A Rule-based System for Tuning
Chord Progressions

by Robert Asmussen, PhD

Online Examples

• Heuristics for Tuning Chords
• Detailed Heuristics

Heuristics for Tuning Chords

Although the following heuristics might possibly account for the majority of chords encountered within a clearly defined key area, they are only the beginning to a much broader picture of tonality. It is quite possible that every chord the human auditory system is capable of identifying as a tonal entity can be accounted for by recursively applying only a fairly limited number of such rules.

Detailed Heuristics

1. Tonic (I) major triad? If yes,
   1. The frequency of the root will be an octave equivalent of 2.
   2. The frequency of the third will be an octave equivalent of 5.
   3. The frequency of the fifth will be an octave equivalent of 3.
2. Tonic (i) minor triad? If yes,
   1. The frequency of the root will be an octave equivalent of 2.
   2. The frequency of the third will be an octave equivalent of 3/5.
   3. The frequency of the fifth will be an octave equivalent of 3.
3. Tonic major (I⁷) chord? If yes,
   1. The frequency of the root will be an octave equivalent of 2.
   2. The frequency of the third will be an octave equivalent of 5.
   3. The frequency of the fifth will be an octave equivalent of 3.
   4. The frequency of the seventh will be an octave equivalent of 15.
4. Tonic minor (i⁷) chord? If yes,
   1. The frequency of the root will be an octave equivalent of 2.
   2. The frequency of the third will be an octave equivalent of 3/5.
   3. The frequency of the fifth will be an octave equivalent of 3.
   4. The frequency of the seventh will be an octave equivalent of 9/5.
5. Supertonic (ii) minor triad? If yes,
   1. The frequency of the root will be an octave equivalent of 5/9.
   2. The frequency of the third will be an octave equivalent of 2/3.
   3. The frequency of the fifth will be an octave equivalent of 5/3.
6. Supertonic ii⁷ minor seventh chord? If yes,
   1. The frequency of the root will be an octave equivalent of 5/9.
   2. The frequency of the third will be an octave equivalent of 2/3.
   3. The frequency of the fifth will be an octave equivalent of 5/3.
   4. The frequency of the seventh will be an octave equivalent of 2.
7. Supertonic half-diminished ii⁷ chord? If yes,
   1. The frequency of the root will be an octave equivalent of 9/2.
   2. The frequency of the third will be an octave equivalent of 2/3.
   3. The frequency of the fifth will be an octave equivalent of 2/5.
   4. The frequency of the seventh will be an octave equivalent of 2.
8. Mediant (III/i) major triad? If yes,
   1. The frequency of the root will be an octave equivalent of 3/5.
   2. The frequency of the third will be an octave equivalent of 3.
   3. The frequency of the fifth will be an octave equivalent of 9/5.
9. Mediant (iii/I) minor triad? If yes,
   1. The frequency of the root will be an octave equivalent of 5.
   2. The frequency of the third will be an octave equivalent of 3.
   3. The frequency of the fifth will be an octave equivalent of 15.
10. Mediant (iii⁷/I) minor seventh chord? If yes,
    1. The frequency of the root will be an octave equivalent of 5.
    2. The frequency of the third will be an octave equivalent of 3.
    3. The frequency of the fifth will be an octave equivalent of 15.
    4. The frequency of the seventh will be an octave equivalent of 9.
11. Subdominant (IV) major triad? If yes,
    1. The frequency of the root will be an octave equivalent of 2/3 multiplied by the frequency of the root of the chord to which it is a subdominant.
    2. The frequency of the third will be an octave equivalent of 5/3 multiplied by the frequency of the root of the chord to which it is a subdominant.
    3. The frequency of the fifth will be an octave equivalent of 2 multiplied by the frequency of the root of the chord to which it is a subdominant.
12. IV⁷ major seventh chord? If yes,
    1. The frequency of the root will be an octave equivalent of 2/3 multiplied by the frequency of the root of the chord to which it is a subdominant.
    2. The frequency of the third will be an octave equivalent of 5/3 multiplied by the frequency of the root of the chord to which it is a subdominant.
    3. The frequency of the fifth will be an octave equivalent of 2 multiplied by the frequency of the root of the chord to which it is a subdominant.
    4. The frequency of the seventh will be an octave equivalent of 5 multiplied by the frequency of the root of the chord to which it is a subdominant.
13. Subdominant iv minor triad? If yes,
    1. The frequency of the root will be an octave equivalent of 2/3 multiplied by the frequency of the root of the chord to which it is a subdominant.
    2. The frequency of the third will be an octave equivalent of 2/5 multiplied by the frequency of the root of the chord to which it is a subdominant.
    3. The frequency of the fifth will be an octave equivalent of 2 multiplied by the frequency of the root of the chord to which it is a subdominant.
14. iv⁷ minor seventh chord? If yes,
    1. The frequency of the root will be an octave equivalent of 2/3 multiplied by the frequency of the root of the chord to which it is a subdominant.
