Developing a Theory of Chords from Harmonic Principles...

Let us say a consonant chord is a set of notes characterized by

1. some notes share one or more low order harmonics
2. there are no strongly dissonant clashes between the other pairs of low order harmonics.

We can form 4 types of consonant chords:

1. A root tone and other tones matching harmonics of the root by octave equivalence ( Like C major, shown below).
2. All tones sharing a common harmonic (Like C minor).
3. Notes matching harmonics not by octave equivalence (like F = c-f-a).
4. Matching some harmonics by octave equivalence and some not. (This also characterizes a minor triad!)

There seem to be two types of chords in practice: those which are based on harmonic principles (e.g. C major and C minor) and those which are based on chosing a pattern of notes on the keyboard (e.g. C-Augmented or C-Diminished-6). Here is a list of C chords, drawn in circle of 5ths diagrams and a catalog of some of their properties. In the present discussion, a companion to the one just linked to, the goal is to study what chords are possible from simple acoustical principles with harmonics.

The figure that I conceived of below is useful for graphically showing the relationships between the harmonics; I have fancifully called it the Harmonic Tower:

The vertical axis is marked by the harmonics of a note of a fixed frequency, call it c. The horizontal axis is marked by a listing of all fractions between 1 and 2 in terms of increasing denominator (luckily they can quickly stop being listed because they become irrelevant, either by only matching too high a harmonic of c, or by being too dissonant against the 1st harmonic of c, the fundamental). It makes sense to list the fractions along the horizontal axis like this because they correspond to a string vibrating as a whole (1), as halves (3/2), as a segment of one third its length (4/3),and a two thirds segment (5/3), and so forth. This is the way that harmonic bodies vibrate!

The green strip, of width about a minor 3rd, roughly marks the so-called critical bandwidth . (If you follow the link, you will see the graph of a bump function representing dissonance). Imagine sliding the strip up and down to compare the harmonics of 2 notes, lining its bottom edge up with one of the harmonic horizontal tic marks in the column of the note that you are interested in. If the harmonic marking in another column lies inside the strip (as opposed to on the edge), then the two notes will produce unpleasant beats when sounded together and thus be dissonant.

Remember that consonant chords are characterized by matching some harmonics. As we shall see, some chords match on some harmonics very nicely, but almost match on others, which means these near misses lie in the critial bandwidth and the pleasing quality of the matches is nullified by the unpleasantness of the clashes; thus the chord cannot be called consonant.

• Look at the notes of C major (c-g-e) and notice how they match the 3rd and 5th harmonics of c by octave equivalence (Yellow boxes on diagram).
• Look at the notes of C minor (c-g-e^b) all having a match on the same line as the 6th harmonic of c.
• Observe the triad C-sus (c-g-f), and how it is poised to be a wonderfully strong consonant chord (the harmonic matching is so low) but this is offset by its higher dissonance (due to the clash of the 1st harmonics of g and f).

Let us try to identify all the combinations and see what chords we get: We require that the note c always be present, thus we always have the harmonic 1. We extended the list of harmonics all the way up to the 9th, so that we could get b^b = 9/5 and d = 9/8. Note that we are missing b = 15/8 , because it is just too implausible to go that high and match the 15th harmonic of c with the 8th harmonic of b (high harmonics are typically inaudible).

Indexes	Matching Harmonics	Note Names	Chord Name	Fractions	Dissonance
(0,1)	(1, 3* )	c - g	5th	1 3/2	0.1595
(0,2)	(1, 4 )	c - f	4th	1 4/3	0.3669
(0,3)	(1, 5 )	c - a	6th	1 5/3	0.2348
(0,4)	(1, 5* )	c - e	major 3rd	1 5/4	0.4796
(0,5)	(1, 7* )	c - 7/4		1 7/4	0.3956
(0,6)	(1, 6 )	c - eb	minor 3rd	1 6/5	0.4798
(0,7)	(1, 7 )	c - 7/5		1 7/5	0.5524
(0,8)	(1, 8 )	c - ab	minor 6th	1 8/5	0.5212
(0,9)	(1, 9 )	c - bb		1 9/5	0.4037
(0,10)	(1, 7 )	c - 7/6		1 7/6	0.6648
(0,11)	(1, 8 )	c - 8/7		1 8/7	0.6813
(0,12)	(1, 9 )	c - 9/7		1 9/7	0.6249
(0,13)	(1, 9* )	c - d		1 9/8	0.7065

Duplicates are omitted (for example (1,3) and (1,6) are both c-g). Asterisks symbolize harmonics that are matched by an octave equivalent note and are thus more strongly matched. In the theory espoused by Terhardt (see References ) when 2 or more notes are sounded together there are two (sometimes competing) components of harmony:

• The amount of sensory dissonance from beating harmonics
• The amount of identification of a root tone, via the auditory phenomena of virtual pitch

Some observations and summary of results and open questions:

1. We certainly get some expected chords C and Cm, F and Fm, and Am.
2. We notably don't get G, because we don't get the note b (its harmonic relationship to c is just too distant).
3. Curiously, A-dim = c-e^b-a shows up and with quite a low dissonance value.
4. Is C-aug = c-e-a^b? If so, it is in the list, and we have constructed the augmented and diminished chords from harmonic principles! But is C-aug = c-e-g^# really?
5. We also get A^b major, which is a little suprising.
6. Evidently, good consonant chords can be made using 7/4 such as c-f-7/4 and c-e^b-7/4. This may have implications for what I call the propagation problem below.
7. The chord F major = c-f-a is made with no octave equivalence matching...to c. It certainly does match f's harmonics by octave equivalence. Is that always the case for triads made by all non-O.E. matching? In other words, is what I called "case 3" at the beginning of this page actually not a case at all?
8. I am probabaly not worrying enough about inversions like c-e-g, e-g-c, g-c-e not having the same dissonance
9. The dissonance values are still not "right", since for example F minor = c-f-a^b has a higher dissonance than c-f-b^b on the line below it.
10. The "Propagation Problem": Western music is based on the major chord. What other chords could form scales in a similar manner and possibly allow similiar harmonic functions, like cadences? This would first mean solving what I call the propagation problem: How to start from the C major triad and generate the C major scale and the other 11 standard keys in some kind of a way that is mathematical and optimal. I attempt to do this here, but it is not complete.

Theorem If you create a scale, say the C major scale, by major chords linked at the 3rd harmonic, like F - C - G, then you automatically have created minor chords as well. Likewise, 3 linked minor chords create major chords.

4-Chords

Indexes	Matching Harmonics	Note Names	Chord Name	Fractions	Dissonance
(0,1,2,3)	(1, 3*, 4, 5)	c - f - g - a	F6	1 - 4/3 - 3/2 - 5/3	0.5637
(0,1,2,5)	(1, 3*, 4, 7*)	c - f - g - 7/4		1 - 4/3 - 3/2 - 7/4	0.5637
(0,1,2,9)	(1, 3*, 4, 9)	c - f - g - bb	C7 sus	1 - 4/3 - 3/2 - 9/5	0.5637
(0,1,3,4)	(1, 3*, 5, 5*)	c - e - g - a	C6	1 - 5/4 - 3/2 - 5/3	0.5535
(0,1,3,6)	(1, 3*, 5, 6)	c - eb - g - a	Cm6	1 - 6/5 - 3/2 - 5/3	0.5535
(0,1,4,5)	(1, 3*, 5*, 7*)	c - e - g - 7/4	Tetrad	1 - 5/4 - 3/2 - 7/4	0.5020
(0,1,4,9)	(1, 3*, 5*, 9)	c - e - g - bb	C7F	1 - 5/4 - 3/2 - 9/5	0.5082
(0,1,5,6)	(1, 3*, 7*, 6)	c - eb - g - 7/4		1 - 6/5 - 3/2 - 7/4	0.5020
(0,1,6,9)	(1, 3*, 6, 9)	c - eb - g - bb	Cm7	1 - 6/5 - 3/2 - 9/5	0.4798
(0,2,8,9)	(1, 4, 8, 9)	c - f - ab - bb		1 - 4/3 - 8/5 - 9/5	0.5212
(0,3,6,7)	(1, 5, 6, 7)	c - eb - 7/5 - a		1 - 6/5 - 7/5 - 5/3	0.5845
(0,4,8,9)	(1, 5*, 8, 9)	c - e - ab - bb		1 - 5/4 - 8/5 - 9/5	0.5643
(0,5,6,7)	(1, 7*, 6, 7)	c - eb - 7/5 - 7/4		1 - 6/5 - 7/5 - 7/4	0.5845
(0,6,7,8)	(1, 6, 7, 8)	c - eb - 7/5 - ab		1 - 6/5 - 7/5 - 8/5	0.5845
(0,6,7,9)	(1, 6, 7, 9)	c - eb - 7/5 - bb		1 - 6/5 - 7/5 - 9/5	0.5845
(0,6,8,9)	(1, 6, 8, 9)	c - eb - ab - bb		1 - 6/5 - 8/5 - 9/5	0.5212
(0,7,8,9)	(1, 7, 8, 9)	c - 7/5 - ab - bb		1 - 7/5 - 8/5 - 9/5	0.5524

1. The table for the 4 note combinations is so big (286 rows) that it is on its own page: 4 note chords table.
2. One chord which is notably missing is C-maj-7 = c-e-g-b and this is again because we don't generate the note b with our scheme of low order harmonics.

Perl source code to generate these tables.