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Οι ακόλουθοι όροι αναζήτησης έχουν επισημανθεί: aristoxenus harmonikoi
Cambridge University Press
9780521879514 - The Science of Harmonics in Classical Greece - by Andrew Barker
Index
Index of proper names
Academy, 326
Adrastus of Aphrodisias, 366n, 388n, 444
on classes of ratio 342n
on ratios of pitches 29n, 444n
Adrastus (legendary), 89
Aelianus 376n, 381n
Aetolians, and diatonic music, 70, 71
Agathocles, 73, 78, 87–8
Agenor, 40n, 52, 55, 78, 81–3
Akoumenos, 89
Alcidamas 69n
Alexander Polyhistor 85n
Alexanderson, B. 432n
Alypius, on notation, 61–6
Anacreon, 82
Anaxagoras, 327
Anaximenes, on ‘dense’ and ‘diffuse’ 25n
Anderson, W. D. 11n, 47, 47n, 48n, 51n
Annas, J. 311n
Apollo, 72
Archestratus 437n
on notes in the pyknon 217n
Archilochus, 85–6
Archytas, 113, 115, 143n, 143–5, 269, 287–307, 311, 321, 327, 329, 332–4, 336–7, 342, 343–4, 361, 364, 375, 390, 409, 440–2
and cosmology 305–6, 306n
on the Delian problem 304n
his divisions of the tetrachord, 288, 292–302, 352, 403–4
and genera, 38, 292–302
and mathematical (‘rational’) principles 288–307
as a mathematician, 304
and means and proportions, 284, 302–3, 332, 410
and musical practice, 294–302, 306–7, 443
personal qualities and achievements, 287
and Plato 287, 307, 316n
Ptolemy on, 287–302
references to instruments, 26
on sound and pitch, 27–9, 305–6, 322, 373, 419–21
his theorem on epimoric ratios, 272, 303–5, 351, 356, 382
Arimnestus, and the monochord 81n
Aristides Quintilianus, 397n, 398n, 400n, 435n, 442–3
on an ‘ancient’ notation, 64
Aristoxenian and Pythagorean elements in 441, 442n
on dynamis, 184, 188
on enharmonic 39n
on melopoiia, 230
the three ēthē of, 255–6
on notes in the pyknon 217n
on the paiōn epibatos 245n
on pre-systematic harmoniai, 45–52, 83, 393, 398
sources and transmission of his evidence, 45–8
on systematised harmoniai 44, 44n
on tonoi 216n, 225n, 226, 226n, 442n
Aristonikos 82, 98n
Aristophanes 6n, 83, 100n, 316n, 324n
Aristotle, 11, 33, 34n, 46, 114, 328–63, 364, 389n, 390, 409, 414n, 421, 426
Analytica posteriora 58n, 97, 105–12, 153, 167, 168, 173, 193n, 193–4, 198, 200, 292n, 353–61, 364, 377n, 410n
Analytica priora, 106
De anima, 28n, 173n, 241n, 332, 373n, 376n, 429–30
De caelo, 271n, 282, 283n, 286n, 323
Eudemus 329, 329n
frag. 47 Rose, 329–38
frag. 48 Rose 332n
De generatione animalium, 28n, 184n, 419
De generatione et corruptione 340n
Metaphysics 6n, 34n, 193n, 269n, 282n, 286, 349–53, 375n, 390n, 424n
Nicomachean Ethics 242n
De philosophia 329, 329n
Physics 112, 159, 379n
Poetics 6n, 235n
Politics 97, 100n, 234n, 244n, 245n, 248n, 251, 252, 253, 328,
Aristotle (cont.)
337–8, 393, 433
De sensu 34n, 97n, 289n, 291n, 338–48, 349–50, 362, 375n, 376n, 425n
On the Timaeus and Archytas, 333–4
and Archytas, 329, 332–4, 336–7
on concords and colours, 289n, 338–48, 362
on contemporary musical experts, 328
on dialektos 241n
on the diesis 349–53, 389n, 424, 425n
on dynamis, 184, 329
on ēthos, 245
fragment on harmonia, 329–38
its affinities with work of other writers, 331–3
analysis of, 334–8
its contexts in Aristotle and in [Plut.] De musica, 329–34
as expounding Pythagorean theories, 333–4
on harmoniai, 253, 292, 393
on harmonics, 328–63
empirical, 58n, 97, 104, 168, 292
mathematical 97, 167, 168, 286, 292, 328, 329–48, 410n
mathematical and empirical, relations between, 353–61
its methods and concepts, 328–9, 349–61
its purposes, 362
on the ‘harmony of the spheres’, 283n, 323
on hierarchies of skills 242n
on ‘mixed’ sciences, 353–61
on musical ethics 72, 248n
on music’s social utility, 251
names for ‘harmonics’, ‘metrics’ 6n
on nature (physis), 159–60
on Olympus 100n, 245n
on percept and concept, 173
on Philoxenus 244n
and Plato’s lecture on the good, 251
on Pythagoreans, 263, 282–3, 286, 329–38, 362
on ratios of concords 289, 291n
on scientific demonstration, 105–12, 168, 193–4, 198, 353–61, 377–8
silent on genera, 38
on sound and pitch, 28n, 373, 417, 419, 429–30
use of term harmonikos, 37
[Aristotle] De audibilibus 367n, 373n, 374n, 429n
[Aristotle] Problemata, 97n, 276n, 291n, 345–6, 348, 375n, 429n, 429–30
Aristoxenus, 12–16, 321, 327, 328–9, 331, 364, 373, 377–8, 388n, 390–1, 400, 404–6, 409, 410, 413, 432n, 434
On auloi 115n
Comparisons, 115
Elementa rhythmica, 7n, 115, 238
On instruments 115n
On Music, 115
On the piercing of auloi 115n
Praxidamantia, 115
Sympotic Miscellanies, 115
Elementa harmonica
composition and structure of, 113–35
as comprising Principles and Elements, 116n, 130, 134–5, 163, 164, 227
concepts, methods and aims, in Book I, 136–64
in Book II, 165–96
in Book III, 197–228
contexts and purposes of, 229–59
digressions in, theoretical and non-theoretical, 125–9, 133, 165, 213–14
discussed and quoted by Porphyry, 130n, 134–5
incompleteness of, and the missing portion, 115, 197, 200, 207–8, 215–28
as lectures, 122, 123
‘overlapping’ passages of Books I–II, 122, 124–34
programmes of Books I–II, 117–21
of Book II, 165
relations between the three books, 115–35
between Book III and the others, 121–2
as remnants of two treatises, 116–17
relations between them, 121, 123–34
as a single treatise 116n
and training for composers, 229–33
and Aristotle’s Lyceum, 114, 122, 123, 229, 231
and Aristoxenians, discussed by Ptolemaïs and Didymus, 438–40
on the harmonikoi see Agenor, Eratocles, Pythagoras of Zakynthos, and
General index s.vv. harmonics (empirical), harmonikoi
life and writings, 113–15
on mathematical harmonics, 143, 166–8, 390–1
musical writings, 115
as a musician, 113, 230
his musical conservatism, 46, 102, 230, 393
his names for ‘harmonics’, ‘metrics’, ‘rhythmics’, ‘organology’ 6n
and the nature and goals of harmonics, 180–3
in Nicomachus, 441
non-musical writings, 114–15
on percept and concept, 173–4
as a philosopher, 113–14, 258–9
on his predecessors, see General index s.v. harmonikoi
his public lectures, 231–3
and Pythagoreans, 113–14, 115, 166–8
contrasts and combinations with, 440–2
as source for Aristides Quintilianus, 46–8
and his students, 122, 123, 132–3, 251–2
and Theophrastus, 422–4, 426n, 427–8, 431, 435–6
Artemon, 74, 82
Athena, 83
nomos of 179n
Athenaeus of Athens (composer), 15n, 100n, 398–9
Athenaeus of Naucratis (author) 74, 78n
on Lamprocles, 83
on Stratonikos, 75
Avezzù, G. 69n
B Club 338n
Bacchius, 400n, 442–3
on harmoniai 44, 44n
on notes in the pyknon 217n
on types of composition 255n
Barbera, A., 366, 366n, 376n, 380–1, 382n, 383n, 386n, 391–2, 393, 394, 403
Barker, A. 7n, 21n, 36n, 48n, 80n, 85n, 97n, 101n, 145n, 224n, 225n, 232n, 271n, 288n, 291n, 296n, 301n, 306n, 308n, 317n, 325n, 330n, 342n, 344n, 347n, 350n, 367n, 373n, 377n, 392n, 408n, 414n, 430n, 434n, 442n, 444n
Barker, J. A. 223n, 295n
Barnes, J. 106n, 108n, 112, 155n
Bélis, A. 116n, 117, 119n, 120n, 134, 134n, 296n
Blumenthal, A. 98n
Blumenthal, H. von 438n
Boccadoro, B. 11n
Boethius, 382, 398, 437–8
and Nicomachus, 445, 446–7
on Philolaus, 271–5, 277–8, 281–2
credentials of his evidence, 271
Bolton, R. 106n, 155n
Bowen, A. C., on Archytas 29n
Bower, C. M. 281n, 382n, 398n, 447n
Brancacci, A. 69n
Burkert, W. 20n, 81n, 263, 304n
on Archytas 29n, 269n, 299n
on Hippocratics 281n
on numerology 282, 282n
on Philolaus 263n, 264, 271n, 272n, 273, 278n, 279n
Burnyeat, M. 317n
Byzantium, 438
Calcidius, 444
Callias, 74
Campbell, D. A. 79n, 80n, 83n
Cassin, B. 70n
Cleonides, 367n, 397n, 400n, 442–3
on concords 376n
on dynamis, 184, 188
on harmoniai 44, 44n
on melopoiia, 230
the three ēthē of, 255–7
on modulation 15n
on notes in the pyknon 217n
quotes Ion 98n
on tonoi 216n, 217–22, 226n
Creese, D. E. 27n, 81n, 306n, 409n, 442n
Damon 47, 72–3, 73n, 74, 87, 252, 310, 434n
Damoxenus 77n
Da Rios, R. 185n, 193n, 232n
Delian problem 304n
Delphi, 100n, 326, 398–9
Didymus (musical theorist) 288n, 442n
on distinctions between ‘schools’ of harmonics, 438–40
his divisions of the tetrachord, 293n, 296n, 438
and the monochord, 438
and Ptolemy, 438, 440n, 443–4
Diocles (historian of philosophy) 173n
Diocles (Pythagorean), 114
Diogenes Laertius, 333
on the ‘last’ Pythagoreans, 114
on persons named ‘Pythagoras’ 75n
on Theophrastus, 411
Dionysius of Halicarnassus, 437
Dionysus, 72
Dolopes, and diatonic music, 70
Dracon, 78
Düring, I. 217n
Duris, on Simos and the monochord 26n, 81n, 409n
Echecrates, 114
Einarson, B., and P. H. De Lacey 244n, 259n, 335n
Empedocles
on colour 142n
on ‘dense’ and ‘diffuse’ 25n
on harmonia, 265
Epicurus, Epicureans 77n, 170, 185n
Epigonus, 52n, 78, 80, 81, 103
Eratocles, 40n, 43–55, 74, 78–9, 83–4, 88, 150, 152–4, 194–5, 223, 224
his treatise, 54
Eratosthenes, 306n, 437n, 438n, 442n, 445
Erinyes 282n
Eryximachus, 72n, 89, 279n, 280n, 345
Euclid, 107n, 197, 367–8, 384
[Euclid] Sectio canonis, 46, 107n, 358–60, 364–410
its aims, sources, strategies and readers, 382–4, 406–10
its author, 367–8
[Euclid] Sectio canonis (cont.)
