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Five and Seven Degree Calculation Lines for Stick |
(or Perfect-fourth and Perfect-fifth reference lines) |
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Before reading this be forewarned that I use Commonsense String Numbers in all explanations.
The Five and Seven Degree Calculation Lines (P-4th and P-5th ref lines)
The Cipher System's translation tables of musical materials and formula are useful by themselves for the insights they provide. To understand, at last, that a Major-13th interval spans 21 frets-worth of pitch is a major revelation for most of us. That's a good start, but it’s only the beginning. We still need to hear all of those musical materials, and we need to know how to locate them on the Stick fretboard. The Five and Seven Degree Calculation Lines are the fretboard navigation tool that will help us do just that.
The Calculation Lines are the key component of the Cipher System. They are the device’s that allow us to transfer and apply the Cipher System’s translated number formula to both halves of the Stick fretboard (the bass side tuned in 5ths and the melody side tuned in 4ths). But the Calculation Lines are, and do, much more than that. The Calculation Line’s are also the key to the Stick fretboard itself. They are map-generator’s, combination’s of movable schematics and counting-grid-overlays that reveal the fretboard's otherwise invisible anatomy with unequaled clarity and speed. They reveal the where, why, and how, of all Stick fretboard patterns.
Figure 1 (top) shows the naturally occurring pattern of successive Perfect-Fourth intervals on the melody side of the Stick fretboard in standard tuning.ţ Each pair of adjacent dots (isolated from the full pattern at the far left) form and are Perfect-Fourth intervals spanning five half-steps, five frets, or five degrees of pitch.
Figure 1 (bottom) shows the naturally occurring pattern of successive Perfect-Fifth intervals on the bass side of the Stick fretboard in standard tuning. Each pair of adjacent dots (isolated from the full pattern at the far left) form and are Perfect-Fifth intervals spanning seven half-steps, seven frets, or seven degrees of pitch.
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Figure 1
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Five Degree Calculation Line (top of Figure 1): The pattern can be viewed two ways, in parts (as above top) or as a whole, i.e. one continuous pattern, as follows. Given that Perfect-Fourth intervals encompass five half-steps of pitch each, the pattern of P-4ths can be approached additively from its beginning to end (i.e. 5 + 5 = 10, 10 + 5 = 15 etc.). Numbered that way, the pattern can be used as a Five Degree Calculation Line. See Figure 2 (right side of).
Seven Degree Calculation Line (bottom of Figure 1): The pattern can be viewed two ways, in parts (as above bottom) or as a whole, i.e. one continuous pattern, as follows. Given that Perfect-Fifth intervals encompass seven half-steps of pitch each, the pattern of P-5ths can be approached additively from its beginning to end (i.e. 7 + 7 = 14, 14 + 7 = 21 etc.). Numbered that way, the pattern can be used as a Seven Degree Calculation Line. See Figure 2 (left side of).
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Figure 2
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Five Degree Calculation Line Figure 2 (right side of): As the pattern progresses (left to right, Fs to D string, first to fifth string [using commonsense String Numbering order]) each successive dot represents a five half-step, five fret, or five degree raise in pitch (relative to the dot the precedes it). From the tone at the zero degree starting point, we can jump across the pattern in five degree increments, continually widening the interval to a maximum of 20 half-steps — just less than two octaves (two octaves = 24 degrees).
Seven Degree Calculation Line Figure 2 (left side of): As the pattern progresses (Right to Left, C to E string, sixth to tenth string, each successive dot represents a seven half-step, seven fret, or seven degree raise in pitch (relative to the dot the precedes it). From the tone at the zero degree starting point, we can jump across the pattern in seven degree increments, continually widening the interval to a maximum of 28 half-steps — just over two octaves (two octaves = 24 degrees).
The Five and Seven Degree Calculation Lines are only the baselines or reference-lines of larger device’s. Their ultimate function is to help us identify the number values (interval widths) of all points (neighboring tones) above and below the lines, linking and servicing any area of frets. The Calculation Lines (therefore) become the center or baseline of counting-grids. See Figure 3.
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Figure 3
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The Calculation Lines are more than a single thing:
- First, the Calculation Lines are movable vertically. They can be positioned (visualized/overlain) at any fret-line. i.e. anywhere up and down the length of the fretboard.
