Einstein field equations

Tonic and subtonic in C My name: Lindsey Gavin
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"'Backdoor' ii-V" in C: ii-Template:MusicVII⁷-I My name: Lindsey Gavin
My age: 28
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]

In music, the subtonic is the scale degree below the tonic or, more specifically, the flattened seventh (Template:MusicVII): the lowered or minor seventh degree of the scale, a whole step below the tonic, as opposed to the leading tone, which is only a half step below the tonic. The distinction between leading tone and subtonic has been made by theorists since at least the second quarter of the 20th century.

The subtonic appears in three forms: as the scale degree, Template:Music${\hat {7}}$, melodically and as the chord Template:MusicVII in both Template:MusicVII-I cadence and in modulations harmonically. The word is also used as an English translation of subtonium, the Latin term used in Gregorian chant theory for the similar usage of a tone one whole step below the mode final in the Dorian, Phrygian, and Mixolydian modes.

For example, in the A minor scale (white keys on a piano, starting on A), the subtonic is the note G (in C major this would be BTemplate:Music); and the subtonic triad consists of the notes G, B, and D (in C: BTemplate:Music-D-F). In music theory, the subtonic chord is symbolized with the Roman numeral Template:MusicVII for a major triad built on the note, or Template:Musicvii for a minor triad; in a minor key, the flat symbol is sometimes omitted by some theorists because the subtonic note appears in the natural minor scale, but the flat symbol is usually used for the major scale because the subtonic is a non-scale note.

A I chord, C major, followed by a Template:MusicVII chord, BTemplate:Music major, borrowed from the parallel minor, C minor, with the key signature of C major followed by C minor shown.

In jazz, the flattened seventh is also used as a substitute for the dominant, V, especially in the Backdoor cadence, ii-Template:MusicVII7-I, where the subtonic is used for the dominant seventh. Template:MusicVII is in this case a pivot chord borrowed from the parallel minor (its dominant seventh). V7 and Template:MusicVII7, the subtonic seventh chord, have two common tones, in C: GBDF and BTemplate:MusicDFATemplate:Music.

However, while, "the leading-tone/tonic relationship is axiomatic to the definition of common practice tonality," especially cadences and modulations, in popular music and rock a diatonic scalic leading tone (i.e., Template:Music${\hat {7}}$-${\hat {1}}$) is often absent. In popular music, rather than "departures" or "aberrant," the "use of the 'flattened' diatonic seventh scale degree...should not even be viewed as departures".

See also

• ♭VII-V7 cadence

Sources

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• Stell, Jason Travis. 2006. "The Flat-7th Degree in Tonal Music". PhD diss. Princeton: Princeton University.

In computational complexity theory, the potential method is a method used to analyze the amortized time and space complexity of a data structure, a measure of its performance over sequences of operations that smooths out the cost of infrequent but expensive operations.

Definition of amortized time

In the potential method, a function Φ is chosen that maps states of the data structure to non-negative numbers. If S is a state of the data structure, Φ(S) may be thought of intuitively as an amount of potential energy stored in that state; alternatively, Φ(S) may be thought of as representing the amount of disorder in state S or its distance from an ideal state. The potential value prior to the operation of initializing a data structure is defined to be zero.

Let o be any individual operation within a sequence of operations on some data structure, with S_before denoting the state of the data structure prior to operation o and S_after denoting its state after operation o has completed. Then, once Φ has been chosen, the amortized time for operation o is defined to be

$T_{\mathrm {amortized} }(o)=T_{\mathrm {actual} }(o)+C\cdot (\Phi (S_{\mathrm {after} })-\Phi (S_{\mathrm {before} })),$

where C is a non-negative constant of proportionality (in units of time) that must remain fixed throughout the analysis. That is, the amortized time is defined to be the actual time taken by the operation plus C times the difference in potential caused by the operation.