on concords and their ratios, 289–90, 291n, 348, 357–8, 359–60, 372, 375–8, 384–8
on construction of ditones, 93n, 294n, 391–2, 409
its date, and supposedly later additions, 358, 364–5, 366–70, 391–4, 403–6
and diatonic, 276n, 290n, 392, 403–6
and division of the kanōn, 365–6, 369–70, 394–408
and empirical harmonics, 359–60, 389–91
and enharmonic, 369, 388–90, 391–4, 403–6
its introduction, 368–9, 370–8
limitations of, 409–10
and the kanōn, 365–6
division of, 365–6
necessity and ‘reasonableness’ in, 377–8
in Porphyry and Ptolemy, 289–90, 366–70, 391
and Pythagoreans, 373, 410
on sound and pitch, 28n, 370–5, 408, 416–18
structure of, 366
its theorems
harmonic, 359–60, 384–94
logical flaw in 357–8, 368n
mathematical, 356, 378–84
its title, 365
Euripides 39n, 89, 245n, 256n
Orestes score, 398–400
Eurytus, 114, 283
Eutocius 304n
Fortenbaugh, W. W., 411
Gamberini, L. 11n
Gaudentius, 442–3
Aristoxenian and Pythagorean elements in, 441
on experience as prerequisite for the study of harmonics, 181–2
Germain, G. 282n
Gerson, L. 448n
Gibson, S. 232n
Glaucon (in Republic), 23–5, 34, 77, 309–11, 315, 424–7
Glaucus of Rhegium, 78, 84–6, 98
as musical historian and theorist, 85–6
as public performer, 84–5
Gombosi, O. J. 226n
Gorgias, 77
his speeches at festivals 82n
Gottschalk, H. 330n, 367n, 373n, 414n, 429n
Hagel, S. 83n, 189n
Hankinson, R. 353n
Heraclides of Pontus (fourth century BC) 46, 330, 373n
Heraclides of Pontus (first century AD) 373n
Heraclitus, 265, 345
Hesiod, 87
Hesychius, 333
Hibeh Papyrus on music 40n, 69–73, 77, 233, 292, 393n, 426n
Hieron I of Syracuse, 82
Hippasus, 84–5, 306
Hippias, 73, 77
Hippocrates, Hippocratics
on abstract principles 104n
on ‘dense’ and ‘diffuse’ 25n
on harmonia and concords in embryology, 279n, 280–1
on ‘imitation’ 53n
Homer, 85, 87
Homeric Hymn to Hermes 276n, 316n
Huffman, C. A.
on Archytas 29n, 269n, 287n, 293n, 294n, 295n, 298n, 299n, 301n, 302n, 303n, 304n, 305, 305n, 307n, 316n, 333n, 364n
on Philolaus, 263n, 264, 271n, 273, 275n, 279n, 284
Iamblichus, 20, 283–5
Ion of Chios 98, 276n
Irwin, T. 155n
Isocrates 52n, 70n, 82, 255n
Jan, K. 366n, 367n, 371n, 394n
Joly, R. 280n
Juba (king and historian), 78n, 80
Kahn, C. H. 20n, 155n, 263n, 279n, 283n, 287n
Kerferd, G. B. 70n
Kirk, G. S., J. E. Raven and M. Schofield 279n
Knorr, W. 304n, 382n
Laloy, L. 193n, 376n
Lamprocles, 49–50, 78, 83–4, 87
Lamprus, 102
Landels, J. G. 59n
Lapini, W. 69n
Lasus, 19–20, 78, 79–80, 103
Lennox, J. 353n
Levin, F. R. 98n
Limenius, 398–9
Lippman, E. A. 11n, 413n, 414n, 434n
Lloyd, G. E. R. 106n
Long, A. A. 438n
and D. N. Sedley 170n, 173n, 185n
Lycaon 398n
Lyceum, 114, 122, 123, 229, 411, 420
Lysander of Sicyon, 80
Maas, M., and J. M. Snyder 74n
Macran, H. S. 61n, 137n, 169n, 231n, 252n
Mathiesen, T. J. 437n, 447n
Medusa, 347, 348
Melanippides, 79
Melissus, on ‘dense’ and ‘diffuse’ 25n
Metellus (or Megillus), 78
Middle Ages, 437
Mourelatos A. P. D. 315n
Moutsopoulos, E. 11n
Muses, 325
Mysians, of Philoxenus, 244, 253
Netz, R. 183n
Nicias 310n
Nicomachus 20, 326, 376n, 382n, 398, 398n, 408n
and Aristoxenus, 441
and Boethius, 445, 446–7
on concords 344–5, 376n
and definitions of ‘note’ 145n
and Philolaus, 271n, 273, 275, 276–7, 283
his writings on harmonics, 445–7
Olympus (musician), 85, 98–101, 245–6, 297, 405–6
his nomos of Athena, 245–6, 253
and the spondeiakos tropos, 246
survival of his music 100, 100n, 245n
Orpheus, 85
Orphic poetry 182n
Palisca, C. 438n
Panaetius the Younger 414n, 448n
Paquette, D. 74n
Parmenides, on ‘dense’ and ‘diffuse’ 25n
Pausanias, on Pronomus 56n
Pearson, L. 7n, 115n
Pericles, 47, 47n, 73n, 87, 88, 89
Persian Wars 309n
Perusino, F. 40n
Phaenias, on Stratonikos, 75–6, 103
Phanton, 114
Pherecrates 324n
Philochorus 80n
Philodemus, 432n, 437
Philolaus, 114, 263–86, 290, 305, 306, 311, 321, 331, 332–3, 336, 345, 409
Boethius on, 271–5
frag. 6a, 263–71
its argumentative structure, 266–71
and Boethius’ evidence, 273–5, 277–8, 281–2
and diatonic, 276–7
and enharmonic 273, 278, 292n
form of its attunement, 275–8
and other Philolaan fragments, 278–80
its terminology, 264–7
on limiters and unlimiteds 279–80, 312n
and means and proportions, 283–5
his purposes, and cosmology, 275, 278–86, 305–6
Philoxenus, 102, 244, 244n, 247, 248–9, 253
Phleious, Pythagoreans at, 114
Phrynis, 79, 104
Pindar 87, 102, 247, 248–9, 276n
Pisistratids, 82
Plato 11, 18, 269, 291, 306, 308–27, 332–3, 375, 414n
Alcibiades 87n
Charmides, 313
Cratylus 316n
Euthyphro, 313
Gorgias 6n, 139, 312n
Hippias Minor 82n
Ion 100n, 245n
Laches, 36n, 78, 87n, 308n, 310n, 313
Laws, 24n, 72n, 90–1, 96–7, 167n, 239–40, 245n, 250, 252, 309–10, 325, 326
Lysis 173n, 256n, 308n
Meno, 111n, 142n, 313, 314
Minos 100n
Phaedo 84, 308n
Phaedrus, 6n, 36, 88–90, 96–7, 259
Philebus, 33, 35–7, 57n, 90n, 96–7
Protagoras 73, 74, 78, 87, 308n
Republic, 6n, 23–9, 33, 34–7, 42, 45, 47, 72–3, 77, 92, 96–7, 137n, 245n, 248n, 250, 252, 307, 308–18, 325, 326, 328n, 332, 349, 350n, 355, 381n, 390n, 393, 406, 424–7
Seventh Letter 287, 312n
Symposium, 72n, 100n, 279n, 280n, 308n, 316n, 345
Theaetetus 414n
Timaeus, 3, 25n, 28n, 72n, 143n, 143–5, 167, 276n, 285, 289n, 290n, 307, 310, 311, 318–26, 331, 332, 333–4, 342, 344, 346, 352, 373n, 375n, 376n, 381n, 390, 406, 409, 419, 434–5, 444–5
on anamnēsis 111n
and Archytas 287, 307, 316n
Aristoxenus’ biography of, 115
on aulos-playing 57n
on composers and harmonics, 90–1
on ‘correctness’ and ‘excellence’, 239–40, 241
and Damon, 47, 72–3
on dianoia 173n
and diatonic, 276n, 292, 352
distinguishes harmonics, metrics, rhythmics 6n
distinguishes two approaches to harmonics, 23, 310–11, 355
and division of the tetrachord, 290
Plato (cont.)
