VISUAL REFERENCE FOR MUSICIANS - Miles Okazaki · 2) Complement (the negative space ... perfect 4th / perfect 5th tritone minor 3rd / major 6th major 3rd / minor 6th 0 pitches 12

VISUAL REFERENCE FOR MUSICIANS

BY MILES OKAZAKI2014 EDITION

TWO NOTE STRUCTURESThe intervallic relationship between any two different pitches within an octave will form one of these 6 shapes.

THREE NOTE STRUCTURESThe intervallic relationship between any three different pitches within an octave will form one of these 19 shapes.

FOUR NOTE STRUCTURESThe intervallic relationship between any four different pitches within an octave will form one of these 43 shapes.

TABLE OF MELODIC ELEMENTS

& œœœ œœœbb œœœnn# œœœbb œœœnn# œœœn œœœbbb œœœnnn œœœbb œœœnn# œœœb œœœ#n#

& œœœœœœ

œœœ œœœœœœ

œœœœœœ

œœœ

œœœ

œœœ

œœœ

œœœ

& œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ

Pitches are shown as a chromatic circle, ascending in a clockwise direction.One circle represents all transpositions - pitches can be assigned in any rotation.For example:

. . . and so on.

Each pitch collection is shown as a shape formed by connected dots. A major triad looks like this:This figure represents all transpositions of the same intervallic relationships:

All inversions and octave displacements are considered equivalent. The same figure above also represents any of these, in any transposition, or any other larger range of pitches:

Pitch sets can be simultaneous (chords) or sequential (melodic).The same figure above also represents any of these, in any transposition, in any rhythm:

All unique pitch formations in 12-Tone Equal Temperament containing a number of pitches from 0 - 12 are listed.Only one rotation of any formation is listed. Rotation is equivalent to transposition.

For example, is listed, but its rotation is redundant and not listed.

The total of 352 pitch formations are derived from 122 “prime forms” (this includes the empty set, with no pitches).Prime form is the arrangement of pitches starting at “12:00” on the circle that has the smallest intervallic span.A prime form can produce three other formations through these operations:1) Reflection (reversing the intervallic order)2) Complement (the negative space of the Prime Form)3) Reflected Complement (the negative space of the Reflection, or the Reflection of the Complement)Each row of the table shows a Prime Form and its related formations. If one of the operations does not produce a new formation, the space is left blank.

For example, upon reflection makes which is the same as the original formation in rotation. (this is the case with any symmetrical shape)

And the complement of is which, when rotated, is the same as the reflection: (this can only happen with six note formations)

C

F

D

E

G

A

B Db

E b

GbAb

BbC

F

D

EG

A

B

Db

EbGb

Ab

BbC

F

D

E

G

A B

Db

EbGb

AbBb

EXPLANATION:

