Mirrors of the church modes

Mirrors tally ascending scale with descending scale (and vice versa).

For better comprehension few notions has to be kept in mind.

1°) Each scale degree is equivalent to a two fifths skip

with a slight default : F is sharp (#)

To remedy this we have to start our series a fifth below the tonic (F)

Starting on F results in a 7 diatonic notes series bordering the tonality
So C major scale limit are F and B : forming the Triton
2°) A note belongs to 7 tonalities according to an immutable order:
7 3 6 2 5 1 4 . So C is the seventh degree of Db,third degree of Ab and so on….


Notice the fifth series forms the triton Db -G ( triton of D Maj) which is symmetric to F-B and able to resolve on C

The seven fifths series forming a triton on C can be copied on the seven notes of the scale.With a two fifths shift we obtain the following table.

Notice

  • E corresponds to the tonality of E
  • D corresponds to O accidental.
  • The Tonality of C stretches from Gb to F#

3°) Pivots (common notes) appear only with pair number of accidentals scales
From the C major scale, we can write a scale in opposite direction that respects the tones and semitones position (mirror).Any chromatic note can start the mirror scale but only scale with pair number of accidentals(and 0) has common note (pivot) associated with a tonality

Mirrors adhere to the tones and semitones distribution according two options

  • Share the tonality  but that might make finales dissonant
  • Choose the common note (pivot)

Actually the two options rests on : pivots.
Since modes have no accidentals, D (associated with 0 accidentals) as pivot places the two scale in the same tonality.

The mirror of the ionian scale (C maj) is E min (with no accidentals) ( E phrygien) that’s why Vincent d’Indy considered it as « true minor relative »

The reverse is also true

We’ll use two tables to write mirrors.
The first table states the 12 tonality of C with the seven scale notes according to the normal fifths series order

  • First line give the key signature of the mirror scales
  • second line give the tonality of the mirror scales
  • Bottom line indicates the pivots

The prime scale is always in C whatever the mode since mode is only a different departure of the C scale

Table 2 indicates which degree of the tonality the mirror starts on(bottom line)

Pivot is the junction-point of the two scales but usually neither the tonic nor the starting point(finale) of the mirror (Finale)explaining the table 2 necessity.
The two first lines correspond to the ascending C Maj scale and define modes.
The two next lines are the descending C maj starting on E which is the true mirror (same tonality as the prime scale (given by the D pivot )

The bottom line figures are the interval between C and the tonic of the other tonalities.
This table shows how it is built but practically we need only two columns

We can see that each mode has its only one mirror mode

Summing up table

Example :To write the mirror of a Phrygian (E) scale with G as common note

G tallies with the Bb scale (column G) which starts on the first degree( Line E) (Bb).
A practical trick :The tonic of the mirror always correspond to E of the other scale (whatever the mode)
( E is the only pivot with common tonality;

Some more examples

Both examples having the same pivot share the Bb mirror scale but with a different starting note (tonic) coinciding with E of the prime scale

The pivot being D the mirror has no accidentals


Perfect mirror that share tonic and tonality but contradict the E rule

Modal Cadences

Another difficult and confused topic . Ancient church modes were declamatory melodies, sang in unison, which   phrases  ended with fixed melodic and rhythmic formulas  and therefore had no cadence, as we understand with our tonal mind.
The first real cadences (clausules) appeared with polyphony which caused: the disappearance of modes
We can trace back modal harmonic cadences to the 19th century when composers tried to destruct tonality .However,Tonal culture is so deep-rooted in our mind that we want to make modal music a kind of tonal music transcription ( especially due to functional harmony).
Modal cadences are not conclusive as would be tonal cadences and are generally used to « add an transitory color » to a phrase.
But let’s first remind some tonality elements

 

.

  • Tonality is based upon physical law of resonance with ascending harmonics that explain the Major triad.
  • Minor triad is more difficult to account for since there is not a descending resonance (in normal conditions ). However we saw (in the mirrors of the church modes) that an ascending major scale had its specific descending minor scale .( a major third above the major scale)
  • Tonal music consists of two modes based upon the third (IIIth degree) that determines
    -The Major mode ( III Majeur) which have 3 strong Major degrees (I- IV-V)
    -The minor mode (iii mineur) with 3 strong minor degrees strong (I- IV- V). However the 7th scale degree must be half a tone of the tonic to look like the major scale so the V th degree is altered and became a Major Chord.