    2. The frequency of the third will be an octave equivalent of 2/5 multiplied by the frequency of the root of the chord to which it is a subdominant.
    3. The frequency of the fifth will be an octave equivalent of 2 multiplied by the frequency of the root of the chord to which it is a subdominant.
    4. The frequency of the seventh will be an octave equivalent of 3/5 multiplied by the frequency of the root of the chord to which it is a subdominant.
15. Dominant (V) triad? If yes,
    1. The frequency of the root will be an octave equivalent of 3 multiplied by the frequency of the root of the chord to which it is a dominant.
    2. The frequency of the third will be an octave equivalent of 15 multiplied by the frequency of the root of the chord to which it is a dominant.
    3. The frequency of the fifth will be an octave equivalent of 9 multiplied by the frequency of the root of the chord to which it is a dominant.
16. Dominant seventh (V⁷) chord? If yes,
    1. The frequency of the root will be an octave equivalent of 3 multiplied by the frequency of the root of the chord to which it is a dominant.
    2. The frequency of the third will be an octave equivalent of 15 multiplied by the frequency of the root of the chord to which it is a dominant.
    3. The frequency of the fifth will be an octave equivalent of 9 multiplied by the frequency of the root of the chord to which it is a dominant.
    4. The frequency of the seventh will be an octave equivalent of 2/3 multiplied by the frequency of the root of the chord to which it is a dominant.
17. Submediant (VI/i) major triad? If yes,
    1. The frequency of the root will be an octave equivalent of 2/5 multiplied by the frequency of the root of the chord to which it is a submediant.
    2. The frequency of the third will be an octave equivalent of 2/1 multiplied by the frequency of the root of the chord to which it is a submediant.
    3. The frequency of the fifth will be an octave equivalent of 3/5 multiplied by the frequency of the root of the chord to which it is a submediant.
18. Submediant (vi/I) minor triad? If yes,
    1. The frequency of the root will be an octave equivalent of 5/3 multiplied by the frequency of the root of the chord to which it is a submediant.
    2. The frequency of the third will be an octave equivalent of 2 multiplied by the frequency of the root of the chord to which it is a submediant.
    3. The frequency of the fifth will be an octave equivalent of 5 multiplied by the frequency of the root of the chord to which it is a submediant.
19. Submediant (vi⁷) minor seventh chord? If yes,
    1. The frequency of the root will be an octave equivalent of 5/3 multiplied by the frequency of the root of the chord to which it is a submediant.
    2. The frequency of the third will be an octave equivalent of 2 multiplied by the frequency of the root of the chord to which it is a submediant.
    3. The frequency of the fifth will be an octave equivalent of 5 multiplied by the frequency of the root of the chord to which it is a submediant.
    4. The frequency of the seventh will be an octave equivalent of 3 multiplied by the frequency of the root of the chord to which it is a submediant.
20. Diminished leading viio triad? If yes,
    1. The frequency of the root will be an octave equivalent of 15 multiplied by the frequency of the root of the chord to which it is a leading viio.
    2. The frequency of the third will be an octave equivalent of 9 multiplied by the frequency of the root of the chord to which it is a leading viio.
    3. The frequency of the fifth will be an octave equivalent of 2/3 multiplied by the frequency of the root of the chord to which it is a leading viio.
21. Half-diminished vii⁷ chord? If yes,
    1. CONTRADICTION.
22. For any octave equivalent of any perfect fifth within a single sonority, the lower tone will be related to the upper as an octave equivalent of 2:3.
23. For any octave equivalent for any perfect fourth within a single sonority, the frequency of the lower tone will be related to the frequency of the upper tone as 3:4.
24. For any octave equivalent of any major third within a single sonority, the lower tone will be related to the upper one as an octave equivalent of 4:5.
25. For any octave equivalent of a minor third serving as the third and fifth of a major triad, the third and fifth will be tuned, relative to each other, as octave equivalents of 5 and 3.
26. For any octave equivalent of a minor third serving as the root and third of a minor triad, the root and third will be tuned, relative to each other, as octave equivalents of 5 and 3.
27. For any octave equivalent of a minor third serving as the fifth and seventh of a dominant seventh chord, the fifth and seventh will be tuned, relative to each other, as octave equivalents of 27 and 32.
28. For any octave equivalent of any minor second within a single sonority, the lower tone will be related to the upper as an octave equivalent of 15:16.
29. For any octave equivalent of any major seventh within a single sonority, the lower tone will be related to the upper as an octave equivalent of 8:15.

<END OF HEURISTICS>

Robert Asmussen
2008