|
|
|
|
|
|
|
|
|
|
The Five Degree Calculation Line for Ukulele |
(or Perfect-Fourth reference line) |
|
|
Fretboard Interval Calculator |
|
|
|
Before reading this be forewarned that I use Commonsense String Numbers in all explanations.
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 ukulele fretboard. The Five Degree Calculation Line is a fretboard navigation tool that will help us do just that.
The Five Degree Calculation Line is the key component of the Cipher System. Its the device that allows us to transfer and apply the Cipher System’s translated number formula to the ukulele fretboard (or guitar, or bass). But it is, and does, much more than that. The Five Degree Calculation Line is also, and truly, the key to the fretboard itself. Its a map-generator, a combination of movable schematic and counting-grid-overlay that reveals the fretboard's otherwise invisible anatomy with unequaled clarity and speed. It reveals the where, why, and how, of all fretboard patterns.
Figure 1 shows the naturally occurring pattern of successive Perfect-Fourth intervals on the ukulele fretboard in standard tuning. 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
|
|
|
|
|
|
The pattern can be viewed two ways: in parts (as above) 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.
|
|
|
|
Figure 2
|
|
|
|
|
|
As the pattern progresses (left to right, low G to high A, first to fourth 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 15 half-steps — just over one octave (one octaves = 12° degrees).
The Five Degree Calculation Line is only the baseline or reference-line of a larger device. Its ultimate function is to help us identify the number values (interval widths) of all points (neighboring tones) above and below the line, linking and servicing an area as large as eight or ten frets and a time (if need be). The Five Degree Calculation Line (therefore) becomes the center or baseline of a counting-grid. See Figure 3.
|
|
|
|
Figure 3
|
|
|
|
|
|
The Five Degree Calculation Line is more than a single thing:
- First, the Five Degree Calculation Line is movable vertically. It 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 the greater Five Degree Calculation Line. i.e. on any string (1 through 3 or 4).
- By moving the zero-point (horizontally), four variations (numerations) of the Calculation Line emerge. See Figure 4. [note; all variations shown in Figure 4 should be imagined superimposed upon each other — as if taking place within a single two-fret area 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 offset path, and (again) gaining Five Degree raises in pitch with each jump to right. When the zero-point is moved horizontally, any strings and tones to the left 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.
|
|
|
|
Figure 4
|
|
|
|
|
|
Given that there are five variations of the Five Degree Calculation Line — one for each of the four horizontal points (strings) that could be called zero degrees — there are also four variations of counting-grid. Each version of Five Degree Calculation Line generates his own unique counting-grid. See Figure 5.
|
|
|
|
Figure 5
|
|
|
|
|
|
All versions of the Five Degree Calculation Line and their respective counting-grids are fundamentally alike, but they all differ slightly. The cause/reason for their differences lies in the way that the ukulele itself is tuned — specifically, the change in tuning-pattern between the second and third strings. The interval between them, a Major-third, is one half-step (one fret) smaller than the ukulele’s otherwise uniform tuning-pattern of successive perfect-fourths. That single deviation in the tuning-pattern is then reflected in the playing-pattern, causing the “one fret down” offset (at the third string) necessary to perform a completely uniform series of perfect fourth intervals (or to construct a uniform Five Degree Calculation Line) on the ukulele fretboard. In the end, that one deviation/offset in the playing pattern effects almost every possible string-set on the ukulele (i.e. any that include the third string) and, more to the point, almost always effects (complicates) the act of moving any single fretboard-pattern (e.g. chord fingering or scale pattern) horizontally across the fretboard. The reason, then, that there are variations in the Five Degree Calculation Lines and their respective counting-grids is that we are moving a single pattern of sound (successive P-4ths) horizontally on the fretboard. Every time we change the starting point of the pattern we hit the Fifth String Pattern Shift at a different relative spot. That is, two places to right (i.e. into the pattern), three places, etc. So we must continually compensate and re-compensate, change and change again, the fingering/shape of any given fretboard (or sonic) pattern that is moved horizontally on the fretboard.
I hope it's clear, then, that the Cipher System is not creating these patterns and variations. Everything you've seen here and all that follows is natural (innate) to the ukulele fretboard — including the Five Degree Calculation Line (being, simply, the ukulele’s built-in pattern of successive perfect-fourths). The Five Degree Calculation Line(s) and counting grid(s) simply make the ukulele’s natural patterns clearly visible to us, and because those patterns are rendered with counting numbers, they are clearly and immediately understood.
The Five Degree Calculation Line, “zeroed” (rooted and aligned) at a chosen tonic or root, functions as a reference line — the centerline of a counting-grid. The pitch and number-value of tones above and below the centerline 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 Five Degree Calculation Line as the central calculation reference or plotting baseline. Plotting can be done directly on the fretboard (by visualizing the grid 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 Five Degree Calculation Line is straight forward and simple to do.
Navigating the Grids (see Figure 5):
|
|
|
|
- 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 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).
- The Fifth String Pattern Shift. Caution; you must always be attentive of the The Third String Pattern Shift — the one fret down offset — that breaks the straight-line pattern of the Five Degree Calculation Line(s) between strings two and three. That visual and physical change in pattern is natural to the fretboard in standard tuning. Ukulele players must learn to adapt-to and navigate that part of the pattern with or without the Cipher. Remember, while the path of the pattern changes (diverts) at the third string, the pitch change (sonic pattern) remains the same — a Perfect-Fourth (five half-step) change in pitch in either direction.
Having a movable calculation reference-line, with it’s floating zero-point, means that you can build or visualize a counting-friendly grid from (and around) any tone on the fretboard. By combining the Cipher Formula, the Five Degree Calculation Line, and knowledge of the ukulele’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 Ukulele Cipher Demonstrations
Index of The Cipher for Ukulele:
|
|
|
|
|
|
|
Up to Top of page
|
