Harmonic Theory Diagrams

8/13/2019 Harmonic Theory Diagrams

1/63

cThe

ory.com

NumberDiagram

s


2/63

c

Any number on a blue line has a factor of 2.

Any number on a green line has a factor of 3.

Any number in red has a factor of 5 and is in a ratio of 5/6 to the number beneath it. Some numbers can

go on for generations of 5/6ths and still be whole numbers. In these diagrams only the first and second genera-

tions from the root number are shown; the first generation is shown in red, and, if there is a second generation

whole number, it is shown in black just above the number in red.

The number with factors of 2, 3, and/or 5 provide the primary warp and woof of the universal numeric

fabric. These are the numbers used in the diagram 31,104Day in the Life of Brahma. The other numbers used

7, 11, and 37each provide a different colored thread to the fabric but use the same multiples of 2, 3, and/or 5

to interweave with the primary 2-3-5 structure.

Any number on an orange line has a factor of 7. 7 is added in the diagram 15120The Lesser Maze in

Theosophybecause it adds numbers to the canon related to the ideal synodic moon cycle, and the ideal cycles

of the outer planets Uranus, Neptune, and Pluto. 15120 is also the sum of all the plane angles of the five Platonic

solids and the number of years it takes for the equinoctial point to pass through seven signs of the zodiac.

Any number on a magenta line has a factor of 11. 11 is added to the diagram 31680Music and

Cosmology of the Earth and Moon. Astronomically and geometrically, the combined radii of the earth and

i 5040 il A i l ith di f 5040 il h i f f 31680 il ( i 22/7)

Notes for the Number Section


3/63

divisible by 4

divisible by 9

divisible by 10

divisible by 12

divisible by 8

divisible by 5

divisible by 6

divisible by 3

divisible by 2

divisible by 25 - (root of Saturn cycle)

(divisible by 7 - not used in this table)

(divisible by 11 - not used in this table)

This table grows out of the Pythagorean tetractys

1

2

4

8

3

9

27

11-15-2005

675450

225

300

150

200

100

50

25

75

1

4

8

16

32

64

128

6

12

24

48

96

192

18

36

72

144

288

54

108

216

432

162

324

648

486

972

2 3

9

27

81

243

10

20 30

40 60 90

80 120 180 270

160240 360 540 810

5

405

135

45

15

31,104 - Day in the life of BrahmaPrec e s s i o na l c y c l e o f 25 ,920 y ea r s p l u s i t sdawn (2592) and twi l i gh t (2592) = 31,104.311,040,000,000,000 = the number o f yearsi n t h e l i f e o f B rahma , wh i c h r ep r e s en t s t h edu ra t i o n o f t h e So l a r S y s t em i n t h e anc i en t B rahman i c a l T ab l e s .

31,104/25920 i s i n the ra t i o o f 6/5 as are a l lthe pa i rs o f numbers (a number wi th i t s dawnand twi l i gh t .) Mus i ca l l y , the ra t i o o f 6/5 o r 5/6is a m ino r 3rd.

f

z

d

s

ideal radius of Sun

432,000 miles

a


4/63

Total sum = 15120

icosahedron 3600

tetrahedron 720(x2)

dodecahedron 6480

octahedron 1440

cube 2160

11-15-2005

450*

225

300*

100 150

50 75

25

200 350 525 675

175

1

4

8

16

32

6

12

24

48

96

18

36

72

144

288

54

108

216

432

162

324

648

486

972

2 3

9

27

81

243

64

128 192

10

20 30

40 60 90

80 120 180 270

240 360 540 810

5

405

135

45

15

160

14

28

56

112

7

21

42

84

168

336

63

126

252

504

189

378

756

630420280

315210140

10570

35

1134

567

945

224

15120 - The Lesser Maze in Theosophy

f

z

s

d

5 related

3 related

2 related

7 related

k

l ;

Some numbers, like a day in nature, have a dawn and twilightthat is a whole number. This is a theosophical conceptexplained in Mathematics of the Cosmic MInd byL. Gordon Plummer. One can find a number withits dawn and twilight by multiplying the original numberby the overtonal musical ratio of the minor third (6/5)Ex. 90x 6/5 = 108. One can also use the inverse (5/6) x108 to find a day without dawn and twilight. Mathematicallythe decimal equivalent of 6/5 is 1.200 and the decimalequivalent of 5/6 is .833. Mystically, the gematria is:

s

ideal radius of Sun

432,000 miles

a

This table grows out of the Pythagorean lambda and tetractys

1

2

4

8

3

9

27


5/63

4

8

16

32

6

12

24

48

96

18

36

72

144

288

54

108

216

432

864576

2 3

9

27

64

192

384

128

15

5

45

180

60 90

3020

10

55

110 165

330

660

495

990

220

440

120

360

80

240

720480

160

320

40

825550

275

600*

300*

150*

75

400

200

100

50

25

22 33

99

11

44

88

176

352

66

198

396

132

792528

264

1188

594

297

31680 - Music and Cosmology of the Earth and Moon

A square with a side of 7920 miles (2x3960, which is the mean radiusof the Earth), has a perimeter (4 x side) of 31680 miles. 4 x 7920 = 31680.A circle with a radius of 5040 miles (5040 is t he combined radius of the Earth -3960 miles, and the Moon - 1080 miles) has a circumference (using pi=22/7)of 31680 miles. (Circumference = pi x 2r). 22/7(2 x 5040) = 31680

z

f

d

g

27 days = ideal sidereal Moon cycle s90 days = ideal Mercury cycle d216 days= ideal Venus cycle f360 days = ideal Earth cycle z720 days = ideal Mars cycle g

s

1/1 =

256/243 =

16/15 =

10/9 =9/8 =

32/27 =

6/5 =

5/4 =

000000

190448

240000

400000450000

663600

720000

900000

C

Db

Db

DD

Eb

Eb

E

5 related3 related2 related

11 related

*1

ideal radius of Sun432,000 miles

a

Some numbers, like a day in nature, have a dawn and twilightthat is a whole number. This is a theosophical conceptexplained in Mathematics of the Cosmic MInd byL. Gordon Plummer. One can find a number withits dawn and twilight by multiplying the original numberby the overtonal musical ratio of the minor third (6/5)Ex. 90x 6/5 = 108. One can also use the inverse (5/6) x108 to find a day without dawn and twilight. Mathematicallythe decimal equivalent of 6/5 is 1.200 and the decimalequivalent of 5/6 is .833. Mystically, the gematria is:

11-15-2005

1200 = lucnon= Light, lamp832 = qeofilo= beloved by God


6/63

Total sum = 15120

icosahedron 3600

tetrahedron 720(x2)

dodecahedron 6480

octahedron 1440

cube 2160

1

2

4

8

3

9

27 1200 = lucnon= Light, lamp832 = qeofilo= beloved by God

450*

225

300*

100 150

50 75

25

200 675

1

4

8

16

32

6

12

24

48

96

18

36

72

144

288

54

108

216

432

162

324

648

486

972

2 3

9

27

81

243

64

128 192

10

20 30

40 60 90

80 120 180 270

240 360 540 810

5

405

135

45

15

160

37

74

148

296

111

222

444

333

666

555370

185

999

Gematria of Pythagoras and Jesus

f

z

s

d

Puqagora

Kosmo= cosmos

= Pythagoras(Lesser Canon)

en idea = in nature; in idea

olh= whole; perfecth alhqeia = The Truth

h kardia = The heart

Some numbers, like a day in nature, have a dawn & twilightthat is a whole number. This is a theosophical conceptexplained in Mathematics of the Cosmic MInd byL. Gordon Plummer. One can find a number withits dawn and twilight by multiplying the original numberby the overtonal musical ratio of the minor third (6/5)Ex. 90x 6/5 = 108. One can also use the inverse (5/6) x108 to find a day without dawn and twilight. Mathematicallythe decimal equivalent of 6/5 is 1.200 and the decimalequivalent of 5/6 is .833. Mystically, the gematria is:

alhqeia= Truth

ideal radius of Sunkatabolh = Foundation

a

5 related

3 related

2 related

37 related

This table grows out of the Pythagorean lambda & tetractys


7/63

A Pythagorean Value of Pi

P Y TH A G O R A S

P U Q A G O R A S

P

U

Q

A

G

O

R

AS

= 80=400= 9= 1= 3= 70=100

= 1=200

864

Greek numeric values

for the letters of the

name Pythagoras

Added together, the numbers 864 + (864 x 360) + (864 x 360 x 100) = a value of Pi accurate to 7 decimal places.

The Greek gematria for the name Pythagoras is 864.

The two most revered symbols of the Pythagoreans were the sphere and the Tetractys..

. .

. ..

.. ..

