Subject Matter Index
Note 1: The subject matter index finds both specific terms and related concepts. For example, for the term âcomposition,â the index finds both âcompositionâ and cognate material relating to the construction of things. There are many synonyms in the index constituting independent sources for recovering the same material.
Note 2: Page citations followed by ât,â âf,â âq,â ân,â and âApâ indicate, respectively, that the reference is to a table, figure, quotation, note, or an appendix to the study, unless such item is itself the subject of the indexing. If one of these designations follows a range of pages, then the designation is relevant to all pages. Tables, figures, notes, and appendices are further designated by number for the readerâs convenience, e.g., ât28,â âAp7,â etc. Occasionally, to make locator designations more compact, there are combined references, such as 64&n33, rather than 64, 64n33, indicating that relevant material may be found both on a page of text and in a note to the page; 48&q, rather than 48, 48q, indicating that relevant material may be found both on a page of author text and in the quotation of a source on the same page; 75&f6, rather than 75, 75f6, or 109&t24, rather than 109, 109t24, indicating that relevant material occurs both in the text and in a table or figure on that page.
Note 3: References to âthe study,â âthis study,â or âstudyâ indicate this work.
Note 4: âUPS,â âGPS,â and âLPSâ mean, respectively, âUnmodulating Perfect System (also Unmodulating System),â âGreater Perfect System,â and âLesser Perfect System.â
Note 5: References to LFT indicate the âList of Figures and Tables.â Figures and Tables are generically indexed to the LFT, except for those that receive significant repeated reference throughout the text. The latter are indexed to specific pages as subheadings, under the generic main headings, respectively, for âFiguresâ and âTables.â
Note 6: References to GL indicate the âGlossary of Musical Concepts and Terms.â
Note 7: Significant numbers are indexed only to define their special features.
Note 8: Indexed numbers are emboldened whenever distinguishing between the indexed item and a page number reference poses a difficulty. On occasion, page numbers are also emboldened for special emphasis.
Academy (Platoâs)
xiii&n6, 1n3, 8n34, 33q, 33â34q, 64&n33, 124, 285
Aetius, nature of number
34
agitation (of Receptacle)
23q, 50
air
cosmic creation and
18, 46q
genesis of (as element)
19, 28, 35, 40, 42, 48
harmonia (bands of analogy and symmetry) and
49â50q
living creatures and
49, 50
mediation and
18, 48q
octahedron and
40, 42, 42t1
other primary elements, relation to
18, 19, 26q, 42, 42t1
precosmic state and
49â50q
Receptacle (see Receptacle: bodies (primary bodies))
visibility, relation to
46
See also pitch
All Perfect Animal
xiii, xvi, 18, 49, 54â56, 59â60, 271, 285
ambiguity
Aristoxenusâ diatonic arrangement and
5
chromatic invasion, created by
125, 240 (see also Timaeus octave phenomena)
filling intervals, and
double intervals with sesquitertian parts
86t14, 86â87t15, 88, 88â90t16
sesquitertian intervals with sesquioctave parts
102t21, 102â103
triple intervals with sesquitertian parts
86t14, 86â87t15, 88, 91â95t17
musical periodicity, and
189t35, 189t36, 190
Plato and scale ambiguity
xvi, 8â10, 180â181, 181q
Plutarch and scale ambiguity
xv, xvn11, 7, 7â8n33, 8, 9
Receptacle, interpretation of (Ïá½½Ïα), and
20â25
reduction of by cutting fabric, and
246
sesquitertian parts to fill, identification
xxii
studyâs approach to scale ambiguity
generally
xv, 5â6, 7, 8â10, 59, 134, 180â181
primary Timaeus scale
124, 126â127t26, 133â134, 136, 174, 175â178t34, 182â184
Decad, relation to
191
fifth periodicity, relation to
188â189
other scales, relation to
194, 195, 219, 238â239, 243â245, 262
scale calculation (see under ascending scales the different types indicated (in studyâs analysis))
standard/alternative GPS, LPS, UPS and
139â140, 142â143, 308â309Ap3, 313â319Ap3
Timaeus GPS, LPS, UPS and
149â151
Timaean octave strings and (see Timaeus octave phenomena)
analogy
Augustineâs De musica and
xxiii, 53n75
bands of
49, 49q
binding power of
54n76, 63, 63n30
beauty and
47q
cosmic band centers, lack of
253
cosmic speculation and, probable nature of
16
cosmos and exemplar, relation of
16
demiurgeâs use of
47q, 48q, 49, 49q
different cosmic levels and, role of cosmic motion
49, 50, 53
divided line (Republic 509 Dâ511 E) and
67
fifth, use of term in relation to diapente
188n66
genesis, elements and primary solids and
35, 39, 40, 43, 44, 46, 48q, 49
Greek tonoi and Western key system, lack of
310Ap3, 312 Ap3
harmonic cosmogenesis and
13, 13q, 28, 31, 48, 48q, 50
homogenesis and
19, 24, 50
monad/duad movement and
388Ap6
number matrix and
28
Philolausâ interval determination and
3
Platoâs divisions and
39, 258, 263â264
psychogony and twenty-eight stringed lyre, relation
113, 113q, 114q, 114
Receptacle and
24
soul sphere radius, use in computing
259
standard/alternative UPS and
150, 173, 321Ap4, 322Ap4t53, 333Ap4t55, 336Ap4t55
symmetry and
49, 49q
Timaeus GPS and
150, 173
Timaeus, relation to Laws
280, 281
âTriangle of Convergenceâ and
249f17, 249f18
unity and
48
apotomÄ (2187/2048 â1.06787109375)
calculation of various intervals, use in
117, 118
âchiâ (Ï) figure and necessity to
248, 249
chromatic scale, necessity to
118, 219, 248
comma, relation to
226, 556GL
definition
117, 118, 554GL, 556GL
designation as Sâ, assignment
126t26, 153, 431Ap8, 492Ap9
emergence, mode of
126â129t26, 225
leimma, relation to
219n4, 554GL, 556GL
Philolausâ treatment of
135
quarter tones derivable from
223, 224, 225â226
size
218, 219, 219n4, 225, 226, 431Ap8, 554GL
whole tone, relation to
219n4, 554GL, 556GL
Appendix 1
103, 104, 106, 108, 110, 112, 289â304Ap1
Appendix 2
118n13, 305â307Ap2
Appendix 3
139n32, 141, 141n44, 144, 159n56, 174, 321Ap4, 308â319Ap3
Appendix 4
141, 173, 320â345Ap4
Appendix 5
139, 346â384App5, 406Ap6
Appendix 6
139, 346Ap5, 347Ap5, 385â406Ap6
Appendix 7
139, 195, 407â430Ap7
Appendix 8
212, 263, 431â464Ap8, 492Ap9
Appendix 9
219, 220, 465â495Ap9
Appendix 10
221, 223, 225, 228, 263, 496â553Ap10
arbitrariness
absence of
xxii, 50, 76, 123
motion of Receptacle, feature of
20n16, 25, 26, 27, 35, 50
Platoâs harmonic expansion and
xii, 55, 55n2, 57, 58
Archytas
age, relative to contemporaries
2, 5, 118n13
fourth (tetrachord; diatessaron), divisions of
3, 10â11, 305â307Ap2
Greek music, history of
xxâxxi
indivisibility of whole tone
3
Plato, awareness of
12, 12n52, 59
Pythagorean identity of
10, 11
area
cosmic bands, overlapping realms
251f19, 251, 253, 254f20, 254, 255â257, 256t49, 256f21
fabric to cut (number matrix), discrete parts
109t24, 110, 243f15, 244, 266f23
overlapping realms, generally
174
sphere, surface area of
258, 260â261
tonal space, overlapping realms (see Timaeus octave phenomena)
Triangle of Convergence
250â251, 251f19
world body, parts
65
Aristotle
xxq, 21, 21q, 22, 33â34q, 181q
Aristoxenus
age, relative to contemporaries
5, xviiin21, xixn28
diatonic genus, and
xxi, 132n16
Eratocles, theoretical foundation for
xix
fourth century octave types, evidence of
xviii
GPS, history of
140
harmonia, approach to
5, 11, 140â141
âharmonicist,â identity as
xix
octave genera and tonoi, knowledge of
xviiin20, xix, xxi
Ptolemy, and
5, xix
Pythagoreans, difference from
5
quarter tone enharmonic and
131
tonoi, notion of
xix, 309Ap3n6
arithmetic
Crantor matrix after Platoâs means operation
76f7
doubles and triples after Platoâs means operation
78t8, 78t9, 79t10, 79t11, 96t18
4/3 intervals after Platoâs means operation, study
xxi, 71, 72, 71n53, 72t3, 73, 99â100t20
4/3 intervals after Platoâs means operation, traditional
73t4, 75t6, 77t7, 86t14
number kinds, relationships
xiiin6, 31â35, 32n44&45, 33q, 33â34q, 52n73
studyâs method and
60 (see also studyâs method)
armillary sphere, Timaeus interpretation
70â71
arrangements
diatonic scale steps
188â189, 560GL
doubles
86â87t15, 88â90t16, 195t37, 211t41, 222t45, 557GL, 562GLt64
fourth
102t21, 557GL, 305Ap2
fifth
557GL
model octave
56, 133, 316Ap3, 562GL
polis
272q, 273q, 274q, 272â274, 275q
primary Timaeus scale
175â178t34
Timaeus octave strings (see Timaeus octave phenomena)
triples
86â87t15, 91â95t17, 558GL
UPS systems (see Timaeus octave phenomena)
artificer
analogy, use of in creating
48q
armillary sphere and
71
goodness of
17q
harmonia, source of
48q, 71
indissolubility of universe and
48q
number, source of
64q, 64â65
ordo of creation, world soul before body
18, 18q
universal cause and governor
16, 17q, 48q, 51â52q, 60q, 64â65, 64q
universe, creation of as happy god
51â52q
universe, form of body and
51â52q
world soul, commixture and
60q, 64, 64q
See also demiurge
ascending scale
chromatic types (in studyâs analysis)
212â217t42, 218, 220t43, 431â464Ap8
diatonic types (in studyâs analysis)
196â203t38, 203â204, 346â384Ap5, 385â406Ap6, 407â430Ap7
enharmonic types (in studyâs analysis)
229t47, 230, 230â238t48, 496, 496â553Ap10
Handschin, Jacque and
8
Levin, Flora and
8
McClain, Ernest and
8
nature of
xvâxvi
Nicomachus of Gerasa and
6
scale ambiguity (see under ambiguity)
structure of (for all octave species and genera by reverse order inference)
222t45, 562GLt64
α-string
chromatic
212â213t42, 220t43, 432â435Ap8
diatonic
behavior
Appendix 7
408â409, 410â411, 413, 415, 418, 420, 422, 425â429
text
197t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 230â231t48, 501â507Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
asymmetry
90t16, 95t17
Augustine
xiii, xxiii, xxiiin2, xxiv, xxivn3, 53n75
See also Selected Bibliography
bands
xiv, xxii, 13q, 19â20, 28â30, 264, 270
analogy and
49, 49q
band broad (see under construction: cosmos Ï (chi figure))
band narrow (see under construction: cosmos Ï (chi figure))
difference relation to
262, 263â264
generation of
20, 28â30, 30â35, 51
harmonia and
28
harmony of the spheres, and
262
microcosmic plane, relation to
28, 35â49, 50â54
middles of
251f19, 252â255, 256f21, 256t49
motions of
19, 261â262, 263â264, 268â269
octave bands, relation to
243f15, 266f23, 264â269
sameness, relation to
262â263
soul sphere and
257â258, 257f22, 261â271, 266f23, 270f24
soul stuff and
28, 29, 30
subbands of difference and
264â269, 266f23
symmetry and
49, 49q, 253
tetractys (tetraktys), and (see Decad: generation; demiurge: harmonic method of (pattern of))
triadic monad, relation to
28, 29, 30, 258, 263, 264
universe and relation to body
20
Barker, Andrew
Boethius (as Philolaic source) and
134
GPS history, account of
xviin15, xviiiâxxi, xxin35, 134â135, 136n28, 305Ap2, 307Ap2
studyâs dialogue with
134â136, 305â307Ap2
works (see Selected Bibliography)
beauty
xxii, 16, 17q, 47q, 51q, 60, 67q
becoming
analogy, bonds of
47q
being and space, relation to
16
bodies (see bodies: generation)
commixture and
64
cosmogenesis and
19 (see also construction: cosmos)
Decad and (see Decad: generation; demiurge: harmonic method of (pattern of))
dimensionality and
65, 68, 70â71
fittingness and
51q, 64
intermediaries and
18 (see also intermediation; mediation)
medium of
20
number and time, relation to
53q
ordo of (see Decad: generation; demiurge: harmonic method of (pattern of))
precosmic state of
26
Receptacle and
20, 22, 23q
unity and
duad, determination of
14, 22, 28, 29
fount as image of
67
generation, return to
35, 68, 191
limit, association with
14, 22
number, procession of
65
universe, analogy for
48
World Soul and
63 (see construction: cosmos (World soul sphere))
being
becoming, distinction from
16
body and
16
Decad and
xiii, 1n3, 15, 30, 124, 285
harmonia and
12, 13, 13n56, 53n75, 54n76, 55, 60, 124
immutability of
16, 53q
impartibility of
29, 60q, 61, 62
intellect and
16, 17q, 18
intermediaries and
18
number and
1n1, 34, 53q
precosmic state of
26
Receptacle and
20
rhythm and
53n75
sameness of
16
soul stuff and
29, 60q, 61, 62
structure of
12, 13
unity and
66
universe and
12, 25, 51q
World soul and
xiii, 15, 16, 54n76, 59
best of causes
16, 17q
bibliography
564â569Bib
bodies
cause, necessity for
16
celestial bodies
14
chaos and
20n16, 25, 26, 27q
cosmic motions, and
22q, 23, 23q, 26, 26q, 49, 49q, 50
Decad and
35, 35â50
demiurge and
17q, 18q, 28, 48q, 49, 49q, 60q
divisibility and
29, 60q, 61â63
elements of
32
Errant Cause and
20n16, 25, 27q
generation and
35â50, 66, 71
geometry and
28, 35â50
human body
15, 19, 23q, 24, 27q, 50
motion requisites for
27q
pitch and
9
primary bodies, ordo and relations
19, 22q, 23, 23q, 26, 26q, 41â50, 42t1
qualities of
46â47, 47q
Receptacle and
20, 21q, 22q, 25, 26, 26q, 28
universe, body of
20n15, 59n17
analogy, creation by
48q, 24â25q
divinity, relation to
25, 51, 51q, 52, 52q
motion suited to
51â52q
World Soul, relation to
17q, 18q, 20, 27, 27q, 41â42, 52q
Boethius
2, 134, 136
See also Selected Bibliography
bonds
beauty of
47
body/soul relation, relevance to
15, 24q
demiurgic action and
47q, 71q, 76q
harmony as
14, 15, 53
intermediation and
76q
intervals and
76q
means and
33q, 77, 77t7, 86t14
source
47q, 71q, 76q
universe, in relation to
14, 47q, 36, 53
Brisson, Luc
diagram, use of
77n59, 77t7, 78t9, 79t11, 86t14
Timaeus, interpretation of
armillary sphere
70
mathematical model, advocate for
xxii, xxiin39
means, insertion of
71n53, 72n54, 72n55, 73n56, 77
musical interpretation, rejection of
xi, 57â58, 179
scale limitation incoherency of
11, 56n4, 56n5, 58n13
sesquitertian parts
72n55, 77, 77t7, 84â85, 86t14, 84t13
studyâs departure from
xi, 73n56, 77â81, 82â100
Timaeus scale extent of
11
works of (see Selected Bibliography)
β-string
chromatic
213t42, 220t43, 435â438Ap8
diatonic
behavior
Appendix 7
409, 411, 414, 415â416, 418, 421, 422, 425â430
text
197â198t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 231â232t48, 507â513Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
Burkert, Walter
13, 31n43, 32, 33, 33â34q, 82n1
See also Selected Bibliography
Calcidius
21, 60, 61, 65â67
See also Selected Bibliography
celestial bodies
14, 261â270, 270f24
central fire (Philolausâ cosmic theory)
14
CF1-string
chromatic
216t42, 220t43, 453â456Ap8
diatonic
behavior
Appendix 5
347â349, 350â351, 352â353, 354â356, 357â359, 360â362, 364â365, 366â368, 369â370, 372â373, 374â375, 376â377, 377â378, 379â380, 380â383
text
201â202t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 235â236t48, 533â538Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
CF2-string
chromatic
216â217t42, 220t43, 456â459Ap8
diatonic
behavior
Appendix 5
349â350, 351â352, 353â354, 356â357, 359â360, 362â363, 365â366, 368â369, 371â372, 373â374, 375â376, 377, 378â379, 380
text
202t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 236â237t48, 539â544Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
change
chaos and
27
chromatic elements and
59, 263, 264 (see also Timaeus octave phenomena; trihemitones)
difference, relation to
264
Errant cause and
26â27
generated things and
16
generation and
263
indeterminacy and
27
octave genus and
xviii
One (unity), relation to
66â67
order and
26â27
primary bodies and
23, 23q, 26, 26q, 45, 46
Receptacle and
22â23, 23q, 26, 26q
sameness, relation to
263, 264
See also difference; differentiation; generation
chaos (chaotic)
abstract character of
27q
change and
27
chromatic invasion absence in
185â186, 209â210, 209t40
corporeality and
18
cosmic circular motions, relation to
26, 26q, 50
cosmic order, relation to
26, 26q, 27, 50, 59
Errant Cause and
20n
form and
27
heterogeneity and
25
indeterminacy and
25, 27
limits of
27
ordered motion relation to
25, 26, 26q
motion feature of
25, 59
Receptacle and
25, 26, 26q, 27
rectilinear motions and
20n16
chi (Ï)
construction of (see under construction: cosmos)
diagram, chi (Ï) figure
256f21
soul sphere, formation of (see construction: cosmos (World soul sphere))
Timaeus musical data, relation to
xiv, 243f15, 243â244, 246
chromatic element (definition)
59n16, 138
chromatic sequences, Timaeus
apotomÄ (2187/2048 ratio), relation to
118, 219, 248 (see also apotomÄ)
Archytasâ chromatic fourth
305Ap2
chromatic octave occurrences (Timaeus set) (see under Timaeus octave phenomena: α-CNF2 strings)
chromatic octave species (ordo)
211, 211t41, 222t45, 562GLt64, 563GLt65
diatonic sequences, relation to
310â312Ap3
first chromatic fourth, Timaeus set
127t26, 130, 431Ap8
first chromatic octave, Timaeus set
211, 212
method of discovery
211â212
trihemitones and (see trihemitones)
undivided TS and
311â312Ap3
UPS systems and (see Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
circle
analogy, Greek music
240
band of difference, subdivisions and motions
19â20, 210, 241q, 242, 264â265, 268
chi (Ï) figure, construction of
257, 258
circles of same and different, motions of
13q, 19â20, 28, 210, 241q, 242, 261â263
cosmic harmony, use in sounding
246, 261
demiurge and
52q, 241q
equilateral triangle and
37
Ï (pi) and
563GL
planets in orbits
269, 270f24
Platoâs Pythagorean network
15
polygon, limit of
51n70
scalene triangle and
37n54
soul stuff divisions, relation to
36
studyâs analysis, verification of
119
Timaeus scale phenomena
239â240
time and
13, 53q
universe, motions of
13q, 50, 51â52q, 210, 241q, 242, 261â263
See also circuits
circuit
26q, 28, 35â36, 50, 245
circumference
22, 22q, 240
CNF1-string
chromatic
217t42, 220t43, 459â463Ap8
diatonic
behavior
Appendix 6
386â388, 390â391, 392, 393â394, 395â396, 397, 398â399, 399â400, 401, 402â403, 404â405
text
202â203t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 237â238t48, 545â550Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
CNF2-string
chromatic
216â217t42, 220t43, 463â464Ap8
diatonic
behavior
Appendix 6
389, 391, 393, 394â395, 396â397, 397â398, 399, 400â401, 402, 403â404, 405â406
text
