Intonation exercices

This  page  is an illustration of  What does means play in tune on violin

  • Harmonics 
     Harmonics technique yields eight notes out of the twelve chromatic scale notes.The four missing notes are in yellow
  •  Those four missing notes appear  as  Tartini’s third sound technique
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Public Domain Mark

Cette œuvre (Intonation exercices on violin, de Alain lefébure) est libre de restrictions de droits d’auteur connues

What does mean « Play in tune » on violin

Fortunately   or unfortunately we can’t definitely answer : it depends  on context.

The  great french violinist Jacques Thibault  was sometimes criticized   for his « out of tune » intonation but, as pointed out by  one of his famous students,Ivry Gitlis , : »playing the  flat very low »  he added  color

In the art of violin , Laurent Korcia likes the    slightly hoarse sound  of Isaac Stern   who « played out of tune but sounded  in tune ».

Mind you, we are discussing international soloists, not  amateur fiddler . Yet,their intonation was   sometimes dissentious

So,let’s try to make things clearer

Main  acoustic systems have been discussed in Acoustics and are summarized below but 

first let’s remind the deaf acoustician  Joseph Sauveur’s discovery in 1701 on  harmonics sounds ,

Most of the time sounds are compound sounds made of a strong sound named fundamental and other weaker sounds called harmonics which frequency is a integer multiple of the fundamental frequency


Fundamental sound  multiplied By gives
  2 The octave
  3 The fifth
  4 The octave
  5 The Major third

(division by 2  may be necessary to remain in a octave span)

Violinist ,violist ,cellist and   double bassist have three choices

    1. The Pythagorean System

based on a string length division

    • by 2 that yields the octave

and by

  •  3 which produce the  Fifth

(actually a 12 th divided by  2 to get a fifth )

12 fifths slightly  overtake 7 octaves ( B# is higher than C)

    • The octave is perfect (pure) :it is the only  common interval for  all  systems
    • Fifths are perfect (except one)
    • Major third is higher than Natural major third (harmonics 5)

it consist of

  • one  sort of  tone
  • one sort of diatonic semi tone which is very narrow and therefore strongly  attractive.
  • one sort of chromatic  semi-tone  larger than the diatonic semi-tone.
  • one  sort of  comma.
  • The Zarlino’s system

adds division par 5   to the Pythagorean system to get natural Major third of the harmonic  series .

This addition brought great disturbances into the system but enhanced the Major and minor chords  (with no beats)

However this  advantage est limited to 3 thirds  the primary chords in major keys


In this system

  • Octaves are pure
  • so are the thirds in a given tonality
  • only  4 fifths are pure
  • There are two kinds of  tones
  • Two kind of  diatonic semi-tones
  • Two kind of  chromatic   semi-tones (both are smaller than diatonic semi-tones )

Globally  semi-tones are very  large and therefore without or little   attractive power

  • Both  tetrachords are  different
  • Sharps and flats have no fixed value which depends upon the chords progression but ,in any case,sharps  are lower than Pythagorean  sharps ( and flats higher )
  • B#  may be higher or lower than C      according to the chord progression.
  • About 10 sorts of  comma are available.

These  inequalities  make the  system improper to modulations

  • The tempered  system
    We saw that  pure intervals but octave impede  the circle closure  (B can’t go to C)
    Therefore the octave  were divided into  12 equal parts where every interval is a bit out of tone but bearable by the ear. Any modulation is possible but  to the  detriment  of color.
    There is only one sort of  tone and semi-ton (a with two orthographies: diatonic/chromatic) and ,therefore, no comma.

These   3 systems are  different from theory  with its  4 commas diatonic semi-tone and its  5 commas chromatic semi-tone but  values are near  Pythagorean   ones  .

It must be pointed out that those  systems are mainly based upon mathematics,not on physics.

