Plato’s Timaeus and the Missing Fourth Guest

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.

• PrefaceAcknowledgementsList of Figures and TablesIntroduction: Plato’s Missing Fourth Guest1 The Timaeus, the Decad, and the Harmonia: an Overview2 Plato’s Construction of the World Soul: the Text as a Number Generator from 35 A to a Conundrum in 36 B1 Timaeus 35 A2 End of Timaeus 35 A–Beginning of Timaeus 35 C3 Timaeus 35 C and 36 A4 Timaeus 36 A (con’t) and 36 B3 Solving the 36 B Conundrum: Deriving the Set of Sesquitertian Parts to Be Filled by Sesqui-Octave Intervals1 Derivation of the Sesquitertian Parts4 The Sesquioctave Operation within the Sesquitertian Parts1 Deriving Matrix Numbers Not Generable from the 2:8/3 Interval2 Special Mathematical Features of the Number Set Reflected in Table 245 The Musical Significance of Plato’s Number Matrix: the Primary Timaeus Scale1 Numerical Arrangement of the Timaeus Numbers with Key2 The First Cognizable Fourth of Any Kind3 The First Diatonic and Enharmonic Fourth4 The “Model” Octave and the Perfect Disdiapason5 Rise to the Perfect Disdiapason6 First Octave of the Model Diatonic Octave Chain Containing Chromatic Elements7 First Instances of Standard GPS, LPS, Diatonic Unmodulating Perfect System, and Unmodulating Perfect System in All Genera8 First Instance of Properly Timaean GPS, LPS, Diatonic Unmodulating Perfect System, and Unmodulating Perfect System in All Genera9 Possibilities for Modulation among Different Perfect Systems Arising within the Timaeus Numbers10 The Primary Timaeus Scale11 Some Other Modern Interpretations of the Timaeus Numbers and Timaeus Scale12 The Feature of Ascending/Descending Ambiguity in Plato’s Scale13 Significance of the Chromatic Invasion for the Primary Timaeus Scale14 The Orderliness of the Chromatic Invasion within the Primary Scale15 Orderly Rise and Fall of Fifth Periodicity with the Decay of the Primary Scale16 Grammar of Chromaticity in the Rise and Fall of Fifth Periodicity17 Another Look at the Crantor Matrix18 The Decad in the Rise, Wax, and Wane of the Primary Timaeus Scale6 The Further Musical Significance of Plato’s Number Matrix: the Many “Secondary” Timaeus Scales and Asociated Musical Phenomena1 The Many “Secondary” Diatonic Scales Hidden in the Fabric2 The Many Chromatic Timaeus Scales Hidden in the Fabric3 The Many Enharmonic Timaeus Scales Hidden in the Fabric7 The Musical Data of the Timaeus Vis-à-vis the Cutting of the Fabric, the Making of the “Chi,” and the Cosmic Orbits1 Division of the Material2 Forming the χ Figure3 Bending the Arms to Form Circular Shapes4 The Uniform Motion of the Whole without Variation5 Separation of the Arms into an Outer and Inner Circle6 Separation and Definition of the Motions of Same and Different7 Elevation of the Motion of the Same to Primacy8 Sixfold Split of the Inner Movement of the Different, i.e., Octave Movement8 Plato’s Generalization of the Timaean Harmonia in LawsConcluding RemarksAppendicesAppendix 1. Verification of the Diesis Remaining after Insertion of Two Sesquioctave Intervals into a Sesquitertian Part for the Sample Sesquitertian Part 2:8/3Appendix 2. The Archytan Alternative in the Pythagorean SchoolAppendix 3. Greater and Lesser Perfect Systems and Associated QuestionsAppendix 4. Alternative Perfect SystemsAppendix 5. Two Overlapping Sequences of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Chromatic Factors of 1719926784Appendix 6. Two Overlapping Sequences of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Chromatic Nonfactors of 1719926784Appendix 7. Continuously Overlapping and Contiguous Chains of Doubles, Including Coincident Diatonic Octaves within Each, Bounded Entirely by Model Scale Numbers and Their MultiplesAppendix 8. Chromatic Scale TablesAppendix 9. Specification of Trihemitones and Chromatic Scales in Which They ManifestAppendix 10. Enharmonic Scale TablesGlossarySelected BibliographyIndex