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This is from the book "tonal harmony",

the 14th chord is vii half diminished 7; consisting of the notes D# F# A C#, with A in the bass, therefore it is vii half diminished 4 3, second inversion, which makes sense to me

but the 16th chord is what's confusing me, it has notes F# A D#, without a C#, the textbook answer says it's vii half diminished 6, first inversion, but can a triad be half diminished? I thought only seventh chords could be HALF diminished, the 6 clearly states that it's a TRIAD in first inversion, and the slashed circle states that it is half diminished,

so can a TRIAD be half diminished?

(UPDATE) It seems there are multiple errors/typos in this book, I have spotted even more typos after further reading

  • It might be a reminder that a four-note chord built on the 7th step of a major scale is half diminished. But a triad indeed can't be half diminished. I wonder what the "vii°" in the last bar of example 3 refers to? Are they at least consistent within the book? – user1079505 Aug 30 '20 at 23:50
  • They are consistent for the most part – John Aug 31 '20 at 0:50

Your analysis is correct, and this seems to be an error in the book, in which chord 16 in the answer key should include an open circle rather than the slashed circle.

A half-diminished chord is, by definition, a seventh chord -- thus requiring a (minor) seventh, which chord 16 lacks.

Note that the excerpt itself is a correct reduction of the original. Here is the passage in question from the Bärenreiter Urtext.

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A triad can be diminished - with a m3 and a d5. A fully diminished chord will be that triad plus a dim 7,the same sounding note as M6. In key C - C, E♭, G♭ B♭♭.The last sounding like A.

The 'half-diminished' part of a chord comes when a m7 is added, instead of that d7. That's why it's only 'half-diminished' as opposed to 'diminished', or 'fully-diminished'.

So it would appear that there's a typo - or it's just incorrect.

Sometimes, it's easier to understand when it's called m7♭5, as that portrays all the relevant details.

  • In precise terminology it's a fully diminished seventh chord, so called because both the seventh and fifth are diminished. In other words, the distinction between B-D-F and B-D-F-A♭ is the word "seventh," not the presence or absence of any modifiers on the word "diminished." This is of course complicated by the fact that in imprecise terminology people often use "diminished" as shorthand for "diminished seventh." – phoog Aug 30 '20 at 17:22

The triads (there are only triads of 3 tones) are built by 2 minor thirds in root position. (m3dim5)

You are right, a triad ti-re-fa (here D#-F#-A) is diminished, in respect to the diminished 5th.

Half diminished is the VII7-5 tetrad: ti-re-fa-la (or iim7b5 in minor - like Tim says)

A diminished tetrad is built by 3 minor thirds e.g. ti-re-fa-lu or si-ti-re-fa: the vii dim7 chords of the major and minor keys.

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