Academic Journal
On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation
| Title: | On the divisions of the octave in generalized Pythagorean scales and their bidimensional representation |
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| Authors: | Cubarsí Morera, Rafael |
| Source: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Publisher Information: | Informa UK Limited, 2023. |
| Publication Year: | 2023 |
| Subject Terms: | Scale temperament, Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica, 4. Education, Generated tone system, Generalized diatonic scales, Well-formed scales, Matemàtica, Mathematics, Myhill's property, Generated scale, Closure condition, Pythagorean tuning |
| Description: | For well-formed generalized Pythagorean scales, it is explained how to fill in a bidimensional table, referred to as a scale keyboard, to represent the scale tones, arranged bidimensionally as iterates and cardinals, together with the elementary intervals between them. In the keyboard, generalized diatonic and chromatic intervals are easily identified. Two factor decompositions of the scale tones, which are particular cases of duality, make evident several properties of the sequence of intervals composing the octave, such as the number of repeated adjacent intervals and the composition of the generic step-intervals. The keyboard is associated with two matrix forms. When they are mutually transposed, the keyboard is reversible, as in the 12-tone Pythagorean scale. In this case, the relationship between the two main factor decompositions is given by an involutory matrix. This is an Accepted Manuscript of an article published by Taylor & Francis Group in Journal of mathematics and music on published online: 30 Apr 2023, available online at: https://www.tandfonline.com/doi/abs/10.1080/17459737.2023.2194301 |
| Document Type: | Article |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1745-9745 1745-9737 |
| DOI: | 10.1080/17459737.2023.2194301 |
| Accession Number: | edsair.doi.dedup.....5cf8dc18c0e6d4909134c304aadbd8bf |
| Database: | OpenAIRE |
| ISSN: | 17459745 17459737 |
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| DOI: | 10.1080/17459737.2023.2194301 |