root

3rd

5th

PRIME FORM REFLECTION COMPLEMENT REFLECTED COMPLEMENT

empty set chromatic scale

single pitch

semitone / major 7th

tone / minor 7th

perfect 4th / perfect 5th

tritone

minor 3rd / major 6th

major 3rd / minor 6th

12 pitches0 pitches

11 pitches1 pitch

10 pitches2 pitches



minor triad major triad

diminished triad

augmented triad

9 pitches3 pitches 3 pitches 9 pitches




major 7th


all interval tetrachord{0,1,4,6}







diminished 7th octatonic


minor 7th

minor 6th dominant 7th









harmonic major harmonic minor



melodic minor

pentatonic diatonic











whole tone

RANDOMIZED CHROMATIC PITCH SPACE

C

D

C

Db

E bE F

Gb

G AbA Bb

B DC

Db

E b E FGbG Ab

ABb

B D

C

Db

E b EF

GbG AbABbB

D

C

DbE b E

F

Gb G

Ab

A

BbB

C

C

C

C

C

C

C

Db

Db

Db

DbDb

Db

Db

DbD

D

D

D

D

D

D

D

E b

E b

E b

E b

E b

E b

E b

E b

E

E

E

E

E

E

E

E

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

F

GbGb

GbGb

GbGb

Gb

Gb

G

G

G

G

G

G

G

G

G

G

G

G

G

G

G

G

G

G

G

G

Ab

Ab

AbAb

Ab

AbAb

Ab

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

A

BbBb

BbBb

Bb

Bb

Bb

Bb

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

C

C

C

C

C

C

C

C

C

C

C

C

C#

C#

C#

C#

C#

C#

C#

C#

C#

C#

C#

C#

F #

D

D

D

D

D

D

D

D

D

D

D

D

D#D#

D#

D#

D#D#

D#

D#

D#

D#

D#

D#

E

E

E

E

E

E

E

E

E

E

E

E

F #

F #

F #

F #

F #

F #

F #

F #F #

F #

F #

G#

G#

G#

G#

G#

G#

G#

G#G#

G#

G#

G#

A#A#

A#

A#

A#

A#

A#

A#

A#

A#

A#A#

Two diagrams of twelve tone rows arranged in large 12x12 Latin Squares made of smaller 3x4 Latin Rectangles.Each complete row, column, and small 3x4 rectangle formed by dotted lines contains 12 tones.There are 479,001,600 possible 12 tone rows, and there are billions of possible versions of these diagrams.

C

E b

A

GbE

G D

Ab

B

Bb

Db

F C

E b

A

GbE

G D

Ab

B

Bb

Db

F

C

E b

A

GbE

G D

Ab

B

Bb

Db

F

C

E b

A

GbE

G D

Ab

B

Bb

Db

F

C

E

Ab

C

E

Ab

C

E b

A

Gb

C

E b

A

Gb

C

TRIADIC PITCH SPACEThree axis lattice, connecting pitches by Minor Thirds, Major Thirds, and Perfect Fourths.

3rd

Root 5th

Root

3rd

5th

Major Triad Minor TriadSemitones

C

AE bA

F G

BDbGb

Ab

BDb

E

D

F G

E bBb

C

AE bA

F G

BDbGb

BDb

E

D

F G

E bAb

Bb D

Ab

C

AE bA

F G

Bb

BDbGb

Ab

BDbD

EF G

E b

C

AE bA

F G

Bb

BDbGb

BDb

E

D

F G

E bAb

D

Ab

Bb

E

Bb

E

G

Bb

CD

F

AbGb

EA

B

DbE b

E

A

CD

Bb

AbGb

B

E b

F

G

Bb

CD

F

AbGb

EA

B

DbE b

E

A

CD

Bb

AbGb

B

E b

F

G

Bb

CD

F

AbGb

EA

B

DbE b

E

CD

Bb

AbGb

G

Bb

CD

AbGb

E

Db

E

A

CD

Bb

AbGb

B

E b

F

MIXED SYMMETRY PITCH SPACEConnected Whole Tone, Diminished, and Pentatonic shapes. Opposing tonal areas connected in the center by Perfect Fourths.

Name:Axes of Symmetry:

Whole Tone12

Diminished4

Pentatonic1

Ab

C

D

EG

A

B

Db E

A

Gb

DbBb

Db

E b

E b

GbGbAb

Ab

Bb

E bCF

G Bb

Ab

Bb

GbAbE b

Gb

Db

E b

Gb

b Db

E b

G

FE b

GbAb

Bb Db

E b

GbAb

Bb Db

E b

GbAb

Bb Db

E b

GbAb

Bb

G Bb

FC

BbG

E bC

Bb

BA

Gb

C

D

EG

A

A

GbAb

Bb Db

E b

EGC

D

EG

A

A

F

G

GbE b

Gb

C

BbGb

Ab

B

GBb

F

E b

C

C

D

EG

A

E

D

D

CC

A

A

G

C

D

EG

A

C

D

EG

A

E

B

Db DbE

Gb

E

AGbB

Db

A

Gb

GbE b

Db

E b

G

FE b

GbAb

Bb Db

E b

Bb

C

E

D

F

D

E

Gb

G

A

G E

GbAb

Bb Db

E b

B

GbAb

Ab

BbBb

CE b

GBb

G E

AB

A

G

C

Db

C

A

Bb

C

GbAb

Bb

DbB

A

C

A

G

F

E b

Gb

Db

E b

E

D

D

E

C

GGb

C

D

EG

A B

DbE

A

Gb

Gb

Bb Db

E bAb

G Bb

F

E bC

GOLDEN RATIO PITCH SPACEAperiodic Penrose Tiling with Pentatonic pitch assignments (shown in corners).As the number of tiles increases, the ratio of Kite to Dart shapes approaches the Golden Mean.

