Illusions in general
One of the main objectives of science is to demystify illusions,
regardless of the sense involved (sight, hearing, smell, taste, or touch).
Some cases may even involve several senses simultaneously,
like the experiments showing the effect of color on the sensation of heat :
participants felt colder in a blue room than in an orange room,
even if both rooms were at exactly the same temperature.

Optical Illusions
               are by far the most popular and the most developed.
     A facinating book fell into our hands recently,
          THE GREAT BOOK OF OPTICAL ILLUSIONS, by Al Seckel, published by Firefly Books.
     It is truly remarkable to discover how many talented artists, of all periods,
          have delighted in confusing the viewer with an almost infinite variety of optical illusions.
     It is in this book that we found the illustration perfectly suited to this presentation.

Acoustical Illusions
How come so much has been developed and published concerning Optical Illusions
          and so little concerning Acoustical Illusions ?
     Do we rely on the ear more than on the eye ?
          Or would it be more accurate to state that we rely more
               on the interpretation the brain makes
                    of what we hear than of what we see ?

It seems high time to draw attention to acoustical illusions,
not only to the scientifically examined ones,
but, above all, to those pertaining to music
which have not yet been examined.

The Doppler Effect
It is surprising to learn that the effect discovered by Austrian scientist Christian Doppler in 1842
     concerned the blue and red "shift" of the light of stars,
          that the effect on sound waves was only discovered later in 1845,
               and that the effect on electromagnetic waves was discovered in 1848.

A Definition Precaution
Before attacking the acoustical illusions that are pertinent to music,
     it would be wise to remember that the term "acoustical",
          is usually applied to sound rather than to music,
               and all music is sound but not all sound is music.

Knowing, using, and teaching only 2 of the 4 strong modes,
the Diatonic Major Mode and the Chromatic Minor Mode,
the normal illusion is to see the Chromatic Minor Mode
as a parallel version of the Diatonic Major Mode.

Missing Modes
Mode 1 (Diatonic Major) and Mode 4 (Chromatic Minor) have, in the reality of generation, little to do with each other, because there in another mode, either Mode 2 (Diatonic Minor) or Mode 3 (Chromatic Major), between them. Mode 1 generates both Mode 2 and Mode 3 directly, and either Mode 2 or Mode 3 will, in turn, generate Mode 4.

Two generations
There are thus 2 generations between Mode 1 and Mode 4 and the common direction of these two modes, both flattening (descending), is merely the inevitable result of a change of direction at each generation. For this same reason, Mode 2 and mode 3 are also in the same direction, both sharpening (ascending), each the inevitable result of only one generation. The chromatic specifics of Mode 4 must be deduced from the harmony of its DOMINANT/TONIC swing and not from any new, invented scales.

Sun and Moon
The comparison with the sun and the moon, both seen from the earth, is too tempting to overlook. They both seem the same size and a solar eclipse seems to be an eloquent proof of this as the moon completely obliterates the sun as it passes between it and the spectators on planet earth. However, in reality, the moon, being a satelite (child) of the earth, which is in turn a satelite (child) of the sun, is itself a grandchild of the grandfather sun. Another 2-generation relationship.

Triads are probably the most current acoustical illusion in all of music,
creating considerable uncertainty
in both the voice-leading of the chord parts,
as well as their appropriate doubling.

Preferential Chord Size
Piling thirds one on top of the other, on each degree of a scale, gives little if any indication of what the basic, fundamental size of a chord should be.

Non-chordal Tones
When asked "How long should a man's legs be?", Abraham Lincoln had answered "Long enough to reach from his body to the ground". When asked "How many notes should a chord have?", we would answer "As many true, chordal tones as possible". A clear, unambiguous definition of "chordal tone" is evidently the first step. The conclusion then becomes immediate and irrefutable - there are four notes to a complete chord, no more, no less. If there are less, the chord is incomplete and the transformation of Incompleteness has been applied. If there are more, the chord is "overstuffed" and a non-chordal tone has been placed on one member (orbit) of the chord, while the original chordal tone has been kept, usually in the bass voice.

