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www.harmonictheory.com© 2006 by Connie Achilles. All rights reserved.

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 E1 F1 G1 A1 B1 C1 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.)

The 2-3-5 lattice is rooted in the 2nd, 3rd, and 5th harmonics: the octave, fifth, and major third. The

Arithmetic Mean of the perfect fifth generates the same major third that is the 5th harmonic of the overtone

series. The Harmonic Mean of the perfect fifth generates the major third descending, which is the same as the

reflection of the harmonic major third of the overtone series. (The major third descending is the minor third

ascending, and the major third ascending is the minor third descending.)

The 2-3-5-9 lattice is rooted in the the 2nd, 3rd, 5th, and 9th harmonics. The Arithmetic and Harmonic

Means of the major third generates major seconds ascending and descending. The Arithmetic Mean of the major

third generates the same major ninth that is the 9th harmonic of the overtone series. The Harmonic Mean of the

perfect major third generates the major ninth descending, which is the same as the reflection of the harmonic

major ninth of the overtone series. (The major ninth, by octave reduction, becomes the major second. The large

major second descending is the small major second ascending, and the large major second ascending is the small

major second descending.)

Notes for the Music Section

1/1 2/12880

3/2432+2

9/8324+4

27/16486+6

81/64364.5

+8

243/128546.75

+10

729/512410.06

+12

4/3384-2

16/9512-4

32/27341.33

-6

128/81455.11

-8

256/243303.4-10

1024/729404.5-12

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

10/9320-18

5/3480-16

5/4360-14

15/8540-12

45/32405-10

135/128303.75

-8

405/256455.625

-6

[ ]

[ ]

( () )

Harmonic Lattice from D 288 Hz, ratio 1/1 and A at 432 Hz(all notes are in the octave D288Hz–D576Hz, ratios 1/1–2/1)

Notes are given with note names

ratioshertz

& cents + or -

Circles are used for the perfect intervals—the 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 enharmonic spellings, depending on the “intention.”

The Arithmetic mean of the Perfect 5th has a Major 3rd ascending 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 3rddescending 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

NWm3rd des

SEm3rdasc

The AM of D288and A432 is F#360

The HM of D288and 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)

1/1 2/1293.334

0

3/2440+2

9/8330+4

27/16495+6

81/64371.25

+8

243/128556.88

+10

729/512417.66

+12

4/3391.11

-2

16/9521.482

-4

32/27347.655

-6

128/81463.54

-8

256/243309.02

-10

1024/729412.04

-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

10/9325.93

-18

5/3488.89

-16

5/4366.67

-14

15/8550-12

45/32412.5-10

135/128309.375

-8

405/256464.06

-6

[ ]

[ ]

( () )

Harmonic Lattice from D 293.334 Hz, ratio 1/1 and A at 440 Hz*(all notes are in the octave D293.34–D586.667, ratios 1/1–2/1)

Notes are given with note names

ratioshertz

& cents + or -

Circles are used for the perfect intervals—the 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 enharmonic spellings, depending on the “intention.”

The Arithmetic mean of the Perfect 5th has a Major 3rd ascending 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

NWm3rd des

SEm3rdasc

The AM of D293.334and A440 is F#366.67

The HM of D293.334and 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 are calculated from D ratio 1/1 and octave 2/1.

AM of

D-D

AM of

D-A

HM of

D-A

HM of

D-D

HM of

D-F#

AM of

D-F#

AM of

Bb-D

AM of

G-D

HM of

Bb-D

HM of

G-D

AM of

D-D

HM of

D-D

2/1

3/2

4/3

5/4

6/59/8

10/9

9/10

8/95/6

4/5

3/4

2/3

1/2

3/2

4/3

GM of

D-D

GM of

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)

D288

D576

A432

D144

D288

D144

D288

G192

D144

D288

Ab204

D288

D576

G#408

= √ ( )x

b = √(a x c)

= √ ( )x

Geometric MeanGeometric Mean

2 ( )x

+( )=

+( )

2=

ArithmeticMean

HarmonicMean

D288

G384

A432

D576

F#360

E320

Bb230.4

C259.2

D144

Eb306

C256

E324

Ab204

G#408

F345.6

A216

G192

B240

D#306

17/161.0625

1062 = The True Logos

17/161.0625

1062 = ologoı iquı

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 logoı h eikwn tou Qeou

12-16-2005

D

A

Bb

B

B

C

C

C#

C#

D

Eb

EE

Bb

F

F

F#

G

Ab

Eb

G#

F#

A#Bb

C#

Db

B

C

D#Eb

E

F

F#

Gb

G

G#Ab

D

A

The colors come directly from the Hue-Saturation-Brightness color wheel on the Macintosh computer. (I am using the note A as 0° red hue and moving counterclockwise in each of the 3 circles.) The inner wheel numbers show ratios and hertz for 12-tone equal temperament. The numbers just outside the inner circle from 0–1200 in 100 degree increments show the number of “cents” for each note. Notice that the cents are all even round numbers, but the ratios are huge and the hertz are uneven. For example, the note B in equal temperament is 900 cents, ratio 9043/5377 and 484.356 hertz. In the middle circle 17-note octave, both ratio and hertz are simpler with B at 27/16 and 486 respectively. The outer circle includes both the 2-3 limit lattice B of 27/16 (486 hz), and the even simpler 2-3-5 limit lattice ratio of B 5/3 (480 hz) The equal temperament hertz number will always be somewhere in between the the 2-3 limit and the 2-3-5 limit versions of the same note. For example, on the inner wheel, the equal temperament C is 513.158 hz ratio 4679/2626; the 2-3 limit C is ratio 16/9 and 512 hz, lower than the equal temperament C; but the 2-3-5 limit C is ratio 9/5 and 518.4 hz, higher than the equal temperament C.