Relation between amortized and actual time

Despite its artificial appearance, the total amortized time of a sequence of operations provides a valid upper bound on the actual time for the same sequence of operations. That is, for any sequence of operations $o_{0},o_{1},\dots$, the total amortized time $\sum _{i}T_{\mathrm {amortized} }(o_{i})$ is always at least as large as the total actual time $\sum _{i}T_{\mathrm {actual} }(o_{i})$. In more detail,

$\sum _{i}T_{\mathrm {amortized} }(o_{i})=\sum _{i}\left(T_{\mathrm {actual} }(o_{i})+C\cdot (\Phi (S_{i+1})-\Phi (S_{i}))\right)=\left(\sum _{i}T_{\mathrm {actual} }(o_{i})\right)+C\cdot (\Phi (S_{\mathrm {final} })-\Phi (S_{\mathrm {initial} }))\geq \sum _{i}T_{\mathrm {actual} }(o_{i}),$

where the sequence of potential function values forms a telescoping series in which all terms other than the initial and final potential function values cancel in pairs, and where the final inequality arises from the assumptions that $\Phi (S_{\mathrm {final} })\geq 0$ and $\Phi (S_{\mathrm {initial} })=0$. Therefore, amortized time can be used to provide accurate predictions about the actual time of sequences of operations, even though the amortized time for an individual operation may vary widely from its actual time.

Amortized analysis of worst-case inputs

Typically, amortized analysis is used in combination with a worst case assumption about the input sequence. With this assumption, if X is a type of operation that may be performed by the data structure, and n is an integer defining the size of the given data structure (for instance, the number of items that it contains), then the amortized time for operations of type X is defined to be the maximum, among all possible sequences of operations on data structures of size n and all operations o_i of type X within the sequence, of the amortized time for operation o_i.

With this definition, the time to perform a sequence of operations may be estimated by multiplying the amortized time for each type of operation in the sequence by the number of operations of that type.

Example

A dynamic array is a data structure for maintaining an array of items, allowing both random access to positions within the array and the ability to increase the array size by one. It is available in Java as the "ArrayList" type and in Python as the "list" type. A dynamic array may be implemented by a data structure consisting of an array A of items, of some length N, together with a number n ≤ N representing the positions within the array that have been used so far. With this structure, random accesses to the dynamic array may be implemented by accessing the same cell of the internal array A, and when n < N an operation that increases the dynamic array size may be implemented simply by incrementing n. However, when n = N, it is necessary to resize A, and a common strategy for doing so is to double its size, replacing A by a new array of length 2n.

This structure may be analyzed using a potential function Φ = 2n − N. Since the resizing strategy always causes A to be at least half-full, this potential function is always non-negative, as desired. When an increase-size operation does not lead to a resize operation, Φ increases by 2, a constant. Therefore, the constant actual time of the operation and the constant increase in potential combine to give a constant amortized time for an operation of this type.

However, when an increase-size operation causes a resize, the potential value of n prior to the resize decreases to zero after the resize. Allocating a new internal array A and copying all of the values from the old internal array to the new one takes O(n) actual time, but (with an appropriate choice of the constant of proportionality C) this is entirely cancelled by the decrease of n in the potential function, leaving again a constant total amortized time for the operation.

The other operations of the data structure (reading and writing array cells without changing the array size) do not cause the potential function to change and have the same constant amortized time as their actual time.

Therefore, with this choice of resizing strategy and potential function, the potential method shows that all dynamic array operations take constant amortized time. Combining this with the inequality relating amortized time and actual time over sequences of operations, this shows that any sequence of n dynamic array operations takes O(n) actual time in the worst case, despite the fact that some of the individual operations may themselves take a linear amount of time.

Applications

The potential function method is commonly used to analyze Fibonacci heaps, a form of priority queue in which removing an item takes logarithmic amortized time, and all other operations take constant amortized time. It may also be used to analyze splay trees, a self-adjusting form of binary search tree with logarithmic amortized time per operation.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.