English letters

Greek letters

Number of degrees in a circle Number of dots

360 10

360 10

.

. .

. ..

.. ..

The Sphere The Tetractys

PUQAGORAS

864


8/63

Cycles

5 x 10

5 Dawn**

50 5 Twilight

30 Dawn300

30 Twilight

180 Dawn1800 180 Twilight


60

360

2160

25920

6 Dawn

60 6 Twilight

36 Dawn360

36 Twilight

72

432


2592


31104

5 x 60

5 x 360

10 x 2160

25,920 solar years is the number of

years in the Great Year of Plato. (It takes

25920 solar years to complete one

precessional cycle.)

432 Hz is the ideal tuning pitch for A

above middle C at 256 Hz.

360 is the ideal number of days

in a solar year.

2160 is the ideal number of years for

the precessional cycle to move through

one sign of the zodiac.

72 is the ideal human pulse rate

for 1 minute of 60 seconds.

60 is the ideal number of seconds

in one miute.

72 is the number of years for the

precession of the equinoxes to move1.

72 is the ideal number of years for the

human life span.

72 is the ideal number of breaths

drawn in 4 minutes - the time

required for the earth to rotate 1 on

its axis.

25920 x 12 31104 x 10

2160 is the ideal diameter ofsin miles.

432,000 is the ideal radius of theain miles.

Cosmic Cycles, Cosmic Numbers, and the Value of !*


9/63

100 +1000 + 100 = 1200 solar years

(Bronze Age) Dwapara Yuga

(Silver Age) Treta Yuga

(Golden Age) Satya Yuga

(Iron Age) Kali Yuga

200 +2000 + 200 = 2400 solar years

300 +3000 + 300 = 3600 solar years

400 +4000 + 400 = 4800 solar years

12000 solar years

The Yugas ascending*The Yugas descending

Concepts adapted from The Holy Scienceby Swami Sri Yukteswar andMathematics of the Cosmic Mindby L. Gordon Plummer

108 + 1080 + 108 = 1296 solar years

216 + 2160 + 216 = 2592 solar years

324 + 3240 + 324 = 3888 solar years

432 + 4320 + 432 = 5184 solar years

12960 years

The Yugas ascending using the 108, (the sacred, mystical)form of 100

108 + 1080 + 108 = 1296 solar years

216 + 2160 + 216 = 2592 solar years

324 + 3240 + 324 = 3888 solar years

432 + 4320 + 432 = 5184 solar years

12960 solar years

12000 + 12000 = 24000 solar years

12000 solar years

12960 + 12960 = 25920 solar years

25920 years is The Great Year of Plato

25920/12 = 2160 solar years = Platonic month. (2160 is also diameter of moon.)

100 +1000 + 100 = 1200 solar years

200 +2000 + 200 = 2400 solar years

300 +3000 + 300 = 3600 solar years

400 +4000 + 400 = 4800 solar years

2160 /30(days in a month) 72 solar years Platonic day

The Value of !, the Platonic Year, and the Hindu Yugas






(Silver Age) Treta Yuga(Golden Age) Satya Yuga






The Yugas descending using the 108, (the sacred, mystical)form of 100

*Whether using 100 or 108 as the unit, the concept of the Pythagorean tetractys (1 + 2 + 3 + 4) is common to both

.

. .

. ..

.. ..

1

2

3

4


10/6386,400

. . . 311,040,000,000,000, the number of years in the Life of Brahma, which represents the duration of the Solar System as given in the ancientBrahmanical Tables, dating back many centuries before Christ. From Mathematics of the Cosmic Mindby L. Gordon Plummer,p. 34.

The value of !, which expresses the relations of the circumference to the diameter of a circle, was concealed in the great cycle known asthe Age of Brahma, 311,040,000,000,000 years. This age is one hundred years of Brahma, and a Brahmic year is therefore3,110,400,000,000 solar years. This number is again divided by 360, which gives the value of the Brahmanic Day or 8,640,000,000 years.Then by adding together the value for the age, year and the day of Brahma, we have 314159, etc., which is the familiar value of !or very

nearly, 355 divided by 113.From The Kabala of Numbersby Sepharial, p.12.

86,400 seconds = 24 (hours in a day) x 60 (minutes in an hour) x 60 (seconds in a minute)

31,104,000 seconds = 864,400 (seconds in a day) x 360 (days in a solar year)

3,110,400,000 seconds = 31,104,000 (seconds in a year) x 100 (100 solar years)

86,400 seconds = 1 solar day

31,104,000 seconds = 1 solar year

3,110,400,000 seconds = 100 solar years

The Value of !and the Brahmanic Tables

1440 minutes = 1 solar day (86,400/60 seconds per minute)

24 hours = 1 solar day (1440/60 minutes per hour)

518,400 minutes = 1 solar year (31,104,000/60 seconds per minute)

8640 hours = 1 solar year (518,400/60 minutes per hour)

360 days = 1 solar year (8640/24 hours per day)

(read as three billion, one hundred ten million four hundred thousand seconds)

seconds in a solar day


11/63cTheory.com

MusicD

iagrams


12/63

A432 Hz is the reference tuning pitch, not A440 Hz. A432 is used because the ideal radius of the Sun is 432,000

miles.

D288 Hz is the center, not middle C. D 288 is the whole step just above middle C. D is used because it

is the only note that has the same symmetry stepwise both ascending and descending, as will be shown exten-

sively in the diagrams. Generally, in a C scale, C is the bottom note and the scale ascends from from C to C1, as

C D E F G A B C1. In this work, D is the center and extends an octave in either direction, as

D1 E

1 F

1 G

1 A

1 B

1 C

1 D E F G A B C D1

The 2-3 lattice is rooted in the the 2nd and 3rd harmonics of the overtone series, the octave and perfect

fifth respectively. The Arithmetic and Harmonic Means of the octave also generate these perfect fifths, ascending

and descending. The Arithmetic Mean of the octave generates the same perfect fifth that is the 3rd harmonic of

the overtone series. The Harmonic Mean of the octave generates the perfect fifth descending, which is the same

as the reflection of the harmonic perfect fifth of the overtone series. (In relation to the upper and lower bound-

ing notes of the octave, the perfect fifth descending is the perfect fourth ascending, and the perfect fifth ascend-

ing is the perfect fourth descending.)

Th 2 3 5 l tti i t d i th 2 d 3 d d 5th h i th t fifth d j thi d Th

Notes for the Music Section


13/63

1/1 2/1

288

0

3/2

432

+2

9/8

324

+4

27/16

486

+6

81/64

364.5

+8

243/128

546.75

+10

729/512

410.06

+12

4/3

384

-2

16/9

512

-4

32/27

341.33

-6

128/81

455.11

-8

256/243

303.4

-10

1024/729

404.5

-12

512/405364.08

+6

256/135546.12

+8

64/45409.6

+10

16/15307.2

+12

8/5460.8

+14

6/5345.6

+16

9/5518.4

+18

10/9

320-18

5/3

480-16

5/4

360-14

15/8

540-12

45/32

405-10

135/128

303.75-8

405/256

455.625-6

[ ]

[ ]

( () )

Harmonic Lattice from D 288 Hz, ratio 1/1 and A at 432 Hz

(all notes are in the octave D288HzD576Hz, ratios 1/12/1)

Notes are given withnote names

ratios

hertz& cents + or -

Circles are used for the perfect intervalsthe unison, octave, perfect fourth, and perfect fifth.

Ovals are used for the other intervals in the cycle of fourths and fifths from D (those intervals

with divisors of 2 and 3, coming from the 3rd harmonic.)

Diamonds are used for notes with divisors of 2, 3, and 5. There are two versions of each note except for G, D, and A.

Notes in brackets are alternatives of pitches with different intentions. [G#, D#, A#] has almost the same hertz (pitches) as (Ab, Eb, Bb), but the

note names are are different (enharmonic spellings.) Similarly (F#, C#, G#) share almost the same pitches as [Gb, Db, Ab], but with enharmonicspellings, depending on the intention.

The Arithmetic mean of the Perfect 5th has a Major 3rdascending from the root. With the AM pattern repeated,

these ascending Major 3rds form their own spine ofPerfect 5ths.

The Harmonic mean of the Perfect 5th has a Major 3rddescending from the 5th. With the HM pattern repeated,

these descending Major 3rds form their own spine ofPerfect 5ths

The Arithmetic Mean of the octave has a P5thascending from the root.

The Harmonic Mean of the octave has a P5thdescending from the root.

(See chart of AM, HM, GM generated from the numberratios of 1, 2, 3, and 4.)

