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The analysis line shows the resulting harmonies of the individual voices, all notes arranged in one octave.

Harmonic States

A Harmonic State is an extension of the concept of a "key" in music. Typically, a "key" represents the single, clear tonal center that a listener perceives. A harmonic state, however, allows for situations where multiple keys coexist. This creates ambiguity about the "current key." Thus, harmonic states always relate to the perception of one or more simultaneous keys.

The Sauterian Formula

The Sauterian Formula is a way to describe the chords and notes used in a harmonic state. Here’s how it works:

Basic Structure

• The formula shows which notes are taken from the chords of the Tonic (T), Subdominant (S), and Dominant (D) within a key.
• For example: T35D13S1 means the harmony includes the 3rd and 5th of the tonic, the 1st and 3rd of the dominant, and the 1st of the subdominant.

Special Cases

• Rests: A rest (silence) in the music is represented by /.
• Ambiguous Keys: If the harmonic state involves multiple keys, we write ind. (short for "indeterminate").
• Atonal Notes: Notes that do not belong to the tonic, subdominant, or dominant chords are called atonal tones. These are listed as pitch classes after an A. For example, T35A14 means the harmony includes the 3rd and 5th of the tonic, plus two atonal notes at pitch classes 1 and 4. 'A' and 'B' are used for pitch classes 10 and 11.

Simplification Rule

If a note belongs to more than one chord (e.g., it is part of both the tonic and the dominant), we simplify the formula by only listing it with the chord that creates the least dissonance (see below). For instance, T135 is used instead of T135D1S5.

Degrees of Dissonance

In this system, dissonance is categorized into levels based on how many chords (tonic, subdominant, dominant) the notes come from - or, for atonal harmonies, based on the number of atonal notes present in the harmony. The levels progress continuously from least dissonant to most dissonant:

• Consonance (con): All notes belong to a single chord (either tonic, subdominant, or dominant).
• False Consonance (fcon, "Scheinkonsonanz"): A harmony that sounds consonant but only fits as such in a different key. For example, the chord a-minor (T13S3) in the key of C major creates a false consonance.
• Low Dissonance (low): Notes come from two chords: either tonic and subdominant or tonic and dominant. This is a mild dissonance.
• Medium Dissonance (med): Notes come from the subdominant and dominant chords. This is more dissonant than "low."
• High Dissonance (high): Notes come from all three chords (tonic, subdominant, and dominant). This is the most complex type of tonal dissonance.
• Atonality (A1, A2, ...): If the harmony includes notes that do not belong to any of the tonic, subdominant, or dominant chords, it is atonal. Atonal harmonies are even more dissonant than "high" dissonance. For example:
  • A1: Atonal harmony with one atonal pitch class.
  • A5: Atonal harmony with five atonal pitch classes, representing the highest degree of dissonance. Only the harmony with all 12 pitch classes is of this level.

As the levels of dissonance rise from con (consonance) to A5 (most dissonant), the harmony becomes progressively more complex and tense.

Summary of Notation

• /: A rest (silence).
• ind.: An indeterminate harmonic state (multiple keys coexist).
• con to high: Degrees of tonal dissonance (from stable to complex).
• A1, A2, ...: Atonal harmonies, where increasing numbers indicate higher dissonance.