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

256/135546.1215/8

540

243/128546.75

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

1/1288

1/1288

100

16/15307.2

135/128303.75

200

16/15307.2

256/243303.4

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 32/27

341.33

9/8324

400

300

10/9320

9/8324

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

729/512410.06

4/3384

5/4360)

6/5345.6

3/2432

11-10-2005

D

A

Bb

B

B

C

C

C#

C#

D

Eb

EE

Bb

F

F

F#

G

Ab

Eb

G#

F#

A#Bb

C#

Db

B

C

D#Eb

E

F

F#

Gb

G

G#Ab

D

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

0°(180)

208°48(29)

323°16(144)

315°(135)

288°(108)

280°(100)

247°19(67)

240°(60)

216°(36)

180°(0)

95°24(276)

152°16(332)

19°04(199)

145°26(325)

120°(300)

90°(270)

72°(252)

66°36(247)

45°(225)

40°(220)

24°(204)

180°(0)

280°(100)

315°(135)

247°19(67)

322°40(143)

216°(36)

209°31(30)

0°(180)

146°09(326)

151°55(331)

120°(300)

95°06(275)

90°(270)

66°36(247)

45°(225)

24°(204)19°41

(200)

330

300

270

240

210180

150

120

90

60

30

0-360

The innermost circle shows 12 equal divisions of 360°. The colors and degree numbers come directly from the Hue-Saturation-Brightness color wheel on the Macintosh computer. (I am using the note A as 0° hue and moving counterclockwise in each of the three circles.) The middle wheel distinguishes the A#/Bb, C#/Db, D#/Eb, F#/Gb, and G#/Ab, which, in equal temperament, are considered the exact same pitch but with different note names. Recognizing these enharmonic spellings as two separate and distinct notes brings the number of notes in this octave to 17, which is the divsion of the net in the biblical parable of the Fishes in the Net. (See John Michell, Dimensions of Paradise, pp. 174–178 and David Fideler, Jesus Christ, Sun of God, pp. 291–308 for the numeric and geometric symbolism.) The outer circle is based on the 2-3-5 limit lattice harmonic ratios growing out of the 1—the monad. Here the top number is the ratio translated into degrees on a circle and the bottom number in parentheses is the HSB color number. In this outer circle the color numbers begin with 0° as A red, but the musical ratios begin with D as ratio 1/1 and 0° harmonically. D is the harmonic mean of the octave from A to A, the perfect fifth descending. Notice that in 12-tone equal temperament, while the complementary colors of red and cyan are 180° apart, the musical interval is an augmented fourth/diminished fifth – a tritone which has historically been considered a dissonant interval. Using the same number ratios for both color and interval keeps the complementary colors of red and cyan 180° apart, but changes the musical interval from a tritone in the inner circle, to a perfect fifth in the middle and outer circles. The perfect fifth is the most consonant interval next to the perfect unison and perfect octave. If the ratio degrees in the outer circle are added together, they total 3168. To quote John Michell, “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, Plato’s World-soul. The Christian term for this spirit, developed from the number 3168, was Lord Jesus Christ” (Dimensions of Paradise, p. 173).

12-16-2005

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 just above middle C at 256 hz. D is color coded as a cyan circle. The ratio 1/2 is the D₁ at 144 hz, the D below middle C. The ratio 2/1 is the D¹ 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 the perfect fourth below is A₁ 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 “Pythagorean”or “cyclic” intervals. Diamonds indicate the 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 and

diminished fifthare coded with a square and

its diagonal indicating the geometric mean which

is 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 the 17/16 ratio from D to D#/Eb. 17/16 = 1.062.

1062 = “The True Logos”o logoı iquı

11-10 2005

40°(10/9)

240°(5/3)

90°(5/4)

315°(15/8)

146°09'(45/32)

19°41'(135/128)

209°31'(405/256)

95°06'(512/405)

322°40'(128/135)

151°55'(64/45)

24°(16/15)

216°(8/5)

72°(6/5)

288°(9/5)

Total degrees = 3168°57'

120°(4/3)

0°1/1-2/1

180°(3/2)

66°36'(32/27)

280°(16/9)

45°(9/8)

247°19'(27/16)

G384

A432

D288

E320

F#360

G#405

A#455.62

B480

C#540

D#303.75

Eb307.2

F345.6

Gb364.08

Ab409.6

Bb460.8

C518.4

Db546.12

F341.33

C512

E324

B486

The 22 note octave in a 2-3-5 lattice with some form of A B C D E F G in each spine. Total degrees = 3168, which “represents the spirit which passes through and encircles the universe, Plato’s World-soul” (John Michell, Dimensions of Paradise p. 173.)

120°(4/3)

0°1/1-2/1

180°(3/2)

Eb306

F344.25

Gb362.67

Ab408

Bb459

C516.375

Db544

E322.375

F#362.67

G#408

A#459

B483.56

C#544

D#306

66°36'(32/27)

280°(16/9)

45°(9/8)

247°19'(27/16)

213°45'(51/32)

22°30'(17/16)

22°30'(17/16)

150°(17/12)

150°(17/12)

320°(17/9)

320°(17/9)

93°20'(12089/9600)

244°27'(12089/7200)

70°18'(153/128)

285°28'(459/256)

42°57'(2579/2304)

G384

A432

D288

F341.33

C512

E324

B486

93°20'(12089/9600)

213°45'(51/32)


This is what might be called the mediated enharmonic octave. Each 7 note spine has some form of A B C D E F G. F#/Gb, C#/Db, G#/Ab, and A#/Bb are not differentiated (same hertz, i.e. pitches). This 22-octave is based on the 17/16 half step ratio from D to Eb, which has a gematria of “The true ratio”

Total degrees = 3181.3182 = (37x43)2. 37x 43 = 1591 = All humanity. 1591 = Spirit of LifePaı anqropoı Pneuma zwhı

Bb455.11

F#364.5

Eb303.4

Eb Bb F C

C#546.75

Ab404.5

G#410.06

145°26'(1024/729)

19°04'(256/243)

208°48'(128/81)

95°24'(81/64)

323°16'(243/128)

152°16'(729/512)


24°(16/15)307.2

216°(8/5)460.8

72°(6/5)345.6

288°(9/5)518.4

120°(4/3) 0°

1/1-2/1180°(3/2)

66°36'(32/27)

280°(16/9)

45°(9/8)

247°19'(27/16)

E324

F341.33

G384

A432

B486

C512

D288

40°(10/9)

240°(5/3)

90°(5/4)

315°(15/8)

E320

F#360

B480

C#540

In this case the center spine has extended perfect fifths outward in each direction from center D with only 4 notes above and 4 below from the 2-3-5 lattice. This reduces the number of pure thirds, but extends the number of perfect fifths. However, again the total degrees = 3168, which “represents the spirit which passes

through and encircles the universe, Plato’s World-soul” (John Michell, Dimensions of Paradise p. 173.)