8-13-2006

SWM3rd

des

NEM3rd

asc

NWm3rddes

SEm3rdasc

The AM of D288

and A432 is F#360

The HM of D288

and A432 is F345.6

Eb

E B F# C# G# D# A#

Ab Eb Bb F C G D A E B F# C# G#

Gb Db Ab Bb F C

(pattern continues)

(pattern continues)


14/63

1/1 2/1

293.334

0

3/2

440

+2

9/8

330

+4

27/16

495

+6

81/64

371.25

+8

243/128

556.88

+10

729/512

417.66

+12

4/3

391.11

-2

16/9

521.482

-4

32/27

347.655

-6

128/81

463.54

-8

256/243

309.02

-10

1024/729

412.04

-12

512/405370.83

+6

256/135556.25

+8

64/45417.186

+10

16/15312.89

+12

8/5469.33

+14

6/5352

+16

9/5528

+18

10/9

325.93-18

5/3

488.89-16

5/4

366.67-14

15/8

550-12

45/32

412.5-10

135/128

309.375-8

405/256

464.06-6

[ ]

[ ]

( () )

Harmonic Lattice from D 293.334 Hz, ratio 1/1 and A at 440 Hz*

(all notes are in the octave D293.34D586.667, ratios 1/12/1)

Notes are given withnote names

ratios

hertz& cents + or -

Circles are used for the perfect intervalsthe unison, octave, perfect fourth, and perfect fifth.

Ovals are used for the other intervals in the cycle of fourths and fifths from D (those intervals

with divisors of 2 and 3, coming from the 3rd harmonic.)

Diamonds are used for notes with divisors of 2, 3, and 5. There are two versions of each note except for G, D, and A.

Notes in brackets are alternatives of pitches with different intentions. [G#, D#, A#] has almost the same hertz (pitches) as (Ab, Eb, Bb), but the

note names are are different (enharmonic spellings.) Similarly (F#, C#, G#) share almost the same pitches as [Gb, Db, Ab], but with enharmonicspellings, depending on the intention.

The Arithmetic mean of the Perfect 5th has a Major 3rdascending from the root. With the AM pattern repeated,

these ascending Major 3rds form their own spine of Perfect 5ths.The Harmonic mean of the Perfect 5th has a Major 3rd

descending from the 5th. With the HM pattern repeated,these descending Major 3rds form their own spine of Perfect 5ths

The Arithmetic Mean of the octave has a P5thascending from the root.

The Harmonic Mean of the octave has a P5thdescending from the root.

(See chart of AM, HM, GM generated from the numberratios of 1, 2, 3, and 4.)

8-13-2006

SWM3rd

des

NEM3rd

asc

NWm3rddes

SEm3rdasc

The AM of D293.334

and A440 is F#366.67

The HM of D293.334

and A440 is F352

Eb

E B F# C# G# D# A#

Ab Eb Bb F C G D A E B F# C# G#

Gb Db Ab Bb F C

(pattern continues)

(pattern continues)

* Since A440 is the conventional standard base

pitch, I have included this page showing hertz

values using A440 rather than A432. Note that

the ratios and cents are still the same and arecalculated from D ratio 1/1 and octave 2/1.


15/63

AM

ofD-D

AMof

D-A

HMof

D-A

HMof

D-D

HMof

D-F#

AM

ofD-F#

AM

ofBb-D

AM

ofG-D

HMof

Bb-D

HMof

G-D

AM

ofD-D

HMof

D-D

2/1

3/2

4/3

5/4

6/5

9/8

10/9

9/10

8/95/6

4/5

3/4

2/3

1/2

3/2

4/3

GM

ofD-D

GMof

D-D

Overview of the Arithmetic, Harmonic, and Geometric Means

b = (a + c)/2

b =2(c x a)/(c + a)

b = !(c x a)

D

288

D

576

A

432

D

144

D

288

D

144

D

288

G

192

D

144

D

288

Ab

204

D

288

D

576

G#

408= ! ( )x

b = !(a x c)

= ! ( )x

Geometric MeanGeometric Mean

2 ( )x

+( )

=

+( )

2

=

ArithmeticMean

HarmonicMean

D

288G

384

A

432

D

576

F#

360

E

320

Bb

230.4

C

259.2

D

144

Eb

306

C

256

E

324

Ab

204

G#

408

F

345.6

A

216

G

192

B

240

D#

306

17/16

1.06251062 = TheTrue Logos

17/161.0625

1062 = ologo iqu

Total hertz of horizontal row = 5179.25179.2 rounded to 5180

5180 octave reduction to 2590.Greek gematria for 2590 =The Logos: Image of God

O logo h eikwn tou Qeou

12-16-2005


16/63

D

A

Bb

B

B

C

C

Bb

F

F

F#

G

Ab

G#

F#

A#Bb

C#

Db

B

C

E

F

F#

Gb

G

G#

Ab

A

Color Tunings in Hue, Saturation, and Brightness for 12-Note Equal Temperament, 17-Note, and

22-Note 2-3-5 Limit Lattice

A

A#

Bb

B

C

C#

Db

D#

Eb

E

F

F#Gb

G

G#

Ab

15/8540

16/9512

9/5518.4

16/9512

1200

0

1100

1000

900

80027/16486

8/5460.8

405/256455.625

128/81455.11

27/16486

5/3480

8/5460.8

100

200

17843/9452543.672

4679/2626513.158

9043/5377484.356

4813/3032457.172

6064/4813362.857

19723/16585342.492

29798/26547323.269140221/

132351305.125

1/1-2/1288-576

32/27341.33

9/8324

400

300

10178/6793431.512 19601/

13860407.294

6793/5089384.434

4/3384

5/4360)

512/405364.08

500

3/2432

45/32405

64/45409.6

600

700

1024/729404.5

32/27341.33

81/64364.5

729/512

410.06

4/3384

5/4360)

6/5345.6

3/2432

11-10-2005


17/63

D

A

Bb

B

B

C

C

C

Bb

F

F

F#

G

Ab

G#

F#

A#Bb

C#

Db

B

C

Eb

E

F

F#

Gb

G

G#

Ab

A

Color Tunings in Hue, Saturation, and Brightness for 12-Note Equal Temperament, 17-Note, and

22-Note 2-3-5 Limit Lattice

A

A#Bb

B

C

C#

Db

D#Eb

E

F

F#

Gb

G

G#

Ab

20848

(29)

288(108)

280(100)

24719(67)

240(60)

216(36)

180(0)

9524(276)

15216(332)

14526(325)

120(300)

90(270)

72(252)

6636(247)

180(0)

280(100)

315(135)

24719(67)

216(36)

20931(30)

14609(326)

15155(331)

120(300)

9506(275)

90(270)

6636(247)

45(225)

330

300

270

240

210

180

150

120

90

60

30

0-360

12-16-2005


18/63

Legend for Harmonic Diagrams

Circles indicate the perfect intervals. Expressed as numbers, these are the simplest ratios of 1/1, 1/2, 2/1, 2/3, 3/2, 3/4, and 4/3. For these diagrams, 1/1 is D at 288 hz, the D justabove middle C at 256 hz. D is color coded as a cyan circle. The ratio 1/2 is the D!at 144 hz, the D below middle C. The ratio 2/1 is the D!at 576 hz, the D above the C above

middle C. Both these Ds are also color coded with cyan circles.

Ratios 3/2 and 3/4 indicate the perfect fifth above (the 3/2) and the perfect fourth below (the 3/4). In relation to D at 288 hz, the perfect fifth above is A at 432 hz, and theperfect fourth below is A!at 216 hz. Ratios 4/3 and 2/3 indicate the perfect fifth below (the 2/3) and the perfect 4th above (the 4/3). In relation to D at 288 hz, the perfect fifth

below is G at 192 hz, and the perfect fourth above is G at 384 hz. A is color coded as a red circle and G is color coded as a magenta circle.

Vertical ovals indicate the major and minor intervals with ratios that can be divided by 2 and 3, often called the Pythagoreanor cyclic intervals. Diamonds indicatethe major and minor intervals with ratios that can be divided by 5, and 2 or 3, often called the just or harmonic intervals. Horizontal ovals indicate mediated

intervals whose pitches are between those with 2-3 or 2-3-5 divisors.

Ab

Eb

Bb G#

A#

Bb

B

C

Db

D#

Eb

E

F

G#

Gb

Ab

C#

F#

(in relation to D)the perfect intervals

are coded with circles

(in relation to D)

the augmented fourth anddiminished fifth

are coded with a square andits diagonal indicating

the geometric mean whichis the square root.

G# Ab

(in relation to D)the major and minor intervals

with 2-3 divisors are coded

with ovals

(in relation to the center spine)the major and minor intervals

with 2-3-5 divisors

are coded with diamonds

G

D

D

A

F

C#

C

E

B

F#

D#

A#

Gb

Db

Gb

Db

F

C

Ab

Eb

Bb

F#

G#

A#

C#

D#

E

B

the mediated intervals are based on the17/16 ratio from D to D#/Eb. 17/16 = 1.062.

