We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we describe similarity concatenation compositions, which are special compositions constructed via similarity. We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament. As well as the standard solution via 12 tone equal temperament, we present a solution via 31 tone equal temperament. We then introduce the notion of a connected triheptad, generalising the tonic, subdominant and dominant of the major scale. We next present some examples of the notions previously introduced. Example 1 features a connected triheptad, and Example 2 features a connected triheptad, a similarity concatenation composition, and an intersecting accompaniment. There follows a section on cubist sets, featuring a returning similarity concatenation composition. We then move a conceptual level higher: we consider the concatenation of similarity concatenation compositions via similarity. This is reminiscent of higher dimensional algebra, and there follows a formal approach to higher dimensional relations, together with an example in 31 tone equal temperament using the formalism described earlier. We use the formalism of braids for our higher dimensional relations. We end with a section on musical applications of paths in graphs, generalising the chromatic scale.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 7, Issue 2) |
| DOI | 10.11648/j.ijtam.20210702.12 |
| Page(s) | 30-39 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Scale Subsets, Similarity Composition, Harmonics
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| [15] | W. Turner. Subsets of scales in compositions constructed by similarity. Example 1, 2, 3, 4, 5, 6, 7, 8. http://homepages.abdn.ac.uk/w.turner/pages/ |
APA Style
Will Turner. (2021). Subsets of Scales in Compositions Constructed by Similarity. International Journal of Theoretical and Applied Mathematics, 7(2), 30-39. https://doi.org/10.11648/j.ijtam.20210702.12
ACS Style
Will Turner. Subsets of Scales in Compositions Constructed by Similarity. Int. J. Theor. Appl. Math. 2021, 7(2), 30-39. doi: 10.11648/j.ijtam.20210702.12
AMA Style
Will Turner. Subsets of Scales in Compositions Constructed by Similarity. Int J Theor Appl Math. 2021;7(2):30-39. doi: 10.11648/j.ijtam.20210702.12
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Download@article{10.11648/j.ijtam.20210702.12,
author = {Will Turner},
title = {Subsets of Scales in Compositions Constructed by Similarity},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {7},
number = {2},
pages = {30-39},
doi = {10.11648/j.ijtam.20210702.12},
url = {https://doi.org/10.11648/j.ijtam.20210702.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20210702.12},
abstract = {We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we describe similarity concatenation compositions, which are special compositions constructed via similarity. We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament. As well as the standard solution via 12 tone equal temperament, we present a solution via 31 tone equal temperament. We then introduce the notion of a connected triheptad, generalising the tonic, subdominant and dominant of the major scale. We next present some examples of the notions previously introduced. Example 1 features a connected triheptad, and Example 2 features a connected triheptad, a similarity concatenation composition, and an intersecting accompaniment. There follows a section on cubist sets, featuring a returning similarity concatenation composition. We then move a conceptual level higher: we consider the concatenation of similarity concatenation compositions via similarity. This is reminiscent of higher dimensional algebra, and there follows a formal approach to higher dimensional relations, together with an example in 31 tone equal temperament using the formalism described earlier. We use the formalism of braids for our higher dimensional relations. We end with a section on musical applications of paths in graphs, generalising the chromatic scale.},
year = {2021}
}
TY - JOUR T1 - Subsets of Scales in Compositions Constructed by Similarity AU - Will Turner Y1 - 2021/05/27 PY - 2021 N1 - https://doi.org/10.11648/j.ijtam.20210702.12 DO - 10.11648/j.ijtam.20210702.12 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 30 EP - 39 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20210702.12 AB - We present techniques in musical composition based on subsets of scales, built on theoretical notions, together with a number of examples. The techniques we describe are for constructing compositions with reference to memory, via similarity. We begin with some technical elements: after introducing the technique of intersecting accompaniments, we describe similarity concatenation compositions, which are special compositions constructed via similarity. We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament. As well as the standard solution via 12 tone equal temperament, we present a solution via 31 tone equal temperament. We then introduce the notion of a connected triheptad, generalising the tonic, subdominant and dominant of the major scale. We next present some examples of the notions previously introduced. Example 1 features a connected triheptad, and Example 2 features a connected triheptad, a similarity concatenation composition, and an intersecting accompaniment. There follows a section on cubist sets, featuring a returning similarity concatenation composition. We then move a conceptual level higher: we consider the concatenation of similarity concatenation compositions via similarity. This is reminiscent of higher dimensional algebra, and there follows a formal approach to higher dimensional relations, together with an example in 31 tone equal temperament using the formalism described earlier. We use the formalism of braids for our higher dimensional relations. We end with a section on musical applications of paths in graphs, generalising the chromatic scale. VL - 7 IS - 2 ER -
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