Theorem of (upwards) Harmonic Subdivision

PDF (English)
Veröffentlicht: Dec 5, 2018
DOI: https://doi.org/10.12681/eml.19245
Demetrios E. Lekkas
The Hellenic Open University, Patras Greece
Abstract

The purpose of this paper is to study the theoretical outcome of an aliquot equidistant fingering / fretting pattern on a string, within standard formality, a. by stating and proving a theorem expressing the exact resulting intervallic structure mathematically –a set of interval ratios between adjacent terms of a mirrored arithmetic sequence, b. by pursuing the complete mathematical properties of this structure, c. by discussing its various applications, with emphasis put on drillings on flutes, d. by calling upon practice in order to spot actual historical manifestations of the theorem and its discussion, e. by hypothesizing the rôle of the particular conditions implicit in the theorem towards the historical emergence and evolution of two primeval prototypes of the most prominent and universally disseminated intonational systems / “ur-scales”: an anhemitonic pentatonic one ([C, D, E¼↑, G, A+]) and the lower fifth of a “smooth” heptatonic one ([D, E¼↓, F+, G, A+]).

Article Details
  • Rubrik
  • Publishing partner
Creative-Commons-Lizenz
Dieses Werk steht unter der Lizenz Creative Commons Namensnennung - Nicht-kommerziell 4.0 International.

Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution Non-Commercial License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g. post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

Authors are permitted and encouraged to post their work online (preferably in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.

Downloads
Keine Nutzungsdaten vorhanden.
Autor/innen-Biografie
Demetrios E. Lekkas, The Hellenic Open University, Patras Greece

Ex Academic, The Hellenic Open University, Patras Greece

Literaturhinweise
Aristoxenus, 1977, Harmonic elements vol. B, series: Ancient Harmonic Writers (Athens Greece, Georgiades publications – Library of the Hellenes) [in Greek].
Boethius, Anicius Manlius Torquatus Severinus, De institutione musica liber III; manuscript: Cam-bridge, Trinity College, c. 505 a.D. [in Latin]. English Transl.: Bower Calvin D., 1989, Fundamentals of music (New Haven Connecticut, Yale University Press).
Chrysanthus of Madytus, Archbishop of Dyrrhachium, 1832, Theoreticum magnum musicæ (Tri-este Italy, Michele Weis Typography) [in Greek].
d’Erlanger, Baron Rodolphe, 1935/2001, La musique Arabe volumes 1-6, (Paris, Paul Geuthner) [in French].
al-Fârâbî, Abû Nasr Mohammad ibn Farûkh, Kitâb al-mûsîqî al-kabir. Transl. into French as Grand traité de la musique. In: d’Erlanger vol. 1.
Helmholtz, Hermann L.F., 1954, On the sensations of tone as a physiological basis for the theory of music, translated, revised, corrected and expanded by Ellis Alexander J. in 1885 (2nd English edition New York: Dover Publications, Inc.) [1st publication in German: 1863].
ibn Sînâ (Avicenna), Abû ’Âli al-Husayn ibn ’Abd-Allâh, Kitâbu’ š-šifâ’. Transl. into French as Avi-cenne mathématiques. In d’Erlanger vol. 2, 103-end.
Jedrzejewski, Franck, Avicenne et le tempérament [in French]. Available online at:
http://perso.club-internet.fr/daschour/micromegas/jedrzejewski/avicen.html (accessed 1 June 2006)
Lekkas, Demetrios, 2003, “On the drilling of flutes”, text # 13 in the collective volume Accompany-ing texts for the course “Arts II: Overview of Greek music and dance” ELP 40, 1-9, (Patras Greece: The Hellenic Open University) [in Greek].
Lekkas, Demetrios, 2006, “The diatonic basis of Byzantine music; systemic structural approach”. In Polyphonia, 8, 7-35 (Athens Greece) [in Greek].
Monzo, Joe, “Philolaus”. Available online at: http://tonalsoft.com/enc/p/philolaus.aspx (accessed 1 June 2006)
Nicomachus Gerasenus, Arithmētikē eisagōgē. Transl. into English in D’Ooge Martin Luther, 1926, Nicomachus of Gerasa, introduction to arithmetic, with notes by F.E. Robbins and L.C. Karpinski, (New York: The MacMillan Co., repr. 1938, Ann Arbor: University of Michigan Press).
Plato, Timæus, 1939, ancient text with introduction, translation (into Modern Greek) and com-ments by Th. Vlyziotis and Ch. E. Papanastasiou, Library of Ancient Greek Writers and Poets, (Athens Greece: Ioannis and P. Zacharopoulos) [in Greek; dialogue composed c. 360 b.C.].
Ptolemy, Claudius, Harmonics. In Barker Andrew (Ed.), 1989, Greek musical writings vol. ii: “Harmonic and acoustic theory”, ch. 11, 270-291, Cambridge Readings in the Literature of Music (Cambridge University Press) [in Greek].
Rameau, Jean-Philippe, 1971, Treatise on harmony, translated, with an introduction and notes, by Philip Gossett (Dover Publications Inc.) [1st publication in French: 1722].
Sachs, Curt, 1943, The rise of music in the ancient world: east and west (New York: W.W. Nor-ton).
Schlesinger, Kathleen, 1939, The Greek aulos: a study of its mechanism and of its relation to the modal system of ancient Greek music, followed by a survey of the Greek harmoniai in survival or re-birth in folk-music (with an introduction by J.F. Mountford), (London: Methuen & Co. Ltd.).
Tannery, Paul (Ed.), 1940, Quadrivium de Georges Pachymère, texte revisé et établi par R.P. Elpidios Stéphanou A.A., Studi e Testi 94, (Città del Vaticano: Biblioteca Apostolica Vaticana) [in Greek].
West, Martin L., 1992, Ancient Greek music (Oxford University Press).
Am häufigsten gelesenen Artikel dieser/dieses Autor/in