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From: | Lukas-Fabian Moser |
Subject: | Re: Naming question: \function |
Date: | Thu, 9 Jul 2020 10:08:28 +0200 |
User-agent: | Mozilla/5.0 (X11; Linux x86_64; rv:68.0) Gecko/20100101 Thunderbird/68.10.0 |
But seriously, do you have a suggestion what to do when the "item" the
command is referencing *is* a function?
Another good synonym for "function", especially if you passing it as an argument, is "callback"
I think there's a misunderstanding here that is worth pointing out.
The "functions" Urs is working on are not functions in the computer science sense (and neither in the mathematical sense, although some theorists disagree). It's about "harmonic functions" in the sense of a certain theory of harmony that is common especially in German-speaking countries.[1]
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. :-)
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