- Second, the zero-point is movable horizontally across the fretboard. Zero-degrees (the tonic or root) can be moved and placed at any point along the path-line of either greater Calculation Line. i.e. on any string.
- By moving the zero-point (horizontally), variations (numerations) of the Calculation Lines emerge. See Figure 4. [note; all variations (in each half) shown in Figure 4 should be imagined superimposed upon each other — as if taking place within a single fret-line simultaneously]. Where-ever the zero-point is moved to, that tone becomes the new tonic/root (zero), (the count begins anew), and the pattern continues as before — following the same straight line path, and (again) gaining five or seven degree raises in pitch with each jump to right or left. When the zero-point(s) are moved horizontally, any strings and tones to the other side of zero are (for that moment) “off the grid”/out of action. That is, until the zero-point is moved again to any tone residing on those (unused) strings.
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Figure 4
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Given that there are five variations of each Calculation Line — one for each of the five horizontal points (strings) that could be called zero degrees in each logical half of the Stick — there are also five variations of counting-grid in each half. Each variation (starting point) of either Calculation Line generates a unique (but fundamentally identical) counting-grid. See Figure 5. Notice that all versions of both the the Five-Degree and Seven-Degree Calculation Lines and their respective counting-grids are essentially the same. This is due to the symmetrical or uniform tuning patterns in both logical halves of the Stick, being all 4ths or all 5ths. So all smaller string-sets within each tuning-half of the Stick will exhibit the same shapes, same patterns of interval, scale, or chord-tone dots or numbers.
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Figure 5
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Everything you've seen here and all that follows is natural (innate) to the Stick fretboard — including the Five and Seven Degree Calculation Lines (being, simply, the Stick’s built-in pattern of successive perfect-fourths and perfect-fifths). The Calculation Line’s and counting grid’s simply make the Stick’s natural patterns clearly visible to us, and because those patterns are rendered with counting numbers, they are clearly and immediately understood.
The Calculation Line’s, “zeroed” (rooted and aligned) at a chosen tonic or root, functions as reference line’s — the center-line’s of a counting-grid’s. The pitch and number-value of tones above and below the center-line’s are gauged and determined relative to the tone at zero-degrees and the other tones on the reference line. The numbers of any Cipher Formula are then plotted on the fretboard's natural grid of coordinates, vertical strings and horizontal frets, with the Calculation Line’s as the central calculation reference or plotting baseline. Plotting can be done directly on the fretboard (by visualizing the grid’s and either counting mentally or using your finger) or drawn on paper-grid facsimiles of the fretboard. [Sheets of blank 4 string fretboard grids are provided on the Free page in high quality PDF format for printing.
Tone-plotting with the Cipher Formula and the Calculation Line’s is straight forward and simple to do.
Navigating the Grids (see Figure 5):
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- Vertical movement within any grid (i.e. up or down the neck on any string) changes the pitch and number-value of a given tone (or interval) one half-step (or one degree) per fret of movement in either direction.
- Horizontal movement across any grid (i.e. parallel with a fret-line and following the path of the Five or Seven Degree Calculation Line) from one string to another changes the pitch of a given tone (or width of an interval) and it’s number value by a quantity of:
five — five half-steps, five degrees, five frets-worth of pitch change per jump i.e. per string or per horizontal move (in either direction) when using the Five Degree Calculation Line . . or . .
seven — seven half-steps, seven degrees, seven frets-worth of pitch change per jump i.e. per string or per horizontal move (in either direction) when using the Seven Degree Calculation Line.
Having movable calculation reference-line’s, with their floating zero-points, means that you can build or visualize counting-friendly grids from (and around) any tone on the fretboard. By combining the Cipher Formula, the Calculation Lines, and knowledge of the Stick’s Pattern of Unisons and Octaves, one can explore, manipulate, and understand, the full range of fretboard patterns with ease and confidence.
[Note, some time later you should see the section on the Descending Five Degree Calculation Lines and counting grids (right-handed guitar version only). At this early stage they really aren’t necessary and they could confuse you a little. But you should know, at least, that they exist.]
Please turn to the Stick Cipher Demonstrations
Index of The Cipher for Stick:
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