Incompleteness and Triads
The concept of Triads as incomplete Tetrads will require a change of order in the harmony curriculum. Complete Tetrads come first, then the transformations of Incompleteness which produce two distinct and very different kinds of Triads which behave quite differently. Real Triads all behave the same way and Deceptive Triads all behave the same way, in both voice-leading and doubling. This is clearly exposed in Triad Harmony, and explains the illusion which Triads have always been.

Phases of the Moon
Primitive societies might very well have imagined that what they saw in the different phases of the moon was all that was left of it and that it was constantly changing size. It is still currently believed that chords are the same way, constantly changing size.

What happens to the voice-leading and the doubling,
which seem to have gone beserk,
when the chord of IV, more precisely IV⁶,
progresses to the chord of V, more precisely V⁷ ?

The 2-Generation Approach Again
A chord does not necessarily have the same function with respect to the chord which precedes it as it has with the chord which follows it. A man does not have the same function with respect to the father which precedes him (that of son) as he does with respect to the son which follows him (that of father).

Metamorphosis 4
In the chord pattern I⁷ - IV⁶ - V⁷ - I, the chord of IV⁶ (composed of the notes C, F, A, D) is transformed into the chord of II⁷ (composed of the notes D, A, F, C), what we call Metamorphosis 4. The same four notes have one function with respect to the preceding chord and another function with respect to the following chord. All the regularity and reliability of the voice-leading of Tetrad progressions by fifth have been preserved, perfectly intact.

Moving Around
These metamorphoses (there are four of them) are what allow Harmony to really move around without ever affecting the rigors of voice-leading which remains constant at all times.

Is the "Blues Chord" really a seventh chord,
even if it resolves to the chord a fifth higher rather than a fifth lower,
and even if the seventh itself resolves upward rather than downward ?
What kind of illusion is this ?

The Illusion of Enharmony
Let's start with a brief definition of enharmony. The notes G# and Ab have exactly the same pitch in equitempered tuning and are only a comma apart in Pythagorian tuning. However, they do not have the same degree of "sharpness" or "flatness" which we call Chrominicism. The G# (a sharp version of the note G) is a very sharp note with a tendency to rise to the note A, while the Ab (a flat version of the note A) is a very flat note with a tendency to fall to the note G. For all intents and purposes, one could say that enharmony is a desire to confuse these two notes and use one instead of the other.

Back to the Blues Chord
In the Key of C, the Blues Chord is commonly labeled "F⁷" with the note Eb as its seventh. When it resolves to the chord of C, its seventh Eb resolves to the third of the next chord, the note E. It seems quite evident that what is commonly called (and written) "Eb" is in reality a D#, the augmented sixth of the chord of F, in reality a chord of F⁺⁶ composed of the notes F, C, A, D#.

Placing It Elsewhere
It seems a bit curious, and somewhat disappointing, that no one seems interested in placing this "Blues Chord" sonority on the C chord when it resolves to the G⁷ chord. This would produce the complete chord pattern (one chord per bar) C / C // C / C⁷ / F⁺⁶ / F⁺⁶ // C / C⁺⁶ / G⁷ / F⁺⁶ // C / C .

Other Examples
There could be many other examples of Chromatic Illusions, mostly involving enharmony, which could be presented here.

"Stretch Chords", with their "stretched" frames,
create another level of triad illusion.
In reality, they are Real Triads
that sound (and look) like Deceptive Triads.

They Act Like Real Triads
"Stretch Chords" are Real Triads, in which a non-chordal tone "stretches" the frame of the chord. Despite their Deceptive Triad sound, the voice-leading, the doubling, and the choice of direction remain those of the Real Triad (without seventh or sixth).

