17

If you look at the kinds of thirds and fifths, you immediately see six obvious combinations:

  • minor third/diminished fifth
  • minor third/perfect fifth
  • minor third/augmented fifth
  • major third/diminished fifth
  • major third/perfect fifth
  • major third/augmented fifth

Four of these comprise the complete set of so-called triads:

  • m3+d5 = a diminished triad
  • m3+P5 = a minor triad
  • M3+P5 = a major triad
  • M3+A5 = an augmented triad

Of the two which remain, the minor third/augmented fifth combination results in another major triad in 1st inversion. However, the last remaining combination—major third/diminished fifth—is simply ignored, overlooked, or perhaps purposely excluded from triadhood. Why?

I'm sure someone is going to think, "Ah, but triads are constructed by stacking thirds, and the distance from a major third to a diminished fifth is a major second."

And I would say, "Well, if you look at an example like C-E-G♭, it seems to me that E to G♭ is a diminished third, not a major second."

♭5 chords are a thing, particularly in the "triadic" portion of dominant chords, so I must guess that there is some historical reason why they aren't considered triads. (Perhaps because classical theory was well-established before jazz came about?) What is it? Or am I missing something?

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    I agree, there's no really meaningful argument why the diminished chord should be considered a triad but the ♭5 chord not. It's just historic. – leftaroundabout Jan 31 '18 at 16:54
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    @leftaroundabout -- I am not sure that there needs to be any meaningful explanation; it's just nomenclature. Triads are formed by stacking major and minor thirds, not by adding a third and a fifth to a root or stacking intervals other than major and minor thirds. As to why the nomenclature is the way it is, I guess that is historic. – David Bowling Jan 31 '18 at 17:16
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    @DavidBowling the definition of triads as stacked thirds is backwards however, because fifths are clearly the more fundamental interval. In fact, a (consonant / post-Pythagorean) minor third is constructed as the difference between a major third and a fifth, so you can't really construct any of the triads except the augmented using only this “standard definition”. – leftaroundabout Jan 31 '18 at 22:26
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    A very thought-provoking question. Well, it made me think deeply! +1. – Tim Jan 31 '18 at 22:32
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    Somewhat related: music.stackexchange.com/questions/56332/… – Richard Mar 25 '18 at 16:37
18

The triads found in traditional harmony are built by stacking major and minor thirds. The list of triads generated in this way is short:

    major 3rd + minor 3rd  -->  major triad
    minor 3rd + major 3rd  -->  minor triad
    major 3rd + major 3rd  -->  augmented triad
    minor 3rd + minor 3rd  -->  diminished triad

Diminished, doubly diminished, augmented, and doubly augmented thirds are just not used to build the harmonic structures called triads in traditional harmony. That does not mean that C, E and G♭ can't be used together, and you can make up your own "triads" if you wish. These days some people talk about sus4 triads and Lydian triads, for example.

"♭5 chords are a thing, particularly in the 'triadic' portion of dominant chords." Sure, ♭5 chords are a thing, but here the ♭5 is considered an alteration. Such a chord may be written as C7(♭5), or C-7(♭5) for example. Here, the underlying chord quality (major, minor, diminished, augmented) comes first, then the highest degree of upper structure (7th, 9th, 11th, 13th), and finally added pitches or alterations (sus4, add9, ♭5, ♯9, etc.). Here, I don't think that you should say that the ♭5 belongs to the "triadic portion of the chord", but rather that it is an alteration of the underlying chord quality.

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    So far, this is the only answer that both makes sense and seems consistent, to me. Stacking thirds other than major or minor and calling them triads would lead to all kinds of "chords" that I might call "degenerate", in the sense that they would have wildly different harmonic functions from what we currently call triads. – Todd Wilcox Jan 31 '18 at 17:56
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    I don't think it's consistent... thirds and fifths both play a role in determining the quality of a triad. So saying a ♭5 is integral for diminished triads and merely an alteration for a ♭5 chord is anything but consistent. I agree with this and other answers that if we started building "triads" off diminished or augmented thirds, we'd end up with a mess. I see now that stacked thirds (of the major and minor variety) are the traditional way to look at triad construction. Continued... – trw Jan 31 '18 at 19:55
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    However, I feel that looking at the combination of third and fifth would make a lot more sense. I also know my feelings aren't going change anything. I hoped there was a self-consistent explanation that I was overlooking, but I don't think there is one. – trw Jan 31 '18 at 19:55
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    "So saying a ♭5 is integral for diminished triads and merely an alteration for a ♭5 chord is anything but consistent." The trouble is, in a traditional context triads are not defined in this way. They are defined in terms of stacked major and minor thirds. There are many other note collections, but this is just a matter of how we name different types of collections. "Triad" is just a name. It refers to something specific, which in common use may differ from what formalized music theory says, or even from what may intuitively make sense. – David Bowling Jan 31 '18 at 20:03
8