1062 = The True Logos

o logo iqu

11-10 2005


19/63


20/63


21/63

Eb

307.2

E

322.375

E

320

Eb

306

E

324

F

345.6

F#

362.67F

344.25

F#

360

F

341.33

Gb

364.08

G#

408

G#

405

G

384

Gb

362.67

Ab

409.6

A#

459

A#

455.62

A

432

Ab

408

Bb

460.8

B

483.56

B

480

B

486

Bb

459

Db

273.06

D#

306

Db

272

D#

303.75

D

28817/18 17/16

17/16

+5

12089/9600

-.9

17/12

+3

51/32

+6.9

9/8

+4

12089/7200

2 8

17/18

+1

17/16

+5

153/128

+8.8

12089/9600-.9

17/12

+3

51/32

+6 9

Ionian Ionian Dorian Phrygian Phrygian

The Arithmetic and Harmonic Mean of the Perfect Fifth and the Geometric Mean of the Enharmonic Octave with Ratios and Cents + or - for Tuning

1/10

2579/2304

+4.8

32/27

+-6

4/3

-2

3/2

+2

27/16

135/128-8

10/9

-18

5/4

-14

45/32-10

405/256

-6

5/38/5

+14

64/45

+10

512/405+6

6/5

+16

16/15

+12

128/135+8

10-11-2005


22/63

216 288 384

3/4

256 324

logaioChosenPath

stiboh pnohThe Breath

216

The Breath

288

Path384

Chosen

A perfect fourth below D is A216 and a perfect 4th above D is G384. A216 is the Arithmetic Mean (AM) of D144 and D288. G384 is the Harmonic Mean (HM) of

D288 and D576. A, D, and G are the three notes of the seven (A B C D E F G) that are bounded on each side by whole steps. As the perfect intervals, they areassigned the perfect shape of the circle.

true Pure; Sacred; Holy

We begin with the musical letters of the alphabet A B C D E F G. D is in the center - it is assigned the ratio of 1/1 and cycles per second of 288 hz. Its Greek

gematria is Path (stibo*). All numbers below are actual hertz numbers.

A perfect fifth above A216 is E324 (ratio 3/2 and AM of A216 and A432). A perfect fifth below G384 is C256 (ratio 2/3 and HM of G192 and G384). The combined

total = 1468. 1468 = 1150 Beloved (Eufile) + 318 Sun (Hlio)

D

DA G

The combined total A216 + D288 + G384 = 888. For those of the Orphic tradition 888 = Olen (Wlhn), the Greek bard and Apollos first speaker of oracles. For those in the

Christian tradition, 888 = Jesus (Ihsou)

How A B C D E F G are Generated from Center D through the Arithmetic and Harmonic Means of the Octave

4/3

12-16-2005


23/63


24/63


25/63


26/63


27/63


28/63

We have dealt with the Arithmetic and Harmonic Means generated from the perfect intervals: the octave (1/2, 2/1), the perfect fifth (2/3 3/2), and theperfect fourth (3/4 4/3). Thus far we used only the numbers 1, 2, 3, and 4. We will now add the number 5. The ratio 5/4 is the major third, which isthe 5th partial in the overtone series. The number 5 in relation to 4 also gives us the AM and HM of the perfect fifth. A perfect fifth below D288 isG192 and a perfect fifth above D288 is A432. B240 is the AM of G192 and D288. F#360 is the AM of D288 and A432. Bb230.4 is the HM ofG192 and D288. F345.6 is the HM of D288 and A432.

192288 432

2/3 3/2

katabolh

FoundationPath

stiboMariam

Mary

D AG1

B F#

5/4

AM of

G - Dis B

AM of

D - Ais F#

5/4240 360

2/3 3/2

The AM of G192 and D288 is B240

The musical gematria of the triad G192 + B240 + D288 = 720

nou720 = Mind; understanding

sproro720 = Seed; birth

Qeion Pneuma720 = Spirit of God

The musical gematria of the triad D288 + F#360 + A432 = 1080

to Pneuma Agion1080 = The Holy Spirit

1080 = Ideal number of breaths taken in an hour

1080 = Ideal radius of the moon in miles

The AM of D288 and A432 is F#360

G1 B1 D D F# A

The Arithmetic and Harmonic Mean of the Perfect Fifth Above and Below D288

The Arithmetic Mean of the Perfect Fifth below and above D288 with musical gematria

The Harmonic Mean of the Perfect 5th below and above D288 with musical gematria

720 = The Holy Spirit o Agio Anemo 1080 = Threefold trisso

11-12-2005


29/63

The Triads of Pythagoras (Bb-D-F) and Jesus (B-D-F#)

192 288 432

2/3 3/2

katabolh

FoundationPath

stiboMariam

Mary

D AG1

B F#

5/4

AM of

G - D

is B

AM of

D - A

is F#

5/4240 360

2/3 3/2

B D F#

240 + 288 + 360 = 888

888 = Jesus Ihsou

888 = Olen Wlhn

Maintaining D288 as the center and adding the arithmetic mean below (B240) and the arithmetic mean above (F#360) results in the minor triad B D F#.

B240 + D288 + F#360 = 888B D F#

Maintaining D288 as the center and adding the harmonic mean below (Bb230.4) and the harmonic mean above (F345.6) results in the major triad Bb D F.Bb230.4 + D288 + F345.6 = 864

Bb D F

(the Greek bard and Apollos first speaker of oracles)

11-12-2005


30/63


31/63


32/63

We have dealt with the Arithmetic and Harmonic Means using the numbers 1, 2, 3, 4, and 5 as divisors. 6 is the octave of 3 (6/3 = 2/1). 7presents a whole other dimension and is not used as a divisor in this system. 8 is the octave of 4, so 9 is the first new number to use beyond 5. 9

gives us the 9th partial above the fundamental which is also the interval of a ninth. This partial with its octave reduction is both the ninth and themajor second.

The Arithmetic Mean of D288 and Bb230.4 is C259.6. C259.6 is the small major second belowD. The Harmonic Mean of D288 and Bb230.4 isC256. C256 is the major second belowD. The Arithmetic Mean of D288 and F#360 is E324. E is the 9/8 major second aboveD. The

Harmonic Mean of D288 and F#360 is E320. E320 is the small major second aboveD. I am awed by the clarity of the gematria.

230.4 288360

h nikh alhqeiathe victory of truthPath

stibooikionHouse, temple

D F#

9/8

10/9

AM of

Bb - D

is C259.6

AM of

D - F#

is E324

9/8

10/9

259.6 324

8/9

9/10

8/9

9/10

Bb

259.6 + 288 + 324 = 871.6

C E

EC

256 320

HM of

D - F#

is E320

HM of

Bb - D

is C256

256 + 288 + 320 = 864

C D E

C D E

259.6 + 288 + 324 = 871.6256 + 288 + 320 = 864

C D E

C D E

Arithmetic and Harmonic Means of the Major Third

11-12-2005


33/63

G

A

B

F

153.6 192

240

katabolh

FoundationPath

stibo

9/8

10/9

AM of

Eb - G

is F172.8

AM of

G - B

is A216

9/8

172.8 216

8/9

9/10

172.8 + 192 + 216 = 580.8

170.66

HM of

Eb - G

is C256

Eb

F

F G A

580 ollumiTo destroy

581 Deadly; of death qanasmo

581 Reality upar

581 Wheel ampu

F G A

172.8 + 192 + 216 = 580.8

243

B

The Arithmetic and Harmonic Means of the 2-3-5 major third from the two other perfect intervals of G and A

Arithmetic and Harmonic Means of the Major Third

9/10

11-12-2005

There is only one A because it is one of the 3 perfect intervals A, D, and G.


34/63


35/63


36/63


37/63


38/63


39/63

cTh

eory.com

Geome

tryDiagrams

11-15-2005

apolloN APOLLON- The Pythagorean Definition of Godbased on an article by the lateA l f h b k


40/63

a

p

l

oo

l

xx

x

x

n

3

4 5B

E

G The Lambda RectanglesEach is in the proportion of 3units (width) to

4units (height). Musically the ratio of 3 : 4isthe Perfect fourth.The cithara (from which the guitar

is descended) was the instrument of Apollo.

It was tuned from the top string down, and theinterval from the top string to the

2nd string was a Perfect fourth, as E to B.

l

The Pythagorean 3 - 4 - 5 Triangles

The diagonal of the 3 x 4 rectangle is 5, (fromthe Pythagorean theorem , a^2 + b^2 = c^2.3^2 = 9. 4^2 = 16. 5^2 = 25. 9 + 16 = 25.

