R L     Chromatic Numbers Explained Page   1    2   3         Section Plan The components of the Cipher System can be divided into two main groups: The Group One components are covered on 3 pages here broadly titled “Chromatic Numbers pages 1, 2, and 3” [you are on page 1 now]. These three pages provide any needed background and explanations for the Cipher System’s Number Formula Translation Tables of intervals, scales, and chords. The Group Two components (Fretboard Components) are covered on their own respective pages. Group One components – New numbers and number-formula Components 1 - 4 Parts that relate to the process or results of translating/converting music’s standard diatonic number-formula to the Cipher System’s equivalent chromatic number-formula: 1. The chromatic number-line 2. Interval number-formula 3. Scale number-formula 4. Chord number-formula Group Two components – Fretboard components Components 5 - 7 Parts that relate to the guitar fretboard — it’s innate pattern structure, and the method/device used to transfer the Cipher System’s chromatic number-formula to it: 5. String numbering order 6. The Five Degree Calculation Line 7. The Pattern of Unisons and Octaves [Note: Two final Advanced Components of the Cipher System, the Master Charts and a universal Speller/Transposer, are available as as well. The Master Charts and Speller/Transposer are the Cipher System’s comprehensive translation tables. They are necessary to complete the Cipher System but too detailed to discus here. They would otherwise be included within the Group-One components above.] The Cipher System uses the chromatic scale and its numbered half-steps or semitones as it’s primary tone-numbering referent. If you don’t know what half-steps or semitones are, please see “Half-Steps Review” and then return here. There are two possible approaches to translating the stuff of music theory — two data-types to choose between: we could focus either on music’s letters and letter-spellings (of its common materials: intervals, scales, and chords), or we could focus instead on music’s numbers and number-formula. Eventually, we must examine both topics. One of those mediums, however, is clearly better suited to our purpose. Numbers are easier to work with, and the information they provide is more useful. For (only) the first few minutes here we will use letters (the letters of the C chromatic scale) to help explain how the Cipher System works. By starting with letters, translating some standard letter-data to the Cipher System’s chromatic numbers, you’ll be able to see how and where the Cipher System began, and you’ll also see that the Cipher System can be used to translate music’s letter-data if that’s what you want to do. After these necessary first examples though, and confined to component #1 The Chromatic Number-Line, we will change direction, ignore letter spellings, and focus thereafter on numbers, that is; translating music’s diatonic number-formula to the Cipher System’s equivalent chromatic number-formula. Core components of the Cipher System Group One – New numbers and number-formula Components 1 - 4 Parts that relate to the process or results of translating/converting music’s standard diatonic number-formula to the Cipher System’s equivalent chromatic number-formula: 1. The chromatic number-line (on this page) 2. Interval number-formula (on page 2) 3. Scale number-formula (on page 3) 4. Chord number-formula (on page 3) 1. The chromatic number-line — tone-numbering referent Zero-based chromatic numbers: 0° – 12° for one octave of chromatic scale-tones. 0° – 24° for two octaves of chromatic scale-tones. The first octave The Cipher system begins with a zero-based chromatic number-line (0° – 12°) marking the twelve lettered tones of any chromatic scale. Following is the step by step procedure used to translate music theory’s letters or letter-formula to the Cipher System’s chromatic numbers and number-formula. First, select any chromatic scale and write down its complete letter sequence including any flats or sharps. Below those letters, number the octave of chromatic scale steps zero through twelve degrees. Figure 1 illustrates this using the lettered tones of the C chromatic scale. That is, the chromatic scale having a C tonic or root.     Figure 1           That chromatically numbered letter-line can be used to determine the true intervalic distance and true numeric values, in half-steps or semitones, of any lettered tone(s) of a C tonic or root. That is; any interval rooted on C, any scale (Major, minor, etc.) that begins on C, or any chord (Major, minor, dominant, etc.) that begins with C. [Note; for compound intervals, and chords larger than sevenths, a second octave of lettered and numbered chromatic scale-tones and letters will be needed (Figure 6). Also note, many tones have two or more possible letter-names called enharmonic spellings. E.g. D# (numbered 3° in Figure 1) can also be called Eb. In order to spell all materials of a given tonic/root some of those alternate enharmonic letter spellings would be needed. This is a simplified chromatic letter-line showing no enharmonic spellings. For more about enharmonic spellings and a more completely rendered chromatic letter-line please see this sidebar page. More complete still is this chromatic scale rendering spanning two octaves and including both letters and numbers. Use your browser’s back button to return here.] For an example of how this chromatically numbered letter-line is used to obtain a chromatically numbered diatonic scale formula we’ll use the letters of the C Major scale. The Major scale with a C tonic is spelled: C, D, E, F, G, A, B, C (Figure 2). In standard procedure, the lettered tones of such scales are all numbered simply (1) through (8), with accidental marks (flats and sharps) added as needed (Figure 3). But those numbers are neither enlightening nor instructive. The true distances (i.e. real numeric values) of C Major’s lettered tones can only be found by consulting our chromatically numbered C chromatic letter-line (Figure 1). There, in the chromatic context, we find, for example, that tone E lies “four degrees” or four “half-steps” above C. Translating the entire C Major scale letter-line, in like fashion, produces the following string of corresponding new chromatic numbers (Figure 4).     