203t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 238t48, 550â553Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
coagulation
22, 26, 50
comma (komma)
226â227, 556GL
commensurability
49, 255, 305Ap2n2
commentator (Platoâs works). See individual entries
commixture
54n76, 63
commotion
22â23
See also change; difference; construction: cosmos (motions of)
complexity
cosmic bands, and
28, 30
dodecahedron and
42n58
generation and
28, 40â41, 285
sesquioctave operation and
112
soul stuff, and
28
studyâs analysis, and
114
Timaeus number set
114
umbrella concept, generation
42â46, 42t1
world soul, monadic composition of
28, 30
composition
artificer, action of (see demiurge)
bodies, mode of
35 (see also construction: bodies; construction: cosmos (plane figures; primary bodies))
chi (Ï) and (see under construction: cosmos)
Decad, role in (see construction: Decad, role in))
dodecahedron
20n15, 37â38, 41â42, 42n58, 44n59, 59n17
doubles and
86â87t15, 88â90t16, 195t37, 211t41, 222t45, 557GL, 562GLt64
fourth and
102t21, 557GL, 305Ap2
fifth and
557GL
harmonic unit and
259
number fabric, and (see construction: cosmos (number matrix (fabric))
plane figures and
(see construction: cosmos (plane figures))
polis, and
272â284
primary bodies and (see construction: cosmos (primary bodies))
primary Timaeus scale and
175â178t34
soul sphere and planetary orbits, and (see construction: cosmos (World soul sphere))
soul stuff and
28, 38
Timaeus octave strings (see under Timaeus octave phenomena)
Timaeus scale perfect disdiapason
133
trihemitones (see trihemitones)
triples
86â87t15, 91â95t17, 558GL
universe, body of
19, 20, 20n15, 41â42, 50, 51q, 59n17
UPS systems (see under Timaeus octave phenomena)
See also construction; patterns; extensions
configuration
19n13, 28
See also construction; Decad; patterns; Timaeus octave phenomena; trihemitones
confusion
17q, 18, 48
congruency
36
See also equilateral constructions; isosceles figures
consonance
epimoric ratios and
305Ap2, 305Ap2n2, 306Ap2â307Ap2
intervals, comparison of
305â307Ap2
Laws, interpretation of
276â284, 276t50, 278â279t51, 282â283t52
major sixth and
56n4, 69n46
musical
1n2, 69, 305Ap2
Pythagorean mathematics, and
11
superparticularity and
xvii, 305Ap2, 305Ap2n2, 306Ap2
See also individual entries for musical intervals
construction
bodies
four species of animals
24, 24q
human beings
19, 23â24, 23â24q
plane figures (see construction: cosmos (plane figures))
primary bodies (see construction: cosmos (primary bodies))
mode of
35
Receptacle and
20, 21q, 22q, 25, 26, 26q, 28
universe, body of
19, 20, 20n15, 41â42, 50, 51q, 59n17
cosmos
artificer, action of (see demiurge)
motions of
disordered (Receptacle)
20n16, 23q, 25, 26, 26q, 27, 27q, 50
ordered
13q, 50, 51â52q, 210, 241q, 242, 261â263
bodies, relation to
22q, 23, 23q, 26, 26q, 49, 49q, 50
circles, same/different
13q, 19â20, 28, 210, 241q, 242, 261â263
divisions, band of difference and
19â20, 210, 241q, 242, 258, 264â269, 270f24
human thought, relation to
50
number matrix (fabric) and
figures and tables
66f5, 75f6, 76f7, 85f8, 98f9, 106f10, 107f11, 108f12, 109t24, 111t25, 126â129t26, 192f13, 243f15, 247f16
text
55â123, 241q, 289â304Ap1
plane figures and
35â39, 37f2, 38f3
primary bodies and
40â49, 42t1
Ï (chi figure)
chromatic factors, importance of
xiv, 243f15, 243â244, 246
cosmic bands
xiv, xxii, 13q, 19â20, 28â30, 264, 270
broad
band of different, identity with
262, 263â264
chi (Ï) formation and
250, 251f19, 251â253, 254f20, 256t49, 256f21, 255â257
divided character of
9â20, 210, 241q, 242, 264â269
doubles, relation to
263â264
motion of and musical potential
261â262, 264
nonchromatic numbers, relation to
266f23
narrow
band of same, identity with
262â263
chi (Ï) formation and
250, 251f19, 251â257, 254f20, 256t49, 256f21
chromatic numbers, relation to
243f15, 266f23
motion and musical nonpotential of
261, 262, 264
primacy of
28, 264
triples, relation to
262
undivided character of
264
superimposition of middles
252â255, 251f19, 256f21, 256t49
fabric ratios, effect on direction
251â252
triadic monad and
28â29, 30
Decad, role of
29â31
diagram, chi (Ï) figure
256f21
fabric of numbers, related to
109t24, 126â129t26, 243f15, 247f16
fabric operations necessary to
249f17, 251f19, 254f20, 256f21, 256t49
soul stuff and
28
10077696
251f19, 252â255, 256f21, 256t49
13824
251f19, 252, 256f21, 256t49
Timaeus musical data, relation to
xiv, 243f15, 243â244, 246
Triangle of Convergence and
249f17, 251f19, 249â252
World soul sphere
chi endpoints, joinder
257f22, 257â258
division broad and narrow bands
broad band consequences
doubles, relation to
263â264
motion of and musical potential
261â262, 264
narrow band consequences
motion and musical nonpotential of
261, 262, 264
triples, relation to
262
further divisions, broad band
264â269, 266f23
planetary orbits
268â269
resulting cosmos
270f24
triadic origin
258
universe, body of
19, 20, 20n15, 41â42, 50, 51q, 59n17
Decad, role in
xiii, xiiin6, xviiâxviii, 1n3, 15, 285
Ï (chi figure)
29â31
chromatic progression and
186â187
cosmic disdiapason
54, 55â56, 59, 60, 124
fifth periodicity and
189t35, 189t36, 191, 193
plane figures and primary bodies
30â50, 45f4
trihemitones and
465Ap9
varieties of mediating number, in relation
31â35, 31n43, 32n45, 33q, 33â34q, 34n51, 35n53
ideal polis, and
272â284
musical scales and systems, and
rules
doubles
86â87t15, 557GL
fifths
557GL
fourths (avoiding UDOP)
102t21, 204â205, 205t39
triples
86â87t15, 558GL
scale patterns
(see Timaeus octave types)
strings: octaves α-CNF2 (see Timaeus octave phenomena: α-CNF2 strings)
Timaeus scale primary
175â178t34
UPS systems (see under Timaeus octave phenomena: α-CNF2 strings)
Φ (phi) and
122â123, 122n16, 123n17
whole number ratios approximating irrational intervals
117â118
See also demiurge: harmonic method of (pattern of); Decad: generation)
contiguity
GPS/LPS modulations and
318Ap3
homestead portions, ideal Laws city state
274qq
octave chains and
207, 212, 347Ap5, 385Ap6, 407Ap7 (see also under Timaeus octave phenomena)
possible absence of
240
primary Timaeus scale, data concerning
175â178t34
UPS systems, data concerning (see under Timaeus octave phenomena: α-CNF2 strings)
continuity
alternative GPS and LPS, presence of
317Ap3
Laws octaves, discernment of
280â281
primary, Timaeus scale
174â175
Pythagorean framework and
306Ap2n6
standard GPS and LPS, lack of
317Ap3
See also contiguity; Timaeus octave phenomena
continuum, dyad of more and less
47n65
cooperation
50, 113â114, 114q
See also harmonia
Cornford, Francis MacDonald
Pythagorean doctrine of numbers and
1n1
Timaeus interpretation and
armillary sphere
70
corporeal elements vis-Ã -vis world soul
precosmic state
18n8
departures of this study from
xi, 61â63, 70â71, 179
dodecahedron, shape of universe
20n15, 59n17, 61
harmonic analysis, rejection of
xii, 55n2, 56n4, 57â58, 57n10, 179
harmonic means and
72n54
Platoâs project, opinion
70
Plato Pythagorean sources
1n1, 70
Receptacle
21, 21q, 27, 27q
world soul mixture
61
corporeality
17â18, 25
See also bodies; construction: cosmos (plane figures; primary bodies)
cosmic bands. See bands; see also under construction: cosmos Ï (chi figure)
cosmogenesis. See demiurge; construction: cosmos
cosmogony
xxii, 7, 13, 15, 19, 82
See also demiurge; construction: cosmos
cosmology
cosmogony and
xxii, 19
human soul and body, relation to
19
Laws
14n64
âMyth of Erâ
14n64
Timaeus
xiâxii, 14â15, 14n64, 58, 112, 132, 271
See also demiurge; construction: cosmos
cosmos
construction of (see demiurge; construction: cosmos)
motions of (see construction: cosmos (motions of; demiurge: cosmic motions))
musical paradigm and
Cornford, rejection of
xii, 55n2, 56n4, 57â58, 57n10, 179
Brisson rejection of
xi, 57â58, 179
finitude of universe, and
191
human realm, relation to
xiii, 15â16, 35, 272â284
Pythagorean influence on
1n1, 15, 70
studyâs argument and
xiâxvii, 1â15
See also demiurge; construction: cosmos; universe
Crantor matrix. See construction: cosmos (number matrix (fabric))
Crantor, Timaeus, interpretation of
8n34, 64, 64n33
creation
Augustine and
xxiii
bands and
xvi
doubles and triples and
29n40
four elements and
46q
four species, animals and
24, 24â25q
myths and
284
order of knowledge, relation to
35n53
soul stuff and
63â64, 64q
universe and
1, 18, 20
World soul and
27q, 55
See also construction: cosmos; construction: Decad, role in; demiurge: harmonic method of (pattern of)
Creator. See demiurge
Creese, Donald
ancients, pitch differences and
8â9, 131n7
GPS/LPS and
140
harmonic theory, fourth century developments and
xviin15, 140, 140n36
Timaeus, interpretation of
10, 132â133
Critias (Timaeus interlocutor)
16
cube
dimensionality and
65, 67â68q, 68â69
number (cube root, perfect cube)
109t24, 126, 126t26
primary body and
40â46, 40n56, 42n58, 42t1, 44nn59â60, 45f4, 46n61, 47â49, 115n10
cycle
ancient Near Eastern music
xxi, 132
Decad and
34, 187 (see also construction: Decad, role in; demiurge: harmonic method of (pattern of))
Eratoclesâ octave analysis and
xvii, xix
generative cycles (see demiurge: harmonic method of (pattern of); Decad: generation; Decad: pattern of)
Timaeus fifth periodicity and
187â193, 189t35, 189t36
Timaeus octave cycles and (see Timaeus octave phenomena)
Damon of Athens
14
Decad
All Perfect Animal, relation to
xiii, 54, 55, 56, 59, 60
harmonia and
xvi, 54â56, 59â60, 124, 285
analytical method and
3, 48, 48q
Augustineâs De musica, relation to
xxiii, 53n75
binding power of
54n76, 63, 63n30
various generational schemes
homogenesis
19, 24, 50
plane figures
39
primary solids
40, 46, 48q, 49
(see also Decad: God-given method; demiurge: harmonic method of (pattern of))
cosmic limit
191
cosmic order and
59, 187
demiurge and
404Ap6
harmonic method of (see demiurge: harmonic method of)
dialectic and
47n63, 49n68
dodecahedron and
59n17
duad (dyad) and
xiiin6, 187, 191
eightfoldness and
xvi
exemplar (paradigm) and
xvi, 1n3, 32, 191, 285 (see also Decad: All Perfect Animal, relation to)
fifth periodicity and
189t35, 189t36, 191, 193
figurate number, relation to
32
fourfoldness, and
xvi, 15, 46, 55, 56, 186, 191 (see also Decad: tetractys; Decad: tetrad)
generation and
xiii, xiiin6, xviiâxviii, 1n3, 15, 285
Ï (chi figure)
29â31
chromatic progression and
186â187
cosmic disdiapason and
54, 55â56, 59, 60, 124
fifth periodicity and
189t35, 189t36, 191, 193
plane figures and primary bodies
30â50, 45f4
trihemitones and
465Ap9
varieties of mediating number, in relation to
31â35, 31n43, 32n45, 33q, 33â34q, 34n51, 35n53
God-given method and
49, 49n68
grammar and
35, 39, 40, 42, 45, 191, 465Ap9
harmonia and
30â31, 31n43, 31n43q, 35, 285
fourfoldness and
55
plane figures and
39
primary bodies and
40, 43
primary Timaeus scale and
55â56, 60, 187, 191, 285
human perception and
xiii, 34, 35
ideal number and
31â35
immutability of
30, 34 (see also Decad: ideal number)
mathematical number and
33, 34â35
number, as nature of
34
missing fourth guest and
1
octave, ancient association with
31n43, 31n43q
pattern of
emanation and return and
35
exemplary character and
xvi, 404Ap6
fourfoldness and
55
generation and
xiii, 30, 39, 42, 43, 49, 285
primary solids and
45f4, 46
immutability and
34
mathematics and
35
Pythagorean number theory, objective
34n51
structure and
31, 31f1, 124
transcendence and
32, 34
primary Timaeus scale and
xvi, 55â56, 59, 124, 180, 126â129t26
duadic self-return
187, 191
fifth periodicity and
189t35, 189t36, 191, 193
fourfoldness of
56, 191
incomplete octaves, and
187, 191, 195
triad and
180
trihemitones and
465Ap9
Pythagorean thought and
All Perfect Animal and
xiii, 59
fourfoldness and
46, 46q
number, nature of
34
pattern and
34n51
sacredness and
46, 46q
tetractys and
31n43, 31n43q, 32, 32n45, 180
Speusippus and
xiii, 1n3, 124, 285
structure and
31, 31f1, 186
âten-nessâ and
32, 34, 56
tetractys (tetraktys) and
31, 31n43, 32
tetrad and
xvi, 30, 32n45, 285
Timaeus, limiting concept
379Ap5
time/eternity distinction and
35, 54
transcendence and
30, 33, 34
trihemitones and
465Ap9
Twenty-seven and
120
decay
xv, 51q, 187â193, 189t35, 189t36
See also degeneration
degeneration
cosmic bands, relationship to
264
enharmonic octaves and
132
Laws octaves, prediction
280, 281
primary Timaeus scale
xv, 12, 29n40, 56, 128â129t26, 130, 175, 184
chromatic invasion and
185â186, 187
fifth periodicity and
187â193, 189t35, 189t36
secondary scales and
195, 207
chromatic invasion and
207 (see also Timaeus octave phenomena: α-CNF2 strings)
study, as concern of
194
Timaeus UPS systems, and
185
δ-string
chromatic
214t42, 220t43, 440â443Ap8
diatonic
behavior
Appendix 7
409â410, 411, 414, 416â417, 418, 421, 423, 425â428
text
199t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 233â234t48, 516â521Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
Î-string
chromatic
215â216t42, 220t43, 450â453Ap8
diatonic
behavior
Appendix 7
408, 410, 412â413, 414, 418, 420, 422, 425â427
text
196t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion, pattern
209t40
enharmonic
229t47, 220t43, 496â501Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
demiurge (artificer; Creator; divinity, father; god)
alternative names, action under
artificer
16, 17q, 48q, 51â52q, 60q, 64â65, 64q, 71
Creator
62q
father
16, 18, 50, 52q
god (or divinity)
17q, 18, 18q, 24â25q, 47â48q, 49q, 51â52q
analogy and
48â49
chaos and
27q
cosmic motions and
28â30
Decad and
404Ap6
fabric of numbers and
28
form of universe, similarity to
50
harmonia and
3, 28
harmonic method of
Augustineâs De musica, relation to
xxiii, 53n75
binding power of
54n76, 63, 63n30
pattern of
birdâs eye view
28â31
Decad and
39, 40 (see also Decad: pattern)
fourfold progression of
xiv, 43, 45, 47â48q
harmonic pattern/principle of
13, 13q, 15, 28, 31, 39, 42, 43, 47â48, 47â48q, 50
mediation as feature
47qâ48q
same/difference, relation within
30, 39, 45, 285
specific examples
chromatic invasion
186â187, 209t40
Laws octaves
277, 280, 281
plane figures and primary bodies
36â39, 39â42, 45â47, 68, 68q
trihemitone patterns
495Ap9
generative pattern and
28 (see also Decad: generation; Decad: pattern; demiurge: harmonic method of)
number and
49
ordo of creation, World soul before body
18, 18q
primary bodies and
28, 49
Receptacle and
25, 27, 28
soul stuff commixture and divisions (before sesquioctave operation)
3, 28, 61â64, 71
spherical shape, similarity to
50
symmetry and
49
Twenty-eight stringed lyre and
113
World soul/body connection and
20, 27q
derivation
Archytan fourths
11, 305Ap2
chi (Ï) (see construction: cosmos (number matrix (fabric); construction: cosmos (Ï) (chi figure))
chromatic genus from diatonic
310â312Ap3
cosmic orbits (see construction: cosmos (World soul sphere))
doubles
86â87t15, 88â90t16, 195t37, 211t41, 222t45, 557GL, 562GLt64
enharmonic genus, from diatonic
310â312Ap3
fabric of numbers (see construction: cosmos (number matrix (fabric))
harmonic proportions, from Crantor lambda
68â70
matrix numbers (fabric) (see construction: cosmos (number matrix (fabric))
model octave
56, 133, 316Ap3, 562GL
perfect disdiapason (first in set)
136â138, 137â138t27
Philolaic scale
135â136
primary Timaeus scale
175â178t34
sesquitertian parts to fill (36B)
figures and tables
84t12, 84t13, 85f8, 86t14, 86â88t15, 88â90t16, 91â95t17, 96t18, 97â98t19, 98f9, 99â100t20
text
83â100
seven tonoi in all genera (see Timaeus octave types)
steps of diatonic scale
57n7, 70n50, 188, 559GL, 560GL
Timaeus octave strings (see under Timaeus octave phenomena)
triples
86â87t15, 91â95t17, 558GL
UPS systems (see under Timaeus octave phenomena: a-CNF2 strings)
World soul sphere (see construction: cosmos (World soul sphere))
descending scale
chromatic types (in studyâs analysis)
212â217t42, 218, 220t43, 431â464Ap8
diatonic types (in studyâs analysis)
196â203t38, 203â204, 346â384Ap5, 385â406Ap6, 407â430Ap7
enharmonic types (in studyâs analysis)
229t47, 230â238t48, 496â553Ap10
Handschin, Jacque and
8
Levin, Flora and
8
McClain, Ernest and
8
nature of
xvâxvi
Nicomachus of Gerasa and
6
scale ambiguity (see under ambiguity)
structure of (for all octave species and genera)
222t45, 562GLt64
diameter (soul sphere)
258, 260â261
dianoëtic energy
52q
diapason (definition)
xvi, 86â87t15, 195t37, 211t41, 222t45, 555GL, 557GL, 562GLt64
See also octave
diapason diapente (definition)
69â70, 86â87t15, 555GL, 558GL
diapente (definition)
69, 126t26, 188n66, 554GL, 557GL
See also fifth (diapente; sesquialter part)
diatessaron (definition)
69, 126t26, 554GL, 557GL
See also fourth (diatessaron; sesquitertian part)
diatonic scale steps
188â189, 560GL
diatonic sequences, Timaeus
Archytasâ diatonic fourth
305Ap2
diatonic octave occurrences (Timaeus set) (see under Timaeus octave phenomena: α-CNF2 strings)
diatonic octave species (ordo)
195t37, 211t41, 222t45, 562GLt64
first diatonic fourth, Timaeus set
126t26, 127t26
first diatonic octave, Timaeus set
56, 126t26, 128t26, 133, 316Ap3, 562GL
leimma (256/243 ratio), relation to
2â3, 556GL
method of discovery
195
other genera, relationship to
310â312Ap3
primary Timaeus Scale
175â178t34
reciprocal character (seven octave species)
563GL
unaccustomed patterns (UDOP)
204â205, 205t39
specific instances
Appendix 5: 346â384Ap5; Appendix 6: 385â406Ap6; Appendix 7: 407â430Ap7
UPS systems and (see Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
diesis (leimma; 256/243 â 1.05349794238683)
ancient disputes concerning
4, 5, 7
apotomÄ, relation to
219, 219n4, 554GL
Archytasâ alternative
3, 11
Aristoxenus and
5
definition
2, 7, 101, 118, 126t26, 211, 218, 219, 226, 555GL, 556GL
fifth, relation to
69
fourth, relation to
2, 69
indivisibility of
2, 149, 222â223, 224â225
octave, relation to
2â3, 554GL, 556GL
1719926784, relation to
226â227
Philolausâ measure
4â5, 135
Pythagorean scale and
4