Musically the ratio of4 : 5is a Major3rd B : G,

2nd to 3rd string on the cithara.Musically the ratio of 3 : 5 is a Major 6th, E : G

1st (top) to 3rd string on the cithara.

4

3

apolloNAnne Macaulay, from the bookHomage to Pythagoras. The samebook is also published with thetitle Rediscovering Sacred Science

a

p

o

l

A - Alpha-1st letter of Greek alphabet.The Alpha circle, (the big encompassing circle)

diam = 10 units (10 x 6 = 60 ticks)circum = 10 x 22/7 = 31.43 units = 188.58 tickscircum =188.58 rounded = 189 ticks = 3^3 x 7

O - Omicron-16th letter of Greek alphabet.circum = 16 units (16 x 6 = 96 ticks)

diam = 16 x 7/22 = 5.09 unitsRounded diam = 5 units =30 ticks

(5 units is also the diagonal of the 3 by 4lambda rectangles.)

P - Pi-17th letter of Greek alphabet.diam = 17 x 7/22 = 5.41 units

diam = 5.41 x 6 = 32.46ticks rounded = 32 ticks

circum = 17 units = 102 ticks

L - Lambdathe 12th letterof the Greek alphabet.

Symbolized by a 3 x 4 rectangle.(3 x 4 = 12)

Width = 3 x 6 = 18 ticks.Height = 4 x 6 = 24 ticks.

(Diagonal, shown at bottomleft = 5 x 6 = 30 ticks

= 5 units.)

n

N - Nu-14th letter of Greek alphabet.circum = 14 units = (14 x 6 = 84 ticks)

diam = 14 x 7/22 = 4.45 units = 26.70 ticksRounded diam = 4.5 units = 27 ticks

Dimensions and Musical Ratios from those Dimensions in Apollo Geometry


41/63

A B

E

FCD

K P

L

M NQ

R S

UT

O1 O2

O3

O4

x x

x

x

CD/DA=18/24=3/4=P4th

AD/AC=24/30=4/5=M3rd

CD/CA=18/30=3/5=M6th

AD/DF=24/36=2/3=P5th

RS/DC=12/18/2/3=P5th

DC/AE=18/36=1/2=P8ve

RS/BC=12/24=1/2=P8ve

radiusO4/AB=16/18=8/9=M2nd

CB/CP=24/27=8/9=M2nd

diameterO4/CT=32/45=A4th

diameterO1/DF=30/36=5/6=m3rd

CP/diameterO4=27/32=m3rd

radius O4/CP=16/27=M6th

BA/PK=18/20=9/10=M2nd

CD/diameterO4=18/32=9/16=m7th

QP/(DC/2)=5/9=m7th

QP/(radiusO4/2)5/8=m6th

radius04/diameterO1=16/30=8/15=M7th

radiusO4/radiusO1=15/16=m2nd

radiusO1/radiusO2=15/15=1/1=PU

not included

128/243

243/256

45/64

81/128

64/81

BA=BE=CD=CF=3 units (6x3=18 ticks)

CB=FE=DA=4 units (6x4=24 ticks)

CA=CE=5 units (6x5=30 ticks)

CT=7.5 units (6x7.5=45 ticks)

CP=4.5 units (6x4.5=27 ticks)

CQ=3.67 units (6x3.67=22 ticks)

PK=PL=3.33 units (6x3.33=20 ticks)

QP=.83 units (6x.83= 5 ticks)

RS=2 units (6x2=12 ticks)

TU=9 units (6x9=54 ticks)

diameter O1 and O2=5 units (6x5=30 ticks

radius O1 andO2=2.5 units (6x2.5=15 ticks)

diameter O3=10 units (6x10=60 ticks)

radius O3=5 units (6x5=30 ticks)

diameter O4=5.33 units (6x5.33=32 ticks)

radius O4=2.67 units (6x2.67=16 ticks)

p

6-5-2005 Dimensions And Musical Ratios In ApollonGeometry - Octave 5 - 10 units, 30 - 60 ticks


42/63

6

810

B

G

E

Eb

Bb

G

6

108

5/5

(G)

5

30

16/15

(Ab)

5.33

32

6/5

(Bb)

6

36

4/3

(C)

6.66

40

8/5

(Eb)

8

48

9/5

(F)

9

54

10/5

(G)

10

60

n nu circle

a alpha circles

Eb48(5 x8/5)

C 40 (5 x 4/3)

G30(5

x10/10)

G30(2.5x10

/5)

Ni F# 32 (10 x 8/15)

Db 32 (5 x16/15)

D 40 (10 x 2/3)

B48

(10 x4/5)

A 54 (10 x 9/10)

E 36 (10 x 3/5)

F 54 (5 x 9/5)

Bb 36 (5 x 6/5)

G30

(2.5x10

/5)

G30(5

x10/10)

A

G

E

FCD

K P L

O1 O2

O3

O4

x x

x

x

O5

H

diam O1 and O2 = 5 units = 30 ticks

diam O4 = 5.33 units = 32 ticks

DF = AE = GH = 6units = 36 ticks

KL = 6.66 units = 40 ticks

p

o

(I have called this the feminineoctave because it contains thekomal or soft intervals. Like the

Moon, which is reflected light,these might be called reflectivesounds. They are the reflected

intervals of the masculine overtonalintervals which are heard directly.)

DG = FH = 8units = 48 ticks

diam 05 = 9 units = 54 ticks

diagonals DH = FG = 10units = 60 ticks

diameter O3 = 10units = 60 ticks

omicron circles

pi circle

The Masculine Octave

The right side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 5 - 10 units or 30 - 60 ticks

string length ratios(denominator larger than numerator)

ex. 10 x 8/15 = 5.33 units5.33 x 6 = 32

(I have called this the masculineoctave because it contains theovertonal or direct intervals. Like theSun, whose light is direct, these arethe overtonal intervals which can be

heard directly above the fundamental tone.)

The Feminine Octave

The left side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 5 - 10 units or 30 - 60 ticks

vibrational ratios(numerator larger than denominator)

ex. 5 x 16/15 = 5.33 units5.33 x 6 = 32

5/5

(G)

5

30

8/15

(F#)

5.33

32

3/5

(E)

6

36

2/3

(D)

6.66

40

4/5

(B)

8

48

9/10

(A)

9

54

10/10

(G)

10

60

Dimensions And Musical Ratios In ApollonGeometry - Octave 2.5 - 5 units, 15 - 30 ticks


43/63

3

45

B

G

Eb

Bb

G

3

54

5/5

(G)

2.5

15

16/15

(Ab)

2.67

16

6/5

(Bb)

3

18

4/3

(C)

3.33

20

8/5

(Eb)

4

24

9/5

(F)

4.5

27

10/5

(G)

5

30

Eb24

(2.5 x8/5)

C 20 (2.5 x 4/3)

G30(5

x10/10)

G30(2.5x10

/5)

F# 16 (5 x 8/15)Db 16 (2.5 x16/15)

D 20 (5 x 2/3)

G15

(2.5

x5/5)

B24

(5 x4/5)

A 27 (5 x 9/10)

E 18 (5 x 3/5)

F 27 (2.5 x 9/5)

Bb 18 (2.5 x 6/5)

G15(5x5/5)

G30

(2.5x10

/5)

G30(5

x10/10)

A27

(5 x9/10)

F27

(2.5 x9/5)

AB

E

FCD

K P L

O1 O2

O3

O4

x x

x

x

O5

radius O1 and O2 = 2.5 units = 15 ticks

radius O4 = 2.67 units = 16 ticks

AB = BE = DC = CF = 3units = 18 ticks

KP = PL= 3.33 units = 20 ticks

DA = CB = FE =4units = 24 ticks

p

a

p

l

o

n

o

ll

(I have called this the feminineoctave because it contains thekomal or soft intervals. Like the

Moon, which is reflected light,these might be called reflectivesounds. They are the reflected

intervals of the masculine overtonalintervals which are heard directly.)

CP = 4.5 units = 27 ticks

radius 05 = 4.5 units = 27 ticks

diagonals CA = CE = 5units = 30 ticks

diameter O1 and O2 = 5units = 30 ticks

radius 03 = 5units = 30 ticks

omicron circles

pi circle

omicron circles

alpha circle

lambda rectangles

lambda rectangles

nu circle

lambda rectangles

The Masculine Octave

The right side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 2.5 - 5 units or 15 - 30 ticks

string length ratios(denominator larger than numerator)

ex. 5 x 8/15 = 2.67 units2.67 x 6 = 16

(I have called this the masculineoctave because it contains theovertonal or direct intervals. Like theSun, whose light is direct, these arethe overtonal intervals which can be

heard directly above the fundamental tone.)

