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Derivative and Parallel

Core Location: Theory and Mapping

Also included in the Nucleus.

Every tone means something unique to every tone. The numbering system we use can first be derived using the tones of the Major scale. We can use a learning tool such as 221-2221 to derive the tones in the scale. They are then numbered 1-7. This is the derivative side of the street.

What is derivative is inside a key center, whether single tones, scales, or chords. Derivative tones, chords, etc. are often called diatonic. Diatonic means "across the tones of a key". Diatonic tones are the tones that are in the Major key; they are inside.

This leaves 5 tones to name/number [not in the key]. These are the outside tones. At every 2 in the Major scale pattern, a tone is skipped. These five tones that were 'not included' by pattern are called non-diatonic tones.

When we compare these tones to what is 'normal' [inside], we are paralleling. Parallel names are created by comparing to the derived tones (1-7). We typically [most commonly] parallel to the Major, yet we can also parallel to minor or other previously paralleled entities.

C Major's Diatonic Tones & Gaps

c major scale with gaps

Every 2 in the Major scale Pattern creates a 'gap', since tones were 'skipped', 'eliminated', or 'not included' at those points.

D is C's 2, and only C's 2. The tone that sits in the first gap is called D♭ [a flat lowers a tone one half-step]. Therefore, it can be called a ♭2 [flat 2]. We've compared a non-diatonic tone to a diatonic tone and based its name on the difference.

To the tone C, D is 2, and D♭ is the ♭2.

In a key, the 2, 4, & 6 tones are the same tones as the 9, 11, & 13 respectively. The 2 = 9, & therefore, the ♭2 is also the ♭9. The ♯2 = ♯9, etc.

When we keep going, beyond the octave [the 8th tone of the scale - the C], we find D again, yet this time in position 9 [1-2-3-4-5-6-7-8-9 = C-D-E-F-G-A-B-C-D]. Same thing applies to the 11 & 13.


2 = 9: 1-2-3-4-5-6-7-8-9 = C-D-E-F-G-A-B-C-D

4 = 11: 1-2-3-4-5-6-7-8-9-10-11 = C-D-E-F-G-A-B-C-D-E-F

6 = 13: 1-2-3-4-5-6-7-8-9-10-11-12-13 = C-D-E-F-G-A-B-C-D-E-F-G-A

C Major's Complete Inventory

c tone complete tone inventory

The top row of numbers are the half steps [I call this Numera].

The second row are the derivative and parallel names in the traditional music theory system.

C Major is the simplest inventory. Since all the derivative tones are naturals, the sharp and flat non-diatonic tones are the sharp and flat versions of the scale members. The flat names are shown above.

The D♭ could also be called C♯ [♯1 - this is not common]; the E♭ could also be called D♯ [♯2 or ♯9], the G♭ could also be named F♯ [♯4], the A♭ could also be called G♯ [♯5], and the B♭ could be also called A♯ [♯6 or ♯13 - this is also not common, though ♯13 can appear].

The non-diatonic tones for C are the F♯/G♭ Major pentatonic scale  (D♯ minor/E♭ minor pentatonic). We favor the tonal spellings of E♭m/G♭ pentatonic: E♭ G♭ A♭ B♭ D♭ for the non-diatonic tones.

We now know what all tones mean [are named] to C [we call this a tone inventory]. These tones hold a specific melodic & harmonic space to C. And, these tones only mean these things to the tone C.

In a different key, the tone D will be in a different position depending on the root. It is important to know the tone inventories for all keys.

Derivative = diatonic = inside = 'normal'.
Parallel = comparing to what is normal which create what are called formulas.

Formulas are ways to describe components in a system or harmonic situation. The Major chord's formula is R-3-5 [C-E-G is C Major]. When we compare a minor chord to this we get R-♭3-5 [C-E♭;-G is C minor]. We compared/paralleled [how are they different] and now have a working formula for a minor type chord.