On the Triangular Properties of Musical Chords

Yeo Doglas (Singapore, Independent Researcher)

Published in musi.philica.com

Observation
Firstly, I will assume that there are 4 kinds of triangles:
1) Right-angled triangles
2) Isosceles Triangles
3) Equilateral Triangles
4) Other triangles (Scalene)

My observation is that Musical Chords fall into one of the above categories, and hence can be compared to triangles. I believe this links music to art/design.

Without scientific proof on the harmonics of different chords, I will now list down the chords belonging to the above triangles.

1) Right-angled Chord.
Example: C major.
The C major 4 note chord doubling the root is CEGC’. One will notice that there are 4 semitones between C and E, 3 between E and G, and 5 between G and C’.
I would write CEGC’ = (3,4,5).

One can see that 3,4,5 is a Pythagorean triple. (3*3+4*4=5*5). Hence, C major is a Right-angled Chord. So is C minor. In general, all perfect major chords and minor chords are right-angled in nature.

2) Isosceles Chords
An example of an isosceles chord is G7=(4,3,3).

3) Equilateral Chords
An example of one is CEbF#A=(3,3,3)

4) Scalene Chords.
There are many of these chords.

Now, looking only at groups 1, 2, and 3. (ignoring 4 because it is too big a group) I would hypothesise, that 1, 2 and 3 are in increasing magnitude of dissonance.
That is to say, right-angled chords sound the best, followed by isosceles, and then equilateral chords.

This may prove that Man prefers “artificial music” compared to natural music which includes bird song, wind, etc. This is because right-angled triangles are rather man-made, in fact discovered by Pythagoras. While isosceles and Equilateral triangles can be said to be more ‘natural’.

This may also show why Man is using synthesisers, and electric guitars nowadays, which supports the above points.

(This observation refers only to 4-note chords, in SATB form, for soprano, alto, tenor, bass)

Peer-review ratings as of 17:30:52 on 17th Dec 2017 (from 1 review, where a score of 100 is average):
Originality = 100.00, importance = 25.00, overall quality = 50.00

Published on Wednesday 9th August, 2006 at 03:51:52.

Peer review added 1st October, 2006 at 23:34:19

One notices that the “right-angled chord” C-F#-D-C has intervals 6-8-10, though the chord is not one most would consider consonant. This implies that being a “right-angled chord” is not a sufficient condition for consonance.

One also notices that the “equilateral chord” consisting of three intervals of 12 half-steps (i.e C-C-C-C) is a quite consonant chord.

This suggests that this means of categorizing chords doesn’t make useful predictions regarding dissonance. Further, as you noted, all major and minor chords are “right-angled” by definition, suggesting you have replaced two useful categories (major and minor chords) with one subsuming category that is less useful.

Constructive criticism: One should always consider the utility of a particular scheme of categorization. As your goal seems to be to provide a way of determining the relative consonance/dissonance of chords, you should consider all members of the category to see if it does indeed capture the relevant details.

Author comment added 2nd October, 2006 at 10:53:33

Thank you very much for your objective review.

I admit that my method of classification has some inexplicable exceptions, and I will try to reconcile that.

I am now working on a theory on 3-note chords, instead of 4 note chords, as I hypothesise that 3-note chords may be easier to analyse than 4-note chords.

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