Since we make extensive use of Chord Symbols in this work, it might be appropriate to define them as accurately as possible and especially to specify how we use them.

Definition
Chord symbols may be -
     absolute, defining specific notes and thus particularly suited
               to atonal (non-tonal) music, such asFree Harmony, or
     relative, defining relationship
          to the tonality (in the form of scale degrees) and/or
          to the chord itself (in the form of its components) and thus particularly suited
               to tonal music, such asMelodic Harmony.

Roman Numerals

Roman Numerals represent the scale degree on which a chord is built
     (traditionally by the superimposition of thirds, major or minor).
This relative system has the advantage of being transposable -
     In the usual descending modes,
          theReal TriadofIwill always be the TONIC chord,
          the Real Triad ofVwill always be the DOMINANT chord,
          the Real Triad ofIVwill always be the COUNTER (sub-dominant) chord,
               and so on, regardless of the key involved.
To these Roman Numerals were eventually grafted
     suffixes (in Arabic numerals - also relative)
          which came from the traditional Figured Bass
               (used extensively in the 17th and 18th centuries) and which specified -
     1. the possible "inversion" of the triad
               (indicating which component of the chord would be in the Bass voice)
           V ₆- in "first inversion" (with the third in the Bass),
           V ₆₄- in "second inversion" (with the fifth in the Bass),
     2. the possible presence of a seventh in the chord, with its "inversions",
           V ₇- in "root position" (with the root in the Bass),
           V ₆₅- in "first inversion" (with the third in the Bass), and so on.
The numerals of these suffixes were placed vertically in the Figured Bass
     but that is not the most convenient disposition for word processors.
We shall use this system of Roman Numerals occasionally,
     mostly inHARMONY - BASIC MATERIALS
     and inHARMONY-TRANSFORMATIONS-INCOMPLETENESS.

American Chord Symbols

American Chord Symbols are mostly used in popular music.
This sytem is in extensive use today and we will use it far more often than Roman Numerals.
However, one must remember that it was conceived for performance only,
     and that it does not, unfortunately, always give a clear insight as to
          what a chord means (what degree of the scale it is built upon),
          where it comes from (what it was originally before being transformed),
          where it is going (to what chord it will resolve), and
          how it gets there (with what Voice-leading).

We will attempt to use it in such a way as to
     give as much information as possible, as clearly as possible,
     without deviating from the original operation of the system.

How do American Chord Symbols work?

The American Chord Symbol consists of -
1 - The absolute note name of the root of the Triad (Ex -  C, Bb, F#)
     It is understood that the fifth of the Triad is a perfect fifth.
          Both notes of theFRAMEare thus clearly defined.
2 - To this is added the symbol "m", only when the chord is minor (Ex - Cm)
     major chords, by default, requiring no special indication (Ex - Bb, F#)
          This tells us if theMEDIANof the chord is major or minor.
3 - Following this, we often find a superscript Arabic numeral  "⁷" or "⁶" (a relative symbol)
     which indicates theMOTRIXof the complete Tetrad:
          the seventh (Ex - G⁷, when the chord resolves down to C) or
          the sixth (Ex - Dm⁶, when the chord resolves up to Am)
4 - Then, any modifications which chromaticism or incompleteness
     bring to theFRAMEof the Tetrad, the fifth (5) and the root (1),
          (these superscripts are also relative),
     and possibly to theMOTRIX.
5 - And finally the /Bass, which indicates
     that a note other than the root is placed in the Bass voice.

Chromaticism

All of these superscript Arabic numerals (⁷), (⁶), (⁵), (¹), may be prefixed by
     "⁺", sharper, or "^-", flatter, so that it is now possible to have -
          the minor seventh (⁷), the major seventh (⁺⁷), the diminished seventh (^-7),
          the major sixth (⁶), the augmented sixth (⁺⁶), the minor sixth (^-6),
          the perfect fifth (⁵), the augmented fifth (⁺⁵), the diminished fifth (^-5),
          the original root (¹), the "sharpened" root (⁺¹), the "flattened" root (^-1).
It seems preferable to use these "+" and "-" prefixes, which are relative,
     rather than "#" and "b", which are absolute and already used in the Chord Name.
We are well aware of the fact that the "-" symbol is occasionally used to replace the "m",
     a confusion between relative and absolute symbols
          that should preferably be avoided.
Note that the following are enharmonically equivalent (of the same "size") -
     the diminished seventh (^-7) and the major sixth (⁶),
     the minor seventh (⁷) and  the augmented sixth (⁺⁶),
     the minor sixth (^-6) and the augmented fifth (⁺⁵),
          and are often confused with each other in the use of Chord Symbols.
A superscript Arabic numeral will affect only the note in question, none of the others -
     thus "G^-7" will indicate that -
          the seventh of the Tetrad is diminished, but
          the Triad of G major remains intact, with major third and perfect fifth.
     (The diminished seventh chord would be indicated "Gm^-7-5".)