The Feminine Octave

The left side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 2.5 - 5 units or 15 - 30 ticks

vibrational ratios(numerator larger than denominator)

ex. 2.5 x 16/15 = 2.67 units2.67 x 6 = 16

5/5

(G)

2.5

15

8/15

(F#)

2.67

16

3/5

(E)

3

18

2/3

(D)

3.33

20

4/5

(B)

4

24

9/10

(A)

4.5

27

10/10

(G)

5

30

The Octave11 15 2005


44/63

2000 = The Perfected Work + The Gnosis2/1 = 2.000

729 Euergesia +1271 h Gnwsi

C

B

RS/BC=12/24=1/2 P8ve

R S12

24

x

xx

BC/Rs=24/12=2/1 P8ve

D C

A E

DC/AE=18/36=1/2 P8ve

18

36

x

xx

AE/DC=36/18=2/1 P8ve

1/2 - .500

500 = the number

o ariqmo

1/2 - .500

500 = Thou art + IEOU (Gnostic name of God)

15 Ei +485 IEOU

11-15-2005

500 = Goddess + The all-holy

1/2 - .500

15 Qea+485 o panagio

x x

x

30 30

60

O1 O2

O3

diamO1 (or O2)/diamO3=30/60=1/2=P8vediamO3/diamO1 (or O2)=60/30=2/1=P8ve

The Perfect Fifth and Perfect Fourth

11-15-2005


45/63

D F

A E

AD/DF=24/36=2/3 P5th

24

36

x

xx

DF/DA=36/24=3/2 P5th

D C

R S

RS/DC=12/18=2/3 P5th

12

18

x

xx

DC/RS=18/12=3/2 P5th

3/2 = 1.500

1500 = Light, life

fw

3/2 = 1.500

1500 = To know the God Apollo

85 oida +o Qeo Apollwn

3/2 = 1.500

1500 = Beloved lyre1150 Eufile +350 kiqari

4/3 = 1.333

1333 = Beloved Hestia (var)

810 fili +523 Estih

4/3 = 1.333

1333 = Apollo + Harmony

1061 Apollwn +272 Armonia

3/4 = .750

750 = Helios(Sun) + Foundation (radius of Sun)

318 Hlio +432 katabolh

D

A

AD/diam O4=24/32=3/4 P4th

24

32x

xx

damO4/AD=32/24=4/3 P4th

O4

2/3 = .6666

666 = the heart, soul, mind

h frhn

D C

A

DC/DA=18/24=3/4 P4th

24

18

x

xx

AD/DC=24/18=4/3 P4th

2/3 = .6666

666 = Whole; perfect + wisdom

108 olh +558 mhti

The Major Third and Major Sixth11-15-2005


46/63

D C

A

DA/AC=24/30=4/5 M3rd

24

30

x

xx

CA/AD=30/24=5/4 M3rd

4/5 = .800

800 = Faith

pisti

4/5 = .800

800 = Of the Universe

Kosmon

800 = Lord; God

4/5 = .800

Kurio

5/4 = 1.250

1250 = Divine (divine = 25. 25^2 x 2 = 1250)qeia

D C

A

DC/CA=18/30=3/5 M6th

30

18

x

xx

AC/CD=30/18=5/3 M6th

3/5 = .600

600 = Thou

su

h qeoth

600 = The Godhead

3/5 = .600

5/3 = 1.666

1666 =Beloved Demeter

1198 prosfilh +468 Dhmhthr

1666 =Beloved Hestia

5/3 = 1.666

1150 eufule +516 Estia

16/27 = .5925

592 = Divinity

agaqoth

27/16 = 1.6875

1687 = Consciousness + Kosmos

1087 Suneidhsi +600 Kosmo

1687 = Consciousness + the Godhead

27/16 = 1.6875

1087 Suneidhsi +600 h Qeoth

1687 = Consciousness + Thou

27/16 = 1.6875

1087 Suneidhsi +600 Su

C

P

radius 04/PC=16/27= 16/27 M6th

27

16x

xx

CP/radius 04=27/16=27/16 M6th

O4

600 = Kosmos

kosmo

3/5 = .600

11-15-2005


47/63

The Minor Third and Minor Sixth 5/8 = .625625 = to learn; to know friendship; amity

74 daenai +551 filia

5/8 = .625

625 = divine + cosmos

25 qeia +600 kosmo

5/8 = .625

625 = divine + Thou

25 qeia +600 Su

8/5 = 1.600

1601 = Lights; wisdom

fwta

P

QP/(radius O4/2)=5/8= 5/8 m6th

5

8x

xx

(radius 04/2)/PQ=8/5=8/6 m6th

Q

O4

D F

A

AC/DF=30/36=5/6 m6th

30

36

x

xx

DF/AC=36/30=6/5 m6th

P

PC/diamO4=27/32= 27/32m6th

27

32x

xx

diamO4/PC=32/27=32/27 m6th

C

O4

5/6 = .833

832 = beloved by God

qeofilo

6/5 = 1.200

1200 = Light, lamp

lucnon

27/32 = .8437

843 = joy + kithara (lyre)

702 cara +141kiqara

32/27 = 1.185

1185 God + Mind; understanding

284 Qeo +901 gnwmh

32/27 = 1.185

1185 flute + path

701 Aulo +484 dromo

C

The 8/9 9/8 Major Second and the 9/16 16/9 Minor Seventh


48/63

P

AD/PC=24/27=8/9M2nd

2724

x

xx

PC/AD=27/24=9/8M2nd

C

A

D

8/9 = .888

888 = Olen (Apollos first prophet)

Wlhn

888 = Jesus

Ihsou

8/9 = .888

9/8 = 1.125

1125 = to lead to a place

katagw

9/8 = 1.125

1125 = Apollo + Truth

1061 Apollwn +64 alhqeia

9/16 = .5625

562 = verily

dhpou

16/9 = 1.777

1777 = the treasure + soul; understanding

1058 o qhsauro +719 qumo

1777 = the treasure + purity

1058 o qhsauro +719 kaqarioth

16/9 = 1.777

radius O4/CF=16/18=8/9M2nd

18

16x

xx

CF/radiusO4=18/16=9/8M2nd

C F

O4

P

ADx2/PC=48/27=16/9m7th

2748

x

xx

PC/ADx2=27/48=9/16m7th

C

A

D

diamO4/CF=32/18=16/9m7th

18

32x

xx

CF/diamO4=18/32=9/16m7th

C F

O4

The 32/45 45/32 Tritone, the 9/10 10/9 Major Second and 5/9 9/5 Minor Seventh


49/63

9/10 = .900

901 = mind, understanding

gnwmh

10/9 = 1.111

1110 = the microscosm

o Mikro Kosmo

1110 = the macrocosm

o Kosmo o olo

10/9 = 1.111

5/9 = .5555

555 = foundation; image

idruma

9/5 = 1.800

1800 = breath, spirit, inspiration +the creation of God

576 pneuma +1224 ktisi Qeou

9/5 = 1.800

720 = mind; understanding +the Holy Spirit

720 nou +1080 to pneuma agion

P

(DC/2)/PQ=9/5=9/5m7th

5

9

x

xx

PQ/(DC/2)=5/9=5/9m7th

Q

CD

P

AB/KP=18/20=9/10M2nd

20

18

x

xx

KP/AB=20/18=10/9M2nd

B

K

A

32/45 = .7111

711 = Muse

Mousa

45/32 = 1.406

1407 = Vital, full of life

zwtiko

diamO4/CT=32/45=32/45tritone

32

45

x

xx

CT/diamO4=45/32=45/32tritone

T

C

O4

The 15/16 16/15 Minor Second, and the 8/15 15/8 Major Seventh


50/63

15/16 = .9375

938 = compassion

sumpaqh

16/15 = 1.066

1065 = understanding, knowledge

sunesi

8/15 = .5333

533 = Sun + Jupiter

318 Hlio +215 Dia

15/8 = 1.875

1875 = Apollo + Pythagoras

1011 Apollw +864 Puqagora

15/8 = 1.875

1875 = Apollo + the divine wisdom

1061 Apollwn +814 h qeia sofia

radiusO1/radiusO4 =15/16=15/16m2nd

16

15

x

xx

radiusO4/radiusO1=16/15=16/15m2nd

O4

O1

AC/diamO4 =30/32=15/16m2nd

32

30

x

xx

diamO4/AC=32/30=16/15m2nd

A

C

O4

radiusO4/AC =16/30=8/15M7th

16

30

x

xx

AC/radiusO4=16/30=15/8m2nd

A

C

O4

diamO4/diamO3 =32/60=8/15M7th

32

60xx

diamO3/diamo4=60/32=15/8m2nd

x

x

O4

O3

8/15 = .5333

533 = the love + foundation

101 h agaph +432 katabolh

11-15-2005


51/63

There is an excellent explanation of the mathematical geometry of the vesica piscis and the fishes in the net in bothJohn Michells Dimensions of Paradiseand David FidelersJesus Christ, Sun of God. The same story is is related about

both Pythagoras and Jesus knowing the exact number of fishes in the net. In the bible, the exact number of fishes isgiven as 153. In the Pythagorean version, the actual number is not givenonly that Pythagoras knew exactly how

many fish were in the net, and when he was found to be correct, asked that the fishes be freed back into the water.

