A knowledge of the traditional interval naming system is important to this work. This system names intervals by quality and step-type; for example, the interval ``major third up'' has the quality ``major'' and the step-type ``third up.''
The basic qualities are major, minor, diminished, and augmented and are abbreviated `M,' `m,' `d,' and `Å' respectively. (``Augmented'' is abbreviated `Å' because the more traditional abbreviation, `A,' can lead to confusion because it is a pitch class letter.) Diminution and augmentation can have degrees associated with them, for example ``doubly diminished.'' These degrees will be abbreviated as superscripts, for example `d2' for doubly diminished. The degree of one is implicit.
Large step-types such as ``tenth up'' are valid. Intervals with step-type ``prime'' are valid, but there are no such step types as ``prime up'' or ``prime down'' since their existence would lead to two names for the same interval. (Some readers may be more familiar with the term ``unison,'' which means the same thing as ``prime.'') Step-types are abbreviated by their associated integer supplemented with a sign for direction. For example, a third up is indicated by +3. As with numbers, the positive sign is often left implicit, especially when discussing interval magnitudes, which are always positive. When one says ``these two pitches form a major third,'' one is discussing an interval magnitude; when one says ``the interval from this pitch to that one is a major third up,'' one is discussing an interval. In other words, intervals other than primes are directed.
When the abbreviations for quality and step-type are put together, the sign floats to the beginning; for example, a major third up would be abbreviated as `+M3.'
Be aware that the direction of an interval cannot be relied upon to determine the direction of the frequency ratio it is tuned to. (A frequency ratio greater than one is directed upwards, a ratio less than one is downwards, and a ratio of 1 has no direction.) It is the direction of the frequency ratio an interval is tuned to that determines whether that interval will be heard as ascending, descending, or neither.
The following are examples of situations in which an interval's direction may not correspond to its heard direction, the direction of the frequency ratio to which it is tuned. Primes do not have direction, yet some of them need to be tuned to ascend and descend. For example, Å1 should ascend and d1 should descend. Also, there are upward intervals that should be tuned to descend and downward intervals that should ascend. For example, +d22 (a doubly diminished second up, such as C to D) should descend. Finally, there are intervals that can be tuned to ascend, descend, or do neither, according to the particulars of the tuning. The best example of this is +d2. In 12TET, +d2 (a diminished second up, like C3 to D3) neither ascends nor descends, but there are tunings that make it ascend, and tunings that make it descend.
There is no such interval as a ``half step'' or ``whole step'' since these phrases are ambiguous. Unambiguous descriptions such as Å1 or +m2 for half step and M2 for whole step should be used.
Interval expressions can be formed using addition and negation. (Formally, intervals under addition form an Abelian group [14, 78].) For example, the expression ``'' equals P4, and ``'' equals Å5. (Integer scaling as in the expression ``'' is allowed since it is simply a shorthand for ``.'') The rules for evaluating such expressions will not be given here since many readers can evaluate such expressions intuitively and, in addition, the most convenient formalization of these rules is for intervals in fifth-register vector form, to be presented later in this work. One aspect of such expressions does deserve a brief mention, though. Although primes do not have direction, they can be negated in expressions. For example, ``'' is not an interval; it is an expression that equals the interval . The situation is analogous to that with the number zero; ``-0'' is not a number; it is an expression that equals the number zero. The point here is merely to distinguish between intervals and interval expressions.