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As to its usefulness, Cope provides the example of the final tonic chord of some popular music being traditionally analyzable as a "submediant six-five chord" (added sixth chords by popular terminology), or a first inversion seventh chord (possibly the dominant of the mediant V/iii). According to the **interval** root of the strongest **interval** of the chord (in first inversion, CEGA), the perfect fifth (C–G), is the bottom C, the tonic.

Although intervals are usually designated in relation to their lower note, David Cope and Hindemith both suggest the concept of **interval** root. To determine an interval's root, one locates its nearest approximation in the harmonic series. The root of a perfect fourth, then, is its top note because it is an octave of the fundamental in the hypothetical harmonic series. The bottom note of every odd diatonically numbered intervals are the roots, as are the tops of all even numbered intervals. The root of a collection of intervals or a chord is thus determined by the **interval** root of its strongest interval.

The **interval** dimension of a partial order can be defined as the minimal number of **interval** order extensions realizing this order, in a similar way to the definition of the order dimension which uses linear extensions. The **interval** dimension of an order is always less than its order dimension, but **interval** orders with high dimensions are known to exist. While the problem of determining the order dimension of general partial orders is known to be NP-hard, the complexity of determining the order dimension of an **interval** order is unknown.

**Interval** cycles, "unfold [i.e., repeat] a single recurrent **interval** in a series that closes with a return to the initial pitch class", and are notated by George Perle using the letter "C", for cycle, with an interval-class integer to distinguish the interval. Thus the diminished-seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0–11 to indicate the lowest pitch class in the cycle.

An **interval** variant of Newton's method for finding the zeros in an **interval** vector can be derived from the average value extension. For an unknown vector applied to, gives :.For a zero \mathbf{z}, that is f(z)=0, and thus must satisfy :.This is equivalent to .An outer estimate of can be determined using linear methods.

The **interval** corresponds exactly to the fluctuation range for the stage \mu_i.

Pitch **interval** - Pitch-interval class

In musical set theory, a pitch-interval class (PIC, also ordered pitch class **interval** and directed pitch class interval) is a pitch **interval** modulo twelve.

These methods only work well if the widths of the intervals occurring are sufficiently small. For wider intervals it can be useful to use an interval-linear system on finite (albeit large) real number equivalent linear systems. If all the matrices are invertible, it is sufficient to consider all possible combinations (upper and lower) of the endpoints occurring in the intervals. The resulting problems can be resolved using conventional numerical methods. **Interval** arithmetic is still used to determine rounding errors.

In each step of the **interval** Newton method, an approximate starting value is replaced by and so the result can be improved iteratively. In contrast to traditional methods, the **interval** method approaches the result by containing the zeros. This guarantees that the result produces all zeros in the initial range. Conversely, it proves that no zeros of f were in the initial range if a Newton step produces the empty set.

"Walk-back sprinting" is one example of **interval** training for runners, in which one sprints a short distance (anywhere from 100 to 800 metres), then walks back to the starting point (the recovery period), to repeat the sprint a certain number of times. To add challenge to the workout, each of these sprints may start at predetermined time intervals - e.g. 200 metre sprint, walk back, and sprint again, every 3 minutes. The time **interval** is intended to provide just enough recovery time. A runner will use this method of training mainly to add speed to their race and give them a finishing kick.

Triangular **interval** - Triangular **Interval** Syndrome

Triangular **Interval** Syndrome (TIS) was described as a differential diagnosis for radicular pain in the upper extremity. It is a condition where the radial nerve is entrapped in the triangular **interval** resulting in upper extremity radicular pain. The radial nerve and profunda brachii pass through the triangular **interval** and are hence vulnerable. The triangular **interval** has a potential for compromise secondary alterations in thickness of the teres major and triceps. It is described based on cadaveric studies that fibrous bands were commonly present between the teres major and triceps. When these bands were present, rotation of the shoulder caused a reduction in cross sectional area of the space. Normal resting postures of humeral adduction and internal rotation with scapular protraction may be speculated as a precedent for teres major contractures owing to the shortened position of this muscle in this position. In addition, hypertrophy of this muscle can occur secondary to weight training and potentially compromise the triangular **interval** with resultant entrapment of the radial nerve. Shoulder dysfunctions have a potential for shortening and hypertrophy of the teres major. Shoulders that exhibit stiffness, secondary to capsular tightness, contribute to contracture and hypertrophy of the teres major. Hence, restricted external rotation can encourage adaptive shortening and thickening of the internal rotators of the shoulder principally the teres major and subscapularis. One may speculate that the lateral arm pain presented in shoulder dysfunctions may be of a nerve origin secondary to adverse neural tension of the radial nerve. The triceps brachii has a potential to entrap the radial nerve in the triangular **interval** secondary to hypertrophy. The presence of a fibrous arch in the long head and lateral head further complicates the situation. Repeated forceful extension seen in weight training and sport involving punching may be a precedent to this scenario. The radial nerve is vulnerable as it passes through this space, for all of the reasons mentioned above.

**Interval** arithmetic can also be used with affiliation functions for fuzzy quantities as they are used in fuzzy logic. Apart from the strict statements x\in [x] and, intermediate values are also possible, to which real numbers are assigned. \mu = 1 corresponds to definite membership while \mu = 0 is non-membership. A distribution function assigns uncertainty, which can be understood as a further interval.

An **interval** can also be defined as a locus of points at a given distance from the centre, and this definition can be extended from real numbers to complex numbers. As it is the case with computing with real numbers, computing with complex numbers involves uncertain data. So, given the fact that an **interval** number is a real closed **interval** and a complex number is an ordered pair of real numbers, there is no reason to limit the application of **interval** arithmetic to the measure of uncertainties in computations with real numbers. **Interval** arithmetic can thus be extended, via complex **interval** numbers, to determine regions of uncertainty in computing with complex numbers.

and can be calculated by **interval** methods. The value corresponds to the result of an **interval** calculation.

To work effectively in a real-life implementation, intervals must be compatible with floating point computing. The earlier operations were based on exact arithmetic, but in general fast numerical solution methods may not be available. The range of values of the function for and are for example. Where the same calculation is done with single digit precision, the result would normally be [0.2, 0.9]. But , so this approach would contradict the basic principles of **interval** arithmetic, as a part of the domain of would be lost. Instead, the outward rounded solution [0.1, 0.9] is used.

An **interval** graph is called p-improper if there is a representation in which no **interval** contains more than p others. This notion extends the idea of proper **interval** graphs such that a 0-improper **interval** graph is a proper **interval** graph.

An **interval** graph is called q-proper if there is a representation in which no **interval** is contained by more than q others. This notion extends the idea of proper **interval** graphs such that a 0-proper **interval** graph is a proper **interval** graph.

Proper **interval** graphs are **interval** graphs that have an **interval** representation in which no **interval** properly contains any other interval; unit **interval** graphs are the **interval** graphs that have an **interval** representation in which each **interval** has unit length. A unit **interval** representation without repeated intervals is necessarily a proper **interval** representation. Not every proper **interval** representation is a unit **interval** representation, but every proper **interval** graph is a unit **interval** graph, and vice versa. Every proper **interval** graph is a claw-free graph; conversely, the proper **interval** graphs are exactly the claw-free **interval** graphs. However, there exist claw-free graphs that are not **interval** graphs.

For a result **interval** r to intersect our query **interval** q one of the following must hold:

**Interval** arithmetic can be extended, in an analogous manner, to other multidimensional number systems such as quaternions and octonions, but with the expense that we have to sacrifice other useful properties of ordinary arithmetic.

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