## Chords - Frequency Ratios

A chord is three or more different notes played together. Michael Keith (see ref below) computed that for the equal tempered scale there are "351 essentially different chords."

The interval between adjacent notes on the chromatic scale is referred to as a half step. The number of half steps between adjacent notes in common three and four note chords, and their frequency ratios, are shown in the table below.

Chord Half Steps between notes Freq. Ratios
Major 4-3 4:5:6
Minor 3-4 10:12:15
Diminished 3-3 160:192:231
(approx. 20:24:29)
7th 4-3-3 20:25:30:36
Min. 7th 3-4-3 10:12:15:18
Maj. 7th 4-3-4 8:10:12:15

The fundamental beat frequency associated with a chord can be determined by looking at the repeat period - that is, for the frequency ratios given above (which are reduced to the lowest possible integer values), the repeat period for the major chord is 4 times the period of the lowest note in the chord. For the 7th, it is 20 times that of the lowest note. Since f = 1/T, the fundamental beat frequency for the major chord is 1/4th the frequency of the lowest note, and for the 7th, it is 1/20th the frequency of the lowest note. If you listen carefully, you can hear the beat frequency as an additional unplayed note.

What makes a chord sound consonant or dissonant depends upon human physiology and psychology. One "rule" is based on work by Helmholtz and relies on "overlapping harmonics." A nice explanation is contained in the article by Jan Wild listed below. Basically, for each pair of notes in the chord, find the lowest harmonics which match. If it is the 8th or less in every case, the chord is consonant. For example, the major triad has frequency radios of 4:5:6. The harmonics of the lowest note are then 4, 8, 12, 16, 20, 24, etc. and the harmonics of the second are 5, 10, 15, 20, 25, etc. The fifth of the lower matches with the fourth of the upper so this interval should be consonant. (i.e. they are both less than the 9th harmonic). One gets a similar result for 4 and 6, and 5 and 6. Hence, the entire major triad is consonant.

If you try to use the rule of eight and the equal tempered scale, you will have to consider harmonics which "almost match" since none of them, except the octaves, will ever exactly match.

Based on this "rule of 8" the "nice" three note chords which start on middle C are:

plus those where the notes are related by octaves. These are (in order) C-minor, Ab-major, C-major, A-minor, F-minor, and F-major chords.

The rule of 8 is, of course, not an absolute rule but only serves as a guideline.

Chord progressions are the basis of most western music. If a tune stays in one key, then the basic chords are triads starting on each of the different notes of the scale. Since these chords are often expressed in terms of the root of the chord, this fact is not always clear. Many tunes will use just the three chords based on the fundamental (I), the fourth (IV) and the fifth (V). Sometimes a fourth note is added which is a third above the highest note of the triad. This gives a 7th chord. When a minor third is used it can result in the use of a note which is not actually part of the original scale. For example, the C major triad is CEG and a minor third above the G is Bb, which is not part of the C major scale.

Simple Triads for C Major scale
Notes of Chord Name of Chord Name Relative to Root*
CEG C Major I
DFA d minor ii
EGB e minor iii
FAC F Major IV
GBD G Major V
ACE a minor vi
BDF b minor dim. viio

* The roman numeral indicates the starting note of the triad, and upper and lower case signify whether it's a major or minor triad respectively. The b minor triad is B, D, and F#. When played with B, D, and F it is referred to as a "diminished" chord.

Particularly common is the I-IV-V progression. For some simple chord progressions and examples of tunes which use them, see, for example, Olav Torvund's site.

The semitone is always dissonant. Michael Keith concludes that the only 30 of the possible 351 chords have no semitone intervals, and that in fact there are only 12 which are "musically distinct."

Some references:

J. Wild, "The computation behind consonance and dissonance," Interdisciplinary Science Reviews, Vol 27, No. 4, p 299 (2002).

H. Helmholtz, "On the sensation of tone as a physiological basis for the theory of music," translated by A. J. Ellis, (Dover, NY, 1954).

Michael Keith, "From Polychords to Polya: Adventures in Musical Combinatorics," Vinculum Press, Princeton, N.J., 1991, ISBN 0-9630097-0-2.