I'm trying to learn more about Music Theory, and I'm currently looking into modulation, more specifically CTM (Common Tone Modulation).
As far as I understand it, the concept is pretty straight forward that (at least) one tone has to be common or sustained between 2 chords.
What I have a problem with understanding is a CTM example that was made on a Lynda.com tutorial (Music Theory for Songwriters: Harmony) and is also stated on Wikipedia in the Modulation: CTM section, that for a given chord one could possibly modulate via CTM to 12 different, other chords.
Wikipedia states this example here:
"Starting from a major chord, for example G (G-B-D), there are twelve potential goals using a common-tone modulation: G minor, G♯ minor, B♭ major, B major, B minor, C major, C minor, D minor, D major, E♭ major, E major, E minor".
but clearly - this is the part that confuses me - when just looking to keep one tone in common, there are more than 12 potential keys one can modulate to from a G major triad (G,B,D)... so I'm missing something.
***** added information *****
via the Wikipedia example from above using CTM to modulate to other keys from a G maj triad G,B,D in key of G major (targeting major/nat minor scales only):
(1) the note of G is found in 17 keys in the following position:
Ab Major 7th
A Natural Minor 7th
A# Major 6th (as F##) (duplicate to Bb Major)
Bb Major 6th
B Natural Minor 6th
C Major 5th
C Natural Minor 5th
D Major 4th
D Natural Minor 4th
D# Major 3rd (as F##) (duplicate to Eb Major)
Eb Major 3rd
E Natural Minor 3rd
F Major 2nd
F Natural Minor 2nd
G Major 1st (duplicate, original key)
G Natural Minor 1st
G# Major 7th (as F##) (duplicate to Ab Major)
--> leaving a total of 13 unique keys as a CTM target for the note of G
(2) the note of B is found in 19 keys in the following position:
Ab Natural Minor 3rd (as Cb) (duplicate to G# Minor)
A Major 2nd
A Natural Minor 2nd (duplicate, already listed)
B Major 1st
B Natural Minor 1st (duplicate, already listed)
C Major 7th (duplicate, already listed)
C# Natural Minor 7th
Db Natural Minor 7th (as Cb) (duplicate to C# Minor)
D Major 6th (duplicate, already listed)
D# Natural Minor 6th
Eb Natural Minor 6th (as Cb) (duplicate to D# Minor)
E Major 5th
E Natural Minor 5th (duplicate, already listed)
F# Major 4th
F# Natural Minor 4th
Gb Major 4th (as Cb) (duplicate to F# Major)
Gb Natural Minor 4th (as Cb) (duplicate to F# Minor)
G Major 3rd (duplicate, original key)
G# Natural Minor 3rd
--> leaving a total of 8 additional unique keys as a CTM target for the note of B
(3) the note of D is found in 17 keys in the following position:
A Major 4th (duplicate, already listed)
A Natural Minor 4th (duplicate, already listed)
A# Major 3rd (as C##) (duplicate to Bb Major)
Bb Major 3rd (duplicate, already listed)
B Natural Minor 3rd (duplicate, already listed)
C Major 2nd (duplicate, already listed)
C Natural Minor 2nd (duplicate, already listed)
D Major 1st (duplicate, already listed)
D Natural Minor 1st (duplicate, already listed)
D# Major 7th (as C##) (duplicate to Eb Major)
Eb Major 7th (duplicate, already listed)
E Natural Minor 7th (duplicate, already listed)
F Major 6th (duplicate, already listed)
F# Natural Minor 6th (duplicate, already listed)
Gb Natural Minor 6th (as Ebb) (duplicate to F# Minor)
G Major 5th (duplicate, original key)
G Natural Minor 5th (duplicate, already listed)
--> leaving a total of 0 additional unique keys as a CTM target for the note of D
The resulting 21 unique keys are (incl. resulting triads w/ note positions):
Ab Major 7th (G,Bb,Db)
A Major 2nd (B,D,F#), 4th (D,F#,A)
A Natural Minor 2nd (B,D,F), 4th (D,F,A), 7th (G,B,D)
Bb Major 3rd (D,F,A), 6th (G,Bb,D)
B Major 1st (B,D#,F#)
B Natural Minor 1st (B,D,F#), 3rd (D,F#,A), 6th (G,B,D)
C Major 2nd (D,F,A), 5th (F,A,C), 7th (B,D,F)
C Natural Minor 2nd (D,F,Ab), 5th (G,Bb,D)
C# Natural Minor 7th (B,D#,F#)
D Major 1st (D,F#,A), 4th (G,B,D), 6th (B,D,F#)
D Natural Minor 1st (D,F,A), 4th (G,Bb,D)
D# Natural Minor 6th (B,D#,F#)
Eb Major 3rd (G,Bb,D), 7th (D,F,Ab)
E Major 5th (B,D#,F#)
E Natural Minor 3rd (G,B,D), 5th (B,D,F#), 7th (D,F#,A)
F Major 2nd (G,Bb,D), 6th (D,F,A)
F Natural Minor 2nd (G,Bb,Db)
F# Major 4th (B,D#,F#)
F# Natural Minor 4th (B,D,F#), 6th (D,F#,A)
G Natural Minor 1st (G,Bb,D), 5th (D,F,A)
G# Natural Minor 3rd (B,D#,F#)
--> the only way to cut this to 12 results is by only considering keys that have the notes G, B or D in either position 1, 3 or 5...
WHY is the note position a requirement for CTM ? The note of G (when sustained between 2 chords) is the note of G, independent of it's position within a scale...
Please enlighten me !
In addition - example 2:
if we want to use CTM from a G major triad in key of B Nat Minor, which has G in 6th position ergo the triad contains notes in positions 6,1,3 would we then only target keys that contain G,B,D at position 6,1 or 3 ?
Thanks !