(o logoı iquı).

5/2-3 spine

2-3 spine

2-3/5 spine

11-10-2005

F isHMof

D–A

F# isAMof

D–A

Bb isHMof

G–D

B isAMof

G–D

A isGM of

D#–D#

A isGM of

Eb–Eb

WholeStep

E324

D288

F341.33

Gb362.67

G384

Ab408

A432

Bb459

B486

C512

D576

Db544

D#303.75

E324

Db272

Eb306

F#360

G#405

A#455.62

C#540

D#607.5

E320

B480

F344.25

C516.37

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

1/2Step

1/2Step

1/2Step

1/2Step

1/2Step

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

Eb612

F#362.67

G#408

A#459

C#544

D#306

E322.37

B483.56

D#612

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

1/2Step

Db273.06

F345.6

Eb307.2

Gb364.08

Ab409.6

Bb460.8

C518.4

Db546.12

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

1/2Step

Stepwise arrangement of Spine of P5ths from

D#306 – D#612.the GM of

D#306 – D#612is A432.749

Stepwise arrangement ofSpine of P5ths

from D288 – D576.The AM of D – D is A.The HM of D – D is G.

432,000 is the idealradius of the Sun (inmiles). 432 in Greek

gematria is “foundation”

WholeStep

Eb612

Ionian Ionian Dorian Phrygian Phrygian

Stepwise arrangement of Spine of P5ths from D#303.75 – D#607.5.

The AM of D#– D# is A#.The HM of D# – D# is G#.


Eb306 – Eb612.the GM of

Eb306 – Eb612is A432.749

Stepwise arrangement of Spine of P5ths from Db273.06 – Db546.12.

The AM of Db – Db isAb.The HM of Db – Db is Gb.

katabolh

The Arithmetic and Harmonic Mean of the Perfect 5th and the Geometric Mean of the Enharmonic Octave (D#306 – Eb306)with the Three Modes that have the Same Pattern in both Upper and Lower Tetrachords

10-11-2005

Eb307.2

E322.375

E320

Eb306

E324

F345.6

F#362.67

F344.25

F#360

F341.33

Gb364.08

G#408

G#405

G384

Gb362.67

Ab409.6

A#459

A#455.62

A432

Ab408

Bb460.8

B483.56

B480

B486

Bb459

C518.4

C#544

C516.375

C#540

C512

Db546.12

D#612D#612

D#607.5

Db544

D576

Db273.06

D#306

Db272

D#303.75

D288

Eb612

17/18 17/16

17/16+5

12089/9600-.9

17/12+3

51/32+6.9

17/9+1

17/8+5

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

459/256+10.8

17/9+1

17/8+5

o logoı iquı


D#306 – D#612.The ratio of 17/16 in

decimal form is 1.062. Gematria for 1062 is

“The True Ratio”

Stepwise arrangement ofSpine of P5ths from

D288 – D576.This is the spine of 5ths in a 2-3 lattice.

432,000 is the idealradius of the Sun (inmiles). 432 in Greek

gematria is “foundation”

Ionian Ionian Dorian Phrygian Phrygian

Stepwise arrangement of Spine of P5ths from D#303.75 – D#607.5.

This is the harmonic spine of 5ths in a 2-3-5 lattice.

Ratios and cents + or - are shown.Stepwise arrangement

of Spine of P5ths from Eb306 – Eb612.

the GM of Eb306 – Eb612

is A432.749idealized to A432

Stepwise arrangement of Spine of P5ths from Db 273.06 – Db546.12.

This is the harmonic reflection spine of 5ths in a

2-3-5 lattice. Ratios and cents + or - are shown

katabolh

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+6

16/9-4

2/10

135/128-8

10/9-18

5/4-14

45/32-10

405/256-6

5/3-16

15/8-12

135-128-8

9/5+18

8/5+14

64/45+10

512/405+6

6/5+16

16/15+12

128/135+8

128/135+8

10-11-2005

216 288 384

3/4

256 324

3/2

logaioıChosenPath

stiboıh pnohThe Breath

216

h pnohThe Breath

288Path

stiboı

384

logaioıChosen

alhqjhı agnoı

243 341.33

3/4

216

h pnohThe Breath

256 324

agnoıPure; Sacred; Holy

384

logaioıChosen

288Path

stiboıWorthy 341

axioı

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 are

assigned 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. It’s Greek gematria is Path (stiboı*). 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 (Eufileı) + 318 Sun (Hlioı)

A perfect fourth above C256 is F341.33 (ratio 4/3 and and HM of C256 and C512). A perfect fourth below E324 is B243 (ratio 3/4 and AM of E162 and E324).The combined total = 2052.33. 2052 = 1240 mystical (mustikoı) + 812 song (wdh).

alhqjhıtrue

korban

Offering

B

E

E

D

D

D

D F

A

A

A

C

C

G

G

G

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

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

4/3

2/3

4/3

12-16-2005

The notes that are not perfect intervals from D are assigned an oval shape. Musically they are “imperfect intervals,” and geometrically they are ovals which are imperfect circles. These are the notes B, C, E, and F. The interval between B and C is a half step as is the interval between E and F. Here I have a G at the bottom and an A at the top to show the whole steps bounding A and G. (Again, it should be noted that only the ovals are bounded by

half step intervals on one side.)

The modes come from using each of these 7 notes as a starting note which changes the pattern of whole and half steps giving a different flavor to each mode. Listed below are the 7 possible patterns with their Greek names as used in jazz today. (For a clear explanation of modes and their ancient and

modern assignment of names, see Counterpoint by Knud Jeppesen, pp. 59–62.)

