Ranking dominant chord alterations by dissonance

Question

In a jazz chord progression, there are many ways to alter the extensions and notes in a dominant chord. Reasons for altering dominant chords are to add harmonic variety, to construct elegant and efficient voice leading schemes, and to 'fine-tune' levels of dissonance for expressive purposes. This question is about the last purpose: tuning dissonance.

To exemplify the diversity of alterations to dominant chords, instead of playing tones from the mixolydian mode of the major scale, one could play tones from either the altered scale (7th mode of the melodic minor scale), the lydian dominant scale (4th mode of the melodic minor scale), the mixolydian b6 scale ( 5th mode of the melodic minor scale), the half-whole octatonic scale, modes from harmonic major and minor scales, and perhaps many other scales with dominant 7th chords.

Is there a theory for ranking these kinds of altered dominant chords by dissonance?

That is, can we construct an ordering or ranking like:
least dissonant > mixolydian > lydian dominant > altered > half-whole octatonic > most dissonant, and if so, what is the order?

If the answer is "no", what is a better way of thinking about the network of alterations to dominant chords?

Answer 1

An objective ranking from most dissonant to least dissonant probably doesn't exist. That said, there are two principles that are pretty widely accepted:

1. Many people find the diminished scale to be extremely dissonant (perhaps the most dissonant).
2. The more alterations you add to the V7 chord, the more dissonant it's going to sound if you're resolving to a major I chord.

So what else can we use, then, when picking and choosing alterations for the dominant 7th chord? I think Tim's comment is on point: the surrounding harmonies will often dictate which alterations make sense. For example, imagine we have a ii-V7-i, and the i chord is Cmin6, which contains an A♮. For continuity, we could make the V7 chord G^{7(♮9,♭13)}, which will similarly contain the A♮. The harmonic function can also be important. For example, I have this gut feeling that lydian dominant is much more frequently used over a II7 chord (à la "The Girl from Ipanema") or maybe over a I7 chord than a V7 chord. By contrast, alterations to the 5, 13, and 9 are much more common when the dominant 7th chord has a V7 function. However, this isn't a hard-and-fast rule. In Latin music, it's fairly common to see the progression II^{7(♭9, ♭13)}-V7^{7(♯9, ♭13)}-i⁷ (à la Armando's Rhumba by Chick Corea).

Another guiding principle is to look at the melody. This is probably the most common factor for determining what alterations are appropriate. If we're dealing with a song like Pent Up House, we see that the melody dictates a ♯5 (and a ♮9 if we want continuity with the E from the prior bar):

And as David Bowling points out in a comment, voice leading can be a crucial part of the decision process. For example, when a comping instrument is reading a lead sheet and sees a G7♭9 chord, they'll immediately think that the ♯9 is fair game too (and vice versa for a G7♯9 chord--the ♭9 is an option). If the pianist is moving from G7♭9/♯9 to CMaj13, then she might choose G7♭9 → CMaj13 if she's looking for the A♭ → A upward resolution/movement. Or she might choose G7♯9 → CMaj13 if she wants the B♭ → A downward resolution/movement.

But in many scenarios, there is more than one good option for the dominant 7th chord.

Answer 2

I would be surprised if someone hasn't tried to construct such a ranking order, but I don't know how useful it would be -- David Bowling

I have measured consonance using a weighted algorithm of dominant seventh chords based on the dimensions below. It might not be that useful since it's subjective, but it does provide an overview of dominant seventh chords.

