### Harmonic Analyses

Each of us (TdC and DT) created a text file with a harmonic analysis of every song in the RS 200. The songs are analyzed in Roman numeral notation, showing the relationship of each chord to the current key. We used a recursive notation (described in detail below) that allows a repeated pattern or section to be encoded as a single symbol. We can "expand" such a reduced analysis into a list of chords by using a computer program that we wrote for this specific purpose. Our reduced analysis contains some information (via the variable names) as to the form of the song; these form labels are absent in the expanded version.

We did the analyses on our own, by ear, without consulting each other or any printed sources (e.g., lead sheets). When we were done, we compared our analyses. Any differences in meter or barlines were resolved (in this respect, the two sets of analyses are identical). Other differences were mostly not resolved. When a difference was clearly due to an error on one of our parts, we corrected it; but differences that reflected a real difference of opinion about the harmony (or key or form) were left standing. For more detail on our analytical process, see our 2011 Popular Music article.

Our analyses of the RS 200 can be downloaded here in a few formats. (The original author is indicated via a "dt" or "tdc" label at the end of the filename.)

- Individual versions of our harmonic analyses
- A zipped file containing all our harmonic analyses (raw)
- A zipped file containing all our harmonic analyses (expanded)
- A zipped file containing all our harmonic analyses (as timed chord lists)

An explanation of the "expanded" format can be found below in the overview of our notation. The "timed chord list" format provides absolute times (in seconds) for each chord; a full description of this format can be found on our programs page (under add-timings.pl). For unexpanded harmonic analysis files, we use the suffix .har; for timed chord lists, we use .clt.

For those researchers familiar with the Python programming language and the music21 toolkit for computer-aided musicology, note that MIT student Beth Hadley has written a parser for our "unexpanded" harmonic analysis format.

### The Harmonic Analysis Notational System

> 1. Overview
> 2. Basic Syntax
> 3. Chord Symbols
> 4. Special Symbols
> 5. Measures and Dots

A. Overview

Our analyses use a recursive notation: the analysis of a section may be defined with a single symbol, and that symbol may then be used in a higher-level expression. For example, an analysis (for a hypothetical song) might look like this:

    VP: I IV |
    Vr: $VP $VP I ii | V |
    Ch: I V | vi IV |
    S: [C] $Vr $Ch $Vr $Ch $Ch I |

"VP" is a short (one-measure) harmonic progression, consisting of the chords I and IV; "Vr" (verse) contains two repetitions of VP, and some other chords; "Ch" (chorus) likewise contains a series of chords; and "S" (the entire song) contains a pattern of alternating verses and choruses, ending with a I chord. "[C]" indicates a key of C.

The program expand6 (which we call the expander) takes such a reduced analysis and expands it like this (for the reduced analysis above):

    [C] I IV | I IV | I ii | V | I V | vi IV | I IV | I IV | I ii | V | I
    V | vi IV | I V | vi IV | I |

We refer to this format as the "expanded" version of the harmonic analyses. This is the format used in the expanded harmonic analysis files mentioned above.

The expander can also output the above representation as a list of chords, on a timeline defined by measures. The following shows the "chord list" for the beginning of the song defined above. (The integers at right are explained in the explanation of expand6 on our programs page.)

     0.00  0.50   I    0   1   0   0
     0.50  1.00   IV   5   4   0   5
     1.00  1.50   I    0   1   0   0
     1.50  2.00   IV   5   4   0   5
    (etc.)

We also provide tools for extracting aggregate data from such a list.

In this documentation, we explain the syntax we use, how the expansion works, and the tools for extracting aggregate data.

B. Basic Syntax

An analysis file is a text file consisting of a series of rules, one on each line.

A rule consists of a left-hand-side (LHS) and a right-hand-side (RHS). The LHS consists of a string, followed by a colon. (Unless otherwise indicated, a "string" here implies any series of letters, numbers, or punctuation symbols, except for a few symbols with special meanings, described below.)

The RHS is a series of nonterminals, defined measures, and key/meter symbols. A nonterminal is a string preceded by "$"; each nonterminal in an RHS must be defined somewhere else as the LHS of a rule (here the $ must be omitted). A "defined measure" is a series of one or more terminals (chord symbols or special symbols) followed by a barline "|". So in the third rule of the hypothetical song above (restated here)

    Vr: $VP $VP I ii | V |

"I ii |" constitutes one defined measure; "V |" constitutes another. The following rule is invalid

    Vr: | $VP $VP I vi | V

for two reasons: 1) it starts with a barline (which is not a defined measure or part of one), and 2) it ends with a harmonic symbol not followed by a barline (which is not a defined measure or part of one).

One exception is that, following a defined measure, another defined measure may be indicated with a single barline (meaning that the previously stated chord continues through the entire measure). So this is a valid rule:

    Vr: $VP $VP I vi | V | |

A key/meter symbol is surrounded by square brackets and indicates key or meter; these will be discussed further below.

The top level symbol (representing the entire song) is assumed to be "S". The expander searches for the rule with S as the LHS, and then outputs its RHS, recursively expanding any nonterminals.

Note that the names of nonterminals are arbitrary; they have no meaning for any of the programs described below. However, we try to use meaningful symbols such as Vr for verse and Ch for chorus; in this way, the definition of S becomes a kind of formal analysis of the song.