From this summary ,we can draw some evidences

  • In solo playing ,Pythagoras is the most  natural since the tuning is based upon the fifths (or fourths for double bass).In another hand, the  attractive power of the diatonic semi-ton allows an expressive melodic line .
  • In a duet with piano, the tempered  system is obviously the best choice since  piano is tuned in equal temperament to restrict the number of keys ( C#Db is the same key for example)
  • In a quartet  Zarlino  is partly an option since viola-violin  form  a  sixth (13 th) C-A too low and a third(17th)  C-E too high with open strings, furthermore music is partly chordal -but also melodic!

So  « partly »is here an important word  since we can’t use this  system at any time .

Threfore, performer has  permanently to make a choice

        • strategic  choice according to playing set up (solo-duet and so on).
        •  stylistic choice: for  example choose   Zarlino for its no  attractive  semi-tones to play a modal melody  or  Pythagoras to play a brilliant  melody.
        •  technique choice : Paganini tuned the G string  a quarter tone higher to soften the thirds/sixths (opening of  Caprice 21  for instance) oh ! we are getting out of the subject.
          But choice also means  criticism
        • Choosing tempered  system is the safest but without personification : Again in the art of violin,Isaac Perlman seems to regret the former generation of  soloists who sounded all  differently.

However , acoustic systems in the intonation work is relative


  • These  systems are based on  calculation that produces  different  results according to the reckoning approach.
  • The ear work s on  logarithmic function  and endures small frequencies difference.The discrimination is around one  Hertz for a trained ear  but dissonance only occurs beyond a 5 Hertz span.
  •  Quasi systematic use of Vibrato obliterates   the gap between frequencies and most kind of commas which enter the rear discrimination  zone.
  • Dynamics may theoretically  spoil  the intonation  but it’s negligible in the scope of usual  melody frequencies.

Besides  acoustical systems ,the most important thing  to discuss.

    • harmonics:Usage of open string is common to control intonation. Natural harmonics can be used in the same way but  I would like to point out that harmonics of a given string produce a just intonation major chord  since harmonics come from division of the string length. A given note played  in  tune with an harmonics  sounds sympathetically  with other string. See intonation exercicesNotice only four notes out of twelve ( with octave skips since fifth of a string is root of the next string)                 are missing  but the


  •   Tartini’s third sound  technique makes good the lack.
    It may be useful to to play different interval each after other, especially alternating sixth -third to perceive the ghost tone . This Leopold Mozart’s advise demonstrates the hardness to hear it. The  double stops progression  for a given ghost tone reminds us the 6


As for  harmonics 7,  Tartini’s third tone is not really  scientific  since it is not easily reproducible  but it’s a mean to search a clean intonation  and a pretty sound.

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.


  • 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


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 .
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.


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



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


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


                                                    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)


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)




The Saxhorns group is the example  of confusing writing   previously discussed

This group has historically been  subjected to addition and subtraction of instruments but the most common range includes

Flugelhorn Eb (or sopranino)

Flugelhorn Bb ( soprano or contralto )

Alto Saxhorn Eb ( or Ténor Horn)

Baritone Bb  ( or Tenor Horn)

Basse Bb

Contrabass Eb
Contrabass BBb

Their theoretical range is similar to the trumpet one .Actually only Flugelhorn (Bb) and Baritone have a satisfying full range   when played by a concertist.

All flugelhorns and Alto Saxhorn (Eb) share the same standard range but the Eb small flugelhorn is really efficient in a smaller range (One octave (E-E) .

Since these instruments are dedicated to the upper parts they logically use the treble clef

Bass and contrabass saxhorns (standard range below) playing the bass line are logically written with bass clef.

However Baritone (also  named tenor and therefore written with treble clef) sharing Bass register a single sound are differently written ,while Bb Bass using the same written note and clef  as BBb contrabass sound differently .


