Why Zero?     The Major scale’s distance numbers The Major scale’s distance numbers, i.e. its whole-step formula numbers are zero-based numbers — they start at zero not one. The ideal vehicle for showing that (all of) music theory’s distance numbers (it’s whole-steps or half-steps) are zero-based numbers is the chromatic scale, with its twelve uniform half-steps, and we will turn to that environment shortly. But first, I want to give you some quick proof that zero-based numbers exist even in music theory’s everyday standard mechanics, and that means the Major scale and its whole-step units of measure. The phrase “whole-step units of measure” is our clue. The thing we need to understand (in five steps) is that: distance numbers are used to perform and record measurements i.e. measurements of pitch distance or separation between tones measurement involves computation (arithmetic operations) computation presumes the use of zero-points and zero-digits consequently, all units of measure (of all kinds) must and do begin at and with zero, not one — check your ruler, measuring cup, or stop-watch. so music theory’s distance measurement numbers must be and are zero-based numbers — although that fact is not immediately clear nor obvious. Zero — the invisible key Once you understand when and where in the world you should expect to find zero (e.g. when measurements are being made), seeing zero is simply a matter of having the object correctly and completely drawn and numbered. [We will visit this issue again when we examine the guitars built-in (and invisible) zero-based fret-numbering scheme.] This matter of incomplete or faulty drawings explains why zero’s presence in music’s everyday materials may have surprised you. Figure 1 gives a more accurate rendering of the Major scale’s whole-step formula, that is, its unsummed zero-based distance numbers. Here, the zero-point is properly inserted, and the formulas digits are properly aligned with the Major scale’s letters — flush left, not offset or in-between.     Figure 1       The Guitar’s numbers We’re always told that a chromatic octave contains twelve tones and twelve half-steps. We are also told that the octave note of any open (unfingered) guitar string falls on/at the twelfth fret. There are no conflicts in those statements — everything works right and seems to make sense. But there’s more there than meets the eye. The fact that things “work right”, i.e. that the fretboard’s octave-fret is numbered twelve, and not thirteen, should not be dismissed so quickly and taken for granted. It took planning and a little magic (an arithmetical trick) to achieve that sync. The nut of the guitar, the peg-head bridge-point, serves the same function as a fret — it stops the string at a fixed point to produce a definite and crisp tone. But because the nut is made of a different material and shape than the other normal frets, and because we tend to think of the nut solely in terms of its bridging function (e.g. its role in determining string action) we usually don’t think of it as being a fret, nor do we consciously count it as one. But the nut is, by function, a fret, and we do in fact count it. We count the nut as zero. The nut is the zero-point and zero-fret of the guitar — the starting point and calibration reference. The next fret (the first metal fret) is then called fret 1 (and so on), just like the numbers printed on any correctly rendered measuring stick — which is, after all, what the fretboard is! The guitar then, has a built-in zero-based fret and tone numbering scheme — 0 through 12 for the first complete octave of tones. In the same way that zero makes half-step numbers read and sync properly (i.e. making the twelfth increment read “twelve” and not “thirteen”), the fretboard’s use of zero places its (open) octaves squarely on/at the fret called twelve, not thirteen. So zero is the thing that makes the fretboard work right. It’s the key. One important consequence of the guitars built-in zero-based numbering scheme is that the fretboard is extremely friendly to and naturally compatible with half-steps, i.e. half-step numbers. That, for example, is why guitarists can perform simple everyday actions and instructions, like the following, without hesitation or confusion. Do this: put any finger on any string anywhere on the fretboard. To raise the pitch of that tone two half-steps you simply count (and slide your finger up) two frets. And you’re done. In other words, the phrases two half-steps and two frets mean virtually the same thing — they’re numerically in sync and interchangeable. Notice further that no matter where your finger was when you began, that starting-point would have to be called zero. Test this by counting frets (or tones) backwards from where you landed,.... count back... 2, 1, 0. So the principal of using zero as your starting-point applies everywhere on the fretboard, not just at the nut. Zero is a universal law of the fretboard. It’s the first rule of all fretboard activity — all navigation, all pattern construction, and all pattern movement. Understand, this stuff about zero is not new. I’m not making it up, just drawing attention to it (see Figure 2, a three hundred year old drawing clearly marking zero across the nuts of many fretted string instruments of the day — zero is clearly visible in the blowup inset). Zero has existed on the guitar fretboard, though quietly, for as long as the guitar itself has existed (see Figure 2), and guitarists have always, though unconsciously, relied on the fretboard’s affinity for half-steps and zero. Also realize that this kind of thing (using zero) is not likely to happen nor be maintained by accident. Someone familiar by trade with arithmetic and measurement, e.g. a luthier, either knew instantly and instinctively how to number the guitars frets, or sat down and thought about it. But either way, that choice was not an accident. That unidentified person would have known the difference and the conflicts between one-based and zero-based numbering and counting schemes and intentionally made allowance for zero at the nut. Thus, the foundation of the fretboard’s zero-based physiology, its innate pattern structure, which then continues throughout the fretboard, was set for eternity. And thank our lucky stars it was. So a conscious and educated choice was made, and in time became fixed as convention. [That act, by the way, is one thing in the whole of music’s now standard system-mechanics, notations, and nomenclature, that our musician ancestors did right the first time. If they had chosen instead to use a one-based numbering scheme for the guitar fretboard we would have been forever sunk. We wouldn’t have had a single intelligently designed component in music to build upon. The Cipher System, for example, could never have evolved because we would never have had an instrument that “worked right” arithmetically.]     Figure 2       The fact that the fretboard’s zero-based design foundation is rarely spoken of (except by luthiers) and has never been exploited before is a mystery and a shame, but facts are facts; zero is the dominant digit of the fretboard, and it’s the only key that can unlock it. There’s another even more fortunate consequence of the fretboard’s built-in zero-based tone numbering scheme that we still have to talk about. I call it the Five Degree Calculation Line. Like zero, the Five Degree Calculation Line is a natural part of the fretboard (when conventionally tuned), but it’s even less visible than zero was — and it also needs zero to work right. With it we can generate a counting-number grid of coordinates, a schematic of the fretboard, that quickly and simply explains the why and where of all fretboard patterns.           Up to Top of page