The work is known variously as Aristoxenou (or Aristoxenoy) Armonika (or Harmonika) Stoicheia i.e. Aristoxenou Armonika Stoicheia, Aristoxenou Harmonika Stoicheia etc. All of these translate as The Harmonics of Aristoxenus. Elementa harmonica translates as Elements of Harmonics. The work is otherwise rendered as The Elements, or Elements, the latter translates into Greek as Στοιχεία.
The Elements is held to be the work founding a tradition of the study of music based on practice, which is, to understand music by study to the ear. Musicology as a discipline achieved nascency with the systematic study undertaken in the work, which treated music independently of those prior studies which held it in a position of something purely and only in relation to an understanding of the kosmos. In-as-much, the Elements is the first and earliest work on music in the classical Greek tradition. Earliest considerations arose within the Pythagorean school c.500 and thinking dwelled on the mathematical nature of harmonia. Aristotle, whose Peripatetic school Aristoxenus belonged to, addressed the subject in his work On the Soul. Dewhitt thinks Aristoxenus treatment of the subject was essentially to attempt to describe and locate the elements of the soul, and provide mathematical proofs for these. Aristoxenus is thought contrary to the position of the Pythagoreans, he favoured an intellectual treatment of the subject which Aristotle had set out in his work, which is the exercise of inductive logic with attention to empirical evidence.
Aristoxenus is thought the first to consider music in this respect, as a separate subject, due to this work.
The work is a theoretical treatise concerned with harmony and harmonics, and thus pertains to a burgeoning theory of euphonics. The study of harmonics is especially concerned with treating melody in order to find its components (the Greek word for melody is μέλος).
In the first sentence of the treatise Aristoxenus identifies Harmony as belonging under the general scope of the study of the science of Melody. Aristoxenus considers notes to fall along a continuum available to auditory perception. Aristoxenus identified the three tetrachords in the treatise as diatonic, the chromatic, and the enharmonic.
The general considered attitude of Aristoxenus was to attempt an empirical study based therefore upon observation. In-as-much his writing contains criticisms of predecessing appreciations and attitudes, of the Pythagorean and harmonikoi, on the problems of sound percptable as music.
Editions were published by Meibom, Marquard (1868) Aristoxenou harmonikōn ta sōzomena: Die harmonischen fragmente des Aristoxenus, published in Greek and German translation, and Westphal. Henry Stewart Macran edition was published during 1902 by Clarendon Press, Oxford. An edition was published in Latin during 1954, and another in the same year in Italian, by Typis Publicae Officinae Polygraphicae.
History of scholarship
The Elements was studied seriously and earnestly during the Renaissance, by theoreticians and musicians, because of the necessary choice which Renaissance intellectuals and thinkers had to make of deciding where to make concordance with, of the reality of the theory on music made by either Pythagoras or Aristoxenus. All the events belonging to the Renaissance Period as an approximate whole occurred within a time some time prior to and including the 15th and 16th centuries 
W.B.Stanford' The Sound of Greek (1967) cites the work.
Landels' Music in Ancient Greece and Rome (1999) deals with intervals in The Elements.
Kuntz (2000) thinks Aristoxenus to have provided a superior understanding to the Pythagorean treatment of the harmonic problem.
D Creese 2012 work concerns itself with Aristoxenus' consideration of the perfect fourth.
The work comprises 3 books. Book II seems not to follow from Book I, and it is quite widely but not unanimously assumed that Book I is a separate work from Book II & III.
(1) The Genera - the ways in which the differences between these are determined
(2) Distantia (Intervals) - the distinction of how these are differentiated
(3) Notes - dynameis
(4) Systēmata - enumerating and distinguishing the types, and explaining how they are put together out of Notes and Intervals
(5) Tonoi (Modes) - including the relations between them
(7) Construction / Composition
The term dynamis seems to have been originated by Aristoxenus. Dynamis (dynameis) are conventionally understood to have, amongst other meanings, power and potentiality. Sidoli contends in his review (c.f. ref.) that the initial use of the concept by Aristoxenus was rather "elusive" in the context of the meaning intended by him.
