Annex 4

11. Symmetrical modes and chords

Two study groups are distinguised: the symmetry between chord/scale classes and the symmetrical internal structures in one chord or scale.

11.1 Symmetry between chords

For simplicity, when referring to chords, I mean 'chord classes'.

It is commonly used the term 'inversion' to speak of the symmetrical chord (or mode); but we know that this word also has another meaning to differentiate chords. This duplicity of meanings can cause confusion; in this book whenever the word 'inversion' appears we are referring to the traditional term used in books of harmony to distinguish the different dispositions of the chords according to the note that appears in the bass. Here I will use the term 'symmetric' instead of 'inversion'.

In figure 80 we have some examples of symmetrical chords (symbolically it would be like a chord and its image in a mirror), they are formed by the same intervals in inverse order:

Fig. 80

A characteristic of symmetrical chords is that the symmetry of any inversion of a chord class is at the same time an inversion of the symmetric chord class. So, a chord class has only one symmetrical chord class, despite the notes layout.

Table 9 shows the relation between symmetric chord classes —following the numeration on the Chord/scale class tables (two first numbers)—. The third numeration (third column, from 3-note group) is the correspondence with the Set Theory, specifically with the numbering given by Allen Forte (PC-Set) in his book The Structure of Atonal Music (1973). In brackets there is the real number of different symmetries —shown by an asterisk— (the symmetries that do not have an asterisk are just a repetition of a symmetry already marked —in order to facilitate their searching—). The chord classes with two asterisks are doubly symmetric, they have internal

Set Theory, which enjoys a certain diffusion, could be considered as a dualistic theory carried to its extreme since it considers a chord and its symmetrical one as a «same class» of chord. For example, it establishes an equivalence or parallelism between the major triad and the minor triad, between the dominant seventh chord and the Tristan chord, etc. This equivalence does not fit with the functionalities of the chords in our theory (since the symmetry of harmonics does not appear in nature) nor do I think it could be applicable to tonal music but it can be an interesting system of analysis and a method to compose atonal music, for which it was thought. Because, although it does not come from a natural phenomenon, we do not rule out that the brain is capable of perceiving these relations (although we do not affirm it a priori).

(January 2020): to calculate PC-Set numbers see Chord Analyzer online.

If we consider this new equivalence between chords (equivalence between symmetric chords), a new level of equivalence could be added in figure 73, which we could call N₈ (equivalent chords discarding order between notes, transpositions, inversions and symmetries), and the possible numbers of chords —for each group of notes— are those that appear in parentheses in Table 9, i.e.: 1, 6, 12, 29, 38, 50, 38, 29, 12, 6, 1, respectively, for the groups of 1 to 11 notes.

Curiously, as happens at the N₅ and N₆ levels of equivalence (see Figure 73 of Annex 1), the number of symmetric chord classes (or PC-Sets) is also symmetrical (1, 6, 12, 29, 38, 50, 38, 29, 12, 6, 1), and the same symmetry is obtained in the number of double symmetrical chord classes (1, 6, 5, 15, 10, 20, 15, 5, 6, 1). Among other things this fact is a manifestation that the symmetric chord class of the complementary chord class is the complementary chord class of the symmetric chord class. Understanding as a complementary chord of a chord x the new chord that has (only) all the notes that are missing to the chord x.

1 -note (1)

**1 1

2 notes (6)

**1 1

**2 2

**3 3

**4 4

**5 5

**6 6

3 notes (12)

**1 1 3-1

**2 2 3-6

*3 14 3-5

*4 6 3-4

**5 5 3-12

6 4

**7 7 3-9

*8 12 3-11

*9 17 3-7

*10 18 3-8

**11 11 3-10

12 8

*13 19 3-2

14 3

*15 16 3-3

16 15

17 9

18 10

19 13

4 notes (29)

*1 11 4-16

*2 21 4-5

*3 17 4-19

**4 4 4-7

*5 27 4-4

*6 25 4-14

**7 7 4-1

**8 8 4-6

**9 9 4-23

*10 23 4-22

11 1

**12 12 4-21

*14 37 4-13

*15 31 4-27

*16 32 4-Z29

17 3

18 13

**19 19 4-24

*20 28 4-2

21 2

*22 43 4-12

23 10

**24 24 4-20

25 6

**26 26 4-8

27 5

28 20

*29 39 4-18

**30 30 4-9

31 15

32 16

**33 33 4-17

**34 34 4-3

**35 35 4-26

**36 36 4-10

37 14

*38 41 4-Z15

39 29

**40 40 4-28

41 38

**42 42 4-25

43 22

5 notes (38)

