Page   1   2   3    4     Number-names — Intervals — page 3     Compound Intervals While most compositions encompass numerous octaves of pitch, the melodic lines of individual parts usually proceed by small intervals. Melodic jumps exceeding twelve half-steps (or one octave) are rare. There is a place, however, where intervals that exceed twelve half-steps are common, that is, in chords. Specifically, when constructing or analyzing the larger chords: ninths, elevenths, and thirteenths. For these, an additional (higher pitched) octave of scale tones, running contiguously with the first, is needed. Intervals that span one octave of pitch or less are called simple intervals. Intervals having widths greater than one octave 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. To obtain the number-names of the compound intervals, extend (or repeat) the C Major letter line to include an additional higher octave of Major scale tones. Number as before with uniform Arabic numerals, continuing this time into the next higher octave through to the number fifteen (Figure 17).           Figure 17       Tones in the second octave (for that matter, any octave) are essentially repeats of those in the first. That is, they use the same letter names and exhibit the same pattern of intervals major, minor, chromatic, etc. — but raised one octave. In the event that it isn't clear; any two tones having the same letter-name and residing in adjacent octaves lie exactly one octave apart (Figure 18).     Figure 18       Being additional Major scale material, these compound intervals use the same prefixes as their first octave counter parts — Major or perfect. As such, they're included among (and complete) music theory's set of reference intervals.     Figure 19       Octave equivalents. Associating the number-names of related simple and compound intervals to each other. All compound intervals have a related simple interval and visa versa. The intervals are related because they both use similarly lettered tones. The difference between them is only a matter of an octave. The letter-names of both interval’s  tones are identical and they won’t change when either interval is converted to the other. But the octave relationships do change, and the number-name(s) of the interval(s) will change accordingly. There is an easy way to associate and remember the number-names of related simple and compound intervals. 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 20             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. Associating the names of compound intervals to their first-octave counterparts — more how to Interval number-names consist of two parts: a prefix word (or its abbreviation) and a number. When converting interval number-names from simple to compound (or the reverse) both name parts must be considered. We'll discuss each separately, beginning with prefixes. Prefix portion (of interval number-names) An interval widened or narrowed by one or more octaves retains the prefix of the former. Thus (in C Major), D in all octaves is marked “M” or “Maj.”, and the intervals C to the G in either the first or second octave (Figure 19) are both “perfect” intervals, i.e. P-5th and P-12th respectively. Number portion (of interval number-names) While simple and compound intervals retain the prefix words (Major, minor, perfect, etc.) of their other octave counterparts, their number values increase or decrease by a constant value, a factor of one — octave. Octaves are the key to associating the number-names of simple and compound intervals. Again, the number 7 is the octave multiplier for all seven tone scales and their materials — intervals. Given that music theory is based on a seven tone scale —seven parted octave — its systemology exhibits characteristics of a “base seven” number system. That is, every seventh thing begins to repeat a cycle or pattern.     Figure 21       Also, from any point within the series, one will find a tone of the same kind (letter name) seven places forward or back (Figure 22).     Figure 22             Raising an intervals number-name by one octave, then, means, to increase it by seven. [We're not talking about pitch increase by seven (of anything e.g. whole-steps or half-steps), but rather, interval number-name change (increase) by seven.] Figure 23     Figure 23           Figure 24             The technique of deriving number-names of compound intervals by adding seven to the number-names of simple intervals can be performed in reverse. That is, subtract seven from the number-name of any compound interval to find it’s first octave counterpart. Figure 25.     Figure 25             Minor, diminished, and augmented compound intervals: All things previously learned about the number-names of compound intervals apply equally to the non-Major scale compound intervals, most importantly these: the qualifying prefixes- minor, diminished, and augmented, given to first octave intervals are used again by their related compound constructions. add or subtract seven to/from an intervals number-name to identify its (one octave) wider or narrower counterpart. Interval number-name summary — simple and compound, two octaves Figure 26 summarizes the names of all intervals, simple and compound, spanning two octaves.     Figure 26                   Half-step values or true intervalic distances (next page)       Page   1   2   3    4         Up to Top of page