on dynamis, 184
his early writings, 308
on empirical theorists, 23–5, 34–7, 68–9, 77, 92, 96–7, 104, 310–11, 349, 424–7
on ēthos, 245
and genera, 38, 406
on harmoniai, 45, 47, 292, 330, 393
his harmonic theories, 311, 315–27, 381
their intended audience, 326–7
their philosophical background, 311–15
on harmonikoi and mousikoi, 88–90
on knowledge and reality, 312–15
his lecture on the good, 251
life and writings 113–15, 312n
on mathematical theorists, 25–9, 307, 310–11
and means and proportions, 284, 285, 410
on melos 137n
on mimēsis 240n
on musical ethics, 72–3, 248n, 250, 308–11
on musical judgement, 90–1, 239–40, 241
and philosophy, 258–9
on musical sophists, 73
names for ‘harmonics’ 6n
on numbers as the subject of harmonics, 315–18
on Olympus 100n, 245n
in [Plut.] De musica, 330, 331
and Pythagorean harmonics, 29n, 310–11, 318
on ratios of concords, 289
on sound and pitch, 28n, 373, 417, 419
[Plato] Sisyphus, 76–7
Platonism, and harmonics in later antiquity, 444–5
Pliny 82n
Plotinus, 447–8
Plutarch, 326
on enharmonic in tragedy 40n, 393n
on the Timaeus 444, 444n
[Plutarch] De musica
Aristotelian fragment in, 329–38
chapters surrounding it, 330–3
on Aristoxenus, 46
his views on musical judgement, 235–59
on contemporaries of Aristoxenus, 65–6, 91–6
on enharmonic in tragedy 40n, 393n
and ‘following’ music, 173, 175
and Glaucus of Rhegium, 78, 85–6, 98
on Lamprocles and Mixolydian, 49, 78, 83–4
on mathematical harmonics, 330, 331–3
on musical ‘inventors’, 76
on Olympus, 98–9
on the ‘New Music’, 79
on Plato, 330, 331
on poiētikē, 139
on Pythoclides 87n
on spondeion and spondeiazōn tropos, 92, 100, 101n, 399–400
Pöhlmann, E., 62n, 64
and West 15n, 39n, 100n, 321n, 399n
Pollux, on the epigoneion 80n
Polymnastus, 114
Polystratus, 185n, 188
Porphyry 20, 28n, 122, 124n, 264n, 375n
on Archestratus 217n
on the Aristoxenian conception of vocal movement 144n
and Aristoxenus’ Elementa harmonica 134–5, 137n
quotes Aristoxenus on tetrachordal divisions 130n
his commentary on Ptolemy, its nature and purposes, 447–9
on Philolaus, 270
quotes Ptolemaïs and Didymus, 438–40
on Ptolemy’s ‘plagiarism’, 438, 440n, 448
on schools of harmonic theory 52n, 79n
and the Sectio canonis, 366–70
on Theophrastus, 411–14, 433, 436
Porter, J. 20n, 41n, 79n, 235n
Potiron, H. 61n
Pratinas, 102, 247
Presocratics, 24–5, 53, 265, 279, 318–23, 327, 345
Priscian, on Theophrastus 429n
Privitera, G. A. 19n, 79n
Proclus, 326, 367n, 444
Prodicus, 66, 77
Pronomus, and auloi 56n
Prophrastus 398n
Protagoras, 73, 77, 87
Psellus, on enharmonic in tragedy 40n, 393n
Ptolemaïs 388n, 437n, 443n
on ‘schools’ of harmonics, 438–40, 443
Ptolemy 306n, 332, 361, 388n, 397n, 429n, 430n
on Archytas, 287–302
on Aristoxenus, 441–2
and astronomy, 444
on auloi 57n
on chromatic 299n
on concords and ratios 342–4, 347, 375n, 377n
and Didymus, 438, 440n, 443
on dynamis, 184
and thesis, 188
on enharmonic 39n
and experimental tests, 407–8, 443–4
on melodic accuracy 412n
on modulation, 219
on the monochord 407n
and musical practice, 443–4
on perfect systems 16n
on pitch 28n
Porphyry’s commentary on, 447–9
principles of his analyses
rational, 288–9, 291
rational and perceptual, 441–2, 443–4
on Pythagoreans, 342–4, 367, 441–2
and the Sectio canonis, 289–90, 367–70
on the soul, 444
and the theorem on epimoric ratios 304n
on tonoi, 216n, 224–7
on types of composition 255n
on the vocal mechanism 412n
on weights and pitches 22n
Pythagoras of Samos (philosopher), 19–20, 446–7
Aristoxenus on, 114, 115
and the ‘harmonious blacksmith’ 22n
and the monochord 26n
Pythagoras of Zakynthos (musician and theorist), 40n, 52, 55, 74–5, 77, 78, 81–3
Pythagoreans, 11, 20, 22, 33, 37, 40, 58–9, 88, 263–86, 287–307, 329–38, 354, 367, 373, 375
see also Archytas, Eurytus, Hippasus, Philolaus
and acoustics, 80
and Aristoxenus, 113–14, 115, 146, 167
contrasts and combinations with, 440–2
on concords and ratios, 342–4
discussed by Ptolemaïs and Didymus, 438–40
extremists and moderates, 440, 443
fundamental theses of, 59
and the ‘harmony of the spheres’ 283n, 286n
and later harmonic theory, 445–7
and measurement of intervals, 25–30
in mainland Greece, 114
and musical practice, 295
and number-symbolism, 281–3
Plato and Pythagorean harmonics, 25–9, 310–11, 318
Pythoclides, 73, 78, 87–8
Rankin, H. D. 70n
Renaissance, 438
Robbins, F. E. 368n
Rocconi, E. 71n, 267n, 269n
de Romilly, J. 70n
Rossi, L. E. 11n
Sachs, C. 81n
Sappho 87n
Sectio canonis see [Euclid]
Sedley, D. N. see Long
Sextus Empiricus 397n
Sicking, C. M. J. 412n, 414n, 415n, 417n, 419n, 422n, 423–4, 426n, 427n, 433n, 434n
Sifakis, G. M. 82n
Simonides, 87
Simos, 26n, 81
Smith, R. 106n
Snyder, J. M. see Maas
Socrates, 23, 34, 35–6, 47, 66–7, 74, 76–7, 88–90, 90n, 142n, 250, 307, 308n, 309–11, 315–16, 327
Aristoxenus’ biography of, 115
see also Plato
Solomon, J. 218n, 226n, 367n
Sophocles, 89
Sorabji, R. 350n
Stesichorus, 83n, 85–6
Stobaeus, 273, 279
Stoics, on percept and concept, 173–4
Strasburger, H. 98n
Stratonikos, 75–7, 78
and harmonic theory, 75–6, 86–7, 103
Strauss, R. 254n
Striker, G. 438n
Suda
on Aristoxenus 114, 114n
on Lasus, 79
Tannery, P. 273, 371n
Taras (Tarentum) 113, 296n
Tarrant, H. 448n
Telephanes, and auloi 56n
Telesias, 102, 247–9, 257–8
Telestes, 115
Terpander, 85–6
Thaletas, 85–6
Thebes, auletes of, 102
Theon of Smyrna, 3, 326, 366n, 397n, 408n, 444
on the characters of the genera 179n
on Lasus, 79
on the theorem on epimoric ratios 304n
Theophrastus 46, 114, 374n, 411, 448n
and Aristotle, tactful treatment of, 421
and Aristoxenus, 422–4, 426n, 427–8, 431, 435–6
on attunement, emmeleia and melōidia, 431–3
on empirical harmonics, 97, 421–8
his Harmonics, 411, 413n, 435–6
on harmonikoi, 74, 422–7
life and works, 411
on mathematical harmonics, 97, 364, 412–21
his On Music, 411
fragment from, 411–36
on music and soul, 411–13, 433–6
Theophrastus (cont.)
