Derivative & Parallel
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 nondiatonic 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
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
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♯ [#♯], 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.