Descending Five Degree Calculation Line     (or descending perfect-fourth reference line)     Before reading this be forewarned that I use Commonsense String Numbers in all explanations. 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 guitar fretboard. The descending Five Degree Calculation Line For most purposes the normal or ascending (low to high) Five Degree Calculation Lines are all you’ll need. They’re sufficient to reveal the things we need to know about fretboard patterns and for plotting chromatic number formula. Once a given pattern or formula on the fretboard is known in it’s ascending direction you can simply ignore all numbers of any kind (standard or Cipher) and reverse the direction of playing to achieve the descending version of the pattern.   Occasionally however, or even just for curiosity sake, you’ll want to know how descending plotting works using the Five Degree Calculation Lines. These descending Calculation Lines and counting grids are worth seeing at least once, providing you don’t let them confuse you too much. They might be a little risky to tackle unless you’re already very comfortable with the normal ascending Calculation Lines and counting grids. That said, they do reveal more things and a larger picture conceptualization of how the fretboard works. And remember, this is the real stuff, this is how the fretboard actually works. If you ever wanted to become really proficient and automatic with descending movement on the fretboard (literally to know it forwards and backwards) this is what you’d find out and need to see. So again, The Cipher is the best tool for that job. These descending calculation lines are very useful for plotting descending intervals, at least the first octave of descending intervals. And we will discus that in a few minutes. However, don’t expect to use these descending calculation lines regularly nor for plotting much other than intervals. That is, just like with standard diatonic numbers and number formula for scales and chords, a given chromatic number formula executed in the descending direction (meaning like a mirror reflection) will not produce the same result as the original ascending construction did. Take the Major scale for example. Figure 1 shows a mirrored Major scale. It just so happens that a mirrored Major scale produces a scale many will recognize as a Phrygian mode scale. But the important thing to understand is that applying the ascending Major formula in the descending direction from a tonic-root-zero does not give you another Major scale. And the same holds for the formula for all other kinds of scales and chords. This is why I said that descending plotting is pretty much restricted to intervals only, and then usually just those intervals in the first octave (because those are the ones you use every day). The main application for the descending plot is just to better understand (and to illustrate on the fretboard) statements like “up a fifth, down a fourth”. Meaning, to further the elementary but fundamental  understanding of intervals, their inversions, and their relation to the octave interval.     Figure 1       Figure 2 shows the naturally occurring pattern of successive Perfect-Fourth intervals on the guitar 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. This applies in both directions, left to right or right to left, meaning in either the ascending or descending direction.     Figure 2       Again, 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. Previously, we numbered this pattern from left to right (low E string to high E string) meaning in the ascending direction of pitch gain. Here, we’ll do the reverse. Start your count at the high E string (string 6 in the Cipher System) and move in the descending direction. See Figure 3.     Figure 3       As the descending pattern progresses (right to left) high E to low E, sixth to first 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 25 half-steps lower in pitch — just over two octaves (two octaves = 24 degrees). Again, 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. The Descending Five Degree Calculation Line, just like the ascending version, becomes the center or baseline of a counting-grid. Here we’ll compare both the ascending and descending versions side by side so you can see what’s happening. Both versions have been aligned to share the same single rooting center and corresponding Pattern of Unisons and Octaves. Figure 4 shows both versions of Five Degree Calculation Line, ascending and descending, with their respective full counting grids. Figure 5 is the same drawing but I’ve isolated the shared rooting center and points of the Pattern of Unisons and Octaves (encircled 0°s, 12°s, and 24°s). These are the points I usually indicate with black diamonds in other drawings. You’ll notice of course that when both versions are aligned to share a common rooting center (two points of a Pattern of Unisons and Octaves that fall on the same fret-line, one on string-one and the other on string-six marked 0° and 24°) the descending calculation line is shifted one fret back towards the nut. Again, this is caused by the Fifth String Pattern Shift and reflects how the fretboard truly works. Oddly enough though, all Octave and Unison points (the encircled 0°s, 12°s, and 24°s) in both versions still fall at exactly the same spots, whether you ascend or descend. Meaning, we can’t dispute what we see. We have the arithmetical proofs. If the octaves work it’s got to be right.     Figure 4     Figure 5       The descending Five Degree Calculation Line is also more than a single thing: First, the descending  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 in the descending direction as well as the normal ascending direction. Zero-degrees (the tonic or root) can be moved and placed at any point along the path-line of the greater descending Five Degree Calculation Line. i.e. on any string (six through two shown here). By moving the zero-point (horizontally), five variations (numerations) of the descending Calculation Line emerge. See Figure 6. [note; all variations shown in Figure 6 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 widening in pitch lower with each jump to left. When the zero-point is moved horizontally in the descending direction, any strings and tones to the right 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 6       Given that there are five variations of the descending Five Degree Calculation Line — one for each of the five horizontal points (strings) that could be called zero degrees — there are also five variations of descending counting-grid. Each version of descending Five Degree Calculation Line generates a unique counting-grid. See Figure 7. Note, you’ll rarely need to plot in the descending direction more than one octaves worth of pitch, meaning 12° lower in pitch than a given root. I’ve shown large area counting grids here (meaning more than just one octave of descend) just for reference purposes.     