Deriving the Primordial Pentatonic Scale:
outward radiating symmetrical fourths vs upwards ascending fifths
The best way to approach this subject is to begin at square one.
Fundamental = 1st harmonic = A110
1st overtone = 2nd harmonic = A220
2nd overtone = 3rd harmonic = E330
3rd overtone = 4th harmonic = A440
As is known from the above, A220 is perfectly consonant with E330
(the 5th above),
and E330 is perfectly consonant with A440 (the 4th above).
Choosing our central point of reference to be D, it is obvious from
the abovementioned laws of physics that the two notes which are perfectly
consonant with D are the A (5th above or 4th below) and the G (4th
above or 5th below).
This is an indisputable fact.
Since 4ths are the inversions of 5ths (and vice versa), we have the
following 3-note pitch set which represents the primary level of perfect
consonances with D (with D as its central point of reference):
A D G
The primary consonants of A (which are E and D) and the primary consonants
of G (which are D and C) are derived using the same indisputable laws
of physics as referred to above, and form the secondary level of consonance.
The marriage of the two abovementioned 3-note consonant pitch sets
of E A D and D G C forms this radially symmetrical 5-note pitch set:
E A D G C
This radially symmetrical pitch collection of the first and second
levels of consonance, when rearranged, creates the 5-note radial symmetrical
A C D E G
This pitch set is THE primordial music scale, commonly called "The
Minor Pentatonic Scale".
This 5-note scale is arrived at in only two short steps via radial
There are those who would have us believe that an ascending order
of consecutive intervals of a fifth offers the most scientifically
sound basis upon which to structure an objective theory of music.
Let's examine that opinion more closely.
By stacking four perfect 5ths upwards beginning at a bottom note C,
we arrive at the following pitch set:
C G D A E
Which rearranged (with C at the bottom) yields the scale most commonly
known as the "Major Pentatonic Scale":
C D E G A
(Note that both the A minor pentatonic scale and the C major pentatonic
scale contain exactly the same collection of pitches)
When comparing arguments of deriving the "minor and major pentatonics"
via radial symmetry emanating outward from a central point versus
via ascending 5ths originating at a bottom point, one should constantly
bear in mind Ockham's Razor: "All things being equal,
the simplest solution tends to be the best one".
1. Number of Levels
Deriving the pentatonic scale via radial symmetry requires only
In order to arrive at the same pitch collection via upwardly ascending
stacked 5ths requires four levels.
2. Inherent Limiting Factor
When constructing a scale via ascending 5ths, there is no clear-cut
rationale as to why the number of notes in a scale should be limited
to five. One could just as easily posit a 4-note scale, a 6-note
scale (or any number of notes for that matter).
In the case of deriving the minor pentatonic scale via radial symmetry
there is an inherent limiting factor which makes it obvious as to
why there are exactly five notes in the scale - BECAUSE THAT'S WHERE
THE SECOND LEVEL OF CONSONANCE TAKES US TO.
3. Indirect Implication
In the case of the C major pentatonic constructed via ascending
5ths, there is no doubt that a C directly implies a G.
It is also true that a G directly implies a D. But the C does not
directly imply a D.
One could say, however, that the C INDIRECTLY implies a D (via the
By extension, the D also implies an A (which means that G indirectly
implies an A, and therefore for a C to imply an A requires TWO degrees
of indirect implication to get there).
For C to imply an E requires then THREE degrees of indirect implication!
In contrast, in the radial symmetrical method only has ONE degree
of indirect implication to arrive at the same pitch set.
4. Harmonics Generating Harmonics
Related to point 3 above, is a hypothesis that harmonics generating
harmonics will directly imply the notes in the ascending stack of
While it IS true that harmonics generate harmonics, keep in mind
that each respective harmonic becomes progressively weaker and weaker
and therefore less detectable AND less influential.
Beginning at middle C (C261.63Hz), its third harmonic G (the 5th)
would be at a frequency 3 times that (G784.89Hz).
3 times G784.89Hz (G's 3rd harmonic D) = D2354.67Hz (D "the 2nd")
3 times D2354.67Hz (D's 3rd harmonic A) = A7064.01Hz (A "the 6th")
3 times A7064.01Hz (A's 3rd harmonic E) = E21,192.03Hz (E "the 3rd")
To arrive at this dubious conclusion requires 4 relatively complex
calculations (as opposed to the radial symmetrical method's simple
Note that because these calculations above are NATURAL harmonics,
they contain big intonation problems due to the pythagorean comma.
Also note that E21,192.03Hz exceeds the range of human hearing.
5. Overtone Series Elements
Deriving a scale with the tonic of C by stacking fifths makes C
the de facto tonal center.
As regards the elements within the C major pentatonic scale, we
have a major 2nd (9th harmonic/8th overtone), a major 3rd (5th harmonic/4th
overtone), a perfect fifth (3rd harmonic/2nd overtone), and a major
6th (27th harmonic/26th overtone).
In order to imply relationships between C and the other notes in
the scale by using the overtone series, one is obliged to reference
higher harmonics (all the way up to the 27th).
This requires a great degree of chicanery to arrive at the conclusion
that C somehow directly implies an A (the 27th harmonic), as well
as a rather large leap of faith.
In contrast, the radial symmetrical derived version of the pentatonic
scale need only concern itself with the very simplest of harmonic
relationships - ie. the perfect 4th and the perfect 5th.
AND, in the radial symmetrical system, the A (C's 6th) occurs in
the VERY FIRST level of consonance.
5. Cadential Tendencies
The three strongest cadences in music are called "The Alpha Cadence"
(V-I), "The Beta Cadence" (IV-I) and "The Gamma Cadence" (bVII-I).
While the C major pentatonic scale contains the one melodic element
necessary to effect the Alpha Cadence (G > C [V-I]), the A minor
pentatonic scale contains ALL of the elements necessary to effect
ALL three of the abovementioned cadences:
Alpha Cadence: E > A (V-I)
Beta Cadence: D > A (IV-I)
Gamma Cadence: G > A (bVII-I)
In addition to the above three essential cadences, the movement from
the bIII to the I also has a strong cadential feel (qv. Cream's "Spoonful").
In contrast, the C major pentatonic scale only exhibits ONE possible
cadence, and motion from any of the other notes to the C is not considered
to be a cadence:
D > C (not a cadence)
E > C (not a cadence)
A > C (not a cadence)
The significance of the comparison of the abovementioned cadential
tendencies is that
ALL ROADS LEAD TO THE RADIALLY SYMMETRICAL
"A MINOR PENTATONIC SCALE" TONIC
as opposed to the non-symmetrical C major pentatonic scale (which
only has one cadence leading to its tonic).
This confirms the A minor pentatonic scale's predominance
over the C major pentatonic scale, and makes it conclusively evident
that the C major pentatonic is a subordinate mode of the A minor pentatonic
(and not the other way around).
The above demonstrates clearly that the most scientifically
sound basis upon which to construct scales in music is by means of
radially symmetrical derived structures (not by an ascending order
of consecutive intervals of a fifth).
While the evidence presented here only deals with pentatonic
scales, the extremely simple principles of the laws of physics involved
in musical radial symmetry remain true right across the board with
no exceptions (as can be seen in the music theory book "Modalogy
- scales, modes & chords: the primordial building blocks of music"
(qv. pg 9 'The Evolution of the Major Scale' and pg 40 'The
Derivation of Radially Symmetrical Altered Scales').
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copyright ©2008/2013 Jeff Brent