quarter tones, derivable from
149, 222, 222â223, 224â226
Timaeus number set occurrences
126â129t26
Timaeus scale occurrences (see under Timaeus octave phenomena)
trihemitones, relation to
219, 219n3
various calculations, use in
117, 118
whole tone, relation to
219n4, 554GL
diezeugmenon
142, 146t28, 558â559GL
difference
band(s) of (generally)
xiv, xxii, 19â20, 28â30, 241q, 242, 264, 270
broad band and (see under construction: cosmos (Ï (chi figure))
chromaticity and
125, 240
Crantor matrix, various stages (see construction: cosmos (number matrix (fabric))
fifth periodicity and
30
generation and, levels of (see demiurge: harmonic method of (pattern of))
Greek tonoi and modern key systems
310â312Ap3
harmony of the polis and
273â284
harmony of the spheres and
270
ideal numbers and
33q
levels of the universe, relationship
49, 50
musical intervals and
80, 86â87t15, 102t21 (see also individual entries for different types)
names, sesquiterian parts
103â106, 104t22, 105t23, 106, 289â304Ap1
number fabric, parts
109t24, 243f15, 266f23
number, kinds of
31â35, 52n73
octave, ancient approaches to (see entries for individual commentators, notably Archytas)
octave and fifth periodicities
187â191
octave genera and species, kinds of
70n50, 195t37, 211t41, 222t45, 562GL64
octave strings and (see under Timaeus octave phenomena; see also separate entries for α-CNF2 strings)
plane figures and
36â39
primary bodies and
39â42, 45â47
primary triangles and
36â39, 41, 46
sameness, relation to
13q, 19â20, 28â31, 36â39, 39â42, 45â47, 124
cosmic narrow band and, (see under construction: cosmos (Ï (chi figure)))
six rectilinear motions and
51q
soul stuff and
28, 60q, 61â63, 64q
Timaeus interpretations and (see separate entries for individual commentators)
Timaeus octave strings, differences among (see Timaeus octave phenomena: α-CNF2 strings)
trihemitones, differences among and (see trihemitones)
UPS systems, differences among (see under Timaeus octave phenomena: α-CNF2 strings)
World soul sphere, role in creation (see under construction: cosmos (cosmic bands; World soul sphere))
differentiation
30, 36â41, 125, 186, 210, 240
See also difference; construction; demiurge: harmonic method of: pattern of
Dillon, John
xiii, 1n3, 13, 30, 34n51
dimensions
of figures and bodies
40â41, 65, 68â69, 71
of octave strings
240 (see Timaeus octave phenomena: α-CNF strings)
of UPS systems (see under Timaeus octave phenomena: α-CNF strings)
disdiapason
chromaticity and
primary Timaeus scale
186â187, 187â191
secondary Timaeus scales
209â210, 209t40
cosmic structure and
xviâxvii, xxiii, 31, 55, 186, 126â129t26
Decad and
xvi, 31, 39, 124, 186â187
definition
xvi, 70, 555GL
dispondee, comparison to
xxiii
dodecahedron and
59n17
first perfect Timaeus disdiapason
128t26
female voice and
316â317Ap3
fourfoldness and
55
generation and
xviâxvii, 43, 44, 46, 55, 186â187
GPS (standard), importance to
139, 142, 308â309Ap3, 313Ap3
perfect designation, meaning of
133
Laws, occurrences in
277, 280, 281
LPS (standard), relation to
140, 142â143, 317Ap3
perfect disdiapasons (secondary Timaeus scales)
210, 408Ap7, 410Ap7, 411Ap7
Timaeus octave strings and (see Timaeus octave phenomena: α-CNF strings)
UPS (standard), relation to
140, 143â149, 145â149t28, 317â318Ap3
vocal range, relation to
140â141
disjunction
diapasons, possible internal feature of
69, 139, 555GL, 557GL
diezeugmenon and
558GL
GPS (standard), feature of
142, 143, 144, 558â559GL
LPS (standard)
feature of
142
GPS, relation to
142, 151
primary bodies, relations of
40, 40n56, 49
synemmenon difference from
560GL
Timaeus UPS
GPS
150
GPS/LPS relation and
143, 151â152
standard system, difference from
143, 150
Twenty-eight stringed lyre and
114q
disorder
chromaticity and
16, 58, 59, 125, 185â191, 240
demiurgic action and
17q, 18, 49, 49q
order, relation to
27, 49, 49q, 59, 185â191, 186, 187
âoriginally visible,â motion of
48
Timaeus scale (relation to order)
ixv, xv
universe, motions of
20n16, 23q, 25, 26, 26q, 27, 50
dissimilarity
band of difference and
241q
beauty and
51q
octave genera derivations (for different octave species)
311Ap3
demiurge and
51q, 241q
planetary speeds and
241q, 269n6
related octave strings
268, 269n6
distortions
chromatic phenomena and
xiv, 16, 58, 59, 125, 185â191, 240
band of same and
xiv
chromatic numbers, relation to
243f15, 244, 266f23
identity with narrow fabric segment
262â263
fifth periodicity and
187â191
LPS, necessity to
143, 317Ap3
primary Timaeus scale, effect on
xvi, 126â129t26, 185â187, 174, 175â178t34
Timaeas octave strings, effect on
209â210, 209t40 (see also individual entries for octave strings α-CNF2)
UPS systems, effect on
170, 173 (see also under Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
harmony of the spheres, reasons for freedom from
210, 246
perfect model disdiapason, best choice and
179
36 A triples division and
29n40, 30
distributions
householders (within polis)
272â273, 272â273q, 275
intervals and, within octaves species, by genus (see under Timaeus octave types)
octaves (within strings α-CNF2) (see under Timaeus octave phenomena)
240
tonal ranges, UPS systems (see under Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
trihemitones (see trihemitones)
ditone (81/64 ratio)
80, 131, 222t45, 253n2, 562GLt64
divinity
appearances and revelations
273q
creation, relation to (artificer, Creator, demiurge, father, god)
17q, 19, 47â48q, 49q, 50q, 51q
demiurgic intellect and
51q, 63n31
exemplar, as feature of
19, 50
mathematical character of
52
human soul, relation to
24q
ideal number and
xiiin6, 1n1, 46q, 52n73, 63n31
divisibility
band of same and
263
bodies, nature divisible about
29, 60â61, 60q, 62â63
chromatic nonfactors 46656 and 41472, relation to
120â121, 123
cosmic creation, relating to (see construction: cosmos; demiurge: harmonic method of)
cosmic size and
258â261
cube, dimensions of and
65
difference and
29, 60q, 61, 62, 63
divided line, analogy of Republic 509 Dâ511 E
67, 122n15
doubles (see under construction: musical scales and systems)
fabric (of numbers) and (see under construction: cosmos (number matrix (fabric))
fifths (see under construction: musical scales and systems (rules))
5040, factors of
276, 276t50
fourths
11, 305Ap2 (see also under construction: musical scales and systems (rules))
musical scales and systems relating to (see under Timaeus octave phenomena; trihemitones)
nine and
277, 280, 281, 284
octave and
2, 4â5, 12, 134â136, 224
octave chains, α-CNF2 (see Timaeus octave phenomena: α-CNF strings)
octaves (see Timaeus octave types; Timaeus octave phenomena)
1719926784
108, 109t24, 110, 115, 120, 123, 226â227
polis, harmonic roots of
276â284
primary Timaeus scale (see under Timaeus octave phenomena)
quarter tone and
149, 222, 222â223, 224â226
sameness and
263, 264
semitone and
2, 149, 222â223, 224â225
sesquitertion parts, and
101â123, 102t21
stability, relation to
264
trihemitone sequences, and (see trihemitones)
triples and (see under construction: musical scales and systems (rules))
UPS systems (see under Timaeus octave phenomena: α-CNF strings)
whole tone and
2, 3, 116, 224, 312Ap3
dodecahedron
20n15, 37â38, 41â42, 42n58, 44n59, 59n17
Dorian octaves, Timaeus
form (all octave genera)
222t45, 562GLt64
in α-CNF2 strings (see Timaeus octave phenomena: α-CNF2 strings)
in primary Timaeus scale
175â178t34
double. See diapason; octave
duads
broad band subdivisions, method
267
chromaticity and
185â187, 187â191
cosmic bands, as
30
cube and
45â46
Decad, relation to
32, 186â187, 191
demiurgic creation (plane figures)
36
diapasons and
191
diatonic octaves, alternative articulations
346â384Ap5, 388Ap6, 401Ap6
disdiapason and
191
equilateral triangle, relation to
36â37, 37f2
fifth periodicity and
187â191, 189t35, 189t36
fourfoldness and
191
harmonia, and
28, 29
identical tone sequences, alternative articulations and
349â372Ap5, 388Ap6, 401Ap6
indeterminacy, relation to
14, 28
limiter, relation to
22, 14, 28
monad, relation to
xvi, 14, 28, 29, 187, 190
diatonic octaves, alternative articulations
346â384Ap5, 388Ap6, 401Ap6
octave ratio and
xvi, 28, 191
order and
187, 185â187, 187â191
ordo of degeneration and
187, 185â187, 187â191, 285
ordo of generation and
plane figures
36, 37, 38, 39
primary bodies
40â46, 45f4, 47â49, 47â48q
primary Timaeus scale and
xvi, 185â187, 191
sameness/difference, relation to
30, 36
self-return, examples
39, 191
square and
38, 39
subdivisions, band of difference and
30
tetrad and
45, 191
Timaean fifth periodicity and
xvi, 191
triad and
29, 30, 36
unity, relation to
22, 28, 29
visibility/tangibility, Millerâs view
46â47
Ïá½½Ïα, relation to
22
earth
cosmic creation and
18, 46q
cube and
40, 42, 42t1
genesis of (as element)
19, 28, 35, 40, 42, 48
harmonia (bands of analogy and symmetry) and
49â50q
living creatures and
49, 50
mediation and
18, 48q
other primary elements, relation to
18, 19, 26q, 42, 42t1
precosmic state and
49â50q
Receptacle (see Receptacle: bodies (primary bodies))
tangibility, relation to
46
Egyptian Ï (pi) (256/81) (3.160493827 )
258, 258n3, 259â260
8 (number of primary sequence)
65t2, 66f5, 75f6, 114q
eightfoldness (duadic self-return)
xvi
81/64 (1.265625; measure of the enharmonic ditone)
80, 96, 132
82944 (begins and ends cycles of combined standard UPS in Timaeus set)
145t28, 149t2
elements (fire, air, water, earth). See fire; air; water; earth; tetrahedron; octahedron; icosahedron; cube; primary bodies
endpoints
Ï arms
257â258, 257f22
GPS and LPS, defining relation of
332t54, 344t56
number generation, importance to
97â98n3, 104, 105t23
quarter tone intervals, use in computing
223
10368 (Timaeus scale traditional)
56, 56n5, 179, 192f13
393216 (Timaeus, study)
124, 174â175, 178t34, 192f13
Timaeus octaves, use in defining
126â129t26, 346Ap5, 349Ap5, 382Ap5, 385Ap6, 387Ap6, 406Ap6
20736 (Timaeus Scale, alternative traditional)
57, 57n9, 179, 192f13
enharmonic sequences, Timaeus
Archytas enharmonic fourth
305Ap2
diatonic sequences, relation to
310â312Ap3
enharmonic octave occurrences (Timaeus set) (see under Timaeus octave phenomena: α-CNF2 strings)
enharmonic octave species (ordo)
222t45, 562GLt64
first enharmonic fourth, Timaeus set
127t26, 131, 132
first enharmonic octave, Timaeus set
126t26, 134, 230t48
ditone (81/64 ratio), size and role of
80, 253n2, 131
method of discovery
221â224, 496â553Ap10
quarter tone, as feature of
131â132, 221â224
reciprocal character, lack of
227, 563GLt65
semitone enharmonic and, QT enharmonic, relation to
131â132, 222
UPS systems and (see Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
epimoric (superparticular) ratios
11, 305â307Ap2, 305Ap2n2
equilateral constructions
36â39, 37f2, 38f3
square
composition of
38f3
generation of
37â38
other plane figures, relation to
37â39
triangle
composition of
36â37, 37f2
generation of
36
other plane figures, relation to
36â38
equilateral idea
41, 42, 45, 45f4
Eratocles
xvii, xviii, xix, xx, 140
Errant Cause (associated with Receptacle)
20n16, 25, 26â27, 27q
essences
eternal kind, timelessness and
52â53, 53q
fifth periodicity, comparison
29â30
impartible reality and
29, 60q, 61â63
Calcidiusâ identification as mind or intellect
60â61
number and
31n44, 32
sameness of being and
29, 60q, 61â63
soul stuff commixture and
28, 29, 60q, 61â63, 62q, 64q
world soul and
68q
ε-string
chromatic
214â215t42, 220t43, 444â447Ap8
diatonic
behavior
Appendix 7
410, 411â412, 414, 417, 419, 421, 423â424, 425â430
text
199â200t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 234t48, 521â527Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
eternity
25, 35, 52, 52â53q, 52n73, 53, 53n75, 54
Euclid
xvii, 11
Eulogius, Favonius
xiii, xxiii, xvii, xxiv, 186
exemplar
xvi, 16, 17, 19, 52, 52n73, 52q, 52â53q, 285
See also Decad: All Perfect Animal, relation to; All Perfect Animal
expansions
cosmic limit and
xxiii, 113, 258
harmonia and
258
Crantor matrix and
85f8, 98f9, 106f10, 123
Decad and
31, 56, 186
disdiapason as
187
monad/duad relationship and
187
octave possibilities and
346â384Ap5, 421Ap7
primary bodies, relations defining
44
Timaeus scale limit and
55, 56, 57, 113
triples, patterns for filling
95
Twenty-eight stringed lyre and
113
world/soul sphere measurement
harmonic unit and
258, 259, 260, 261
See also extensions
extensions (applications to, kinds of or limits to)
Brisson table and
77n59
cosmic motion and
25, 28, 269
cosmogenesis, relation to homogenesis
23, 23â24q
cosmos and
124, 187, 227
Crantor (number) matrix (fabric)
figures and tables
(see construction: cosmos (number matrix (fabric))
fractions, mode of finding in matrix
73
limits of
76, 107, 109t24, 113, 115, 119
1719926784, as last number
109t24, 115
related limits
116, 121, 123, 124
text, pertaining to matrix derivation
xi, 55â123, 99, 119, 241q, 289â304Ap1
Decad
extension of
31f1
other extensions, relation to
xiii, xiiin6, xviiâxviii, 1n3, 15, 285
Ï (chi figure)
29â31
chromatic progression and
186â187
cosmic disdiapason
54, 55â56, 59, 60, 124
fifth periodicity and
189t35, 189t36, 191, 193
plane figures and primary bodies
30â50, 45f4
trihemitones and
465Ap9
varieties of mediating number, in relation to
31â35, 31n43, 31n43q, 32n45, 33q, 33â34q, 34n51, 35n53 (see also Decad: pattern)
doubles and triples after 35A
77, 78t8, 78t9, 79, 79t10, 79t11, 84
double intervals, limits to
86â87t15, 557GL
fabric (of numbers) and
243, 252 (see also construction: cosmos (number matrix (fabric))
fifth intervals, limits to
557GL
fourth intervals, limits to (avoiding UDOP)
102t21, 204â205, 205t39, 305Ap2
GPS, range of
308Ap3, 558GL
human analysis and
35
intervals (musical) and
556 (see also separate entries for different kinds)
LPS, range of
559GL
model metaphysical harmony, Platoâs application of
15
octave strings (see Timaeus octave phenomena: α-CNF2 strings)
1719926784 and
chromatic factors and
108, 109t24
chromatic nonfactors and
109t24, 110, 120, 123
irrational values and
226â227
primary sequence, relation to
115
pitch, relating to vibrations
10
Platoâs life and
131
Platoâs primary sequence
65t2, 66, 133
primary Timaeus scale
175â178t34
endpoint, 393216 (Timaeus, study)
124, 174â175, 178t34, 192f13
traditional interpretations, difference from
58, 178, 179, 180
shapes, relation to number
32
studyâs method, factoring exercise
59
Timaeus scale and, traditional limits
56, 69n46
triple intervals, limits to
86â87t15, 558GL
10368 (traditional endpoint)
56, 56n5, 179, 192f13
20736 (alternative traditional endpoint)
57, 57n9, 179, 192f13
Timaeus style number extensions, limits, generally
116
Timaeus text and, reflective character of
19
Triangle of Convergence and
249
universe, extension of
cosmic disdiapason
54, 55â56, 59, 60, 69n46, 124
diatonic scale as symbol
227
finitude of
59
UPS systems and (see Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
World soul and
20, 52q
fabric (see construction: cosmos (number matrix (fabric))
cosmic bands and (see under construction: cosmos (Ï (chi figure)))
cosmic construction and (see construction: cosmos (number matrix (fabric)); see also, demiurge: number)
difference and (see under construction: cosmos (Ï (chi figure): cosmic bands))
divisions of (see under construction: cosmos; see also Timaeus octave phenemena)
musical scales/systems, construction (see under construction: cosmos; see also Timeaus octave phenomena)
musical significance of (see Timeaus octave phenomena)
number matrix (see construction: cosmos (number matrix (fabric)))
sameness and (see under construction: cosmos (Ï (chi figure): cosmic bands))
Ï (chi) formation (see construction: cosmos (Ï (chi figure)))
fabrication
16, 17q, 47q, 48q, 51q, 52q
See also construction; demiurge: harmonic method of (pattern of)
factoring
number generation and
Appendix 1
289n1, 289â304
text
59, 97â98t19, 99, 103â105, 104t22, 105t23, 111â112t25
studyâs method and
97â98t19, 99, 103â105, 104t22, 105t23, 111â112t25, 289n1
factors
5040 and
276t50, 278â279t51, 282â283t53
1719926784 and
110
chromatic factors and
108, 109t24
chromatic nonfactors and
109t24, 110, 120, 123
studyâs method, relevant considerations
19, 96, 101, 112
father (artificer, Creator, demiurge, divinity, god)
16, 18, 50, 52q
See also demiurge
female, suppression of
316, 316Ap3, 317Ap3
fifth (diapente; sesquialter part)
alternative UPS systems and
308Ap3, 313Ap3, 314Ap3, 315Ap3, 316Ap3, 320â321Ap4, 321Ap4, 322â332t53Ap4
apotomÄ, relation to
219n4, 554GL, 556GL
chromatic number diatonic strings, relation among
347Ap5, 386Ap6
composition of (see fifth (diapente; sesquialter part): definition)
cosmic motion, relation to
261, 264
Decad and
31n43
definition of
69, 126t26, 188n66, 554GL, 557GL
diapente, as ancient term for
188n66
diatonic octaves, alternative formulations
346â384Ap5, 385â406Ap6
diatonic scale rise, relation to Timaeus set
137â138t27
divisions, means operation and
71q
fifth periodicity and (see fifth periodicity)
fourth, relation to
3, 4â5, 11, 139, 306Ap2
generation and
43â44, 69
harmonia and
3, 4, 31n43
leimma and
2, 3, 5, 6
major sixth and
56n4, 57
musical twelfth, relation to
xiv
octave, relation to
3, 4â5, 11, 139
ordo of (see fifth (diapente; sesquialter part): definition)
Platoâs primary divisions and
64q, 65t2, 69
sesquioctave interval (whole tone) and
3, 4, 5, 86â87t15, 102t21, 139
standard Timaeus interpretation and
55, 56, 57, 56n4, 69n46, 178â180, 562GL
tetractys and
31n43
Timaeus number set occurrences
126â129t26
Timaeus scale occurrences (see fifth periodicity; Timaeus octave phenomena)
fifth century (musical history)
xvii, xviii, 9, 14, 130, 131, 132n16, 134, 496n1
fifth periodicity
analogy and
29
chromaticity and
29n40, 125, 190â192
Decad, relation to
191
definition of
187
difference, relation to
124
generative order, place in
30
Handshin Jacques and
