On implementing the augmented fifth

HertzDevil · August 7, 2014, 5:40pm

Continuing the discussion from Some more suggestions:

[quote=“Ryan, post:4, topic:76”]“Missing” chords: This is an important yet tricky issue. Hooktheory tries to strike a balance of maintaining the simplicity of the user interface while at the same time allowing users to compose/analyze with a robust and often complex set of harmonic tools, which sometimes requires tradeoffs. From the beginning, it was always important point for us that we build a framework that prioritizes chord function. We could have designed HookPad to allow users to put in arbitrary chords (e.g., entering chords by chord name and quality) the way that most other music software does. However, by encouraging our community to think about chord progressions functionally, we hope to to answer questions like, “Why do these chords sound good together?” rather than, “How do I play this song on my guitar?”. We also want to show people that music theory isn’t quite as scary as it seems .

When we were first designing the backend to Hooktheory, we supported “embellishments” that included ♭5, ♭9, #9, #11, etc. These were mainly designed to accommodate so-called alt. 7 chords, that come up from time to time. Here’s an early example from Sara Bareilles’s Bottle It Up:

We eventually took this away, because they cluttered up the interface and felt their function was too limited. That being said, I personally find a few omissions particularly inconvenient. Without augmented chords, classics like Blackbird, and Stairway to Heaven can’t be properly analyzed.[/quote]
The different triads built on the natural minor scale and the harmonic minor (HM) scale are: (using popular music style for Roman numerals)
Natural: ♭III, v, ♭VII
Harmonic: ♭III⁺, V, vii^o
The V/ and vii^o/ secondary chords were most likely implemented to allow use of these harmonic minor functions in the Major mode in Hookpad before modes and borrowed chords were available, as extensive use of V/vi and vii^o/vi are seen in older Theorytabs. (IV/ was probably due to the subtonic ♭VII instead) On the other hand, ♭III⁺ serves little harmonic function in tonal music as a chord built on the HM scale. When the augmented triad is not employed in chromatic passages, it usually appears instead as ♭III⁺⁶ = V⁺, whose function as a substitute for V in both classical and contemporary music, as well as ability to resolve to five other major or minor chords by moving one note by a semitone, has been well established. Examples of progressions using the augmented triad will be given below.

There are two ways to implement the augmented triad: through implementing the HM scale with the existing 7 diatonic modes, or through adding an “aug” embellishment option. From this point ⁺ instead of ^(♯5) will be used to refer to the augmented fifth since this post is not concerned with the other alterations (♭5, ♭9, ♯9, ♯11 etc.) as mentioned in the original post. Should augmented fifth be added to Hookpad, the plus symbol should also be used instead of (♯5).

The method of adding augmented fifths through the ability to borrow chords from the HM scale has been suggested before. The idea is that, by adding the harmonic minor scale as a “mode” that can be chosen in the key menu, a variety of chords containing ♯5 can be natively inserted in Hookpad and referenced by the Trends API; it is also possible at the same time to add modes of the HM scale itself. However, this could further complicate the Hookpad interface by simply introducing these new modes and, unlike the diatonic scale, the HM scale differs from each mode of itself by more than one note, that no simple transformation akin to sharpening or flattening the diatonic scale could apply to the HM scale. Some of the immediate consequences include:

Mode VII of the HM scale is inconsistent with the Mixolydian mode because of the raised fifth degree, and consequently the HM scale becomes supermodal in the relative Mixolydian, yet its mode VII corresponds to the supermode containing 8 flats;
By sharpening one scale degree of the diatonic scale, the harmonic scale alterations for the S(5) and S(6) supermodes contain double accidentals in F♯♯ and C♯♯ respectively;
Before the Trends API is redone to adopt the parallel Major instead of the relative Major, a lot of chords from modes of the HM scale would take a long time to appear frequently enough to show up in the Trends page and Magic Chord;
If chords can only borrow from only the HM scale but not its modes, there would not be enough transpositions to allow chords built upon it to be used in all 12 pitches (e.g. +M7, Madd♭9, +7 via the nonstandard ♭III^+6(add9)).

