Harmony/Structure of Pitch/Just Intonation/Diagrams

Tuning Procedure 1


Commonly called Pythagorean Tuning
          but which we will call Trunk Tuning, for reasons to be presented shortly,
     it is tuned uniquely with the interval of Perfect Fifth - (proportion 3/2)


This interval of Perfect Fifth is applied indefinitely, both upward and downward,
     to produce what we call the series of fifths.

The line encompassing the seven natural notes (called the "Window")
     indicates that we are in the natural diatonic system
          (with only natural notes, no sharp or flat notes).
This is an arbitrary choice because this Window could just as well be
     moved up (indicating the systems of 1#, 2#, 3# ...) or
     moved down (indicating the systems of 1b, 2b, 3b ...).
All notes inside the Window are considered diatonic
     (belonging to the diatonic system) and
all notes outside the Window are considered chromatic
     (not belonging to the diatonic system).
Being considered diatonic or chromatic must not be confused
     with being natural, sharp,or flat,
          (In the Key of G, the F# is diatonic and the F is chromatic).

Trunk Tuning (Pythagorean) generates the two notes of the FRAME of each chord,
          commonly called the root and the fifth,
     which we call the COMMON TONE and the PROPER TONE,
               depending on the role which they play in the chord progression.
          As its name implies, the COMMON TONE is common to both chords, and
               the PROPER TONE is specific to each chord.
          These two notes are also called the Primary Notes of the chord.
     The two other notes of each complete Tetrad (4-note chord),
               the third, which we call the MEDIAN, placed in the middle of the FRAME, and
                    the seventh or sixth, which we call the MOTRIX, the source of movement,
                         placed outside the FRAME,
               borrow their notes from the generated Primary Notes.
          The MEDIAN and the MOTRIX are called the Secondary Notes.
     See Harmony - Basic Materials

In this form of tuning, both diatonic modes (major and minor) share the same notes,
     tuned exactly the same way.
          This will not be the case in more complex forms of tuning.

Trunk Tuning (Pythagorean) has only melodic application.
     It is used naturally by any performer, vocal or instrumental,
          when playing a melody, especially when it is unaccompanied.
     It might even be used when an accompaniment is using
          another more complex tuning procedure.

The reason it is restricted to melodic use is that the interval of major third,
          placed between the borrowed MEDIAN and the FRAME,
     is too complex and too dissonant to be held simultaneously
          (a proportion of 81/64, the result of four fifths, each of a proportion of 3/2).

Proportion - Ratio
This might be a good place to clarify this notion of "proportion" (also called “ratio”). The proportion of an interval is the frequency of the upper note (in vibrations per second called "Hertz") divided by the frequency of the lower note. Interval proportions are always multiplied (when we add intervals) or divided (when we subtract intervals) never added or subtracted.

The Perfect Fifth
The perfect fifth has the proportion 3/2 (or 1.5) because the frequency of the upper note divided by the frequency of the lower note is always 3/2 (1.5).

2 Fifths
The distance between notes 2 fifths away (say G and A) will be of proportion 3/2 multiplied by 3/2, or (3/2)2, = 9/4, the proportion of the major ninth. Removing the octave (dividing by 2), = 9/8, the proportion of the major second.

3 Fifths
The distance between notes 3 fifths away (say C and A) will be of proportion 3/2 X 3/2 X 3/2, or (3/2)3, = 27/8, the proportion of the major thirteenth. Removing the octave (dividing by 2), = 27/16, the proportion of the major sixth. Placing the bottom note C an octave higher (16X2), above the A, = 32/27, the proportion of the minor third.

4 Fifths
The distance between notes 4 fifths away (say C and E) will be of proportion 3/2 X 3/2 X 3/2 X 3/2, or (3/2)4, = 81/16. Removing 2 octaves (dividing by 4), = 81/64, the proportion of the major third.

In Trunk Tuning (Pythagorean), for purposes of simultaneous Harmonic use,
     the perfect fifth (3/2) is considered ideal,
     the major second (9/8) is considered excellent,
     the minor third (32/27) is considered acceptable, but
     the major third (81/64) is considered too complex and discordant.

