As young Mozart said in a letter to his dad, “You tune the sharps flat and the flats sharp”. (*)
The augmented sixth has a characteristic flatted sixth scale step and sharped fourth scale step, in most cases expanding to an octave on the fifth scale step. The tuning of the augmented sixth is considerably different than the tuning of the apparent enharmonic equivalent, the minor seventh.
The brief audio recording introduces the three augmented sixth chords with a C, the tonic; the dominant seventh is introduced by a Db tonic. The German sixth is given with its most common resolution, to a I 6-4; it could also resolve directly to a dominant chord.
And here is a pair of Shenkerian speculations: in Shenker’s terms, the arrival at 2 could not occur on a ii triad because that particular 2 should be tuned low; the correct 2 is tuned high as in a V chord. OR! as in an augmented sixth, even the German sixth, which doesn’t even have a 2 in it.
(*) paraphrased by memory from an English translation of Mozart’s letters, last read over thirty years ago, sitting on the floor of the Sibley library stacks …
I’m curious about your combining of Shenkerian analysis with precise intonation. It seems to me that how individual intervals are tuned would be much too granular to be of concern in Shenkerian analysis; which, to my understanding, is concerned with how counterpoint can define the basic structure of a musical work.
Secondly, it has been my experience as a barbershop singer and playing Partch instruments that the Ger+6 chord is tuned the same as a dominant seventh chord (i.e. so that individual notes of the chord correspond to the overtone series generated by the root of the chord, in this case the lowered 6 scale degree, or Ab in the key of C). Although there may be an F# in the score, it would be tuned exactly the same as the Gb of an Ab7 chord.
Jude, my Shenkerian hypothesis is just a bright idea: I’m not a specialist.
In barbershop singing, EVERYTHING is tuned to the harmonic series of the bass, or sometimes, the melody note, so that’s not a particularly good argument against the tuning I show for the German sixth. And, imo, the single problem with Partch’s ideas is that they fix intonation, whereas in my theory intonation shifts depending on harmonic context: not randomly, but among a small lattice of possibilities.
I have no analyzed recorded parts to confirm this, but has been my experience with quasi-blind testing and by testimony of experienced barbershoppers–whose sense of pitch discrimination and pitch memory can be quite uncanny–that barbershop tunes to the harmonic series of whomever has the root of the current chord. Those roots are then placed into the context of the prevailing key, so that the tonic note’s frequency should never be changed. So that in barbershop the third of the Ger+6 chord (the tonic) would dictate the frequency of the other members of the chord. As an aside, this is also the same method used by Partch in his compositions (when he uses traditional tonal resources).
In actual practice with less experienced a cappella singers, the key will often drift. So it can be difficult to corroborate this hypothesis with human voices.
Of course, one can chose to tune a chord however they want. It all depends upon the sonic goals of the musicians involved. I know from my extensive experience of singing barbershop that when I sing the +6 of a Ger+6 (which is a common chord in barbershop) I sing the septimal minor seventh of the b6 scale degree, which is a 7/5 ratio.
Finally, what ratios are used in your examples (if C = 1/1)? It’s quite possible I’ve just misunderstood your article.
The augmented sixth is, as you say, related to the tonic. In my example that’s C. A dominant seventh on Ab would be related to THAT different tonic, Db. The seventh in a dominant seventh, say of Db, must be prepared, in traditional terms, by the fourth scale step, which is a perfect fifth away from tonic, and NOT the high seventh of the overtones from Ab. The difference is the syntonic comma.
By “prepared, in traditional terms, by the fourth scale step” do you mean that the Gb would be approached from the F natural?
And yes, I think I was confused of your article’s main point. Are you trying to say that the Ger+6 chord of C major, would be tuned differently that a Ab7 chord that modulates the music to the key of Db?
Thanks, Jude, your comments made me realize that the correct dominant seventh for comparison against the German sixth in C would be a V 6-5 of vi. I’ll make a new recording soon.
Preparation of the seventh in Db would mean that the Gb would be introduced as a consonant tone in another chord, IV or ii, before the V7 sounds. And, yes, the two chords are tuned differently, though only in the seventh itself (the F#, for the Ger +6 and the Gb for the Ab7). The tuning of the V6-5 of vi is altogether different; of course, the root is different as well (E vs D).