【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。更多还原

【Abstract】 The textbook Harmony by Walter Piston has pointed out that: 1) a diminished 7thchord on the sharped II and VI degrees functions as a "non-dominant"; 2) most of the augmented 6th 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 following calculations in temperamental mathematics: The ratio series of a minor 9^th 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 totri-tones. As in chromatic progress,the half-tones within a diminished 7^th chord present in three 1 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-dominant"diminished 7this 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 " × 16/17"and"×20/21"respec-tively. This is the unique nature of the functional motivation in "non-dominant "diminished 7^thchord. Further,there are two counter-functional approaches to derive out augmented 6thchords: 1)to raise the root of the sub-dominant minor a semitone（3/8×17/18=17/48）to generate a strengthenedaugmented-6thin 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 astrengthened augmented 6^th in the ratio series of 1/17:1/24:1/30 in the secondary dominant function.更多还原