Tuning Procedure 2

Commonly called Natural Tuning but which we will call Branch Tuning,
          for reasons to be presented shortly.
     It is tuned with the interval of the Natural Major Third - (proportion 5/4)

This will be divided into 2 sections -
     2A - Short Branch Tuning,
          In which the interval of Natural Major Third will be used only once, and
     2B - Long Branch Tuning,
          In which the interval of Natural Major Third will be used twice.

Section 2A will in turn be divided into 2 sub-sections -
     2A1 - In which only diatonic MEDIANs will be generated, and
     2A2 - In which chromatic MEDIANs will also be generated.

Section 2A1 - The Natural Major Third used once
generating only diatonic MEDIANS


Here again, the Window indicates that we are
     in the natural diatonic system, an arbitrary choice.

     are still in Trunk Tuning (Pythagorean),
          (they are generated in the central column).

- The MEDIAN is now also generated,
     in Short Branch Tuning (Natural), indicated
          "-" on the left, when it is major, and
          "+" on the right, when it is minor.

Notice that MEDIANs are not related to each other,
as are the FRAME notes (like a string of pearls)
but that each MEDIAN is related uniquely
to the FRAME to which it belongs
(like a branch to the trunk of a tree).

This is why the terms
     "Trunk Tuning", for the FRAMEs of the central column, and
     "Short Branch Tuning", for the MEDIANs beside it,
          seem appropriate.

The interval of Branch major third,
          of a simpler and more consonant proportion
               (5/4 or 80/64),
     is slightly, but appreciably, smaller
          than the Trunk major third (81/64).
The difference between the two (81/80)
     is called a syntonic comma
          (approximately 1/5 of a semi-tone).
For this reason, using fundamental Trunk Tuning
          as point of reference,
the Branch major MEDIANs
          (generated up from the root of the chord)
     will be indicated "-" (being lower than Trunk Tuning), and
the Branch minor MEDIANs
          (generated down from the fifth of the chord)
     will be indicated "+" (being higher than Trunk Tuning).

- The MOTRIX (a seventh or a sixth)
     will (in general) still be borrowed
          from the Trunk central column.
     Major sevenths and minor sixths
          present a far more complex situation
               and will be seen in Section 2B.

Intervals between two notes of similar tuning, usually Trunk Tuning, will be classified as "normal", whereas Intervals between two notes of different tuning will be classified as either "small" or "large".

With this combination of Trunk and Branch Tuning,
     each diatonic mode (major and minor) has its own tuning,
          the major mode in the bottom of the Window and
          the minor mode in the top.
The minor mode is a comma higher than the major mode.
     Only the central note (D) remains unchanged, common to both modes.

The major mode consists of F A- C E- G B- D*
The minor mode consists of D F+ A C+ E G+ B
* This is the tuning of the major scale proposed by Zarlino in the XVIth century.

Adjustments must thus be made when passing from one section of the Window to the other.

What new proportions have we here?

1. The Branch, small, major third (between E- and C)
          which we have just seen and which generates all the other Branch intervals,
     is of proportion 5/4 (80/64), smaller than the Trunk major third (81/64).
               In Harmony, only the small major third is used.

2. The Branch, large, minor third (between G and E-)
          which is a perfect fifth (3/2) minus a small, major third (5/4), thus 3/2 X 4/5,
     is of proportion 6/5 (162/135), larger than the Trunk minor third (32/27 or 160/135).
               In Harmony, both of these minor thirds are used.

3. The Branch, small major second (between E- and D)
          which is a small major third (5/4) minus a normal major second (9/8), thus 5/4 X 8/9,
     is of proportion 10/9 (80/72) , smaller than the Trunk major second (9/8 or 81/72).
               In Harmony, the large major second is generally used.

The performer has interest in being aware of what tuning is required at any specific moment.

See the

Section 2A2 - The Natural Major Third used once
generating both diatonic and chromatic MEDIANS


Here again, the Window indicates that we are
     in the natural diatonic system,
          an arbitrary choice.

We have here both BranchMEDIANs for each chord FRAME,
     the major MEDIAN on the left,
          diatonic at the bottom of the Window,
               chromatic at the top, and
     the minor MEDIAN on the right,
          chromatic at the bottom,
               diatonic at the top.

What new proportions have we here?

In the melodic sequence F, E, Eb, D,
          (on the chords G7, C, Cm6, G)
     it is interesting to compare the sizes of
          diatonic and chromatic semi-tones,
               in different forms of tuning.

