In this case, your introduction in E major for a piece in A minor is not particularly unusual. E major is not a distant key to A minor. E major is the dominant of both A minor and A major.
My purpose in my statement about E major as an introduction to A minor was because Christian keeps thinking the he must go through all sorts of steps to get from E major to A minor when in fact he has everything he needs. To get from E major of the final cadence of the introduction to A minor is as simple as a finger slipping from an "E" above the bass to "D" forming the dominant 7th of A major or minor.
Let's look at the situation being discussed. The intent is to go from E major to A minor, not from A minor to E major as a secondary key of the piece. This represents a V - I harmonic movement, the most tonally decisive movement in music. In a Schenkerian point of view, this would simply represent an extended unfolding of the E major triad and could even be thought of as an extended dominant upbeat. Schoenberg's "regional" view offers similar conclusions. Everything between the initial E and the final E is just fortification or embellishment of the dominant harmony. It is no more than a basic gesture found over and over in the music of the "Common Practice Era", V - I or Dominant to Tonic.
The brighter major mode inflection of the introduction delays and provides contrast to the darker minor mode of the main body of the piece. One may want to introduce the "C" of A minor into the E major introduction to hint of what is to come. That can be done as simply as substituting the minor form of the IV of E (A minor) through modal interchange. This usage of the "foreign" A minor is nothing exceptional and involves no modulation.
E major is not a distant key to A minor
I'm afraid Scott is quite erroneous regarding this matter. (Sorry Scott
) The chord
of E major is diatonic to A Harmonic
minor due to the altered raised 7th; this is in fact the very reason for the Harmonic minor form: to have a Major dominant chord. But the key
of E major, is very remote from the key of A minor.
First, I did not use the term "closely related", I chose the wording "not distant" specifically. The simplistic view of "closely related keys" as being those that differ by one sharp or flat derives from the theories of music presented by Rameau in 1722 and further reiterated and repackaged by Reimann in the early 20th century and from which the theory texts pervasive through the first 3 quarters and on into the last quarter of the 20th were derived.
To put Rameau's work into perspective, Bach would not enter service at Leipzig for another year, C.P.E. Bach was 9 years old, Style Galant was just beginning, and Haydn would not be a gleam in anyone's eye for another 10 years. There was a lot of "Common Practice Era" music to be written.
Walter Piston (whose "Harmony" is a derivative Rameau / Reimann) recognizes an additional force at play in key relationships, interchangeability of modes. In his book (1959 edition page 82) under the section "Related Keys" he states ""The relationship of keys has two aspects of definition. The first conception is based on the number of tones in common between two keys." He then continues using the same description that you have given concerning the one sharp more or less of the key signature.
During the next section "Interchange of Modes" he gives the second aspect. I quote "During the nineteenth century, as more interest in harmonic color developed, the latent potentialities of another aspect of key relationship were exploited. This is the closeness of the major mode to the minor mode having the same tonic. Under the principles just outlined ("Related Key Section that I just described"), the keys C major and C minor are rather distantly related since there is a difference of three flats in the signature. We have seen, however, that these two keys are practically identical, having as they do the same tonal degrees and really differing only in the third degree. Practice in the nineteenth century, and much individual practice in the eighteenth, tends to regard the two modes as simply two aspects of one tonality, so the 'family' of keys is greatly enlarged."
Unfortunately, other than introducing a close relationship between parallel major and minor, Piston gives little more information about how this works. But, Schoenberg does give us a chart that includes the parallel modes.
Schoenberg, in his "Structural Functions of Harmony" relates the keys through "regions". In is chapter "Regions in Minor" (pg. 30, 1983 ed.) he shows the relationships of keys through a chart, first a general description and then in a minor (conveniently).
G em E c#m C#
em C am A f#m F#
fm F dm D
(there is a polygon in the shape of a "cross" that encompases "am", the keys above and below ("em" and "dm") and the ones immediately to the right and left ("C major" and "A major") defining the closest relationships). You will note that he places E major (the dominant) in the same relationship as G major (a key that is admitted by the above "one sharp, one flat difference system").
This diagram indicates that Schoenberg believes that the closest relationships are those in the 3 x 3 square around "am", though he does indicate two different degrees of closeness. Those keys further from that square are more distant. But in this reckoning, E major is not "very remote" (
But the key of E major, is very remote from the key of A minor. You will recall that I mentioned that related keys differ in their signatures by one, in either dirrection.
Schenker views the tonality as a "major-minor" complex, thus simply bypassing the need for "interchangeable modes". In his system, he is most concerned with the root/tonic relationship by 5ths to tonic. In this case, "E major-minor", being the "first 5th" in relationship to "A" is the closest -- it represents the dominant - tonic relationship (V - I).
You will recall that I mentioned that related keys differ in their signatures by one, in either dirrection. A Minor, sharing the same key signature as C Major, has zero (0) sharps or flats in the key signature. To go towards more sharps (or less flats), we go to 1 sharp and have the keys of G major/E Minor. To go in the other direction or to go towards more flats (or less sharps, again from zero sharps/flats) takes us to 1 flat and the keys of F major/D minor. These encompass the Closely Related Keys. E Major is 3 more steps away than G Major. Another way to view it is that E major has F#, C#, D#, (and G#) where A minor has F, C and D (and maybe G too). If you compare closely related keys, their Key Signatures differ only by one (1) step, and their scales have one (1) note different: compare G Maj (1 sharp) with C Maj (0 sharps) or D Maj (2 sharps). The Beethoven Sonata in C major, Op.53 (Waldstein), 1st movement, is exemplary precisely because of its distant modulation from C major (the relative of A minor) to E major for the second theme of the Exposition; an unusual modulation to a very distant key! I wish to be helpful here and not pedantic. I so love teaching theory and appreciate anytime that I can share concepts on this forum. I hope it helps.