xivâxv, 29n40, 187
Laws octaves and prediction
280, 281
rise and fall, symmetry of
58, 187â191, 189t35, 189t36
studyâs interpretation and
xvi, 12, 29, 30, 58, 124, 125
termination of, in triadic monad
58
Timaeus octave periodicity, relation to
xivâxvi, 58, 187â191, 189t35, 189t36, 191â192
fifth Platonic solid. See dodecahedron
54 (factor of 1719926784; also sum of its modern digits)
115
figurate number
31â32, 32nn44â45, 34n51
Figures 1â24
Fig. 12, Dot chart showing pattern of Crantor matrix emerging from the Timaeus
107, 108, 191
Fig. 15, Fabric to Cut
242, 243, 244, 245, 246, 261, 265
Fig. 16, Dot chart for Ï operation showing pattern of matrix
247, 248, 249, 250, 251
Fig. 19, Cut, slide, and rotation operation
250, 251, 252, 253, 255
Fig. 21, the Ï figure
255, 256, 257
See also LFT, xxvi
fire
cosmic creation and
18, 46q
genesis of (as element)
19, 28, 35, 40, 42, 48
harmonia (bands of analogy and symmetry) and
49â50q
living creatures and
49, 50
mediation and
18, 48q
other primary elements, relation to
18, 19, 26q, 42, 42t1
precosmic state and
49â50q
Receptacle and (see Receptacle: bodies (primary bodies))
tetrahedron and
40, 42, 42t1
visibility, relation to
46
576 (beginning number of the first standard diatonic GPS sequence in the Timaeus set)
142
531441/524288 (ancient comma â1.013643264)
226
5040 (number householders, ideal polis)
272
formation
human soul and music education
xi, 14
lambda and
64
See also construction
form
bodies and (see under construction: bodies)
chaos and (see chaos; construction: cosmos (motions of))
chi (Ï), 256f21 (see also construction: cosmos (Ï (chi figure)))
Decad and 31f1 (see also Decad: pattern of; Decad: generation)
demiurge and (see demiurge: harmonic method of (pattern of))
doubles (see under construction: musical scales and systems (rules))
fifth and (see under construction: musical scales and systems (rules))
fourth and (see under construction: musical scales and systems (rules))
lambda and
66f5
mediation and (see construction: Decad (role in); Decad: generation; Decad: pattern of)
number and (see construction: Decad (role in); Decad: generation; Decad: pattern of)
number matrix (see construction: cosmos (number matrix (fabric)))
octave species and types (see Timaeus octave types)
octave strings (see Timaeus octave phenomena: α-CNF2 strings)
plane figures (see construction: cosmos (plane figures))
primary bodies and (see under construction: cosmos (primary bodies))
primary Timaeus scale (see under Timaeus octave phenomena)
Receptacle and (see Receptacle: bodies; forms)
trihemitones (see trihemitones)
triples and (see under construction: musical scales and systems (rules))
universe, body of (see under construction: cosmos (World soul sphere))
UPS systems (see Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
World soul/ body sphere 270t24 (see also construction: cosmos (World soul sphere))
41472 (square root of 1719926784; bounds combined standard UPS in Timaeus set)
116, 118, 119, 120, 145t28, 149t28
46656 (product of primary sequence; 9/8 ratio with 41472)
115, 119, 120
4 (number of primary sequence)
65t2, 66f5, 75f6, 114q
fourfold relations
ambiguity and
86â87t15
chromatic invasion and
xiv, 186, 187, 191
Decad and
xvi, 15, 30, 31, 46, 55, 56, 187, 191 (see also Decad: pattern of)
duad and
xviâxvii, 187, 191
elements and
15, 42â43, 42t1, 44, 45
generation, ordo of
xiv, xiv, 255
cosmos
xviâxvii, 15, 30, 31, 55
planar figures
35â39
primary bodies
42â43, 42t1, 44, 45
harmonia and
31
perception, ordo of
species of animals and
49
symmetry and
44
Timaeus octaves and
56
trihemitones and
495Ap9
See also construction: cosmos; Decad: generation; Decad: pattern of; demiurge: harmonic method of (pattern of)
4478976 (apex of Triangle of Convergence)
249f17
fourth (diatessaron; sesquitertian part)
apotomÄ, relation to
219n4, 554GL, 556GL
composition of (see; fourth (diatessaron; sesquitertian part: definition of))
Decad and
31n43
definition of
69, 126t26, 554GL, 557GL
diatonic scale rise, relation to Timaeus set
137t27
divisions
71q, 76q, 99â100t20, 305Ap2
fifth, relation to
3, 4â5, 11, 139, 306Ap2
Timaean diatonic octaves, alternative formulations
346â384Ap5, 385â406Ap6
first chromatic fourth, Timaeus
127t26, 130, 431Ap8
first diatonic fourth, Timaeus
126t26, 127t26
first enharmonic fourth, Timaeus
127t26, 131, 132
generation and
43â44, 69
harmonia and
3, 4, 31n43
leimma and
2, 3, 5
octave, relation to
3, 4â5, 11, 139
ordo of (see fourth (diatessaron; sesquitertian part): definition)
Platoâs primary divisions and
64q, 65t2, 69
sesquioctave interval (whole tone), relation to
3, 4, 5, 139
sesquioctave division of fourths
76q, 86â87t15, 102t21
tetractys and
31n43
Timaeus number set occurrences
126â129t26
Timaeus scale occurrences (see Timaeus octave phenomena)
fourth guest
xxii, 12
4/3 (1.3 ; ratio of fourth) (see fourth)
4608 (ends 1st Timaeus set standard diatonic LPS and standard diatonic UPS sequences)
143, 144
γ-string
chromatic
213â214t42, 220t43, 438â440Ap8
diatonic
behavior
Appendix 7
409, 411, 414, 416, 418, 421, 423, 425, 426
text
198â199t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion pattern
209t40
enharmonic
229t47, 232â233t48, 513â516Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269
generation (as in creation)
analogy and
48q, 50
ambiguities and, Receptacle
25
artificer and
17q, 18, 52
cause, relation to
16
cosmic bands and
20, 28, 35, 263, 264
being and
16
bodies and
26q, 28, 37, 38, 42, 44n59, 48â49
cosmos and (see construction: cosmos)
Decad and
xiii, 30, 33, 34, 35, 39, 40, 49, 285, 465Ap9
dodecahedron and
20n15, 44n59
diatonic scale and
xiii, 70n50
different and
39, 46, 48, 264, 285
disdiapason and
31, 39, 46
duad and
349Ap5, 353Ap5
fourfoldness of
xiii, xiv, 15, 31, 495Ap9
god-given method and
49n68
harmonia and
31, 39, 40, 49, 53n75
polis and
272, 274, 276, 277, 284
mediation and
40, 42, 48â49
mode of (see demiurge: harmonic method of (pattern of); Decad: generation; Decad: pattern of)
monad and
39, 119
musical phenomena, scales and systems, generally, and
xiii, 31, 263, 264, 312Ap3, 349Ap5, 351Ap5
Eratocles and
xviii, xx
Laws octave scales and
280â281
musical intervals and
29n39, 53n75
trihemitones
465Ap9
number and
xxii, 29, 33, 34, 35, 52, 52n73, 53q, 70n50
Laws octave scales and
280â281
Platoâs divisions and
73, 75, 83, 125, 138
political harmonia and
277
Pythagoreans and
xxii
1719926784 and
119
Thomas Taylor and
67â68, 67â68q
ordo of
39, 40, 41, 48, 191, 263, 264, 265, 285
pattern of (see demiurge: harmonic method of (pattern of) Decad: generation; Decad: pattern of)
same and
39, 46, 264, 285
seven, special feature of
65
studyâs method and
xxii, 55, 59n16, 64, 76, 85, 97n3, 99, 103â107, 114, 124
tetrad and
39
time and
53q
triad and
39, 264
unity and
48
universe and
16, 17q, 18, 19, 31, 48, 48q, 51q, 52q, 60q
world soul and
60q
See also construction: cosmos (World soul sphere)
generator
numerical method of study
xxii, 55, 59n16, 64, 76, 85, 97n3, 99, 103â107, 114, 124
See also demiurge: harmonic method of (pattern of); Decad: generation; Decad: pattern of
genesis
chromatic elements and
59
cosmos and (see construction: cosmos)
four kinds of living creatures and
49
harmonia and
15
mediation and
49
ordo of
49, 264
solids (primary bodies) and
15, 19, 49
soul sphere and
264
terms, 1719926784 based Φ ratio sequence
122
See also construction; demiurge: harmonic method of
genus (music)
Archytan fourths and
305Ap2
kinds
xxâxxi, xxin35
history of
xviii, xxâxxi, xxin35, 132, 132n16
pitch and
559GL
structural relationships among
222t45, 562GLt64, 310â312Ap3
tonoi system, ancient Greeks
309â312
octaves, relationship to
xviii
Philolaus calculations and
3
primary Timaeus scale and
175â178t34
effect of chromaticity on
59n16
Timaeus, diatonic primacy in
xx, 124, 132
Timaeus scale patterns (see Timaeus octave types)
Timaeus strings of octaves: α-CNF2 (see Timaeus octave phenomena)
geometric means
67, 121, 122
geometric progressions. See geometric means
geometry
atoms and molecules
19n13
bodiliness and
23, 23â24q
chi (Ï) and
247, 256f21
cosmic bands and
20, 252â253
cosmology and
15
geometric means and progressions
66f5, 67, 109t24, 121, 122
metaphysics and
15
primary bodies and
28, 42t1, 47, 49
Pythagoreans
15, 192f13, 243f15
Gerson Lloyd
22
Glossary of Musical Terms and Concepts
554â563GL
god. See demiurge
God-given method
xiii, 46, 46n63, 49, 49n68
golden section
122, 122n15
GPS. See Greater Perfect System
grammar
chaos and form and
27
chromaticity and
190â191
Decad and
35, 39, 40, 42, 45, 191, 465Ap9
fourfold progression and
191
generation and
38, 39, 40, 42, 45, 190â191
harmonic and rhythmic intervals
53n75
perception and
35
world soul sphere and
39
See also fourfold relations; Decad: generation; Decad: pattern of; demiurge: harmonic method of
great and small, the (the dyad)
67, 187
Greater Perfect System (standard)
definition
141, 181, 558â559GL
first instance (standard) among Timaeus numbers
142, 313Ap3
first instance (Timaeus) among Timaeus numbers
150
fourth century and
xvii, xviin15, 140n36, 140n37
GPS/UPS alternatives (see under Timaeus octave phenomena: α-CNF2 strings)
history of
xvii, xviin15, 114, 139â140, 140n36, 140n37
Aristoxenus and
140
Eratocles and
xix, 140
Euclid (Sectio canonis) and
xvii
harmonia, relation to
xx
Plato and
31, 141
standard GPS
145â149t28
formation of
141, 142, 308â310Ap3
LPS, relationship to
140, 143, 317Ap3
UPS relationship to
140, 143â149, 145â149t28
study and
xvii, xx
Timaeus GPS
formation of
150â151, 180, 313â314Ap3
LPS, relationship to
151â152
UPS relationship to
152â163, 158â159t29, 160â163t30, 164â167t31, 167â170t32, 170â173t33
Timaeus primary scale and
175â178t34
See also construction: musical scales and systems (rules)
Greek music (ancient)
554â563GL
ambiguity, role in (see ambiguity)
apotomÄ (see apotomÄ)
basic ratios of
69, 276 (see also diapason, diapente, diatessaron, fourth; fifth; octave; whole tone)
diapason (see diapason; octave)
diapason diapente (see diapason diapente)
disdiapason and
31 (see also disdiapason)
Dorian octave, primacy of
181, 205, 316â317Ap3 (see also Dorian octaves, Timaeus)
female voice, suppression of
316, 316Ap3, 317Ap3
fourth, divisions of
102t21, 204â205, 205t39, 305Ap2
history of
xiâxxi, 1â15
GPS/UPS, pertaining to
xvii, xviin15, 114, 139â140, 140n36, 140n37
key figures in (see separate main entries for figures below)
Archytas; Aristoxenus; Damon of Athens; Eratocles; Eratosthenes; Euclid; Hippasus of Metapontum; Philolaus; Pronomos; Plato; Pythagoreans
key Platonic texts relevant to and mentioned
Laws and
xi, xxii, 12, 15, 271, 272â284, 14n64
âMyth of Erâ (Republic) and
xi, 13, 14n64, 181
Timaeus
xiâxxii, 1â563
monochord
8
other key texts (see under entries for individual key figures above in this entry)
Timaeus scale interpretation, pertaining to generally
xvn11 (see also, individual entries for modern scholars and ancient figures)
leimma (see diesis)
major sixth and
56n4, 69n46
modulation and
xviiâxviii
ânew musicâ and
130
octave genera and species of
125, 222t45, 562GLt64
historical preferences of
xxi, 130
Phrygian diatonic disdiapason, special characteristics
205, 318Ap3
quarter tone and
149, 222, 222â223, 224â226
semitone, indivisibility of
2, 149, 222â223, 224â225
study and
xvii, xxiv, 57n7, 173, 184, 195, 211, 240
Timaeus scale interpretation, generally xvn11 (see also, individual entries for particular scholars and for ancient figures)
tonoi system, ancient Greeks
8, 309â312
whole tone, indivisibility of
2, 3, 116, 224, 312Ap3
See also individual entries for each interval and octave type
Haar, James
xx
Hagel, Stephen
Archytas, divisions of fourth
307
modulating music in ancient Greece
Aristoxenus
xxi, 131
chromatic music, interest in
131
diatonic genus, origins and age
xxi
enharmonic genus, interest in
xxi, 131, 496Ap10n1
Eratocles
xviii, xix
GPS/UPS, development of
xviii, xix, 140, 140n41
Greece, origins
xvii, 132
octave species of 5th and 4th century
xviii, xix
Philolaus and
xxin35
Pronomos
xviii
Pythagoras of Zacynthus
xviiâxviii
Pythagorean framework, perdurance
306Ap2n6
seven tonoi system
xviiin20, xix
Timaeus interpretation
132
Platoâs diatonic interest
132, 305Ap2n1
half tone. See apotomÄ; diesis
Handschin, Jacques
ancient thinkers and
8, xvin13
Timaeus Scale and
arbitrary limits of
58n13
ascending/descending scale character
xvin13, 8, 10, 181
chromatic elements and
xiv, 29n40, 59n16
diatonicism, limits and
xiv, 188q
fifth periodicity and
xiv, xv, 29n40, 187
first number of
8
Platoâs octaves and musical twelfths, relation of
xiv, 138, 188q
Platoâs purpose and
xii, 13, 58
world soul stuff, effects of divisions
xiv, 138, 188q
harmonia
construals of
Archytas
10, 12
Aristoxenus
5
Creese, Donald
10
Eulogius, Favonius
xvii, xxiii
McClain, Ernest
13n56
Philolaus
12, 135
Plato
1, 5, 10, 12, 187, 274, 275
cosmogenesis and
13, 13q, 28, 31, 48, 48q, 50
Augustineâs De musica
xxiii, 53n75
binding power of
54n76, 63, 63n30
other generational schemes
homogenesis
19, 24, 50
plane figures
39
primary solids
35, 40, 43, 44, 46, 48q, 49
cosmos, structure of being and
55, 59, 60, 124, 134, 275, 276, 284
Decad and
30â31, 31n43, 31n43q, 35, 285
generation and
31, 39, 40, 49, 53n75
polis and
272, 274, 276, 277, 284
planets, music of
113, 271
study and
1, 124, 134, 193
term, ancient semantic usages of
xxq
Twenty-eight stringed lyre and
113â114, 113â114q
harmonicists
xix
Harmonicorum (Ptolemy)
11
harmonics
xii, xviin15, xxin35, 132â133, 136n28, 305Ap2n1
harmony. See harmonia
Hermocrates (Timaeus interlocutor)
16
heterogeneity, Receptacle and
25, 26
See also change; chaos; difference; Receptacle: chaos; Errant Cause; heterogeneity; motions of
Hippasus of Metapontum
3â4
history (ancient Greek music). See under Greek music (ancient): history of
Huffman, Carl
5, 63n30, 14n64
human body
15, 19, 23â24, 23â24q, 50
human happiness
50
human music, relation to cosmic ordo
31
human perception
xiii, 23q, 34â35, 181q
human social relations
harmonia (polis) and
272, 274, 276, 277, 284
Laws and
272â284
human soul
15, 19, 23â24, 23q, 23â24q
hypate. See hypate hypaton; hypate meson
hypate hypaton (standard)
148t28, 559GL
hypate meson (standard)
148t28, 559GL
hypaton (standard)
142, 143, 145â149t28, 558â559GL
hyperbolaion (standard)
142, 145â146t28, 558â559GL
Hypodorian octaves, Timaeus
form (all octave genera)
222t45, 562GLt64
in α-CNF2 strings (see Timaeus octave phenomena: α-CNF2 strings)
Hypolydian octaves, Timaeus
form (all octave genera)
222t45, 562GLt64
in α-CNF2 strings (see Timaeus octave phenomena: α-CNF2 strings)
Hypophrygian octaves, Timaeus
form (all octave genera)
222t45, 562GLt64
in α-CNF2 strings (see Timaeus octave phenomena: α-CNF2 strings)
Iamblichus, missing fourth guest and
12
icosahedron
40â46, 40n56, 42n58, 42t1, 44nn59&60, 45f4, 47â49
ideal number
xiii, 31â32, 33, 33q, 33â34q, 52n73, 272
identity
alternatively formulated identical octave sequence, meaning of
388Ap6
model octave, determination of
56, 133, 316Ap3, 562GL
primary Timaeus Scale, determination of
xiv, 175â178t34
Timaeus UPS, determination of
160â163t30, 164â167t31, 167â170t32, 170â173t33, 313â314Ap3
identity proportion, primary bodies and
43, 44, 45
images
Calcidius and
67
Decad and
32, 35
disdiapason and
31
figurate number and
32
fount, in relation to unity
67
harmonia and
53, 59
human social relations and
271, 272â284
human soul/body relationship and
23
indeterminacy and
25
life and
52
movement and
25, 35, 49, 52â53q
number and
52, 52q
persuasion and
25
time and
52â53q
music and
53
participation and
25
Receptacle and
21, 23, 25
rhythm and
53n75
square, in relation to right triangle
38
universe and vis-Ã -vis immutable exemplar
16, 17, 18
unity and
35, 67, 69
wholeness and
69
immovability, eternity and
53q
See also immutability; impartibility; indivisibility
immutability
xvii, 11, 16, 17, 28, 34
See also immovability, impartibility, indivisibility
impacts on air (vibration(s)). See pitch: ancient approaches (air impacts)
impartibility (world soul commixture)
29, 60â61, 60q, 62â63
See also indivisibility
indeterminacy
25, 26â27
indivisibility
cosmic motion and
19â20, 241q
narrow cosmic band and
210, 243f15, 266f23, 262â263, 264
nonmusical numbers (Laws), in relation to nine
277, 280, 281, 284
octave, de facto six part indivisibility
2, 12, 224
semitone and
2, 149, 222â223, 224â225
TS interval, chromatic scale
311â312Ap3
TT interval, enharmonic scale
80, 131
whole tone and
2, 3, 116, 224, 312Ap3
world soul commixture and
61
infinity
beauty and
51q
cosmic size and
227, 258
dyad, great and small
67
ideal number and
33q
octave and, division of
Philebus and
80, 82, 85
polygonal solid and
51n70
Receptacle and
50
similarity and
51q
string lengths, relating to true whole tone, semitone
227
studyâs method and
80, 82, 85
soul sphere and
261
In somnium Scipionis
xvii, xxiii, xxiv, 186
intellect
demiurgic
25q, 63n31
fourth guest, inclination to
12
generation and
17q, 18, 67â68q (see also demiurge: harmonic method of (pattern of))
happiness and
54n76
harmonia and
54n76
impartible essence and
60
mediation and
16, 17q, 18
number and
63n31, 67â68q
Plato, musical cosmology and
271
Timaeus interpretation
armillary sphere and
71
world and
16, 17q, 18, 51q, 67â68q
interference
chromatic elements and
xiv, 56, 59n16, 125, 138
fifth periodicity, effect on
xvi, 58, 29n40, 124, 125, 187â191, 189t36
octave periodicity, effect on
29n40, 56, 59n16, 125, 138
proliferation of octave species and types
59n16
effect of chromaticity upon (examples)
346Ap5, 386Ap6, 387Ap6, 390Ap6, 384Ap5
effect upon original octaval order (examples)
346, 387Ap6, 390Ap6
resultant fifth periodicity, relation to
29n40, 125
cosmic order, relation of fourths and disdiapasons
