• 主题
高级搜索  >
历史搜索 清空历史
首页> 外文期刊>Advances in Pure Mathematics >Music as Mathematics of Senses
【24h】

Music as Mathematics of Senses

机译：音乐作为感官数学

获取原文
           
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

It is often said that music has reached its supreme and highest level in the 18th and 19th centuries. One of the main reasons for this achievement seems to be the robust structure of compositions of music, somewhat remindful of robust structure of mathematics. One is reminded of the words of Goethe: Geometry is frozen music. Here , we may extend geometry to mathematics. For the Middle Age in Europe , there were seven main subjects in the universities or in higher education. They were grammar, logic and rhetoric—these three (tri) were regarded as more standard and called trivia (trivium), the origin of the word trivial . And the remaining four were arithmetic, geometry, astronomy and music—these four (quadrus) were regarded as more advanced subjects and were called quadrivia (quadrivium). Thus for Goethe, geometry and mathematics seem to be equivocal. G. Leibniz expresses more in detail in his letter to C. Goldbach in 1712 (April 17): Musica est exercitium arithmeticae occultum nescientis se numerari animi (Music is a hidden arithmetic exercise of the soul, which doesn’t know that it is counting). Or in other respects, J. Sylvester expresses more in detail: Music is mathematics of senses. Mathematics is music of reasons. Thus , the title arises. This paper is a sequel to [1] and examines mathematical structure of musical scales entailing their harmony on expanding and elaborating material in [2] [3] [4] [5] , etc. In statistics, the strong law of large numbers is well-known which claims that This means that the relative frequency of occurrences of an event A tends to the true probability p of the occurrences of A with probability 1. In music, harmony is achieved according to Pythagoras’ law of small numbers, which claims that only the small integer multiples of the fundamental notes can create harmony and consonance. We shall also mention the law of cyclotomic numbers according to Coxeter, which elaborates Pythagoras’ law and suggests a connection with construction of n -gons by ruler and compass. In the case of natural scales (just intonation), musical notes appear in the form 2 ~( p ) 3 ~( q ) 5 ~( r ) (multiples of the basic note), where p∈Z , ? q=-3, -2, -1, 0, 1, 2, 3 and r=-1, 0, 1 . We shall give mathematical details of the structure of various scales.
机译：人们常说音乐在18和19世纪达到了最高和最高的水平。取得这一成就的主要原因之一似乎是音乐创作的稳健结构，多少使人想起了数学的稳健结构。人们回想起歌德的话：几何是冻结的音乐。在这里，我们可以将几何扩展到数学。在欧洲的中世纪，大学或高等教育中有七个主要科目。它们是语法，逻辑和修辞学，这三个（tri）被认为是更标准的，被称为trivia（trivium），这是琐碎单词的起源。其余四个是算术，几何，天文学和音乐-这四个（四合一）被认为是更高级的科目，被称为Quadrivia（四合一）。因此，对于歌德来说，几何学和数学似乎是模棱两可的。莱布尼兹（G. Leibniz）在1712年（4月17日）给戈德巴赫（C. Goldbach）的信中更详细地表达了这一观点：音乐是一种算术运算，它是灵魂的一种隐藏的算术练习，它不知道它在计数）。或者在其他方面，西尔维斯特（J. Sylvester）更详细地表达：音乐是感官的数学。数学是音乐的原因。因此，标题出现了。本文是[1]的续篇，研究了在[2] [3] [4] [5]等中扩展和精心制作的音阶的音乐结构的数学结构。在统计中，大数的强定律是众所周知，这意味着事件 A发生的相对频率趋向于以概率1发生 A的真实概率 p。在音乐中，根据毕达哥拉斯的小数定律，该定律声称只有基本音符的小整数倍才能产生和声与和声。我们还将提到根据Coxeter提出的圈数数定律，该定律详细阐述了毕达哥拉斯定律，并提出了与用尺子和罗盘构造 n边形的联系。在自然音阶（正调）的情况下，音符以2〜（p）3〜（q）5〜（r）（基本音符的倍数）的形式出现，其中p∈Z，？ q = -3，-2，-1、0、1、2、3和r = -1、0、1我们将给出各种尺度结构的数学细节。

著录项

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号