• fit dominant quality (prefer 7 over 7b5, exclude 7#5 since it's an aug7)
• number of alterations (prefer 7#9 over 7#9b13)
• preferred alterations (prefer 7b9 over 7#9)

appendix

- pitch class set                  interval set               name                properties              quality          prime set               
 - 0,4,7,10                        1-3-5-b7                      C7                  C7                      Dominant7        0,3,6,8                 
 - 0,4,6,10                        1-3-b5-b7                     C7b5                C7 b5                   Dominant7        0,2,6,8                 
 - 0,1,4,7,10                      1-b9-3-5-b7                   C7b9                C7 b9                   Dominant7        0,2,3,6,9               
 - 0,4,7,8,10                      1-3-5-b13-b7                  C7b13               C7 b13                  Dominant7        0,3,4,6,8               
 - 0,3,4,7,10                      1-#9-3-5-b7                   C7#9                C7 #9                   Dominant7        0,2,5,6,9               
 - 0,4,6,7,10                      1-3-#11-5-b7                  C7#11               C7 #11                  Dominant7        0,2,3,6,8               
 - 0,1,4,6,10                      1-b9-3-b5-b7                  C7b9b5              C7 b9 b5                Dominant7        0,2,3,6,8               
 - 0,3,4,6,10                      1-#9-3-b5-b7                  C7#9b5              C7 #9 b5                Dominant7        0,2,5,6,8               
 - 0,1,4,7,8,10                    1-b9-3-5-b13-b7               C7b13b9             C7 b13 b9               Dominant7        0,1,3,5,6,9             
 - 0,1,4,6,7,10                    1-b9-3-#11-5-b7               C7#11b9             C7 #11 b9               Dominant7        0,2,3,6,8,9             
 - 0,3,4,6,7,10                    1-#9-3-#11-5-b7               C7#11#9             C7 #11 #9               Dominant7        0,1,3,4,7,9             
 - 0,3,4,7,8,10                    1-#9-3-5-b13-b7               C7b13#9             C7 b13 #9               Dominant7        0,1,4,5,7,9             
 - 0,4,6,7,8,10                    1-3-#11-5-b13-b7              C7b13#11            C7 b13 #11              Dominant7        0,2,3,4,6,8             
 - 0,1,3,4,7,10                    1-b9-#9-3-5-b7                C7b9#9              C7 b9 #9                Dominant7        0,2,3,5,6,9             
 - 0,1,3,4,6,10                    1-b9-#9-3-b5-b7               C7b9#9b5            C7 b9 #9 b5             Dominant7        0,2,3,5,6,8             
 - 0,1,4,6,7,8,10                  1-b9-3-#11-5-b13-b7           C7b13#11b9          C7 b13 #11 b9           Dominant7        0,2,3,4,6,8,9           
 - 0,3,4,6,7,8,10                  1-#9-3-#11-5-b13-b7           C7b13#11#9          C7 b13 #11 #9           Dominant7        0,1,3,4,5,7,9           
 - 0,1,3,4,6,7,10                  1-b9-#9-3-#11-5-b7            C7#11b9#9           C7 #11 b9 #9            Dominant7        0,2,3,5,6,8,9           
 - 0,1,3,4,7,8,10                  1-b9-#9-3-5-b13-b7            C7b13b9#9           C7 b13 b9 #9            Dominant7        0,1,3,5,6,8,9           
 - 0,1,3,4,6,7,8,10                1-b9-#9-3-#11-5-b13-b7        C7b13#11b9#9        C7 b13 #11 b9 #9        Dominant7        0,1,3,4,6,7,8,10   

Answer 3

3.5 years later, I'm going to attempt to answer my own question and order dominant seventh chord extensions by dissonance using the results of some Python code I have written. I believe the following analysis is comprehensive given the assumptions I state upfront.

Disclaimer: I'm not coming from an academic music theory background so I expect at least some of this may seem unorthodox, either methodologically or by choice of terminology. And as these results were generated by a software program inspected only by me, there may be errors. I humbly look forward to any feedback or corrections.

Harmonic assumptions

Assumption 1: All dominant seventh chords have a root, a major third, and a flat seventh.

I define a non-extended 7th chord as the root, 3, and b7. In pitch class notation, this is [0, 4, 10]. Allowing the fifth to be altered allows for a more comprehensive analysis that includes 7#5, 7b5, etc and their extensions.