Tonal chord analysis is based around the idea of stacked thirds that naturally occur in some diatonic scale. 1-3-♭5 doesn't naturally occur. This is also why 1-♭3-♯5 doesn't exist, not because it's enharmonic to another major triad.

Of course, you can write anything you want, but once you get away from stacked diatonic thirds, you get far enough away from classical tonal theory that it doens't make sense to use the same notation system. Within tonality, you're most likely to encounter this combination of pitches as 1-3-♯4, which would get analyzed differently.

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    I like this explanation, but where, in chords built off diatonic scales, does one encounter an augmented chord? (My understanding of "diatonic scale" is "the major scale and its modes".) – trw Jan 31 '18 at 17:04
  • @trw Some claim it's a variant of III in the harmonic minor scale, but that's bunk. III+ doesn't exist, as far as I'm concerned. – Richard Jan 31 '18 at 18:18
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    I think that the harmonic minor scale, treated as a melodic device, doesn't really exist (at least in tonal music). But the harmonic minor scale as a set of pitches--a musical alphabet--totally exists, and so it makes sense to describe chords as being a combination of those pitches. – MattPutnam Jan 31 '18 at 19:26
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    I have come across III+ in minor pieces. I will try to find an example. Pretty sure it was Bach. I have also used it in my own composition, but I probably would not have used it if I hadn't heard it somewhere else and tucked it away in my musical subconscious. III+ is a pretty cool chord. However, analysis could say that the altered note (say a G# in CEG#) is some kind of passing tone instead of calling the set of notes it's own chord. In analysis, different people can argue over the meaning of the same thing. Thankfully, it is not analysis that produces music. – Heather S. Oct 1 '18 at 10:04
  • @HeatherS. I'd be interested to see an example of III+ in "diatonic harmony" for sure (something baroque especially), please do post if you think of one! – Some_Guy Oct 1 '18 at 20:52
2

Major triad = maj3 +min3.

Minor triad = min3 +maj3.

Diminished triad = min3 + min3.

Augmented triad = maj3 + maj3.

All these common stacked thirds use only maj or min 3s.

A dim3 ends up as the same note as a M2 (in sound, in 12et), while an aug3 ends up, in similar manner, as P4. So, I guess there's no real point in calling them another third, to be stacked, as the notes will make another named chord.

C>E (M3) + A (P4) is Am.

C>D (M2) + G (P4) is Csus2.

C>F (P4) + G (M2) is Csus4.

C>F (P4) + Bb (P4) is Bbsus2.

C>E (M3) + F# (M2) is C b5

C>Eb (m3) + Bb (P4) is Cm7.

C>F (P4) + A(M3) is Fmaj.

C>F (P4) + Ab is Fm.

C>Eb (m3) + F (M2) is F7 (most of!)

Think I've covered most if not all eventualities, hoping to show that 'stacked' dim and aug thirds (aka M2 and P4) already have names, mostly, and would duplicate things unnecessarily.

Wouldn't surprise me if this gets downvoted - it's somewhat avant-garde, so please attach reasons!

  • Would describing the interval as a diminished 3rd that is not major or minor be clearer than saying it's enharmonic to a major 2nd? If it's spelled C-E-Gb then it's not actually a 2nd but some kind of 3rd. – Todd Wilcox Jan 31 '18 at 17:54
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    No downvotes here; I once asked a question related to your answer, pointing out that they're largely enharmonic with other chords: music.stackexchange.com/questions/56332/… – Richard Jan 31 '18 at 18:19
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    @ToddWilcox - yes, but partly that was my point. I was saying that if we agreed that dim and aug 3rds could be used in 'stacked thirds', they would sound like other chords that already have other names. For example, if I played C-E-Gb (with a dim3), and you played C-E-F# (with a M2), they would sound pretty similar, I guess, so here, I've tried to rationalise by giving effectively enharmonic names. I could have gone the other way- maybe I should have - but the offering here took a bit of working out, and I'd got a tired brain by then. – Tim Jan 31 '18 at 22:27
  • @Richard -- the question and discussion you linked to was interesting and seems quite relevant here; I hope that OP takes a look at it. – David Bowling Feb 1 '18 at 2:08
2

You're overthinking this. If we only allow 'triads' to be derived from major and minor scales, or built from major and minor thirds, it's a modified triad. It's only a label.

