Pitch Shift From Change In Loudness
by Johan Liljencrants (031217 MMDigest)

With reference to the MMD thread, "Apparent Frequency Change With Intensity", of  2003.12.12.04 and following, the attached samples illustrate one way to see how loudness affects pitch.

Intensity level and sound pressure level are physical measures that tell how 'strong' a sound is, the 'volume' of it, most often on a standardized decibel (dB) scale. Idiosyncrasies of hearing physiology combined with physical phenomena like resonance in the ear canal make up for a complicated relation between intensity and the perceptual measure of loudness level, measured in Phons. This relation is generally described by the famous Fletcher-Munson curves. The affix 'level' implies these are logarithmic measures. Furthermore there is a perceptual quantity of comparative loudness, measured in Sones, a linear measure. Twice the number of Sones, twice as loud perceptually. It is noteworthy that this over most of the hearing range this corresponds to about 9 Phon and 9 dB - to make your hi-fi twice as loud you have to increase its power by a factor of 8, not only 4.

Frequency is a physical measure that tells the number of cycles per second in a periodic signal. Frequency can usually be measured in Hz with high accuracy, using appropriate instruments. For a contrast, pitch is a psychological measure, how you perceive frequency. This is not so easy to find out because human individuals differ a lot, and also do not have very tangible pointers or digital displays. Instead you have to use indirect methods, and also take an average of results obtained from many persons.

This is also complicated by the fact that you perceive pitch two different ways, in what we may roughly call a musical context as opposed to a non-musical one. The prototype example of a non-musical tone is a sinusoid, a signal having one specific frequency component. The present illustrations are of this type, representing some 'lab' conditions. A prominent finding is that by varying loudness you can cheat the perception of pitch corresponding to one semitone or more.

But a continuous musical note generally is complex, it has a fundamental plus several harmonics sounding simultaneously. Musical pitch is established in octaves (frequency ranges with factors 1:2), each divided into 12 semitone intervals with names from A through #G. For microscopic work one semitone interval can further be divided into a range of 100 cents. With such complex sounds the ear is remarkably sensitive, you usually can perceive deviations to the order of a few cents.

The following three illustrations are at frequencies 100, 200, and 400 Hz respectively. Each has six pairs of tones. In each pair there is first 1 second of a weak tone, then 1 second of a strong tone, 30 decibel higher level. The pairs are interlaced with 0.5 second of silence. Ideally they should be reproduced at about 56 and 86 dB sound level respectively, the latter pretty loud, so probably you should listen to them over headphones, rather than small computer loudspeakers.

  100 Hz (30 kb)
  200 Hz (30 kb)
  400 Hz (30 kb)

In one sample of six pairs, the weak first tone in a pair is always exactly the same. But the following strong tone is successively rising in frequency, by 0, 20, 40, 60, 80, and 100 cents respectively. Listening to the first pair you will probably perceive that the strong tone has a lower pitch than the leading weak one. But somewhere along the sequence the weak and the strong one should have the same pitch. The ordinal number of this pair determines your perceptual pitch shift at a 30 dB level change, in terms of cents.

A special remark is that winds and strings in an orchestra tend to raise their frequencies a few cents at loud passages, presumably to conserve pitch.

Johan Liljencrants
Stockholm, Sweden
19 Dec 2003 00:54:08 +0100

19 December 2003