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Platoâs Timaeus and the Missing Fourth Guest
Finding the Harmony of the Spheres
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In Plato's Timaeus and the Missing Fourth Guest, Donna M. Altimari Adler proposes a new Timaeus scale structure. She finds the harmonic cosmos, mathematically, at 35 A-36 D, regarding the text as a number generator. Plato's primary number sequence, she argues, yields a matrix defining a sophisticated harmony of the spheres. She stresses the Decad as the pattern governing both human perception and the generation of all things, in the Timaeus, including the World Soul and musical scale symbolizing it. She precisely identifies Plato's "fabric" and its locus of severance and solves other thorny problems of textual interpretation.
Donna M. Altimari Adler received her M.A. and Ph.D. (systematic and philosophical theology) from the University of Notre Dame. She is also a graduate of Northwestern University Law School and holds an M.A. (Divinity) and B.A. (Linguistics) from the University of Chicago. She has taught, in different capacities, at St. Xavier University, Loyola University, DePaul University, Benedictine University, and the Institute for Lay Formation of the University of St. Mary of the Lake, all in the Chicago area, and has contributed at numerous academic conferences. She is preparing other work for publication.
Preface Acknowledgements List of Figures and Tables
Introduction: Platoâs Missing Fourth Guest
1 The Timaeus, the Decad, and the Harmonia: an Overview
2 Platoâs Construction of the World Soul: the Text as a Number Generator from 35Â A to a Conundrum in 36Â B â1âTimaeus 35Â A â2âEnd of Timaeus 35Â AâBeginning of Timaeus 35Â C â3âTimaeus 35Â C and 36Â A â4âTimaeus 36Â A (conât) and 36Â B
3 Solving the 36Â B Conundrum: Deriving the Set of Sesquitertian Parts to Be Filled by Sesqui-Octave Intervals â1âDerivation of the Sesquitertian Parts
4 The Sesquioctave Operation within the Sesquitertian Parts â1âDeriving Matrix Numbers Not Generable from the 2:8/3 Interval â2âSpecial Mathematical Features of the Number Set Reflected in Table 24
5 The Musical Significance of Platoâs Number Matrix: the Primary Timaeus Scale â1âNumerical Arrangement of the Timaeus Numbers with Key â2âThe First Cognizable Fourth of Any Kind â3âThe First Diatonic and Enharmonic Fourth â4âThe âModelâ Octave and the Perfect Disdiapason â5âRise to the Perfect Disdiapason â6âFirst Octave of the Model Diatonic Octave Chain Containing Chromatic Elements â7âFirst Instances of Standard GPS, LPS, Diatonic Unmodulating Perfect System, and Unmodulating Perfect System in All Genera â8âFirst Instance of Properly Timaean GPS, LPS, Diatonic Unmodulating Perfect System, and Unmodulating Perfect System in All Genera â9âPossibilities for Modulation among Different Perfect Systems Arising within the Timaeus Numbers â10âThe Primary Timaeus Scale â11âSome Other Modern Interpretations of the Timaeus Numbers and Timaeus Scale â12âThe Feature of Ascending/Descending Ambiguity in Platoâs Scale â13âSignificance of the Chromatic Invasion for the Primary Timaeus Scale â14âThe Orderliness of the Chromatic Invasion within the Primary Scale â15âOrderly Rise and Fall of Fifth Periodicity with the Decay of the Primary Scale â16âGrammar of Chromaticity in the Rise and Fall of Fifth Periodicity â17âAnother Look at the Crantor Matrix â18âThe Decad in the Rise, Wax, and Wane of the Primary Timaeus Scale
6 The Further Musical Significance of Platoâs Number Matrix: the Many âSecondaryâ Timaeus Scales and Asociated Musical Phenomena â1âThe Many âSecondaryâ Diatonic Scales Hidden in the Fabric â2âThe Many Chromatic Timaeus Scales Hidden in the Fabric â3âThe Many Enharmonic Timaeus Scales Hidden in the Fabric
7 The Musical Data of the Timaeus Vis-Ã -vis the Cutting of the Fabric, the Making of the âChi,â and the Cosmic Orbits â1âDivision of the Material â2âForming the Ï Figure â3âBending the Arms to Form Circular Shapes â4âThe Uniform Motion of the Whole without Variation â5âSeparation of the Arms into an Outer and Inner Circle â6âSeparation and Definition of the Motions of Same and Different â7âElevation of the Motion of the Same to Primacy â8âSixfold Split of the Inner Movement of the Different, i.e., Octave Movement
8 Platoâs Generalization of the Timaean Harmonia in Laws
Concluding Remarks
Appendix 1. Verification of the Diesis Remaining after Insertion of Two Sesquioctave Intervals into a Sesquitertian Part for the Sample Sesquitertian Part 2:8/3
Appendix 2. The Archytan Alternative in the Pythagorean School
Appendix 3. Greater and Lesser Perfect Systems and Associated Questions