2:3

2:5

4:3

2:7

3:5

4:5

3:7

8:3

4:7

6:5

5:7

8:5

6:7

8:7

7:8

7:6

5:8

7:5

5:6

7:4

3:8

7:3

5:4

5:3

7:2

3:4

5:2

3:2

POLYPULSEThe 28 possible polypulses created by using subdivisions of the beat 2-8, shown in reciprocal pairs.

Large dots are sounded events, small dots are rests, and arcs show subdivision groupings.(turn the page upside down to reverse orientation)

.˙ œ œ .˙œ œ œ œ œ œ œ

2:7(3.50)

˙ œ jœ ˙ jœ œ ˙3 3

œ œ œ œ œ œ œ œ3:8(2.67)

˙ œ œ ˙œ œ œ œ œ

2:5(2.50)

˙ jœ œ œ œ jœ ˙3 3

œ œ œ œ œ œ œ3:7(2.33)

œ .œ œ œ œ œ œ œ .œ œœ œ œ œ œ œ œ

4:7(1.75)

œ œ jœ œ jœ œ œ3 3

œ œ œ œ œ3:5(1.67)

œ .œ œ œ rœ œ œ rœ œ œ .œ œ5 5 5 5

œ œ œ œ œ œ œ œ5:8(1.60)

POLYPULSEThe 28 possible polypulses created by using subdivisions of the beat 2-8,

showing pulse ratio and size of upper note in relation to lower note.

œ œ œ œœ œ œ2:3

(1.5)

œ œ .œ œ rœ œ rœ œ .œ œ œ5 5 5 5

œ œ œ œ œ œ œ5:7(1.40)

œ jœ œ œ jœ œ3 3

œ œ œ œ3:4(1.33)

œ œ .œ œ œ .œ œ œœ œ œ œ œ4:5

(1.25)

œ rœ œ œ .œ .œ œ œ rœ œ5 5 5 5

œ œ œ œ œ œ5:6(1.20)

œ œ œ œ jœ œ .œ .œ œ jœ œ œ œ œ6 6 6 6 6

œ œ œ œ œ œ œ6:7(1.17)

œ rœ .œ œ œ œ .jœ œ œ .jœ œ œ œ .œ rœ œ

7 7 7 7 7 7

œ œ œ œ œ œ œ œ7:8(1.14)

. .œ œ .œ œ œ œ .œ œ œ .œ œ œ œ .œ œ . .œœ œ œ œ œ œ œ8:7

(.875)

.œ rœ œ rœ jœ œ .jœ .jœ œ jœ rœ œ rœ .œ7 7 7 7 7 7

œ œ œ œ œ œ7:6(.857)

œ œ œ œ jœ .œ .œ jœ œ œ œ œ6 6 6 6 6

œ œ œ œ œ6:5(.833)

œ rœ .œ œ œ .œ rœ œ5 5 5 5

œ œ œ œ5:4(.800)

.œ œ œ œ œ .œœ œ œ4:3

(.750)

œ œ œ .jœ œ rœ œ œ œ œ .jœ œ œ œ7 7 7 7 7

œ œ œ œ œ7:5(.714)

œ jœ jœ œ3 3

œ œ3:2(.667)

œ œ .œ œ œ œ œ œ œ œ œ œ œ .œ œ œœ œ œ œ œ8:5

(.625)

.œ œ œ .œ œ œ .œ5 5 5

œ œ œ5:3(.600)

œ .jœ rœ œ jœ jœ œ rœ .jœ œ7 7 7 7

œ œ œ œ7:4(.571)

.œ .œ œ œ .œ œ œ .œ .œ7 7 7

œ œ œ7:3(.429)

œ œ œ œ œ œ5 5

œ œ5:2(.400)

.œ .œ œ œ .œ .œ œ œ .œ .œœ œ œ8:3

(.375)

œ œ œ œ œ œ œ œ7 7

œ œ7:2(.286)

FOUR BEAT RHYTHMIC MODES

3 (1 - 6)2 (1 - 4) 3 (7 - 12)

EXPLANATION:

The vertical lines show a four beat space.The numbered box shows the beat subdivision.The number beside the box counts the modes below.

Modes are created by grouping subdivisions in two ways:

Short (2 subdivisions, indicated by an upward arc)Long (3 subdivisions, indicated by a downward arc)

Rhythmic modes fill the space entirely, without gaps.

All possible four beat modes are shown for the first four prime number subdivisions of the beat (2,3,5 and 7).