*1 49 5-20

*2 9 5-6

**3 3 5-15

*4 37 5-21

*5 17 5-Z18

**6 6 5-Z17

*7 54 5-Z38

**8 8 5-Z37

9 2

**10 10 5-22

*11 47 5-27

*12 24 5-29

*13 18 5-4

*14 45 5-2

*15 35 5-5

*16 53 5-Z36

18 13

*19 63 5-31

*20 41 5-3

**21 21 5-35

*22 48 5-23

*23 25 5-24

24 12

25 23

**26 26 5-34

*27 39 5-30

**28 28 5-33

*29 38 5-26

*30 60 5-25

*31 62 5-10

*32 66 5-28

**33 33 5-1

*34 43 5-9

35 15

*36 52 5-14

37 4

38 29

39 27

*40 46 5-13

41 20

**42 42 5-8

43 34

*44 59 5-16

45 14

46 40

47 11

48 22

49 1

**50 50 5-Z12

*51 64 5-11

52 36

53 16

54 7

*55 58 5-7

*56 61 5-32

*57 65 5-19

58 55

59 44

60 30

61 56

62 31

63 19

64 51

65 57

66 32

*1 5 6-Z19

*2 79 6-Z43

*3 4 6-Z44

4 3

5 1

**6 6 6-Z29

*7 37 6-31

*8 13 6-5

*9 46 6-Z17

*10 68 6-27

*11 76 6-30

**12 12 6-Z13

13 8

*14 18 6-Z11

**15 15 6-1

*16 66 6-Z3

**17 17 6-Z6

18 14

*19 21 6-33

*20 24 6-Z24

21 19

*22 23 6-34

23 22

24 20

**25 25 6-Z23

*26 27 6-9

27 26

*28 55 6-Z12

*29 30 6-22

30 29

**31 31 6-Z4

*32 33 6-2

33 32

*34 35 6-Z10

35 34

**36 36 6-Z45

37 7

*38 48 6-15

*39 50 6-21

**40 40 6-Z49

*41 60 6-14

**42 42 6-Z28

*43 47 6-Z39

**44 44 6-Z37

**45 45 6-Z48

46 9

47 43

48 38

50 39

**51 51 6-35

*52 65 6-Z25

*53 73 6-Z46

*54 59 6-18

55 28

**56 56 6-Z50

**57 57 6-32

*58 67 6-Z40

59 54

60 41

**61 61 6-20

**62 62 6-Z38

**63 63 6-Z26

64 49

65 52

66 16

67 58

68 10

**69 69 6-8

*70 72 6-Z47

*71 74 6-Z36

72 70

73 53

74 71

**75 75 6-Z42

76 11

*77 80 6-Z41

**78 78 6-7

79 2

80 77

7 notes (38)

**1 1 7-35

**2 2 7-34

*3 4 7-32

4 3

*5 8 7-30

**6 6 7-33

**7 7 7-22

8 5

*9 24 7-14

*10 47 7-11

*11 40 7-24

*12 19 7-25

*13 60 7-23

*14 48 7-29

**15 15 7-Z12

*17 32 7-9

*18 27 7-7

19 12

20 16

*21 42 7-27

*22 64 7-Z38

**23 23 7-Z37

24 9

*25 33 7-13

*26 41 7-Z36

27 18

*28 57 7-21

*29 30 7-6

30 29

*31 45 7-20

32 17

33 25

*34 55 7-2

*35 56 7-4

*36 43 7-5

*37 61 7-26

*38 66 7-28

**39 39 7-15

40 11

41 26

42 21

43 36

*44 59 7-Z18

45 31

*46 53 7-16

47 10

48 14

**49 49 7-8

*50 51 7-31

51 50

*52 63 7-10

53 46

**54 54 7-1

55 34

56 35

57 28

*58 65 7-3

59 44

60 13

61 37

**62 62 7-Z17

63 52

64 22

66 38

8 notes (29)

*1 10 8-22

**2 2 8-21

*3 4 8-27

4 3

*5 22 8-Z15

*6 36 8-Z29

*7 20 8-18

**8 8 8-9

*9 21 8-13

10 1

*11 34 8-16

**12 12 8-6

*13 28 8-11

*14 32 8-2

**15 15 8-8

**16 16 8-3

**17 17 8-10

**18 18 8-26

**19 19 8-17

20 7

21 9

22 5

**23 23 8-23

*24 41 8-4

*25 39 8-14

**26 26 8-24

*27 31 8-19

28 13

*29 42 8-12

**30 30 8-25

31 27

32 14

*33 35 8-5

34 11

35 33

36 6

**37 37 8-20

**38 38 8-7

39 25

**40 40 8-1

41 24

42 29

**43 43 8-28

9 notes (12)

**1 1 9-1

*2 6 9-8

**3 3 9-6

*4 12 9-11

*5 15 9-7

6 2

**7 7 9-10

*8 18 6-2

*9 11 9-5

*10 16 9-3

11 9

12 4

*13 17 9-4

**14 14 9-9

15 5

16 10

17 13

18 8

**19 19 9-12

10 notes (6)

**1 1

**2 2

**3 3

**4 4

**5 5

**6 6

11 notes (1)

**1 1

12 notes (1)

**1 1

There are four different kinds of symmetry in an octave (a, b, c and d in Figure 81): when the symmetric core goes through one or two notes and when the symmetry contemplates (or not) the closing of the octave. Each symmetry generates in another mode of the same scale another symmetry that can be of the same class or not. The numbers —in Tables 10 and 11— show the number of notes on the scale and, separated by a hyphen, there is the numbered classified scale class according to the Chord/scale Tables.

Fig. 81

On Table 10, there are represented all the modes with symmetry from types (a) and (b) of Figure 22, transposed to C, that is, in the boundary on an octave. On table 11, there are all the modes till 7 notes with symmetry of type (c).¹

Fig. 82

Bibliography