Assumption 2: Dissonance is related to harmonic versatility or lack thereof.

For my ordering I am defining dissonance as the inverse of harmonic versatility. Harmonic versatility is the number of compatible parent modes. E.g., if an extended dominant chord is a subset of multiple jazz modes, I say that chord is inherently less dissonant than an extended dominant chord that is a subset of a single jazz mode.

For a concrete example, the dominant seventh chord with a natural 5th and a natural 9th (pitch classes: [0, 2, 4, 7, 10]) is a subset of the following three modes: Mixolydian b6, Mixolydian and Lydian Dominant. In contrast, the chord 7b9b5 (or "equivalently" 7b9#11) defined by pitch classes: [0, 1, 4, 6, 10] is only a subset of the Altered scale. Thus, this chord is more dissonant. To be comprehensive, this last chord is also a subset of the octatonic half-whole mode, but I explain why I'm not considering this mode in the Results section below.

Assumption 3: Dissonance is related to brightness and darkness.

Let's briefly consider modes of the major scale and the dark -> bright spectrum: Locrian -> Phrygian -> ... -> Ionian -> Lydian .

Chords exclusive to bright modes such as Maj7sus4 in Ionian or Maj7#4 in Lydian are very different from dark chords like minb5 that is only in Locrian. You may say the chord min#6 chord, exclusive to the neutrally bright Dorian mode, is less dissonant than the aforementioned 4-note chords exclusive to Locrian and Lydian mentioned above. You may respectfully disagree, but this quality is important to me. Regardless, the brightness / darkness attribute can also be calculated for the modes of some other parent jazz scales besides the major scale (see Results and Methods).

Aside: It is challenging to disentangle brightness/darkness and harmonic versatility. These two properties perhaps exist as separate dimensions for every chord, however, I want to include this brightness/darkness information for its utility in harmonic composition.

Results

Parent modes

In this investigation of dissonance, I considered chord membership in all modes of the following jazz scales with seven notes: major,melodic minor, harmonic minor, and harmonic major. This leads to 48 eligible jazz modes that I call "parent modes". I am not examining the octatonic half-whole mode and the whole tone scale that both include [0, 4, 10] simply because they do not have 7 notes and they are less straightforward to characterize in terms of brightness and darkness (see Methods below).

Extensions

To add extensions to three notes [0, 4, 10], I considered all other notes in the chromatic scale: [1, 2, 3, 5, 6, 7, 8, 9, 11]. This leads to 2^9 = 512 unique sets of extensions. Adding subsets from this pool of possible extensions to the [0,4,10] chord encompasses all alterations to dominant 7th chord in twelve tone.

The last step of my algorithm intersects these 48 modes with the pitch classes. If an extended 7th chord is not a proper subset of one of the 48 modes, it is discarded. For example, some of the 512 extensions are very "out there" -- like the Maj7 extension [11] (a "dominant" seventh with a Major 7, represented by pitch classes [0, 4, 10, 11]). This exotic alteration includes three consecutive half steps and does not exist as a subset of any of the 48 modes and so it is not in the table below.

Extended 7th chords, ranked by harmonic versatility and absolute brightness

My intersection algorithm takes all possible extensions to the dominant seventh chord and looks for subset-based membership across 48 parent modes. The algorithm yielded 59 total dominant seventh extensions. The 7 parent modes supporting dominant seventh chords and their extensions were identified to be Mixolydian, Altered, Phrygian b4, Phrygian #3, Mixolydian b6, Mixolydian b2, and Lydian Dominant.

Note 1

I am not attempting to name all of these extensions, so I am leaving them in pitch class form, the second column.

Note 2:

To prevent confusion, the pitch class of [7] is the natural 5th, not the dominant 7th, which is [10] and occurs implicitly in every row. The root, major third, and flat seventh are implicit to each row. The extension [] is the non-extended seventh (pitch classes [0, 4, 10])

Table 1: dominant seventh chords, sorted from most consonant to most dissonant.