and Plato’s Republic, 424–7
on reeds of auloi 56n
his theory of pitch, 413, 428–31
Thrasyllus 366n, 375n, 400n, 404, 444, 445, 448n
his division of the kanōn 395–8, 400, 401n, 401–3, 404n, 444n
on ratios of pitches 29n, 408n
Timaeus of Locri, 445
Timotheus, 102, 104, 247
Vitruvius, 437
Wallace, R. 47n, 234n
Weil, H. and T. Reinach, 335
West, M. L. 15n, 18, 47n, 48n, 56n, 59n, 61n, 62n, 63n, 64n, 69n, 73n, 80n, 81, 82n, 86, 86n, 98n, 100n, 182n, 269n, 393n, 399n, 400n
and Pöhlmann see Pöhlmann
Winnington-Ingram, R. P. 48n, 51n, 64n, 101n, 226n, 294n, 298n, 398n
World-Soul see General index s.v. soul
Xenocritus, 85–6
Xenophilus, 114
Yamamoto, T. 116n
Zanoncelli, L. 182n, 366n, 445n
Zedda, S. 319n
General index
accompaniment, 7n, 101n, 240, 241n, 246, 399–400
acoustics, 7n, 79–80, 84, 141, 146, 193, 305–6, 370–5, 407–8
see also pitch, sound
agōgē (‘consecution’) 132
anamnēsis 111n
apodeixis see demonstration
apotomē 272–4, 275, 277–8, 351–2
ratio of, 272, 352
‘appearances’ (phainomena)
as subject-matter of harmonics, 141–3, 148, 149–50, 166–8, 193n, 428, 435–6
and principles, 168, 193
see also perception
appropriateness (oikeiotēs), 241, 242, 243–59
arithmētikē (‘number theory’), 7n, 315, 353–6
arrangement (schēma)
of concords see species
in enharmonic genus, 39
astronomy, 10, 11, 315, 323, 334
and harmonics, ‘sisters’ 332n
attunement
and emmeleia and melōidia, 431–3
and melody 7–8, 172n
and scale 8n
‘ten-step’ 98
see also harmonia, systēmata
aulos, 26, 27, 28, 420
articulation and concordance of 241–2
pitching of notes on, 57
range of, 126
ratios of sounding-lengths, 58–9
syrinx of, 126
and theories of the harmonikoi, 55–6, 57–60
axioms, 108, 112
see also principles
beats, 297–8
bull-roarer see rhombos
character, human
music’s effects on, 250, 251–4, 255–6, 257, 309
see also soul (human)
see also ethics, ēthos
choruses, 90, 91
chromatic see genus
comedy, 327
harmonics in, 77
commensurable, incommensurable, 288–91, 292, 339–42, 343–5
competitions, musical, 239
composers, 10, 90–1, 102, 244–5
training of, 75–6, 86–7, 102–3, 104, 229–33, 247–9, 257–8
composition (as an activity)
melodic (melopoiia), 66, 120, 139, 140, 230–3, 252
and the telos of harmonics, 230–3
three ēthē of, 255–7
of music in all its aspects (poiētikē), 138–40
compositions
Dorian, 246–7
ēthos of see ēthos
evaluation of see judgement
harmonic analysis of, 11, 90–1, 102–3, 175, 236–43
particular examples, 100–1, 244, 245–6
as musical ‘ends’, 241–2
styles of, 175, 178, 232, 245–6, 247–9, 297
concords (symphōniai)
in accompaniments 101n
blending of notes in, 298, 344–5, 372, 375–6, 419
and colours, 289n, 338–48, 362
added to concords, 127
added to octave, 126, 127
in construction of discords (‘method of concordance’), 65–6, 93, 94, 121, 157, 158, 294n, 299, 321, 445
definition of, in Aristoxenus, 189–91
concords (symphōniai) (cont.)
distinction from discords, problems in, 346–8, 357–8
in embryology, 280–1
forms of see species
largest and smallest, 119, 125–7
little or no variation in magnitude, 93, 189–91
and mathematical means see means
notes of, equal in power, 419, 420
number and identities of, 125–7, 348
‘in numbers’, ‘numbers in’ see numbers
ratios of see ratios
recognition of, 65–6, 156, 190, 294n, 297–8
relative measurement of, 25
species of see species
see also fourth, fifth, octave, symphōnos
conjunction (synaphē), 12, 14, 53–4, 98, 186, 191–2, 198, 199, 208–9
constant and changing elements in music, 170–2, 245–6
continuity (to syneches, synecheia)
and the harmonikoi, 162
of melodic sequences, systēmata, 42, 103, 118, 130, 161–4, 238–9
of rhythmic sequences, 238–9
of sequences of speech-sounds, 161, 238–9
see also succession
cosmology, 265, 275, 278–86, 305–6, 318–23
criterion, of judgement, 438–40
see also judgement, perception, reason
criticism, musical see judgement
cyclic rearrangement of intervals see harmoniai (Eratoclean)
demonstration (apodeixis)
in Aristotle, 105–12, 193–4, 328–9, 353–61
and explanation, 107
logical form of, 106
premises of, 106–11
colontruerunontrue
cannot be demonstrated 106–7, 194, 358, 377–8;
true of subject ‘in itself’, 108–9
principles of, 106–7, 108, 110–12, 353–61, 377–8
prohibition of ‘kind-crossing’, 109–10, 167–8, 353–61
see also ‘same domain’ rule
in Aristoxenus’ harmonics, 105, 122, 124–34, 135, 150, 152–5, 166, 167–8, 192–6, 197–228, 328–9, 377–8
and ‘agreement’, 193, 195–6
hierarchy of, 154, 193, 194–5
and perception, 154–5, 166, 167–8, 193–4
preconditions of, 192–6
in mathematical harmonics, 167
neglect of by harmonikoi, 41, 104, 166
principles of, in harmonics see principles
see also theorems
determinacy, of propositions in harmonics, 186, 213–14
diagrams
of harmonikoi, see harmonikoi
in Sectio canonis, 380–1, 395
of Stratonikos, 75–6
dialektos, of instruments 241n
dianoia (‘thought’), its role in Aristoxenus’ harmonics, 168–9, 171–5
non-discursive, 173–4
as ‘meaning’ 173n, 184n
as simultaneous with perception, 173–4, 237–8
training of, 171
diaschisma, 273, 274, 278
diastellein and systellein, meanings of, 256
diastēma
in Archytas and Aristotle 379n
in Philolaus, 270
in Sectio canonis, 378–9
see also intervals, ratios
diatonic see genus
diatonos hypatōn, in Sectio canonis, 395, 397, 398–400
diesis
in Aristotle 349–53, 389n, 424, 425n
two dieses in mathematical harmonics, 350–3
in aulos-playing, 269–70
chromatic, 156, 157
enharmonic, 91–3, 157
successions of, 42–3
as unit of measurement, see measurement
see also quarter-tone
in Philolaus, 264, 266n, 268–70, 272–5, 276–8, 281, 321
as minor semitone, 269
its ratio, 269, 272
see also semitone, leimma
as unspecified small interval, 269
discords, 372, 386
in accompaniments, 99–100
constructed through concords, see concords
variable in magnitude 93, 190, 294n
discs, used as instruments, 27, 84–5
disjunction (diazeuxis), 12, 14, 43, 53–4, 98, 186, 191–2, 198, 199, 206, 208–9, 320
in Mixolydian, 49–50
dithyramb 79, 83, 244, 244n
ditone
composite and incomposite, 101n, 177
constructed through concords, 92n, 93n, 294n, 391–3, 409
counterparts of, in chromatic and diatonic, 205n, 205–6, 210–13
in enharmonic, 38, 43, 99, 177, 235, 296–8
progressions from, 206
relations with pyknon, 205, 208, 209–10
relations with tone, 205–6
sequences of, 101n, 205, 209–10, 212
use of the term, in El. harm. Book, 3 205n, 205–6, 210–12
division
of kanōn see kanōn
of tetrachord see tetrachord
dynamis
of appropriateness, 244
of harmonia, in Aristotle, 329
kritikē 238
see also judgement
as melodic ‘function’, in Aristoxenus, 116, 120, 121, 122, 123–134, 171, 175, 183–92, 213–16, 427
bearers of, 169n, 191–2
and hearing, and dianoia, 168–9
occurrences of in El. harm., quoted, 185–6
and perceived ‘character’, 192, 196
non-musical meanings, 184–5
in other musical writers, 184, 188
and thesis, 188
see also notes
ear, see hearing
education, 324
of children, 309, 311, 330
see also ethics
of musicians, see musicians
of philosophers, 311–15, 318
elements (stoicheia), the term’s use in Aristoxenus, 134–5, 163, 164, 227
elite and popular taste, 234, 259
emmelēs and ekmelēs (‘melodic’ and ‘unmelodic’), 101n, 160, 234, 420, 431–2
emotion
expression of, 248–9, 250, 253–4
release from, 411
empiricists see harmonics (empirical)
enaulos kitharisis, 80
enharmonic see genus
enumeration and analysis, 52, 81, 150–2
epagōgē see induction
epideixeis (‘displays’), 69, 71, 73–5, 76–7, 82, 104, 232
epigoneion, 80–1
epogdoic (epogdoos), in Philolaus, 264, 266–7, 270
see also tone
equality
in Aristotle’s Politics, 337–8
of intervals see intervals
‘of measure’ and ‘in number’, 334–5
ethics, 11–12, 70–1
in Aristoxenian theory, 239, 249–59
and musical structure, 72–3
in Plato’s musical theories, 250, 308–11
and harmonics, 309–11
ēthos
arising from composers’ treatment, 244, 245–7, 254–5
of compositions and performances, 241, 242, 243–59
of definable musical elements, 245, 254–5
diastaltikon, systaltikon, hēsychastikon, 255–7
and the genera, 179–80, 244
and human character, 249–59
see also ethics
and mixis and synthesis, 245–6
as a musical or ‘aesthetic’ category, 179n, 251, 253, 257–9
evaluation
of music see judgement
in Plato’s harmonics, 317–18
in Plato’s philosophy see values
experience (empeiria), 111
as prerequisite for the study of harmonics, 179–83, 203
facts and explanations, in Aristotle, 353–61
see also demonstration
fifth
names of, 22, 264–5, 280–1
quantified by Philolaus, 268
size of, 359, 360, 388–90
species of, see species
see also interval, ratio
‘following’ music, 103, 172–5, 236–9
see also understanding
form (eidos, schēma), of concords see species
forms, in Plato, 313, 314, 318
fourth
divisions of, 38
names of, 22, 264–5, 280–1
size of
according to Aristoxenus, 121, 127, 129, 190, 359, 390
according to Philolaus, 268
in Sectio canonis, 359, 360, 388–90
species, arrangements of, see species
see also interval, ratio, tetrachord
‘fringe’ performances, 82
function, melodic see dynamis
genus (genos), 14, 38, 103, 118, 120, 125, 150, 170, 175, 176, 198, 199
chromatic
in Archytas, 293–4, 295, 298–9
colontruerunontrue
construction of, from diatonic, 298–9
contains a pyknon, 200–1, 202–3
indeterminacy of, 70–1
‘hemiolic’ 94, 158n
in later mathematical harmonics, 444–5, 446
perceived character of, 179–80, 192, 196
genus (genos), chromatic (cont.)