Figure 2           Figure 3           Figure 4           Our primary concern is to translate music theory’s common generic and reusable number-formula to the Cipher System’s (also) generic and reusable number-formula. That is; to catalogue single examples of musical materials (single number-formula), standard and Cipher, that represent all incarnations (keys, tonics, roots) of the given materials regardless of specific letter spellings. We have just generated one of those sets of reusable generic formula — Standard procedure’s and the Cipher System’s number-formula for any Major scale, in any key (Figure 5).     Figure 5             The second octave — extending the number-line To construct and then translate intervals that exceed one octave in width (called compound intervals) a second and higher pitched octave of scale-tones is required. In this case, scale-tones means lettered and numbered chromatic scale-tones. The second octave of scale-tones replicates the first: the letter-line of the first octave is duplicated and repeated once, to the right of the first octave, and the chromatic number-line is likewise continued (to the right) for another octave, adding the numbers thirteen through twenty-four degrees to our zero-based chromatic number-line (Figure 6). The second octave of scale-tones is used primarily when constructing large chords. It supplies (among other things) the 9th, 11th, and 13th, intervals (standard numbers) needed to construct 9th, 11th, and 13th, chords.     Figure 6           The zero symbol — meanings and uses. In the Cipher System the number zero always represents the tonic, key-center, or root-tone of any musical construction or formula. Zero, not one, marks the starting point of any scale, key, or chord. As a mnemonic aid you can also think of zero as being the letter oh, as in the words home or home-tone — also meaning tonic or root. For example; you could say (to a child), “the tone you want lies four degrees (or four half-steps) from home”. In either case, the circle symbol should be seen as an empty container, a neutral and universal starting point, that can be filled (and, if you want, literally filled-in) with any letter naming a specific or the current tonic or root (Figure 7). For example, if you want to specify a particular key, tonic, or root, take any of the Cipher System’s all-purpose number-formula and write the letter-name of any tonic or root inside the circle of the zero symbol.     Figure 7           Zero will also be used as a form of octave symbol in this book. We commonly think of octave tones as being functionally identical to each other — they are near replicas that sound almost the same and occur in exact multiples of pitch frequency (or distance) e.g. The tones A-220 cps., A-440 cps., and A-880 cps., being octaves of each other, are arithmetically related. — the octaves (higher or lower) of any tone are found by multiplying or dividing the frequency of the original tone by 2. The Octave phenomenon has always invoked the metaphysical; The beginning and the end of an octave are one and the same thing. Any tone and all of its octaves are so alike that musicians are satisfied to give them all the same letter-name — hence there are many C notes and many D’s, etc. As a rule then, octave tones always share the same name no matter what kind of name we use — letter, number, or other. In standard procedure’s nomenclature, the boundaries of an octave run from “tonic-to-tonic”, “root-to-root”, “C to C”, or “1  to  1”. In the Cipher System, octaves run from “tonic-to-tonic” or “root-to-root”, but numbered “zero-to-zero” rather than “1” to “1”. — or that’s one way we’ll number them. I’ll explain. In the Cipher System the tonic/root and its octaves can be numbered in either of three related ways: 0°, 12°, and 24° — reflecting the fact that octaves are multiples of each other [in our case multiples of twelve because we’re using the chromatic octave] 0°, 0°, and 0° — indicating the “sameness” of octave tones, and highlighting the “repeat-in-pattern” points .....or...... — an over-striked composite notation, combining the previous two methods, that superimposes zero on top of 12° and 24°, and ensures that the tonic/root and its octaves are symbolically and visibly connected to each other, catching our eye and reminding us.       With this third method, then, zero is clearly functioning as a form of octave symbol. Notice here, because of zero, the Cipher System’s octave-numbers work right. The Cipher System’s numbers confirm that there are twelve tones per chromatic octave, as generally claimed, while standard procedure, with its one-based numbers, must use the number thirteen to mark the octave — contradicting itself and confusing us. Notice too, that the Cipher System’s octave numbers (0°, 12°, and 24°) are multiples of twelve and multiples of each other, as they should be. By contrast, standard procedure’s chromatic octave numbers for any tonic/root are (1, 13, and 25). Their arithmetic doesn’t work because they don’t use zero. One final note; our discussion of octaves and octave numbers has thus far been focused solely on the octaves of the tonic or root. But you should know that in the Cipher System the octaves and octave numbers of any tone can be found by adding or subtracting the number “12” to/from the value of the original tone. e.g. The next higher octave of the tone at 4° (i.e. the Major-third) is 16°. Zero is movable Zero and the chromatic number-line, our tone-numbering and counting referent, is movable. All (letter-spelled) chromatic scales can be thought of as being portions of one large and continuous chromatic letter-line. Zero, therefore, can be repositioned at any point within any chromatic scale and so be realigned for any new key, tonic, or root. When you move the zero-point everything will still (and always) work the same way. You can move the zero-point anywhere you want, and use the same chromatic number-formula over and over again. Figure 8 shows some examples of moving the zero-point within any chromatic letter-line.     Figure 8             Turning to numbers (or good-bye letters) next page       Page   1    2   3         Up to Top of page