On the Properties of Original Triads Regarding Dissonance

Yeo Doglas (Singapore, Independent Researcher)

Published in musi.philica.com

Abstract
This idea presents a surprising result that the series of original triads is 50% dissonant, and 50% non-dissonant.

Contents:

(A) Definition of Terms

(B) Main Article

(D) References

(A) In this paper, the following terms are defined:

Original: starting from the origin, which is any arbitrarily defined note.

Triad: any 3-note chord, not necessarily the root, the third, and the fifth, without repetition of notes.

Dissonance: the "unpleasant" sound produced by a chord that has one pair of notes one semitone apart. Eg. CEF, where E & F are one semitone apart.

(B)

This paper will explore the properties of original triads, which are 3-note chords starting from the origin. Original triads are chosen to be studied for this paper because firstly, there is a finite number of such triads (when an origin is fixed). Secondly, by the Principle of Relativity of Music (in another article), by choosing a different origin, all possible triads can be covered.

For convenience, we will represent notes by numbers as such:

Using C natural as the origin, we have:

C=0

C#=1

D=2

D#=3

E=4

F=5

F#=6

G=7

G#=8

A=9

A#=10

B=11

As there is to be no repetition of notes in original triads, we do not have to worry about same notes in different octaves.

Hence, using this notation we can list down all the original triads:

1) (0,1,2) ie. C, C#, D

2) (0,1,3) ie. C, C#, D#

3) (0,1,4) ie. C,C#,E

4) (0,1,5)

5) (0,1,6)

6) (0,1,7)

7) (0,1,8)

8) (0,1,9)

9) (0,1,10)

10) (0,1,11)

11) (0,2,3)

12) (0,2,4)

13) (0,2,5)

14) (0,2,6)

15) (0,2,7)

16) (0,2,8)

17) (0,2,9)

18) (0,2,10)

19) (0,2,11)

20) (0,3,4)

21) (0,3,5)

22) (0,3,6)

23) (0,3,7)

24) (0,3,8)

25) (0,3,9)

26) (0,3,10)

27) (0,3,11)

28) (0,4,5)

29) (0,4,6)

30) (0,4,7)

31) (0,4,8)

32) (0,4,9)

33) (0,4,10)

34) (0,4,11)

35) (0,5,6)

36) (0,5,7)

37) (0,5,8)

38) (0,5,9)

39) (0,5,10)

40) (0,5,11)

41) (0,6,7)

42) (0,6,8)

43) (0,6,9)

44) (0,6,10)

45) (0,6,11)

46) (0,7,8)

47) (0,7,9)

48) (0,7,10)

49) (0,7,11)

50) (0,8,9)

51) (0,8,10)

52) (0,8,11)

53) (0,9,10)

54) (0,9,11)

55) (0,10,11)

A quick check using the Basic Counting Principle will verify that there are indeed 55 original triads.

0 _ _

No. of ways=1 x 11 x 10 x 0.5= 55 (verified)

(Note: the x0.5 factor is to avoid duplication caused by rearrangement of the last two notes in the chords.)

Now, using the defintion of dissonance in (A), we can conclude that there are:

27 dissonant triads: No. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 27, 28, 34, 35, 40, 41, 45, 46, 49, 50, 52, 53, 54, 55

Of which 2 are doubly dissonant (have two pairs of notes one semitone apart): 1 and 55

28 non-dissonant triads: No. 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 36, 37, 38, 39, 42, 43, 44, 47, 48, 51.

Hence, approximately half of the original triads are dissonant (considering that 2 are doubly-dissonant), and the other half are non-dissonant, which is a rather surprising result. (given that the percentages are very close to 50%)

This result can be applied to other triads with another origin, not necessarily C natural. and hence it covers the entire spectrum of possible triads.

1. This result shows that dissonant triads make up a large proportion (approx. 50%) in music, and hence they are not be be neglected. With this result in mind, the music of Stravinsky, Schoenberg and other "atonal" composers begin to make more sense, as their compositions utilise the other half of the often neglected dissonant triads.

2. One disadvantage of the idea of original triads is that they only work for music in which the smallest interval is a semitone. Microtones, such as half a semitone, could not fit into this theory of original triads. I predict that a musical theory involving microtones must involve calculus, to deal with the infinitesimally small microtones that may occur in music of the future.

3. Another disadvantage is of course, that the definition of dissonance in (A) is subjective. However, it must be noted that it is true that whenever two notes one semitone apart are sounded in a chord, their harmonics "clash" to produce an unpleasant sound.

(D) References:

- Essentail Dictionary of Music, Lindsey C. Harnsberger, Alfred Publishing co.