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4.11.10

On The Mathematics Of Harmony and Music

Mathematics of Harmony


Harmonic function

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of R^n) which satisfies Laplace's equation everywhere on U.


Subharmonic function


In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.

Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is below the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the boundary of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also inside the ball.

Superharmonic functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharmonic functions.


Properties of Harmonic, Subharmonic
and Superharmonic Functions

1.  A function is harmonic if and only if it is both subharmonic and superharmonic.

2.  If PHI is C^2 (twice continuously differentiable) on an open set G in R^n , then PHI is subharmonic if and only if one has ΔPHI >= 0 on G where Δ is the Laplacian.

3.  The maximum of a subharmonic function cannot be achieved in the interior of its domain unless the function is constant, this is the so-called maximum principle.

4.  Subharmonic functions are upper semicontinuous, while superharmonic functions are lower semicontinuous.

5.  A subharmonic function is at any point no greater than the average of the values in a circle around that point, a fact which can be used to derive the maximum principle.

If f is a holomorphic function, then

PHI(z) = log |f(z)|

is a subharmonic function if we define the value of PHI(z) at the zeros of f to be −∞. It follows that

[P_{a}](z) = |f(z)|^a

is subharmonic for every a > 0. This observation plays a role
in the theory of Hardy spaces, especially for the study of H^p
when 0 < p < 1.

In the context of the complex plane, the connection to the convex functions can be realized as well by the fact that a subharmonic function f on a domain that is constant in the imaginary direction is convex in the real direction and vice versa.

Let PHI be subharmonic, continuous and non-negative in an open subset Omega of the complex plane containing the closed unit disc D(0, 1). The radial maximal function for the function PHI (restricted to the unit disc) is defined on the unit circle by

(MPHI)(z) = sup PHI(z) where z = rexp([i]theta), r <= 0 < 1,
i is the imaginary unit sqrt(-1) and theta is a suitable argument for z (theta = arg z)

If P_r denotes the Poisson kernel, it follows from subharmonicity that:

0 <= PHI(z) = PHI(rexp([i]theta))

<= (1/2*(PI)) Integral_{0}^{[2*(PI)]} {(P_r)(theta - t) times
PHI(exp([i]t)) dt, r < 1.

It can be shown that the last integral is less than the value at
exp([i]theta) of the Hardy-Littlewood maximal function PHI* of the restriction of PHI to the unit circle T,

PHI*(exp([i]theta))
= sup_{0 < a <= PI} { (1/2a) times
Integral_{theta - a}^{theta + a} {PHI(exp([i]t) dt } },

so that 0 <= MPHI <= PHI*.

It is known that the Hardy-Littlewood operator is bounded
on [L^(p)](T) when 1 < p < infinity.

It follows that for some universal constant C,

(MPHI^2)_{[L^2](T)}

is less than or equal to

(C^2)*Integral_{0}^{[2*(PI)]} { (PHI[exp([i]theta)])^2 d(theta) }.

If f is a function holomorphic in Omega and 0 < p < infinity, then the preceding inequality applies to PHI = f^(p/2).

It can be deduced from these facts that any function F in the classical Hardy space H^p satisfies:

Integral_{0}^{2*(PI)} { ( sup_{0 <= r < 1}
{ F(rexp([i]theta))} )^(p) d(theta) }

equals

(C^2)*( sup_{0 <= r < 1} Integral_{0}^{2*(PI)}
{ F(rexp([i]theta))}^(p) d(theta) }

With more work, it can be shown that F has radial limits

F(exp([i]theta)) almost everywhere on the unit circle, and
(by the dominated convergence theorem) that (F_r), defined
by (F_r)(exp([i]theta)) = F(rexp([i]theta)) tends to F in [L^(p)](T).


Subharmonic functions on Riemannian manifolds

Subharmonic functions can be defined on an arbitrary Riemannian manifold.

Definition: Let M be a Riemannian manifold, and f : M --> R an upper semicontinuous function. Assume that for any open subset U of M, and any harmonic function f_{1} on U, such that f_{1} <= f on the boundary of U, the inequality f_{1} <= f holds on all U. Then f is called subharmonic.

This definition is equivalent to the one given above. Also, for twice differentiable functions, subharmonicity is equivalent to the inequality

Δf >= 0

where Δ is the usual Laplacian.


I found this reference (in PDF format) on the Internet.  Those who might be interested to study the theory of harmonic functions will probably want to have a look at it.


Mathematics of Music


Diatonic Function

A diatonic function (also chord area), in tonal music theory, is the specific, recognized role of each of the 7 notes and their chords in relation to the (diatonic) key. "Role" in this context means the degree of tension produced by moving away to a note, chord or scale other than the tonic; and at the same time "how" this musical tension would be eased ("resolved") towards the stability of returning to the tonic chord, note, or scale (namely, "function").

"Each degree of the seven-tone diatonic scale has a name that relates to its function. The major scale and all
three forms of the minor scale share these terms." (Benward & Saker)

Three general and inseparable essential features of harmonic function in tonal music are:

1. Position within a gamut (the available collection) of notes determines a note's function
2. Each note within the gamut is a generator and collector of other notes in the gamut, in other words both the root and its chord exercise function, and
3. Exercise and identification of function depends on musical behaviour or structure.

