Page   1   2    3     Chromatic Numbers (page 2)     Turning to numbers (or good-bye letters) Translating music’s letter-data to the Cipher System’s chromatic numbers isn’t difficult to do, but it’s time consuming — or it would be without a Speller-Transposer. Every musical material (interval, scale, or chord) must be translated twelve to fifteen times — once for every letter of our twelve keys. Standard procedure itself favors using numbers over letters for this very reason. While spelling related chores are time consuming, the only knowledge required to do your own translations is knowing how to spell all twelve chromatic scales — something you can easily learn to do, and need to be able to do anyway. Remember too, for quick reference, use the Speller-Transposer — there’s no need to reinvent the wheel. But again, don't worry about spellings now. What we really need is better numbers. Converting music's standard number-formula to the Cipher System's chromatic number-formula is the best way to obtain the information we want, and that's the approach we’ll take. From here-on, we’ll focus only on numbers. Following, then, are the Cipher System's Number-Formula Translation-Tables for music’s everyday materials: intervals, scales, and chords. Take a look at  the tables (if you haven’t yet) and then return here for any background and explanatory information you might need.       2. Intervals — number-name translation tables In music, most instructions and all musical formula are communicated using numbers. Every number used, single or series, refers to and is an interval. Understanding intervals and their number-names is the key to understanding (and translating) all musical number-formula. Once we translate music’s standard interval number-names to the Cipher System’s chromatic-number equivalents the resulting translation table will be used to translate all other larger standard number-formula, e.g. scale-formula and chord-formula. Note, because of their singular importance, I’ve included a large stand-alone primer on intervals and inversion of intervals for your perusal. Four Interval Number-name Translation-Tables are included here. The four tables contain and repeat the same information, but each has a different focus. Following is an explanation of those tables. Some basics Whole-steps and half-steps (or tones and semitones) are the basic units of pitch measurement in Western music. The distances measured with those whole-steps and half-steps are called intervals. An interval is a measurement of pitch separation between any two tones. [Whole-steps and half-steps themselves are intervals — Major-second and minor-second intervals respectively.] Intervals are always measured from the lower pitched tone up to the higher pitched tone. The starting tone of any interval (any measurement) is called the tonic or root. The target tone’s number-name is determined by its distance (in whole-steps or half-steps) from the given tonic/root. The number-name of the upper tone (the target) becomes the name of the interval itself. Two variations of interval number-name In standard procedure the number-names of intervals are made-up of two parts: an Arabic numeral plus a qualifying prefix. The prefixes can appear either as words [e.g. Major, minor, augmented, or, diminished], or as symbols, i.e. accidental marks, e.g. . The meanings and use of these two types of qualifier are similar but not identical — we’ll discuss their differences in a moment. In any event, we must be able to translate both kinds of prefixed numbers. I’ve included a separate translation table for each of the two types of prefixes (word and accidental-mark) and a third summary table combining both. The two forms of interval number-name, employing either word prefixes or symbol prefixes, are used for different purposes. Word prefixes, like Major, or minor, communicate the particulars of intervals while remaining neutral with regard to specific combinations of accidental marks present at any given time. Word prefixes, then, are collective and neutral qualifiers. For example; the following pairs of lettered tones are all Major third intervals. They all span the same distance, 2 whole-steps: B-D#, Cb-Eb, B#-Dx,  C-E,  C#-E#, Db-F. Each spelling of this one interval displays a different combination of accidental marks (or none), yet all of them can share the same collective number-name (Major third) because that name incorporates the neutral qualifying word Major rather than some single and specific accidental mark. Interval number-names that incorporate accidental-mark prefixes (e.g. b3 and #5) are more compact than their word-prefixed counterparts and they’re also more precise. This form of interval number-name is used primarily as a kind of musical notation, a written shorthand, and it’s the preferred form for cataloguing musical formula. For example; in standard procedure, chord formula are typically written like this: R, b3, 5, b7 = Minor-Seventh chord. Each digit of such formula (with or without accidental marks) represents and names an interval of a specific width in whole-steps or half-steps. Those digits, those interval number-names, are the things we must translate to real (counting) numbers. Interval number-name translation tables Table-One: standard word-prefixed interval number-names to chromatic or half-step equivalents. Music theory’s standard interval number-names, e.g. Major-third and perfect-fifth, are usually explained using a second set of equally vacant numbers, namely whole-step numbers. A typical pair of such numbers would read; P-5th = 3½. That is, “A Perfect-Fifth interval equals three and one-half whole-steps”.  