1

Your minor third/augmented fifth combination is not ignored: it is indeed a thing: it is 3 notes of the French sixth chord.

Wikipedia's example is A♭CDF♯. Transpose down a major 2nd and you get G♭B♭CE, which includes your chord's 3 pitches. When used as a French 6th, it is a predominant to B♭ minor (or B♭ major, if either there's a tierce de Picardie or the G♭ was just a chromatically lowered G).

0

well, if you look at an example like C-E-G♭, it seems to me that E to G♭ is a diminished third, not a major second.

Triads are formed by 'stacking' major or minor thirds, as you already noted. So, whether or not a major second seems like a diminished third is irrelevant.

Think of it this way: if we 'stacked' diminished thirds, then, C-D-E would be considered a triad.

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    E to G♭ is absolutely a diminished third. – David Bowling Jan 31 '18 at 15:33
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    E to G♭ does not seem like a diminished 3rd; it is a diminished 3rd, and not a major 2nd. – David Bowling Jan 31 '18 at 16:11
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    @DavidBowling "E to G♭ does not seem like a diminished 3rd" That is why I put it in italics. I was simply making use of the language that he had chosen. – DougRisk Jan 31 '18 at 16:24
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    @Tim: Going up or down a diminished third from D would yield B# and Fb. If diminished thirds could be part of a triad, that wouldn't allow C-D-E, but would allow B#-D-Fb, which would sound identical. – supercat Jan 31 '18 at 18:19
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    C-D-E would be considered a triad ?!? This is flat out wrong: C-D-E is 1/2/3. A triad is 1/3/5 - two 3rds of some sort. You only have one 3rd in C-D-E : C->E. – Stinkfoot Feb 2 '18 at 21:43
0

If we have a look at the status of contemporary qualities, as a theorist, we are a bit in the twilight zone. Triads are built upon stacking min/maj thirds, while sevenths can be subdivided into qualities built upon staking min/maj thirds, and those that also include dim/aug thirds. But there is additional inconsistency, not all sevenths stacked the above way have a unique quality.

Traditional stacking: stack min/maj thirds

Contemporary stacking: stack min/maj/dim/aug thirds as long the notes approximate the main intervals (m/M3, d/p/A5, m/M7). One exception to this rule is the very symmetrical m3-d5-d7.

The following altered qualities are left in nomandsland, but deserve their own quality in my opinion: maj(b5), m7(#5), mΔ7(#5), 7(b5), Δ7(b5)

Triads

Traditional stacking

- quality  i1  i2   3th 5th
- dim      m   m    m   d      diminished triad
- min      m   M    m   p      minor triad
- maj      M   m    M   p      major triad
- aug      M   M    M   A      augmented triad

Contemporary stacking

- quality  i1  i2   3th 5th
- min(#5)  m   A    m   A     inversional equivalent of major triad
- maj(b5)  M   d    M   d     <nameless quality!>

Sevenths

Traditional stacking

- quality  i1  i2  i3   3th 5th 7th
- o7       m   m   m    m   d   d      diminished seventh
- ø7       m   m   M    m   d   m      half-diminished seventh
- m7       m   M   m    m   p   m      minor seventh
- mΔ7      m   M   M    m   p   M      minor-major seventh
- 7        M   m   m    m   p   m      dominant seventh
- Δ7       M   m   M    m   p   M      major seventh
- +Δ7      M   M   m    m   A   M      augmented-major seventh

Contemporary stacking

- quality  i1  i2  i3   3th 5th 7th
- oΔ7      m   m   A    m   d   M     diminished-major seventh
- m7(#5)   m   A   d    m   A   m     <nameless quality!>
- mΔ7(#5)  m   A   m    m   A   M     <nameless quality!>
- 7(b5)    M   d   M    M   d   m     <nameless quality!>
- Δ7(b5)   M   d   A    M   d   M     <nameless quality!>
- +7       M   M   d    M   A   m     augmented(-minor) seventh

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