Using American Chord Symbols in this manner
enables us to describe any chromaticized chord, no matter how complex, without losing the reality of its origin, and without using any kind of enharmonic equivalence (and thus clearly distinguishing between e.g. G# and Ab).

Examples
Let us consider a few misleading examples -
     In the section Chromaticism-Diminution, in the Blues progression, "C, C⁷, F⁺⁶, C", the F⁺⁶chord is usually, and erroneously, written "F⁷", as if the D# were an Eb, which creates confusion concerning the progression itself (F⁷progresses normally to Bb, not to C), and the Voice-leading (the seventh Eb resolves normally to D, not to E).
     In the same section, you will also find "Cm^+6+1-5" (instead of the enharmonic versions F#⁶or Ebm⁷, take your pick) and "G^-7-5+1" (instead of the enharmonic version C#m⁷).
     Well aware of the fact that most performers will prefer reading F⁷rather than F⁺⁶, it is not suggested to replace the traditional notation with what is proposed here. This new concept is mainly recommended for theoretical clarity.

Non-chordal Tones

Definition
Let us first define what is meant by a Chordal Tone and a Non-chordal Tone and how they differ from each other. A Chordal Tone is an integral part of the chord and cannot be resolved independently of the chord. A Non-chordal Tone is not an integral part of the chord and can be resolved independently (ahead) of the chord.

Example
Let's look at G⁷and G⁹-
on the chord of G⁷one cannot resolve the F before the resolution of the chord but
on the chord of G⁹one can resolve the A before the resolution of the chord.
The ear tells us immediately! The F is a Chordal Tone and the A is a Non-chordal Tone.

Since chords are composed of 4 Chordal Tones, no more, no less,
ninths, elevenths and thirteenths are all Non-chordal Tones.

The sus symbol
Since it is possible to place Non-chordal Tones on any member of a Tetrad,
     it seems preferable to forget all about using the numerals "9", "11", and "13",
     and, instead, to expand the use of the "^sus4" symbol ("^s4")
          which indicates that the fourth of the chord is suspended
               (usually carried over from the preceding chord)
          and replaces the third,
               exactly what a well-behaved Non-chordal Tone should do.

The chord C is composed of the notes C, E, and G.
The chord C^s4is composed of the notes C, F, and G,
     the Non-chordal Tone F replacing the E.

Extension of the sus symbol
If we can do this for the "^s4" which replaces the third (instead of using "11"),
     why not do it for -
"^s2", which would replace the root  (instead of using "9"),
"^s6", which would replace the fifth (instead of using "13"),
          not to be confused with "⁶" (theMOTRIXwhich cohabits with the fifth),
     and why not
"^s7" which would replace the sixth
          not to be confused with "⁷" (theMOTRIX),
     and possibly even
"^s1" which would replace the seventh.

In all of these cases, we are dealing with superior Non-chordal Tones which eventually resolve down to the Chordal Tone just below it. These suspended (superior) Non-chordal Tones can be designated by the abbreviation "sus" since it only applies to Non-chordal Tones.

The push symbol
We shall designate inferior (pushed) Non-chordal Tones by the abbreviation "push".
     Why the term "push"? Because -
          The major diatonic mode is essentially a descending mode,
               in which the progressions and theMOTRIXes are predominantly descending.
          Descending (superior) Non-chordal Tones resolve far more easily than
               ascending (inferior) Non-chordal Tones
                    which must "push againt the current" to rise to their resolution.
This being the case, the letter "p" will stand for a push -
     a Non-chordal Tone which resolves up to the Chordal Tone just above it -
"^p7", which would replace "1",
"^p2", which would replace "3",
"^p4", which would replace "5",
     and why not
"^p5" which would replace "6",
     and possibly even
"^p6" which would replace "7".

We are well aware of the fact that "^s2" ("sus2")
is used by some to indicate an inferior Non-chordal Tone (replacing "3"),
as if Non-chordal Tones of "3" were the only ones available.
It seems preferable to clearly indicate the direction of the Non-chordal Tone
and use both "sus" and "push" at all times.