Extensions are sorted by two measures of dissonance: first by the number of compatible parent modes, then by the absolute value of the parent-mode-averaged brightness.

	extensions (pitch classes)	compatible modes	parent scale types	brightness
0	[]	Altered; Phrygian b4; Phrygian #3; Mixolydian b2; Mixolydian b6; Mixolydian; Lydian Dominant	melodic_minor; harmonic_major; harmonic_minor; harmonic_major; melodic_minor; major; melodic_minor	[-4, -3, -1, 0, 0, 1, 2]
1	[7]	Phrygian b4; Phrygian #3; Mixolydian b2; Mixolydian b6; Mixolydian; Lydian Dominant	harmonic_major; harmonic_minor; harmonic_major; melodic_minor; major; melodic_minor	[-3, -1, 0, 0, 1, 2]
2	[5, 7]	Phrygian #3; Mixolydian b2; Mixolydian b6; Mixolydian	harmonic_minor; harmonic_major; melodic_minor; major	[-1, 0, 0, 1]
3	[5]	Phrygian #3; Mixolydian b2; Mixolydian b6; Mixolydian	harmonic_minor; harmonic_major; melodic_minor; major	[-1, 0, 0, 1]
4	[1]	Altered; Phrygian b4; Phrygian #3; Mixolydian b2	melodic_minor; harmonic_major; harmonic_minor; harmonic_major	[-4, -3, -1, 0]
5	[8]	Altered; Phrygian b4; Phrygian #3; Mixolydian b6	melodic_minor; harmonic_major; harmonic_minor; melodic_minor	[-4, -3, -1, 0]
6	[2, 7]	Mixolydian b6; Mixolydian; Lydian Dominant	melodic_minor; major; melodic_minor	[0, 1, 2]
7	[2]	Mixolydian b6; Mixolydian; Lydian Dominant	melodic_minor; major; melodic_minor	[0, 1, 2]
8	[7, 9]	Mixolydian b2; Mixolydian; Lydian Dominant	harmonic_major; major; melodic_minor	[0, 1, 2]
9	[9]	Mixolydian b2; Mixolydian; Lydian Dominant	harmonic_major; major; melodic_minor	[0, 1, 2]
10	[1, 7]	Phrygian b4; Phrygian #3; Mixolydian b2	harmonic_major; harmonic_minor; harmonic_major	[-3, -1, 0]
11	[7, 8]	Phrygian b4; Phrygian #3; Mixolydian b6	harmonic_major; harmonic_minor; melodic_minor	[-3, -1, 0]
12	[1, 8]	Altered; Phrygian b4; Phrygian #3	melodic_minor; harmonic_major; harmonic_minor	[-4, -3, -1]
13	[1, 5, 7]	Phrygian #3; Mixolydian b2	harmonic_minor; harmonic_major	[-1, 0]
14	[1, 5]	Phrygian #3; Mixolydian b2	harmonic_minor; harmonic_major	[-1, 0]
15	[2, 5, 7]	Mixolydian b6; Mixolydian	melodic_minor; major	[0, 1]
16	[2, 5]	Mixolydian b6; Mixolydian	melodic_minor; major	[0, 1]
17	[5, 7, 8]	Phrygian #3; Mixolydian b6	harmonic_minor; melodic_minor	[-1, 0]
18	[5, 7, 9]	Mixolydian b2; Mixolydian	harmonic_major; major	[0, 1]
19	[5, 8]	Phrygian #3; Mixolydian b6	harmonic_minor; melodic_minor	[-1, 0]
20	[5, 9]	Mixolydian b2; Mixolydian	harmonic_major; major	[0, 1]
21	[6]	Altered; Lydian Dominant	melodic_minor; melodic_minor	[-4, 2]
22	[2, 7, 9]	Mixolydian; Lydian Dominant	major; melodic_minor	[1, 2]
23	[2, 9]	Mixolydian; Lydian Dominant	major; melodic_minor	[1, 2]
24	[1, 7, 8]	Phrygian b4; Phrygian #3	harmonic_major; harmonic_minor	[-3, -1]
25	[1, 3, 8]	Altered; Phrygian b4	melodic_minor; harmonic_major	[-4, -3]
