Tonal Relationships: Key Signatures

In Studies in Music Theory by Harrison, Martin and Fink, it is not until Lesson 6 that the term ‘key signature’ is even mentioned, and it is not until Lesson 15 that the discussion of this topic is completed. In Lesson 6, the authors state: ‘The word key refers to a specific transposition of the major or minor tonal system; it accordingly implies the identification of a tonal center. One speaks of the ‘key of C major,’ or the ‘key of A minor’. The term key signature, therefore, is misleading: there is nothing in the one-flat symbol, for example, that specifically indicates F major or D minor. Signatures do imply ‘families’ of tonal systems, however.’

The illustration above is a composite of examples from Lessons 14 and 15. It shows the typical approach to key signature notation: the sharps in increasing numbers of accidentals from 1 to 7, and the same for the flat keys. C major and A minor are not included. Each family (or pair) of tonalities is indicated directly below its key signature.

Traditional thinking is that there are 24 keys - 12 major and their relative minors, in other words, a major and a minor key for each of the 12 chromatic pitches. This is not the complete picture, however. The illustration shows 14 major and 14 minor keys, to which we add the keys of C major and A minor, for a total of 30 keys. What therefore is the explanation for this discrepancy? The answer obviously lies in the three pairs of keys that are duplicated by their enharmonic equivalents: 5 sharps/7 flats, 6 sharps/6 flats, and 7 sharps/5 flats.

A frequently asked question arises: why do the keys of 7 sharps - C sharp major and A sharp minor, and 7 flats - C flat major and A flat minor exist? Wouldn’t it be just as expedient to use their enharmonic keys instead - D flat major, B flat minor, B major and G sharp minor, respectively - with only 5 accidentals in the signature? [The vast majority of instances do, in fact, appear in the 5 sharp or flat keys]. The answer centers around the concept that every key is part of a spectrum of 6 diatonically related keys. Let us consider C major as a ‘home’ key. To derive the 5 related keys, we simply name the key with the same key signature as C major, the 2 keys with 1 sharp more [or 1 flat less], and the 2 keys with 1 flat more [or 1 sharp less]. The resulting family of keys is: C major - A minor - G major - E minor - F major - D minor. Using A minor as the home key, the outcome is the same as that of C major, owing to the fact that C major and A minor share a key signature.

Taking traditional thinking a step further, it should be observed that, because each of the 4 keys with 7 sharps or flats in their key signatures have a set of 5 diatonically related keys of their own, there exist yet 4 additional keys. These 4 keys do not have key signatures, but rather, they appear within pieces as secondary to the home key. They are G sharp major and E sharp minor [8 sharps, diatonically related to C sharp major and A sharp minor], and F flat major and D flat minor [8 flats, diatonically related to C flat major and A flat minor]. To go one step further, there are actually examples in the literature written in keys with more than 8 sharps or flats.

Where are some of the famous places the ‘8-sharp’ and ‘8-flat’ keys appear? G sharp major and E sharp minor: Bach, Das Wohltempierte Klavier, both C sharp major fugues; G sharp major: Chopin, 1st Piano Concerto, 2nd movement; F flat major: Schubert, Moments Musicaux, No. 4; D flat minor: Schubert, Impromptu in A flat major, op. 142, no. 2. For an example of a key with 9 sharps, we refer again to the Chopin slow movement [E major]: when G sharp major modulates to its dominant, the result is D sharp major.

The example is quoted by permission of the authors.

Rhythmic Grouping: Asymmetry

The excerpt quoted above is the third and final verse of Schubert’s song, ‘Heidenröslein’ (The Wild Rosebud), 2/4 meter. The 3 verses of 16 measures each are identical, except, of course, for the text. In the autograph, Schubert wrote the music out only once, with the first verse of the poetry under the vocal line, while the text for verses 2 and 3 appears below the music in block form.

By far, the most typical rhythmic grouping for a 16-measure musical unit is (8 + 8), with each 8-measure phrase subdividing into (4 + 4). Perfect balance, proportion and symmetry operate within these parameters. [See the previous post - ‘Rhythmic Grouping: Symmetry’ - April 4, 2008].

The rhythmic grouping of ‘Heidenröslein’ is asymmetrical: [4 + 6 + 6], subdivided as [(2 + 2)] + [(2 + 2) + 2] + [(2 + 2) + 2]. The asymmetry exists on 2 levels: there are 3, not 2 (or 4) phrases, and the second and third phrases are expanded by way of a 2-measure extension. Clearly, the structure of the text has everything to do with the asymmetry of the music.

Other structural features of the song are worth noting: 1] a modulation to the dominant key takes place from measure 5 to 12; 2] there is a beautiful array of nonharmonic tones in the vocal line, including appoggiatura, passing tone, accented passing tone [on 2 different metrical levels], escaped tone, and accented neighboring tone; 3] strong dissonances occur in measure 9 between the vocal part and the top note of the piano part; 4] the highest note, G5, occurs in the vocal line of the following measures: 4 - root of the tonic, 10 - 7th of the dominant 7th of D major, 12 - root of the tonic, and 13 - 5th of the IV chord.

