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Building a Pattern of Unisons and Octaves
The member tones (points, dots) and any Pattern of Unisons and Octaves can be rendered using numbers only, and (when acting as the pattern of tonic/root tones) using only the numbers 0°, 12°, 24°, and 36°. But which of those numbers are signed to which points of the pattern? We'll answer that question using an eight-plate series of illustrations.
The first thing to understand and prepare for is that number assignments are relative and changing. Specific numbers, whether 0°, 12°, 24°, or 36°, are assigned to tonic/root tones on a moment to moment, case by case, basis, depending on the immediate circumstances and needs. And circumstances change. A given point in the pattern might be numbered twelve-degrees one time, then later be numbered zero or twenty-four degrees. In each case the number assignment will be true and correct for the given moment. Assigned numbers are relative — relative to where zero has been fixed or set. You can and will change the location of the zero-point (the overall context) often, to an alternate point in the pattern that is either unison with the original zero-point or a different octave of it. Each time you change the location/octave of the zero-point the octave number of every other tonic/root in the pattern changes accordingly. That is, to reflect their new octave relationships relative to the new zero-point, its location and its octave.
With that said, we'll ask again; which tone(s), which points, in any tonic/root Pattern of Unisons and Octaves can be assigned the number zero? The short answer is that any and all of them can be called zero degrees, but only one at a time. That is, only one point, one tone, plus any other points that are unison with it. All points within the pattern are tonic/roots, and they're all functionally the same tonic/root — having the same letter-name. So any and all of them could rightly be numbered zero-degrees at some time or another. But, they are not all the same octave of the given lettered tone, so they can't all be numbered zero-degrees (at the same time) and still accurately reflect their octave differences. If all points had the same number value at the same time, they would all appear to be unison with each other, which they are not. What we have to do, then, is to approach the pattern one point at a time. That is, select one tone to be zero-degrees, observe the counting-grids and number values that arise and apply relative to that specific zero-point, then change the zero-point (i.e. select another point in the pattern) and repeat the experiment. After comparing the results of different trials, it will become clear that the number value of any single point will change, alternating between 0°, 12°, 24°, and 36°, relative to its relationship to the current (and changing) zero-point. Stated concisely; Assigned number-values are relative to the changing zero-point.
Knowing now, that the number values of octave tones are relative and changing, and that the pattern must (therefore) be explored one point at a time (with the results cataloged and compared) we have only to decide where to begin, which point in the pattern to call zero-degrees first.
Absolute Zero
Among all possible zero-points within any Pattern of Unisons and Octaves there is one convenient (and probably the best and most proper) place to start. We'll call that point absolute-zero. Absolute-zero of any Pattern of Unisons and Octaves lies at the rooting-center of the pattern. Specifically, it is the tone on string-one. It, you recall, is the lowest pitched version of the given lettered tone available on the guitar fretboard — hence the distinction absolute. All other points in the pattern are either higher pitched octaves of that tone or they’re unison with it.
We will, then, begin at the rooting-center, using absolute-zero as our first zero-point, and then gradually and methodically build a complete twelve-fret pattern of tonic/root unisons and octaves, one tone (one zero-point) at a time. Each new tone added to the pattern will be numbered more than one way. For example, by the time the pattern has grown to include three octave points, that third tone will be numbered twenty-four degrees (relative to absolute-zero, the first tone in the pattern), then twelve degrees (relative to the second point in the pattern) and finally zero degrees (being in and of itself a potential zero-point.), see Plate 8. So, as each new tone is added to the pattern we will find multiple new ways to analyze and relate all tones to one another, and their number-values will constantly be changing as octave relationships are continually defined and redefined. To provide some constant reference point, so we don't get lost amidst all that change, new tones added to the pattern will always be referred (at least) to absolute-zero. That is, throughout the upcoming eight plate series of illustrations each plate used to illuminate each new tone begins by numbering/gauging that new tone relative to absolute-zero, our starting point, back at the rooting-center, no matter how impractical that distance may at first seem. Each plate and then concludes (at it's far right) with a second form of reference constant; a continually updated cumulative summary of the various number value identities (0°, 12°, 24°, 36°) that all points have assumed/exhibited up to that point.
Realize and remember that this exercise, these number games, are exploratory — a one time only tool of discovery and an analysis. It’s purpose is to familiarize you with the lay of the land, the big picture of how the fretboard works, and is meant to be played primarily on paper — just enough to help you understand what's going on and why.
When the last tone is added to the pattern we will have accumulated six tonic/root tones, at six different locations on the fretboard. By implication then, we should have acquired six unique perspectives, six points of reference, and six possible ways to analyze, number, and relate to all points to one another. Overall, that prediction will prove to be true, but needing qualification. That is, not all points in the pattern are entirely unique and different from each other. While their locations are unique, and we will have six unique optional places to construct and play any given musical material, roughly half of those six points are redundant. i.e. they are unisons of other tones in the pattern. Being duplicates, their presence represents a lessor difference, bringing less change and having less effect upon number values previously determined. The effect of their inclusion is more to extend by repetition an earlier environment (numbering scheme), making it available elsewhere on the fretboard, rather than to change the environment. In the end, we will have only three or four truly different octave tones, plus two or three unisons of each. So identifying the locations of unison tones and the consequences of their existence (in terms of redundant numbers and patterns on the fretboard) is also a large part of our objective here.
We will be doing it at least a half-dozen things at once within the following eight-plate series of illustrations, among them these:
- Learning everything we can about the Pattern of Unisons and Octaves.
- Discovering how octave tones work on the fretboard. i.e. how their number-values change.
- Building a complete Pattern of Unison Octaves, one tone at a time.
- Locating unison tones, as well as octave tones.
- Inadvertently identifying the locations and number-values of all the in-between points (e.g. 1° through 11°, 13° through 23°) — the raw material of all musical formula. [Each number/point is a distinct interval. See Interval Number-Name Translation Tables] That is, we’ll not only be learning about unison and octave tones, but discovering where everything occurs and why — the ultimate full-fretboard big-picture revelation of how the fretboard works (or least the giant leap in that direction).
- And, of course, to do all of that, we will be linking our two primary reference patterns: the Five Degree Calculation Line and the Pattern of Unisons and Octaves.
How and where the Five Degree Calculation Lines are incorporated into the following plates deserves some special attention. The Five Degree Calculation Lines are the facilitators of this entire exercise, but they might get lost in the drawings unless a point is made of seeing them.
In the upcoming plates there are generally five columns per plate — one plate for each new tone added to the pattern. The left-most column in each plate is the place-keeper showing which tone is to be explored next (encircled diamond marks the current tone). The last column in every plate is a summary. The center three columns, then, are the heart of the matter, the meat of each plate. One or more Five Degree Calculation Lines is present in each of the three center columns, but you’ll have to look for them. The Five Degree Calculation Lines are woven seamlessly into each drawing. You are already familiar with their visual and numeric pattern: they begin at any zero-point (tonic/root), form a (nearly) straight line across the fretboard, and read in multiples of the number five. e.g. (0°, 5°), (0°, 5°, 10°), (0°, 5°, 10°, 15°), (0°, 5°, 10°, 15°, 20°), or (0°, 5°, 10°, 15°, 20°, 25°). The straight-line pattern is interrupted by a one-fret-down offset whenever the pattern crosses the fifth string. [To refresh your memory, review the plate, all possible Lefty Five Degree Calculation Lines and Counting Grids].
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