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In linear algebra , a minor of a matrix A is the determinant of some smaller square matrix , cut down from A by removing one or more of its rows and columns.

Minors obtained by removing just one row and one column from square matrices first minors are required for calculating matrix cofactors , which in turn are useful for computing both the determinant and inverse of square matrices.

If A is a square matrix , then the minor of the entry in the i -th row and j -th column also called the i , j minor , or a first minor [1] is the determinant of the submatrix formed by deleting the i-th row and j-th column.

This number is often denoted M i,j. To compute the minor M 2,3 and the cofactor C 2,3 , we find the determinant of the above matrix with row 2 and column 3 removed.

Minor of order zero is often defined to be 1. For a square matrix, zeroth minor is just the determinant of the matrix. Also, there are two types of denotations in use in literature: In this article, we use the inclusive definition of choosing the elements from rows of I and columns of J.

The complement, B ijk The complement of the first minor of an element a ij is merely that element. In other words, the cofactor expansion along the j th column gives:.

The matrix formed by all of the cofactors of a square matrix A is called the cofactor matrix also called the matrix of cofactors or comatrix:.

Then the inverse of A is the transpose of the cofactor matrix times the reciprocal of the determinant of A:. The transpose of the cofactor matrix is called the adjugate matrix also called the classical adjoint of A.

The above formula can be generalized as follows: Then [ citation needed ]. A simple proof can be given using wedge product.

We will use the following notation for minors: The distinctive sound of the harmonic minor scale comes from the augmented second between its 6th and 7th scale degrees.

While some composers have used this interval to advantage in melodic composition, others felt it to be an awkward leap, particularly in vocal music , and preferred a whole step between these scale degrees for smooth melody writing.

To eliminate the augmented second, these composers either raised the sixth degree by a semitone or lowered the seventh by a semitone. The melodic minor scale is formed by using both of these solutions.

In particular, the raised 6th appears in the ascending form of the scale, while the lowered 7th appears in the descending form of the scale. Traditionally, these two forms are referred to as:.

Using these notations, the two melodic minor scales can be built by altering the parallel major scale.

Composers have not been consistent in using the two forms of the melodic minor scale. Just as often, composers choose one form or the other based on whether one of the two notes is part of the most recent chord the prevailing harmony.

In jazz , only the ascending form of the scale is usually used. In modern notation, the key signature for music in a minor key is typically based on the accidentals of the natural minor scale, not on those of the harmonic or melodic minor scales.

Major and minor keys that share the same key signature are relative to each other. For instance, F major is the relative major of D minor since both have key signatures with one flat.

Since the natural minor scale is built on the 6th degree of the major scale, the tonic of the relative minor is a major sixth above the tonic of the major scale.

For instance, B minor is the relative minor of D major because the note B is a major sixth above D. The figure below shows all 12 relative major and minor keys, with major keys on the outside and minor keys on the inside arranged around the circle of fifths.

Sometimes scales whose root, third, and fifth degrees form a minor triad are considered "minor scales". In the Western system, derived from the Greek modes , the principal scale that includes the minor third is the Aeolian mode the natural minor scale , with the minor third also occurring in the Dorian mode and the Phrygian mode.

The Dorian mode is a minor mode with a major sixth, while the Phrygian mode is a minor mode with a minor second. The Locrian mode which is very rarely used has a minor third but not the perfect fifth, so its root chord is a diminished triad.

From Wikipedia, the free encyclopedia. For the simulated nuclear detonation, see Minor Scale. By convention, in modern notation and for tonal music written since the common-practice period , key signatures are typically only based on a major Ionian mode or minor natural minor or Aeolian mode key, not on modes like the Dorian mode.

Diatonic scales and keys. Tonal Harmony 5th ed. Its Theory and Practice , pg. Tonal Harmony , p. Holt, Rinhart, and Winston.

Modes in Western music. Dorian Phrygian Lydian Mixolydian. Hypodorian Hypophrygian Hypolydian Hypomixolydian. Retrieved from " https: Heptatonic scales Minor scales Modes.

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Minor of order zero is often defined to be 1. For a square matrix, zeroth minor is just the determinant of the matrix. Also, there are two types of denotations in use in literature: In this article, we use the inclusive definition of choosing the elements from rows of I and columns of J.

The complement, B ijk The complement of the first minor of an element a ij is merely that element. In other words, the cofactor expansion along the j th column gives:.

The matrix formed by all of the cofactors of a square matrix A is called the cofactor matrix also called the matrix of cofactors or comatrix:.

