How does one establish a key? What are the criteria?

Question

I am reading 'Guide to the Practical Study of Harmony' by Tchaikovsky. On page 63, he says

to pass from one key into another it is not sufficient merely to employ a number of chords not belonging to the main key—we must clearly establish the new key; that is to say, we must insert a chord, that can belong only to the key we wish to reach and which, forming the bridge between the two keys, clearly defines the new from the fold.

What exactly does it mean to "establish" a key?

To provide an example, I recently wrote a song involving only two pitch classes C and G. I wrote this song explicitly to study the nature of key. Interestingly, I could instantly change the key of my song by inserting a pulse into the background. Specifically, if I played an E note over and over again, the song sounded major, but if I played a E flat, then the song sounded minor.

Answer 1

To change keys, many theories of tonal music maintain that you need to have a cadence in the new key. In the case of an authentic cadence, this will involve some predominant chord (e.g. IV or ii), a dominant-function chord (such as V) and a tonic chord (I). Such a progression orients you in the new key by identifying the function of various scale degrees (most importantly, scale degree one in the new key). It also uses all of the seven diatonic pitches of the new key which identifies the note collection of the new key.

Your piece that uses just the pitches C and G sounds minimalistic. That's not a value judgement, it's just not the sort of music that Tchaikovsky's quote you mentioned was intended to refer to. Also, many people believe that going from a major key to its parallel minor is not technically a modulation because the tonic pitch (in your case C) does not change. Ultimately what constitutes a change of key and how that change is accomplished in the music is a style-dependent question. My answer here and Tchaikovsky's quote apply to common-practice tonal music.

Answer 2

The book (on line at http://imslp.org/wiki/Guide_to_the_Practical_Study_of_Harmony_(Tchaikovsky,_Pyotr), page 60 of the PDF) answers the question (to Tchaikovsky's satisfaction at least) in the next section (52), which states that "a dominant 7th chord can only belong to one key" (presumably he's ignoring the difference between major and minor keys here!) and therefore a dominant 7th chord establishes a new key.

That sort of statement is fairly standard in hack 19th-century harmony textbooks. Tchaik was a pretty good composer, but there's not much evidence he was ever a serious theoretician. Being employed as a teacher in one of the newly founded Russian conservatoires set up to emulate the European system of music education, he most likely thought it was a good idea to write a Russian-language book re-hashing the content of European texts - and the book was then was translated into English via German. (A nice marketing ploy, of course - buy the book and you can be as good a composer as Tchaikovsky.)

It's easy to pick holes in the 19-century logic, though - for example

1. There is a passage in one of Mozart's piano sonatas (last movement of K.311 or K.284c, depending which version of the K numbers you use) where he "establishes" several keys in succession with no dominant 7ths at all - he just writes a sequence of I-II-V-I cadences, four block chords in each key.

2. In the contemporary theory of common practice harmony, secondary dominants (which may be dominant 7ths) are not considered to be "modulations that establish new keys". Tchaik has a rather ambivalent statement about that in section 57, PDF page 66, but uses the 19th-century term "passing modulations" rather than "secondary dominants".

3. The so called "French" and "German" augmented 6th chords (https://en.wikipedia.org/wiki/Augmented_sixth_chord) are enharmonically identical to dominant sevenths, but nobody has ever claimed they result in modulations.

4. Tritone substitutions are another nail in the coffin - https://en.wikipedia.org/wiki/Tritone_substitution

Answer 3

We have an innate sense of tonality, and you can establish a key by playing notes in that key, with the tonic and the dominant being the most important notes to set our perception of diatonic functions. By playing C and G and nothing else you give our perception a strong hint that C is the tonic and G the dominant. Since this tonal context (us currently perceiving the music as being in the key of C) is established, we perceive subsequent notes (like the E and E flat) within that context, and which one you play (E or E flat) further informs our perception of scale.

Changes that are at odds with the established tonal context can change our tonal perception, it can be subtle like diatonic functions changing places (the dominant can start to be used as the tonic), it can be a simple modulation (like everything suddenly shifting up one semitone), or as in your case the tonic and dominant can remain the same but the rest of the scale changes, like by going from minor to major. It can be very unsubtle like a shift up 6 semitones (as in Prokofiev's Lieutenant Kijé's romance), in which case it's quite noticeable and you can feel how it takes a couple of seconds for your tonal perception to adjust to this new tonal context.

Answer 4

Seems like there are two different things going on here. The first is Tchaikovsky's quote, which to me seems kind of straightforward. If I am playing in the key of C and I want to move to the key of G, then the bridge chord should be a D major, which is found in the key of G and not in the key of C.

As for the second part if you play an E against a C & G then naturally it's going to sound major because you've constructed a C major chord or arpeggio. An Eb will give you a C minor chord or arpeggio and will thus sound minor.