| 翻訳と辞書 |
| Rothenberg propriety ： ウィキペディア英語版 |
Rothenberg propriety
In music, Rothenberg propriety is an important concept in the general theory of scales which was introduced by David Rothenberg in a seminal series of papers in 1978. The concept was independently discovered in a more restricted context by Gerald Balzano, who termed it ''coherence''. The notion belongs to the class of concepts often, but misleadingly, termed diatonic set theory; in fact, as with most concepts of diatonic set theory, it applies far more widely than simply to the diatonic scale.
== Definition of propriety ==
Rothenberg defined propriety in a very general context; however for nearly all purposes it suffices to consider what in musical contexts is often called a ''periodic scale'', though in fact these correspond to what mathematicians call a quasiperiodic function. These are scales which repeat at a certain fixed interval higher each note in a certain finite set of notes. The fixed interval is typically an octave, and so the scale consists of all notes belonging to a finite number of pitch classes. If ''β''''i'' denotes a scale element for each integer i, then ''β''''i''+''℘'' = ''β''''i'' + ''Ω'', where ''Ω'' is typically an octave of 1200 cents, though it could be any fixed amount of cents; and ℘ is the number of scale elements in the Ω period, which is sometimes termed the size of the scale.
For any ''i'' one can consider the set of all differences by ''i'' steps between scale elements class(''i'') = . We may in the usual way extend the ordering on the elements of a set to the sets themselves, saying ''A'' < ''B'' if and only if for every ''a'' ∈ ''A'' and ''b'' ∈ ''B'' we have ''a'' < ''b''. Then a scale is ''strictly proper'' if ''i'' < ''j'' implies class(''i'') < class(''j''). It is ''proper'' if ''i'' ≤ ''j'' implies class(''i'') ≤ class(''j''). Strict propriety implies propriety but a proper scale need not be strictly proper; an example is the diatonic scale in equal temperament, where the tritone interval belongs both to the class of the fourth (as an augmented fourth) and to the class of the fifth (as a diminished fifth). Strict propriety is the same as ''coherence'' in the sense of Balzano.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
■ウィキペディアで「Rothenberg propriety」の詳細全文を読む
スポンサード リンク
| 翻訳と辞書 : 翻訳のためのインターネットリソース |
Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.
|
|