31
Decad and
30, 31
fifth periodicity and
29n40, 30
fourths and
xiv, 29n40, 30, 56, 59n16, 125, 138
generation and
30
gods, worship of
273
Laws octaves, prediction
280, 281
legislators and
273
means operation (36 A) and
xiv, 29n40, 30, 59n16, 125, 138
mediation and
30
motion and
270
octave basic arrangement and (examples)
387Ap6, 388Ap6
Timaeus octave chains, purgation from
264, 270
triples (musical twelfths) and
xiv, 29n40, 30, 59n16, 138
intermediation (mediation)
16, 17q, 18, 61â63, 85, 85n2, 349Ap5, 386, 387
See also mediation
intermixture
28, 77
See also commixture; mixture
interpenetration, triads and primary bodies
45, 45f4
See also areas: overlapping realms
interval(s). See individual entries for different kinds
inventory
Ap. 5â10
346â384Ap5, 385â406Ap6, 407â430Ap7, 431â464Ap8, 465â495Ap9, 496â553Ap10
studyâs method, centrality to (examples)
86â95, 99, 202â204, 220, 227, 228, 493Ap9t58
Tab. 2â8, 10, 12, 13â21, 24â26, 37â48, 50â52, 57â58, 64 (see LFT)
textual (not table or catalogue) summaries of Timaeus scales
chromatic octaves
218, 238â240
diatonic octaves
203â204, 205, 238â240
enharmonic octaves
238â240
tones, octave strings (see Timaeus octave phenomena; trihemitone)
isosceles figures
non-right triangle, relation to
36â37, 37f2
equilateral as
36
circularity, relation to
37
composition of
36â37, 37f2
isosceles type for pentagonal construction
37
origination of triangles and
36
Receptacle and, relation to shapes and numbers
28
right triangle and
36â41, 45, 46, 49
congruency and
36
definition
36
dodecahedron, relation to
37
duad and
36â39, 41, 45
monad and
36â39, 45
ordo of generation and
36â41, 46
pentagon relation to
37
plane figures, relation to
36, 37, 38, 38f3
primary solids, relation to
28, 40, 40n56, 41, 45, 46, 49
sameness/difference and
36â39, 41, 46
association with sameness
36
scalene triangle, relation to
36â39, 37n55, 40n56, 41, 45, 46, 49
sesquitertian ratio and
39
tetrad and
38
triad, relation to
36â37
triangularity and
38
tangibility and
46
visibility and
46
Jupiter
269, 269n6, 270f24
komma. See comma
lambda
64&n33, 66â67, 66f5, 69, 70, 75&f6, 119
Lasserre, François
14
laws
harmony and
3, 15
nature and
66
state and
272â274
Laws (Plato)
Decad and
285
harmonia and
12, 272, 274, 276, 277, 284
gods and
273&q, 274&q, 275&q, 276, 277
Laws octave scales and
280â281
Laws octaves, prediction
280, 281
Moutsopoulos, Evanghélos, harmonic interpretation of
14n64
polis and
xi, 272, 273, 276, 284
Timaeus interpretation, relation to
xxii, xxv, 12, 15, 271, 272n1, 272â284, 285
leimma (diesis; 256/243 â 1.05349794238683). See diesis
Lesser Perfect System (standard)
chromatic elements, dependence upon
143, 151
definition
140, 142, 559GL
first instance standard LPS among Timaeus numbers
143
first instance Timaeus LPS among Timaeus numbers
151
GPS/UPS alternatives, relation to (see under Timaeus octave phenomena: α-CNF2 strings)
history of
xvii&n15, xix, xx, 140, 140n36
modulation (standard)
140, 143â144
standard GPS, relation
140, 143â144, 145â149t28
standard UPS, relation to
145â149t28
Timaeus LPS, difference of formation in relation to
151â152, 153â157
Timaeus UPS, relation to
160â163t30, 164â167t31, 167â170t32, 170â173t33, 313â314Ap3
LPS. See Lesser Perfect System
lichanos (standard). See lichanos hypaton; lichanos meson
lichanos hypaton (definition)
148t28, 559GL
lichanos meson (definition)
147â148t28, 559GL
limit
chaos, and
27
cosmic expansion and
xiv, xxiv, 191
Crantor matrix and
76, 109t24, 126â129t26
chromatic nonfactors, indicators
120â121
1719926784 as endpoint, indicators
115â123
duad and
22, 191
monad and
14, 22, 31
polygonal solid and
51
Receptacle and
22, 23
sameness within difference
30
sesquitertian part derivation, use of
83
Ten as, in relation to seven
65
36A means operation, double and triple extensions and
78, 79
Timaeus octave periodicity and
188
Timaeus octave strings, α-CNF2 (see Timaeus octave phenomena: α-CNF2 strings)
Timaeus scale, primary, study
175â178t34
Timaeus scale, traditional
56
Twenty-seventh row and
115â123
unity as
14, 22
ζ-string and
206
See also endpoints; extensions; construction
Lippmann, Edward
13
See also Selected Bibliography
locus
chi (Ï) arm crossing
257
harmonia and
54n76
quarter tones, alternative perfect systems
323Ap4t53, 324Ap4t53, 325Ap4t53, 336Ap4t55, 337Ap4t55
Receptacle and
21
semitones, alternative perfect systems
324Ap4t53, 336Ap4t55
Calcidius and
21
See also area
Lydian octaves, Timaeus
form (all octave genera)
222t45, 562GLt64
in α-CNF2 strings (see Timaeus octave phenomena: α-CNF2 strings)
lyre
music, generally, and
xxi, 132, 141, 141q
pitch and
5, 6, 142, 560GL
positions on
5, 6, 142, 560GL
tuning and
131, 134â136
twenty-eight strings and
113â114
Macrobius
130n5, 132n16
macrocosm
differentiation, pattern of
30
Decad and
30â31, 31f1, 49, 285
disdiapason and
31 (see also: demiurge: harmonic method of (pattern of); Decad: generation; Decad: pattern of)
divisions related to
28â31 (see also construction: cosmos)
generation of (see macrocosm: differentiation, pattern of)
harmonia and
10, 28, 134, 285
microcosm, relation to
19, 28, 35, 49 (see also demiurge: harmonic method of (pattern of); Decad: generation; Decad: pattern of)
major sixth
ancient Greek music and
xii, 56nn4â5
traditional Timaeus limit, and
56â57
ancient commentators and
8, 56n5, 57n9, 58, 59, 58n13, 562GL
male voice range, connection with
55
modern commentators and
xii, 55&n2, 56n4, 57â58, 57nn6&9, 58n13, 179
other possibilities
xii, 55â56, 58â59, 113â114
10368, traditional limit
56
20736, alternative traditional limit
57
male voice (Greek music)
55, 316Ap3
female voice, relation to
316â317Ap3
Manuale harmonicum (á¼Î³ÏειÏίδιον á¼Ïμονικá¿Ï; Nicomachus of Gerasa)
113
Mars
269, 269n6
mathematical number
xiiin6, 31â32, 33&q, 33â34q, 34â35, 52n73
mathematici
82
mathematics
analogy, harmony and rhythm
xxiii
cosmic construction and (see construction: cosmos)
Egyptian pi (Ï) and
258, 258n3, 259, 260, 563GL, 563GLn35
epimoric (superparticular) ratios and
xvii, 305â307Ap2
figurate number
31â32, 31n43, 32n44
harmonic units and dimensionless number
258, 258â261
ideal number
30â35, 52n73
mathematical number
35
means
67, 71&q, 71n53, 72, 73n56, 74t5
musical scales and (see construction: musical scales and systems (rules); Timaeus octave phenomena)
1719926784 and whole number approximations to irrational values
224â227
Plato and
7, 14, 181q
corporeal entities and
25 (see also plane figures; primary bodies)
cosmic order and
50
disdiapason significance of
187
divinity of
52
eternal paradigm and
30, 35, 52â53q
Ptolemy and
307Ap2
Pythagoreans and
1, 5, 11, 14, 15, 82, 34n51, 46q
studyâs method and
xii, xxii, 15, 35, 58, 85, 97â98n3 (see also studyâs method)
Timaeus, model of the universe and
xxii, xxiin39, 70â71, 85
Timaeus number set derivation and
(see construction: cosmos (number matrix (fabric)))
Timaeus number set special features
113â121, 122â123
Mathiesen, Thomas
ancient musical theorists and
xix, xxi, 4â5, 6, 11, 113, 114
ascending/descending Timaeus scale ordo
6â7
Greek tonoi, history of
xix, xxi
studyâs reliance upon for standard UPS
145, 321Ap4
tonoi, pitch assignments and
309â310Ap3n6
twenty-eight stringed lyre and
113â114, 113â114q
matrix. See construction: cosmos (number matrix (fabric))
McClain, Ernest
âMyth of Erâ and
14n64
Timaeus scale and
xiv, 7&n32, 8, 13, 57nn6&9, 64n33, 102, 138, 179
means
67, 71, 71q, 71n53, 72, 73n56, 74t5
means operation (36 A)
71q, 71â76, 72t3, 73t4, 74t5, 75t6, 75f6, 76f7
new doubles and triples, consequence of
76â81, 78t8, 78t9, 79t10, 79t11
measure
musical intervals (see separate entries for different kinds)
octave string extension (see Timaeus octave phenomena: α-CNF2 strings)
soul sphere and
258â261
mediation
chromatic STN diatonic octave chains and
347Ap5, 386Ap6
generation and
15, 30â31
harmonia and
63 (see also demiurge: harmonic method of (pattern of); Decad: generation; Decad: pattern of)
number and
28â29, 35, 47â48q, 48, 49, 49q, 52â53q, 262 (see also, generation, pattern of; Decad: pattern of)
plane figures
35â39
Plato, emphasis upon
85
primary solids and
40â49, 42t1, 45f4
soul and
61
3/2 proportion
262
See also intermediation
mediator
octahedron as
46
3/2 proportion as
262
See also intermediation
mese (standard)
147t28, 559GL
meson (standard)
142, 143, 147â148t28, 558â559GL
metaphor
musical, for cosmos
xii, 13, 58, 76
Receptacle, containment and
24
method of study
mathematical character of
xii, xxii, 15, 35, 58, 85, 97â98n3
Philebus and
60, 80â81, 82, 85, 85n2, 187
Meyerstein, Walter
xi
microcosm
generative pattern, Decad and
28, 35, 49
macrocosmic plane, relation to
28, 35, 285
polis and
272â284, 285
middles
alternative GPS, and
314Ap3, 315Ap3, 318Ap3
analogy to unity and
48
arithmetic mean and
71
chi (Ï) formation and
cosmic bands and
241q, 247, 250, 251f19, 252â255, 256f21, 256t49, 257
cosmic bonds and
47q
cosmic soul/body relation and
52q
fifth periodicity and
29
geometric means and
121
harmonic unit and
259
icosahedron
44
ideal octave symmetry and
205
octahedron
44, 46
Platoâs primary sequence and
66
primary bodies and
48q, 44n59
proportion and
48
quarter tones
223
semitones and Lydian enharmonic
312Ap3
sesquitertian parts and
101
soul stuff commixture
28, 60q, 61
sphere of universe and
51q
standard GPS and
141, 142, 308Ap3
standard LPS and
142, 144
studyâs method and
22
10077696
252
13824 and
252
Timaeus diatonic UPS and
152
Timaeus GPS and
150, 152, 313Ap3, 314Ap3
Timaeus LPS and
151
twenty-eight stringed lyre, representation of
114q
universe, body of
52q
wholeness and
48
See also intermediation; demiurge: harmonic method of (pattern of); Decad: generation; Decad: pattern of
Miller, Mitchell
xiii, 46â49, 47n65, 49n68
âmissing fourth guestâ
xxii, 12
misstep (crucial, common error in Timaeus analyses at 36B)
xxi
Mixolydian octaves, Timaeus
form (all octave genera)
222t45, 562GLt64
in α-CNF2 strings
(see Timaeus octave phenomena: α-CNF2 strings)
mixture
chromatic elements and
346â347Ap5, 385Ap6
harmonia and
54n76, 63
number matrix and
109t24, 243f15
soul stuff and
28, 61â63
model octave and
xvi, 56, 133, 180â181, 8n33, 316Ap3, 562GL
alternative sequence of (768, 864, 972, 1024, 1152, 1296, 1458, 1536)
316Ap3, 562GL
Î-string, locus of
208f14, 266f23, 267â269 (see also Î-string)
1536 (ends first perfect Timaeus disdiapason and diatonic GPS)
127â128t26, 133, 136, 137â138t27, 150
pattern of (384, 432, 512, 576, 648, 729, 768)
56, 133, 316Ap3, 562GL
primary Timaeus Scale
126â129t26, 175â178t34
reciprocal, Lydian/Dorian character of
appendices, glossary
317Ap3, 561GL, 563GL, 563GLt65
book main text
xvâxvi, 9â10, 55, 58, 133, 181, 181q
scale direction, studyâs approach
xv, 5â6, 7, 8â10, 59, 134, 180â181
768 and
128t26, 151, 152, 316Ap3, 562GL
6144 and (number above which 10368 is a âmajor sixthâ; determinable by deduction)
57
(10368, 11664, 13122, 13824, 15552, 17496, 19683, 20736) and
56â57
384 and
127â128t26, 133, 136, 150, 175t34, 316Ap3
3072 (ends perfect disdiapason, first articulation of diatonic Timaeus UPS)
128t26, 152
12288 (number above which 20736 is a âmajor sixthâ)
57
moon
269, 269n6
monad
diatonic octaves and, in reversal of fourth and fifth
346â384Ap5, 388Ap6, 401Ap6
CNF2 as
203t38, 208
equilateral triangle as
36â37, 37f2
fifth periodicity and
58, 190, 191
4478976
249f17
γ-string as
200t38, 201t38, 203t38, 207, 208, 208f14, 267
ideal numbers as
32
number fabric as
30
1719926784 as
119
ordo of generation and
36
Decad and
xvi, 30â31, 31f1, 32, 39, 124, 187, 285
disdiapason and
xvi, 31, 39, 187
double expansion of
31
duad and
xvi, 30â31, 31f1, 32, 36, 45, 285
chromaticity and
187, 191
fourfold progression
xvi, 31
harmonic proportions and
30â31, 124
monadic self-return and
36, 37, 39
nonequilateral plane figure/equilateral plane figure
41
octave periodicity/fifth periodicity
xvi, 124
sameness/difference, relation to
28, 37, 39, 45, 285
scalene/isosceles differentiation
36
self-return of
36, 37, 39, 187
square as
38
tetrad and
30â31, 31f1, 32, 39, 45, 285
triad and
28â29, 30â31, 31f1, 32, 45, 285
2:1 proportion and
39
unity and
28â29, 30â31, 31f1
primary equilateral solid as
45, 45f4
âquintessential triangleâ as
36, 39
soul stuff as
28â29, 39, 63
3 as, in relation to model octave scale
58
âvery idea of a triangleâ as
39
monochord
11
Moralia (Plutarch)
Timaeus scale directionality and
xvn11
more and less, the
47n65, 187
See also great and small, the
motions (cosmic). See construction: cosmos (motions of); construction: cosmos (World soul sphere)
Moutsopoulos, Evanghélos
14, 14n64, 15
See also Selected Bibliography
movement
band of difference
242, 264â269, 266f23
cosmic motion, relation to human perception
xiii
diagonal direction of
28, 29, 263
fourfold movements, Platoâs emphasis on
255
generation and
31, 35, 39, 55, 255
harmony of the spheres and
271
monad/duad progression, octave variants
346â384Ap5, 388Ap6, 401Ap6
narrow band triples and
29, 242, 263
order and indeterminacy
26â27
planetary variety, role as heavenly clock
13q
Receptacle and shaking movement
28
uniform movement of the whole
261, 262, 269, 270
multiple repetition. See Decad: generation; Decad: pattern of; fifth periodicity; demiurge: harmonic method of (pattern of); Timaeus octave phenomena; trihemitones
multivalency
11, 34, 125, 240
musical scales. See construction: musical scales and systems (rules); Timaeus octave phenomena
musical twelfth. See diapason diapente
âMyth of Erâ
11, 13, 14n64, 181
nature
analogy and
14, 47â48q, 49, 49q
bodily and
14, 47, 47q, 60q
bonds of in universe
14, 47â48q, 49, 49q, 60q, 63
cosmic bands, difference and
29
difference and
28, 29, 60q, 61, 62, 63, 241
double, as nature of
2, 36
exemplar (paradigm)
25q, 52q, 53q
generated copy, exemplar, medium of
19
human beings and
19
laws of seven and
65â66
mediation and
28â29, 35, 47â48q, 48, 49, 49q, 52â53q, 262
nete (neate). See nete diezeugmenon; nete hyperbolaion; nete synemmenon
nete diezeugmenon (standard)
146t28, 559GL
nete hyperbolaion (standard)
145t28, 559GL
nete synemmenon (standard)
146t28, 559GL
new music (chromatic)
130
Nicomachus of Gerasa
3, 4â5, 6â7, 8, 52n73, 113â114, 113â114q
9 (number of primary sequence; harmonic key to Laws octaves)
65t2, 66f5, 75f6, 114q, 277, 280, 281, 284
9/8 (1.125; whole tone ratio) (see whole tone)
98304 (ends a Timaeus combined primary scale UPS; begins another)
160t30, 163t30, 175, 175t34, 178t34
nodes (octave)
constituent fourth or fifth, internal bounds
387Ap6
means operation (36 A) and
tables exhibiting
77t7, 78t9, 84t12, 84t13, 86t14
text relating to
71, 77â80, 82â85, 88, 95
primary bodies, occurrence in generation of
44
number
34
Decad and
30â35
Divine intellect, power of
63n30
harmonic unit and
258â261
mediation and
28â29, 35, 47â48q, 48, 49, 49q, 52â53q, 262
musical proportions and
66f5, 69, 75f6, 114q
one and
66â67, 67q, 68q
origin of
66â67
Φ and
67, 122, 122n15
primary bodies, harmonically bound rectilinearity re
35
Pythagoreans and
1, 5, 11, 14, 15, 34n51, 46q, 82
Receptacle and
20, 25
right angled triangle and
38
right triangle and
36, 37
sameness and
28, 29, 60q, 61, 62, 63, 241, 263
scalene and
36
shapes and
28, 31â32, 31f1
significant numbers (see significant numbers)
soul stuff and
28, 29, 60q, 61, 62, 63
studyâs method and
xii, xxii, 15, 35, 58, 85, 97â98n3
symmetry and
49q
Timaeus number set (see construction: cosmos (number matrix (fabric)))
Timaeus octaves and (see Timaeus octave phenomena)
time and
53q
triple as nature of
3, 36
universe and
16, 51q, 52q
octahedron
37, 40â46, 40n56, 42t1, 44n59, 45f4, 47â49
octave (2:1)
cosmic limit, relation to
xviâxvii, xxiii, 31, 55, 126â129t26, 186
Decad, relation to 3 and
xvi, 31, 39, 124, 186â187
definition
xvi, 86â87t15, 195t37, 211t41, 222t45, 555GL, 557GL, 562GLt64
diapason as designation
3, 4, 69, 557GL
fourfoldness, relation to
55
generation and
xviâxvii, 43, 44, 46, 55, 186â187
harmonia and
69
indivisibility and
2, 12, 224
Laws, occurrences in
277, 280, 281
other intervals, relation to
apotomÄ (2187/2048 â1.06787109375), necessity to chromatic scale
118, 219, 248
disdiapason
70, 555GL
fifth
3, 4â5, 11, 139
fourth
3, 4â5, 11, 139
leimma (256/243 â 1.05349794238683)
2â3, 554GL, 556GL
whole tone (9/8; 1.125)
86â87t15, 102t21
quarter tone
222, 223, 224, 222t45
Timaeus fifth periodicity and (see fifth periodicity)
Timaeus octaves chains, composition (see Timaeus octave phenomena: α-CNF2 strings)
trihemitones (see trihemitones)
2:1 measure
39, 44, 191, 347â384Ap5, 385â387Ap6, 389â405Ap6, 557GL
octave bands. See Timaeus octave phenomena
octave genera, relation of
310â312Ap3
octave periodicity
ambiguity of in relation to Timaeus fifth periodicity (see fifth periodicity)
as degenerating to fifth periodicity in the Timaeus (see fifth periodicity)
octave species in genera (form). See Timaeus octave types
octave strings
octave types (patterns of) (see Timaeus octave types)
pitch relations and (see Timaeus octave phenomena; trihemitones)
planets
115n10, 269, 269nn5â6
polis
272â284, 272â273q, 273â274q, 274q, 275q
primary sequence and
66f5, 75f6, 114q
primary Timaeus scale
175â178t34
Receptacle, cosmic circular motions, relation to
25, 26, 26q, 50
standard UPS (see under Timaeus octave phenomena: α-CNF2 strings)
Timaeus UPS (see under Timaeus octave phenomena: α-CNF2 strings)
time and
52â53q
number and
52â53q
tone relations and (see Timaeus octave phenomena; trihemitones)
trihemitone sequences (see trihemitones)
1 (begins Timaeus primary sequence; endpoint Ï arms; linked to monadic unity)
28, 65t2, 66f5, 75f6, 114q, 257f22
1743392201/1719926784 (comma)
226
1719926784 (last number of Timaeus set; endpoint of Ï arms)
114â123, 129t26, 224â227, 257f22
191102976 (endpoint of Ï arms; last scale element)