1 Flugelhorn Eb -2 Flugelhorne Bb 3 Alto (tenor Eb horn)                                          4-Contrabass Eb 5-Contrabasse BBb


An historical  ambiguïty   results  from  the old classification;   Baritone  beeing   considered as a  Bass should had been written with  bass clef

Tonalities and transposing instruments

Music notation software have solved most transposition problems. However the choice of a piece tonality according to a transposing instruments panel remains an  issue; so does the choice of a transposing instrument according to the tonality of the piece.

Transposition is a simple transfer of the fifths cycle.Let’s imagine a sliding rule:The fixed (yellow) part reads tonalities of the piece while the sliding (blue) part reads the instrument tonalities.

So willing to write an obsolete Db piccolo part for an Bb (2b) melody

I place the instrument C facing Db(piccolo tonality) on the fixed part ;to Bb (piece tonality) tallies (A) instrument(blue) tonality.



This needless  method for a single instrument becomes a  precious tool  when several  transposing instruments are involved.

First of all it should be reminded that the writing confines tonalities  to 7 accidentals(From  Cb(7b) to C#(7#)) in order to keep a  relative legibility; Beyond  those limits we  use enharmonic writing




« Forbidden » zones are in red but enharmonic choice  is also advisable  if an easier alternative tonality is available.

For  example , let’s take an  Ab Maj (4b) piece

For an A  clarinet the chart reads     7b  key signature

The best choice would be to switch to a Bb clarinet (2b)


But ,should we use an A clarinet for any reason ,we ‘d choose the  B Maj (5#) enharmonic writing.

Below is a correspondence chart

Neglecting the classical  double flats/sharps notation we  suppose an unlimited number of flats/sharps instead.

The result is always  12.



Double sharps equivalence



Notice the reverse direction of increasing flats and sharps

Transposing instruments

It easy to explain transposing wood wind instruments   but ambiguities  occur  when brass instruments are concerned.

This post attempts  to clarify  brass instruments.


Reminders: As  far as  transposing  instrument is concerned, the eye  (l’ oeil)   and  ear (l’oreille)  doesn’t perceive the same thing .

For example  a  trumpet player   reads  then plays  « C »  but audience hears B flat.
The aim of transposing  instruments  is  to keep the same fingering  with different sounding instruments.

For  example with  saxophone.saxotenor

In both cases written notes and fingerings are the same but sounds differently

Range of woodwind instruments is clearly established  in the lower register.

There are few exceptions
-One key added on bari sax that can reach A instead on Bb on other saxophones.

-one missing key on English horn which cannot play the Bb of the oboe.

For any instrument, range in upper register   is subject to variations due to  player ability.

This  simplicity is due to the wood wind instrument sophistication through holes and keys which  allow   to select   partials (or harmonics).

Difficulty  of brass instrument  paradoxically results from their  simple construction

Brass are, historically speaking, a  simple flared  tube  with an  embouchure  which allow the  emission of a  note ,the  fundamental or pedal tone  , and its  partials or  harmonics.Those added sounds are simply, if I may say, selected by the way the performer blows in the tube.



Sound  1 Fundamental or  Pedal sound depends on  the length  of the tube.

A mere sight  of the series  reveals a first group of problems.

1-Lower notes progress by skip up to the seventh sound

2- The following conjunct notes are not necessary diatonic to the first sound.

3-  Some diatonic sound are out of tune in our equal temperament  system.

4-The  series doesn’t provide all chromatic sounds.

Our modern instruments are 

-The association of several  tubes (trumpet, saxhorn) to supply the missing notes or the nearest notes of the equal temperament system.

This is done by mean of valves or slide that  broaden the length of the initial tube

-or by fusion  of two  instruments,  as seen in the french horn (F+Bb) or F-attached tenor trombone . The   switching is made with a key.

Three valves provide seven combinations corresponding to  seven bugles covering a tritone chomatically.


Therefore, a Bb trumpet is made of a (no valve) Bb trumpet, an A trumpet, an Ab trumpet.. and so on up to E (the lowest)

Those seven combinations are similar to the  7 positions of the slide of the trombone.