and Philolaus, 273, 278
‘soft’ 94, 158n, 211
in Ptolemy 299n
structure of 38, 101n
‘tonic’ 212
treatment of, in El. harm. Book, 3 205n, 205–6, 212–13
concept of, and Aristoxenus, 404–5
development and precursors of the concept, 71n, 404–6
diatonic
in Archytas, 293, 294, 298–9
assimilation to chromatic, 70–1
its counterparts for ditone and pyknon, 206, 212–13
lacks the pyknon, 179n, 200–1, 202–3
in later mathematical harmonics, 444–5, 446
in Philolaus, 276–7, 292
in Plato, 276n, 321, 406, 444–5
in Sectio canonis, 276n, 392, 403–7
‘soft’ 158n
structure of 38, 294n
divisions of tetrachords in see tetrachords
earliest allusions to, 38–9, 70–1
earliest analysis of, 292
enharmonic
in Archytas, 293, 296–8
admired by Aristoxenus, 40, 91, 128, 297, 393
by ‘fraudulent’ harmonikoi, 70–1
by Olympus, 99
contains a pyknon, 200–1, 202–3
difficulty of, 39–40
‘distorted’ in fourth century, 39, 128, 235, 297–8
earliest form of, 98–101, 297
in fifth century, 40
and the harmonikoi, 37–41, 48, 100, 236, 292, 392–3
in later mathematical harmonics, 444–5, 446
and Olympus, 98–101, 245–6, 297, 405–6
and Philolaus 273, 278, 292n
and Plato and Aristotle, 393, 406
rejected by some of Aristoxenus’ contemporaries 65–6, 91–6, 297n
in the Sectio canonis, 369, 388–90, 391–4, 403–5
structure of, 38
in tragedy, 40, 70, 71, 393
ethical attributes of, 70–1, 72–3
as first topic in harmonics, 122, 124, 134
mixtures of, 118, 125
modulation of, 177–8
see also modulation
non-technical discrimination of, 72
and notation, 62, 175
not recognised as such, 404–6
origin of the term 405n
perceived characters of, 171–2, 178–80
relative ages of 99, 160n
‘shades’ (variants) of, 14, 171–2, 178–80, 200, 206, 212
see also tetrachords (divisions of)
geometry
and Aristoxenian harmonics, 169–70, 203
in Plato, 315
‘golden age’ of music 309n
harmonia
‘body’ of, 335
in cosmology, 265
divine, 326, 329
in embryology, 280–1
as enharmonic genus, 37–8, 236n, 393
as musical attunement, 37–8
and the number 10 (the decad), 282–3
as octave attunement, 264, 265, 268, 329, 332–3
see also octave
its ‘parts, magnitudes and excesses’, 334–7
in Philolaus, 264, 265, 268, 279–80, 345
in plural, harmoniai 38n
Aristotle on, 253
and education, ethics, 97, 309–11
Eratoclean (‘cyclic’), and Aristides’ pre-systematic forms, 43–55, 74, 83–4, 153, 194–5, 224
Dorian, 49, 54n, 398, 399–400
Hypolydian 54n
Iastian (‘Ionian’), 50–1
Lydian, 50, 51
Mixolydian, 49–50, 54n, 83–4, 87
names of 44, 44n, 45n
Plato on, 45, 47, 239, 250, 309–11, 330
pre-systematic, sources of Aristides’ evidence, 45–8
and the senses, 332
harmonics
agenda of, 6–11, 244
Aristoxenian and Pythagorean, contrasted and combined, 440–2
and arithmetic, 353–6
autonomy of, in Aristoxenus, 149–50, 434, 435–6
and composition see composers, composition
and Dorian compositions, 246–7
empirical, 10–11, 23–5, 29–30, 33–67, 68–104, 105, 136–64, 165–96, 197–228, 229–59, 349–51, 443–7
Theophrastus’ criticisms of, 413, 421–8
see also appearances, harmonikoi, perception
as evaluative, 317–18
in later centuries, 5, 46–7, 259, 389, 391, 404, 410, 437–49
its relations with earlier harmonics, 437
treatises on, Aristoxenian, 442–3
mathematical, 443–7
manuals of, 392
mathematical, 25–30, 59, 79, 84, 143, 166–8, 193, 263–86, 287–307, 315–27, 328, 329–48, 350–3, 364–410, 435n, 443–7, 449
complete treatises in, 364
mixed resources of, 384–5
Theophrastus’ criticisms of, 412–21
see also mathematics, numbers, ratios, reason
mathematical and empirical
compared, contrasted, 9, 18, 29–30, 37, 52n, 58–9, 96–7, 143, 166–8, 291–2, 310–11, 389–91, 440–2
mixed approaches, 271, 272, 274–5, 440–2
relations between, 353–61, 362–3, 389–91
see also harmonics (‘schools’ of)
and musical judgement, 233–59
and musical practice, 4, 11, 40, 48, 55–6, 66, 69–70, 74–5, 78–96, 97–104, 125–6, 129, 133, 275–6, 294–302, 306–7, 326, 443–4, 446–7
nature and goals of, in Aristoxenus, 180–3, 229–59
objectivity of, 170
origins of, 19–20
Plato’s treatment of, 308–27
and later harmonics, 443, 444–5
its intended audience, 326–7
its philosophical background, 311–15
in Republic, 311, 315–18
in Timaeus, 311, 318–26
principles of see principles
its relations with other disciplines, 4, 5, 7, 10, 11, 149–50, 409
see also acoustics, arithmētikē, astronomy, cosmology, geometry, history, linguistics, mathematics, medicine, metrics, natural science, philosophy, rhythmics
‘schools’ of 52n, 79n
distinguished, 438–42
see also harmonics, mathematical and empirical
scope and limits of, in Aristoxenus, 103, 236–43
sources of evidence on, 5–6, 19–20, 33–4
status of, 7, 104, 105, 137–8, 229, 259, 291–2
terminology of, 9, 20–3, 24–5
unity of, in Aristoxenus, 192, 195–6, 228
harmonikoi (predecessors of Aristoxenus), 37–67, 68, 310, 350n, 354, 390–1
Aristoxenus’ attitude to, 41, 136
Aristoxenus’ essay on, 33n, 39n, 122, 123
as composers and performers, 78–96, 97–104
their diagrams, 39, 41–3, 45–8, 65, 74, 75–6, 86–7, 141, 153, 162
differences between, 40–1
exclusive concern with enharmonic, 37–41, 48, 100, 236, 292, 392–3
failures of enumeration, 52, 81
failures of observation, 153, 166–7
and fraudulent ‘harmonikoi’, 69–73
historical works of, 49, 83, 98–101
see also Index of proper names s.v. Glaucus
incompetence of, 39
and instruments, 55–6, 57–60
see also aulos, epigoneion, tripous
and knowledge of ratios, 58–9
and measurement see measurement
methodological failings of, 150, 152–4, 155–6, 157
and mousikoi, 88–90, 98–9
and musical practice, 40, 48, 78–96, 97–104, 326
and musicians’ training, 75–6, 86–7, 102–3, 104, 229
names of, listed, 78
neglect of demonstration and principles, 41, 52, 55, 57–8, 104, 153, 166–7
and notation, 56, 60–7, 68, 74, 77
and octave systems, 39, 43–4, 52, 127n, 321
on Olympus, 98–101
and other practical skills, 89, 104
and philosophers, mathematicians, intellectuals, 34, 66–7, 79, 96–7, 104
public profile of, 68–78, 82–3, 85, 103–4
punning use of the term, 37–8
purposes of, 96–104
and sophists, 68–78, 87–8
Theophrastus on, 422–4
on the tonoi, 55–6, 216
see also Index of proper names s.vv. Agenor, Eratocles, Pythagoras of Zakynthos
harmonikos, uses in Plato and Aristotle, 36–7
harmonious blacksmith 22n, 446n
harmony, 7
‘of spheres’, 149, 283, 286n, 318, 323
hearing (akoē)
and analysis of compositions, 100
as criterion, 23–4, 29, 30, 86, 424–5
its objects, 168–9, 174
and perception, in Aristoxenus, 169n, 174–5
purpose and workings of, 325–6, 332
its role in Aristoxenian harmonics, 168–9, 172–3, 174–5
see also perception
hermēneia 242, 411n
see also performance
to hērmosmenon (‘that which is attuned’), in Aristoxenus, 161, 240
history, musical, 49, 83, 85–6, 98–101, 125–9, 133
hypatē (hypata), in Philolaus, 264, 265–6
hyperhypatē 398–400
independent of genus, 398–9
see also diatonos hypatōn
hyperochē (mathematical ‘excess’), 270
imitation
of composers’ styles, 85–6
see also mimēsis
impacts in sound-production
coincident and non-coincident, 374–5
frequency of, 370–5, 408, 416–18
induction (epagōgē), 112, 143, 149, 378
and composition, 249
instruments
construction of, and the tonoi, 55–6
eight-stringed, seven-stringed, 276
eleven-stringed, 98
experimental use of, 407–8
makers of, 59, 60
in Plato, 309
power used in note-production, 418–19
range of, 126, 321
and theories of harmonikoi, 55–6, 57–60, 69–70, 81, 82–3
used in mathematical harmonics, 26–7
see also kanōn, monochord
‘voice’ of, see voice
see also aulos, epigoneion, kanōn, kithara, lyre, monochord, Panpipes, rhombos, simikion, tripous, vessels, zither
intervals (diastēmata), 118, 120
as causes of difference in pitch, 422–7, 428
composite and incomposite, 101n, 118, 122, 123, 132, 175, 177, 199, 202
content of, 423–7, 432–3
