piano pieces op.11 of Shoenberg.
Before we demonstrate the practical use of the approach by Y. Kon, must become acquainted with what is happening in conventional harmonic cadence IV- VI- I and II m7- V7- I maj7.
1. Analysis of chords in the usual harmonic cadence IV-VI-I
One should pay attention to the big jump in the degree of dissonant in dominant seventh , while the surrounding chords balanced.
2. Cadenza II m7-V7-I maj7. Dominant chord, although here the most dissonant , but creates less contrast with the neighboring seventh chords - a known phenomenon to jazz piano players.
3. Therefore it is necessary to add additional tension pitch.
New II - V – I : An example of real chords "cooking", based on ear, and using the approach of Y. Kon. Functionality is preserved only in the bass line.
Here's a practical example of applying the method Y. Kon - in free jazz . Over the piano voicings marked degree of their dissonant. Chords were found by ear, but the voice-leading, ups and downs tensions serve as a guide for selection of selection of these sounds :
Traditional harmonization in tonal music - including traditional jazz - based on the functional relationship between the degrees of major and minor scales and chords built on these degrees, ;and performing the main principles of stability and instability, of tension and resolution .So the choice of chords depends on an understanding or sense of tonal relationships within the melody, if the melody was created before the harmony . This is the first and the main power acting in harmony and chords .
When you select the chords in accordance with the harmonic functionality is required to link them with each other using voice-leading. Hence - the use of closed and open positions. Also among the chords support the principle of instability - stability, tension - resolution, Hi-Lo, etc. In connection between chords and melody, melodic notes are or part of the chord, or embellishment of chord tones, or are built on the scale of chord .
All this is elementary and well known .
But what happens in chords, when tonal context falter, become more vague, or completely dissolve? What happens to the building, which has destroyed the foundation?
The presence of three functions in tonality - not from God; He gave us just the tonic.Тhat occurred in the era of the old modality.
The disappearance of chord logic based on the functional relationship does not mean the disappearance of logic in general, but its transfer to other tracks, Y. Kon raises another property - chords density. In his view, this property has always existed, but has always remained in the shadows - because of functionality. "With the weakening or disappearance of functional chord density, its variation and ripple" become a factor in determining the movement of harmony.
The density of the chord consists of:
1. set of intervals
We used a method of Yuzef Kon, where we calculated a measure of the density and dissonant intervals between each pair of chords voices, and then taking stock total for each voicing. Were obtained the following results:
In cadenza Dm7 – G7 – CMaj7 ——------- [67.5 - 75 - 67.5]
In cadenza D Dorian – G mixo – CIo —— [76.5 - 72 - 50.5 ]
In cadenza Dm7/b5 – G7/b9 – Cm6 ——- [45 - 52 - 44 ]
In cadenza D Loc- G phryg – C mel min —[82 - 62.5 - 81.5]
The results showed the following: standard conventional voicings in tonal cadence 2-5-1 proportional and balanced with each other ; relevant principles tension-resolution , in which the most intense and unstable chord is located in the middle of the pattern.The density of the first and third voiсing either identical or very close.
In the modal cadence, however, the situation is different. First, it is clear that the chords are found by ear intuitively. And yet, in the second modal cadences first and third voicing balanced; in first modal cadence - doesn't . However, in both modal cadence middle voicing is weaker than the first .
What is it - accident or intention?
To create a modal chord progression will need to be assessed their constituent intervals in terms of the degree of their dissonant. So, I place here a few tables with the system increases dissonant intervals + bass in the 7 diatonic modes.
In the Ionian and Dorian modes:
Added voicings in Phrygian and Lydian , as well as a small example of the modal comping for improvisation:
Signed numbers - determining the density of each assonance - in accordance with the tables above. They were found by ear, but the horizontal logic is clear, and if you need more full chords, the above tables show, where they can search.
It is still preferable, than blindly poking fingers into keyboard ...
Method Y. Kon’s analysis also includes take account of the register (also numerical) , without which chords evaluation can’t be complete. What determines the index - the location of the lower note .
This evaluation system registers constructed as follows:
For sub contra octave and contra octave------------- 3
For great octave ---------------------------------------4
For small octave ---------------------------------------5
For One-Line Octave-----------------------------------6
For Two-Line Octave ----------------------------------7
For Three-Line Octave---------------------------------8
For Four-Line Octave ---------------------------------9
For Five-Line Octave---------------------------------10
The end result in evaluating the chords density
consists of dissonant minus factor of the chord spreading , divided by a factor of the register:
...........(I - S)
D = --------------
where D - density of the chord, I - factor of all intervals dissonant in a chord , S - factor of chord spreading , R - factor of register.
Analysis of a fragment of Monk's Crepuscule :
Pay attention to detail use changes in the density of chords - in connection with the climax points in the melody, as well as in connection with the rhythm (very important!).
Since the beginning of a more rapid movement is characterized by repeated low-density, which increases very balanced for accentuation.The highest density - in the second bar at two points in the most longest and accented chords.
Next, the overall density of jumping up and down - but elected within between 3 and 13 (secondary climax in the fifth bar ), while changes logical and balanced. The density of primary climax in bar 8 is balanced by comparison with secondary climax.
Only the last chord of fragment returns precisely primary density.
Amazing ear and intuition!