Appendix 4. Alternative Perfect Systems
Appendix 5. Two Overlapping Sequences of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Chromatic Factors of 1719926784
Appendix 6. Two Overlapping Sequences of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Chromatic Nonfactors of 1719926784
Appendix 7. Continuously Overlapping and Contiguous Chains of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Model Scale Numbers and Their Multiples
Appendix 8. Chromatic Scale Tables
Appendix 9. Specification of Trihemitones and Chromatic Scales in Which They Manifest
Appendix 10. Enharmonic Scale Tables
Glossary Selected Bibliography Index
Introduction: Platoâs Missing Fourth Guest
1 The Timaeus, the Decad, and the Harmonia: an Overview
2 Platoâs Construction of the World Soul: the Text as a Number Generator from 35Â A to a Conundrum in 36Â B â1âTimaeus 35Â A â2âEnd of Timaeus 35Â AâBeginning of Timaeus 35Â C â3âTimaeus 35Â C and 36Â A â4âTimaeus 36Â A (conât) and 36Â B
3 Solving the 36Â B Conundrum: Deriving the Set of Sesquitertian Parts to Be Filled by Sesqui-Octave Intervals â1âDerivation of the Sesquitertian Parts
4 The Sesquioctave Operation within the Sesquitertian Parts â1âDeriving Matrix Numbers Not Generable from the 2:8/3 Interval â2âSpecial Mathematical Features of the Number Set Reflected in Table 24
5 The Musical Significance of Platoâs Number Matrix: the Primary Timaeus Scale â1âNumerical Arrangement of the Timaeus Numbers with Key â2âThe First Cognizable Fourth of Any Kind â3âThe First Diatonic and Enharmonic Fourth â4âThe âModelâ Octave and the Perfect Disdiapason â5âRise to the Perfect Disdiapason â6âFirst Octave of the Model Diatonic Octave Chain Containing Chromatic Elements â7âFirst Instances of Standard GPS, LPS, Diatonic Unmodulating Perfect System, and Unmodulating Perfect System in All Genera â8âFirst Instance of Properly Timaean GPS, LPS, Diatonic Unmodulating Perfect System, and Unmodulating Perfect System in All Genera â9âPossibilities for Modulation among Different Perfect Systems Arising within the Timaeus Numbers â10âThe Primary Timaeus Scale â11âSome Other Modern Interpretations of the Timaeus Numbers and Timaeus Scale â12âThe Feature of Ascending/Descending Ambiguity in Platoâs Scale â13âSignificance of the Chromatic Invasion for the Primary Timaeus Scale â14âThe Orderliness of the Chromatic Invasion within the Primary Scale â15âOrderly Rise and Fall of Fifth Periodicity with the Decay of the Primary Scale â16âGrammar of Chromaticity in the Rise and Fall of Fifth Periodicity â17âAnother Look at the Crantor Matrix â18âThe Decad in the Rise, Wax, and Wane of the Primary Timaeus Scale
6 The Further Musical Significance of Platoâs Number Matrix: the Many âSecondaryâ Timaeus Scales and Asociated Musical Phenomena â1âThe Many âSecondaryâ Diatonic Scales Hidden in the Fabric â2âThe Many Chromatic Timaeus Scales Hidden in the Fabric â3âThe Many Enharmonic Timaeus Scales Hidden in the Fabric
7 The Musical Data of the Timaeus Vis-Ã -vis the Cutting of the Fabric, the Making of the âChi,â and the Cosmic Orbits â1âDivision of the Material â2âForming the Ï Figure â3âBending the Arms to Form Circular Shapes â4âThe Uniform Motion of the Whole without Variation â5âSeparation of the Arms into an Outer and Inner Circle â6âSeparation and Definition of the Motions of Same and Different â7âElevation of the Motion of the Same to Primacy â8âSixfold Split of the Inner Movement of the Different, i.e., Octave Movement
8 Platoâs Generalization of the Timaean Harmonia in Laws
Concluding Remarks
Appendices
Appendix 1. Verification of the Diesis Remaining after Insertion of Two Sesquioctave Intervals into a Sesquitertian Part for the Sample Sesquitertian Part 2:8/3
Appendix 2. The Archytan Alternative in the Pythagorean School
Appendix 3. Greater and Lesser Perfect Systems and Associated Questions
Appendix 4. Alternative Perfect Systems
Appendix 5. Two Overlapping Sequences of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Chromatic Factors of 1719926784
Appendix 6. Two Overlapping Sequences of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Chromatic Nonfactors of 1719926784
Appendix 7. Continuously Overlapping and Contiguous Chains of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Model Scale Numbers and Their Multiples
Appendix 8. Chromatic Scale Tables
Appendix 9. Specification of Trihemitones and Chromatic Scales in Which They Manifest
Appendix 10. Enharmonic Scale Tables
Glossary Selected Bibliography Index
Scholars in the fields of Classics, Ancient Philosophy, Ancient Greek Music, Aesthetics, and Music, Musicology, and Philosophy, as well as academic libraries, will be interested in this book.