A crook is  another possibility to lengthen the  tube . We can,for example, turn a Bb trumpet into A trumpet but with many defects.

A fourth  valve  extends the tritone range to an octave (12 combinations) which ease the reach of pedal tones especially on  saxhorns.


However, the fourth valve doesn’t necessary mean a larger range. It may only  provide a better intonation but  also bring new problems which possibly require more valves to solve the problem.

A second group of  problems is due to the impossibility to cover the overall series with a single  instrument.

Only instrument with large diameter can reach pedal tones.

Overcoming the harmonics 7  require a small diameter instrument

The  Ratio diameter/length of the  tube is also  important to reach the pedal tones.

The shape of embouchure  is  another factor to select partials which determine the timbre. It may help to reach some  pedal tone but too weak in strength to be efficient.

For instance a bass saxhorn (tuba  ) can reach the contrabass ‘ lower notes  but they are weaker, so the  larger range of the bass   is at the expense of the strength of lower notes.


However this instrument is only effective in the following rangesaxhorane

It must not be confounded with the Bb low  bass Tuba which uses a French horn embouchure. Its range is

bastfsounding a second below.

That’s why  we can hardly express the range of brass instruments. Furthermore,instrument makers have evolved lot of devices to enhance the playing which broaden  the range and panel of instruments.

A third  group of  problems   comes under instruments  making and nomenclature.

The first difficulty is

-The name of  instrument which doesn’t reflect  the   register. For example, the  saxhorn Bass  has the range of  contrabass Eb but lower notes haven’t their strength.

-A given  name  correspond to different  instruments   according to countries For  example  » bombardons »   mean   either  contrabass  or include tenor bass. In an other hand the word tuba refer either  to a non-homogeneous group of low brasses  or a specific name for bass-saxhorn  (in C or Bb) or Bb saxhorn  contrabass.

–Confusion of register  and  instrumental group.  Especially    US Baritone (saxhorn) and euphonium  Bb  which is not a saxhorn; its timbre is  different .

The complete  range of   Jupiter euphoniums should stops the confusion.

Another difficulty  is to muddle  tonality of the  original tube  and the concert tone.

For  example, the  trombone is a  Bb  instrument  because  in first   position ( the shortest tube) it  emits a Bb  and its  harmonics but it’s a C instrument  (meaning  non transposing)   because the    trombonist reads and hears Bb  and so does the audience.

Caution:Modern  tubas follow this  logic contrary to former saxhorn- tubas that are transposing instruments.

At least  some  puzzling writing  conventions

-No accidentals on the score for old instruments   (Naturals)

-Treble clef for   « tuben » according to the saxophones logic  to  keep a common fingering for  different (lower Major  sixth (Eb instrument ) or  lower Major  ninth(Bb instrument) Therefore  transposing instruments). This practice is found in the salvation army brass band where all part but bass trombone part are written in treble clef

But the same part may be written in bass key sounding a higher minor third   (Eb instrument ) or  a  lower major second(Bb) instrument

or -written in bass clef with concert tone ( non transposing  instruments).

The problem often occurs with euphonium. The reason is more or less historical  when trumpet players,  only reading  treble clef and playing three valve instrument, switched to euphonium especially with 4 valves.

Fingering of  trumpet Bb  is similar to the one of euphonium sounding a ninth lower.(An Eb euphonium would sound a sixth below)



(Figures refer to valves)

Another historical reason is  the octaves determination according to Organ pipes length.

Sound threshold   tallies with a 64 feet pipe

-32 feet pipe corresponds with lower C of a 5 strings double bass

-16 feet  correspond with lower C  of cello

-8  feet correspond  with lower C  of viola

Since Tenor voice was written an octave above its sound,(baritone  saxhorn) is written in treble clef  but sounds as a 16 feet pipe (an octave lower)  On the other hand  a bass  saxhorn  is written (as a bass)  in bass clef ; Being a 16 feet pipe its sound register is similar to the baritone’s one.

