Ternary Subdivisions of Level 0

The 3/4 Bar

Ternary subdivisions are produced by removing the second of a group of four.

To transform a 4/4 bar into a 3/4 bar,
      we would remove the second and keep beats 1, 3, and 4.
          In order to subdivide into 4 parts, we must evidently use 2 binary levels,
               in this case, Level -2 and Level -1,
                    fusing the two into one ternary level, Level -2-1.

In the section on more elaborate footsies we saw
     that it was possible to have the feet beat 2 levels at the same time,
          by placing the feet farther apart, and
          by having each foot touch the ankle of the other foot before striking the ground.
     This would give the following sequence for the four beats of a 4/4 bar -
          (a) the right foot strikes the ground,
          (b) the left foot touches the right ankle,
          (c) the left foot strikes the ground, and
          (d) the right foot touches the left ankle,
               which we could abbreviate : R-T-L-T, for R(ight), T(ouch), L(eft), T(ouch),
                    each T(ouch) produced by the foot which follows it.
     At the larger of the 2 levels - (a) R is the Beat and (c) L is the Off-Beat, and
     at the smaller of the 2 levels - (b) and (d) Ts are the Off-Beats,
          (a) R and (c) L being the Beats.
     Now if we remove (b), the second beat of a 4/4 (binary) bar, we have the 3/4 (ternary) bar -
          (a) the right foot strikes the ground,
          (b) the left foot strikes the ground, and
          (c) the right foot touches the left ankle,
               which we could abbreviate : R-L-T.
               The left foot, used for antecedent elements (on the Off-beat), has lost a member,
                    and we realize that a 3/4 bar is more consequent-oriented than the original 4/4 bar
                         which is perfectly balanced.
We recommend you practice this for a while and really get the feel of it.
     Sing a waltz (any waltz) while you move the feet,
          so as to move the feet quickly enough.

The 6/8 Bar

To transform a 4/4 bar into a 6/8 bar,
     we would remove the second from each of the two groups of ,
          in other words, from each ,
     using the two binary levels Level -3 and Level -2,
          fusing them into one ternary level, Level -3-2.

The ternary footsie sequence would be used twice, one movement for each ,
          R-L-T - R-L-T for each bar.
     In the 6/8 bar, the left foot has lost two members,
          and, like the 3/4 bar, it is also more consequent-oriented than the 4/4 bar.
This should not require too much practice because
     most waltzes are really in 6/8 and should preferably be written as such.

Applications

It will not be as easy to identify and place ternary levels as it was for binary levels.
     1. We must first identify if a level is binary or ternary (now that we have a choice of either).
          We will find it often quite useful to use the sound of the words to help us
               (more details on this as we see individual cases).
     2. We must then see if a level starts
          - on R, on L, or on T,
               by seeing if we end, as we always should, on R.

Those on a guided Tour should click on in the Navigation Bar below.

Those browsing might like to try our first song Rock-A-Bye, Baby.
This is really the fourth song. The first three, in Basic Materials, were completely binary.
There will be four new songs exploring ternary levels.
The other three are
Oh, Where, Oh, Where, Is My Little Dog Gone?
Happy Birthday
Down In The Valley.
Enjoy !

Incomplete Fusion - the 7/8 Bar

To transform a 4/4 bar into a 7/8 bar,
     we would remove the second from only one of the two groups of ,
          in other words, from only one , either the first (3+4) or the second (4+3)
     still using the two binary levels Level -3 and Level -2,
          but only fusing half of the bar into one ternary level, Level -3-2, leaving the other half intact.

The ternary footsie sequence would only be used once, one movement for each ,
          either R-L-T - R-T-L-T or R-T-L-T - R-L-T, for each bar.
     In the 7/8 bar, the left foot has lost only one member,
          and is less consequent-oriented than the 3/4 or 6/8 bars.
You will find very little music with this bar size,
     but this septernary structure is occasionally found at larger levels,
          as in the song America ("My country 'tis of thee"), in which (at Level 0),
               the first section is composed of 3 cells and the second section is composed of 4 cells.

Fusion of Several Levels

Three Levels - the 5/8 bar

Rebound Grouping and Pick-up Grouping
The definition given for Ternary subdivisions was appropriate, convenient, and efficient at the time,
          but is no longer applicable to the fusion of a larger number of levels.
     Removing the second beat of a 4/4 bar and keeping beats 1, 3, and 4,
               was simple and easy to apply,
          all the more so that, in Meters, we did not remove any material
               but condensed ("squeezed") all of the material in the first half of the bar to 1/2 its size.
     This simplified process was based on Rebound Grouping (with Rebound Association).
On the other hand, the authentic procedure of ablation,
          which we must now use
     is one based on Pick-up Grouping (with Pick-up Association),
          in which we remove the FIRST quarter of each group.
When we produced 3/4 bars, the result would have been the same with either procedure,
     if ONLY s were involved,
          (a) removing the SECOND in Rebound Grouping, () '/ () ', or
          (a) removing the FIRST in Pick-up Grouping, () / '() / ',
               always the second (beat) of each bar,
     but NOT if s were involved,
          the old, convenient procedure removing the THIRD and FOURTH s of each bar,
               in () '/ () ' (Rebound Grouping),
                    where each Off-beat is associated to the preceding Beat, and
          the new, authentic procedure removing the SECOND and THIRD s of each bar
               in () / '() / ' (Pick-up Grouping),
                    where each Off-beat is associated to the following Beat.

From now on, let's stick to Pick-up Grouping
What would happen if we started with the complete 4/4 bar and applied both
          the ablation to a 3/4 bar, and
          the ablations to a 6/8 bar?
     In other words fused -
          Level -2 and Level -1 together,
               the process for producing a 3/4 bar, and
          Level -3 and Level -2 together,
               the process for producing a 6/8 bar,
                    thereby producing a 5/8 bar, the fusion of Level -3-2-1.