D Dorian

E Phrygian

F Lydian

G Mixolydian

A Aeolian

B Locrian

C Ionian

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step Whole

StepWholeStep

1/2Step Whole

StepWholeStep

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step Whole

StepWholeStep

1/2Step

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

1/2step

1/2step

G GA AB C D E F

D F D

D

D

D

D

D

D

G

G

G

G G

G

G

G

A

A

A

A

A A

A

A

F

F F

F

F

F

F

E

E E

E

E

E

E

E

C

C

C

C

C

C

C C

B

B

B

B

B B

B

B

12-16-2005

The pattern of whole and half steps defines the mode, not the starting note. If I want to use the same pattern of notes as the Dorian mode but I want to start on the note G rather than D, I will need to change the B natural to a Bb.

G Mixolydian

D Dorian

G Dorian

If I want to use the same pattern of notes as the Dorian mode but I want to start on the note A rather than D, I will need to change the F natural to a F#.

D Dorian

A Aeolian

A Dorian

Let’s look more closely at half steps. The half steps occurring between B and C and E and F are the naturally occuring half steps in the diatonic scale of A B C D E F G. Because there is already a half step between B and C, I can only add a half step below the B and a half step above the C.

Similarly, I can only add a half step below the E and and a half step above the F. I will name these Bb, C#, Eb, and F#respectively, as the half steps that occur on either side of the natural half steps B, C, E, and F.

CB E F

Bb

C#

Eb

F#

With the perfect interval notes A, D, and G, because they are bounded diatonically by whole steps on either side, chromatically I can add a half step on either side, so A can have both an Ab and A#, D can have both a Db and a D#, and G can have both a Gb and a G#.

D GA

G#

Ab

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

WholeStep

D F DG AE CB

G A DCB F GE

G A CBb F GED

D F DG AE CB

A DCB F GE A

A DCB F# GE A

A#

Db

D#

Gb

12-16-2005

GbD F GA ECBBb

F#C# G#D#A#

EbAb Db

In equal temperament, each note is 100 cents, beginning with C at 0 as shown below. Many electronic keyboards now have the capability to tune notes individually. Pitches are calculated in cents + or - from the 12 equal divisions.

Ab

G#AG

GbD F GEC B

Bb

F#C# G#D# A#

EbDb AbA

0

100

100

200300

300

400 500

600

700

800

800

900

1000

1000

1100

600

In equal temperament, the enharmonic notes of G#/Ab, D#/Eb, A#/Bb, F#/Gb, and C#/Db are treated as the same pitch. These are like the black keys on a keyboard instrument. I have shown them below to demonstrate the symmetry outward in both directions from D. D is the only note that has this symmetry. It happens also to be the center note of A B C D E F G, making it the only possible center of this group of notes.

C

1200

If I go back to the Arithmetic and Harmonic Means of A B C D E F G and keep extending them outward in either direction, I also generate the enharmonic notes of C#/Db, D#/Eb, F#/Gb, G#/Ab, and A#/Bb, but the outcome is quite different. First, here is how the pitches are generated from the ratio of 1/1, and by using only the ratios of 1/2, 2/1, 2/3, 3/2, 3/4, and 4/3. For a complete explanation of Aritmetic and Harmonic Means of octaves, see the diagram “Aritmetic (AM) and Harmonic (HM) Means of Octaves descending and ascending from A (A=432 Hz and ratio 1/1). For musicians looking at the illustration below, it is the circle of fifths ascending and fourths descending from D outward to the right, and the circle of fourths ascending and fifths descending from D outward to the left.

fifths ascending 3/2

b = (a + c) / 2

Example 288 = (144 + 576) / 2

Formula for Arithmetic Mean

269.67 202.27 303.4 227.55 341.33 461.32615.09410.06546.75364.5486324432288384256

DF G A EC BBb F# C# G# D# A#EbAbDb

179.79

Gb

3/4

3/2 3/2 3/2 3/2

3/4 3/4 3/42/3

4/3

2/32/32/3

4/34/34/3

D

D144

576

fifths descending 2/3 fourths descending 3/4

fourths ascending 4/3

b = 2ac / (a+c)

Example 288 = 2 x 216 x 432 / (216 + 432)

A216

Formula for Harmonic Mean

b

a a

c

cb

D D D= +( )₁ ₁

2

D =2 x A ₁ x A

₁

₁

₁

( A ₁ + A

b ca

a

a

b c

c

12-16-2005

Now I have all 17 notes in hertz generated from the Arithmetic and Harmonic Means of the octave. Next I convert the hertz into cents. (I use a program called Justonic Pitch Palette to translate the hertz into cents.) The results are shown below.

269.67 404.5 303.4 455.11 341.33 461.88615.84410.56274.37364.5486324432288384256


359.56

Gb

588.10 90.06 791.99 294.09 996.08 498.03 0 701.96 203.91 905.87 407.82 1109.76 611.73 113.69 817.74 319.69 1021.65Cents

Hertz

Then I round the cents values up or down to whole numbers. The cents values + or - are given comparative to their equal tempered divisions. Example — Gb in equal temperament is 600 cents. Gb from the ratios of numbers 1, 2, 3, and 4 is 588 cents, 12 cents

below the equal temperament Gb.

90-10

792-8

294-6

996-4

498-2

1022+22

320+20

818+18

114+14

612+12

1110+10

408+8

906+6

204+4

702+2

0

DF G A EC BBb F# C# G# D# A#EbAbDb588-12

Gb

588.10 90.06 791.99 294.09 996.08 498.03 0 701.96 203.91 905.87 407.82 1109.76 611.73 113.69 817.74 319.69 1021.65

Cents

Cents+ or -

GbD F GEC B

Bb

F#C# G#D# A#

EbDb AbA

0

100

100

200300

300

400 500

600

700

800

800

900

1000

1000

1100

600

C

1200

GbD F GEC B

Bb

F#C# G#D# A#

EbDb AbA

0

114+14

90-10

204+4

320+20

294-6

408+8

498-2

612+12 702

+2

818+18

792-8

906+6

1022+22

996-4

1110+10

588-12

C

12000

Here again is the 12-tone chromatic scale with 12 equal divisions of 1200 cents, treating C# and Db as the same pitch, as well as D# and Eb, F# and Gb, G# and Ab, and A# and Bb.

