Editing Ptolemy (section)

===''Harmonics''===
[[File:Epogdoon.jpg|thumb|upright=.8|A diagram showing [[Pythagorean tuning]].]]
{{see also|Ptolemy's intense diatonic scale}}
Ptolemy's ''Harmonics'' ({{Langx|el|Ἁρμονικόν}}) is a work in three books on [[music theory]] and the mathematics behind musical scales<ref>
{{cite book
 |last=Wardhaugh |first=Benjamin
 |date=5 July 2017
 |title=Music, Experiment, and Mathematics in England, 1653–1705
 |publisher=Routledge
 |isbn=978-1-351-55708-5
 |location=London, UK / New York, NY
 |page=7
 |url=https://books.google.com/books?id=BzcrDwAAQBAJ&pg=PA7
}}
</ref>

''Harmonics'' begins with a definition of harmonic theory, with a long exposition on the relationship between reason and sense perception in corroborating theoretical assumptions. After criticizing the approaches of his predecessors, Ptolemy argues for basing musical intervals on mathematical ratios (as opposed to the ideas advocated by followers of [[Aristoxenus]]), backed up by empirical observation (in contrast to the excessively theoretical approach of the [[Pythagoreans]]).<ref>
{{cite journal
 |last=Barker |first=A.
 |year=1994
 |title=Ptolemy's Pythagoreans, Archytas, and Plato's conception of mathematics
 |journal=[[Phronesis]]
 |volume=39 |issue=2 |pages=113–135
 |doi=10.1163/156852894321052135
 |jstor=4182463  |issn=0031-8868
}}
</ref><ref>
{{cite journal
 |last=Crickmore |first=L.
 |year=2003
 |title=A re-valuation of the ancient science of ''Harmonics''
 |journal=[[Psychology of Music]]
 |volume=31 |issue=4 |pages=391–403
 |doi=10.1177/03057356030314004
 |s2cid=123117827
}}
</ref>

Ptolemy introduces the ''harmonic canon'' (Greek name) or ''[[monochord]]'' (Latin name), which is an experimental musical apparatus that he used to measure relative pitches, and used to describe to his readers how to demonstrate the relations discussed in the following chapters for themselves. After the early exposition on to build and use monochord to test proposed tuning systems, Ptolemy proceeds to discuss [[Pythagorean tuning]] (and how to demonstrate that their idealized musical scale fails in practice). The Pythagoreans believed that the mathematics of music should be based on only the one specific ratio of 3:2, the [[perfect fifth]], and believed that tunings mathematically exact to their system would prove to be melodious, if only the extremely large numbers involved could be calculated (by hand). To the contrary, Ptolemy believed that musical scales and tunings should in general involve multiple different ratios arranged to fit together evenly into smaller [[tetrachord]]s (combinations of four pitch ratios which together make a [[perfect fourth]]) and [[octave]]s.<ref>
{{cite journal
 |last=Barker |first=A.
 |year=1994
 |title=Greek musicologists in the Roman Empire
 |journal=[[Apeiron]]
 |volume=27  |issue=4  |pages=53–74
 |doi=10.1515/APEIRON.1994.27.4.53
 |s2cid=170415282
 |url=https://www.degruyter.com/document/doi/10.1515/APEIRON.1994.27.4.53/html
}}
</ref><ref>
{{cite book
 |last=West |first=Martin Litchfield |author-link=Martin Litchfield West
 |year=1992
 |title=Ancient Greek Music
 |place=Oxford, UK
 |publisher=[[Oxford University Press]]
 |isbn=0-19-814975-1
}}
</ref>
Ptolemy reviewed standard (and [[enharmonic scale|ancient, disused]]) musical tuning practice of his day, which he then compared to his own subdivisions of the [[tetrachord]] and the [[octave]], which he derived experimentally using a [[monochord]] / harmonic canon. The volume ends with a more speculative exposition of the relationships between harmony, the soul (''psyche''), and the planets ([[Musica universalis|harmony of the spheres]]).<ref name=Feke-2012>
{{cite journal
 |last=Feke  |first=J.
 |year=2012
 |title=Mathematizing the soul: The development of Ptolemy's psychological theory from ''On the Kritêrion'' and ''Hêgemonikon'' to the ''Harmonics''
 |journal=Studies in History and Philosophy of Science Part&nbsp;A
 |volume=43  |issue=4  |pages=585–594
 |doi=10.1016/j.shpsa.2012.06.006
 |bibcode=2012SHPSA..43..585F
 |url=https://www.sciencedirect.com/science/article/abs/pii/S0039368112000428
}}
</ref>

Although Ptolemy's ''Harmonics'' never had the influence of his ''Almagest'' or ''Geography'', it is nonetheless a well-structured treatise and contains more methodological reflections than any other of his writings. In particular, it is a nascent form of what in the following millennium developed into the scientific method, with specific descriptions of the experimental apparatus that he built and used to test musical conjectures, and the empirical musical relations he identified by testing pitches against each other: He was able to accurately measure relative pitches based on the ratios of vibrating lengths two separate sides of the same [[monochord|single string]], hence which were assured to be under equal tension, eliminating one source of error. He analyzed the empirically determined ratios of "pleasant" pairs of pitches, and then synthesised all of them into a coherent mathematical description, which persists to the present as [[just intonation]] – the standard for comparison of consonance in the many other, less-than exact but more facile [[meantone temperament|compromise tuning]] systems.<ref>
{{cite journal
 |last=Barker |first=A.
 |year=2010
 |title=Mathematical beauty made audible: Musical aesthetics in Ptolemy's ''Harmonics''
 |journal=[[Classical Philology]]
 |volume=105 |issue=4 |pages=403–420
 |doi=10.1086/657028 |s2cid=161714215
}}
</ref><ref>
{{cite journal
 |last=Tolsa |first=C.
 |year=2015
 |title=Philosophical presentation in Ptolemy's ''Harmonics'': The ''Timaeus'' as a model for organization
 |journal=Greek, Roman, and Byzantine Studies
 |volume=55 |issue=3 |pages=688–705
 |issn=2159-3159
 |url=https://grbs.library.duke.edu/article/view/15395
}}
</ref>

During the [[Renaissance]], Ptolemy's ideas inspired [[Kepler]] in his own musings on the harmony of the world (''[[Harmonices Mundi|Harmonice Mundi]]'', Appendix to Book V).<ref>
{{cite book
 |last=Hetherington |first=Norriss S.
 |date=8 April 2014
 |title=Encyclopedia of Cosmology
 |series=Routledge Revivals
 |volume=Historical, Philosophical, and Scientific Foundations of Modern Cosmology
 |publisher=Routledge
 |isbn=978-1-317-67766-6
 |page=527
 |url=https://books.google.com/books?id=EP9QAwAAQBAJ&pg=PA527
}}
</ref>