Harmonic distance in intervals and chords

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 low...

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Bibliographic Details
Published in:Journal of mathematics and music (Society for Mathematics and Computation in Music) Vol. 13; no. 1; pp. 85 - 106
Main Author: Cubarsi, Rafael
Format: Journal Article Publication
Language:English
Published: Baton Rouge Taylor & Francis 02.01.2019
Taylor & Francis Ltd
Subjects:
05 Combinatorics
05C Graph theory
11 Number theory
11A Elementary number theory
54 General topology
54E Spaces with richer structures
Anàlisi matemàtica
chords
Chords (geometry)
Classificació AMS
consonance
dissonance
Espais topològics
Geometria
Grafs, Teoria de
Graph theory
Harmonic analysis
harmonic distance
Instrument tuning
Matemàtica discreta
Matemàtiques i estadística
melodic distance
Nombres, Teoria dels
Number theory
Sensory perception
Teoria de grafs
Tonnetz
Topological spaces
Tuning
tuning systems
Àrees temàtiques de la UPC
ISSN:1745-9737, 1745-9745, 1745-9745
Online Access:Get full text
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Summary: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.
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content type line 14
ISSN:1745-9737
1745-9745
1745-9745
DOI:10.1080/17459737.2019.1608600