Richter, Lehrbuch der Harmonie

Ernst Friedrich Richter was a composer who taught music theory in the Leipzig Conservatory, where his Lehrbuch der Harmonie was used for instruction. His views on the extended chords were conservative, even though his attitude toward practical harmonic theory was progressive, even radical, for the early 1850s, in that he not only adopted Weber's Roman numeral labels but specifically rejected figured bass symbols as chord labels, using them only as adjuncts to the Roman numerals (in other words, in the manner still widely used today). In the Preface (I am citing from the third edition [1860], Richter says that the study of music theory by means of thorough bass "has long since been replaced by a more rational method" (ix-x), but that he still finds them useful for instruction in voice leading.

Richter strongly emphasizes practical harmony, which works to the disadvantage of the extended tertian chords. He does not acknowledge the Ramellian heritage of these chords but instead associates them with thorough bass: "The ninth chord . . . is a vestige of the old so-called general-bass, which taught every vertical formation of tones, no matter how accidental they were, as individual chords, without bringing all these accidental formations into an orderly system, and thereby made instruction in harmony much harder and quite complex" (1860, p. 77; my translation).

He divides chords into two groups: in Part 1, basic harmonies (Grundharmonien) , in Part 2, accidental chords (Zufällige Akkordbildungen), to which are added notes foreign to the harmony (that is, passing tones and the various accented and unaccented neighbor notes). The basic harmonies are the usual triads and seventh chords of the key, and he does include a short chapter on the 9th, 11th, and 13th chords in Part 1, but like many other authors, he does so only by way of explaining that they don't really belong there. He allows the dominant ninth chord only (no ninth chords on other scale degrees) and only with preparation (see below, his Figure 157). Otherwise he places it, along with the eleventh and thirteenth chords, among suspensions and the category of "accidental chords" (these are largely the same as Kirnberger's non-essential chords—Richter even includes passing chords (Durchgehende Akkorde) in Part 2).

The views which may be entertained of the above chords are various, but they all tend to one practical result. It may be taken for granted that these are either real chords, such as the chord of seventh, in which case they must be considered and treated as such, or that they belong to the list of suspensions, [that is] occur accidentally when one or more parts remain stationary. In the first case their explanation, and especially that of their inversions, would be extremely prolix, and moreover the chords themselves would often be difficult to recognize, inasmuch as in four-part harmony one or more of their intervals must always be omitted. If, however, considered as suspensions or accidental chords, their explanation becomes very simple. In order to obtain a clear insight into their nature, we will now proceed to examine their construction, &e. 

In pure harmonic progression it is necessary that either the ninth or the root should be prepared the following example, therefore, in which both root and ninth enter freely, is objectionable on account of its [stiffness (Steifheit)] and want of connection [Verbindungslosigkeit].  (1864, p. 62) 

Finally, from Part 2, here are three suspensions involving the ninth. The first and third of these form the dominant ninth chord (1860, p. 99, from Fig. 198):

Reference:
Ernst Friedrich Richter, Lehrbuch der Harmonie (1853). The third edition (1860) was available to me. Source: IMSLP; digital facsimile of a copy in the Sibley Library, Eastman School of Music.
Ernst Friedrich Richter, Treatise on Harmony, trans. Franklin Taylor (1864). Source: Internet Archive; digital facsimile of a copy in the National Library of Scotland.

Gottfried Weber

Gottfried Weber's influence on German music theory and its pedagogy is substantial, if for no other reason than he introduced the system of Roman numeral harmonic notation still widely in use even today. The book is in the typical two parts: (1) "general musical instruction,"* including tones, notation, intervals, and rhythm/meter (this last being unusually long); (2) "theory," which has chapters on "single series of tones considered as such," harmony, keys, and modulation.

* Throughout I am quoting from the English translation (see "reference" below).

Although he introduces the system of Roman numeral labels under "keys" (Chapter 3, section Vb, §151, p. 288), Weber only begins using those labels in earnest under "modulation," one of the earliest examples being the following, one of several to demonstrate key "inertia," so that it is unlikely one would hear the fourth chord below as V7/Bb (from Chapter 4, section IIIb, §193, p. 338).

This example is of course of interest because it includes a dominant ninth chord, but note that Weber designates it as V7. He follows the model according to which basic harmonies are triads and seventh chords, that is, the third-based chords found within an octave compass. He does not object to dominant ninth sonorities, but, as this figure shows, he reads the ninth as an added note, a note "foreign to the harmony" and therefore not included in an analysis. His discussion of such notes is in Chapter 2, whose headings are:

Section 1, "Of harmonic combination in general"
Section 2 "Fundamental harmonies"
Section 3, "Transformation"
Section 4, "View of the equivocalnesses arising from the transformations of the fundamental harmonies"
Section 5, "Consonances and dissonances"
Section 6, "Preparation"

Under Section 3, "Transformation":

A. Transformation of position
     1. Transposition in general
     2. Inversion
B. Doubling
C. Omission
D. Tones foreign to the harmony
     1. Independent ninth
     2. Elevation and depression of an interval
     3. Transitions
     4. Other tones foreign to the harmony

Ninth chords are independent when the ninth is not prepared (as it must be in suspension figures) and may possibly not be left by step, either. Here is Weber's first example of the two types of ninth chord (p. 195):

After this, he offers several examples from the contemporary repertoire (pp. 196-97):

Immediately thereafter, Weber writes that "If now, as is undeniably the fact, these examples in part have a very perceptible harshness of sound, it is, on the other hand, quite remarkable, how soon this harshness disappears, the moment the fundamental tone is omitted" (p.197; emphasis in original), as here:

This mitigation of harshness is all the explanation we ever get. Instead, Weber goes on for several pages on various ways to distinguish the ninth without its root from the seventh chord that looks just like it. (In keeping with a central tenet of his theory, ambiguity—or "equivocalness," as in one of the headings above—he says at one point that sometimes you can't tell them apart.)

Reference:
Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst (1817-21). English translation of the 3d edition by James F. Warner as An Attempt at a Systematically Arranged Theory of Musical Composition (2d ed, 1842). Source: IMSLP; digital facsimile of a copy in the Yale University Library.