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Harmonic distance in intervals and chords
Journal of Mathematics and Music ( IF 1.1 ) Pub Date : 2019-05-15 , DOI: 10.1080/17459737.2019.1608600
Rafael Cubarsi 1
Affiliation  

The harmonic distance between two pure tones, in the sense used by Tenney, is generalised to chords whose pitches are harmonic fractions. In the tonal graph generated by the harmonics involved in a chord, which for n-TET systems has its equivalent in the Tonnetz, the melodic distance between the lowest common ancestor and the lowest common harmonic of the pitches composing the chord is a measure of the relative dissonance. This notion, rooted in just intonation, is extended to Pythagorean tuning and is used as an approximation for equal temperament scales. Harmonic distance and sensory dissonance are compared and discussed for chords in different tuning systems. It is borne out that proximity of chords in the Tonnetz is not exactly related to the harmonic distance.



中文翻译:

间隔和弦的和声距离

在Tenney使用的意义上,两个纯音之间的谐波距离被广义化为音高为谐波分数的和弦。在和弦中所包含的谐波所产生的音调图中,对于n-TET系统,其等效于吨数,构成该和弦的音高的最低共祖先和最低共和谐波之间的旋律距离是对弦的度量。相对不和谐。这个概念植根于语调,并扩展到毕达哥拉斯调律,并被用作相等气质音阶的近似值。比较和讨论了不同调音系统中和弦的谐波距离和感觉失调。可以证明,吨位中和弦的接近度与谐波距离并不完全相关。

更新日期:2019-05-15
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