A Rule-based System for Tuning
Chord Progressions

by Robert Asmussen, PhD

Online Examples

Chord Progression Example

The traditionally notated score below illustrates the familiar chord progression I-vi-ii-V-I. Beneath each note is a fraction representing the frequency relative to tonic. To obtain the frequencies in Hertz, simply multiply each such fraction by 65.406, the octave equivalent of C2. The rules, referenced in the comments of the Csound Score below, by which to obtain the fractions representing relative frequencies are given in Heuristics for Tuning Chords.

 

Traditionally Notated Score

chord_prog_pluck.MP3 (250KB)

 

Notice that the low D in the bass of measure one, beat three, is tuned as 20/9. In the same measure but in the treble and on beat four, there is another D tuned as 9/1, which is not an octave equivalent of 20/9. Such redefined notes are known as "mutable tones". Mutable tones do not signify a weakness in just intonation; on the contrary, they indicate a need for more than twelve tones, even in a simple and common two-chord progression such as ii-V.

It may seem unnatural to have two different tunings for octave equivalents of D's in adjacent chords; however, it is absolutely necessary in this particular passage when the following conditions are assumed:

  1. The C is a common tone between the first two triads (C major and A minor).
  2. The A is a common tone between the second two triads (A minor and D minor).
  3. The G in the fourth triad (G major) is related to tonic C as an octave equivalent of 3:1.
  4. The relative frequencies within a major triad are always octave equivalents of 1, 3 and 5.
  5. The relative frequencies within a minor chord are always octave equivalents of 3, 5 and 15.

To find freely available Csound software and tutorials, the Csound Front Page is a good place to start.

 

Csound Orchestra

sr = 44100
kr = 441
ksmps = 100
nchnls = 1

instr 1
a1 pluck 9000, (p5/p6)*65.406, (p5/p6)*65.406, 0, 1, 0, 0
out a1
endin

instr 2
a1 pluck 9000, (p5/p6)*65.406, (p5/p6)*65.406, 0, 1, 0, 0
out a1
endin

instr 3
a1 pluck 9000, (p5/p6)*65.406, (p5/p6)*65.406, 0, 1, 0, 0
out a1
endin

instr 4
a1 pluck 9000, (p5/p6)*65.406, (p5/p6)*65.406, 0, 1, 0, 0
out a1
endin

 

Csound Score

; "Chord Progression Example"
; Key: C Major
; 1/1 = C2 (C2 = 65.406 cps)
t 0 30 ; tempo 30 beats per minute

; Measure 1 Beat 1 Chord Type: I
i1 0 1 0 10 1 ; (E5, Rule: 1.b)
i2 0 1 0 6 1 ; (G4, Rule: 1.c)
i3 0 1 0 6 1 ; (G4, Rule: 1.c)
i4 0 1 0 2 1 ; (C3, Rule: 1.a)
; Measure 1 Beat 2 Chord Type: vi
i1 1 1 0 8 1 ; (C5, Rule: 18.b)
i2 1 1 0 20 3 ; (A4, Rule: 18.a)
i3 1 1 0 5 1 ; (E4, Rule: 18.c)
i4 1 1 0 10 3 ; (A3, Rule: 18.a)
; Measure 1 Beat 3 Chord Type: ii
i1 2 1 0 32 3 ; (F5, Rule: 5.b)
i2 2 1 0 20 3 ; (A4, Rule: 5.c)
i3 2 1 0 20 3 ; (A4, Rule: 5.c)
i4 2 1 0 20 9 ; (D3, Rule: 5.a)
; Measure 1 Beat 4 Chord Type: V
i1 3 1 0 9 1 ; (D5, Rule: 15.c)
i2 3 1 0 15 2 ; (B4, Rule: 15.b)
i3 3 1 0 6 1 ; (G4, Rule: 15.a)
i4 3 1 0 3 1 ; (G3, Rule: 15.a)
; Measure 2 Beat 1 Chord Type: I
i1 4 4 0 10 1 ; (E5, Rule: 1.b)
i2 4 4 0 8 1 ; (C5, Rule: 1.a)
i3 4 4 0 6 1 ; (G4, Rule: 1.c)
i4 4 4 0 2 1 ; (C3, Rule: 1.a)
e


Robert Asmussen
 2008