Scale-step

In Schenkerian theory, a scale-step (German: Stufe) is a triad (based on one of the diatonic scale degrees) that is perceived as an organizing force for a passage of music (in accordance with the principle of composing-out). In Harmony, Schenker gives the following example and asserts that

A scale-step triad is designated by an uppercase Roman numeral representing the scale degree of the root, much as in traditional "harmonic analysis" (see chord progression). Thus, in the above example (which is in G major), the G major triad that Schenker claims we perceive through the first two measures would be labelled "I". However, unlike traditional harmonic analyses, Schenker's theory is not concerned with the mere labelling of such chords, but rather with discerning hierarchical relationships among tones. For Schenker, the chords occurring in a passage need not be of equal import. As he explains:

Furthermore, in terms of Schenker's mature theory, the question of whether a given triad possesses scale-step status depends on the structural level under discussion. Indeed, it follows from Schenker's concepts that, at the highest level, a tonal composition possesses only one scale step, since the entirety of the work may be understood as an elaboration of its tonic triad (i.e. scale-step I).

Stufe (algebra)

In field theory, the Stufe (/ʃtuːfə/; German: level) s(F) of a field F is the least number of squares that sum to -1. If -1 cannot be written as a sum of squares, s(F)=. In this case, F is a formally real field. Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.

Powers of 2

If then for some .

Proof: Let be chosen such that . Let . Then there are elements such that

Both and are sums of squares, and , since otherwise , contrary to the assumption on .

According to the theory of Pfister forms, the product is itself a sum of squares, that is, for some . But since , we also have , and hence

and thus .

Positive characteristic

The Stufe for all fields with positive characteristic.

Proof: Let . It suffices to prove the claim for .

If then , so .

If consider the set of squares. is a subgroup of index in the cyclic group with elements. Thus contains exactly elements, and so does . Since only has elements in total, and cannot be disjoint, that is, there are with and thus .