Double bass is written  like a bass but as a 32 feet pipe sounds an octave lower .

To summarize

Baritone-Bass____ Different Clef —Same sound

Bass -Double bass—Different sound—-Same  clef


« Real » tuba  player  read bass clef and play concert pitch with a specific fingering which is different  according to the tuba tonality). In open, trumpet plays G (treble clef) while tuba plays F'(bass clef). the 1-3 valve combination yields D  to the trumpet but C to the tuba that can  possibly replace the 1-3 by 4

Treble clef writing presents  an ambiguous  zone  due to the large compass  of euphonium



In the above example , are the framed notes written in concert pitch for a tuba player or in transposed notes written for a trumpet player?

So I advise to use the tenor clef to write those notes in concert pitch.


Euphonium is Bb according to the trombone logic explained above and therefore non transposing in bass clef

Nowadays  tuba  scores are generally written in concert pitch (Non transposing)

Since tuba range is very broad, professional tuba players, king of the transposition  choice the best tuba for the given tonality .


For the composer  the difficulty concerns writing for wind orchestra especially  for amateur,unaware of transposition .

From the trumpet range.


The three valves  saxhorn baritone is written  an octave lower

F  contrabass  is written a fifth below the baritone and the Eb   contrabass  a sixth  below the baritone  but start on G

Bb bass and contrabass are written an octave below the baritone but start on G with a smaller span (2 octaves).

Tenor trombone shares baritone range while ‘euphonium and bass trombone add a tritone to the lower register


French horn covers the range of the trumpet and of the   baritonecor-range

However the whole range should theoretically  be divided up into low horn  (II-IV)  and high horn (I-III)




Actually  , the standard writing  is as follow


Below is  the advisable range (étendue) of brass instruments (in concert tone)




Avoid pedal tones  in  writing for  trumpet and tenor  trombone . Pedal tone  should be assigned to bass trombone

French horn can emit pedal tones only through the Bb tube but only two pedal tones  (B-A)  are reliable.

When writing for a specific instrument

-it’s advisable to consult the player.

-Whatever the skill of the player one must never forget the general melodic  rules: remain  in the median register with a progressive  reach by step of extreme notes .

More about transposition
Tables of brass instruments (notation and concert pitch equivalence


What makes that music Masonic?

Free masonry and music so deeply entwine that we can hardly describe the linkage. It can be divided up into two situations
1°) Music serves masonry:
Some music is specifically written for ritual purposes. For example Mozart wrote
“An die Freude”(to joy); a song based on a Masonic poem dedicated to the union chain
– Some Instrumental pieces such as “Adagio K411” intended to processional entrance for lodge or ”Maurerische Trauermusik” (Masonic funeral music) for funeral or  (possibly for initiation to the third degree)
“Gesellenreise” (journey of the companions) sung during the initiation to the second degree.
However, the situation is not always so clear. For instance “Auld lang syne”,sometimes  sung during the union chain, is a Masonic adaptation by Robert Burns of an old folk song that broadly oversteps masonry. Furthermore, lots of rituals are supplemented with non Masonic music.
In an other hand ,some Sibelius’ music pieces are labeled “Church Music” though there were specifically intended to Masonic ritual.

2° Masonic symbolism is part of the music material for the composer.
“Die Zauberflöte” (the magic flute), by Mozart and Schikaneder (both masons) is certainly the most convincing example but most of the time the situation is rather unclear; the danger being to see symbol where there is not. The most striking example is the number 3 which commonly occurs in classical sonatas and others pieces .
The most evident situation is therefore songs based on an explicit Masonic text even though some doubts occur as far as Beethoven is concerned.

Occurrence of numerology or symbol in musical piece may be fortuitous or remains the, not necessary a Free Mason, composer ‘secret. Usage of the golden number is so frequent nowadays that it is no longer a secret.

A third situation embraces the two first ones.