26	[1, 3]	Altered; Phrygian b4	melodic_minor; harmonic_major	[-4, -3]
27	[3, 8]	Altered; Phrygian b4	melodic_minor; harmonic_major	[-4, -3]
28	[3]	Altered; Phrygian b4	melodic_minor; harmonic_major	[-4, -3]
29	[1, 5, 7, 9]	Mixolydian b2	harmonic_major	[0]
30	[1, 5, 9]	Mixolydian b2	harmonic_major	[0]
31	[1, 7, 9]	Mixolydian b2	harmonic_major	[0]
32	[1, 9]	Mixolydian b2	harmonic_major	[0]
33	[2, 5, 7, 8]	Mixolydian b6	melodic_minor	[0]
34	[2, 5, 8]	Mixolydian b6	melodic_minor	[0]
35	[2, 7, 8]	Mixolydian b6	melodic_minor	[0]
36	[2, 8]	Mixolydian b6	melodic_minor	[0]
37	[1, 5, 7, 8]	Phrygian #3	harmonic_minor	[-1]
38	[1, 5, 8]	Phrygian #3	harmonic_minor	[-1]
39	[2, 5, 7, 9]	Mixolydian	major	[1]
40	[2, 5, 9]	Mixolydian	major	[1]
41	[2, 6, 7, 9]	Lydian Dominant	melodic_minor	[2]
42	[2, 6, 7]	Lydian Dominant	melodic_minor	[2]
43	[2, 6, 9]	Lydian Dominant	melodic_minor	[2]
44	[2, 6]	Lydian Dominant	melodic_minor	[2]
45	[6, 7, 9]	Lydian Dominant	melodic_minor	[2]
46	[6, 7]	Lydian Dominant	melodic_minor	[2]
47	[6, 9]	Lydian Dominant	melodic_minor	[2]
48	[1, 3, 7, 8]	Phrygian b4	harmonic_major	[-3]
49	[1, 3, 7]	Phrygian b4	harmonic_major	[-3]
50	[3, 7, 8]	Phrygian b4	harmonic_major	[-3]
51	[3, 7]	Phrygian b4	harmonic_major	[-3]
52	[1, 3, 6, 8]	Altered	melodic_minor	[-4]
53	[1, 3, 6]	Altered	melodic_minor	[-4]
54	[1, 6, 8]	Altered	melodic_minor	[-4]
55	[1, 6]	Altered	melodic_minor	[-4]
56	[3, 6, 8]	Altered	melodic_minor	[-4]
57	[3, 6]	Altered	melodic_minor	[-4]
58	[6, 8]	Altered	melodic_minor	[-4]

Methods

Mode brightness calculation

To calculate the brightness of a mode, I am using the following formula:

brightness = sum (pitch classes (mode_x)) - sum (pitch classes (Dorian))

This formula successfully ranks major scale modes in the correct order from darkness to brightness with the darkest Locrian having -3, the neutral Dorian having value 0, and the bright Lydian having a value 3. After manually inspecting modes of the melodic minor, harmonic major, and harmonic minor and their assigned brightness, I concluded this quantitative measure of mode brightness feels reasonable. A notable exception are the modes Phrygian #3 and Phrygian b4 that are assigned to be darker than Dorian yet contain a major third.

Modes of scales with more or fewer than seven notes, such as half-whole octatonic and whole tone, have very high and very low brightness scores, respectively. This extreme brightness or darkness is related to the number of pitches in each scale and does not reflect the net behavior resulting from one-to-one voice leading from the other 48 modes. Thus, these two scales were not included in my analysis.

I am not asserting this is the best way to do this, but I cannot think of another appropriate quantitative approach.