differences between, 125, 150, 151, 154–5
diversity of, 8
equal and unequal, 43, 162, 178
familiar, 156
identified by reference to notes, 177, 199, 206, 427
to positions in the system, 264–5, 268, 274
irrational, 95
and larger numbers, 29–30
largest and smallest 34n
‘left out’, 423–7, 432–3
as linear distances, 23–5, 29, 51, 59
magnitudes of, see magnitudes
measurement of, see measurement
melodic synthesis of, 118, 131, 150, 154–5, 160–1
minimal, 23, 34–5, 42, 92, 155–6, 162, 424–6
see also diesis, measurement (unit of)
odd and even, 65–6, 94, 95
quantitative descriptions of
in El. harm. Book III, 198, 200, 210–12
Theophrastus’ critique of, 412–28
as ratios see pitch-relations, ratios
sequences of see succession
sizes of see magnitudes
as spatial, spaces, 144n, 422n, 423–4
see also pitch-relations, ratios
judgement, musical, 139–40, 233–59
of ends and means, 241–3
and ethical, 249–59
and philosophy, 258–9
and social utility, 251
see also appropriateness, ēthos
kalos (‘fine’, ‘beautiful’)
in Aristotle 235n
in Aristoxenus, 234–5, 247, 257–8
in Plato, 239–40
in Pythagorean theory, 342
kanōn, 365–6
division of in Sectio canonis, 365–6, 369–70, 394–408
constructing the ‘immutable’ systēma, 394–401
constructing the movable notes, 401–3
and diatonic, 403–7
enharmonic impossible, 403–4
geometrical conception of, 407
knowledge presupposed by, 406–9
mathematical and musical bases of, 394, 395–403
physical basis of, 407–8
use of instrument in, 407–8
division of by Thrasyllus, 395–8, 400, 401n, 401–3
see also monochord
katapyknōsis, 41–3, 56n, 162
keys see tonoi
kithara 74, 80, 87, 398n
knowledge
and reality, in Plato, 312–15
scientific, 105–12
see also demonstration
komma, 273, 274, 277–8
kritikos, in Aristoxenus, 240–1, 254
see also judgement
law of fourths and fifths (abbreviated as L), 131, 160–1
and Aristoxenus’ theorems, 198–9, 201–2, 204, 209, 212
see also principles
leimma, 294n, 299, 321, 351–2
division of, 445n, 446
see also diesis, semitone
lichanos
a ditone below mesē, 128, 235, 297
dynamis of, 188
identity of, 128
infinite in number, 129, 130, 158, 211
melodic role of, 163
omission of 99–100, 100n
positions and ranges of, 128, 129–30, 142–3, 144, 158–9
limit and the unlimited
in Aristotle, 332
in Philolaus, 279–80
linguistics, 66
lyre
movements in its sound-box, 430
nine-stringed 398n
seven-stringed, 276
tuning of, 15, 44, 264–5
magnitudes (‘sizes’)
of intervals, 125, 264–5, 267, 388–90
and Aristoxenian theory, 61, 66, 158, 164, 168–9, 171, 175–80, 185–6, 206, 207, 427–8
grasped by hearing, 168–9, 174
incomposite, 132
and notation, 61–5, 175
in Philolaus, 264–5, 268, 271, 272–5, 285–6
in theories of harmonikoi, 61
of pitched sounds, 418
of systems, 39
see also measurement
mathematical harmonics, see harmonics (mathematical)
mathematics, 7n, 197–8
and Archytas’ harmonics, 288–302
evaluative aspects of, 317–18, 342–8
in Plato, 312–13, 314, 317–18
the five mathematical disciplines, 315
see also harmonics (mathematical)
means, mathematical
in Archytas, arithmetic, geometric, harmonic, 302–3, 320, 331–3, 336–8
in Aristotle frag. 47 Rose, arithmetic and harmonic, 329, 334–8
in Aristotle’s Politics, arithmetic and harmonic, 337–8
geometric, 337–8
and construction of concords, 302
and Philolaus, arithmetic, 283–4
harmonic, 283–4
in [Plut.] De musica, 331
in Sectio canonis, 380
in the Timaeus, arithmetic and harmonic, 319–20, 331–2
geometric 320n
see also proportion
measurement
of auditory phenomena, by ear, 23–4, 30, 34–5, 155–9, 349–51, 424–5
of intervals, 9, 20–30, 65–6, 155–9, 168–9, 349–53
linear, 23–5, 274
see also magnitudes
by ratios, 25–9, 266–7, 268, 269–71, 272–5, 350–3
by division of ratios, 352–3
see also ratios
by reference to the tone, 156–9
two approaches to, 9, 23
compared, 29–30
mixed, 271, 272, 274–5
unit of
in harmonics, 23–4, 29, 34–5, 42, 65–6, 92, 94, 155–9, 349–53, 389, 424–6
as indivisible, 349–50, 350n, 424
in ratios, 288–9, 344, 350–3
in rhythms, 349
medicine, medical writers 11, 89, 104, 104n
harmonia and concords in embryology, 279, 280–1
melodic and unmelodic see emmelēs
melody, 7, 118
analysis of, 100–1, 102–3
as coming-into-being, 172–3
common to the genera, 125
and genus, 38, 125
inspirational, 100
nature of see melos
notation of, see notation
orderliness of, 142–3, 228
and scales, attunements, 7–8, 62–3
melōidia (melodic singing), 163, 431–3
see also singing, voice
melopoiia see composition, melodic
melos
meanings of, 62–3, 137–8
nature of (and of vocal movement, of to hērmosmenon, etc.), 118, 146n, 149–50, 155, 159–64, 166, 189, 192, 228
see also melody
memory
in establishment of principles, 111
its role in Aristoxenian harmonics, 172–3
mesē
dynamis of, 214
melodic role of, 163
in Philolaus (messa), 264, 265–6
‘method of concordance’ see concords
methodology
Aristotle on, 349–61
Aristoxenus on, 120, 121, 122, 123, 135, 136–59, 164, 165–96
condition of completeness, 150–2, 153
metrics, 6, 176
agenda of, 6
mimēsis (‘imitation’), 53, 90–1, 240n, 309, 325, 326
mixture (mixis)
of musical elements, 238, 254
and ēthos, 245–6
and synthesis 238n
modulation
melodic 12, 15, 15n, 16n, 55, 87, 118, 120, 122, 124n, 171, 175, 177–8, 215–28, 244, 338n
through concords and tones, 218, 222, 225
through the octave, 219
rhythmic, 245–6
monochord, 26, 81n, 365–6, 394, 407–8, 409n, 438
see also kanōn
mousikos
in Aristoxenus, 138, 139–40, 233, 240
uses of the term, 36, 88–90, 98–9, 139–40
movement see singing, sound, voice
melody-making see soul (human)
music
as coming-into-being, 172–3
evaluative attributes of, 234–5, 238, 239–40, 243, 257–8
and appropriateness and ēthos, 243–59
and ethical attributes, 249–59
see also judgement
and movement of the soul see soul (human)
musical and unmusical
as an evaluative distinction, 317
as a mathematical distinction, 291–2, 317
objectiveness of the distinction, 10
see also composers, composition, musicians
musicians, 296n, 409
and musical judgement 139n
training of, 75–6, 86–7, 102–3, 104, 229–33, 247–9, 257–8
their uses of harmonic theory, 96–104
see also composers, harmonics (and musical practice)
vocabulary of, 9, 264–5, 267, 268, 269–70, 274
natural science, and Aristoxenian harmonics, 149–50, 229
nature (physis)
in Aristotle, 159–60
and artefacts, 159–60
of melos and related conceptions see melos
nētē (neatē), in Philolaus, 264, 265–6
nētē synēmmenōn, in Sectio canonis, 395, 397–8, 399–400
‘New Music’, 79, 102, 103, 309, 324
nomos, 85
of Athena, 245–6
non-quantitative attributes, 175–92, 212–15
see also dynamis, pitch, quantification
notation
melodic
Alypian and non-Alypian, 61–6
determinacy of, 66–7
of melodies and structures, 62–3
and musical practice, 66
reveals magnitudes only, 61–5, 175, 176
and theories of the harmonikoi, 56, 60–7, 74, 77
see also scores
metrical, 62–3, 176
notes, 118, 120, 141
as having ‘breadth’, 80, 144
coinciding in pitch, 207–8, 210, 214–27
composed of parts, 371–2
defined 119, 145n
diatonic and enharmonic, names of, 404–6
and dynameis, 120, 185–92, 213–15
see also dynamis
as dynameis and as pitches, 185, 187
fixed, 13–14, 38, 128, 397, 398–9, 400
flattened, 95
as ‘high’ and ‘low’, 21, 144
higher as more conspicuous, 429
in identification of intervals and progressions, 177, 199, 206, 213–15, 222
identities of, 12, 128, 163–4
melodic roles of, 163–4, 177
see also dynamis
moveable, 13–14, 38, 118, 397, 401–3
earlier conception of, 404–5
positions of, 128, 129–30
ranges (‘spaces’) of, 128, 150, 158–9
their limits, 142–3
as movements, 141
names of, 12–13, 163, 187n, 187–8, 404–6
by dynamis and thesis, 188
omitted in certain styles, 246
as pitches, pitched sounds 185, 187, 416n
as points without extension 145n
and positions in the pyknon, 207–8, 210, 214–19, 227
