Undertones

If one turns the overtone series in this score sample
on its head, one gets the following undertone series:


There is one very important distinction between overtones and undertones or the superharmonic and subharmonic spectra: whilst overtones are physically present, undertones are not. So in a sense they are mental constructions.

Overtones and undertones are dialectically related to each other: while the overtone spectrum clearly presents a major triad, the undertone spectrum presents a minor triad, which in the score sample below is a C minor triad.
There is a very widespread misconception that undertones exist as physical entities. This misunderstanding is probably due to a confusion between difference tones (or combination tones in general) and undertones.


Combination tones

Combination tones are tones produced by the human ear when two or more pitches are played simultaneously. When this happens, one hears one or more tones over and above the original acoustic signal. In other words, these combination tones are actually formed in the ear, though we have not yet fully discovered the exact mechanism behind this phenomenon. From the listener's point of view, they cannot be distinguished from true sounds coming from outside'.


Difference tones
The most common combination tone is a difference tone. If we listen to two notes on the frequencies 528 and 440 Hz, the difference tone will be on frequency 528 Hz - 440 Hz = 55 Hz. Other combination tones can exist in such cases as: 2 * 440 Hz - 528 Hz = 352 Hz; or, 3 * 440 Hz - 2 * 528 Hz = 264 Hz; or, 4 * 440 Hz - 3 * 528 Hz = 176 Hz.

There are considerable individual differences as to whether particular combination tones are heard or not. In general terms, they are best heard if the sounds are loud.

Nørgård intentionally uses difference tones to emphasise a low fundamental tone. In the example with 528 Hz and 440 Hz, we have in fact partials 5 and 6 in an overtone spectrum with the fundamental tone on 55 Hz. A difference tone between two notes that follow each other in the overtone series will point to a low fundamental tone. (This also occurs as the result of yet another mechanism in the ear known as residual tones - see, for example, Jørgen Mortensen: Musikteoretiske og akustiske aspekter ved konsonans og dissonans (Music Theory and Acoustic Aspects of Consonance and Dissonance), Dansk Årbog for Musikforskning, 1981).

The following score sample shows difference notes related to the simplest intervals in the overtone series. Note that the difference tones together actually form a subharmonic spectrum:



One of the most beautiful passages in which Nørgård intentionally uses difference tones is in Symphony No. 3, at the beginning of the first movement, where there are a large number of falling lines in the overtone spectrum, pointing downwards towards low fundamentals.