Figure 7       Like their ascending counterparts, all versions of the descending Five Degree Calculation Line and their respective counting-grids are fundamentally alike, but they all differ slightly. The cause/reason for their differences lies (again) in the way that the guitar itself is tuned — specifically, the change in tuning-pattern between the fourth and fifth strings. The interval between them, a Major-third, is one half-step (one fret) smaller than the guitars 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 fifth string) necessary to perform a completely uniform series of perfect fourth intervals (or to construct a uniform Five Degree Calculation Line) on the guitar fretboard. In the end, that one deviation/offset in the playing pattern effects almost every possible string-set on the guitar (i.e. any that include the fifth 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 descending pattern we hit the Fifth String Pattern Shift at a different relative spot (if we hit it at all). That is, one place to left (i.e. into the pattern), or dead on if you begin with zero on string five, or never if you descend from any point on strings four, three, or two. So we’ll have to compensate and re-compensate, change and change again, the fingering/shape of any given fretboard (or sonic) pattern that is moved horizontally in the descending (lower in pitch) direction on the fretboard — just like we do in the ascending direction. Again, I remind you that the Cipher System is not creating these patterns and variations. Everything you see here is natural (innate) to the guitar fretboard — including the Five Degree Calculation Line(s) (being, simply, the guitars built-in pattern of successive perfect-fourths).  The Five Degree Calculation Line(s) and counting grid(s), both ascending and descending, simply make the guitars natural patterns clearly visible to us, and because those patterns are rendered with counting numbers, they are clearly and immediately understood. Just like the ascending version, the descending 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. But this time notice the reversed direction of the descending numbers compared to the normal ascending calculation lines. Study Figure 4. The numbers of any Cipher interval Formula (but only intervals) are then plotted on the fretboard's natural grid of coordinates, vertical strings and horizontal frets, with the descending 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 fretboard grids are provided here, or in high quality PDF format here. PC users right-click and “save target as”. Mac users, click-hold and select “download link to disk” from the pop-up menu.] Interval plotting with the Cipher Formula and descending Five Degree Calculation Line is, of course, the reverse of the ascending direction and ascending version of Five Degree Calculation Line. Navigating the descending grids (see Figure 7):     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. Notice, however, the reversed direction of the descending numbers compared to the normal ascending calculation lines. Look at Figure 4 again if you didn’t catch that the first time. 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. Again, you must be attentive of the The Fifth String Pattern Shift — the one fret  offset (one fret up in this case) — that breaks the straight-line pattern of the descending Five Degree Calculation Line between strings five and four. That visual and physical change in pattern is natural to the fretboard in standard tuning. Guitarists 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 fifth 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 three tools: the Cipher Formula, the Five Degree Calculation Line(s) (all versions; ascending and descending, plus their respective variations that depend on starting point string), plus your knowledge of the Pattern of Unisons and Octaves, you can explore, manipulate, and understand the full range of fretboard patterns with confidence.   Interval plotting with the descending Five Degree Calculation Line (restricted to one octave of descend) As we said earlier, these descending calculation lines are very useful for plotting descending intervals, at least the first octave of descending intervals, being the ones we use most frequently. That’s what we’ll explore here. Figure 8 is very similar to the previous large plate of descending calculation lines and counting grids. The only difference here is that here I’ve deleted almost all numbers higher than 12°, the octave. There are a few grayed out numbers larger than 12° scattered around, e.g. some 13°, 14° and 15°s. Those are there just to show a linkage with the number 15°  in some nearby Five degree Calculation line, and are meant to help you see how an instance of 12°, the octave, wound up where it is. For practical and illustrative purposes, I limited these grids to approximately 10 or 11 fretsworth of pattern with a centrally located main Five Degree Calculation Line plus five frets above and five frets below the calc line. You can of course move as far up or down the neck as you want, need, or can do. There is no inherent five fret up or down limit. There is some redundancy in these large (meaning long) counting grids. In practice you’d probably narrow your area down some, perhaps ignoring the top and bottom two frets worth of grid. If you did that you’d still find at least one instance of every number from 0° through 12°, meaning any simple (or first octave) interval. Nevertheless, I thought you’d want to see this larger area picture via the long grids. Descending movement and interval plotting is limited when begun from Strings One or Two. You’ll see this at the far right of the illustration showing the practical limits of descending interval widths commenced from String Two. I didn’t even bother showing String One for this same reason. You can only go backwards towards the nut on that single string. But depending on where you start (meaning how high up on the neck) you could still descend (melodically) quite a ways, even an octave or more if you start at the twelfth fret or higher. Again, consult the full Interval Number-NameTranslation Tables or this one octave only version in Figure 9.     Figure 8           Figure 9           Up to Top of page