Modern harmonic modality can be approach from different point of view.
The most intuitive is the minor mode without leading tone like the Aeolian mode with a minor Vth degre) for both ascending and descending scales resulting in change in degrees numeration . Cadence traces tonal cadence (v-I) for ascending scale or plagal (iv-I) for descending scale

We may consider semi tones as  leading tone : ascending LT B-C and descending LT F-E

According to the slope direction , two possible dominantes and two possible leading tones are available.

Remark :In tonal music progression B -C, B is leading tone because the semitone leads to a strong degree (The tonic ) while F is not a leading tone because it leads to a weak degree (III).

In the tonal progression F -E, F is not a leading tone because the semitone lead to a weak degree however E,as  descending mirror, is the  tonic and therefore a strong degree : thus F becomes a descending leading tone .

Let’s place a triad on the Vth degree of each scale in ascending and descending forms


Among those V degree chords let’s select those with one of the two possible leading tone F or B which is not at the bass

Besides the ascending Aeolian mode,only two chords appear

  • ii from the original C maj scale(Dm)
  • Dominant 7 chord ( GBD et B D F)

Dominant 7 chord appears under two forms GBD and BDF
BDF can’t be used due to the triton that would resolve to C maj.However, adding the sixth from the descending Aeolian scale enable the BDF chord to form a common dominant chord (DFAB) to Ionian(CM )and Aeolian (am) scale.

Notice that the sixth (B) placed on the( II chord ( dfa) is the leading tone of the ascending scale.
Let’s generalize the method by adding ascending leading tone to descending scale and reversely

  • Ionian Mode GBD ,resolving to CM, is ruled out and FAB as well since it adds a second dissonance to the triton leading to C major). The solution is FAC: creating a plagal cadence
  • Lydian Mode  is particular because we can neither use BDF which lead to C Maj scale nor CEG that would resolve to the F Major scale -The solution is GBDwhich contains the characteristic note and the leading tone altogether
  • Mixolydian Mode : The only possible chord is DFA with added sixth (B) DFAB

    • Aeolian Mode: we keep DFAB for its dominant chord characteristic in a plagal progression IV

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    We can of course use EGB but it’s less characteristic

  • Dorian Mode: Two chord progression are available (ACF) III6– and (GBD)IV-I
  • Phrygian mode The CM titron rules out the BDF chord) so we keep the ACF chord with the so called phrygian cadence II6-I

Another approach

As discussed in old church modes, Octave were divided into two tetrachords. There are different kinds of tetrachord:Four are part of old church modes (3 with perfect fourth and 1 with augmented fourth) and one perfect fourth tetrachord that does’nt belong to church mode:The harmonic tetrachord

The place of the semitones gives the direction of the slope,ascending on the right,descending on the left ,no slope in the middle
Considering Modes as two associated tetrachords

Lydian and Locrian modes do not allow two disjunct tetrachords and therefore those modes are the association of a pentachord and a tetrachord
Notice

    • -The central position of the semitones in minor tetrachord
    • -Ascending slope of the major tetrachord
    • -Descending slope of the phrygian tetrachord .

Association of two opposed sloped tetrachords

Lydian and Locrian modes concentrate semitones in a tetrachord
It should be observed

      • -The Dorian scale is a pivot scale with central position of semitones making this scale ambiguous with no slope
      • – The symmetry of th Ionian,Dorian and Phrygian modes E
      • -The inverted mirror ot Mixolydian and Aeolian modes

Conclusion

This last approach helps to choose progression according of the slope of the scale and a tool for counterpoint which is by essence related to mode

Another  hint

Let’ s write our scale as triads , not as a succession of triads but rather like a three voices counterpoint to take into account the respective location of semitones

The three minor scales share those three chords


The three Major scales share those three chords
However those chords do not necessary content the characteristic tone of the mode.
Adding a 6 th , 7th or 9th make things very interesting provide they don’t create the triton F-B which suggest the C scale.
We also may use medieval cadences discussed in
disappearance of church modes and the Baroque cadences which are similar to the medieval cadences but they may lead to a third interval (instead of octave or fifth)

 

 

Last Hint

We can make a tracing of functional Harmony according to two schools

1-The  Niedermeyer-Ortig’s french school  which uses tonal progression but chords  building abides by the mode (so V can be a minor chord)

2- The Persichetti’s american school which makes a distinction between  three kinds of chords

-The primary chords :The tonic chord and chords that contains the distinctive note of the mode.