192f13, 257f22
192 (beginning of first complete diatonic tetrachord, number set)
126t26, 127t26
165888 (end of a complete combined standard UPS sequence in the Timaeus set)
149t28
â1.03125 (1773674496/1719926784; QT from whole tone) (see under whole tone)
â1.033935547 (76230/73728; one of quarter tones derivable from apotomÄ) (see under apotomÄ)
â1.032821724 (78732/76230; one of quarter tones derivable from apotomÄ) (see under apotomÄ)
â1.026748971 (998/972; one of quarter tones derivable from leimma) (see under diesis)
â1.026052104 (1024/998; one of quarter tones derivable from leimma) (see under diesis)
1536 (ends first perfect Timaeus disdiapason and diatonic GPS)
127â128t26, 133, 136, 137â138t27, 150
1152 (begins first standard diatonic UPS sequence in Timaeus set)
144
1728 (beginning of first standard diatonic LPS system in Timaeus set)
143
opposite(s). See reciprocity; reciprocal relations
orbit(s)
construction of (see under construction: cosmos (World soul sphere))
identification of
115n10, 269, 269nn5â6
ordo (order)
change, relating to
26â27
chaos and
cosmic order
26, 26q, 27, 50, 59
ordered motion, relation to
25, 26, 26q
chromatic invasion and
185â186, 209â210, 209t40
cosmic motions, relating to
13q, 50, 51â52q, 210, 241q, 242, 261â263
bodies, relation to
22q, 23, 23q, 26, 26q, 49, 49q, 50
circles, same/different
13q, 19â20, 28, 210, 241q, 242, 261â263
divisions, band of difference
19â20, 210, 241q, 242, 264â265, 268
human thought, relation to
50
Decad and
cosmic order
xiii, 59, 187
generation and
xiii, xxin6, xviiâxviii, 1n3, 15, 285
human perception and
xiii, 34â35 (see also Decad: pattern of; ordo: number)
diatonic scale steps
188â189, 560GL
eternity and
harmonic perspective, cosmic order and
53n75
rhythmic perspective, relation to
53n75
number and
52, 52n73
universe, as moving image
25
Decad and
35, 54
music and
53, 54
number and
52, 52â53q, 53
time, relation to
52, 52â53q, 53, 54
fifth, divisions of
557GL
fifth periodicity and (see fifth periodicity)
rise and fall, symmetry of
58, 187â191, 189t35, 189t36
fourth, divisions of
Archytas
305Ap2
Timaeus analysis (avoiding UDOP)
102t21, 204â205, 205t39
generation, pattern of (see demiurge: harmonic method of (pattern of))
GPS tetrachords
142, 558â559GL
human perception and
34â35
intervals, filling
doubles with fourths, fifths
86â87t15, 557GL
fifths with whole tones
557GL
fourths with whole tones
102t21, 204â205, 205t39
triples with fourths, fifths
86â87t15, 558GL
knowledge, relating to
47n63, 49n68
LPS tetrachords
143, 559GL
number, and
Decad and (see Decad: pattern of)
eternity and
52â53q
generation and
mediation and
28â29, 35, 47â48q, 48, 49, 49q, 52â53q, 262
time and
52â53q
overlap. See areas: overlapping realms, generally; interference; Timaeus octave phenomena
paramese
147t28, 559GL
paranete. See paranete diezeugmenon; paranete hyperbolaion; paranete synemmenon
paranete diezeugmenon (standard)
146t28, 559GL
paranete hyperbolaion (standard)
145â146t28, 559GL
paranete synemmenon (standard)
146t28, 559GL
parhypate. See parhypate hypaton; parhypate meson
parhypate hypaton (standard)
148t28, 559GL
parhypate meson (standard)
148t28, 559GL
participation
corporeality and, in Timaeus
25
limitedness of notion
25
harmonic circulation of the whole
49, 263, 268â269
Receptacle and
20, 25, 27, 49q
right triangular idea and
36
right triangular nature and
36, 37, 38
triangular idea and
46
parts (sesquitertian)
selection for filling with 9/8 intervals
82â100, 99â100t20
sesquioctave operation on
101â123, 289â304Ap1
figures, relating to
106f10, 107f11, 108f12, 192f13, 243f15, 247f16
tables related to
109t24, 111t25, 126â129t26
See also fourth
passions, human, cause of
50
patterns
All Perfect Animal and
xiii, xvi, 54â56, 59â60, 285
chromatic invasion
185â186, 209t40
Crantor matrix (see construction: cosmos (number matrix (fabric)))
Decad and (see Decad: pattern of)
diatonic scale, rise of
137â138t27
diatonic scale steps
188â189, 560GL
doubles and
86â87t15, 88â90t16, 195t37, 211t41, 222t45, 557GL, 562GLt64
exemplar and (see patterns: All Perfect Animal)
fifth and
557GL
fifth periodicity, rise and fall
symmetry of
58, 187â191, 189t35, 189t36
figurate number and
31â32, 32nn44â45, 34n51
fourth
102t21, 305Ap2, 557GL
generation and (see demiurge: harmonic method of (pattern of))
ideal number and
30â35, 32nn44â45, 34n51
Laws, harmonia and
277, 280, 281
model octave (see model octave)
number fabric (see construction: cosmos (number matrix (fabric)))
octave species by genera, and (see Timaeus octave types)
polis and
272q, 273q, 274q, 272â274, 275q
primary Timaeus scale and
175â178t34
Timaeus octave strings (see Timaeus octave phenomena: α-CNF2 string)
trihemitones (see trihemitiones)
triples
86â87t15, 91â95t17, 558GL
Unaccustomed diatonic patterns
205t39
UPS systems (see Timaeus octave phenomena: α-CNF2 strings (UPS system comparison))
pentagon
37â38, 59n17
perfect cubes
perfect squares relation to
109t24, 126â129t26, 243f15
Timaeus numbers, identification of
109t24, 126â129t26, 243f15
perfect squares
1719926784
twenty-seventh row missing squares and
118â119
196, number of Timaeus set terms
108
perfect cubes relation to
109t24, 126â129t26, 243f15
Timaeus numbers, identification of
109t24, 126â129t26, 243f15
perfect systems. See Greater Perfect System; Lesser Perfect System; Unmodulating Perfect System
periodicities. See Timaeus octave phenomena; fifth periodicity
persuasion and
governance of the universe, mode of
25
Phrygian mode and
317Ap3
phi (also Φ; 1.6180339887 â¦)
67, 121â123, 122n17
Philebus (Plato)
duad of more and less, unity of
187
god-given method
46n63
harmonia and commixture
54n76
knowledge, order of
35n53
number and (for Divine Intellect)
63n31
studyâs method and
60, 80â81, 82, 85, 85n2, 187
Unlimit and
47n65
Philolaus
age of, relative to others
2, 3â4, 12, 14
harmonia and
3, 63n30, 305Ap2n2
leimma and
2, 3
music history and
xx, xxin35
musical scale of
4, 5, 10, 134â135
triad, relation to
136n28
âMyth of Erâ and
14n64
Pythagorean tradition association with
59
schisma and
556GL
seven-stringed lyre, attunement and
134â136
source for other theorists
4, 11, 134, 305Ap2
Timaeus scale and
10
Phrygian octaves, Timaeus
form (all octave genera)
222t45, 562GLt64
in octave strings α-CNF2 (see Timaeus octave phenomena: α-CNF2 strings)
physics
21
Physis (Philolaus)
3
pi (also Ï)
258, 258n3, 259â260
See also Egyptian Ï (pi) (256/81) (3.160493827)
pitch
ancient approaches
air impacts
xv, 8, 9â10, 130â131, 134, 136, 308Ap3, 309Ap3
ancient intervals and
554â556GL
Plutarch on Timaeus and
8n33
Timaeus diatonic scales lists, mention of
408â425Ap7
sixth century, pitch measurement and
9
string lengths and
xv, 4â5, 8â9, 116
lyre strings and, positions of
5, 6, 57n7, 142, 560GL
modulating music and
xviiâxviii
secondary Timaeus systems
173â174, 317Ap3â318Ap3, 319Ap3
standard system
140, 143, 144, 145, 317Ap3
Timaeus primary system
152
names and ordo of
188â189, 559GL, 560GL
pitch ordo of Timaeus scale
xv
ancient, modern debates
xv, 5â10, 7â8n33
studyâs position
xv, 7, 9â10, 59, 133, 134, 180â181
tonoi
pitch range assignments, artificiality of
309â312Ap3
variation ancient means of achieving
9
plane figures (see construction: cosmos (plane figures))
planets
as heavenly clock
13
cosmic bands, relation to
xiv, 13, 20, 269
harmony of
57â58, 113, 115n1o
orbits of
269
Proclusâ planet groupings
269n6
sun, association with 729
269n5
Plato (commentators). See separate entries for individual scholars and ancient figures
Platonists
Pythagorean ratio theory, knowledge of
181q
World soul
27q
Plotinus (on matter and space)
22
Plutarch
De animae procreatione in Timaeo
7â8n33, 8
ascending Lydian Timaeus scale, presupposition of
7â8n33, 8
primary sequence, remarkable numbers and
114
lambda arrangement
attribution to Crantor
64n33
sesquitertian parts, mode of filling
102
Timaeus scale interpretation and
alignment of sun with 729
115n10, 269n5
ascending Lydian, affirmation of
xv, xvn10, 7, 7â8n33, 9, 59, 134
Platoâs pairings in primary sequence
115
primary sequence, remarkable numbers
114
384, beginning of
8, 8n34, 59
Porphyrius (on Euclid)
11
precosmic state (debate, Receptacle)
25, 27
primary bodies (see construction: cosmos (primary bodies))
primary divisions
65t2, 66f5, 75f6, 114q
primary Timaeus scale
175â178t34
Proclus
chromatic fourth, composition of
219n3
planetary orbits and
115n10, 269, 269n5, 269n6
Plato, Pythagorean influence on
xiiin6, 1n1
Severus, source for
562GL
Timaeus scale interpretation
ascending/descending ambiguity and
openness to either
xvin13, 133n21, 181
384 as beginning of
8, 56n5, 59
whole tone
apotomÄ and leimma, relation to
219n4
progressions. See extensions; demiurge: harmonic method of (pattern of); construction: cosmos; construction: musical scales and systems; construction: cosmos (number matrix (fabric)); Decad: generation; Decad: pattern of; Timaeus octave phenomena; Timaeus octave types
Pronomos, modulating aulos and
xviii
proportion
among cosmic bands
264
among cosmic orbits and
268
among octave types in octave strings (see Timaeus octave phenomena)
ascending to descending scales (see Timaeus octave phenomena)
basic harmonic ratios, definition
554â556GL
chi (Ï) arm endpoints, and
257f22, 257â258
Φ (phi) and
67, 121â123
primary bodies and
42t1, 43â45, 45f4
trihemitones, proportion of octaves using
219â220
proslambanomenos (standard)
149t28, 559GL
Psellus, Michael
xvin13, 133n21, 181
Pseudo-Plutarch (que)
3, 14n64, 180â181
Pseudo-Timaeus
xvin13, 133n21, 181
psychogony (Timaeus)
4, 7, 15, 113q
Ptolemy
xix, 3, 5, 11, 306â307Ap2, 306Ap2n6
pyknon (definition)
556GL
pyramid (primary body). See tetrahedron
Pythagoras
1n1, 3, 4, 31n44, 32
Pythagoras of Zacynthus (lyre tuning accomplishments)
xviiâxviii
Pythagoreans
Aetius (see separate entry)
Archytas (see separate entry)
Aristoxenus, (see separate entry)
cosmology and
13q
Decad and
31n43, 32, 32n45, 34, 46q
divisions of the fourth
102t21, 204â205, 205t39, 305Ap2
Euclid and (see separate entry)
geometry, cosmic models and
15
harmonia and
3, 4
basic concords
305Ap2
epimoric ratios and
305Ap2
numerical conception, uniqueness of
13
roots of
1n1
Hippasus of Metapontum (see separate entry)
octave, discovery of
3, 4
Nicomachus of Gerasa (see separate entry)
number and
Decad, doctrine of
32n45, 34
harmonic proportions, esteem for
31n43
pattern as objective
34n51
planets as heavenly clock
13
ratio theory, reciprocal relations
181
real being of all things, number as
1n1, 31n44
tetractys and
31n43
whole numbers, esteem for
82
world soul, harmonic divisions of
13
Philolaus (see separate entry)
Plato, influence upon
consensus of many
xii, 1n1, 13, 59
cosmogony, planets as heavenly clock
13q
harmonia as cosmic bond
14
mathematical nature of
14
psychogony, applicability to
15
numbers and being, Pythagorean doctrine of
as root of Platonism
1n1
Decad, triads, tetractys, evident familiarity with
180
Platoâs divine science and
xiiin6
Pythagorean diatonic, Platoâs acceptance, xix
Plato, knowledge of
xiii, xiiin6, xxii, 11â12, 59
Section Canonis (Euclid) (see separate entry)
study and
xii, xx, 3, 7, 11, 15, 82, 125
quarter tone
definition/derivation of
apotomÄ, as source
calculation, mode of
224
de facto inexpressibility as whole number ratio
224, 225â226
1.033935547
223
1.032821724
224
other quarter tones, comparison
226
whole number ratio approximations and 1719926784
224, 226
leimma, as source
calculation of
149, 222â223
de facto inexpressibility as whole number ratio
224, 225â226
1.026748971
149
1.026052104
149
other quarter tones, comparison
149
whole number ratio approximations and 1719926784
223, 226
whole tone, as source
calculation of
223
de facto inexpressibility as whole number ratio
224
1.03125
223, 224, 501Ap10, 507Ap10, 515Ap10, 516Ap10, 532Ap10, 544Ap10
whole number ratio approximations and 1719926784
223
enharmonic scale, use in
222, 222t45
history of Greek music
131, 149, 163
studyâs analysis, occurrences of (see Timaeus octave phenomena: α-CNF2 strings (enharmonic scales))
true measure of
225â226
Quinitilianus, Aristides
113â114, 309â310Ap3n6
quintessential triangle
36, 38, 39
See also Very idea of a triangle
radius, soul sphere and
51q, 258, 259, 260
ratio. See specific indexed ratios
Receptacle (Ïá½½Ïα)
abstraction and (chaos)
27, 27q
agitation as feature of
23q, 50
analogy and
49, 50, 49â50q
becoming and
20
being and relationship to
20
bodies and
appearance in
21q
contraction in
22, 22q, 26, 26q, 50
formation in
28
geometrical characteristics of
23, 23q
motions and transformations in
23, 23q
similarity to
35
tangibility, relationship to
21q, 46
universe
19, 20, 20n15, 41â42, 50, 51q, 59n17
visibility, relationship to
20, 21q, 46
chaos and
20n16, 27
containment and
20, 23, 24, 25q
demiurgic action and, imposition of harmonia on
25, 28, 28, 49, 50, 49â50q
duad and
22
Errant Cause and
20n16, 25, 26
eternity, relationship to
20
form and
20
geometry and
28
harmonia, imposition on
28, 49, 50, 49â50q
heterogeneity and
25
human happiness and
50
human soul and body, effect on
50
indeterminacy and
25, 26
intelligible, relationship to
20
interpretation, ambiguities of
22, 25
limiter(s) and
22
matter (á½Î»Î·) and
21, 21q, 22
motions of
agitation
23q, 50
chaos and
25, 26, 26q, 27, 27q
commotion and
23
cosmic circular motions, relation to
25, 26, 26q, 50
rectilinear motion and
20n16, 35
names for (and associated ideas)
container
20, 23, 24, 25q
foster mother
24
imprint bearer
20, 24
matrix
20
Mother itself
24
nurse
20, 24
place
21, 23, 24
space
20, 21, 22, 24
universal recipient
24
winnowing basket
24
nature of
20 (see also Receptacle: names for; Receptacle: qualities of)
number and
25, 28
ontological status of
20, 22
participation and
25
precosmic state of (speculation)
25â26, 27
qualities of
20
everlastingness of
20
indestructibility of
20
invisibility of
20
kinetic potentiality of
22â23
receptivity of
20
tangibility of
21
role of
20, 21, 22, 23, 23q, 24, 26
shape and
28
space and
20, 21, 22, 23
unity and
22
universe, and
body of
19, 20, 20n15, 41â42, 50, 51q, 59n17
inseparability from
25, 27, 28
reciprocal relations
chromatic scales, lack of
211, 563GLt65
diatonic scales
xv, xvi, 10, 55, 58, 133, 181, 181q, 239, 315Ap3, 317Ap3, 561GL, 563GL, 563GLt65
Phrygian scale special character
315Ap3, 318Ap3, 409Ap7, 421Ap7
Unaccustomed pattern, special character
442Ap8
enharmonic scales, lack of
227, 563GLt65
mediation construction of primary bodies
44
sesquitertian parts, sesquioctave operation
101, 102
reciprocity (musical thesis)
181, 181q
rectilinear motion
20n16, 35, 275q
rectilinear surfaces, primary bodies
35, 36
relationships
apotomÄ to
comma, leimma, octave (chromatic), quarter tone, whole tone (see apotomÄ)
chromatic element set/whole Timaeus number set
109t24, 210, 243f15, 266f23
chromatic STN diatonic octave chains (mediation among)
347Ap5, 386Ap6
cosmic bands (see under construction: cosmos (Ï (chi figure)))
Decad/harmonic generation (see Decad: generation and)
diatonic, chromatic, enharmonic, octave species 310â312Ap3 (see also Timaeus octave types)
doubles/triples
65t2, 66f5, 75f6, 78t8, 78t9, 79t10, 79t11
endpoints, chi (Ï) arms
257â258, 257f22
fabric of numbers, parts
109t24, 210, 243f15, 266f23
female/male voice
316â317Ap3
fifths to
apotomÄ, basic harmonic ratios, fifth periodicity, leimma, musical twelfths (see under fifths; see also fifth periodicity)
figurate number, and
31â32, 33
forms/sensible things
20, 22, 25, 27, 32
fourths to
apotomÄ, Archytan alternative, basic harmonic ratios, leimma, quarter tone (enharmonic) (see under fourth)
GPS (standard) and
GPS alternatives, GPS Timaeus (see under Greater Perfect System (standard))
harmonia and
cosmic structure
55, 59, 60, 124, 134, 275, 276, 284
Decad
30â31, 31n43, 31n43q, 35, 285
ideal number
figurate number
31â32
mathematical number
33
intervals, cosmic bands
doubles/triples; triples/sesquialter intervals (see under construction: cosmos: Ï (chi figure): World soul sphere)
Laws/ennead
284
Laws octaves/Timaeus primary scale
277, 280, 281
leimma to
apotomÄ, basic harmonic ratios, quarter tone (see diesis)
Limit/Unlimit
14, 47n65
LPS (standard) and
LPS alternatives; LPS Timaeus (see under Lesser Perfect System (standard))
mathematical/ideal number (see relationships: ideal number)
monad (ordo of generation) to
duad, tetrad, triad (see under monad)
motions, cosmic bands (see under construction: cosmos (Ï (chi figure)))
motions, Receptacle/whole universe (see construction: cosmos: motions of)
musical phenomena (see Timaeus octave phenomena; trihemitones)
numbers, Crantor matrix (fabric) and (see construction: cosmos (number matrix (fabric)))
octave/fifth periodicity, Timaeus
xivâxvi, 58, 187â191, 189t35, 189t36, 191â192
octave genera (derivation)
310â312Ap3
octave harmonia, construals
Archytas
10, 12
Aristoxenus
5
Creese, Donald
10
Eulogius, Favonius
xvii, xviii, 186
McClain, Ernest
13n56
Philolaus
12, 135
Plato
1, 5, 10, 12, 187, 274, 275
octave to
apotomÄ, basic harmonic ratios, leimma, quarter tone types (see under octave; other intervals, relation to)
1719926784 to
irrational numbers, approximations
223â227
missing squares, twenty-seventh row
118â119
Φ geometric series
121â123
pitch/impacts on air
(see pitch: ancient approaches (air impacts))
plane figures (see construction: cosmos (plane figures))
Plato (with Pythagoreans)
xiii, xiiin6, xxii, 11â12, 59
polis/cosmic harmonia
272, 274, 276, 277, 284
primary bodies and (see construction: cosmos (primary bodies))
quarter tone intervals to
octave, leimma, apotomÄ, various possibilities (see under quarter tone)
reciprocity, diatonic octaves (see under reciprocal relations)
sameness/difference