153

153

153

153

153

153

153

153

153

153

153

153

153

153

153153

153 Hz306 Hz

216 Hz

EbEb-1

108 Hz

GM

GM

A-2 A-1


52/63

180(0)

20931

(30)

216(36)

24719(67)

280(100)

315(135)

32240(143)

0

(180)

1941(200)

24(204)

45

(225)

6636

If the enharmonic notes used in equal temperament (like the black keys on a keyboard)are separated out and treated as different notes, the total number of notes is 17 rather than the

12 notes of 100 cents each in equal temperament. This is just the right number of notes needed tofill in the diamond net of the vesica piscis. There are 16 small diamonds and the one big diamond

which is the outer frame. The ratio of 17/16 in its decimal form is 1.062.

What makes this remarkable is that 1062, in Greek gematria, is o logo iqu,the true ratio

A

A#

Bb

B

C

C#

Db

D

D#

Eb

E

A

A#

Bb

B

C

C#

Db

D

D#

F#

F

E

E

Gb

G

G#

Ab

11-15-2005


53/63

!

!

!

!

!

!

Light, though really threefold, has traditionally also a sevenfold nature. In the scheme of geometric representation, thefirst emanation from, or manifestation of, the Absolute, in the series we are now considering, is figured as a Cube developed from anoriginal point, which is one of its angles, and this is portrayed in its visible aspect of symmetry as a Hexagon with six internal linesradiating from the center.

The . . . New Creation is symbolized by the Cube, for a true knowledge of which a sense beyond that of mere physical vision isrequired. By this the presence of the seventh ray is revealed and the Eight points are manifest in their true relation. Thus from Oneare seen to proceed Seven and the Seven are connected with the parent One by rays or lines of Three several lengths (or which arenow seen to be of three several lengths), wheras in the flat representation only six were seen, and they were apparently of equallength.

Theoretically there is a seventh radius but never more than six are visible because the seventh approaches the eye and unites thecentral visible point with the eighth point lying exactly behind it.

The six outward points, with their connecting lines represent the Kosmos, or manifestation of the Divine Power, and the seventh,the central point, is the Divine Source, the revelation of God to man.

But the eighth or invisible original is required to complete the figure of the cube, which is only apprehended by the higher reason.

From Gematria, A Preliminary Investigation of The Cabala

by Bligh Bond and Simcox Lea

11-15-2005


54/63

!

!

!

!

!

!

!

There are, as will readily be seen, three rays of this first order emanating from any one point in the cube, and connecting that pointwith the three lying nearest to it in the cube. Secondly there are to be seen three more connecting lines which are the diagonals on thesquare sides of the cube, linking the first point with the three next in order of remoteness, and these lines, on the scale now

determined, measure each 100 x !2, or 141.42their total, 424.26, being integrally expressed as 424 or 425 by substituting unityfor the fraction. Finally, there is the One ray leading diametrically across the cube to the opposite point, and the measure of this is100 x !3, or 173.205. Call this 173. thus we have the total of the Seven rays

!

!

!

!

!

!

!

!

!

!

!

!

!

!

side of square = 100100 x 3 = 300

diagonal of square side of cube100 x !2 = 141.42141.42 x 3 = 424.26

(424 or 425)

diagonal across cube100 x !3 = 173.205

Thus we have for the total of the Seven rays the sum of 300 + 424.5 + 173, which is 897 or 898 and may with equal propriety be takenas either, as the actual figure is halfway between. Hence we find in Gematria that whereas 897 gives AIZ EN THASIA EKKLHSIAI,

898 is the number by Gematria ofOI Z ASTERES, the seven stars or rays which are the angels of the seven Churches.


55/63

The whole series of light-rays comprises those whose wave-length varies from a little more than 7600 tenth-metres in the extreme Redto a little less than 3900 in the extreme Violetaltogether about an octave of light. Now if we take the Green ray at about line E or5270 in measure then our series must be as follow:

Red 7453 . . . . or 5270 x !2

Green 5270

Violet . . . . or 7453 x (1/!3)

and this is remarkable, because 7453 brings us to the intense Red close to Frauenhofers A line (7604) and between it and B, whilst 4302

for the Violet brings us to the point of the most intense deep Blue, and practically on to Frauenhofers line G which is 4307.

We must be content to leave our instructed readers to judge as to the real nature of so striking an approximation. If, on furtherinvestigation, the parallel appears based on undeniable physical fact, then a few more such instances may go far to rehabilitate the lostwisdom of Antiquity. And if it be true that the data of the colour-scalewhich are derived from molecular motionsare indeed foundedupon harmonics whose source is not, as in the case of sound and other physical measure, based upon the interaction of forces havingwhole-number proportionals, but upon those mysterious entities, the Roots of Two and Three, then we are face to face with a conditionpointing to a genesis of motion in a region of space unknown to us, and wherein the dynamic laws operate in a relation quite unfamiliar.

It looks as if a fourth and interiordimension must be assumed, and is this not precisely what some physicists say of the Atom, that itmust possess some substance in a fourth dimension?

From GEMATRIA, A Preliminary Investigation of the Cabala, by Frederick Bligh Bond and Thomas Simcox Leapublished by Research into Lost Knowledge Organization Trust

Blue 43027453 x (1/!3)Frauenhofer

Red 74535270 x ! 2Frauenhofer

Green 5270Frauenhoferline E (5270)


56/63

cTh

eory.com

AstronomyDiag

rams

Beginnings of Music of the Spheres

11-15-2005


57/63

MOON1080miles

3960miles

5040miles

perimeter of square 31680miles

circumfe

renceof circle = 31680miles

The Earths mean diameter is 7920 miles. (2 x 3960).The Moons mean diameter is 2160 (2 x 1080).

The combined radii of the two bodies are equal to3960 + 1080 = 5040 miles.

A circle with radius 5040 has a circumference(usingpi=22/7) of 31680miles.

(Circumference = pi x 2r)22/7 (2 x 5040) = 31680A Square with a side of 7920 miles (2 x 3960)

has a perimeter (4 x s) of31680miles4 x 7920 = 31680

256/24316/15

10/9

9/8 15/8

1/1-2/1

1/1 =

256/243 =

16/15 =

10/9

000000

190448

240000

400000

C

DbDb

D

(radius of earth)(radius of earth and moon combined)

From the point of view of numbers, the divisionof the modal octave must necessarily be oftwenty-two intervals. This division is not pecularto music, but represents a universal law. Indeed,the relation of those twenty-two intervals withthe seven notes of the scale is the only one thatcan give to music a complete and logical substructure.p. 75 Music and the Power of Sound by Alain Danielou

Beginnings of Music of the Spheres -The Number 3168

00

19

24

40

243/128

1 x 2 x 3 x 4 x 5 x 6 x 7 = 50407 x 8 x 9 x 10 = 5040

The mean circumference of theStonehenge sarsen circle is 316.8 ft.or a hundredth part of 6 miles.

The general character of the number 3168,

as conveyed by its position in ancient cosmo-logical diagrams and the phrases associatedwith it through gematria, is that it representsthe spirit which passes through and encirclesthe universe, Platos World-soul. The Christianterm for this spirit, developed from the number3168, was Lord Jesus Christ. p. 173The Dimensions of Paradise by John Michell

L o r d J e s u s C h r i s t

K u r i o I h s o u C r i s t o 2

0

4

0

0

1

0

0

1

0

7

0

2

0

0

2

0

0

2

0

0

7

0

7

0

2

0

0

2

0

0

4

0

0

1

0

1

0

8 3

0

0

1

0

0

6

0

0800 888

800 + 888 + 1480 = 3168

B

B

C

C+

C--

C-

C++

DD

Db

DbDb

Db+

ASC CCW C toG

Ci l ti f th C th P th P i d th V di S l


58/63

To quote from Danielous chapter on the Music of the Greeks: The Theory of the Scale. The Pythagorean theory can be very briefly expressed as

follows; the demiurge had divided the composition [of the soul of the world] into seven parts, which relate to one another like the terms of

two geometrical progressions, one of ratio 2,(1, 2, 4, 8) and the second ratio 3,(1, 3, 9, 27). With the help of these two progressions, the

demiurge has formed a single progression [of] 1, 2, 3, 4, 9, 8, 27.

