The symmetry built into the Diatonic system has many manifestations. One of the most compelling is the phenomenon of modal common tones.
Deep Symmetry
The modes form palindromic interval groupings ...
Ionian and Phrygian,
Locrian and Lydian,
Aeolian and Mixolydian,
... and Dorian mode ...
The intervals within each mode are part of the overarching Diatonic symmetry. Study the interval structure of the modes ... notice the palindromic pairings:
Common Tones connect seven keys in a modal continuum ...
Modal Common Tones
When the notes of Major scale (Ionian mode) each become the tonic of a new key there is but one note common to them all. In the Key of C are the notes C D E F G A B ... the scale will include the complete octave — C D E F G A B C.
Make each of these natural tones the Tonic, and you have seven keys, and within each of them is the note 'E' ... the III of the original key. There will be no other note to find a place in all seven keys.
That's interesting enough, but there's more!
Do this with all of the seven Modes, and you'll find the correlation between Mode's interval symmetry and the Common Tone Counterpart groupings.
Common Tone Counterparts are those notes that link the diatonic symmetry due to their corresponding key degrees.
Just as Ionian and Phrygian are a palindromic pair, the notes C and E are positioned symmetrically around the central note of the key: D ... as are B and F, as well as A and G ... and their respective modes.
These images show the common tones in each mode. In each case — within seven keys which have Natural tones as their tonics:
(The tonics are always found in the column with C at the top and bottom, i.e.: the tonics for the Phrygian mode above are C Db Eb F G Ab Bb. The tonics for Aeolian mode below are C D Eb F G Ab Bb.)
The reciprocal relationships between modes and their counterpart common tones is profound!
... It reinforces the centrality of the Tritone ...
... and of the Dorian mode which stems from the 2nd Key Degree ...
All of this plays out on the fretboard with absolute consistency.
Notice the rotational symmetry of Key Degree relationships on the fretboard within the upper and lower string groups:
I is opposite III
VII is opposite IV
VI is opposite V
and II is opposite II
Just how the fact of this symmetry can factor into music in practice is a complex subject. Save to say that awareness of a thing naturally generally precedes one's understanding of it.
The Modal Common Tone can be a pivot between keys. It has a different function in each, but ... there it is. Is there a difference between key transitions with or without common tones?
There must be a mathematical formula which expresses all this. Since I'm about as good a mathematician as I am a brain surgeon, I'll leave the math to others.
Also of interest is how these commonalities align with the Circle of 5ths and 4ths.
On the left side of the circle are flat keys (keys containing one or more flat) and on the right side are sharp keys (keys with one or more sharp). The interval symmetry of counterpart modes plays out in the incidence of flat and sharp keys expressed by each mode. Since the key of C is all natural tones, the remaining keys are an equal number, with F# and Gb each occurring only once — through Lydian and Locrian respectively — among the common keys ...
Common Keys
C Ionian: C, D, E, F, G, A, B
E Phrygian: C, Db, Eb, F, G, Ab, Bb
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F Lydian: C, D, E, F#, G, A, B
B Locrian: C, Db, Eb, F, Gb, Ab, Bb
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A Aeolian: C, D, Eb, F, G, Ab, Bb
G Mixolydian: C, D, E, F, G, A , Bb
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D Dorian: C, D, Eb, F, G, A, Bb
Among keys which are not found within each mode is the following array ...
Non-common Keys:
C Ionian: Bb, Eb, Ab, Db, and Gb or F#
E Phrygian: B, E, A, D, and Gb or F#
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F Lydian: Bb, Eb, Ab, Db, F
B Locrian: B, E, A, D, G
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A Aeolian: B, E, A, Db, and Gb or F#
G Mixolydian: B, Eb, Ab, Db, and Gb or F#
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D Dorian: B, E, Ab, Db, and Gb or F#
A careful analysis of the balance of sharp keys and flat keys within counterpart modes will reveal a clear symmetry vis-a-vis the Circle of 5ths as they are equally distributed.
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What do we do with this knowledge?
That question can best be answered by asking another question:
"What do we do with nouns and verbs?"
On the most basic level, these ideas are simple matters of fact ... like the Circle of 5ths. The interlocking relationships between common modes across multiple keys builds upon the Major key-Minor key paradigm usually associated with the Circle of 5ths. What applies to Ionian and Aeolian modes also factors into connections between all the other modes, and that's not limited to I - IV - V patterns.
The common tone connection brings chromaticism into the mix. How does that relate to, say, Melodic and Harmonic Minor ?
Recapping: Modal Logic
Any mode, when used as the source of seven tonics (or finals), will share with all of them one — and only one — common tone. That common tone will be the final of its palindromic counterpart mode.
For example:
The common tone of Ionian mode is its 3rd degree. That note is the final of Phrygian mode, which has a common tone on its 6th degree ... which is the final of Ionian mode.
The final of Lydian mode is its 4th, and the final of Locrian mode is its 5th.
The final of Aeolian mode is its 7th degree, and the final of Mixolydian mode is its 2nd ...
Since Dorian mode is its own palindrome, its common tone is its own final ... note number 1.
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Secondary Source Keys*
The Secondary Source Key (or 'Parallel Source Key') for each modal group is the major key built from the modal scale with the note C as its final. There is an inverse relationship between the place of the tonic of each Secondary Source Key and the degree of the mode itself in the major scale overall. (*This terminology is original. I'm unaware of any existing term for this precise phenomenon.)
The Secondary Source Key for each mode:
C Ionian (tonic mode) = C ... its own tonic.
C Phrygian (mediant mode) = Ab ... it's own submediant.
C Locrian (leading tone mode) = Db ... its own supertonic
C Lydian (subdominant mode) = G ... its own dominant
C Aeolian (submediant mode) = Eb ... its own mediant
C Mixolydian (dominant mode) = F ... its own subdominant
and for C Dorian (supertonic mode) = Bb its own subtonic.
Here are the diagrams with the major tonics indicated by yellow highlights. Notice that the major tonic of the secondary source key, in each case, is where the two lines converge:
These layers of diatonic symmetry — each layer being balanced within itself — form overlapping asymmetries which in turn reveal interesting commonalities. Though the modal common tones follow the intervalic mirroring of mode pairs, the secondary source key relates the degree of the mode final within the original source key, to its own internal degrees.
Put another way; the diatonic place of each mode's final has an inverse interval relationship to its relative major tonic. II is the inverse of bVII, III is the inverse of VI, IV is the inverse of V ... I is, of course, is own inversion.
The particular aspects of modal commonalities across keys delineated here are by no means the only possibilities, but they represent a way into this realm harmonic complexity.
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