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 pitch set:

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 symmetry.

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

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 two steps.

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 G).

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 5ths.

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 TWO steps).

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)