In Equi-tempered tuning,
     the 3 distances are evidently the same
          (100 cents, by definition)
     for a total of 300 cents for the tempered minor third F-D.

In Trunk Tuning,
     the two diatonic semi-tones (F to E and Eb to D) are the same,
          5 fifths (3/2)5 minus 3 octaves (1/8) and invert = 256/243,
               a proportion of 1.0535 (90.225 cents)
     the chromatic semi-tone (E to Eb) is larger,
          7 fifths (3/2)7 minus 4 octaves (1/16) = 2187/2048,
               a proportion of 1.0679 (113.685 cents)
     for a total of 294.135 cents
          slightly smaller than the tempered minor third
               (which is too large).

In Short Branch Tuning,
     the two large diatonic semi-tones (F to E- and Eb+ to D)
          are the same,
               F to C (3/2) plus C to E- (5/4) minus 1 octave (1/2)
                    and invert = 16/15
               a proportion of 1.0667 (111.7308 cents)
     the small chromatic semi-tone (E- to Eb+) is considerably smaller
          E- to C (5/4) minus C to G (2/3) plus G to Eb+ (5/4) = 25/24
               a proportion of 1.0417 (70.673 cents)
     for a total of 294.135 cents (the same Trunk minor third).

Don't shoot the performer
One can well imagine the dilemma of the performer -
     Is this phrase to be considered a melody and tuned Trunk,
          without listening to the accompanying voices, or
     is it to be considered a line of Harmony and tuned Branch,
          by listening closely to the FRAME notes in the other voices
               while tuning the MEDIANs?


Section 2B - The Natural Major Third used twice
generating the Augmented Triad with its augmented fifth


Here again, the Window indicates that we are
     in the natural diatonic system,
          an arbitrary choice.

Let us make a short review
     of the preceding sections -

1. The notes of the FRAME,
          the COMMON TONE and
          the PROPER TONE,
     are generated
          in the central column of Trunk Tuning.

2. The MEDIANs, major and minor,
          diatonic and chromatic,
     are generated
          on the sides of the central column with
               Short Branch Tuning,
          a comma lower ("-") or higher ("+")
               than those of the central column.

3. Most MOTRIXs,
          those a whole tone away from the COMMON TONE,
     are borrowed from the central column
          of Trunk Tuning.

Now, using the Natural, small, major third twice,
          will generate the
     doubly small augmented fifth ("--" or "++").
This is the chromaticized
     substitution of the MOTRIX.
          See Augmentation

The proportion of the interval between
     the chromaticized PROPER TONE (D#) and
     its original diatonic version (D)
          is 25/24,
     the same as that between the major and minor
          MEDIAN in the same FRAME (F#- and F+).

This form of tuning will also be used for the
          major seventh (minor sixth)
     as well as for the
          diminished seventh (augmented sixth)
     which we will see in detail in

Now that we have a complete picture of Functional Tuning generation,
     we can stand back a little and observe elements of interest -

1. The central column of Trunk Tuning is essentially diatonic.
     All the notes within the Window are diatonic and we even surmise that
          this is where the concept of "diatonicism" might be born and
               subsequently imposed elsewhere.

2. On each side of this central column,
     Short Branch Tuning generates MEDIANs
          which are indifferently chromatic or diatonic, major or minor,
               depending on the lateral relationship with the central Window.
     The notes within the horizontal extension of the Window are
          diatonic if they correspond to what exists in the central Window,
               otherwise they will be chromatic.
          They are the only notes the diatonic system will ever use
               in this form of tuning.
     This form of tuning could be considered ambivalent,
          without preference for either diatonicism or chromaticism.

3. On each side of Short Branch Tuning, we find Long Branch Tuning
     which is essentially chromatic.

Another recap
We could recap all this another way -

     The Primary Notes are essentially diatonic and in Trunk Tuning.

     The MEDIANs are chromatic or diatonic
               only with respect to the Trunk Window.
          The great majority of the chromaticism we use is in Short Branch Tuning,
               evidently MEDIANs because they are the only ones there.
          In other words, this chromaticism is very normal to MEDIANs,
               far more than to any other chordal function.

     When MOTRIXs are diatonic, they are in Trunk Tuning,
          However, when they (or their substitutions) are chromatic,
                    or if they are major sevenths (or minor sixths),
               they are in Long Branch Tuning (not in Short Branch Tuning).