ratios of, 27–9, 371–2
see also ratios
as ‘sharp’ and ‘heavy’, 21, 144
see also pitch
successive, 42, 80, 163, 177
see also continuity, succession
as ‘tense’ and ‘relaxed’, 21–2, 144
see also lichanos, mesē, parhypatē, pitch, pitch-relations
numbers
concordant, 306–7, 315, 316–17
‘in concords’, 25, 29, 315
and limit, 279–80
in ratios, and pitch, 322, 408
‘related under a single name’, 372, 375–8
as the subject of Plato’s harmonics, 315, 318, 322
and sizes of intervals, 29–30
symbolic meanings of, 274–5, 281–3
terms ‘in numbers’, ‘not in numbers’, 340–1
in Theophrastus see quantification
see also ratios
observation see perception
octachords, seven, 43–4, 52, 223
octave
added to concord 126, 127, 388n
basic structure of, 13, 265–6, 329, 331–2
division of, by means, 302, 331–2, 335–8
names of, 22, 264, 265, 268, 329, 332–3
not equally divisible, 305, 388–90
octave systems see systēmata
its ‘parts, magnitudes and excesses’, 334–7, 338
quantified by Philolaus, 264, 268, 277
size of, 359, 360, 383, 388–90
species of, see species
symmetry in see symmetry
see also interval
oikeios, oikeiotēs see appropriateness
opposites, harmonisation of, 345
organikoi (instrument-teachers), 439
paean
of Athenaeus, 15n, 100n, 398–9
of Limenius, 398–9
Panpipes, 26, 27, 28
paramesē 265
parhypatē, positions and ranges of, 129–30, 144–5, 158
perception (aisthēsis)
accuracy or inaccuracy of, 37, 169–70
of concords and discords, 93
as criterion of judgement, 438–40
and demonstration see demonstration
of differences in genus, 171–2
and hearing, in Aristoxenus, 169n, 174–5
and musical ‘meaning’, 174–5
of microtones, 91–3
of octave plus fourth, 387–8
of phenomena studied by harmonics, 141–3, 148, 149–50, 151–2, 153, 428
of pitch, 28
and principles, 111–12, 167–8, 358, 441–2
its role in Aristoxenian harmonics, 168–75, 439–40
and scientific knowledge, 107–8
simultaneous with dianoia, 173–4, 237–8
of simultaneous musical elements, 173n, 237–8
as ‘spark’ for reason, 439
training of, 40, 169–70, 171, 240
and understanding, 90, 91, 170–5
‘of the universal’, 112
see also hearing
perfect systems see systēmata
performances
contexts of, 82–3
ēthos of see ēthos
as musical ‘ends’, 241–2
phainomena, phenomena see appearances, perception
philosophy
and empirical harmonics, 96–7
and musical judgement, 258–9
and Nicomachus’ harmonics, 446–7
and Plato’s harmonic theories, 311–15
and Porphyry, 449
phōnaskikoi (voice-trainers), 439
physis see nature
pitch (tasis), pitches, 12, 118
abstract and concrete conceptions of, 147–8, 149
Aristoxenus’ discussion of, 146–8
attributes of, 29, 144
oxytēs and barytēs, 21, 144, 146–7, 148, 280, 429–30
continuum of, 21
and force or vigour, 28, 418–21
and frequency of impacts, 370–5, 408, 416–18
and key see tonoi
linear conception of, 24–5, 29
and motionlessness of voice, 143, 146, 147–8
and notes of the pyknon, 207–8, 210, 215–27
perception of, 28
physical and auditory conceptions of, 30, 146, 166–8, 390–1, 428
as points lacking extension, 144–5
as qualitative, 413, 430–1, 448–9
quantitative conceptions of, 27–9
criticised 412–28
and shape, 429–31
and speed, 28, 146, 305–6, 322, 373, 374, 381, 416–21
as ‘tension’, 21–2, 147
tension and relaxation of (epitasis and anesis), 144, 146–7
pitch (tasis), pitches (cont.)
as a topos (‘space’), 140–1
see also notes, pitch-relations
pitch-relations
as linear distances, 23–5, 29, 140–1
the theory criticised, 413
measurement of, see measurement
as ratios, 27–9, 371–5
the theory criticised, 412–21
representations of, 20–3, 144
see also intervals, pitch, ratios
poiētikē see composition
point-line-plane-solid 271n
principles
of Aristotelian demonstration, see demonstration
of harmonics, 9
of Archytas’ harmonics, 288–307
of Aristoxenian harmonics, 41, 101n, 104, 120, 121, 122, 123, 131–4, 149, 154–5, 167–8, 189, 193, 194, 198–9, 204, 228
see also law of fourths and fifths
governing concords, 288–91, 342–8
governing synthesis of intervals, 160–1
of mathematical harmonics, 167–8
of Theophrastus’ harmonics, 431, 435–6
progressions
geometrical, 320
melodic, 186, 189, 191–2, 197–228
from incomposite intervals, 206, 212–13
from notes in the pyknon, 206–7, 213–15, 222
of notes, rhythmic durations and letter-sounds, 236–9
proportion, 410
musical, 283–5
in the Timaeus, 285
see also means
psychology see character, ēthos, soul
pyknōma, 23, 24–5, 42, 425
pyknon, in tetrachords, 42, 129, 200–1, 211, 390
not equally divisible, 359–60, 388–90, 391–4
as an indissoluble unit, 202–3, 205–6
and notes in chromatic and enharmonic, 206, 217
notes differently positioned in, 207–8, 210, 214–19, 227
perceived character of, 178–9, 180, 190, 192, 196
progressions from, 186, 191–2, 206
from its notes, 206–7
relations with ditone and tone, 205–6, 208–10
sequences of, outlawed, 201–4, 205
as a systēma 119n
pyknos (‘dense’, ‘compressed’), 24–5, 178, 180
quantification
Aristoxenus’ attitude to, 158, 164, 171–2, 175–92, 198, 200, 210–12, 222
Theophrastus’ critique of, 412–28, 435–6
see also measurement
quantity and quality, terms for 414n
quarter-tone, 38
absent in earliest enharmonic, 99–100
as a melodic interval, 157, 158, 162
recognition of, 156–8
rejected by certain theorists, 65–6, 91–3, 157
successions of, 42–3, 162
see also diesis, genus (enharmonic), intervals (minimal)
ratios
absent from Hippocratic De victu, 281
Adrastus’ classification of 342n
in Aristides Quintilianus, 441
in Aristotle frag. 47 Rose, 334–8
in aulos-construction, 58–9
better and worse, and terms ‘in no ratio’, 339–48
commensurability of terms see commensurable
of concords, 25, 267, 268, 288–91, 302, 316, 338–48, 372, 375–8, 384–8
lacking mathematical distinctiveness, 346–8, 357–8, 387
and the tetraktys, 282
of doubled concords, 345–6
of double octave, 348
epimeric (‘number to number’, superpartient), 289n, 321, 342–8, 372, 393
epimoric (superparticular)
and Archytas’ divisions, 288–91, 295–6, 301n, 297–302
and concords, 288–91, 342–8, 372, 375–8, 384–8
construction through chains of, 300–1, 301n, 302–3
defined, 289n, 375
not equally divisible, 272
Archytas’ proof, 303–5, 341, 351, 356, 382
in Sectio canonis, 379–82, 389
and mathematical means, 302–3
and melodic intervals, 288–91
in Plato, 321
of small numbers, 298, 300–1
of frequencies of impacts, 374–5
and string-lengths, 408
in Gaudentius, 441
interpretation of, 26–9
larger and smaller, 29–30
of the leimma (diesis in Philolaus), 269, 272–5, 321
mathematical theorems on, 378–84
multiple, 289, 290, 291, 321, 342–8, 372, 375–8, 379–82, 384–8
defined, 289n, 375
of octave plus fifth, 348
of octave plus fourth, 290, 348, 377, 387–8
principles governing, 288–91, 342–8
basis of, 291–2, 344–5
consequent difficulties, 290, 348
roles of, in Philolaus, 266–7, 268, 269–71, 272–5, 285–6
of string-lengths, 26, 407–8
and frequencies of impacts, 408
see also kanōn
terms of, and pitch, 322, 408
of thicknesses, 84
in the Timaeus see soul (of the universe)
of the tone, 267, 268, 320
words designating, 266, 360n, 376
see also harmonics (mathematical), intervals, measurement, pitch-relations
reason
as criterion of judgement, 438–40
in Nicomachus, 446–7
and ‘rational principles’, 288–302, 441–2
its role in Aristoxenus’ harmonics, 439–40
see also harmonics (mathematical), mathematics
rhetoric, rhetoricians, 66, 89
rhombos (‘bull-roarer’), 28
rhythm, 236–9, 244, 245–6, 349
Plato on 250, 309, 310n
rhythmics, 6, 7, 86, 240
agenda of, 6–7
routes (hodoi), from a given note or interval see progressions
‘same domain’ rule
in Aristotle, 109–10, 168, 353–61
in Aristoxenus 110, 167–8, 258n
see also demonstration
scales see systēmata
schisma, 273, 274, 277–8
sciences, domains of, 110
scores, 15n, 39n, 86, 321, 398–400
see also notation
