Which is one more argument for *not* naming the command \function.
Thank you for the following explanations which should make it clear for everyone what we are talking about.
I have the impression I have no choice but to follow Carl's suggestion and add a clarifying adjective, although that makes for quite "expansive" user interface. E.g. \harmonicFunction might be the best bet so far.
I think "Bassstufe" could be translated to "scale step" or "scale degree" and could therefore be used as a command like \scaleDegree. However, people having written roman numeral analysis code (I know of David Nalesnik and Malte Meyn so far) used \scaleDegree for the roman numerals.
The latter would handle the fact that it's used in German contexts only anyway. And it would nicely deal with the triple "s" ;-)
However, since we're still in a computing environment I'm afraid the reference to roman numerals might be similarly problematic as "function". What do you think?
General idea
In that theory, some of the chords usually denoted by Roman
numerals have special namens and symbols (now called "Funktionen"
= functions): I is "T" (for "tonic"), IV is "S" (for
"subdominant"), V is "D" (for "dominant"). But the more important
half of the story is that in this theory, these three "functions"
are the _only_ basic chords, from which all other chords are
derived in some way. For instance, a vii° is regarded as a D⁷ with
root omitted, a ii⁶ is (most often) interpreted as a S with its
fifth replaced by a sixth, and so on.
The term "function" can, I think, be interpreted in two different
ways here:
- In the mathematical sense that these functions map from the set
of key areas to the set of actual chords ("dominant(f major) = c
major-triad") [but this applies for roman numerals as well!]
- In the musical sense that chords tend to express a "function"
for the harmonic progression of a piece: tonic chords have the
function of "being at home", so to speak, while dominant chords
express the function of "being only one step away from home", and
so on.
Strenghts and weaknesses
As can be expected, a theory with such a strong focus on harmonic
interpretation of chords has its strengths and weaknesses.
For an example of what I consider a strengh, if you compare
cadence formulas ii⁶ V I and IV V I, it can be argued that it
might make more sense to "hear" the ii⁶ as a "kind of major" chord
since the major third f-a is the same in both progressions.
"German" function theory caters for this by writing S⁶.
For examples of what I consider as weaknesses:
- While a vii°⁶ quite often has the "function" of "wanting to
resolve to a tonic", it's highly awkward to explain it as a
"dominant seven with root omitted". First, from a historical
perspective, V and vii°⁶ both occur much earlier than an actual
V⁷, so the theory explains an old and well-known phenomenon from
(at the latest) early baroque music as being derived from
something basically unknown at that point in time. Second, from
the point of view of classical voice-leading, the seventh of a V⁷
has restrictions for its voice leading (the rule of moving
downwards by a step, for instance) that are completely unknown for
the same note as part of a vii°6. (And let's not forget that even
the standard designation of vii°⁶ in roman numeral analysis has
the flaw of explaining a very old "primary" phenomenon as being
the first inversion of another phenomenon virtually unknown at
that time.)
- A mediant iii (in a major key context) has to be explained
either as a relative of V or as leading-tone exchange chord of I
(the corresponding German function theory symbols are "Dp =
Dominantparallele" and "Tg = Tonikagegenklang"), but more often
than not, if a iii actually occurs somewhere, it gets its peculiar
and interesting sonic quality from being in some sense "neither
tonic nor dominant".
Where is this used?
In German-speaking countries, some very popular
(mid-20th-century) textbooks made this "Funktionstheorie" standard
- to such a degree that "harmonic analysis of music" was
considered equivalent to "using the theory of functions" (and this
notion can still be found up until today sometimes).
For other countries, the situation is different: As far as I can
see, in English-speaking countries, it seems to be standard to use
roman numerals (which itself comes in different flavors, just
think of ii⁶ vs. IIb!). But in my teaching (at an Austrian music
university with lots of international students), I always ask my
students about the harmonic theories they have learned in their
native countries; my impression is that in eastern-european,
northern-european and far-asian countries, there are harmonic
theories being used the are to a certain degree a mixture between
"German" function theory and "international" Roman Numeral
analysis. (A Chinese student once explained to me that he had
learned to write something like S-ii-56, which means: function
theory, roman numerals, thoroughbass, all in one.)
Lukas
[1] This theory was basically invented by 19th century
musicologist Hugo Riemann, but has been simplfied and streamlined
very much during the first half of the 20th century. Funnily, the
word "theory of functions" also appears in the mathematical field
of complex analysis, with one of its most important contributors
being Bernhard Riemann. The two Riemanns are not related (as far
as I know), and the theories are completely unrelated. :-)