Here I’m thinking to the opening of Mozart’s quartet K465 which expresses the impression of the apprentice’s initiation passing from dark to light. This piece « containing many mistakes that offend the good taste« * gave rise to to many critics and were largely rejected by his contemporaries . This is an illustration of the difficulty to export  masonic idea outside from the lodge.

Conclusion: intricacies of symbolism do not allow perceiving a clear border between profane and sacred music.

*Prince Krazalkovicz

Two-part writing insight

Disclaimer: this paper is a personal view on two- part writing way off the academic approach.


Two simultaneous notes form an interval while a greater number of simultaneous notes creates either chords or other terms such as Cluster that depend on school and music eras.

In a nutshell, counterpoint uses interval while harmony is based on chords; so Two-part writing is logically the concern of counterpoint.However,counterpoint is prior to tonal music and its rules, justified by Pythagorean intervals, are somehow obsolete with equal temperament. In an other hand, tonal  harmony stems from counterpoint and therefore shares some rules with it but  its at least three voices setting induces inaccuracy in two-part writing. That might be  the reason of the  scattered number of published methods

Hindemith is one of the most prominent contributors to tonal renewal of counterpoint but his sophisticated  method  might be hard to start with.

This paper is an attempt to draw some functional  implications of   intervals from tonal considerations .

1°) Scales

Tonal music consists of two modes: the Major mode and the  minor mode which can appear  under three aspects :

-The natural  scale or descending melodic scale similar to eolian mode with a subtonic instead of leading tone.

-The harmonic minor scale ;a leading tone replace the subtonic.

-The ascending melodic scale with a sharped sixth degree to correct the augmented interval created by the leading tone.

2°) Triads

Stacks of thirds over each scales degrees yield four kinds of triads

  1. Major triad made of a Major third and a minor third (from bottom to top)
  2. Minor triad formed by a minor third and a major third
  3. diminished fifth triad that consist of two minor thirds (to differentiate from consonant harmony)
  4. Augmented fifth triad made of two major thirds (which does not belong to tonal harmony)

3°)Scale degrees

Scale degrees are not equal in importance

The most important are I-IV-V known as primary or tonal  chords which support either a major or a minor triad according to the mode.

others chords are secondary chords that may be of any kind according to the scale.

4°) Chord inversion.

Any triad has two inversions but some are uncommon . As a rule, only primary chords can non-restrictively  be inverted. II and VII can be  inverted under some conditions.Other  chords are normally  not inverted but in sequence

Inversions result in chord structure changes.

5°) Omission of note

When  an  incomplete chord is needed the fifth  is generally omitted. Therefore,  chords  in root position appear under a third form (either Major or minor).

The third  is obviously omitted  in diminished fifth chord.

In tonal harmony the  third can be omitted

-in the Dominant chord and the Tonic chord if the key is well established. Thus, a perfect  fifth means V or I

in sixth  chords so first inverted chords are  under the form of a Sixth interval 

The fourth can be omitted  in 4/6  cadential  chord or the  passing  VII (b) on a weak beat

Removal of the sixth  may occur  in I6 ,V6  and  VII4/6 which occur under a minor third structure


3 means either 5/3 or 6/3.  Thus,  CE can CEG or CEA

6 is 6/3 or 6/4. EC can be EGC  or EAC

Primary chords prevails when interpreting  an Interval .Therefore, with CE  in CM we choose CEG=I5  but in minor we’d choose CEA=I6

The precedence is

I >VI>III  so EG  is I6 ,not III5 (In CM)

V>III GE is I4/6,not III5


but IV is equal II

From all those considerations  we can establish  the following guideline

7°) Rhythmic position

is the last consideration  to take into account: Especially  secondary chords should  be on weak beat.


Although the logic  is different from the classical taught  counterpoint, usage shows  a similar  result with  the classical writing rules so  Much ado about nothing but I did try.

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.


  • 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




    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

    • -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


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



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