Chromaticized Non-chordal Tones
"^s2", "^s4", "^s6", "^s7", "^s1", being all superior descending Non-chordal Tones,
     could very well become
"^s-2", "^s-4", "^s-6", "^s-7", "^s-1", when chromaticized,
     and
"^p7", "^p2", "^p4", "^p5", "^p6", being all inferior ascending Non-chordal Tones,
     could very well become
"^p+7", "^p+2", "^p+4", "^p+5", "^p+6", when chromaticized.

"G^7s-2" would be our way of writing "G^7b9",
     clearly indicating that the Ab is a Non-chordal Tone which replaces the G, and
"G^7p+4" would be our way of writing "G^7aug11",
     clearly indicating that the C# is a Non-chordal Tone which replaces the D.

Incompleteness

Why not simply "strikethrough" the numerals involved?
     For Real Triads,
          G ⁷ would be more complete than G because,
               even though the seventh is absent in both cases,
          G ⁷ indicates the direction of the chord, and specifies that
               G is theCOMMON TONEand that D is thePROPER TONE.
     For Deceptive Triads, we could indicate -
          G^{7 1} , C^{6 5} , and so on, conserving the notion of the precious original Tetrad.

Bass Note

One last thing - at the very end, after the numerals, may be added the /Bass symbol
     which indicates if a note of the chord other than the root
          is placed in the Bass, specifying the "inversion" of the chord
     - "C/E" - a triad of C major (C, E, G) with theMEDIANE in the Bass,
          "first inversion",
     - "G/F" - a chord of G major (G, B, D) with theMOTRIXF in the Bass,
          "third inversion".
               This is really a chord of G⁷, but it is better to write it "G/F" because
                    "G7/F" would indicate that the F is played both in the Bass and in the upper parts,
                         certainly not a desirable doubling.
The same applies to writing
     - "C/A" rather than "C⁶/A" with its undesirable doubling of theMOTRIX.

Our chord symbols are thus complete and capable of defining any possibility.

Wrap-up

          And now a short wrap-up in which we will concentrate on Abbreviations, Logical Procedures, and the Musical Structure of the chords.
          Important abbreviations are present at the very beginning. The chord symbol "C" not only signifies the root C but also the perfect fifth G, the completeFRAMEwith its 2 Primary Notes. The chord symbol "Cm" signifies the presence of the minor third Eb (the minorMEDIAN), and the absence of the "m" signifies the presence of the the major third E (the majorMEDIAN) the"M" which is omitted being the default of this Boolean (2-way) possibility. Now back to the chord symbol "C" which now signifies not only the root C and the perfect fifth G, but also the major third E (the majorMEDIAN), a considerable abbreviation which does not seem to jeopardize in any way the precision of the symbol.
          It is important to note that, so far, the symbol for Triads gives no indication of the direction which the chord is to take, which also means no indication of the respective functions of the Primary Notes, (theCOMMON TONEand thePROPER TONE) which are so important in the transformations.
          The addition of the first numeral ("⁷", or "⁶") is thus of the greatest significance. It completes the Tetrad, we now have the 4 Chordal Tones of the chord, with its direction of resolution and its distinctive Primary Notes (theCOMMON TONEand thePROPER TONE). The chord symbol "G⁷" signifies the presence of theCOMMON TONEG and thePROPER TONED, theFRAMEof the chord, the majorMEDIANB, and theMOTRIXF, a lot of information for only 2 characters ! 3 of these 4 notes were already specified by the "G" and the "⁷" specifies the note F.and the specifics of theFRAME.
          There seem to be few, if any, abbreviations in the other numerals.
          However, a lot of new material is logically derived from the existing usage.
          The choice of using "+" and "-" instead of "#" and "b" seems important. Notice that both are common usage ("b9" instead of "-9", for example) but that "#" and "b" are already used to specify the root name (as in Bb). Thus, the symbol "Bb9" (especially without superscript)  could mean (a) a chord of Bb with a major ninth, or (b) a chord of B with a minor ninth, whereas "Bbs2" and "Bs-2" cannot be confused with each other (even without superscript).
          The choice of extending the expression "sus4" ("s4"), instead of using "9", "11", "13", also seems justified because it permits a far clearer and more extended presentation of Non-chordal Tones and especially of the removal of the Chordal Tones which they replace.
          Remember that these new options are not meant to replace the common usage which tries to offer ease of reading at the price of analytic understanding. These new options are rather destined for those who wish to understand "what makes music tick".

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