- Rotman Institute of Philosophy Archived 2015-04-21 at the Wayback Machine Western University [Retrieved 2015-05-04]
- M.C. Howatson (22 Aug 2013). The Oxford Companion to Classical Literature. Oxford University Press (reprint). p. 73. ISBN 978-0199548552. Oxford Paperback Reference
- Aristoxenus, Henry Stewart Macran (1902). Harmonika Stoicheia (The Harmonics of Aristoxenus). Georg Olms Verlag. ISBN 978-3487405100. Retrieved 2015-05-04.(and World Cat)
- The Perseus Catalog - Elementa Harmonica Tufts University , University of Leipzig [Retrieved 2015-05-04]
- A.D. Barker (29 Mar 2012). The Oxford Classical Dictionary (edited by S Hornblower, A Spawforth, E Eidinow). Oxford University Press. pp. 163–164. ISBN 978-0199545568. Retrieved 2015-05-08.
- C.H. Kahn (1 Jan 2001). Pythagoras and the Pythagoreans: A Brief History. Hackett Publishing. p. 69. ISBN 978-0872205758. Retrieved 2015-05-04.
- Sophie Gibson - Aristoxenus of Tarentum and the Birth of Musicology (p.6) Routledge, 8 Apr 2014 Studies in Classics ISBN 1135877475 [Retrieved 2015-05-03]
- C.A. Huffman (2012). Aristoxenus of Tarentum: Discussion. Transaction Publishers, 2012. ISBN 9781412843010. Retrieved 2015-05-03.(p.254)
- H Partch (5 Aug 2009). Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments. Da Capo Press. ISBN 978-0786751006. Retrieved 2015-05-04.
- J. Hawkins (1858). General history of the science and practice of music. [With] vol. of portraits, Volume 1. J.Alfred Novello 1858. Retrieved 2015-05-04.
- Mitzi Dewhitt (7 Sep 2004). Aristoxenus's Ghost. Xlibris Corporation. ISBN 978-1465332059. Retrieved 2015-05-03.[self-published source]
- L.M. Zbikowski Associate Professor of Music University of Chicago (18 Oct 2002). Conceptualizing Music : Cognitive Structure, Theory, and Analysis: Cognitive Structure, Theory, and Analysis. Oxford University Press. ISBN 978-0198032175. Retrieved 2015-05-03. AMS Studies in Music Series
- D.M. Randel (2003). The Harvard Dictionary of Music. Harvard University Press, Volume 16 of Harvard University Press reference library. p. 358. ISBN 978-0674011632. Retrieved 2015-05-04.
- Lilian Voudouri
- D Obbink (9 May 1995). Philodemus and Poetry : Poetic Theory and Practice in Lucretius, Philodemus and Horace. Oxford University Press. p. 140. ISBN 978-0195358544. Retrieved 2015-05-04.(additionally using American Heritage® Dictionary of the English Language, Fifth Edition at thefreedictionary.com)
- Cristiano M.L. Forster - Musical Mathematics : on the art and science of acoustic instruments CHAPTER 10: WESTERN TUNING THEORY AND PRACTICE Chrysalis Foundation [Retrieved 2015-05-04]
- R Katz, C Dahlhaus (1987). Contemplating Music: Substance. ISBN 9780918728609. Retrieved 2015-05-04.(p.273)
- J Godwin (1 Nov 1992). The Harmony of the Spheres: The Pythagorean Tradition in Music. Inner Traditions / Bear & Co. ISBN 978-1620550960. Retrieved 2015-05-04.
- A. Barker (13 Sep 2007). The Science of Harmonics in Classical Greece. Cambridge University Press. ISBN 978-1139468626. Retrieved 2015-05-03.(p.187 "Meibom, Westphal")
- S.J. Livesey (John of Reading) (1989). Theology and Science in the Fourteenth Century: Three Questions on the Unity and Subalternation of the Sciences from John of Reading's Commentary on the Sentences. BRILL. p. 25. ISBN 978-9004090231. Retrieved 2015-05-04.