Staff Notation: Clefs and Key Signatures

Most tonal music is notated with a key signature. It should be noted that ‘key’ and ‘key signature’ are not synonyms. F sharp major is a key; 6 sharps is a key signature. And most tonal music modulates from the home key and returns to it by the end of the piece. When the music moves to a secondary key, accidentals are inevitable, with special exception.

Let us use E flat major as a home key and consider its five diatonically related keys - F minor, G minor, A flat major, B flat major and C minor - in terms of the ‘accidental landscape’ when each of these secondary keys is in force: F minor - D flat, E natural; G minor - A natural, E natural, F sharp; A flat major - D flat; B flat major - A natural; C minor - A natural, B natural.

For minor keys, we shall use B minor, which like all minor keys, resides in a 9-note system including, in the case of B minor, the common accidentals G sharp and A sharp. Its related keys are D major, E minor, F sharp minor, G major and A major. For D major - this is the exception - accidentals disappear; E minor - C natural, D sharp; F sharp minor - G sharp, D sharp, E sharp; G major - C natural; A major - G sharp [and no A sharp].

An obvious conclusion can be drawn from this: as a musical score progresses, the music is not necessarily in the home key, or its attendant key signature, in every measure of the piece.

For your consideration: what are the clefs and key signature of the contrapuntal excerpt above? We can assume that the two clefs appear in order of register from bottom to top - bass, baritone, tenor, alto, mezzo soprano, soprano, treble. We should not assume, on the other hand, that the two clefs aren’t the same. There is more than one correct solution within these assumptions.

Range Limitations: Chopin's Piano

The excerpt quoted above is from Chopin’s Ballade No. 3 in A flat major, op. 47, measures 33 - 36 [6/8 meter, Allegretto]. The entire passage, except for the last two notes in measure 36, is in C major, notwithstanding the key signature of 4 flats. Chopin has modulated to the diatonically unrelated key of the mediant major, or major III. A flat major and C major do have a relationship, however: they are both related to C minor. Also, the common tone C is in both keys. On the downbeat of measure 37, the music is back in A flat major.

What is particularly wonderful about this excerpt is that in the time span of exactly one measure - from the downbeat of m. 35 to the downbeat of m. 36 - Chopin traverses the complete range of the piano of his time. We should try to visualize a keyboard whose range is C1 to F7 - a total of 8 3/8" narrower than today’s 88-key range, and then imagine Chopin presiding over it and knowing that he can’t go any further down or up than its almost 6½-octave range would allow. Chopin probably didn’t give it a second thought. His magnificent oeuvre is a testament to how the range limitations of a musical instrument can be transformed into the most beautiful aesthetic and structural/design enhancements, but only in the hands of a genius.

There are some noteworthy dissonances in the excerpt: m. 33, beat 2 and again on the downbeat of m. 35 - F [RH] appoggiatura against E natural [LH]; m. 34, beat 1 - D natural [RH], also an appoggiatura; beat 2 - five of the seven white key pitches - from bass up - G, C, F, E and A. The logical explanation is: the F in the LH is an accented upper neighboring tone, while the A in the RH is an appoggiatura - all within the prevailing tonic harmony of C major. Chopin probably didn’t give any of this a second thought either.

Tonal Relationships: Related Keys and Modulation

Every major and minor key has 5 diatonically related keys. To illustrate, I am arbitrarily choosing the keys of A flat major and C sharp minor. For A flat major, the related keys are B flat minor, C minor, D flat major, E flat major and F minor. For C sharp minor, they are E major, F sharp minor, G sharp minor, A major and B major. Any two diatonically related keys either have the same key signature or a key signature with one sharp or flat more or less. Another type of key relationship is known as ‘parallel’, for example, B flat major and B flat minor. But this is not a diatonic relationship.

The 231-measure fugue from the Prelude and Fugue in E minor, BWV 548 is one of Bach’s longest. The formula given above would suggest that this fugue might visit the keys of G major, A minor, B minor, C major and D major. Every one of these keys indeed appears. In addition, however, there is an excursion, during the 113-measure middle section, to the unrelated key of F sharp minor. Bach achieves this goal by progressing from D major - a related key to F sharp minor - just before it. Thus, while E minor and F sharp minor are not closely related, D major is related to both of them.

Arriving at F sharp minor is done quite straightforwardly. It is the way in which Bach gets from F sharp minor back to related tonal areas that is remarkable. In the passage quoted above, measures 136-145, a continuous modulatory excursion takes place. Starting in F sharp minor, there is a motion to A major [related to F sharp minor], on to E minor [related to A minor, the parallel key of A major], G major [related], D minor [related to G minor, the parallel key of G major], F major [related to D minor], and C major [related]. The excerpt shows only upto the onset of G major. Bach has gone from F sharp minor to the remotest key of C major in only 10 measures by way of 5 intervening keys. Then from C major, the home key of E minor returns by way of a simple harmonic sequence.

Similar tonal excursions are found throughout the music of Bach.