-The secondary chords :chords that don’t contains the distinctive note of the mode.

-The Chords to avoid because of tritone  that lead to major mode.(It can be a primary chord with an added  7th  or 9th

PERSEC

 

to summarize

Modal music is mostly melodic based on intervals and two functions Teneur (kind of dominant) and Final (Tonic). Its harmony is rather static and baffle functional harmony

The old church modes

Their theoretical constitution is very simple : With C Maj scale for reference (old Ionian mode ) we play one octave starting on each degree of the scale

Table 1:
Names are objectionable due to confusion in their origins.

Thus the mode phrygian (E) mode in the middle age was called  Dorian under Platon .

Futhermore Greek names do not refer to ancient Greek scales which were descending but to ascending scales from probable byzantine origin.
Another difficulty stems from Gregorian songs, from which church mode were born,that are under two forms

  • -The authentic form that begins by the finale (our tonic); but the scale may start on one degree below ; so Dorian mode (from D to D) may extend from C to D
  • -The plagal form that starts a lower fourth below the authentic one. The name is then preceded by the prefix Hypo so Dorian is the authentic and Hypo Dorian the Plagal form( from A to A)  of the same mode

 

It is simpler to consider octave either as the association of

a pentachord and tetrachord that share a common note (authentic mode )

    • or an tetrachord and a pentachord ( plagal mode)

. The finale is the first note of the pentachord in both forms

Table 2 Mode Authente et Mode Plagal

Table 3:Shows ambiguous span

Some modes share their range :The authentic phrygian mode (E ) is   similar to  plagal  Aeolian mode   (hypoéolien) , or authentic mixolydian mode similar to the plagal ionian mode.

The  difference between authente and  plagal rests upon the  teneur (our dominant); but here again things are not so simple:
In  authentic mode , the  teneur   is normally  a fifth  above THE  FINALE  but if  the teneur  is B it becomes C
In the  plagal form, the teneur is  normally a fourth above the finale but it might be a third or a sixth above the finale .

Table 4: Finales and teneurs

To crown it up accidentals (especially F# and Bb)was gradually introduced to set up the semitone tone of leading tones .

Others accidentals might appear in the descending forms . All those changes led to the reduction of the number of modes  to the only major and minor modes of the tonal system.

Remark

The 14 modes previously listed are pure theory and don’t reflect the confused historical reality.
Classically there are only 8 byzantine modes ( D E F G ) under the 2 forms.

 

For simplicity we’ll  keep the nomenclature of scales without   accidentals of table 1, currently used especially in jazz music.

Characteristic Notes and key signature

Each mode has a characteristic note except the Locrian mode that has two characteristic notes
This note is said characteristic because it makes the difference between the tonal scale and the parent C Major scale.

For example the Lydian mode starts on F and runs on one octave without  alteration  so it is different from F major which has a Flat (Bb) . The characteristic note is therefore B=4th degree of the F maj scale.

For the minor modes (dorian, phrygian et locrian) one must reason from the major relative
For example the phrygian mode (mode of E) is a minor scale without alteration  while, E minor is the relative G major scale with a sharp(F#) ),the characteristic note is therefore F: second degree of the E scale

Remark Aeolian mode has not a characteristic note since it is the relative of C major (no accidentals)

Let’s draw a table to enable characteristic note finding and transposition .
1°)From tonal notes CFG and their minor relative ADE, plus B by assimilation to minor mode,our table looks like this

2°) Key signature adaptation
F major scale is a fifth below C Maj To transpose the F major scale to C major one must add a fifth.
The trick is to count the number of degree from the tonic (N1) to C . Reporting the number of degree from C give the new key signature

In minor modes the goal is not C but A

 