     When a FRAME note is chromaticised -
          it subsitutes for a MOTRIX (if it is followed by a MEDIAN), and
               is in Long Branch Tuning, or
          it substitutes for a MEDIAN (if it is followed by a PROPER TONE), and
               is in Short Branch Tuning, or
          it is an Orbit 4 (followed by a MOTRIX), and
               is also in Short Branch Tuning.

In other words, each form of tuning
     has its affinity for certain chord components, and
     has its preference for diatonicism, chromaticism, or neither/both.

There will be far more to find and to add on this subject.

New Terms
Our new terms seem all the more appropriate as we go along:
     "Trunk Tuning" for the central column, the trunk of our generating tree, and
     "Branch Tuning", which uses the small major third, for the branches,
          "Short Branch Tuning" ("-/+") when the major third is used only once, and
          "Long Branch Tuning" ("--/++") when the major third is used twice,
               who knows, possibly abbreviated to "Short Tuning" and "Long Tuning".
Time and usage will tell.

The delicate case of the Major Seventh

It seems impossible to understand the complex operation of the major seventh
          (and its inversion, the minor sixth)
     without attaching primary importance to its generation and to its tuning,
          and this seems as good a place as any to do so.

The tuning is B--
The tuning of the MOTRIX on the chord of C+7 seems to be B--.
     1. A MOTRIX can only be in Trunk Tuning, B, or in Long Branch Tuning, B--,
          and Trunk Tuning, B, is definitely too high.
     2. The vertical distance between the two Secondary notes of C+7, E- and B--, is just right,
               one comma smaller than the normal perfect fifth,
          as it is in minor seventh and major sixth chords.
     3. The horizontal distance of the resolution of the MOTRIX of C+7,
          between B-- and A- (the MEDIAN of the F chord) is also just right,
               one comma smaller than the normal major second,
          as it is in the progression between minor seventh (and major sixth) chords.
     4. The horizontal distance of the procuration of the MOTRIX of C+7 (in Voice-leading C),
          between B- (the MEDIAN of the G7 chord) and B-- is also just right,
               one comma lower than the preceding note of the same name,
          as it is in the progression between minor seventh (and major sixth) chords.

The generation makes this B-- chromatic
The generation of B-- is 2 Major Thirds (5/42) from Eb, the augmented fifth of Eb,
     a chromatic form of Bb, the normal fifth of Eb,
          with a ratio of 25/24 between the B-- and the Bb.

The chromatic (minor) seventh of the chord of C is Bb
The chord of C7 (in C major) is normally considered chromatic
          because the MOTRIX, Bb, is outside (just below) the natural Window.
     We are thus obliged to consider the B-- as doubly chromatic,
               chromatic (in Long Branch Tuning) of the Bb,
                    itself in turn chromatic by being outside of the Window,
          rather than diatonic (by its note-name).

We seem to have several paradoxes here -
     1. The B-- which seems diatonic is, in the reality of its generation, chromatic,
          actually doubly so.
     2. The relationship between the B-- and the Bb, both MOTRIXs on the flattening chord of C,
               seems dangerously ambiguous,
          because these two notes were originally meant to be in the order Bb followed by B--
               and on the flattening chord of C they would be in the order B-- followed by Bb.

     It does not seem advisable to have both the Bb and the B-- on the same flattening C chord.
          If they are in the order Bb followed by B--, the direction is not acceptable to the resolution, and
          if they are in the order B-- followed by Bb, the chromatic precedes the (relatively) diatonic.
     Both the B-- and the Bb are perfectly acceptable individually,
          just that it must be one or the other, not both.
The next step is to verify this hypothesis in the reality of musical activity -
     1. We are aware that both the B-- and the Bb are often found on the same flattening C chord.
     2. On the other hand, we often find only the B-- or only the Bb.
     3. When one wants the C, B, Bb, A line from TONIC to COUNTER,
          one often places a DOMINANT (G7) chord with the B, producing a B- rather than a B--
               much the same way one would place a D7 with the F# in the G, F#, F, E line,
                    with the TONIC, ANTE-1, DOMINANT, TONIC chords.
There is still a great deal to examine before coming to any definite conclusion.

Let us know what you think and what you find.

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