Secondary Dominants

This is a Western music theory thing, the notion being given a key, such as an A key, or A major, one then has a starting note, A, or the Tonic, and a fifth above that (a 3:2 frequency ratio, or roughly so depending on the tuning system) the Dominant.

    A   I   Tonic
    B   II
    C#  III
    D   IV
    E   V   Dominant
    F#  VI
    G#  VII
    A   I

    $ atonal-util pitch2freq a\'
    69      440.00
    $ perl -E 'say 440 * 3 / 2' 
    660
    $ atonal-util pitch2freq e\'\'
    76      659.26

The exact frequencies have varied over time and usually tone generators are spiced up so that there isn't an A440 test tone all the bloody time.

A secondary dominant is when you take other notes of the scale besides the Tonic, treat those other notes as a Tonic, and then figure out what the Dominant of that is. In Roman Numeral notation this is usually written V/II or V/ii or such, so in A major that's to treat the note B as the tonic, go up a fifth from that, F sharp, and then one can have a progression such as I, V/II, II, V,

    $ vov --transpose=a I V/ii ii V
    a cis e
    fis ais cis
    b d fis
    e gis b
    $ vov --natural --transpose=a I VI II V
    a cis e
    fis a cis
    b d fis
    e gis b

which could also be written as I, VI, II, V though a secondary dominant may involve out-of-scale notes. The A sharp of VI (as opposed to the natural vi of F# A C# which is not a Dominant as it's minor not Major) is also part of the Neapolitan chord which is a sidequest yet one probably worth looking into.

The good news is that there are not many scale degrees, so it should be straightforward to figure out all the secondary dominants and then to try them out in various harmonic progressions. One may also spot them when reviewing musical scores, if only one knows what to look for.

    A   I   E - the normal dominant
    B   II  F# A# C#
    C#  III G# ...
    D   IV 
    E   V   ... V/V/V/V walks around the circle of fifths ...
    F#  VI
    G#  VII

Name

The expression, Nebendominanten (secondary dominants), may not be original with me, but (I cannot recall precisely) probably arose in a conversation with a musician to whom I presented the idea of constructing dominants on the secondary triads. Which of us first uttered the expression I do not know, and for that reason I assume it was not I.

— "Theory of Harmony". Arnold Schoenberg. 1922.

Usage of the thus named technique is however older than when it was named, just as the use of grammar is older than being force-fed grammar in middle school.