[Smt-talk] Proximity between Keys

Nicolas Meeùs nicolas.meeus at scarlet.be
Thu Sep 11 00:15:45 PDT 2014


Schoenberg devotes to this question Chapter IX of his "Structural 
Functions of Harmony" (1954), coming to conclusions similar to yours, 
but with five degrees of proximity (Direct and close; Indirect but 
close; Indirect; Indirect and remote; Distant). He had discussed some of 
this, albeit less systematically, in his "Harmonielehre"of 1911, Chapter 
IX on Modulation. He had probably read Richard Stör's "Leitfaden".

To base the proximity on the key signature, as you do, may raise 
problems in minor, if only because the 'natural' dominant is the minor 
one, a point that Schoenberg did not easily accept, but I think he was 
mistaken on this. The idea of a gradation in the proximity is the 
important one; how many degrees one considers, and how exactly one 
defines them, may be a matter of personal taste.

Nicolas Meeùs
Professeur émérite
Université Paris-Sorbonne
nicolas.meeus at scarlet.be

Le 10/09/2014 21:46, Ninov, Dimitar N a écrit :
> Dear Colleagues,
> Concerning modulation, the sole factor which determines the proximity between two keys is the availability of pure diatonic chords shared by those keys. Thus C major and D major have two diatonic chords in common: G and Em. Therefore, it does not make sense to call these two keys "remote", and yet this is how they are categorized in some sources. Throwing the tonalities in two categories only: "closely related keys" and "remote keys" ignores the fact that relationships change gradually, and there are intermediate stages.
> In my teaching, I classify the proximity between keys in three different levels/degrees. This approach is very consequential, for it classifies the proximity based on the availability of diatonic chords. The idea is not mine – it has been introduced  in various harmony books among which "Praktischer Leitfaden Der Harmonielehre" (Richard Stör, Wien, 1908, 9, 11, 17) and "Uchebnik garmonii", by four Russian authors: Dubovskii, Evseev, Spossobin, Sokolov (Moscow, 1955). It is exposed below.
> 1. Fist Degree of Proximity: relative keys and those which differ by one accidental. These tonalities may be called "close" or "closely related". A certain key has five closely related keys, and these are the keys whose tonics are diatonic major or minor triads built on the natural scale of that key. Close to C major are d, e, F, G, and a. Close to C minor are Eb, Fm, Gm, Ab, and Bb. Relative keys share seven diatonic chords, and those which differ by one accidental share four diatonic chords.
>   2. Second Degree of Proximity: keys which differ by two accidentals. I call these "relatively close". A certain key has four relatively close keys. To C major these are D, b, Bb, and g. To C minor these are F, d, Db, and b-flat minor. Relatively close keys share two diatonic common chords.
> 3. Third degree of Proximity: keys which differ by three or more accidentals. These may be called "remote". For example, C and f#. Such keys do not share diatonic chords, but a good common chord may always be found in the realm of borrowed chords (modal interaction).
> Therefore, it becomes clear that there is no "Chinese wall" between "closely related" and "remote", but the relationships unfold gradually. One may even create more levels of proximity, for example, the first level may only include the relative keys, etc. while a fifth level may be based on the opportunities of obtaining most typical borrowed chord relationships, etc. (for instance C major and F minor are keys which are formally remote but practically are closer than other relationships).
> This reminds me that Rimsky Korsakov has a different system of classifying the proximity between keys, placing the minor S in a major key and the major D in a minor key as proximity of degree one. I personally prefer the simpler concept exposed above, for Fm is a great S chord into C major, but the tonality of F minor itself does not reveal the same level of closeness to C as the tonalities of G, e, F, and d. The same comparison may be made between C minor and G major. G is a great actual dominant in C minor, but as tonality, with all of its chords, it does not seem as close to Cm as Bb, g, f, and Ab.
> We shall also keep in mind that, in order to find infallibly all the available purely diatonic chords shared  between two keys, we shall only look into their keys signatures and nothing else. This will instantly reveal the chords built on the natural scale of that key (whether it be major or minor). This process is also spoiled by some authors, as a result of which they exclude legitimate chords from the process and confuse the role of the common chord with that of the modulating chord which follows. But I will have to say more on this in a further email.
> Thank you for your attention. Have a nice weekend!
> Best regards,
> Dimitar
> Dr. Dimitar Ninov, Lecturer
> School of Music
> Texas State University
> 601 University Drive
> San Marcos, Texas 78666
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