This Pythagorean progression, with its characteristic inversion of the fifth and sixth numbers, is to be found in every traditional system

and is, for example, the basis of the scale on which the Samaveda is sung, a scale that in much more ancient times showed the same inversion.

This ancient vedic scale in modern notation would be: Pa(G) ma(F) Ga(E) Re(D) Ni(B) Sa(C) Dha(A). The numbers representing these

notes would correspond to those of the Pythagorean progression (Alain Danielou, Music and the Power of Sound, p. 97).

This diagram is adapted from the book Mathematics of the Cosmic Mind by L. Gordon Plummer; The Greater and Lesser Perfect Systems,

a website on Greek Esoteric Music Theory by John Opsopaus; and Alain Danielous book, Music and the Power of Sound, the Influence of Tuning

and Interval on Consciousness.

On the diagram, If you start at the lower right and read thru the orbits from lower right to upper left (using earths orbit for the Sun); the

orbits from Saturn inward are: Saturn orbit(j), Jupiter orbit (h), Mars orbit(g), Sun-earth orbit(a), Mercury orbit(d), Venus orbit(f),

Moon orbit(s). Using the same planetary order but inserting the note assignment of the planets from Greek Esoteric Music Theory, the

notes are Saturn (G), Jupiter (F), Mars (E), Sun (D), Mercury (B), Venus (C), Moon (A). This sequence of notes, G, F, E, D, B, C, A is a

musical manifestation of a deep mathematical and cosmological truth.

Circulation of the Cosmos, the Pythagorean Progression, and the Vedic Scale

Saturn G PaJupiter F maMars E GaSun D ReMercury B NiVenus C SaMoon A Dha

12 3

4 9

8 27

1 - 2 - 3 - 4 - 9 - 8 - 27G - F - E - D - B - C - A

j

hg

s

a

x

Sun

Mercu

ryOr

bit

VenusO

rbit

Moon

Orbit


59/63

cTh

eory.com

GematriaDiagr

ams


60/63

A a1

B b2

G g3

D d4

E e5

6

Z z7

H h8

Q q9

I i10

K k20

L l30

M m40

N n50

O o70

P p80

!

90X x60

R r100

S s200

T t300

U u400

F f500

C c600

Y y700

W w800

900

Greek Alphabet for Gematria

11 15 2005


61/63

102

204

408

306

612

1224

34

136

36

72

6

3

24

12

2

4

8

1

15351

17

918

68

72 The Truth h alhqeia

306 Trust; faith pisti

408 Worship; Glory seba

408 Hz

612 Zeus Zeu

612 Hz

1224 Fishes icque

1224 The Net to diktuon

The Divisors, Gematria, and Musical Gematria of 1224

katabolh

Foundation

Apollo +

Apollon

Apollo (281) +Apollon (331)

432Hz 864 Hz612 Hz

Pythagoras

Puqagora

Eb612 is the Geometric Mean of A432 and A864 A864 is the Geometric Mean of Eb612 and Eb1224

612 Hz 1224 Hz864 Hz

306 Hz 612 Hz408 Hz

Ab408 is the Harmonic Mean of Eb306 and Eb612

Messenger (540) +the Truth (72)Pompo +

h alhqeiaPuqagora

Pythagoras

seba

Worship;Glory

Trust,Faith

pisti

153 Divine(25) Spirit (128) qhia aiqer

153 Hermes Ermh

612 Pure, sacred, holy (324) + path (288)

Agno +Stibo*

Eb-1

1224 Wisdom (781) + the Logos (443)

Sofia +o Logo

153 Hz

1224 Hz

306 Hz

108 Hz72 Hz 144 Hz72 Hz

The truthh alhqeia

The hearth kardia

Whole; perfectolh

Beautiful (52) Teacher (1172)

kala paidagwgo

A108 is the Arithmetic Mean of D72 and D144

Truly (61) +Friendship (551)

nai +filia

Ab

612 Truly (61) + Friendship (551)

nai +filia

612 Messenger (540) + the Truth (72)

Pompo + h alhqeia

1224 Beautiful (52) Teacher (1172)

kala paidagwgo

Eb

Eb1

Eb2

D-2

D-2

Ab

AbEb Eb1

Eb1

Eb1 Eb2

D-2 D-1A-2

A1A1

A

11-15-2005

11 15 2005


62/63

x

x

x

x

x

x

x

x

x

x x x x x

144 Hz

144 The Heart h kardia

O Qeo Apollwn +Pyqagora +Ihsou

x66

132

264

396

792

1584

22

88

36

72

2

4

8

1

198

528

11

18

44

32

3168

144

176

33

96

48

24

12

6

99

72 The Truth h alhqeia


3168 The God Apollo 1416 + Pythagoras 864 + Jesus 888

72 Hz

16

x

39

288

352

x

1056

x3168

288 Path stibo

288 Hz

D

D-1

D-2

G-3

G-2

C-3

D-2

D-1

D

D-3

G-4

C-4

D-4

The general character of the number 3168, as conveyed by its position in ancient cosmological diagrams and the phrases

associated with it through gematria, is that it represents the spirit which passes through and encircles the universe, Platos

World-soul. The Christian term for this spirit, developed from the number 3168, was Lord Jesus Christ.

John Michell, Dimensions of Paradise p. 173

3168 Light bringing 1184 + Light of God 1984

Selah foro +fw Qeon

1584 Apollo 1061+ Hestia 523

Apollwn +Estih

99 Knowledge; teaching of the mysteries

maqhma

22 Hearth; altar , Hestia estia*

88 Victory, the goddess Nike Nikh

352 Lamp; light, sun lampa

96 Morning; east

96 The victory h nikh

hoih

132. All, whole apan

264. Of truth alhqeia

528. The goddess Hestia Istih

1056. true (256) + faith (800)

alhqh pisti

1056. messenger (540) + Hestia (516)

pompo +Estia

792 Mind, understanding 720 + the truth 72

nou +h alhqeia

396 Pure, sacred, holy 324 + the truth 72

agno +h alhqeia

99 Hestia 29 + the truth 72

Estih* +h alhqeia

198 Hestia 27 + beauty 171

Istia* +kalon

11-15-2005

Th Di i G i d M i l G i f 504011-15-2005


63/63

x

x

x

x

xx

xx

xx

x

x

x

x

x

x

x

x

x

xx x

x

xx

x

x

x

x

x

B-3

G48 is the Harmonic Mean of D144 and D288

36 Hz 48 Hz 72 Hz

The truth

h alhqeia

Earth(9) +Hestia(27)

Immortal(26)+ Hestia (22)

h alhqeiaaiei + Estia*

immortal(26)+ Hestia (22)

aiei+ Estia*

630 Precious; Honored timio

315 The Egyptian God Hadit Adit

prostiroph

1008 Teacher paideuth

126 Heart kear

144 Hz

144 The Heart h kardia

elix

1680 Heaven (891) + the soul (789) Ourano + o qumo

720 Mind; Understanding nou

720 Seed, birth sporo

720 Spirit of God Qeion Pneuma

720 The Holy Spirit Qeion Pneuma

720 Hz

120

60

30

20

10

5

210

840

36

72

6

3

24

12

2

4

8

1

40

15

105

9

18

420

21

42

84

168

336

315

630

1260

2520

5040

16

63 126 252 504 1008

80

35

70

140

280

560

144

1680

240

45

112

4890

180

360

720

5628147

72 The Truth

B-1

F#

F#1

F# -1

F#-2

B-2

B-3

D-2

D-3

D-1

G-3

h alhqeia

504 Divine; Inspired by God Qespi

1008 Prayer

210 Strength; force i

420 The Egyptian Goddess Isis Isi


48 Hz 72 Hz60 Hz

B60 is the Arithmetic Mean of G48 and D72

mageia

Magic

1260 Superior, More excellent upertero

2520 to initiate into the mysteries + Orpheus

meuw+Orfeu

840 Microcosm mikrokosmo

5040 combined radius (in miles) of earth and moon

105 Circle; coil

72 Hz

The truth

Gea + Istia*

1680 Loving (816) + Pythagoras (864) stergh + Puqagora

F#1

D-2

D-1

D-2

D-2D-3 G-3

G-3

5040

B-2 F# -1D-1

144 Hz 180 Hz120 Hz

D144 is the Harmonic Mean of B120 and F#180

This combination of notes - B D F#in the next octave

up is the gematria for Jesus

240 + 288 + 360 = 888

The Heart

h kardia

To be within

eneimi

Beautiful (52) + spirit (128)

kala +aiqhr

888 = Ihsou

336 to worship

sebazomai

The ideal combined radius (in miles) of the Earth and the Moon

11 15 2005

Harmonic Theory Diagrams

Documents

Transcript of Harmonic Theory Diagrams