Composers utilize the double sharp when a note that is already marked with a sharp must be raised further, but when the conservation of interval quantity is imperative.
A simple example is the seventh scale degree of A# minor, D# minor, and G# minor. In natural minor, scale degree seven (known as the subtonic) is a whole step away from scale degree eight. A half step distance between scale degree seven and scale degree eight, however, is so important to Western music that the seventh scale degree of a minor key is often raised by a half step and is called by a new name: the leading tone. The scale is also renamed to harmonic minor.
The subtonic of A# natural minor is G# and composers raise this G# by a half step in order to achieve the leading tone. Scale degree seven of A# must never be called A, so Gx becomes necessary. The leading tones of D# harmonic minor and G# harmonic minor are Cx and Fx, respectively.
A secondary dominant is a chord that has been turned major so it can act as the dominant of another chord. The original chord is usually minor, and in this case the third must be raised by a half step. If the third is already marked with a sharp, the double sharp again becomes necessary.
For example, in the key of F# major, changing the minor VI chord to a major chord involves raising scale degree one (F#) by a half step. This yields Fx.
A secondary leading tone is a chord that has been turned diminished so it can act as the leading tone of another chord. This commonly involves raising the root of a major chord by a half step.
A simple example is when scale degree four of a major scale is raised so that the major IV chord (known as the subdominant) is turned to a diminished chord and acts as the leading tone of the V chord (known as the dominant). In the key of C# major, scale degree four must be raised to Fx.
Illustrating that the leading tone1 of certain major scales must have a double sharp accidental is rather moot because these scales technically do not exist2 except in theoretical situations.
Study the following rectangle of fifths,3 paying close attention to the five keys that span the bottom of the rectangle:
^ ^ ^
notation of enharmonic keys
The pattern of enharmonic notation could be continued both clockwise and counter-clockwise with Ab/G# and Fb/E, but it never is except by sadistic music theory professors, guitar teachers, and the like. In practical use, C# when traveling clockwise and Cb when traveling counter-clockwise are the end of the road; they respectively contain seven sharps and seven flats. The key of G# major requires eight sharps and since there are only seven notes, the odd sharp out piggybacks itself onto the already created F#, resulting in an Fx.4
The scale works, of course, if it is familiar to the musician. In equal temperament, it sounds identical to the Ab major scale. But as mblase has already noted, double accidentals are generally not allowed in key signatures simply because they are too difficult to read. If this clockwise pattern of enharmonics is continued to four more keys, the resulting theoretical key is B# major (enharmonic to C) and contains five double sharps and two sharps! If this scale were to be written utilizing a key signature, there would be no visible accidentals and it would be up to the reader to know which notes are doubly sharped and which are merely sharped (this is quite a task after learning to play or sing with no more than seven sharps).
This fundamental limitation of our key signature system sometimes creates situations for composers and editors. For example, I was a member of a pit orchestra for a musical of which one song was in the already inane key of C# major (a key signature of seven sharps). The music of instruments pitched in the key of Bb,5 however, is transposed up a major second.6 This effectively adds two sharps to the existing key signature. In the concert key of C# major this results in nine sharps and the editor has three choices:
- Publish the music for trumpets in the key of C, which require no transposition. Since the two instruments have a different timbre, however, this definitely undermines the original intent of the composer. In addition, many instruments in the key of Bb do not have counterparts in the key of C or they are scarce or uncommon for most ensembles.7 Thus, the problem would remain for those instruments.
- Attempt to publish the parts in D# major (I say attempt because he or she would likely not get away with it). Remember that the key signature would contain two double sharps and five sharps. A professional musician would understand this concept but would likely have great difficulty reading the piece; the average high school musician, having never seen this before, would be utterly lost.
- Publish the parts not in D#, but in its enharmonic equivalent Eb. In my opinion, this is the best choice and it is also the option that the editor of the trumpet book chose. This is perhaps the strangest choice in that it causes the music to be transposed up not by a major second, but by a diminished third.8 One could say that particular song was written for a trumpet in the key of A#!
1 In "G# major," the leading tone is Fx.
2 More appropriately put, the G# major scale is pointless. This is true in equal temperament, the current tuning standard. In tuning systems such as mean-tone temperament where G# and Ab are not actually enharmonic equivalents, a G# major tonal center could be utilized since its pitches would not exactly match those of an Ab major scale as they do in equal temperament. See Medieval and Renaissance tunings and temperaments for more information about historical tuning systems.
3 Thanks to Soujirou for this brilliant invention :). This particular rectangle only shows major keys, but this one includes minor keys as well.
4 Sharps are applied to the key signature according to the order of sharps, FCGDAEB. After seven sharps have been added, the cycle returns to what is now F# and begins to create double sharps (in theory).
5 This includes some instruments common to a pit orchestra, such as Bb clarinets and Bb trumpets.
6 This practice may seem odd to someone who is not familiar with it, but it actually serves at least two purposes. First, it allows an instrumentalist to pick up a similar instrument constructed in a different key and play without regard to different figerings, provided he or she has music written for the different key. Second, it can help to keep center of the instrument's range in the musical staff. See transposing instrument.
7Regarding a now-corrected error, my fellow musician Powers informed me that clarinets in C do exist and are "one of the least uncommon of the uncommon clarinet pitches." This leads me to believe that most professional clarinetists, if the need were to arise, could furnish one from their own collection or could obtain access to one. It would be pretentious, however, for someone publishing a piece for primary or secondary level music educators to publish music for clarinet in C.
8 A diminished third is one half-step lower than a minor third and is enharmonically equivelent to a major second.
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