They Sound Like Deceptive Triads
These chords have all the liberty of Real Triads even with a Deceptive Triad sound (the same three notes the Deceptive Triads have). Both Stretch Chord forms (stretched at one end or at the other) may be used on the same Real Triad so that the fundamental Real Triad sound is never heard at all (it can be reserved for the last chord of a cadence).

They Can Move In Either Direction
Having neither seventh nor sixth, these Stretch Chords, like the Real Triads, can move in either direction depending on the nature of the substitution. This makes them extremely flexible.

What is commonly called the "Diminished Seventh Chord"
is probably the most flagrant illusion of them all.
It seems high time to challenge
its reputation as master-key to all the other chords.

Generation of the Diminished Seventh Sound
The "Diminished Seventh" is not a real bona-fide chord. It is merely a Sonority, the result of various Transformations : Chromaticism, Non-chordal Tones, or both. Understanding it and using it properly depend, above all, on knowing the basic fundamental Tetrad from which it evolved. Resolution of these "Diminished Seventh Sounds" should be exactly the same as that of the original Tetrad.

Possible Sources
The Diminished Seventh Sound composed of the notes C#, E, G, Bb
     can come from six possible sources (basic material), the Tetrads :
          Eb⁶, C⁷, Gm⁶, A⁷, Em⁶6, C#m⁷.
     The progressions involved would, of course, be :
          Eb⁶-Bb, C⁷-F(m), Gm⁶-D(m), A⁷-D(m), Em⁶- B(m), C#m⁷-F#m.

Transformations
The Transformation involved would be :
     the Chromaticism of Diminution for the Tetrads of Eb⁶ and C#m⁷,
     the Chromaticism of Orbit 4 for the Tetrads of C⁷ and Em⁶, and
     the chromatic Non-chordal Tone of Orbit 0 for the Tetrads of Gm⁶ and A⁷.
          This is commonly called the "minor ninth" (^-9) for the A⁷ Tetrad,
               and would be called the "augmented fourth" (⁺⁴) for the Gm⁶ Tetrad.

Resolution
That this "Diminished Seventh Chord" with its six possible functions can resolve anywhere
          is not as certain as is often claimed.
     The following progression, passing from the flattest end of this spectrum suddenly to the sharpest end,
               does not really sound that well :
          Eb⁶, C#m^-7-5/E, F#m, B⁷, E.
               The passage from C#m^-7-5/E to F#m
                         (especially with the E to F# in the bass voice)
                    is not particularly pleasant.

Is the well known song Dark Eyes
really in 6/8 with its syncopations,
or generated in 3/4 without syncopations,
and then merely performed in 6/8 ?

In 6/8
Bars of 6/8 are defined here as being at Level -1, 2 bars for each cell of Level 0. Each bar of 6/8 could evidently be written as 2 bars of 3/4, which is the way that Dark Eyes is usually written, in a kind of 6/4. With these 6/8 (6/4) bars, the fifth note of each phrase is tied over to the sixth note across the center of the bar as a syncopation, /._'/._ (with s and s).

In 3/4
If the note-values of each 6/8 bar are disposed in 3/4 bars, the syncopations disappear, the fifth note falling exactly on the second beat of the 3/4 bar, /. ' /. (with s).

Where Is The Syncopation ?
There are actually 2 syncopations involved in this song :
     the first syncopation is the dot on the third note which retards the fourth note,
          and the second syncopation is the anticipation of the fifth note.
     The dot "syncopation" appears in both the 6/8 and the 3/4 versions,
          but the anticipated syncopation appears only in the 6/8 version.
The illusion here is in determining where this anticipated syncopation crept in.
     (a) In the 4/4 "original" (/ ' / )
          the anticipated syncopation of the fifth note (/ _' / _)
               placed it ahead of the third of the bar.
          The dot on the third note (/__'/__)
               retarded the fourth note.
          The ablations produced the final 6/8 version /._'/._.
     (b) In the 3/4 version there is always the dot (/. ' /. ),
          but the anticipated syncopation of the fifth note
               appears only when the song is performed in 6/8.