灵敏的和弦功能感所倚蕴的数理秘骨——由辟斯顿对于减七、增六和弦的功能断言引发的和声功能理论探索  

Mathematical Secrets Hidden in Sensitive Chord Functions: An exploration into the functional theory of diminished 7^(th) and augmented 6^(th) raised by Piston

作　　者：赵宋光[1] Zhao Songguang

机构地区：[1]星海音乐学院

出　　处：《星海音乐学院学报》2017年第1期37-54,共18页Journal of Xinghai Conservatory of Music

摘　　要：辟斯顿《和声学》指出:(1)升Ⅱ级、升Ⅵ级减七和弦具有"非属"功能;(2)增六和弦常用的一类是下属和弦的变体,不常用的较高深的一类才是副属和弦的变体。文章用律学数理方法论证了这两个命题的合理性。论文首先剖析了属小九和弦的波长连比式——1/8∶1/10∶1/12∶1/14∶1/17,指出貌似1.5全音的自然音程有四种,貌似3全音的自然音程也有四种。作为减七和弦解决声部进行的半音,在人们熟知的自然半音(16/15)之外还有"特小二度"(81/20)与"紧自然半音"(17/16)两种。在认知音程关系的基础上,文章指出"非属"减七和弦的波长连比式是"20∶17∶14∶12∶10",其中,有上行半音解决强烈倾向的是相当于17、14的两音位,解决音程各自是"×16/17""×20/21"。这正是"非属"减七和弦功能动力的独特本质所在。文章又指出,生成增六度有两种相反功能的衍生途径:(1)由下属小和弦根音升高半音的衍生途径是3/8×17/18×17/48,生成的增六和弦波长连比式是"30∶24∶17",具有副下属强化动力功能;(2)由重属大和弦五音降低半音的衍生途径是16/27×18/17×32/51,生成的增六和弦波长连比式是"1/17∶1/24∶1/30",具有副属强化动力功能。Abstract： The textbook Harmony by Walter Piston has pointed out that： 1 ） a diminished 7＇h chord on the sharped II and VI degrees functions as a ＂non - dominant＂ ; 2 ） most of the augmented 6＇h chords are variations of the sub - dominant chord, and only those not frequently used are variations of secondary dominant chords. This paper expounds the rationality of the two propositions with the fol-lowing calculations in temperamental mathematics： The ratio series of a minor 9th chord is 1/8：1/10：1/12：1/24：1/17,and there are 4 natural intervals similar to 1. 5 tones and 4 other natural intervals similar to tri -tones. As in chromatic progress, the half- tones within a diminished 7th chord present in three ratios, the natural semitone（16/15）,the ＂special semitone＂ （21/20）,and the ＂tight semitone＂ （17/16）.Based on the recognition of these intervals, the essay points out that the ratio series of ＂non - domi- nant＂ diminished 7th is 20 ： 17 ： 14 ： 12 ： 10, and, amongst, the two degrees tending upwards chromatically are at the positions 17 and 14, with their solving intervals in ＂x16/17＂ and ＂20/21＂ respec-tively. This is the unique nature of the functional motivation in ＂non - dominant＂ diminished 7th chord. Further, there are two counter - functional approaches to derive out augmented 6th chords ： 1 ） to raise the root of the sub -dominant minor a semitone （3/8×17/18=17/48） to generate a strengthened augmented -6th in the ratio series of ＂30 ： 24 ： 17＂ , in the function of a secondary sub - dominant;or 2） to lower the fifth of the dominant of dominant chord a semitone （7/6×18/17=32/51） to generate a strengthened augmented 6th in the ratio series of 1/17：1/24：1/30 in the secondary dominant function.

关 键 词：非属减七和弦 常用的增六和弦——下属和弦变体 高深的增六和弦 ——副属和弦变体 

分 类 号：J614.1[艺术—音乐]