After all, the HM scale is clearly not a mode of the natural minor scale, and using HM chords (melodic minor, even) in natural minor or vice versa can hardly be considered mode mixture since the HM scale is per definition an alteration of the natural minor (Dorian, for example, is not per definition Aeolian with ♮6). Hence “Scale” needs to be separated from “Mode” in both Hookpad’s key menu and the borrowed chord menu. But this will not resolve these technical issues - the fact that adding scales that are not the diatonic scale to Hookpad will upset the fundamental manner by which it references all chords using transposed / shifted versions of the major scale.

Ultimately, chord types shall not be evaluated from the scale structure and the chord root’s scale degree (although they can be pre-computed); the diatonic scale simply comprises seventh chords in the form of {M, m, m, M, dom, m, ø}, the HM scale {mM, ø, +M, m, dom, M, °}, but these chord types in turn are not bound to appear on fixed scale degrees of some scale transposed from the current key’s tonic pitch, so that the chord can be unambiguously defined by the chord type and the scale degree, without the need to reference any specific mode to infer its chord type. Such a large update could eventually eliminate many of the problems evident in the Trends API and, of course, enable addition of a lot more scales. (Compromises must be made that supermodal chords may have to be named by mode numbers prefixed by accidentals)

The second way, which I prefer over the HM scale method to be implemented into Hookpad, is to add the augmented fifth as an option on the list of chord embellishments. Though chords unique to the HM scale could no longer be used, e.g. mM7 and °7, it does provide a more direct approach to establishing the functions of the augmented triad.

Somewhat like the behaviour of secondary chords, the “aug” / “+” embellishment will override all triads and use the augmented triad regardless of the chord quality; so ii⁺ is D-F-A♯ = ♭VII⁶ but only II⁺ D-F♯-A♯ is available. Moreover, the augmented fifth can be combined with the “7” to create two types of augmented seventh chords. Besides, the embellishment also works for the three kinds of secondary chords. Inversions of these augmented chords are possible, and they need not be treated as identical to other augmented chords even if they have the same makeup, e.g. VI⁺₄⁶ and IV⁺. Take the chords in the Major key for example:

Standard:
Augmented {0, 4, 8}: I⁺, II⁺, III⁺, IV⁺(/), V⁺(/), VI⁺, VII⁺(/)
Augmented seventh {0, 4, 8, 10}: II⁺⁷, III⁺⁷, V⁺⁷(/), VI⁺⁷, VII⁺⁷(/)
Augmented major seventh {0, 4, 8, 11}: I⁺⁷, IV⁺⁷(/)
Nonstandard:
Second inversion of added fourth minor {0, 5, 8, 10}: II⁺⁷_sus4 etc.
Incomplete fourth inversion of diminished ninth {0, 5, 8, 11}: I⁺⁷_sus4, I⁺⁷_sus2 etc.
Italian sixth {0, 4, 10}: Lydian I⁺₄⁶_sus4

The only drawback I could think of is that the Roman numerals may have a slightly crowded superscript when distinguishing +7 and +M7 in borrowed chords, e.g. between IV⁺⁷ and IV^+♭7 borrowed from Dorian. It is up to the staff to decide whether these augmented fifth chords are recognized in the current Trends API without embellishments (like the suspended chords) or simply neglected (like supermodal chords) before adopting the parallel major method.

The augmented triad accepts no workarounds; replacing its major third or augmented fifth with other intervals will misrepresent the characteristic tension and function of the augmented triad. I hope the Hooktheory staff would carefully consider implementing one of the two options to allow usage of the augmented triad in Theorytabs. Finally here are some demo chord progressions using the augmented triad: (alternate notations are possible)

I - I⁺(III⁺₄⁶ or V⁺₄⁶/vi) - vi⁶
I - VII⁺⁷ - v⁶ - VI⁷
ii - V⁺ - I
ii - VI⁺₅⁶ - IV₄⁶ - V₅⁶
ii^ø7 - V⁺⁷ - i
iv - V⁺ - ♭VI⁷
i - ♭II⁺⁷ - V⁷/v - V⁺₄⁶ - iv - ♭V⁺⁷ - V⁺ - V
i - II⁺₂⁴ - vii^o7
I - ♭III⁺ - ♭V⁺ - VI⁺
i - V⁺₅⁶ - i₂⁴(i^♭7d) - VI⁺(V⁺/ii) - ii
V - I⁺⁷_sus4 - i (nonstandard notation)