Circle of Fifths and minor mode made (almost) simple

The tonal system is supposed to be a simplification  of the  older  modal  system since   it  only consists of two modes namely the Major and minor modes. That  is a kind of « swindle »  since  actually the so called minor mode consists of  three derivative versions  from old minor church modes.It would be more « honest »  to state that key center has two possible tonics a major sixth apart. (CM-am)

It is not easy to define the tonal system which is a slow process so let’s assume few things

        • Tonal system is made of 12 major scales  and 12 minor scales (6 flat and six Sharps). In theory there are 15 scales  from 0 to 7 alterations (sharps or flats)but  3 scales sound similar  with flats and sharps by enharmony thanks to the equal  temperament system.
        • Tonality consists of 7 fifths  according to the Pythagorean Principe of string length division: A string 1.5 shorter sounds a fifth higher.However the fifths series doesn’t yield directly  a scale; Some division of length  and new order are necessary to keep the seven tones within an octave. Example CM fs
        •   The tonal model is the major scale made of two tetrachords that yield a symmetric distribution of tone -semitone C-D-EF  and  G-A-BC
        • Each scale degree supports a triad made of stack of diatonic (that belongs to the scale) thirds. The triads are –
          • Major on I, IV and V
          • -Minor on II,III and VI
          • (all those chords are called perfect because the bottom – top notes form a perfect fifth ) –

          • diminished(the fifth) on VIIth degree
        • The fifths circle is therefore IV-I-V-ii-vi-iii-vii  Notice IV -VII form a tritone: the tonality limits
        • The primary chords which supply all scale tones are grouped on the left.Their roots are the tonal notes whereas their modal notes (thirds) are grouped  on the right.On the middle stands the fifth of the dominante.

notmodmajresu

  • the  direction of the scale is opposite to the circle of fifths direction.

 

 

We can represent the tonal system as a sliding rule with a long series of fifths from Cb(7 flats) to  C#(7 sharps)  on the fixed part of the rule and our seven fifths ( from IV to vii) on the sliding  part of the rule. The tonality is I on the sliding part and the key signature is on the fixed part facing I

tonal2

Notice that  numbered 5-6-7  tones on the right sound like the 7-6-5 ones  on the left thanks to enharmony So the chain is closed forming a circle.

Thank to the sliding,we can see how a chord can easily be reinterpreted for modulation.

Unfortunately minor mode obscures this very simple  system.

If we want to write a mirror scale  (opposite) to the major scale with respect to the key signature and  Tone -semitone distribution we must start with the third degree  of the scale which produces a minor descending scale .
mir
No ascending
 minor scale fits to these  two conditions; none respects the tone -semi tone distribution.

we must  start the scale  on II, III or VI to have a minor third but none would produce a leading tone which require a Major third . Therefore the second tetrachord can’t be symmetric to the first one and  we must add some  alterations.

mmo

Notice  the symmetric D mode (Dorian) which was the most frequent minor mode used during the baroque era.

With our CM scale

II (D) needs C#, III(E)requires D#, and  VI(A) G# which are not the direct continuation of the fifths circle.

Mind you, that is an explanation of the problem but not the reality:Both D and E modes are to be transposed  on A so only G#  and F# (coming from the B of the D  mode) appear.

Adding a leading tone to the a minor scale creates  an unmelodic  augmented  second interval so  the VIth  degree  is altered (the borrowed B from D mode) to enhance the melody

So, the minor scale has three aspects

1-Natural or descending melodic or Aeolian mode A-BC-D-EF-G-A  which does not have a leading tone.

2-Harmonic: A- BC- D -EFG#A  which provides a leading tone and a bad augmented second.

3- Ascending melodic A-BC-D-E-F#-G#A. with a leading tone and an altered VI th degree to  correct the unmelodic  augmented second

The complete  minor scale (natural plus the two ascending scales)produces 13  triads

mimin

 

Why A  rather than D or E?

From the natural minor scale the A scale need F# and G# to complete the ascending scale.

The D (Dorian)scale would request Bb and C# and the E (Phrygian) scale  would need F# ,C# and  D#

So A is the nearest from the Fifths circle.

Since the A minor scale has no alteration it’s associated  to the C major  and our sliding rule can be expanded like this

compslide

So, to find the key signature of a minor scale one must refer to the relative major scale or the third degree of the  minor scale.

It must be  pointed out the reverse image of the major mode

aminotmodmin

                                                    The major numeration is In yellow

Rationale in tonal music is  to consider a scale as a Pentachord and a tetrachord (with a common tone)

pentate

For a given tonic, all notes of the pentachord but the Third are common to all tonal kind of  scales

Tetrachord  has  two movable notes (The sixth and seventh  degrees)