I see this chord spelled as C E E Bb. Having said that, I doubt myself because I see an Eb note in the first chord here, and I am not sure what that means for the E note we are looking at here.
The notation follows the standard convention that accidentals carry through the measure. Thus, the chord is C E♭ E♭ B♭. See also Understanding alternate key signature in Neumeyer's Hindemith analysis.
Adding to my doubt, we have the fact that the harmonic symbol that corresponds to this chord is a "VI" with an upward sloping line crossing through the I.
The VI-with-upward-slash means that the root of the chord is the "naturally occurring" sixth degree of the major scale. A downward slash means that the root comes from the minor scale (i.e., a flat 6 or its enharmonic equivalent). This is illustrated in Example 3.4, in which VI-with-downward-slash is positioned at the 8th semitone in the diagram (i.e., minor sixth above the root), and the VI-with-upward-slash is positioned at the 9th (i.e., major sixth above the root).
I would like to know why this is being called a minor-minor seventh chord
A minor-minor seventh chord, also called just "minor seventh" chord, is spelled m3 M3 m3. Thus, with root on C, the (complete) chord is spelled C E♭ G B♭. Here, however, the fifth is missing from the chord, which is a common alteration. See E7 chord at 5th fret missing a B note.
The name "minor-minor" means "minor triad plus a minor seventh above the root". There is also a "minor-major" seventh chord, which is a minor triad plus a major seventh above the root (C-E♭-G-B); and the dominant seventh chord is sometimes called a "major-minor" seventh, because it's a major triad plus a minor seventh above the root.
The lowest tone in this voicing is the root of the chord, the C note, so what is this about how "[the chord's] correct root lies high"
The asterisk next to the chord symbol in Example 3.12 indicates a chord with a root other than what is apparent. Note the specific mention of minor-minor seventh chords.
A chord with a root other than that of the best interval (for example, certain presentations of the major-minor or minor-minor seventh chords). (p. 54. See also image below.)
Unfortunately, Neumeyer's specific meaning is unclear from based on the material provided. He seems to mean that while the chord presents as a VI chord, the "real" root of the chord is not C, but it's not clear what he believes the "real" root is.
An educated guess in that he considers the root to be E♭; that is, this is functionally a I chord. This is because I-VII-I is a common prolongation of tonic harmony, and since the following chord is IV, a tonic prolongation makes sense.
The alternatives would be either:
- It's some kind of V (B♭) chord, which seems unlikely given that only one chord tone is present and it's unusually for a predominant harmony to follow the dominant, or
- The "true" root isn't present in the chord.