Then the inverse of A is the transpose of the cofactor matrix times the reciprocal of the determinant of A:. The transpose of the cofactor matrix is called the adjugate matrix also called the classical adjoint of A.

The above formula can be generalized as follows: Then [ citation needed ]. A simple proof can be given using wedge product.

We will use the following notation for minors: Both the formula for ordinary matrix multiplication and the Cauchy—Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices.

This formula is a straightforward extension of the Cauchy—Binet formula. A more systematic, algebraic treatment of the minor concept is given in multilinear algebra , using the wedge product: Now consider the wedge product.

Using the properties of the wedge product, namely that it is bilinear and. Thus, for instance, the A natural minor scale can be built by lowering the 3rd, 6th, and 7th degrees of the A major scale by one semitone:.

Because of this, the key of A minor is called the parallel minor of A major. The intervals between the notes of a natural minor scale follow the sequence below:.

The natural minor scale is maximally even. Thus, a harmonic minor scale can be built by lowering the 3rd and 6th degrees of the parallel major scale by one semitone.

Because of this construction, the 7th degree of the harmonic minor scale functions as a leading tone to the tonic because it is a semitone lower than the tonic, rather than a whole tone lower than the tonic as it is in natural minor scales.

The intervals between the notes of a harmonic minor scale follow the sequence below:. The scale is called the harmonic minor scale because it is a common foundation for harmonies chords in minor keys.

For example, in the key of A minor, the dominant V chord the triad built on the 5th scale degree, E is a minor triad in the natural minor scale.

The triads built on each scale degree follow a distinct pattern. The roman numeral analysis is shown below. An interesting property of the harmonic minor scale is that it contains two chords that are each generated by just one interval:.

Because they are generated by just one interval, the inversions of augmented triads and diminished seventh chords introduce no new intervals allowing for enharmonic equivalents that are absent from its root position.

That is, any inversion of an augmented triad or diminished seventh chord is enharmonically equivalent to a new augmented triad or diminished seventh chord in root position.

One chord, with various spellings, may therefore have various harmonic functions in various keys. While it evolved primarily as a basis for chords, the harmonic minor with its augmented second is sometimes used melodically.

In this role, it is used while descending far more often than while ascending. The harmonic minor is also occasionally referred to as the Mohammedan scale [4] as its upper tetrachord corresponds to the Hijaz jins , commonly found in Middle Eastern music.

The Hungarian minor scale is similar to the harmonic minor scale but with a raised 4th degree. The scale also had a notable influence on heavy metal, spawning a sub-genre known as neoclassical metal , with guitarists such as Yngwie Malmsteen , Ritchie Blackmore , and Randy Rhoads employing its use in their music.

The distinctive sound of the harmonic minor scale comes from the augmented second between its 6th and 7th scale degrees.

While some composers have used this interval to advantage in melodic composition, others felt it to be an awkward leap, particularly in vocal music , and preferred a whole step between these scale degrees for smooth melody writing.

To eliminate the augmented second, these composers either raised the sixth degree by a semitone or lowered the seventh by a semitone.

The melodic minor scale is formed by using both of these solutions. In particular, the raised 6th appears in the ascending form of the scale, while the lowered 7th appears in the descending form of the scale.

Traditionally, these two forms are referred to as:. Using these notations, the two melodic minor scales can be built by altering the parallel major scale.

Composers have not been consistent in using the two forms of the melodic minor scale. Just as often, composers choose one form or the other based on whether one of the two notes is part of the most recent chord the prevailing harmony.

In jazz , only the ascending form of the scale is usually used. In modern notation, the key signature for music in a minor key is typically based on the accidentals of the natural minor scale, not on those of the harmonic or melodic minor scales.

Major and minor keys that share the same key signature are relative to each other. For instance, F major is the relative major of D minor since both have key signatures with one flat.

Since the natural minor scale is built on the 6th degree of the major scale, the tonic of the relative minor is a major sixth above the tonic of the major scale.

For instance, B minor is the relative minor of D major because the note B is a major sixth above D. The figure below shows all 12 relative major and minor keys, with major keys on the outside and minor keys on the inside arranged around the circle of fifths.

Sometimes scales whose root, third, and fifth degrees form a minor triad are considered "minor scales". In the Western system, derived from the Greek modes , the principal scale that includes the minor third is the Aeolian mode the natural minor scale , with the minor third also occurring in the Dorian mode and the Phrygian mode.

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