Finally, here is the chromatic scale of 17 notes. This 17-note chromatic scale elimanates enharmonic spellings found in equal temperament, hence 5 new pitches are added. C#/Db at 100 cents becomes C# 114 cents and Db 90 cents. Similarly D#/Eb at 300 cents become D# 320 cents and Eb 294 cents. F# and Gb separate so that instead of both being 600 cents, F# is 612 cents and Gb is 588 cents. G# and Ab separate so that instead of both being 800 cents, G# is 818 cents and Ab is 792 cents. Finally, A# and Bb separate so that instead of both being 1000 cents, A# is 1022 cents and Bb is 996 cents.

12-16-2005

432 648 486

364.5

729 546.75 820.12 615.09

461.32

922.64384512341.33455.11303.4404.5269.67

170.67

227.56179.78

b = (a + c) / 2

Formula for Harmonic Mean

113.7889.89 128 384

288

216 648 243 729 274.37341.33303.4101.13269.67

DF G A EC BBb F# C# G# D#EbAbDbGb

192 432256 324170.67 486227.56 364.5151.7 546.75202.25134.84 615.09

A#

179.78 461.32

Example 288 = (192 + 384) / 2

Example 288 =2 (216 x 432) / (216 + 432)

288 820.12 307.54 922.64

410.06

GMGMGMGMGMGMGMGM

Formula for Geometric Mean

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

Example 288 = √ (384 x 216)

GMGMGM

GMGMGM

GM

GM

b

b

a

a

c

c

512 384 576 432 648 486 729 546.75 820.12 615.09341.33455.11

151.7

404.5

134.84

359.56

256 384 288 432 324 486 364.5 546.75 615.09 461.32341.33227.56303.4202.25269.67

256 192 288 216 324 243 364.5 274.37 307.54170.67227.56

303.4

202.25

269.67

179.78

256 384 288 432 324 486 364.5 546.75 615.09341.33303.4202.25269.67

576

288

Formula for Arithmetic Mean

b = √ (a x c)

256 192 216 324 243 274.37 307.54227.56151.7202.25134.84







The Arithmetic , Harmonic , and Geometric Means generated from ratios using the numbers 1, 2, 3, and 4(1/1 1/2 2/1 2/3 3/2 3/4 4/3)

a c

410.06

410.06

410.06

410.06

b

12-16-2005

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 the perfect 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 is the 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 is G192 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 of G192 and D288. F345.6 is the HM of D288 and A432.

192 288 432

2/3 3/2

katabolhFoundationPath

stiboıMariamMary

D AG1

B F#

5/4

AM of G - Dis B

AM of D - Ais F#

5/4240 360

192 288 432

2/3 3/2


stiboıMariamMary

D AG1

4/5

HM of G - Dis Bb

HM of D - Ais F

4/5

230.4

Bb F

345.6

The AM of G192 and D288 is B240

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

nouı720 = Mind; understanding

Pneuma Agion710 = Holy Spirit

sproroı720 = 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

710 = Willingness, desire proqumia

711 = Wholly; utterly acri

711 = Muse Mousa

sunesiı1065 = Understanding, knowledge

The AM of D288 and A432 is F#360

The HM of D288 and G192 is Bb230.4 The HM of A432 and D288 is F345.6

G1 B1 D D F# A

G1 Bb1 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

The musical gematria of the triad G192 + Bb230.4 + D288 = 710.4 The musical gematria of the triad D288 + F345.6 + A432 = 1065.6

720 = The Holy Spirit o Agioı Anemoı 1080 = Threefold trissoı

1065 = God + Light; splendor 284 Qeoı + 781 feggoı

1066 = true + beloved, friend 256 alhqhı + 810 filoı

11-12-2005

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

192 288 432

2/3 3/2


stiboıMariamMary

D AG1

B F#

5/4

AM of G - Dis B

AM of D - Ais F#

5/4240 360

192 288 432

2/3 3/2


stiboıMariamMary

D AG1

4/5

HM of G - Dis Bb

HM of D - Ais F

4/5230.4

Bb F

345.6

Bb D F

230.4 + 288 + 345.6 = 864

864 =Pythagoras Puqagoraı

B D F#

240 + 288 + 360 = 888

888 = Jesus Ihsouı

864 = Of the saints agiwn

864 = Of the Gods Qewn

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 = 888

B 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 Apollo’s first speaker of oracles)

11-12-2005

E

F A C

B D F#

Bb D F

G1 B1 D

F# C#A

G1 DBb

A C#

A EC

Eb G1 Bb

E G1 B

D F# A

D AF C Eb G1

C G1E

E BG#

F#B D#

F# A# C#F Ab C

FBb Db

Eb Gb Bb

EC# G#

F#D# A#

BG# D#

Gb Bb Db

FDb Ab

Ab C Eb

D A E B F# C#GCFBbEb

B

Bb

E

Eb

CF

F# C#Ab

A#

Db

D#G#

Gb

G# D# A#AbDbGb

These are the only 6 triads that have a perfect interval note as the mean (using D288 hz and ratio 1/1 as center). The perfect interval as the center makes it possible to move from the harmonic (minor-yin-descending from center) side, to the arithmetic (major-yang-ascending from center) side. In each case, the perfect fifth comes from 2-3-5 (the upper or lower spine of fifths), and the center comes from 2-3 (the center spine of fifths). The symmetry becomes even more beautiful because the Major-yang-bright-sharp triad is on the minor-yin-gentle-flat side, and the minor-yin-gentle-flat triad is on the major-yang-bright-sharp side, creating a perfectly centered balance.

192 + 240 + 288 = 720

192 + 230.4 + 288 = 710.4

288 + 360 + 432 = 1080

288 + 345.6 + 432 = 1065.6

216 + 270 + 324 = 810

216 + 259.2 + 324 = 799.2 256 + 307.2 + 384 = 947.2

256 + 320 + 384 = 960

345.6 + 432 + 518.4 = 1296

360 + 432 + 540 = 1332

240 + 288 + 360 = 888

230.4 + 288 + 345.6 = 864153.6 + 192 + 230.4 = 576

160 + 192 + 240 = 592

These are the 4 triads that have perfect intervals as the bounds.Each bounding P5th can generate both an arithmetic and harmonic mean.

These are the 4 triads that have 1 perfect and 1 imperfect interval as the bounds. Each bounding P5th can generate both an arithmetic and harmonic mean.