5 (1 - 20) 5 (21 - 40) 5 (41 - 60)

5 (61 - 80) 5 (81 - 100) 5 (101 - 114)

7 (1 - 20) 7 (21 - 40) 7 (41 - 60)

7 (61 - 80) 7 (81 -100) 7 (101 - 120)

7 (121 -140) 7 (141 - 160) 7 (161 - 180)

7 (181 - 200) 7 (201 - 220) 7 (221 - 240)

7 (241 - 260) 7 (261 - 280) 7 (281 - 300)

7 (301 - 320) 7 (321 - 340) 7 (341 - 360)

7 (361 - 380) 7 (381 - 400) 7 (401 - 420)

7 (421 - 440) 7 (441 - 460) 7 (461 - 480)

7 (481 - 500) 7 (501 - 520) 7 (521 - 540)

7 (541 - 560) 7 (561 - 580) 7 (581 - 600)

7 (601 - 620) 7 (621 - 640) 7 (641 - 660)

7 (661 - 680) 7 (681 - 700) 7 (701 - 720)

7 (721 - 740) 7 (741 - 760) 7 (761 - 780)

7 (781 - 800) 7 (801 - 820) 7 (821 - 840)

7 (841 - 860) 7 (861 - 880) 7 (881 - 900)

7 (901 - 920) 7 (921 - 940) 7 (941 - 960)

7 (961 - 980) 7 (981 - 1000) 7 (1001 - 1020)

7 (1021 - 1040) 7 (1041 - 1060) 7 (1061 - 1081)

TABLE OF RHYTHMIC ELEMENTS

= = = pulse of 2

=

(not listed) (not listed) (not listed)(listed)

= = =

(not listed)(listed)

= (not listed)

(listed)

Explanation: Rhythms are shown as a circle to eliminate redundancies and visualize the shape.On the circular path, large nodes are sounded events, and small nodes are rests.One circle represents all rotations - rhythms can start at any point. For example:

All unique rhythmic figures built from lengths of 1, 2 and 3, with number of events from 2 to 7 are listed, with these exceptions:

== or or or or œ œ ‰ ‰ œ œœ ‰ œ

Rhythms that are equivalent to a constant pulse are not listed. For example:

Only one rotation of a rhythm is listed.For example:

Internally repetitive rhythms are not listed.For example:

Rhythms are independent from any subdivision or time signature. For example, this figure played in different beat subdivisions could be:

œ œ ‰ œ œ ‰

œ œ ‰ œ œ ‰ œ œ ‰ œ œ ‰3 3 3 3

œ œ œ œ œ œ œ œ

œ œ œ œ œ œ œ œ œ œ5 5 5

œ œ œ œ œ œ œ œ œ œ œ œ6 6 6

œ œ œ œ œ œ œ œ œ œ œ œ œ œ7 7 7

œœ®œœ®œœ®œœ®œœ®œœ®œœ®œœ®

=

or

or

or

or

or

or

. . . etc.

The table shows the 505 rhythms withlengths of up to 7 events, where each event can be of length 1, 2 or 3. The number of possible rhythms with this system extended to 12 events would be:

Events:23456789101112

Possible Rhythms:381848116312810

2,1845,88016,10444,220

RHYTHMS WITH TWO EVENTSLengths = 3, 4, 5

RHYTHMS WITH THREE EVENTSLengths = 4, 5, 6, 7, 8

RHYTHMS WITH FOUR EVENTSLengths = 5, 6, 7, 8, 9, 10, 11

RHYTHMS WITH FIVE EVENTSLengths = 6, 7, 8, 9, 10, 11, 12, 13, 14

RHYTHMS WITH SIX EVENTSLength = 7, 8, 9, 10, 11

(columns connected by brackets contain rhythms of the same length)

RHYTHMS WITH SIX EVENTSlength = 12

RHYTHMS WITH SIX EVENTSLengths = 13, 14, 15, 16, 17

RHYTHMS WITH SEVEN EVENTSlengths = 8, 9, 10, 11

(columns connected by brackets contain rhythms of the same length)

RHYTHMS WITH SEVEN EVENTSlength = 12





RHYTHMS WITH SEVEN EVENTSLengths = 17, 18, 19, 20

Documents

VISUAL REFERENCE FOR MUSICIANS - Miles Okazaki · 2) Complement (the negative space ... perfect 4th / perfect 5th tritone minor 3rd / major 6th major 3rd / minor 6th 0 pitches 12