If you will recall, I said:
The idea of closely related keys being those with just one sharp or flat difference does not completely work when discussing minor since minor key signatures are derived through unusual means and have not always been the same as we use today.
I fully understand the key signature of C major and E major. Since my statement referred to minor, I am not sure what the example of the "Waldstein" Sonata was intended for. This example perfectly describes and instance where Beethoven chose to use the key of the major mediant in relationship to a piece in major instead of the textbook more normal dominant. But it really does nothing to prove the distance or closeness of these two keys, only Beethoven's compositional choices. Mind you, this piece also begins with a free interchange of the minor sub-dominant for the major sub-dominant and a half cadence incorporating the minor tonic instead of the expected major tonic all within the first phrase of a piece in C major!!
Though, I might have stated it better, I do believe that my statement clearly indicated that I was referring to the minor mode. So let me try to be clearer about my reasoning.
In the tonal system that developed during the 17th century, major mode was capable of fulfilling several functions using one scale, the major scale. Harmonically it contains the dominant root movement from V - I, and analagous movements from I - IV, II - V, VI - II III - VI, and VII - III. It naturally contains major harmonies on the primary chords of I, V, and IV (Thus making I - IV parallel to V - I). It naturally contains the melodic Leading tone to tonic 7 - 1. This with the 4th degree forms the tritone that resolves to the root and third of the tonic triad. The major dominant triad naturally contains the leading tone. All of this creating the strong dominant to tonic motion that the tonal system has been built on. In fact, tonic can be identified just by announcing dominant and sub-dominant in conjunction without actual reference to tonic.
The dominant also gains additional strength to identify tonic with the additon of the 7th (the 4th degree of the major scale).
Thematically, the lower tetrachord (1 - 4 or, in C major, C D E F) and the upper tetrachord (5 - 1 or, in C major, G A B C) are exactly parallel allowing motivic statements on both tonic and dominant, particularly important in fugue.
In minor, no single form of the minor scale can fulfill all of these purposes, and ,in fact, some are mutually exclusive. Thus, we have several forms of the minor scale, with at least two forms often operating simultaneously. To chose any one form of minor as the overriding form without recognition of the the others forms an incomplete theoretical description.
All three modal minors (aeolian, dorian, and phrygian) contain the root movement by perfect 5th as in major. None of them contain the leading tone or the major dominant required in the tonal harmonic system Aeolian does contain the same quality of chord above each of those roots, minor, for motivic development, but it looses the parallel motion between the lower tetrachord and the upper tetrachord. The lower tetrachord contains steps W H W from 1 to 4 while it contains H W W from 5 - 1.
Dorian, on the other hand, maintains the parallel tetrachords, but replaces the minor IV chord with major. Use of the major IV detracts from the minorness of a piece.
Of course, harmonic minor was devised to allow for the leading tone for the major dominant, a constituent of tonality. It is the driver of the structural harmony in the tonal system. But, it has its own problems melodically, unless the composer wants the sound of the augmented 3rd that is melodically characteristic of this form. It also does not have the requisite parallel lower and upper tetrachords for imitative counterpoint. And of course melodic minor smooths out the melodic movement, but harmonically, it admits two forms of IV and V.
One could further ask, "why aeolian as the natural minor and not dorian, which is just as natural?" In the modal system, aeolian was not even defined until 1542 when the theorist, Glaren, decided to fill in the modal system with modes on A, B, and C. The eight modes in regular use had been the authentic modes -- dorian, phrygian, lydian, and mixolydian -- which ran from final to final (the modal equivalent of tonic) and their plagal form, with the prefix "hypo-" added, that ran from the perfect 4th below to the perfect 5th above the final of the authentic mode. Though hypodorian has the same range as aeolian, its final is in the middle of the range.
There is actually more historic precedence for minor being derived from dorian rather than aeolian since it was in more common usage throughout the modal period. Fux's rules for dealing with dorian minor in his "Gradus ad Parnassum" form what we identify as melodic minor.
At least some aspects of minor in the tonal system may well have derived from the parallel major since the defining difference between a minor mode and a major mode is the third degree. Lower the third in major and you get melodic minor ascending. Lower it in mixolydian and you get dorian.
The only real reason for aeolian as the theoretical parent to tonal minor is that it represents the relationship of a third between the two tonics, vs. the relationship by a second represented by dorian.
So, what is the real key signature for the minor mode in tonal music? Due to the number of notes admitted into minor (nine with the two inflections of the 6th and 7th degrees; 10 if the b2 of phrygian -- which Schenker proposes as the derivation of the neapolitan harmony -- is also admitted.) The convention used for minor key signatures is just that, a convention that got its standing by usage. The only practical consideration for the key signature of the aeolian over that of the dorian is that it represents the tonal relationship between tonics a minor third apart as opposed to the more distant relationship of tonics a major second apart.
There are alternative viewpoints for many concepts in the field of music theory. In the end, music theory should work to describe the practice, not try to fit actual practice into theories that require exceptional instances to work.