341.33 + 409.6 + 512 = 1262.93

227.55 + 273.06 + 341.33 = 841.95

303.4 + 364.08 + 455.11 = 1122.59 324 + 405 + 486 = 1215

243 + 303.75 + 364.5 = 911.25

364.5 + 455.625 + 546.75 = 1366.88

409.6 + 512 + 614.4 = 1536

273.06 + 341.33 + 409.6 = 1024

364.08 + 455.11 + 546.12 = 1365.31

270 + 324 + 405 = 999

405 + 486 + 607.5 = 1498.5

303.75 + 364.5 + 455.625 = 1123.88

These are the 3 triads bounded by a 3/2 P5th, and generating an arithmetic mean.

These are the 3 triads bounded by a 3/2 P5th, and generating an harmonic mean.

These are the 3 triads bounded by a 3-2-5 P5th, and generating an arithmetic mean.

These are the 3 triads bounded by a 3-2-5 P5th, and generating an harmonic mean.

the ineffable, the secret

o aporrhtoı ineffable God

Qeoı aneklalhtoı

1124 the most high upisth

1499 messenger

shmantwr

1367=368 Gentle, kind + 999 Ineffable God

epioı + Qeoı aneklalhtoı

911 joy, grace

cariı 841 = 1 beginning + 40 middle + 800 end

A + M + W

belovedfiloı

800 faithpistiı

divinityagaqothı

breath, spirit,inspiration

pneuma

JesusIhsouı

PythagorasPuqagoraı

1332 Gnostic diety Chnoubiscnoubiı

666 Graeco-Egyptian God Sarapis, Osiris-ApisSerapiı

180 + 216 + 270 = 666

172.8 + 216 + 259.2 = 6481296 = 576 Breath, spirit +720 mind 648 Mysteries, initiation

teleth

the circleto kuklon

1263 Gnosis, wisdomgnwsiı

to purifykaqarizw

1365 shining

faeqwn

1024=256 x 4 = true x 4

alhqhı x 4

nouı720 = Mind; understanding to Pneuma Agion

1080 = The Holy Spirit

711 = MuseMousa sunesiı

1065 = Understanding, knowledge

1536=1008 teacher + 528 The GoddessHestia (var. of)

prostroph + Istih

353 Hermes + 769 Pythios (Apollo at Delphi)Ermehı + Puqioı

1153 Beloved + 62 Athanaeufilhı Aqana

{ } not used in 2-3-5 triads

{ } not used in 2-3-5 triads

pneuma + nouı

HM

AM

AM

HM

12-16-2005

A#

A#

D#

D#

A#

A#

C#

CF

E

BG A

BbEb D A

E

BGCF

DEb Bb

E BA

D A E B

DCF G

GCF F#

F# C#

F#

F#

Ab

Ab

G#

G#

Bb

Bb

CF G A

BbEb D AGCF

DEb Bb

Ab

Ab

E

E

F#E C#B G#

D#F#E C#B G#

D#

F#E C#B G#

F#E C#B G#

F#E C#B

F#E C#B

F

Bb F C

CBb

F

Bb F C

CBb

F

Bb F C

CBb

EbAb

EbAb

EbAb

EbAb

EbAb

EbAb

Db

Db

Db

Db

Db

Db

Gb

Gb

Gb

Gb

288

576

132.71

66.36

246.9 1250

2500

720

360

345.6

172.8

165.88

82.94

447.9

1562.5

3125

432

864

1875

3750

199.06

99.53

259.2

518.4

540

1080

2343.75

4687.5

248.83

124.42

648

1296

2592

5184

671.85

335.92

810

1620

3515.63

7031.25

19212885.33

256 384

88.47

44.24

493.8

833.33

1666.66

240

480

500

1000

230.4

115.2

53.08

26.54

1041.67

2083.33

160

153.6

76.8

320

333.33

666.66

58.98

29.49

164.6109.73 555.55

1111.11

170.66

329.2219.47

370.35

740.71

102.4

51.2

222.22

444.44

462.96

925.92

694.44

1388.88

19.6613.11

39.3226.21

35.39

17.6911.79

23.59 79.62

39.81 59.72

119.43 179.15

89.57 134.36

268.72

149.29 223.95

298.6

373.25

186.62

972

1944

1215

2430

1458

2916

1822.5

3645

56.89

113.77

75.85

151.70101.13

50.57

34.13

68.2745.51

22.76

3888

11664

5832

7776

148.15

296.29

98.76

197.53

This honeycomb pattern has all the ratios which are part of a 2-3 lattice on the front part of a cube with its octave. These are all circles, the perfect intervals G, D, and A; and the ovals of E, B, F#, C#, G#, Ab, Eb, Bb, F, and C.The diamonds have all the ratios which are part of a 2-3-5 lattice and are on the back part of the cubes. These are the 2-3-5 versions of all the notes except for those perfect intervals of G, D, and A. D288 at the very center is both the Arithmetic Mean of the perfect fifth B240 and F#360, and the Harmonic Mean of the perfect fifth Bb230.4 and F345.6. B-D-F# is a minor chord but on the major (sharp spine of fifths). Bb-D-F is a major chord but on the minor (flat spine of fifths). Gematria for B240 + D288 + F#360 = 888 = Jesus. Gematria for Bb230.4 + D288 + F345.6 = 864 = Pythagoras

(Ihsouı) (Puqagoraı)

(Ihsouı)

(Puqagoraı)

11-12-2005

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). 7 presents 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 the major second.