- Aristoxenus, P Marquard - Aristoxenou harmonikōn ta sōzomena: Die harmonischen fragmente des Aristoxenus Weidmann, 1868 [Retrieved 2015-05-04]
- H.S. Macran - The harmonics of Aristonexus The Boston Library Consortium - Northeastern University Libraries [Retrieved 2015-05-04]
- the Internet Archive - Open Library ID : OL14785002M University of Toronto MARC record [Retrieved 2015-05-08]
- I. Kagis McEwen - Vitruvius: Writing the Body of Architecture, MIT Press 2003, p.1 of 493 pages, ISBN 026263306X [Retrieved 2015-12-16]
- D.K.S. Walden - Frozen Music: Music and Architecture in Vitruvius’ De Architectura 2014 Greek and Roman Musical Studies, Volume 2, Issue 1, pages 124 – 145 DOI: 10.1163/22129758-12341255 [Retrieved 2015-05-04]
- J. Prins - Echoes of an Invisible World: Marsilio Ficino and Francesco Patrizi on Cosmic Order and Music Theory BRILL, 28 Nov 2014, 476 pages, History, ISBN 9004281762, Brill's Studies in Intellectual History [Retrieved 2015-12-16]
- P. Van Ness Myers - facsimile of Mediæval and Modern History published by Boston: Ginn and Company, 1905 - p.251 Archived 2015-12-21 at the Wayback Machine published online by Sam Houston State University [Retrieved 2015-12-16]
- Alexander Chalmers - The General biographical dictionary, Printed for J. Nichols 1815, Volume 22 of The General Biographical Dictionary: Containing an Historical and Critical Account of the Lives and Writings of the Most Eminent Persons in Every Nation (Original from the University of Michigan Digitized 31 Aug 2007) [Retrieved 2015-12-16]
- Lilian Voudouri - Music Library of Greece Friends of Music Society (Athens) [Retrieved 2015-05-04]
- M Litchfield (1988). "Aristoxenus and Empiricism: A Reevaluation Based on His Theories". Journal of Music Theory. Journal of Music Theory Vol. 32, No. 1 (Spring, 1988), pp. 51-73 Published by: Duke University Press. 32 (1): 51–73. doi:10.2307/843385. JSTOR 843385.
- N Cazden (1958). "Pythagoras and Aristoxenos Reconciled". Journal of the American Musicological Society. Journal of the American Musicological Society Vol. 11, No. 2/3 (Summer - Autumn, 1958), pp. 97-105 Published by: University of California Press. 11 (2/3): 97–105. doi:10.2307/829897. JSTOR 829897.
- T.G.Rosenmeyer - Elegiac and Elegos (in) California Studies in Classical Antiquity, Volume 1 (p.221) University of California Press & Cambridge University Press (ed. T.S.Brown, W.K.Pritchett)[Retrieved 2015-05-04]
- T.J. Mathiesen (1999). Apollo's Lyre: Greek Music and Music Theory in Antiquity and the Middle Ages. University of Nebraska Press, ACLS Humanities E-Book Volume 2 of Center for the History of Music Theory and Literature Bloomington, Ind: Publications of the Center for the History of Music Theory and Literature. ISBN 978-0803230798. Retrieved 2015-05-04.
- A. Barker - staff profile University of Birmingham [Retrieved 2015-05-04]
- A Briggman (12 Jan 2012). Irenaeus of Lyons and the Theology of the Holy Spirit. Oxford University Press, Oxford Early Christian Studies. ISBN 978-0199641536. Retrieved 2015-05-04.("Distantia & Landels")
- P.G. Kuntz - Whitehead’s Category of Harmony: Analogous Meanings in Every Realm of Being and Culture - part.XX Archived 2015-09-24 at the Wayback Machine Process Studies, pp.43-65, Vol. 29, Number 1, Spring-Summer, 2000. Process Studies is published quarterly by the Center for Process Studies [Retrieved 2015-05-04]
- D Creese - Instruments and Empiricism in Aristoxenus' Elementa harmonica Archived 2015-05-18 at the Wayback Machine Newcastle University - School of History Classics and Archaeology 2012 [Retrieved 2015-05-04]
- Erik Nis Ostenfeld, Plato (The Republic - Ergon and dynamis)- Forms, Matter and Mind Volume 10 of Martinus Nijhoff philosophy library Springer Science & Business Media, 1982 ISBN 940097681X [Retrieved 2015-05-08]
- definitions taken from bible hub - Strong's Concordance & Merriam-Webster [Retrieved 2015-05-08]
- Nathan Sidoli, Andrew Barker (October 2009). "Andrew Barker, The Science of Harmonics in Classical Greece reviewed by Nathan Sidoli, Waseda University". Bryn Mawr Classical Review. Bryn Mawr Classical Review 2009.10.38. Retrieved 2015-05-08.