What is removed, what is left
As we now know, with the authentic Pick-up Grouping, -
     the 3/4 bar would be () / '() / ' (between the 2 bar-lines)
          with the SECOND and THIRD s removed in each bar, and
     the 6/8 bar would be () '() / '() '() / ' (between the 2 bar-lines)
          with the SECOND and SIXTH s removed in each bar.
               The SECOND is rermoved twice,
                    the THIRD is removed once, and
                    the SIXTH is removed once, leaving
                         bars of ' / ' ' / ' (between the 2 bar-lines)
                    with small breaths immediately AFTER the two Beats left in the 5/8 bar.
     The first 2 s and the last 3 s would be flagged together.

What about the Footsies?
In this 5/8 bar, we have two groups which were originally of four s each -
     in the first group of four s,
          only the FIRST and FOURTH s remain, and
     in the second group of four s,
          only the FIRST, THIRD, and FOURTH s remain,
     producing the following footsies,
          R-T - R-L-T, with both T(ouches) produced by the R(ight) foot,
                    the foot which follows the T(ouch),
               an overall ratio of four movements for the R(ight) foot and only one for the L(eft),
                    very consequent-oriented because ablations remove antecedents performed by the L(eft) foot.
There is not very much music composed in 5/8,
          the first thing that comes to mind is the second movement of Tchaikovski's Symphony No. 6,
               (actually written in 5/4),
     but you might enjoy placing the footsies to the clip Red River Limp 2 or its inversion.
          They both have 2 notes pick-up, to which you can footsy L-T
               or you can wait and footsy R-T - R-L-T at the third note.
                    Good luck !

For Red River Limp 2, you might enjoy seeing
the Graphics, Changing Meters, and the Melo-rhythmic Box.

Four Levels - the 8/16 bar

Always using Pick-up Grouping
What would happen if we started with the complete 4/4 bar and applied -
          the ablation to a 3/4 bar,
          the ablations to a 6/8 bar, and
          the ablations to a 12/16 bar?
     In other words fused -
          Level -2 and Level -1 together,
               the process for producing a 3/4 bar,
          Level -3 and Level -2 together,
               the process for producing a 6/8 bar, and
          Level -4 and Level -3 together,
               the process for producing a 12/16 bar,
                    thereby producing a 8/16 bar, the fusion of Level -4-3-2-1.

What is removed, what is left
As we now know, with the authentic Pick-up Grouping, -
     the 3/4 bar would be ( ) / '( ) / ' (between the 2 bar-lines)
          with the SECOND, THIRD, FOURTH, and FIFTH s removed in each bar,
     the 6/8 bar would be () '() / '() '() / ' (between the 2 bar-lines)
          with the SECOND, THIRD, TENTH, and ELEVENTH s removed in each bar, and
     the 12/18 bar would be () '() '() '() / '() '() '() '() / ' (between the 2 bar-lines)
          with the SECOND, SIXTH, TENTH, and FOURTEENTH s removed in each bar.
               The SECOND is removed three times,
                    the THIRD and TENTH s are removed twice, and
                    the FOURTH, FIFTH, SIXTH, ELEVENTH, and FOURTEENTH s are removed once, leaving
                         bars of ' ' / ' ' ' / ' (between the 2 bar-lines)
                    with small breaths immediately AFTER the three Beats left in the 8/16 bar.
     The first 3 s, the next 2 s and the last 3 s would be flagged together.

What about the Footsies?
In this 8/16 bar, we originally had four groups four s each -
          in the first group of four s,
               nothing remains,
          in the second group of four s,
               the FIRST, THIRD, and FOURTH s remain,
          in the third group of four s,
               only the FIRST and FOURTH s remain, and
          in the fourth group of four s,
               the FIRST, THIRD, and FOURTH s remain,
     producing the following footsies,
          R-L-T - R-T - R-L-T, with the three T(ouches) produced by the R(ight) foot,
                    the foot which follows the T(ouch),
               an overall ratio of six movements for the R(ight) foot and only two for the L(eft),
                    once again, very consequent-oriented, a lot of the antecedent (L(eft) material removed).
There is not very much music composed in 8/16,
     but you might enjoy placing the footsies to the clip Bonnie Circle or its inversion.
          The footsies are much slower here because we are dealing with large levels, a fused Level 0+1+2+3.
               For the chorus, grouped in 8 (3+2+3), the footsies are L-T - R-T - R-L-T - R,
                    moving only at each bar of 6/8, and
               for the verse, grouped in 5 (2+3), the footsies are T - R-L-T - R
                    always moving only at each bar of 6/8, followed by
               a repeat of the chorus with footsies L-T - R-T - R-L-T - R.
          Note the Pick-up Grouping in these footsies.
                    Good luck !

Fibonacci numbers
Whenever we have fusion of levels by complete, maximum ablation,
          the result is what we call a "Fibonacci number".
     When we fused 2 levels together, we produced a group of 3,
     when we fused 3 levels together, we produced a group of 5,
     when we fused 4 levels together, we produced a group of 8.
Fibonacci numbers are generated by adding the last two numbers together,
     each new number being the sum of the preceding two -
          0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ...
In musical rhythm, it is possible to use any Fibonacci grouping
     at any small level,
     at any large level,
     one number at one level and another number at a different level,
     several numbers at different levels, or
     at one huge integrated level which includes the whole piece,
          (which has been done quite successfully on several occasions).

Incomplete Fusion
Needless to add that, within the framework of a four-level fusion,
     it would have been possible to produce all sorts of incomplete fusion,
          too numerous to list here.

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