13q, 19â20, 28â30, 36â39, 39â42, 45â47, 124 ((see also under construction: cosmos (motions of); construction: cosmos (World soul sphere))
semitones, generally, various sizes (see under apotomÄ; diesis)
soul sphere poles
258
Timaeus/Laws (order of authorship)
272n1
Timaeus octave scales/chromaticity patterns
185â186, 209t40
Timaeus Scale/Laws scales
277, 280, 281
tonal space relationships (see Timaeus octave phenomena)
triples/doubles (see relationships: doubles/triples)
UPS (standard) and
UPS alternatives; UPS Timaeus (see under Timaeus octave phenomena: α-CNF2 strings)
whole tone to
apotomÄ, basic harmonic ratios, leimma, quarter tone, triples (see under whole tone)
Republic
diatonic scale and
14n64
divided line analogy
67
opposites and
181
music of the spheres
14n64
âMyth of Erâ
xi, 13, 14n64, 181
reciprocal relationships and
181
tonoi
Dorian
xviii, 181
evidence of
xviii, 181, 317Ap3
female/male, attribution
181
Ionian
xviii
Lydian
xviii, 317Ap3
Mixolydian
xviii, 317Ap3
Phrygian
xviii, 317Ap3
Platoâs opinion of
181
Socratesâ opinion of
317Ap3
revolutions
musical scales
275
thought
50
universe
50, 51q
rhythm, harmonia and
xxiii, 53n75
riddle (Timaeus 35Aâ36D)
xiii, xxii, xxv, 81, 180, 181q, 250, 285
right angle
36â37, 38
right triangle. See isosceles: right triangle; scalene: right triangle
right triangularity. See isosceles figures: right triangle; scalene figures: right triangle
sacredness
Decad and
46
fourfoldness and
46
polis, precincts and
273q
Sallis, John (on Receptacle)
21, 23n29
sameness
being and
16
All Perfect Animal and
52â53q
impartible essence
60q, 61â63
chromatic invasion, strings with similar patterns
210
containment analogy
23â24
cosmic motion and
13q, 19â20, 28, 210, 241q, 242, 261â264
cosmic narrow band and
28, 262â263, 264
Decad/exemplar (paradigm) and
All Perfect Animial
xiii, 54, 55, 56, 59, 60
exemplar
xvi, 1n3, 32, 285
diatonic intervals and
diatonic reciprocity (see under reciprocal relations)
Philolaus/Archytas (harmonic framework)
305Ap2
Philolaus/Plato (diatonic attunement)
5
primary triangles and
36â39, 41, 46
Pythagoras/Timaeus diapason
4
Sectio canonis/Timaeus
11
Timaeus/Eratosthenes of Cyrene
11
difference, relation to
bands, generally
xiv, xxii, 13q, 19â20, 28â30, 262, 263â264
generation and
(see sameness: generation)
soul stuff and
60q, 61â64, 64q
six rectilinear motions and
51q
Timaeus octave phenomena and
124, 240 (see also relationships: sameness/difference)
dodecahedron and
42n58
Eulogius, Favonius, identity
xxiiin2
exemplar and
xvi, 1n3, 16, 25q, 32, 285
fabric of numbers, relation to
28
fifth periodicity, relation to
30
generation mode of (across different levels) and (see demiurge: harmonic method of (pattern of))
impartible essence and
29, 60, 61
intellect, relation to
16
Laws octaves, Timaeus primary scale
277, 280, 281
means and
71â72, 71q
number matrix, determining fractions
73â74
pitch, and
xix, 8
planetary groupings, Proclus
269n6
polis divisions and
274q
reason relation to
16
soul stuff commixture and
28, 60q, 61â64, 64q
analogy, motion of same
264
same complexity level as number fabric
28
stereoscopy and
19n13
structures, different levels of universe
49
study and
interval names, redundancy and
85, 97â98n3
study/Brisson presentations, means operation results
77n59, 85
sun and
115n10
Timaeus interpretation and, traditional
8, 57, 102
time (simultaneity) and
artificerâs actions
17q, 29n40, 52q, 64q, 76q, 102q, 242
cosmic fourth/cosmic disdiapason
xvi
harmonia and rectilinear character, primary bodies
35
musical phenomena, Timaeus set
124, 240
perfect squares/perfect cubes
115n10
progressions and
38, 45
tonal space and (see Timaeus octave phenomena)
Triangle of Convergence and, comparison
249
Twenty-seventh row phenomena, and
116, 121
Saturn
269, 269n6
Sayre, Kenneth
24, 25, 35n53
scales (musical) (see Timaeus octave types; Timaeus octave phenomena)
scalene figures
non-right triangle and
36
isosceles non-right triangle, relation to
37
origination of triangles and
36
Receptacle and
shapes and numbers, relation to
28
right triangle and
36
circularity and
37, 37f2, 37n54
congruency, relation to
36
definition
36
dodecahedron, relation to
37
duad and
36â39, 41, 45
isosceles triangle, relation to
28, 36â41, 40n56, 45â46, 49
monad and
36â39, 45
ordo of generation and
36â41, 46
Decad and
46
disdiapason and
46
fourfoldness and
46
pentagon relation to
37
plane figures, relation to
equilateral triangle, relation to
36â39, 37f2, 41, 45
square, relation to
38, 39
primary solids, relation to
28
cube, relation to
40, 45 and
earth, relation to and
46
fire, water, air, relation to
40, 40n56, 45
triangular solids and
40, 40n56, 45
sameness/difference and
36â39, 41, 46
difference, association with
36
sesquitertian ratio and
39
monad and
36, 45
tetrad, relation to
38, 39, 41, 45
tetrahedron and
37
triad and
36, 39, 45, 46
triangularity and
40, 40n56, 45, 46
tangibility and
46
visibility and
46
science (harmonic). See harmonics
schisma (definition)
556GL
secondary Timaeus scales (see Timaeus octave types: Timaeus octave phenomena: α-CNF2 strings)
Sectio Canonis (Euclid)
xvii, 11
Selected Bibliography
564â569
self-return
conversion and
67q
cosmic limit, relation to
191
Decad and
34, 35
duad and
39, 191
emanation and
34, 35, 67q, 67â68q, 67â68
ordo of
67q, 67â68q
harmonia and
68â69
human analysis and
35
monad and
36, 37, 39
number and
34, 67â68, 67q, 67â68q
octave idea and
45
ordo of generation and
36
studyâs method and
194
unity and
35, 68, 69
World soul and
67q, 67â68q
semitone. See apotomÄ; diesis
sensation
23q, 23â24q, 181, 181q
sensible things
apprehension, mode of
16, 23â24q, 181, 181q
bodiliness, relation to
16
bond of analogy and symmetry, role of
49, 49â50q
cause, necessity for
16
forms, relation to
22, 25
numbers, relation to
22, 32
human soul, effect on
23, 23q
universe, as
16
sequences
chromatic invasion patterns and
185â186, 209t40
Crantor matrix, various stages (see construction cosmos: (number matrix (fabric)))
diatonic scale steps
188â189, 560GL
doubles
86â87t15, 88â90t16, 195t37, 211t41, 222t45, 557GL, 562GLt64
Fibonacci sequence (27th row)
122â123, 123n17
fifth
557GL (see also fifth)
fifth periodicity and (see fifth periodicity)
fourth
102t21, 557GL (see also fourth)
generation and (see construction: cosmos; demiurge: harmonic method of (pattern of))
ideal numbers and
32â35, 33q, 34â35
incomplete octaves, Timaeus set
128â129t26
Laws harmonic series
277, 280, 281
limits of, regarding duadâs self-return
191
means operation and (see means operation)
model octave
56, 562GL (see also octave)
natural numbers, and
33, 33q, 34â35
numbers, sesquioctave operation
106f10, 107f11, 109t24, 126â129t26, 289â304Ap1
octave genera and chromatic elements
effect upon
125, 240
chromatic sequences (see chromatic sequences; Timaeus octave phenomena: α-CNF2 strings)
diatonic sequences (see diatonic sequences; Timaeus octave phenomena: α-CNF2 strings)
enharmonic sequences (see enharmonic sequences; Timaeus octave phenomena: α-CNF2 strings)
primary Timaeus scale
175â178t34
UPS systems (see under Timaeus octave phenomena: α-CNF2 strings)
Platoâs primary sequence
65t2, 66f5, 75f6, 114q
primary Timaeus scale
175â178t34
progressions and (see Decad: generation; Decad: pattern of; demiurge: harmonic method of (pattern of))
Timaeus number set
109t24, 126â129t26
Timaeus octave strings (see Timaeus octave phenomena: α-CNF2 strings)
Timaeus scale perfect disdiapason
133
trihemitones (see trihemitones)
tones and (see Timaeus octave phenomena: α-CNF2 strings)
tonoi and (see Timaeus octave phenomena: α-CNF2 strings; Timaeus octave types)
twenty-seventh row, numbers
122â123, 123n17
UPS systems (see under Timaeus octave phenomena: α-CNF2 strings)
series. See sequences
sesquialter part. See diapente; fifth (diapente; sequialter part)
sesquioctave part. See whole tone
sesquitertian part. See diatessaron; fourth (diatessaron; sesquitertian part)
768 (ends first complete Timaeus set octave; begins second perfect disdiapason, first diatonic Timaeus
LPS and UPS; begins model octave in one traditional interpretation)
128t26, 151, 152, 316Ap3, 562GL
768, 864, 972, 1024, 1152, 1296, 1458, 1536 (alternative model octave of standard Timaeus scale interpretation)
316Ap3, 562GL
729 (number attributed by ancients to the sun)
115n10, 269n5
shaking motion, Receptacle
23q, 25, 26, 28
shape
bodiliness and
16, 28
broad band (see under construction: cosmos Ï (chi figure))
circles (see circle)
cube (see cube)
Decad and
31f1, 31â32
dodecahedron (see dodecahedron)
equilateral (see equilateral constructions)
elements (see scalene figures; isosceles figures; primary bodies; air; earth; fire; water; tetrahedron; icosahedron; octahedron; cube)
formation relating to (see construction: cosmos (plane figures)); construction: cosmos (primary bodies)
GPS, fourth century
xvii, xviin15 (see also Greater Perfect System)
icosahedron (see icosahedron)
isosceles triangle (see isosceles figures: right triangle; isosceles figures: non-right triangle, relation to)
narrow band (see under construction: cosmos Ï (chi figure))
numbers and
figurate numbers
31f1, 31â32
Receptacle, presence in
22, 28
octahedron (see octahedron)
plane figures and (see construction: cosmos (plane figures))
primary bodies and (see construction: cosmos primary bodies)
primary Timaeus scale. See primary Timaeus scale
Receptacle and
16, 22
scalene triangle (see scalene figures: right triangle; scalene figures: non-right triangle, relation to)
sphere (see sphere)
square (see square)
standard UPS (see under Timaeus octave phenomena: α-CNF2 strings)
Timaeus octave strings. See Timaeus octave phenomena: α-CNF2 strings
universe
19, 20, 20n15, 41â42, 50, 51q, 59n17
dodecahedron
20n15, 41â42, 59n17 (see also construction: cosmos (World soul sphere))
World soul sphere (see under construction: cosmos (World soul sphere))
significant numbers (analysis)
8 (number of primary sequence)
65t2, 66f5, 75f6, 114q
81/64 (1.265625; ditone measure; associated with the enharmonic scale)
80, 96, 132
82944 (begins one cycle of combined standard UPS; ends another)
145t28, 149t28
54 (factor of 1719926784; also sum of its modern digits)
115
576 (begins first standard diatonic GPS sequence)
142
531441/524288 (ancient comma â1.013643264)
226
5040 (number householders, ideal polis)
272
41472 (square root of 1719926784; 9/8 ratio with both 36864 and 46656; begins complete articulation of combined standard UPS; ends an almost complete articulation)
116, 118, 119, 120, 145t28, 149t28
46656 (product of primary sequence; whole tone ratio with 41472)
115, 119, 120
4 (number of primary sequence)
65t2, 66f5, 75f6, 114q
4478976 (apex of Triangle of convergence)
249f17
4/3 (1.3; ratio of fourth) (see diatessaron; fourth (diatessaron; sesquitertian part))
4608 (ends first standard diatonic LPS and UPS sequences)
143, 144
9 (number of primary sequence; harmonic key to Laws octaves)
65t2, 66f5, 75f6, 114q, 277, 280, 281, 284
9/8 (1.125; whole tone ratio) (see whole tone)
98304 (ends a complete articulation of Timaeus combined UPS in primary Timaeus scale; begins another)
160t30, 163t30, 175, 175t34, 178t34
1 (begins Timaeus primary sequence; Ï arm endpoint; linked to monadic unity)
28, 65t2, 66f5, 75f6, 114q, 257f22
1743392201/1719926784 (comma)
226
1719926784 (last number of Timaeus set; Ï arm endpoint)
114â123, 129t26, 224â227, 257f22
191102976 (Ï arm endpoint; last scale element)
192f13, 257f22
192 (begins first complete diatonic tetrachord, number set)
126t26, 127t26
165888 (ends one complete combined standard UPS sequence)
149t28
â1.03125 (1773674496/1719926784; quarter tone derivable from a whole tone) (see under whole tone)
â1.033935547 (76230/73728; one of quarter tones derivable from apotomÄ) (see under apotomÄ)
â1.032821724 (78732/76230; one of quarter tones derivable from apotomÄ) (see under apotomÄ)
â1.026748971 (998/972; one of quarter tones derivable from leimma) (see under diesis)
â1.026052104 (1024/998; one of quarter tones derivable from leimma) (see under diesis)
1536 (ends first perfect Timaeus set disdiapason and first diatonic Timaeus GPS)
127â128t26, 133, 136, 137â138t27, 150
1152 (begins first standard diatonic UPS sequence)
144
1728 (begins first standard diatonic LPS system)
143
phi (also Φ; 1.618033988 â¦)
122, 123n17
768 (ends first complete octave sequence in Timaeus set; begins second perfect disdiapason, first diatonic Timaeus LPS and UPS; begins alternative model octave)
128t26, 151, 152, 316Ap3, 562GL
768, 864, 972, 1024, 1152, 1296, 1458, 1536 (alternative standard model octave)
316Ap3, 562GL
729 (number attributed by ancients to the sun)
115n10, 269n5
6144 (number above which 10368 is a âmajor sixthâ; determinable by deduction)
57
10077696 (narrow band middle)
251f19, 252, 256f21, 256t49
10368 (a standard Timaeus scale end; begins nearly complete combined standard UPS)
56, 145t28
10368, 11664, 13122, 13824, 15552, 17496, 19683, 20736 (alternative ending octave of standard Timaeus scale interpretation)
57
13824 (broad band middle)
251f19, 252, 256f21, 256t49
36864 (factor of 1719926784 with 46656; square of 192; whole tone ratio with 41472)
119, 120
3 (number of primary sequence; triadic monad comprising source of the primary Timaeus scale)
58, 65t2, 66f5, 75f6, 114q, 137t27
3/2 (1.5; ratio of fifth) (see diapente; fifth (diapente; sesquialter part))
384 (begins first complete octave, first perfect disdiapason, first diatonic Timaeus GPS, and primary Timaeus Scale)
127â128t26, 133, 136, 150, 175t34, 316Ap3
384, 432, 512, 576, 648, 729, 768 (model octave)
56, 133, 316Ap3, 562GL
393216 (ends primary Timaeus scale, study; ends last Timaeus combined UPS in primary Timaeus scale)
163t30, 175, 178t34
331776 (ends last complete articulation of standard combined UPS)
149t28
3072 (ends perfect disdiapason; and first articulation of diatonic Timaeus UPS)
128t26, 152
12288 (number above which 20736 is a âmajor sixthâ)
57
24 (begins first cognizable fourth of the Timaeus set)
126t26, 127t26, 130
24576 (begins a complete articulations of Timaeus combined UPS in the primary Timaeus scale)
160t30, 175, 175t34
27 (ends primary sequence; linked to rise of three dimensionality)
65, 65t2, 66f5, 68, 75f6, 114q
20736 (alternative standard Timaeus scale end; begins a complete combined standard UPS)
57, 145t28
2 (number of primary sequence; associated with duad)
28, 65t2, 66f5, 75f6, 114q
256/81 (3.160493827 ; Egyptian Ï)
258, 258n3, 259, 260, 563GL, 562GLn35
256/243 (â1.05349794238683) (see diesis)
2187 (first chromatic number; Ï arms endpoint)
127t26, 128t26, 257f22
2187/2048 (â1.06787109375) (see apotomÄ)
2304 (ends first standard diatonic GPS sequence; ends first diatonic Timaeus LPS sequence)
142, 151
2:1 (octave ratio) (see diapason; octave (2:1))
21/6 (â1.122462048 [1930552540/1719926784]; sixth root of 2; size of a true whole tone)
2, 224
21/12 (â1.059463094 [1822198952/1719926784]; square root of 2 1/6; size of a true semitone)
2, 225
21/24 (â1.029302237 [1770324486/1719926784]; fourth root of 2 1/6; size of a true quarter tone)
226
similarity. See sameness
simultaneity
artificerâs actions
17q, 52q, 64q, 76q, 102q, 242
cosmic fourth/cosmic disdiapason and
xvi
harmonia/ rectilinear character
35
musical phenomena, Timaeus set
124, 240
perfect squares/perfect cubes and
115n10
progressions and
38, 45
6144 (number above which 10368 is a âmajor sixthâ; determinable by deduction)
57
size
cosmos
31, 257â261
musical intervals (see individual entries for separate kinds)
octave chains (see individual entries for α-CNF2 strings; Timaeus octave phenomena: α-CNF2 strings)
primary Timaeus scale (see individual entry)
UPS systems (see under Timaeus octave phenomena: α-CNF2 strings)
See also extensions
Socrates
xviii, 1, 2, 16, 46â47n63, 317Ap3
solids. See primary bodies; fire; water; air; earth; tetrahedron; octahedron; icosahedron; cube
soul
human soul
xi, 10, 14, 15, 19, 23, 23q, 23â24q, 25
World soul
body, relation to
corporeal elements and
18
universe, body of
17q, 18q, 20, 20n15, 27, 27q, 41â42, 51â52q
harmonic bond to
14, 47â48q, 49, 49q
cosmic ordered motions, and
13q, 50, 51â52q, 210, 241q, 242, 261â263
bodies, relation to
22q, 23, 23q, 26, 26q, 49, 49q, 50
circles, same/different
13q, 19â20, 28, 210, 241q, 242, 261â263
divisions, band of difference
19â20, 210, 241q, 242, 264â265, 268
divisions of
primary
66f5, 75f6, 114q
secondary (see construction cosmos: (number matrix (fabric)))
harmonic construction of
xii, 12, 13q, 14, 28â31
Decad and
xiii, 15, 30 (see also demiurge: harmonic method of: pattern of)
intellect, relation to
16, 18
mediation role in
16, 17q, 18
Receptacle, relation to
20, 25, 26, 26q, 28, 50
shape of
xiv, xxiii, 20, 20n15, 41â42, 241q
Timaeus interpretation, armillary sphere theory
70â71
soul sphere (World soul sphere)
270f24
construction of (see construction: cosmos (World soul sphere))
Egyptian Ï, 256/81 and
258, 258n3, 259, 260, 563GL, 562GLn35
genesis, analogy to soul stuff divisions
258, 264
measure of
31, 257â261
motions of (see construction: cosmos (motions of))
See also construction: cosmos (World soul sphere)
soul stuff
commixture, identity with
28, 54n76, 61â64, 64q
divisions of
demiurgic action
47â48q, 49q
number matrix and associated operations
55â123, 241q, 289â304Ap1
primary divisions
64q, 65t1, 66f5, 75f6, 114q
means operation divisions
71q, 71â76, 76f7
sesquioctave operation divisions
101â123, 109t24, 126â129t26, 289â304Ap1
ultimate result
109t24, 243f15
musical significance of
124â240
species (octave)
form (see Timaeus octave types)
occurrences (see Timaeus octave phenomena)
triad, relation to
28â29
Speusippus
xiii, xiiin6, 1n3, 33â34q, 124, 285
spheres
armillary sphere
70â71
harmony (music) of
13, 270
