The Proper Analysis of Triads

As we have frequently warned at various stages of the development of Harmony -
Beware of Triads !
We feel that it is impossible to repeat this often enough,
and advantageous to offer numerous examples of how insidious and deceiving triads can be.

We will look at what seem to be swings between the Em and F triads.
     1. The distances between the two triads
               both in Pitch (only a semi-tone away)
               and in Chrominicism (at opposite ends of the Window)
           should already alert us to the fact that they cannot both be real triads.
     2. At least one of them must be a deceptive triad
               with Orbit 1 substituting for Orbit 0.
     3. Both triads might be deceptive, as we will soon see.
     4. We will start with three different Tetrad swings,
                written with the orbit colors but without any rhythmic structure,
          and observe how the Triad Sonorities, Em (3-4) and F (4-3), evolve from them,
               written in plain black, but rhythmically structured at Level +1.
     The orbit colors show us how evident Tetrad analysis is, and
          the plain black shows us how difficult Triad analysis can become.

The C Swing

The F6-C+7 Swing, with COMMON TONE C
          which we could call the C Swing without any loss of precision or detail,
     is the flattest swing in the natural diatonic Window, in which
          the F triad is evidently a real triad, part of F6,
               with the PROPER TONE F substituting for the MOTRIX D, and
          the Em triad is evidently a deceptive triad, part of C+7,
               with the PROPER TONE G substituting for the COMMON TONE C.
     Both of these PROPER TONEs are always doubled on the Triads,
          the F on the F triad and the G on the Em triad.
               What seems to be the "third of a deceptive triad" will always be doubled,
                    whether it be major or minor.
     The F triad will be tuned F, A-, C, and the Em triad will be tuned E-, G, B--.

We first have the complete tetrad swing as it was exposed in Voice-leading A
We then have the version with the real triad of F and the deceptive triad of Em.
          Note the PROPER TONE to COMMON TONE movement
               the notes G to C, in the Soprano and Tenor voices.
          Note also that the Soprano and Tenor voices have been inverted
               to give the parallel fifths (F-C and E-B) the most advantageous disposition,
                    between Bass and Tenor, then between Alto and Soprano.

Parallel Fifths
     Academic purists might very well object to these parallel fifths.
     However, exceptions to this rigid interdiction inevitably appear subreptiously
               with examples from our most respected composers.
          Unfortunately, no truly valid reasons are ever presented
               either to condemn the use of parallel fifths in general,
               or to justify these "exceptions".
     The most valid criterion seems that of Voice-leading,
               including proper resolution of Non-chordal Tones,
          and truly accurate Voice-leading is not possible with Deceptive Triads,
               only with complete Tetrads.

A few examples might be in order -

Here is an example in a Spanish "Flamenco" style which cadences to the DOMINANT
     It might actually strike a bell (Bell-6) for aficionados of French chamber music.

Now an example in a Jazz style which cadences either to the DOMINANT or to the TONIC,
     a chord of G7-5-9 (often mislabeled Db7), in which
          the PROPER TONE to PROPER TONE line, transformed by Diminution (Db-C)
               is in the Bass voice,
          the COMMON TONE to COMMON TONE line, with the chromaticized Non-chordal Tone (Ab-G)
               is in the Tenor voice,
          the MOTRIX to MEDIAN line (F-E)
               is in the Alto voice, and
          the MEDIAN to PROPER TONE line (B-C)
               is in the Soprano voice.
     Certainly current practice.

A few more random examples -
     The beginning of Act 3 of La Bohème (Puccini)
     The accompaniment of Pink Panther (Mancini)
     As well as all the Organum from the IX to the XII centuries.
Let the ear be the final judge.

The D Swing

The G6-Dm7 Swing, with COMMON TONE D
          which we could call the D Swing without any loss of precision or detail,
     is the central swing in the natural diatonic Window, in which
          the F triad is evidently a deceptive triad, part of Dm7,
               with the PROPER TONE A substituting for the COMMON TONE D, and
          the Em triad is also evidently a deceptive triad, part of G6,
               with the PROPER TONE G substituting for the COMMON TONE D.
     Both of these PROPER TONEs are always doubled on the Triads,
          the A on the F triad and the G on the Em triad.
               The "third of a deceptive triad" is doubled on both chords,
                    one major and the other minor.
     The F triad will be tuned F+, A, C, and the Em triad will be tuned E, G, B-.

We first see the complete tetrad swing as it was exposed in Voice-leading A
We then see the version with the two deceptive triads of Em and F.
     There are no parallel fifths in this swing because we have 2 deceptive triads,
          and even the most stringent academic purist could hardly dispute the Voice-leading here.
          Note the PROPER TONE to MOTRIX procurations,
               the notes G to C, in the Soprano and Tenor voices, and
               the notes A to E, in the Alto and Bass voices.

The E Swing

The Am-6-Em7 Swing, with COMMON TONE E
          which we could call the E Swing without any loss of precision or detail,
     is the sharpest swing in the natural diatonic Window, in which
          the F triad is evidently a deceptive triad, part of Am-6,
               with the PROPER TONE A substituting for the COMMON TONE E, and
          the Em triad is evidently a real triad, part of Em7,
               with the PROPER TONE B substituting for the MOTRIX D.
     Both of these PROPER TONEs are always doubled on the Triads,
          the A on the F triad and the B on the Em triad.
               The "third of a deceptive triad" is doubled on the F chord,
                    in this case it is major.
     The F triad will be tuned F++, A, C+, and the Em triad will be tuned E, G+, B.

We first see the complete tetrad swing as it was exposed in Voice-leading A
We then see the version with the deceptive triad of F and the real triad of Em.
     Once again, we find parallel fifths (E-B and F-C)
          between Alto and Soprano, then between Bass and Tenor.
     It was not necessary to invert the Soprano and Tenor voices here,
          these "parallel fifths" already being in the most advantageous disposition.
          Note the PROPER TONE to COMMON TONE movement,
               the notes A to E, in the Alto and Bass voices.

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