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

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

230.4 288 360

h nikh alhqeiaıthe victory of truthPath

stiboıoikionHouse, 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 - Dis C256

256 + 288 + 320 = 864

871 acoıPain, sorrow871 Secret, dark skotaioı

871 Web; cloak faroı

871 To purify agnizw

871 against one's will akon

864 Pythagoras

Holy of holies

Puqagoraı

864 agiwn

864 Of the Gods Qewn

864 Throne of Abraxas Qronoı Abraxaı

864 Jerusalem Ierousalhm

864 Joiner's Square;corner stone

gwnia

C D EC D E

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

C D E

C D E

Arithmetic and Harmonic Means of the Major Third11-12-2005

G

A

B

F

153.6 192

240


stiboı

9/8

10/9

AM of Eb - G

is F172.8

AM of G - B

is A216

9/8

10/9

172.8 216

8/9

9/10

172.8 + 192 + 216 = 580.8

170.66

HM of Eb - Gis C256

Eb

F

F G A

580 ollumiTo destroy581 Deadly; of death qanasmoı

581 Reality upar

581 Wheel ampuı

F G A

172.8 + 192 + 216 = 580.8

G

A

B

243

AM of A - C#is B243

10/9216

B

9/8

8/9

C#

9/8

F

9/10

172.8

192 240

270

HM of A - C#is B240

HM of F - A

is G192

192 + 216 + 240 = 648

G A B

648 Initiation, celebration; mysteries teleth

The House of Truth648 domoı alhqeiaı

648 The Truth + Spirit h alhqeia + pneuma

h Basileia eirhnhıThe Kingdom of Peace648

192 + 216 + 240 = 648

G A 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.

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

204.8 256

9/8 9/10AM of Ab - C

is Bb230.4

230.4

8/910/9

204.8 + 230.4 + 256 = 691.2

227.55

HM of Ab - C

is Bb227.55

Bb

Bb

Ab C

364.5

AM of E - G#

is F#364.5

10/9324

9/8 9/10

8/9405

HM of E - G#

is F#360

E

F#

F#

360

G#

BbAb F#E G#

Ab BbE G#F#

C

C

204.8 + 227.55 + 256 = 688324 + 360 + 405 = 1089

324 + 364.5 + 405 = 1093.51093 understanding (901) + that

which is played on the kithara (192)

688 cup

1089 All seeing

1089 Dead

pantopthı

fqitoı

691 knowledge, science,history

691 Power, force, victory, dominion

istoria

kratoı

pothrion

G

F

153.6 192

9/8 9/10

AM of Eb - G

is F172.8

172.8

8/910/9

153.6 + 172.8 + 192 = 518.4

170.66

HM of Eb - Gis C256

Eb

F

A

B

243

AM of A - C#is B243

10/9216

B

9/8 9/10

C#

8/9

240

270

HM of A - C#is B240

216 + 240 + 270 = 726

A B

230.4 288

D

9/8 9/10AM of Bb - D

is C259.6

259.6

8/910/9

Bb

230.4 +259.6 + 288 = 778

C

C

256

HM of Bb - Dis C256

230.4 + 256 + 288 = 774.4

288 360

D F#AM of D - F#is E324

9/8 9/10324

8/910/9

E

E

320

HM of D - F#is E320

775 Messenger of the gods; guide

diaktoroı

778 a sound778 winged

hcoı

pothnoı

288 +320 + 360 = 968

288 +324 + 360 = 972

968 mad frhnoblabhı

972 A vanishing afanisiı

Bb C

Bb C

D

E

E

D

D F#

D F#

B

C#

A C#F GEb

GEb F

216 + 243 + 270 = 729

153.6 + 170.66 + 192 = 516.26

516 The goddess Hestia

Estia

518 Mistake, error examartia

518 sorrow, grief luph

726 Unutterable

726 The Messiah

anaudoı

o Messiaı

729 The Perfected Workeuergesia

729 Cubic StoneKhfaı

729 = 9 x 9 x 9


gnwmh + kiqariama

273.06

9/8 9/10AM of Db - F

is Eb307.2

307.2

8/910/9

273.06 + 307.2 + 341.33 = 921.73

303.4

HM of Db - F

is Eb303.4

Db

Eb

F

341.33

Eb

273.375

AM of B - D#

is C#273.375

10/9243

9/8 9/10

9/8

303.75HM of B - D#

is C#270

270

B

C#

D#

C#

364.08

9/8 10/9

AM of Gb - Bb

is Ab409.6

409.6

8/910/9

404.5

HM of Gb - Bb

is Ab404.5 455.11

410.06

AM of F# - A#

is G#410.06

10/9364.5

9/8 9/10

8/9

455.625HM of F# - A#is G#405

405

F# A#

G#

G#

Gb

Ab

Ab

Bb

F#

F#

B C# D#

G#Gb

Db

Db

Ab

Eb D#B C#

F

Bb

Eb

G#

Ab

F

BbGb A#

A#

273.06 + 303.4 + 341.33 = 917.79 243 + 270 + 303.75 = 816.75

243 + 273.375 + 303.75 = 820.125

364.5 + 405 + 455.625 = 1225.13

364.5 + 410.06 + 455.625 = 1230.19

364.08 + 404.5 + 455.11 = 1223.69

364.08 + 409.6 + 455.11 = 1228.79

918 poppy 816 Loving

816 Mercy

ktisiı Qeou

icqueı

to dikton

1230 Thought; reflection

frontiı

1228 an undertaking, beginningepiceirhsiı

918 Moses

918 The King

mhkwn

Mouseı

o basileuı

sterghı

elehmosunh

820 Strong, mighty

820 Way; direction820 Reverent

820 The sacrifice

820 To overcome, force820 The sphere

puknoı

tropoı

eusebhı

to quma

biazw

h sfaira

921 Rod, staff, law, canonkanwn

1224 The creation of God

1224 Fishes

1224 The net

ktisiı Qeou

icqueı

to dikton

1224 The creation of God

1224 Fishes

1224 The net


The Pattern of Whole and Half Steps in the Three Modes Where the Pattern is the Same in both the Lower and Upper Tetrachord

Eb612

WholeStep

E324

D288

F341.33

Gb362.67

G384

Ab408

A432

Bb459

B486

C512

D576

Db544

D#303.7

E324

Db272

Eb306

F#360

G#405

A#455.6

C#540

D#607.5

E320

B480

F344.25

C516.37

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

1/2Step

256/2431/2Step

1/2Step

1/2Step

9/8WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

WholeStep

Each vertical spine is a stepwise arrangement of perfect fifths. Each has a different pattern of half steps and whole steps.(The center spine grows from Arithmetic and Harmonic Means of D.)

The pattern of the center spine is totally balanced with W - H - W (W) W - H - W. In current modal theory, this pattern is called the Dorian mode.The pattern of the left spines has a sequence of W - W - H (W) W - W - H. In current modal theory, this pattern is called the Ionian mode.