soul sphere (see soul sphere; construction: cosmos (World soul sphere))
square
37â39, 38f3
square roots
ancient interval size and
2, 121, 223â227
perfect squares and
116â119, 120, 126â129t26
stability
CNF1
208
CNF2
208
cube and
46
Î-string
198t38, 207
γ-string and
198t38, 213t42
narrow band motion and
264
octave strings, patterns in
203â204
proportion, terms of 1719926784 based Φ series
121â123
sameness and
264
soul stuff and
264
starting tone number (definition)
57, 57n7
string length
Philolaus scale and
4â5
pitch, basis for
xv, 8â9, 116
pitch ordo of Timaeus scale and
xv
ancient, modern debates
xv, 5â10, 7â8n33
studyâs position
xv, 5â6, 7, 8â10, 59, 134, 180â181
string (octave string) See Timaeus octave phenomena
structures
polis (Laws)
272â273q, 273â274q, 274q, 275q
See also construction
studyâs method
mathematical character
xii, xxii, 15, 35, 58, 85, 97â98n3
Philebus and
60, 80â81, 82, 85, 85n2, 187
subsistence
mediation and
28, 29, 60q, 61
sensible natures and, precosmic state
49q
soul stuff and
28, 29, 60q, 61
time, precosmic absence of
52â53q
sun
115n10, 269, 269nn5â6
superparticularity. See epimoric (superparticular) ratios
surface area (soul sphere)
258, 260â261
symbolism
68
See also symbols
symbols
cosmos vis-Ã -vis All Perfect Animal
49, 51q, 52â53q
cosmos vis-Ã -vis diatonic scale
xiii, 227
disdiapason vis-Ã -vis cosmic limit
xxiii
Decad, vis-Ã -vis exemplar (All Perfect Animal)
xiii, xvi, 54, 55
figurate number vis-Ã -vis Decad
31â32, 31f1
Figure 15, use of
242â245, 243f15
octave string designations (see Timaeus octave phenomena: α-CNF2 strings)
musical model vis-Ã -vis planetary order
271
music vis-Ã -vis time imitating eternity
53
9 vis-Ã -vis planar dimensionality
68
primary sequence and levels of dimensionality
68â69
primary sequence and twenty-eight stringed lyre
114q
Table 15, use of
86â87t15
Table 16, use of
88â89t16
3 and soulâs processions
67
27 and three dimensionality
68, 68q
27th row extension and twenty-eight stringed lyre
113â114
2 and soulâs processions
67
symmetry
cosmic bands and
49, 49q, 253
equilateral notion and
41
fifth periodicity, rise of
58, 189t35, 189t36
filling intervals, patterns in
90t16, 95t17
matrix numbers, Crantor-style display
58, 99, 107
octave genera, varied relations of
311Ap3
octave patterns, role in
205
primary bodies and
42, 44, 45
primary Timaeus scale, rise of
58, 137â138t27
tetrachord patterns, role in
205
trihemitone distributions
492â495Ap9
symphonies
68â69
synemmenon
143, 146â147t28, 559GL
Tables 1â65
Tab. 24
107, 108, 109, 113, 114, 116, 120, 208, 244, 245, 248, 254, 257, 266, 267
Tab. 26
125, 126â129, 130, 136, 138, 142, 152, 153, 163, 167, 174, 183, 185, 188, 218, 219, 244, 245, 269, 346Ap5, 385Ap6, 407Ap7, 414Ap7
See also LFT
xxvi
tangibility
body, quality of
21q, 46, 47, 47q
cube and
46
earth, relation to
21q, 46, 47, 47q
primary bodies, comparative degrees
42t1, 45f4
Receptacle and
20
stability and
46
tetrad, relation to
46
visibility, relation to
46, 48â49
world, quality of
48q
Tarán, Leo
xiiin6, 32, 33q
See also Selected Bibliography
Tarentum
Archytas and
2, 3, 10, 12n52, 59, 118n13
Plato and
xiii, xxii, 11â12, 59
Taylor, A.E. (on missing fourth guest)
12
See also Selected Bibliography
Taylor, Thomas T. (on Platoâs harmonia)
67q, 70â71
See also Selected Bibliography
10077696 (narrow band middle)
251f19, 252, 256f21, 256t49
10368 (a traditional Timaeus scale end; begins a nearly complete combined standard UPS)
56, 145t28
10368, 11664, 13122, 13824, 15552, 17496, 19683, 20736 (alternative standard last Timaeus octave)
57
tetrachord. See diatessaron; fourth (diatessaron; sequitertian part)
tetrad
band of difference divisions, relation to
30, 210
cosmic fourth and
31
cosmic structure and
xviâxvii, 31
Decad, relation to
xvi, 30, 31, 31f1, 31n43, 32, 39, 45, 45f4, 46, 46q, 191, 285
exemplar, relation to
xvi
duad, relation to
31f1, 32, 39, 41, 45f4, 191
diapason and
xvi
chromatic STN diatonic scales, relation to
346â384Ap5, 390Ap6
cube and
45, 45f4
disdiapason and
xvi, 31, 39, 46, 191
earth and
46
figurate numbers and
31â32, 31n43
fourfoldness and
xvi, 30, 31, 46q, 191
harmonia and
39
monad, relation to
xvi, 31, 31f1, 32, 39, 45f4
progressions, role in
xvi, 30, 38, 39, 41, 45, 45f4, 46, 191
sameness/difference, relation to
30, 38, 39, 41, 45â46, 285
square and
38
triad, relation to
30, 31f1, 32, 39, 45f4
tetrahedron
complexity and triangularity, levels of
37, 38, 42t1, 43â46
construction of
40â46, 47â49
Decad and
40, 45, 45f4, 46
fire, association with
42, 42t1
fourfoldness and
43
monad, relation to
45, 45f4
ordo of generation vis-Ã -vis other primary bodies
40, 42t1, 40â46, 45f4
qualities of
46
sameness and difference and
45
scalene triangle and
37, 40, 40n56, 41, 45
triangularity, relation to
37, 45
tetractys
31, 31n43, 32, 180
See also Decad
tetraktys. See Decad; tetractys
13824 (broad band middle)
251f19, 252, 256f21, 256t49
36864 (factor of 1719926784 with 46656; square of 192; 9/8 ratio with 41472)
119, 120
3 (number of primary sequence; triadic monad comprising source of the primary Timaeus scale)
58, 65t2, 66f5, 75f6, 114q, 137t27
3/2 (1.5; ratio of fifth) (see diapente; fifth (diapente; sesquialter part))
384 (begins first complete octave, first complete perfect disdiapason, first diatonic Timaeus GPS, and primary Timaeus Scale)
127â128t26, 133, 136, 150, 175t34, 316Ap3
384, 432, 512, 576, 648, 729, 768 (model octave)
56, 133, 316Ap3, 562GL
393216 (endpoint primary Timaeus scale, study; endpoint, last instance of Timaeus combined UPS in primary Timaeus scale)
163t30, 175, 178t34
331776 (ends last complete articulation of standard combined UPS)
149t28
3072 (ends perfect disdiapason; and first articulation of diatonic Timaeus UPS)
128t26, 152
Timaeus (interpretation). See separate entries for individual commentators
Timaeus number set (see construction: cosmos (number matrix (fabric)))
Timaeus octave phenomena
α-CNF2 strings
chromatic scales and
212â217t42, 218, 220t43, 238â240, 431â464Ap8
diatonic scales and
125, 183â185, 203â204, 205, 207, 209â210, 238â240
scale catalogues
346â384Ap5, 385â406Ap6, 407â430Ap7
tabular summary
196â203t38, 205t39
enharmonic scales and
228t46, 229t47, 230â238t48, 238â240, 496â553Ap10
UPS system comparison
alternative UPS systems
314â317Ap3, 317â319Ap3, 320â321Ap4, 322â332Ap4t53, 332â333t54, 333â343Ap4t55, 344â345t56
standard UPS
145â149t28, 308â310Ap3
Timaeus UPS
160â163t30, 164â167t31, 167â170t32, 170â173t33, 313â314Ap3
UPS system construction
152â157, 308â310Ap3, 313â317Ap3, 322â325Ap4t53, 332â333Ap4t54, 333â337Ap4t55, 344â345Ap4t56
primary Timaeus scale
175â178t34
Timaeus octave strings. See Timaeus octave phenomena
Timaeus octave types
chromatic scales (form)
211t41, 222t45, 562GLt64
diatonic scales (form)
195t37, 211t41, 222t45, 562GLt64
enharmonic scales (form)
222t45, 562GLt64
Timaeus scale
175â178t34
times
ancient, music introduction
xvin12
Archytasâ era and, epimoric ratios
306â307Ap2
Aristoxenusâ era, musical developments
xx, 131
band of difference subdivisions and
20, 30, 241, 265
body/soul relation and
27q
cosmos and
13q, 24q, 27q, 49q, 51q, 52q, 52â53q, 53, 54
Decad and
54
disorder and
49q
eternity and
53, 52â53q, 54
ideal number, relation to
33q
image and
52q
music and
53
number and
33q, 53q
octave genera, opportuneness of
317Ap3
octave genera, time ordo of
xxi
pitch and
8, 9, 10, 131
Platoâs era, musical developments
xi, 1, 4, 10, 140
Plutarchâs era, surviving Timaeus commentary
64n33
precosmic state and
49q
preoccupations of study and
19
Ptolemyâs era, musical developments
306â307
Receptacle and
20, 23q
reciprocal relations and
181, 181q
rhythm and
53n75
Sectio canonis, possible evolution of
11
seven tonoi, chronological emergence of
xix
simultaneity (see simultaneity)
space, relation to
20, 33q
tonal space. See Timaeus octave phenomena
tones. See Timaeus octave phenomena and individual entries for different musical intervals
tonos
Aristoxenus and predecessors
xix, xviiin20, xxq
definition of
xxq
Dorian, association with standard GPS
181, 308Ap3, 313Ap3
evidence of, various kinds
xviii, 181, 317Ap3, 317Ap3
Eratocles and
xix
GPS/UPS, possiblilities and (see under Timaeus octave phenomena: α-CNF2 strings)
modern keys, comparison to
6, 141q, 309Ap3n6, 310â312Ap3
Phrygian, special features
315Ap3, 318Ap3, 409Ap7, 421Ap7
pitch and
8, 159n56
Platoâs opinion of
181
Ptolemy and
xix
relations among
211t41, 222t45, 562GLt64, 310â312Ap3 (see also reciprocal relations)
Socratesâ opinion of
317Ap3
studyâs usage of term
xx, xxq
Timaeus interpretation and
xvin13, 8, 114, 124, 145â149t28, 195, 211, 212, 219
Timaeus tonos phenomena (see Timaeus octave phenomena)
Twenty-eight stringed lyre and
113â114
transpositions
fourths
7, 10, 11, 55, 59
octave species, and
311Ap3, 315Ap3, 316Ap3
perfect systems, point of
140, 181
vocal ranges and
140â141
triadic monad
equilateral triangle and
37f2
fabric/bands
30
Laws diatonic scales, number sets sourcing
277
octave genera
180
quintessential triangle and
36, 39
soul sphere/cosmos
258
soul stuff
28, 29, 39, 63, 263
source of octave scale
58, 136, 137t37, 244
Timaeus set termination
58
triads
bands
210, 264
cosmos as
258
fabric/bands relation
30
Decad, relation to
32, 46
duad, relation to
30, 36, 38, 46, 190
equilateral triangle and
37f2
fifth periodicity and
190
figurate number and
31â32, 31f1
γ-string/Î-string relation and
200t38, 201t38, 207, 208, 208f14
gods (Laws) and
277
Laws diatonic scales, source of
277, 280, 281
monad, relation to
28, 29, 30, 36, 37, 39, 190, 207, 208, 208f14, 244
Platoâs preoccupation with
180
progressions, order and role in
generally
36, 285
harmonia and
39
octave scale
58, 136, 137â138t27, 244
plane figures
36, 37, 38
primary bodies
40, 45, 45f4, 46
sameness/difference, relation to
30, 37, 39, 263, 264
soul sphere
258
soul stuff and
28, 29, 36, 63
tetrad, relation to
30, 39
Timaeus set termination
58
visibility and
46
triangle. See isosceles figures: right triangle; isosceles figures: non-right triangle, relation to; scalene figures: right triangle; scalene figures: non-right triangle, relation to; equilateral constructions: triangle
Triangle of Convergence
249f17, 249â252, 251f19
triangularity
plane figures
35â39
primary bodies
40â46, 42t1, 45f4
See also equilateral constructions: triangle; isosceles figures: right triangle (triangularity); scalene figures: right triangle (triangularity)
trihemitones
chromatic scales, relevance to
218â219
distribution patterns
494Ap9t60, 494Ap9t61, 495Ap9t62, 495Ap9t63
sequence patterns
219, 219n3, 465â492Ap9t57, 492â495Ap9, 493Ap9t58
triple. See diapason diapente
trite. See trite diezeugmenon; trite hyperbolaion
trite diezeugmenon (definition)
146t28, 559GL
trite hyperbolaion (definition)
146t28, 559GL
tumult
50
twelfth (musical)
diapason diapente and1
70, 555GL, 558GL
Platoâs primary divisions and
xiv, 29n40, 138
triple intervals, identity with
xivn8, 29n40, 70, 138, 209, 555GL, 558GL
See also diapason diapente
12288 (number above which 20736 is a âmajor sixthâ)
57
Twenty-eight stringed lyre
113â114, 113q
24 (begins first cognizable fourth of the Timaeus set)
126t26, 127t26, 130
24576 (begins a Timaeus combined UPS sequence in primary Timaeus scale)
160t30, 175, 175t34
27 (ends primary sequence; associated with three dimensionality)
65, 65t2, 66f5, 68, 75f6, 114q
20736 (alternative traditiona Timaeus scale end; begins a combined standard UPS sequence)
57, 145t28
2 (number of primary sequence; associated with duad)
28, 65t2, 66f5, 75f6, 114q
256/81 (3.160493827 ; Egyptian Ï)
258, 258n3, 259, 260, 563GL, 563GLn35
256/243 (â1.05349794238683) (see diesis)
2187 (first chromatic number; endpoint of Ï arms for their joinder)
127t26, 128t26, 257f22
2187/2048 (â1.06787109375) (see apotomÄ)
2304 (ends first standard diatonic GPS sequence and first diatonic Timaeus LPS sequence)
142, 151
2:1 (octave ratio) (see diapason; octave (2:1))
21/6 (â1.122462048 [1930552540/1719926784]; sixth root of 2; size of a true whole tone)
2, 224
21/12 (â1.059463094 [1822198952/1719926784]; square root of 2 1/6; size of a true semitone)
2, 225
21/24 (â1.029302237 [1770324486/1719926784]; fourth root of 2 1/6; size of a true quarter tone)
226
ultimate animal (All Perfect Animal; exemplar)
17
See also All Perfect Animal; Decad; exemplar
umbrella concept (uniting primary bodies)
40, 41, 42
unaccustomed sequences (UDOP)
204â205, 205t39
occurrences, specific instances
346â384Ap5, 385â406Ap6, 407â430Ap7
unity
48, 67, 69
analogy and
48, 67, 69
bonds of universe, relation to
14
Decad and
35
dimension, relation to
65
duad, relation to
22, 28, 29, 67, 187, 190â191
5040 and
273
generation and
66
harmonia and
28, 29
intermediaries, role in relation to
48
limit and
14
limiter, role in relation to
22
members of primary sequence, relation to
65t2, 66
monad, role in relation to
28, 29, 187, 190â191
number and
33q, 65, 66, 67
numerical relationships, and
14
proportion and
28, 48
Receptacle and
22
self-return and
68
2, relation to
14
wholeness and
48
universe (cosmos)
All Perfect Animal and
17, 18, 24q, 49, 52q, 59
analogy and
47q, 48, 49, 49q, 50, 51q, 53
artificer, action of (see demiurge)
beauty of
16, 17q
body, nature and shape of
19, 20, 20n15, 41â42, 50, 51q, 59n17
bonds of
14, 48q, 49q
cause, need for
16
changeable nature of
12, 16
construction of
14, 18, 34, 35, 59, 186, 191, 404Ap6 (see also construction: cosmos)
corporeality, acquisition of
17â18
divine character of
51, 51q, 52, 52q
dodecahedron and
20n15, 41â42, 59n17
doubles and triples, vis-Ã -vis fourths
86â87t15
eternity, image of
16, 25, 52, 52â53q, 53
exemplar, image of
16, 17, 24q, 52q
finitude of
12, 59, 191
fourths, vis-Ã -vis whole tones
102t21
harmonia and
12, 48q, 53, 187, 191, 193
harmonic units and
258, 258â261, 285
human passions and disease, relation to
50
intelligence of
16, 17q, 27q
intermediation (mediation) and
47q, 48, 48q
living character of
16, 17, 17q, 27q
living things, genesis of
51q
motion of
(see construction: cosmos (motions of))
number and
49, 52, 52q
nutriment, manner of
51q
perfection of
51q
plane figures and (see construction: cosmos (plane figures))
primary bodies and
22q, 26, 26q, 41, 47q, 48q, 50q (see also construction: cosmos (primary bodies))
Receptacle and
20, 22, 22q, 25â28, 26q, 50
soul and
18, 18q, 20, 27q, 52q
structure, similarity on different levels
49, 50
symmetry and
49, 49q
three dimensionality and
71
Timaeus account of
creation story and
1
mathematical model for
xxii, xxiin39, 50, 52, 53n75, 70â71
musical paradigm of
xiii, 50, 53, 59, 186, 187, 191, 193
probable character of
xii, 16
time and
52â53q, 53
unity and
48, 53
wholeness and
48
See also construction: cosmos
Unlimit (definition)
14, 47n65
Unmodulating Perfect System (also UPS) (see under Timaeus octave phenomena: α-CNF2 strings)
Venus
269, 269n6, 270f24
very idea of a triangle (quintessential triangle)
39, 40
quintessential triangle and
36, 38, 39
vibration(s) (impacts on air). See pitch (es): ancient approaches (air impacts))
visibility
body, quality of
46
fire, relation to
47
Millerâs analysis
46â49
monad, relation to
46, 49
primary bodies, comparative degrees
40â46, 42t1, 45f4
tangibility, relation to
46, 47
tetrahedron and
42t1, 42n58, 46
triad, relation to
46
Triangle of Convergence and
250
triangularity, relation to
42n58, 46
Vogel, C.J.
13q
volume (soul sphere)
258â260, 261
water
cosmic creation and
18, 46q
genesis of (as element)
19, 28, 35, 40, 42, 48
harmonia (bands of analogy and symmetry) and
49â50q
icosahedron and
40, 42, 42t1
living creatures and
49, 50
matrix of numbers, analogy to
28
mediation and
18, 48q
other primary elements, relation to
18, 19, 26q, 42, 42t1
precosmic state and
49â50q
Receptacle and (see Receptacle: bodies (primary bodies))
visibility, relation to
46
West, M. L
7, 13, 14n64, 31n43, 140
wholeness
48, 68â69
whole tone (9/8 (1.125); also sesquioctave proportion)
apotomÄ, relation to
219, 219n4, 554GL, 556GL
calculation of particular intervals, role
117, 118
chromatic invasion patterns and
186
definition of
2, 11, 115â116, 276, 556GL
disjunct demiurgic fourths and
3, 69
double intervals, relation to
557GL
fifths, relation to
4â5, 557GL
fourths, relation to
aesthetics
205
composition
4, 557GL
Timaeus numbers and
98f9, 101â123, 106f10, 107f11, 108f12, 109t24, 111â112t25, 289â304Ap1
Timaeus sesquitertian parts parts to fill
99â100t20
GPS, relation to
141, 308Ap3, 315Ap3, 317â318Ap3, 558GL
indivisibility of
2, 3, 116, 224, 312Ap3
lambda (λ) representation
70
leimma (diesis), relation to
2â3, 219n4, 554GL, 556GL
major sixth, relation to
57
measure of
2, 134, 225
octave, relation to
3, 4, 5, 86â87t15, 102t21, 139
Philolausâ units and
134
quarter tone, source of
calculation of
223
de facto inexpressibility as whole number ratio
224, 225â226
â1.03125 (1773674496/1719926784)
223, 224, 501Ap10, 507Ap10, 515Ap10, 516Ap10, 532Ap10, 544Ap10
whole number ratio approximations and 1719926784
223, 225â226
significant numbers, special relations
41472/36864
120
46656/41472
115, 116, 120
1719926784
121, 224, 225, 226, 227
studyâs analysis and (see Timaeus octave phenomena)
triple intervals and
558GL
unity, image of
68â69
World soul. See soul: world soul; construction: cosmos (World soul sphere)
World soul sphere. See construction: cosmos (World soul sphere)
Ï (chi). See chi (Ï); construction: cosmos (Ï (chi figure))
Xenocrates
33â34q, 64n33
Ïá½½Ïα. See Receptacle
ζ-string(s)
chromatic
215t42, 220t43, 447â450Ap8
diatonic
behavior
Appendix 7
410â411, 412, 414, 418, 419, 421, 424â430
text
200â201t38, 203â204, 205t39, 206â207, 208f14
chromatic invasion, pattern
209t40
enharmonic
229t47, 235t48, 527â533Ap10
other octave strings relation to
208f14
universal orbits and
266f23, 267â269