A fourth feature is the ambiguity that arises from the use of the same terms to describe functions across all temporal spans of a hierarchical structure from the surface to the deepest level, and that the longer term or deeper functions act as a center for shorter higher level ones and that the functions of each tend to counteract each other.

"Harmonic function essentially results from the judgment that certain chords and tonal combinations sound and behave alike, even though these individuals might not be analyzed into equivalent harmonic classes," for example V and VII.

"Harmonic function is more about...similarity than equivalence".

Pandiatonic music is diatonic music without the use of diatonic functions.


Functional harmony

The term functional harmony derives from Hugo Riemann and his textbooks on harmony in the late 19th century, with roots back to Jean-Philippe Rameau's theoretical works amongst others. His main idea was to create a comprehensive theoretical basis for understanding the principles of harmonic relationships typical for the Baroque, Classical and Romantic periods.

His work had huge impact, especially where German influence was strong. A good example in this regard are the textbooks by Hermann Grabner.

Riemann's basic theories have since been adopted, refined and elaborated upon by many authors of textbooks in harmony, arranging and composition. Functional harmony is being taught as a basic discipline in music theory all over the western world, though different labels are used. Other terms used in the English and American tradition include Common Practice Harmony (stemming from Walter Piston), Tonal harmony (as used by Allen Forte), and Traditional harmony (as used by Gordon Delamont). Vincent Persichetti describes the 19th century harmonic repertoire as "chords evolving around the tonic pillars" (tonic, subdominant, dominant).

Nonfunctional harmony, the opposite of functional harmony, is harmony whose progression is not guided by function.

Diatonic functions of notes and chords

Each degree of a diatonic scale, as well as each of many chromatically-altered notes, has a different diatonic function as does each chord built upon those notes. A pitch or pitch class and its enharmonic equivalents have different meanings. For example, a C♯ cannot substitute for a D♭, even though in equal temperament they are identical pitches, because the D♭ can serve as the minor third of a B♭ minor chord while a C♯ cannot, and the C♯ can serve as the fifth degree of an F♯ major scale, while a D♭ cannot.

In music theory, as it is commonly taught in the US, there are seven different functions. In Germany, from the theories of Hugo Riemann, there are only three, and functions other than the tonic, subdominant and dominant are called their "parallels" (US: "relatives"). For instance, in the key of C major, an A minor (chord, scale, or, sometimes, the note A itself) is the Tonic parallel, or Tp. (German musicians use only uppercase note letters and Roman numeral abbreviations, while in the US, upper- and lowercase are usually used to designate major or augmented, and minor or diminished, respectively.) In the US, it would be referred to as the "relative minor."

As d'Indy summarizes:

1. There is only one chord, a perfect chord; it alone is consonant because it alone generates a feeling of repose and balance;
2. This chord has two different forms, major and minor, depending whether the chord is composed of a minor third over a major third, or a major third over a minor;
3. This chord is able to take on three different tonal functions, tonic, dominant, or subdominant.

In the United States, Germany, and other places the diatonic functions are:

Function                               Roman Numeral


Tonic                                             I
Supertonic                                     ii
Mediant                                        iii
Sub-Dominant                             IV
Dominant                                      V
Sub-Mediant                                vi
Leading/Subtonic                         vii

German                                                           German abbreviation

Tonic                                                                             T
Subdominant parallel                                                    Sp
Dominant parallel/Tonic counter parallel                     Dp/Tkp
Subdominant                                                                 S
Dominant                                                                      D
Tonic parallel                                                               Tp
Incomplete Dominant seventh diagonally slashed        D7 (Đ7)



Note that the ii, iii, vi, and vii are lowercase; this is because in relation to the key, they are minor chords. Without accidentals, the vii is a diminished vii°.

The degrees listed according to function, in hierarchical order according to importance or centeredness  (related to the tonic): I, V, IV, vi, iii, ii, vii°. The first three chords are major, the next three are minor, and the last one is diminished.

The tonic, subdominant, and dominant chords, in root position, each followed by its parallel. The parallel is formed by raising the fifth a whole tone; the root position of the parallel chords is indicated by the small noteheads.


Functions in the minor mode

In the US the minor mode or scale is considered a variant of the major, while in German theory it is often considered, per Riemann, the inversion of the major. In the late eighteenth-early nineteenth centuries a large number of symmetrical chords and relations were known as "dualistic" harmony. The root of a major chord in root position is its bass note, but, symmetrically, the 'root' of a minor chord in root position is the fifth (for example CEG and ACE). The plus and degree symbols, + and °, are used to denote that the lower tone of the fifth is the root, as in major, +d, or the higher, as in minor, °d. Thus, if the major tonic parallel is the tonic, with the fifth raised a whole tone, then the minor tonic parallel is the tonic with the US root/German fifth lowered a whole tone.

Major


Parallel              Note letter in C                 US name

Tp                         A minor                        Submediant
Sp                         D minor                        Supertonic
Dp                        E minor                         Mediant

Minor

Parallel              Note letter in C                 US name

tP                         Eb major                        Mediant
sP                        Ab major                        Submediant
dP                       Bb major                        Subtonic

Sources:
[1]  Harmonic function (From Wikipedia, the free encyclopedia)
[2]  Subharmonic function (From Wikipedia, the free encyclopedia)
[3]  Harmonic Function Theory (2nd edition, by Sheldon Axler, Paul Bourdon & Wade Ramey
[4]  Diatonic function (From Wikipedia, the free encyclopedia)
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