We can now complete our table

    • Mode of Fa :Key signature of G major 1#
    • Mode of Sol:Key signature of F major 1b
    • Mode of Ré: Key signature of E Minor=G major 1#
    • Mode of Mi: Key signature of D Minor=F Major 1b


Notice : characteristic notes are either F or B (the triton of CM)
Locrian mode is  particular since its tonic  fifth is diminished . It’s origin is not a relative major scale like other minor mode but its third being minor it can be assimilated to a minor scale to used our trick ( from B to A = 7 degrees: A +7= G Minor= 2b )

Transposition

1°) Write a mode in another tonality
Our table give the direction to the key signature modification.(Toward more  flats/less sharps or More sharps/less flats)
Examples :
1-to write an A lydian scale ( F mode)
– Lydian=Major mode so we have to adapt A major scale(3#)
-our table reads 1 # to add to to scale
So A lydian is a scale with 4# (starting on A)

2-To write a Eb lydian scale (F Mode )
– Lydien =Major Mode so we have to adapt Eb Major scale(3b)
-Our table indicates +1 # so flats go opposite (-1b)
Eb lydian is a scale with 2 b (begins on Eb)

3-To write a G phrygienne scale (E Mode )
– Phrygien =Minor Mode Mineur – we have to adapt G minor scale –Major relative Bb= 2b
-our table read+1 b
G phrygian is a scale with 3b flats and start on G

Caution the reference minor scale have no accidentals(but the stuctural one of the major relative) and therefore no leading tone

Building a kind of sliding rule will make things easier especially for minor modes.

1°) Writing a mode in another tonality (Key center)
The fixed part( règle fixe) is a fifths series from 6b to 6 # on the upper part and the corresponding tonalities on the lower part
The movable part( règle mobile) is the same fifths series with a colored part indicating the seven modes .

By coinciding the note of Mode from the movable part with 0 accidentals ( C) ,key signature are given for tonalities on the fixed part .

With our former example A Lydian scale ;from the movable part align F= Lydian mode with O on the fixed part;  from A on the movable part A we reads 4# on the fixed part

Same thing for minor mode : D ( Dorian mode ) facing 0 accidentals (CM),  A Dorian is a scale with 1#

2°) Identifying a mode from a tonality
We use the same fixed part of our sliding rule and reverse the fifths series on the movable part,using only the 7 modes (from F to B).

The tonality is on the movable part facing 0 of the fixed part. The movable part lists the 7 modes coinciding with key signature on lower fixed part
Example A scale with 1#.What is the mode?
Align A from the movable part with 0 on the fixed part, the 1# column reads D on the movable part
So A scale with 1# is a A Dorian scale

Disappearance of church modes

Because church modes still prevails in numerous cultures, « Desertion » could be a proper word.

Actually the reducing evolution of mode that give birth to tonal music,based upon only two modes, is a very slow process that makes difficult to find a proper word to describe it.

There are many reasons that lead to the disappearance of modal scales.

    • One of the main reasons results from the way to avoid the triton F-B in polyphonies by lowering B, so it becomes Bb, or altering F into F#.
    • the second reason is a rule of counterpoint: an imperfect consonance(thirds and sixths) followed by a perfect  consonance is better approached by a semi-tone in one part and one tone in the other part
      A or B are better than C

      In three part this rule yields the double leading tones cadence which is of two kinds

      1-Machaut cadence:The two upper parts contain an half tone, opposed to the Bass:

      Adding a cambiata produces a Landini cadence


      2-Phrygian cadence:The two upper parts contain a tone, opposed to the bass

The third factor is the movement to a fifth lower at the cadence which created a forbidden dissonance seventh with the double leading tones cadence. So the authentic cadence is born: a major third resolving into an octave and a movement of fifth but leading to the disappearance of double leading tones cadence the F mode and G mode at the advantage of the C mode due to the necessary adaptation to avoid the dissonance seventh

