Musical Set Theory

Cluster-free

Index

Harmonious focuses on cluster-free set classes, a subset of 124 of the 336 set classes with three to nine notes. Cluster-free set classes do not contain chromatic clusters, or runs of two (or more) consecutive semitone intervals (i.e. three notes or more). More

Cluster-free scales and chords form the basis of jazz theory and include all perfectly even and nearly even chords, all nearly even scales, and most maximally even scales. (See Evenness & Clusters.)

There are 1,420 OPC-equivalent, cluster-free transpositions or pitch class sets, and 6,720 modes of those transpositions. There are 479 cluster-free OTC-equivalent objects.

Less

Ancohemitonic chromatic-cluser-free

3 Notes

3-12

3-11

3-10

3-9

(027)
sus
Quartal-3

3-8

(026)
X7 no 5
Aug 6

(046)
X7 ♭5 no 3
Aug 6 no ♯4

3-5

(016)
Maj ♯11 no 3

(056)
X7(♯9)noR/no5

3-7

(025)
min 7 no 5

(035)
Maj 6 no R

3-4

(015)
Maj 7 no 5

(045)
Maj 7 no 3

3-6

(024)
Maj add 9 no 5
X9(no3/5)

3-3

(014)
XmM7(no5)

(034)
XM7(♯5)no3

3-2

(013)
Diatonic-3

(023)
Diatonic-3

4 Notes

4-28

4-27

(0258)
min 7 ♭5
min 6

(0368)
Dom 7
Gr. 6

4-25

(0268)
X7(♭5)
Fr. 6

4-26

(0358)
min 7
Maj 6

4-20

4-23

(0257)
X7 sus 4
Quartal-4

4-24

(0248)
X7(♯5)

4-16

(0157)
X13 noR/no5

(0267)
X7sus4(add3/no5)

4-19

(0148)
min Maj 7

(0348)
Aug Maj 7

4-22

(0247)
Maj add 9
X9sus4(no5)

(0357)
Maj 6/9 no 5
Maj 9 ♯11 no 5

4-18

(0147)
X7(♯9)noR

(0367)
XM11(♯5)no3/9

4-17

(0347)
Dom ♯9

4-9

(0167)
Maj 11 ♭5

4-Z15

(0146)
All-Interval
X7♯11 no 3

(0256)
All-Interval
X7♯9 no 5

4-Z29

(0137)
All-Interval
X13 no 5

(0467)
All-Interval
XM♯11

4-14

(0237)
XM7♯11(noR)

(0457)
Maj 9 no 3

4-21

(0246)
Dom 9 no 5

4-8

(0156)
XM7♯11(no3)

4-13

(0136)
min Maj 11

(0356)
X7(♭9)no3

4-12

(0236)
X7(♭9)no5

(0346)
XM9(♯5)no3

4-7

(0145)
Hexatonic-4

4-11

(0135)
Diatonic-4

(0245)
Diatonic-4

4-10

(0235)
Diatonic-4

4-3

(0134)
Acoustic-4

5 Notes

5-35

(02479)
Pentatonic

5-34

(02469)
Dom 9
min ♭13 ♭5

5-32

(01469)
X13(♯9)no♭7

(02569)
X7(♯9)

5-30

(01468)
X9♯11(no3)

(02478)
Aug 7 ♯11

5-33

(02468)
X9(♭5)
X9(♯5)

5-22

(01478)
Harm-M/m-5

5-29

(01368)
X7sus4(add3)
min 13 no 5

(02578)
min 6/9
Xm11♭5 no 9

5-31

(01369)
XM11(♯5)no3

(02369)
X7(♭9)

5-28

(02368)
Dom 7 ♯11
X♯11(♭9)no5

(02568)
X13(♯11)no5

5-20

(01568)
Maj 9 ♯11 no 3

(02378)
XM7♯11

5-21

(01458)
Hexatonic-5

(03478)
Hexatonic-5

5-27

5-26

(02458)
min 9 ♭5

(03468)
XM9(♯5)

5-19

(01367)
X13(♯9)♭5noR

(01467)
X♯11(♭9)no3

5-25

(02358)
Maj 6 ♯11
min 13 no 11

(03568)
Dom 13 no 9

5-24

(01357)
XM9♯11 no 5
Dom 13 no 5

(02467)
Dom 11 no 5
X9♯11(no♭7)

5-Z18

(01457)
Harm-M/m-5

(02367)
Harm-M/m-5

5-23

(02357)
Diatonic-5

(02457)
Diatonic-5

5-Z17

(01348)
XAlt(♯5♭9♯9)

5-16

(01347)
Diminished-5

(03467)
Diminished-5

5-Z12

(01356)
Diatonic-5

5-10

(01346)
XAlt(♭9♯9)

(02356)
X7(♭9♯9)no5
min Maj 13 no 5

6 Notes

6-35

(02468t)
Whole-Tone Scale

6-34

(013579)
Aug 9 ♯11
Alt 7 ♭9

(024689)
Dom 9 ♯11
Alt 7 ♯9

6-30

(013679)
X13(♯9)♭5

(023689)
X♯11(♭9)

6-31

(014579)
Harm-Major-6

(024589)
Harm-Minor-6

6-33

(023579)
Dom 13

(024679)
min 13

6-20

(014589)
Augmented (Hex.)

6-32

(024579)
Quartal-6

6-Z29

(023679)
Harm-M/m-6

6-Z50

(014679)
X13(♯9)
X13♯11(♭9)no♭7

6-Z26

(013578)
Maj 9 ♯11

6-Z28

(013569)
Harm-M/m-6

6-Z49

(013479)
X♯11(♯9)
X13♯11(♭9)no5

6-27

(013469)
XAlt13♯11no3/5

(023569)
X7(♭9♯9)
X13(♭9)

6-Z19

(013478)
Harm-Major-6

(014578)
Harm-Minor-6

6-Z25

(013568)
Diatonic-6

(023578)
Diatonic-6

6-Z24

(013468)
XAlt(no♭7)

(024578)
X7(♯5♭9♯9)
Xm11♭5

6-Z23

(023568)
Dom 13 ♯11
X7(♭5♭9♯9)

6-Z13

(013467)
Diminished-6

7 Notes

7-35

(013568t)
Diatonic

7-34

(013468t)
Acoustic

7-32

(0134689)
Harmonic Minor

(0135689)
Harmonic Major

7-31

(0134679)
Diminished-7

(0235689)
Diminished-7

8 Notes

8-28

(0134679t)
Diminished (Oct.)

By Interval