The pattern of the right spines has a sequence of H - W - W (W) H - W - W. In current modal theory, this pattern is called the Phrygian mode.Looking horizontally, the only time that whole steps occur in all five spines is between Gb and Ab, between G and A, and between G# and A#.Mathematically, in terms of pattern, these are the only three possibilities where the pattern of whole and half steps is the same in both the lower

and upper tetrachords.

W1/2W

W1/2W

DorianLower

Tetrachord

DorianUpper

Tetrachord

IonianLower

TetrachordWW1/2

PhrygianLower

Tetrachord1/2WW

IonianUpper

TetrachordWW1/2

PhrygianUpper

Tetrachord1/2WW

DorianIonian PhrygianDb

273.06

F345.6

Eb307.2

Gb364.08

Ab409.6

Bb460.8

C518.4

Db546.12

256/243

256/243

9/8

9/8

9/8

9/8

9/8

Eb612

F#362.67

G#408

A#459

C#544

D#306

E322.37

B482.67

D#612

256/243

9/8

9/8

9/8

9/8

256/243

9/8

9/8WholeStep

9/8WholeStep

9/8WholeStep

256/2431/2Step

9/8WholeStep

11-12-2005

Ab408

A432

Bb459

B486

C512

D576

Db544

A#455.6

C#540

D#607.5

B480

C516.37

WholeStep

WholeStep

WholeStep

1/2Step

1/2Step

WholeStep

WholeStep

1/2Step

W1/2W

DorianLower

Tetrachord

E324

F682.66

Gb682.66

G768

Ab816

A864

E648

Eb612

F#720

G#810

A#911.2

E640

F688.5

WholeStep

WholeStep

1/2Step

WholeStep

WholeStep

WholeStep

1/2WW

PhrygianUpper

Tetrachord

AeolianMixolydian Locrian

Each vertical spine becomes a mixed mode using Ab, A, and A# as starting notes. (A , the starting note of the center spine, is the Arithmetic Mean of D and its octave.)

The pattern of the center spine is a combination of Dorian and Phrygian; W - H - W (W) H - W - W. In current modal theory, this pattern is called the Aeolian mode.(Aeolian also appears as the right spines pattern beginning on G# on the page using Gb, G, and G#.)

The pattern of the left spines are a combination of Ionian and Dorian; W - W - H (W) W - H - W. In current modal theory, this pattern is called the Mixolydian mode.(Mixolydian also appears as the center spine pattern beginning on G on the page using Gb, G, and G#.)

The pattern of the right spines are a combination of Phrygian and Lydian; H - W - W (H) W - W - W. In current modal theory, this pattern is called the Locrian mode.The two modes that have a half step rather than a whole step between tetrachords are Locrian with a flatted or diminished 5th, and Lydian

which has a raised or augmented fourth.

The Pattern of Whole and Half Steps in the Three Modes from Ab, A, and A#, Creating the Mixed Modes of Mixolydian, Aeolian, and Locrian.

(In each case the pattern differs between the lower and upper tetrachord.)

Ab409.6

Bb460.8

C518.4

Db546.12

256/243

9/8

9/8

9/8

F345.6

Eb307.2

Gb364.08

Ab409.6

256/243

9/8

9/8

Eb612

A#459

C#544

B482.67

D#612

9/8

256/243

9/8

F#362.67

G#408

A#459

E322.37

256/243

9/8

9/8

9/8

PhrygianLower

Tetrachord1/2WW

IonianLower

TetrachordWW1/2

DorianUpper

TetrachordW1/2W

LydianUpper

TetrachordWWW

256/2431/2Step

9/8WholeStep

9/8WholeStep

9/8WholeStep

9/8WholeStep

9/8WholeStep

256/2431/2Step

11-12-2005

Eb612

Gb362.67

G384

Ab408

A432

Bb459

B486

C512

D576

Db544

G#405

A#455.6

C#540

D#607.5

B480

C516.37

Eb612

E648

WholeStep

WholeStep

WholeStep

WholeStep

1/2Step

1/2Step

WholeStep

WholeStep

WholeStep

WholeStep

E320

1/2Step

WW1/2

IonianLower

Tetrachord

F682.66

Gb725.34

G768

F#720

G#810

F688.5

WholeStep

1/2Step

WholeStep

W1/2W

DorianUpper

Tetrachord

MixolydianLydian Aeolian

Each vertical spine becomes a mixed mode using Gb, G, and G# as starting notes.(G , the starting note of the center spine, is the Harmonic mean of D and its octave.)

The pattern of the center spine is a combination of Ionian and Dorian; W - W - H (W) W - H - W. In current modal theory, this is called the Mixolydian mode.(Mixolydian also appears as the left spines pattern beginning on Ab on the page using Ab, A, and A#).

The pattern of the left spines are a combination of Lydian and Ionian; W - W - W (H) W - W - H. In current modal theory, this pattern is called the Lydian mode.The pattern of the right spines are a combination of Dorian and Phrygian; W - H - W (W) H - W - W. In current modal theory, this pattern is called the Aeolian mode.

(Aeolian also appears as the center spine pattern beginning on A on the page using Ab, A, and A#.)The two modes that have a half step rather than a whole step between tetrachords are Locrian with a flatted or diminished fifth, and Lydian

which has a raised or augmented fourth.

The Pattern of Whole and Half Steps in the Three Modes from Gb, G, and G#, Creating the Mixed Modes of Lydian, Mixolydian, and Aeolian.

(In each case the pattern differs between the lower and upper tetrachord.)

Gb364.08

Ab409.6

Bb460.8

C518.4

Db546.12

256/243

9/8

9/8

9/8

F345.6

Eb307.2

Gb364.08

256/243

9/8

Eb612

G#408

A#459

C#544

B482.67

D#612

F#362.67

G#408

E322.37Ionian

UpperTetrachord

WW1/2

PhrygianUpper

Tetrachord1/2WW

DorianLower

TetrachordW1/2W

LydianLower

TetrachordWWW

9/8

9/8

256/243

9/8

9/8

256/243

9/8

256/2431/2Step

9/8WholeStep

9/8WholeStep

9/8WholeStep

9/8WholeStep

9/8WholeStep

256/2431/2Step

9/8

11-12-2005

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