Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Reason and perception
- 3 Pitch and quantity
- 4 The ratios of the concords: (1) the Pythagoreans
- 5 The ratios of the concords: (2) Ptolemy's hupotheseis
- 6 Critique of Aristoxenian principles and conclusions
- 7 Ptolemy on the harmonic divisions of his predecessors
- 8 Melodic intervals: hupotheseis, derivations and adjustments
- 9 Larger systems: modulations in music and in method
- 10 The instruments
- 11 The tests
- 12 Harmonics in a wider perspective
- Bibliography
- Index of names
- Index of topics
6 - Critique of Aristoxenian principles and conclusions
Published online by Cambridge University Press: 22 September 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Reason and perception
- 3 Pitch and quantity
- 4 The ratios of the concords: (1) the Pythagoreans
- 5 The ratios of the concords: (2) Ptolemy's hupotheseis
- 6 Critique of Aristoxenian principles and conclusions
- 7 Ptolemy on the harmonic divisions of his predecessors
- 8 Melodic intervals: hupotheseis, derivations and adjustments
- 9 Larger systems: modulations in music and in method
- 10 The instruments
- 11 The tests
- 12 Harmonics in a wider perspective
- Bibliography
- Index of names
- Index of topics
Summary
At the end of 1.7 Ptolemy restates the conclusions he has derived from his hupotheseis, and explains what his next step will be.
From these points we may say in summary that the first multiple and those measured by it are homophones, that the first two epimorics and those composed from them are concordant, and that those of the epimorics that come after the epitritic [4:3] are melodic. The ratio peculiar to each of the homophones and concords has been stated; and of the melodic class the tone has thus simultaneously been shown to be epogdoic [9:8], because of the difference between the first two epimorics and concords. The ratios of the remainder will receive their appropriate definition in the proper places. But now it would be a good thing to demonstrate the clear truth of those that have already been set out, so that we may have their agreement with perception established beyond dispute, as a basis for discussion.
(16.21–31)There is nothing new about his results, of course. What is important is that they have been shown to follow from hupotheseis that are both acceptable to reason and capable of being intelligibly represented as precise, mathematical counterparts of the relevant perceptual impressions. As the final sentence indicates, 1.8 will describe the ways in which the results can be made to display their credentials before the court of perception.
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- Scientific Method in Ptolemy's Harmonics , pp. 88 - 108Publisher: Cambridge University PressPrint publication year: 2001