C. Chord Symbols

Each chord symbol is assumed to have this syntax (using "regular expression" notation):

(RN).*(/RN)?

where .* may not contain a slash. RN is a Roman numeral, which must be one of the following: I #I bII II #II bIII III IV #IV bV V #V bVI VI #VI bVII VII (or the lower-case versions of these). (We assume upper-case for major triads, lower-case for minor triads.)

In other words: A harmonic label must begin with a Roman numeral symbol. After that, anything can happen (as long as it doesn't contain a slash); this is to allow all kinds of additional symbols such as "o", "7", "63", "b9", etc. After that, there may be an optional "/" plus Roman numeral to indicate an applied chord, e.g. "V7/IV".

The portion of the chord symbol after the first Roman numeral (before the slash, if any) can be used to indicate what might be called "subcategorical" information about harmony, such as chord quality, inversion, and extensions. For the most part, our tools for aggregate data extraction look only at root and key, not at subcategorical information. And we did not attempt to fully standardize our treatment of subcategorical information in our analyses. However, we did agree on certain conventions, most of which are quite standard:

> Inversions: 6 = first inversion triad, 64 = second inversion triad; 7 = root-position seventh chord, 65 = first inversion seventh chord, 43 = second inversion seventh chord, 42 = third inversion seventh chord

> Triads: Upper-case for major triads, lower-case for minor triads, lower-case plus "o" for diminished, upper-case plus "a" for augmented.

> Seventh chords: capital Roman numeral plus 7 (e.g. IV7) is a major seventh, lower-case Roman numeral plus 7 (e.g. iv7) is a minor seventh, capital RN plus d7 (e.g. Id7) is a dominant seventh, lower-case RN plus h7 is a half-diminished seventh, lower-case RN plus x7 is a fully diminished seventh. The exception is V7, which indicates a dominant seventh chord. Inversions may be used with any of these: for example, iih65 is a first-inversion half-diminished ii chord.

> Miscellaneous: "s" indicates a suspended note: for example, "Vs4" indicates a triad with a suspended fourth (and no third). V11 indicates a IV triad over 5 in the bass. Other symbols may also be used occasionally.

D. Special Symbols

A few symbols have special meanings.

The asterisk is used to represent repetitions of a nonterminal; for example, "$Ch*3" means Ch three times in a row. This notation may only be used with nonterminals and barlines (e.g. "|*3").

Strings surrounded by square brackets are key/meter symbols. Keys must be pitch names such as C or C#. (Major/minor distinctions are not recognized. For black-note keys, either of the two common spellings may be used, e.g. G# or Ab.) The time signature string must be N/D, where N is an integer 1-12 and D is 2, 4, 8, or 16. The exception is the symbol [0], which refers to a span of music without any clear meter. (Multiple measures of [0] meter are a convenience used to indicate measure-sized divisions in a general way.) The S statement must start with a key symbol; a time signature symbol is optional (if no time signature is stated, 4/4 is assumed). Key and time signature statements may also be inserted in other RHS expressions (at the beginning or in the middle) to indicate changes of time and meter. (Time signature symbols may only occur at the beginning of a measure.) Key/meter symbols stated in a rule apply recursively to all descendant nonterminals, but may be overridden by a symbol stated in a descendant rule; at the end of the descendant span, the key/meter reverts to that stated in the parent rule.

'.' indicates the continuation of the previous chord. See section E below for explanation. This symbol may not be used at the beginning of an RHS.

'R' means a segment of "rest" that seems to have no harmony. This may occur at the beginning of the song (e.g. if there is an intro with just drums) or elsewhere. (In the chord-list output, R's are ignored; the previous chord is assumed to continue over them. However, R's are recognized as taking time at the beginning of the piece, e.g. if the analysis starts "R | I |" then the first chord statement is assumed to start at 1.0.)

'%' means that everything afterwards on that line is a comment. ('%' need not be at the beginning of a line.)

To summarize, the following are the symbols with special meanings:

> ':' must be used after the LHS of a rule, nowhere else.

> '%' means that everything afterwards on that line is a comment.

> '*' may only be used immediately following a nonterminal or barline, and must be immediately followed by an integer.

> '$' must be used at the beginning of a string in an RHS expression that is defined elsewhere; it may not be used anywhere else.

> '[' and ']' may be used around a key/meter symbol, nowhere else.

> '.' (as a complete string) indicates the continuation of the previous chord, and may not be used at the beginning of an RHS.

> 'R' means rest and may be used anywhere that a chord symbol may be used.

E. Measures and Dots

The chords stated in a defined measure are assumed to partition the measure evenly. So this

    I vi IV V |

indicates I in the first quarter of the measure, vi on the second, IV on the third and V on the fourth. (There is currently no check to ensure that the number of divisons of the measure makes sense given the time signature.)

For uneven divisions, the dot may be used, e.g.

    I . IV V |

This implies that the I chord takes up the first half of the measure.

A dot has the same meaning as simply repeating the previous symbol. This may be done anywhere, even when it is redundant (except at the beginning of the RHS). So the following are all legal and equivalent:

    I | |
    I | . |
    I I . . | . I . I |