To summarize we can say the main cause of the change is the growth of polyphony

* X th century with melody accompanied with parallel fifths or fourths.
* XI th century the melody starts and ends with the same note (named final, now tonic) and is accompanied with oblique motion which begins and ends in unison preceded by major second.
* XI-XIIth century introduction of the contrary motion with a major second as pre-cadential dissonance and a fifth added to the tonic
* The third is henceforth a imperfect consonance (formerly a dissonance) XIII th century cadence with double leading tone according to the rule of counterpoint edicted by Jean De Murs) The final chord has no third.
Consonance (8-5-4) evenly appear at the beginning of each « perfection » (group of 3 breves); there are non harmonic tones in short values between groups
* Ars Nova give up the rhythmic modes so consonances may appear on weak beat (XIVth century); the third is included into the fifth chord except in final position. Appearance of the movement V-I at the bass and the movement VII-I at the soprano {G-D-G}-{CGC}
* XVth Century: Instrumental music spread over the sonorous space with non modulating sequences on each degree. The third appears on the final chord so the authentic cadence is completed
*
* XVI-XVIIth centuries is period of great change:alteration to approach consonance by half step resulting in 12 divisions of the octave and the polyphonies that jut out the octave impeded to distinguish the authentic and plagal forms-another cause of disappearance of modes Adoption of the Zarlino’s system with pure third and false fifths will yield new rules of counterpoint such as to forbid direct fifth and change the rule of third and sixth which become perfect consonance.

Mirror of chromatic scales

A chromatic scale may received a mirror whose any note can act as pivot but the mirror has to be in the tonality of the whole tone scale. Only the first half of a chromatic scale appear in one direction as pivot so the original chromatic scale must have to separated part

  • from tonic to triton
  • from triton to tonic
  • In the descending chromatic scale the slope of the pivots line is inverted.

    All tonalities of the mirror are available thanks to another chromatic scale starting on C# that provides the second whole tone scale.

    Caution:E scale as mirror is different from the E chromatic scale.
    Here the E scale is the mirror of a C chromatic scale with D as pivot

    Now E is the mirror of a E chromatic scale with two pivots E and Bb

    Therefore to start a chromatic scale with another degree one must define the whole tone scale and the triton

Mirror of transposed church modes

Don’t get confused ! Up to now we worked on the C Major scale starting on each scale degree(Church modes) mirrored by various scales having each scales degree as common tone (pivot) resulting in different tonalities (transposed scales) that begin with different scale degree (modes).

For example:

Let’s write the mirror of a CM scale starting on F (=Lydian mode)F.G. A.BC.D.E.F

with G as  common tone (pivot)

the mirror is a Bb Maj scale (transposed scale one tone below)which begins ont its 7th degree (A Locrian mode)

Now  we are discussing the transposed original scale (the previously CM scale). The issue is  a mere shifting of the tonal center on the fifths series but carrying it into effect might be difficult due to enharmony.

Let’s go back to C major. The so called tonal center C is flanked by 6 fifths on either side

the outer enharmonic fifths close the fifths circle and share the location of the pivot forming the characteristic Triton of CM (F-B)

Remember   the pivots are the seven fifths of CM starting a fifth below the tonic and distributed  every to fifths (FCGDAEB) placed on Gb-Ab- and so on
In C Maj

Ib Bb

Notice the big zone of enharmony and  common zone that shares the Triton and is a source of difficulty.

Transposition  a tone below is a shift of two fifths to the left

Transposition  a tone above is a shift of two fifths to the right

Notice the how the transposed G dorian mode appears .

  • Some pivots are common to CM SCALE but the associated tonalities are shifted in opposed direction to transposition
  • The relation of the modes are the same

  • Taking transposition into account ,other properties are similar to the CM mirror

  • Example -1
    Mirror tonality similar to original scale

  • The pivot is the tonic of the mirror
  • note that starts the Mirror in the same tonality

    Pivots and related tonalities

    Since scales have common pivots with different associated tonality, we need a mean to know which scale we are in.

    Pivots are the 7 scale degrees in a ascending fifths series form.Each pivot is related to a specific interval with the associated tonality according to their scale function shown in the following table.

    For example let’s take the CM scale

    With G the associated tonality is Bb : a minor third above so G Is the V of the original scale(C).
    Now let’ examine DM scale

    G is associated to Ab (a second minor above) therefore G is the IV of the original scale(D)

    With A as pivot
    – A-D =a fourth so a VIth degree in CM
    -A-C=a third so a Vth degree in DM