All of those so called numbers, however, fail to communicate any real or usable information. In the Cipher System, half-step numbers, i.e. counting numbers, are used to define and describe intervals rather than whole-steps. To convert music theory’s whole-step numbers to half-step numbers, multiply the whole-step values of any interval by 2. e.g. 3½ whole-steps x2 = 7 half-steps. In the Cipher System, then, a Perfect-Fifth interval equals seven half-steps. The number seven (used here) has meaning. It’s a real number, a counting number — like seven fingers or seven frets. It’s information that you can actually use. Details of Table-One Interval Table One shows two octaves of progressively wider intervals. Each successive interval, from the top of the list down, is one half-step (or one chromatic-step) wider than the interval above. The table is divided into two columns, Standard and Cipher: Standard The left column of Table One lists all intervals by their standard word-prefixed interval number-names. See the Abbreviation Key at the bottom of the page. [e.g. Maj. = Major]. Cipher The right column of Table One shows the Cipher System’s half-step number translations of music theory’s standard interval number-names. In this table the half-step numbers are preceded by zero and a dash. Zero represents the tonic or root, and the dash is simply a spacer — just like the one in P-5th. The important numbers here are the ones that follow the dash. They are the half-steps, the counting numbers. The P-5th interval becomes 0° - 7° — spoken oh (like the letter) seven. i.e. an oh-seven interval. The expression 0° - 7° means; “from the tone at root zero, the other tone you want is seven half-steps higher in pitch, or seven frets-worth of pitch away. Remember, all of the following terms and their associated numbers are synonymous and interchangeable: half-steps, chromatic-steps, frets, and degrees. Again, two complete octaves of intervals are shown, numbered zero through twenty-four degrees. The Cipher System’s numbers and number-formula are easy to distinguish from music’s standard number-formula; The Cipher System’s formula numbers always have degree symbols (°) superscript to the right of each formula digit, and they never have flats or sharps. Table-Two: standard accidental-mark prefixed interval number-names to chromatic or half-step equivalents Interval Table Two covers the second type of interval number-name used in standard procedure. Interval number-names that incorporate accidental-mark prefixes (rather than word prefixes) are used to write musical number-formula — strings of intervals meant to be played sequentially (like scales) or simultaneously (like chords). Fundamentals review: understanding music theory’s standard number-formula. Music-theory and its many sub-systems is centered around one thing, the Major Scale, and the C Major scale in particular. Intervals are no exception. In fact, intervals, their names and numbers, is the place in music theory where the Major scale’s influence first takes hold. The number-names of intervals, and the rules of writing musical number-formula (i.e. which numbers and accidental marks to use when and where) are all dictated by the Major scale. The intervals that occur naturally above the tonic of the Major scale are used as reference-intervals in music theory. Likewise, the number-symbols assigned to the Major scale’s intervals are the starting-point of music’s standard system of formula-number symbology. Given that music’s interval number-names and symbols are all based on the Major scale’s intervals we’ll begin there and progress to the number-names and symbols of non Major-scale intervals. We know that standard procedure numbers the tones of any Major scale 1 through 8. Earlier, I referred to those numbers as scale-degree position-numbers — which they are. I used that term to help us distinguish between music’s two tone-numbering requirements, namely: scale degree position and distance.     Figure 1           Note, by definition, intervals are measured distances, and interval numbers are distance numbers — or, they should be. But music’s standard interval numbers do not reflect those definition requirements. That, of course, is the problem with standard procedure’s numbers. In standard procedure, the Major scale’s scale-degree position-numbers (1 through 8) are also used as interval number-names! The problem with that is that those numbers have absolutely nothing to do with the distances between tones. They are not distance numbers at all. In any event, we need to know what those numbers mean when they’re used as interval number-names, so we’ll continue. Standard procedure numbers the tones of any Major scale 1 through 8. Those same numbers are then used as interval number-names — the foundation reference-set of interval number-names. So the intervals that occur naturally above the tonic of any Major Scale are depicted/numbered with plain unqualified Arabic numerals 1 through 8 — free of accidental marks (flats and sharps). They’re the same numbers normally used simply to number-mark the scale-degrees of any Major scale, and they refer to exactly those same Major scale tones. In the mechanics of music’s standard number-formula symbology, then; the unqualified numerals 1 through 8 represent only the intervals of (the first octave of) any Major scale, tonic through octave. So any time you see an unqualified numeral used in any standard number-formula (interval, scale, or chord) you’ll know that the interval it represents belongs to (is a member of) the Major scale — i.e. it occurs naturally above the tonic (first tone) of any Major scale. All of the Major scale’s intervals are classified and named as either Major (type) or Perfect (type) intervals. So the unqualified interval number-names 1 through 8 always and only represent Major or perfect  intervals, as follows (Figure 2):     Figure 2           The Major scale’s intervals (all of them Major or perfect) and the unqualified numerals used to represent them are the starting point of music’s standard system of formula-number symbology. All other intervals, meaning those that do not occur naturally above the tonic of a Major scale (e.g. minor, augmented, and diminished  intervals) are symbolized with numbers by adding accidental marks to the Major scales unqualified interval numerals. The differences of those non-Major-scale intervals are then seen and compared relative-to the Major scale’s (Major and perfect) intervals — the standard referents. For example; the minor-third interval (which does not occur above the tonic of a Major scale) is symbolized b3. In simple language, that number-symbol flat-third means; “a little less than a Major-third”. The referent is the Major-third (i.e. the number 3 with no flats or sharps), and the flat symbol is the qualifier, meaning “narrowed by one half-step or semitone”. Figure 3 shows more examples of how standard procedure numbers these non Major-scale intervals. Notice that all of these interval number-symbols (potential formula-digits) are qualified with flats or sharps.     