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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?

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    The most common changes to dominant chords are to the 5s and 9s. Thus: C7b5, C7aug (C7#5), C7b9, C7#9, C9b5, C9#5, C7b5b9, C7#5b9, C7b5#9, C7#5#9. Could be more or less! I'm tired. There probably isn't a ranking order, it's more dependant on the preceding and following harmonies. – Tim Feb 24 '18 at 18:12
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    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. I tend to think of alterations in terms of voice-leading, or sometimes just in terms of the specific "color" or "feel" of a chord. Is C7(♭9) more or less dissonant than C7(♯9)? They certainly sound different, but ranking this difference on a one-dimensional axis doesn't seem like it would be too helpful, to me at least. – David Bowling Feb 24 '18 at 18:29
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    I think a one-dimensional axis could be helpful, like Brighter > Mixolydian > Dorian > Aeolian > Phrygian > Locrian > Darker – R Tyler McLaughlin Feb 24 '18 at 18:45
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    It seems like you may be actually asking about scales rather than chords. The Mixolydian -> ... -> Locrian range exhibits a clear pattern: each mode has one note lowered from the previous mode. This can be seen as a progression to darker scales (I actually don't find this ranking terribly useful either, but it is common and has a certain logic). You could devise a similar ranking of brightness/darkness for Lydian Dominant, Mixolydian, etc., but I think that the ordering would become unclear, especially when octatonic scales are added. I doubt that "dissonance" is amenable to rankings. – David Bowling Feb 24 '18 at 19:03
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    It has been said/written (by Marc Levine?) words to the effect of, 'The flat seventh in a dominant chord validates any alterations/ tomfooleries'. A corollary: 'alterations sound stinkier over a major seventh chord'. This does not answer the question, but maybe it supports the idea that there is a hierarchy of dissonance. Most would agree that sitting on a piano keyboard produces a chord that is pretty damn dissonant. Depending on buttock dimensionality, you're going to be playing a dozen or so notes, each a semitone away from its neighbour. Is this the benchmark (or butt mark?) – Areel Xocha Feb 24 '18 at 23:11
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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 G7(♮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 II7(♭9, ♭13)-V77(♯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):

enter image description here

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.

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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   

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