Terminology

“°” (written to the left of the symbol) denotes a minor chord

“°a” denotes the minor chord d – f – a and not a – c – e!

So you have to get used to the fact that in the case of a minor chord, the root note is not indicated. Minor chords are always indicated from the fifth downwards:

“+” (written to the right of the symbol) indicates a major chord.

Riemann therefore notates the chord sequence notated in modern notation C – a – d – G as follows:

c+ – °e – °a – g+

A chord that is homonomous to a chord has the same tonal gender, i.e. major in the case of major and minor in the case of minor.
Correspondingly, an antinominal to another chord has the opposite gender.
For example, c+ and d+ are homonomous chords, but a+ and °e are antinomous.

A chord that is homologous to a chord is on a step that is in the direction of the tone row that applies to the chord construction.

If a chord of a certain degree is related to a major chord, the degree is counted upwards; if it is related to a minor chord, the degrees are counted downwards, true to Riemann’s conception of the minor.

The opposite of homologous is antilogous.
The homologous fifth to c+ is, for example, g+, the homologous fifth to °a is °d.

The two pairs of terms -nom and -log can be combined. The homologous and antinomous fourth sound to c+ is °f, the homonomous and homologous third sound to °e is °c, the antilogous and antinomous sixth sound to f+ is °a and so on.

The dictionary of Riemann’s grammar therefore consists of the major and minor triads:

Sounds = {c+, d+, e+, f+, g+, a+, h+, °c, °d, °e, °f, °g, °a, °h}

Riemann extends the dictionary with all chromatic variants and seventh sounds (for dominant chords). However, we will limit ourselves here to a reduced version; after all, it is only a matter of making the principle clear.

The grammar

More complex structures can now be put together from the basic elements using more or less clearly defined rules. Riemann calls the simplest of these theses.

A selection of rules for constructing one-sided theses is:

A thesis is a sound, followed by the homologous homonymous fifth sound, followed by the sound.
Examples: in major: c+ – g+ – c+; in minor: °e – °a – °e

A thesis is a sound followed by the homologous homonymous third sound, followed by the sound.
Examples: in major: c+ – e+ – c+; in minor: °e – °c – °e

Rules for sixth sounds, as well as antilog homonome, antinome antilog and all other variants are defined in the same way. In addition to one-sided theses, two-sided theses can also be constructed with rules. In contrast to the simple one-sided ones, they combine three different sounds:

“Complete cadence”: A thesis is a sound followed by the antilog homonymous fifth sound, followed by the homologous homonymous fifth sound, followed by the sound.
Examples: in major: c+ – f+ – g+ – c+; in minor: °e – °h – °a – °e

The complete cadence in minor is, as already mentioned, a head birth in riemann’s definition and has never progressed beyond the theoretical formulation. To be fair, however, it must be said that this does not detract from the grammar. This could be stated in exactly the same way if one were to define minor as a key parallel to major with the homologous homonomous sixth tone as the fundamental tone and abandon the idea of the undertone series. If you would like to familiarize yourself a little with Riemann grammar, you can do this as an exercise. Have fun.

Riemann also refers to the fundamental sound as the tonic, the homologous homonomous fourth sound as the subdominant, and the homologous homonomous fifth sound as the dominant.

To make the whole thing a little more complex, he defines further types of theses: Those that do not end on the tonic are called open, those that end on the tonic are called closed. Theses that begin on the tonic are called direct theses, those that begin on a different tone are called indirect theses.

Analogous to the rule of the complete cadence, rules can be formulated for open, closed, indirect, direct and all combinations of two-sided theses. For example:

g+ – c+ – f+ – c+ is an indirect, closed thesis, as is

°d – °a – °e – °a

Finally, thesis concatenations are introduced to complete the system. On the whole, this is a modulation theory, or rather a key plan for larger compositional structures, in which the small structures can be reflected in a kind of self-similarity. A chain of theses is for example:

A C major thesis, followed by a G major thesis, followed by a C major thesis.

Another possibility based on tripartite theses:

C major – F major – G major – C major.

Riemann’s grammar and comparable systems of harmony theory adapt their system of rules in such a way that the starting point is usually either a melody to be harmonized or a bass to be suspended. The classical combination of harmony theory and tuning rules takes a chord progression as its starting point. In this case, the basic elements are all possible ways of distributing the notes in a triad or tetrachord. These can be wide or narrow arrangements, different ways of doubling or omitting notes, but also the distribution of the root, third, fifth and seventh to the individual voices.

The rules in this case are, for example:

Two sounds may not be concatenated if there are fifth or octave parallels between two of their voices.

Many of riemann’s ideas are still alive in musical practice and training today, such as step and functional harmony, thinking in cadences and eight-bar phrases. However, his Hegelian terminology and the breaking down of music into theses have not caught on.