semitone
constructed through concords, 157, 219–22
in El. harm. Book III 211n
major see apotomē
minor see diesis, leimma
relations with tone, 205–6
sequences of, 205
undivided in earliest enharmonic, 99–100
sequences see succession
seven-note systems see systēmata
shape
and pitch, 429–31
and sound and hearing 429n
simikion, 81
singing
accuracy in, 412–13, 433
movements generated in, 430
power used in, 418–19
and soul see soul
see also melōidia, voice
sizes, see magnitudes
sophists, 68–78, 87–8, 327
soul
of the universe (‘World-Soul’), 319–26, 406, 432n, 444
and astronomy, 323
circles in, 322–3
its division, 319–23, 331
indeterminate features of, 322
its musical compass, 320, 321–2
and music, 320–2
its ratios, intervals, structure, 320–1
human
and melody-making movement, 411–13, 433–6
musical therapy for, 324–6, 433–5
in the Timaeus, 323–6, 434–5
colontruerunontrue
distortion of its structure 323–4;
its ratios and intervals, 323–4
see also character, ēthos
sound
apparent continuity of, 373–4
direction of travel, 430
distance of travel or perceptibility, 419–20, 429
and impacts see impacts
as movement, 27, 141, 146, 193, 305–6, 322, 370–5, 416–18, 419–21
verbal, 236–9
words designating, 426
see also acoustics, pitch
speaker-hole, 126
specialisation, development of, 327
species, arrangement, form (eidos, schēma)
of fifth, 194–5, 223
of fourth, 122, 123, 194–5, 197, 208, 223
of octave, 43–4, 54n, 83, 153, 194–5
see also harmoniai (Eratoclean)
and tonoi, 223–7
spondeion, spondeiazōn (spondeiakos) tropos, 100, 101n, 399–400
strings
lengths and ratios of, 26, 407–8
see also kanōn
oscillations of, 373–4, 407–8
tensions of, 425
weighted, 27
succession (to hex={e}s)
of intervals, notes and systēmata, 41–3, 118, 121, 122, 123, 130, 132n, 161–4, 177
of equal incomposite intervals, 205
of intervals of one third of a tone, 127–8
of unequal incomposite intervals, 205–6
see also continuity, synthesis, systēmata
superimposed structures, 210, 215–27
see also modulation, tonoi
‘sweetening’ of intervals, 128, 235, 297–8
syllogism
forms of, 106
non-syllogistic arguments, 201
symbainein, meanings of 193n
symboulē 255
symmetry, in the octave
in Aristotle frag. 47 Rose, 336–7
in Philolaus, 277–8, 280, 336
in [Plut.] De musica, 331
symphōnos
meanings of, 316
see also concords
synthesis
of intervals, 118, 131, 160–1
of musical elements 238n
and ēthos, 245–6
syrinx
as device on aulos, 126
as instrument see Panpipes
systēmata (scales, systems), 8–10, 42, 118, 120
and attunements 8n
bounded by concords, 119
continuity of see continuity
diversity of, 8
and demonstration, 152–3, 154–5
‘gapped’ 99–100, 275
as melodic syntheseis of intervals, 131, 150, 154
and melody, 7–8
modulating and simple, 177–8
see also modulation
and the nature of melos, 160
nine-note, 394–401
octave, 13, 15, 39, 43, 321, 331–2
forms or species of see species
pentatonic 100n
perfect, 12–18, 188–191
Greater, 13, 16, 83, 127n, 219
Lesser, 16
Unchanging, ‘immutable’ (ametabolon), 16, 54, 394–401
applications of the expression, 397, 400–1
as a primary topic of harmonics, 138, 150–2
relations between, 9
see also tonoi
seven-note, 15, 98, 275–7
successions of see succession
and tonoi, 223
see also tonoi
two-octave, 12–18, 321, 386–7, 394–404
varieties and attributes of, 151–3, 154
technical disciplines, and critical judgement, 236–43
see also judgement
telos, teleios, atelēs, in Aristoxenus, 230–3, 241–3
tetrachords 12–16, 38n
as basic to melodic structure, 54, 199n, 202–3, 329, 334–5
chromatic, 38, 200
diatonic, 38, 200
divisions of, 94, 171–2, 438
according to Archytas, 288, 292–302
runontruecolontrue
peculiarities of, listed, 293–4
discussed 295–300
according to Aristoxenus, 128–30, 155–9, 200, 292–3, 294, 296–8
see also genus
dynameis of, 175, 185–6, 191
enharmonic, 38, 200
identification of, in Aristoxenus, 191
irregular or defective, 50, 51
names of, 13, 14
not equally divisible, 304–5
omitted in certain styles, 246
pyknon in see pyknon
relations between, 12, 49–50, 131, 191, 198
relations between their intervals, 293–4, 295–9
as systēmata, 119
in the Timaeus, 321
tetraktys, 282–3, 347, 397, 398
theorems
in Aristoxenus, 122, 123, 131, 134, 197–228
arrangement of, 204–8
logical peculiarities of, 199–200, 203, 207–15
and mathematics, 197–8
in missing parts of El. harm., 227–8
negative, 201
non-quantitative interpretation of, 212–15
positive, 204
preliminary propositions, 197, 198–9
problematic features of, 199–200, 208–15
purposes of, 200
and quantification, 198, 200, 210–12
and tonoi, 215–28
two examples discussed, 201–4
underlying assumptions of, 203
in Sectio canonis, 378–94
harmonic, 359–60, 384–94
runontruecolontrue
and ‘bridging principles’ 385;
and division of the kanōn see kanōn;
logical flaw in 357–8, 386–7;
musical and mathematical premises of 384–5, 387–8;
on ratios of concords and tone 384–8;
on sizes and divisions of intervals, 388–94
mathematical, 356, 378–84
colontruerunontruenocommatrue
arithmetical or geometrical 378–9, 380–1, 383, 407;
and constructions 391–2;
on epimoric ratios 304n, 379–82;
on multiple ratios 379–82;
on numerically specified ratios, 383
origins of, 381–3
and prior knowledge, 383–4, 408–9
systematic arrangement of, 409
therapy, musical, 324–6, 433–5
thirds, major, 300
minor, 300
thought see dianoia
tone (interval)
constructed through concords 294n
defined, 93, 156
in disjunctions see disjunction
division of
in Aristoxenus, 119, 121, 127–8
not equally divisible, 272, 277, 304–5, 341, 359–60, 388–90, 391, 441
and quantification of, in Philolaus, 272–5, 277–8
in El. harm. Book III 211n
names for, 266–7
in Philolaus, 266–7, 272–5, 276–8, 281
progressions from, 206
ratio of see ratio
recognition of, 156
as reference-point for measurement, 156–9
relations with ditone and semitone, 205–6
relations with pyknon, 205, 208–9, 214
sequences of, 205, 222
tonoi (‘keys’), 12, 16n, 17, 54n, 65, 118, 120, 122, 124, 215–28
and Aristoxenus’ theorems, 217–28
Cleonides on, 217–22
Dorian, 99, 227
and modern (‘transposition’) keys, 55–6, 215–16, 224, 225–7
names of, 56, 218, 225, 226–7
and octave-species, 223–7
in Olympus’ nomos of Athena, 245–6
in Philoxenus’ Mysians, 244
as a primary topic of harmonics, 138
‘redundant’ 226–7
spacings, irregular, 56
by semitones, 216, 218, 219–22
by semitones and tones, 225
system of fifteen 442n
of seven, 224–7
of thirteen 216, 218, 219–22, 225–7, 442n
treatment by harmonikoi, 55–6, 87, 216
see also modulation
topos (‘space’)
of moveable notes see notes
of voice’s movement see voice
tragedy, 40, 70, 71, 89, 104, 393
tripous, 52n, 74–5, 77, 82–3
tritē (trita), in Philolaus, 264, 265–6, 275
tritone, in Mixolydian, 49–50
understanding (xynesis), 170–7, 181–3, 239
and ‘visible product’, contrasted, 182–3
see also ‘following’ music
unit see interval (minimal), measurement
use of musical elements
and evaluative attributes, 238, 244–5
values, in Plato’s philosophy, 313–15
see also evaluation, judgement
vessels, as instruments, 27
virtues see values
voice, human, 28
mechanism of 412n
its melodic accuracy, 412–13, 433
mathematical explanations of, 412–13
movement of, 150, 441
direction of, 144
melodic (including ‘voice’ of an instrument), 118, 140–50, 160–1, 188
as diastēmatikē (‘intervallic’), 143n, 143–5
imperceptible, 144, 145, 148
natural, 166
its temporal continuity 145n
as represented in perception, 141–3, 145
in speech, 141
as ‘continuous’, 143
non-rational, 412
as persisting subject of movement, 145–6, 148, 160
range of, 126, 321
regions of, 118
stationary, 143, 146, 147–8
topos (‘space’) in which it moves, 140–1, 148–9
weasel, 145
words, 236–9
and musical ēthos, 250
zither, 80
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