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INTRO Since there's no possible that any course anywhere is able to cover all literature existing in depth, it's important to teach then the tools by which one can analyze without being dependent on a teacher or course. This is a small introduction to the subject of harmony analysis and how there are different systems and methods for looking at it. I'm not going to cover every single method in the universe (no reduction analysis for example,) but the most general ones. Everything else is a derivation or a simplification of the same type of principles so it shouldn't be hard to grasp any other system if these are well understood. We'll be talking about "harmony" here in two senses: 1) As any given sounds played simultaneously. 2) As harmonic functions, which are a sequence of sounds played simultaneously. Therefore "harmony" is not simply the vertical aspect of sound composition, but also the sequential art in which to organize vertical sound formations. 1) Vertical thinking, chord anatomy and figured bass With "vertical" we're implying the observation of sounds happening simultaneously across a timeline as opposed to sequentially ("horizontally.") In brief, the transition from the horizontal linear thinking to vertical began happening slowly in the pre-baroque era and was cemented with the conception of the figured bass. The figured bass is most likely the oldest form of pure-harmonic analysis, although it only functions the first sense of harmony as defined above: it simply outlines the anatomy of the chord. The anatomy of the chord is namely out of which intervals it is composed of and their relative positions (often only the bass is given a specific octave register.) For instance, a 6 written under an E in the bass clef would mean that the anatomy of the chord to play is a 6th from that tone, a 3rd and the octave. The reason why nobody writes the entire sequence of numbers is simply out of tradition and to simplify the writing as almost all symbols have similar patterns. The benefit of figured bass is that you can adapt it to literally any kind of harmony or chord anatomy, as all that is needed is simply the given intervals. It's entirely possible to analyze both Bach and Boulez using figured bass, but the practicality of doing this may vary. Because figured bass is so simple in that it only concerns itself with chord anatomy, it also lacks style-specific functionality. It addresses only the first meaning of harmony (chord anatomy) without concerning itself with relationships across a timeline. Hence, from this system two other systems were conceived that do attempt to deal with both aspects of harmony cited above. 2) Harmonic analysis using scale degrees Dating from around 1800, conceived by Jacob Gottfried Weber (1779–1839) and later developed further by Simon Sechter (1788–1867), this system bases itself around the concept of a root tone from which a diatonic scale is built. From this scale, each degree of the scale is then used as the root tone from which a chord (in 3rds) is built. The degrees then are named using roman numerals (I to VII) and used in relation to one another. This system also tackles chord anatomy the same way figured bass does, but the main distinction is the observation of the key in which the relationships take place. Therefore, now it becomes possible to abstractly refer to chord relationships such as I -> V or II -> V, and so within a major/minor tonality framework. Clearly this analysis system then is narrower in scope to figured bass, but because it observes relationships across a timeline as well as vertical chord anatomy we can find its use for analyzing music based on major/minor tonality much more straight forward. Because of this, while it's theoretically possible to analyze music outside of major/minor tonality, it will not benefit from the abstract formulation of chords relationships and of course without a given "key" in the traditional sense it becomes harder to establish from which degree it's being talked about. Analysis by degrees can be seen as a middle ground between attention to chord anatomy (objective analysis (and quantification) of musical material) and interpretation of harmonic context (keys, relationships between chords, and so on.) 3) Analysis by harmonic functions Conceived by Hugo Riemann in 1893 and later developed further by people like Diether de la Motte, Wilhelm Maler and Hermann Grabner, the idea here is almost the complete theoretical abstraction for a form of analysis that observes very specific relationships (called "functions.") In this, not only is it possible to conceive of entire harmonic passages entirely using theoretical constructs, but the degree of interpretation left by the analyst is much greater than by analysis by degrees. This system observes the same idea of scale degrees and root tone from degree analysis, but with a major difference: the interpretation of functions don't always correspond with the actual chord anatomy or musical material. A good illustration of how both systems differ and where their strengths and weaknesses lie is in the analysis of a simple 6-4 5-3 suspension in a perfect cadence (V - I or D - T.) Where the analysis by degrees will consider the suspension chord a root (I) chord with the 5th in bass, functions will consider it dominant (D) by virtue of the cadence context, in spite of the actual notes present. Functions observe the "intention" of the passage rather than the raw musical material, and as such it relies heavily on interpretation. In fact, to consider such a cadence a "suspension" is also by virtue of thinking the "suspension" chord is actually not really the tonic or I, but rather a "suspended" dominant (or V) chord despite having the actual sound and notes from a tonic (or I) chord. The same happens with different types of chords such as the VII6 which is viewed as an altered dominant (V) form, despite being actually a diminished triad where the 5th degree isn't even present (called a "shortened" dominant 7th in functions.) The usefulness of this system lies in that it becomes possible to deal with extremely complex harmonic passages in a way that may highlight a common direction. The weakness is in that there could be several possible interpretations for a given harmonic passage in functions (where in degrees the scope is lessened as they need to be based on the actual tones.) Because of the even narrower scope, functions are very difficult to apply to anything that doesn't rely on major/minor tonality, and even then even late 1800 music contained many segments which were "functionless." Going even further back, even Mozart can be observed to have "functionless" passages when analyzed with this theory, though not nearly in the same degree as later composers. A "functionless" segments are segments where this analysis theory is inapplicable as A) it's difficult to establish a root chord (T) and thus functions are impossible to draw out and/or B) there are jumps not predicted or expected by functions, such as passing chords or chords conceived through entirely chromatic movement. This is where degrees would have a slight advantage, as you could still write down the anatomy and position of the chords in relation to the scale depending on how violent the changes are. 4) Which one to use? For what? As a good rule of thumb, an analysis system is only really useful in analyzing music it was designed to analyze. Surely you can write figured bass marks on Stockhausen, but the benefits are doubtful. Likewise, where it's possible to analyze late baroque works with functions (Bach's chorales and so on) the further you go back the harder it is to analyze anything (Louis Couperin's preludes are often almost entirely functionless!) And of course, the further you go back the less music has to do with vertical thinking and chords to begin with leaving you with a very difficult to interpret horizontal line from which to guess any kind of harmony. There are well-known ways to blur harmony, as well. The less voices used the harder it is to infer detailed harmonic analysis, so you instead have to settle half-way. This is a problem that plagues much of baroque music if vertical harmonic analysis is attempted (Bach's inventions, french suites, etc.) There is also the question of rhythm and identifying different elements that can be purposely hidden (uneven harmonic rhythm and interrupted cadences, etc.) As well as observing and interpreting hierarchies (not confusing harmony-central notes as passing notes and vise-versa.) Another stumbling point is the variety of symbols and standards, where different schools use different ways of analysis within the same system (some schools for instance ignore the "Gegenklang" versions of the parallel chords as being redundant save for interrupted cadence exceptions, others don't. Ugh.) As for modern analysis, such as by Bartok or Hindemith, functions are useless and degrees aren't very helpful either unless you are able to identify elements that relate back to tradition. In music that uses entirely different idioms (Ligeti, Cage) you should resort to organizing the musical elements as they are rather than resorting to any abstract system unless the composition employs organization systems that can be abstracted (Xenakis, Babbitt, Varese.) 5) In conclusion You should've probably picked a different hobby, man.