Figure 3           Enharmonic number-names — the Cipher System’s treatment of Approximately one-half of all available intervals are either Major or perfect — they are natural to the Major scale. The remaining intervals are either minor, augmented, or diminished. But due to standard procedure’s habit of assigning enharmonic names to those non Major-scale intervals (alternate but synonymous letter-names and number-names assigned to any single tone; e.g. #4 = b5 and 6 = bb7) the apparent quantity of non-Major-scale intervals doubles, at least. In other words, it will seem as if there are twice as many non Major-scale intervals as there really are, and there will be twice as many non Major- scale interval names (in fact) to learn about and then translate. Thankfully, there are no enharmonic numbers in the Cipher System. The Cipher System treats each pair of standard procedure’s enharmonic alternates as a single tone (which, in the equal tempered chromatic scale, they are) and assigns a single (real, chromatic) number-value to each such tone. Any of the Cipher System’s single number designates, then, will always encompass any and all of standard procedure’s enharmonic alternates for the given tone. Figure 4 shows some examples of standard procedure’s pairs of enharmonic number-names compared to the Cipher System’s corresponding single number-designate.     Figure 4           For a more detailed review of music theory fundamentals, including music’s standard letter-names, number-names, and information on intervals in general, see Chapter Four (in the book). Details of Table-Two Interval Table Two focuses on music’s interval number-symbols (formula digits) rather than word-names. Like Table One, Table Two shows two full octaves of progressively wider intervals — see the right-hand columns marked All Intervals. Each successive interval, from the top of the list down, is one half-step (or one chromatic-step) wider than the interval above it. The table is divided into two primary columns marked Major Scale Intervals Only and All intervals: Major scale intervals (only) For reference purposes, the left-hand column(s) of Table Two show only the numbered intervals of the Major scale. Notice the absence of accidental marks. All intervals The right-hand column(s) of Table Two list all intervals (i.e. with no preference shown to the Major scale’s tones). This, then, is what real life in standard procedure looks like — complete with accidental-marked numbers and enharmonic number-names. [Note, the quantity of enharmonic number-symbols included here is (still) not exhaustive, but it’s more than enough to start with.] As always, music’s standard interval number-names and symbols are here translated to the Cipher System’s chromatic or half-step number equivalents. Table-Three: summary table. Standard word-prefixed and accidental-mark prefixed interval number-names to chromatic or half-step equivalents. Interval Table-Three combines and correlates all material from tables One and Two. Enharmonic entries — that could and should share a single line — are marked with brackets. Note; the two types of qualifier-prefixes, words and accidental marks, cannot be brought into exact and exclusive one-to-one correspondence. The word qualifier diminished, for example, may correspond to either of two different accidental marks: the flat [b] and double-flat  [bb]. e.g. diminished-fifth intervals use single flats but diminished-sevenths use double flats. For more information regarding the near correspondence of word and accidental-mark prefixes, see Chapter Four Part Two. Table-Four: octave equivalents. Associating the number-names of simple and compound intervals (standard and Cipher). Interval Table Four illustrates how the number-names of intervals change when intervals are widened by one octave. Intervals that span one octave of pitch or less are called simple intervals. Intervals that exceed one octave in width are called compound intervals. Compound intervals are essentially repeats or duplicates of intervals available in the first octave. Compound intervals are exactly one octave wider than their first octave counterparts. For example, if the upper tone of a given simple interval is G you can form a similar but compound  interval by using the next higher G instead. i.e. G in the second octave. The letter-names of the interval’s tones do not change. Only the octave relationships change, and the number-name of the interval changes accordingly. For convenience sake, we usually think of octave tones as being a single same thing. If we treat simple and compound intervals in a similar fashion we can halve our learning curve. The trick is to see the recurring pattern in the number-names (values) of simple and compound intervals — their number-names always increase or decrease by a constant value equivalent to one octave. The value of that constant, the octave multiplier, is different in each of our two tone-numbering environments, diatonic and chromatic. In the diatonic realm (a seven-thing oriented and one-based environment) the number seven is the octave multiplier. In the chromatic environment (zero-based and twelve-thing oriented) the number twelve is the octave multiplier. To associate the number-names of related simple and compound intervals (i.e. to find one from the other or to change one into the other) use these formula:     Figure 5       To simplify this table only the Major scale’s intervals (the standard referents) are included. The principle, however, applies to the number-names of all intervals. Note; when converting standard (diatonic) number-names from simple to compound (or the reverse) always make sure to carry over and duplicate any accidental mark from the original interval’s number-name. For example; if you begin with a minor-third interval (symbolized b3), and then widen it by one octave, the result is a minor-tenth interval (symbolized b10) — the flat is carried over.            next page     Page   1   2    3         Up to Top of page