Microtone (interval size measure)
The microtone is a unit of interval size equal to one one-millionth of a whole tone. It can be considered as sufficiently precise for all thinkable musical and music-science purposes. Besides its high accuracy, it is of high neutrality since it favors neither twelve-tonality nor even the octave. Nevertheless its relevance for practical application in music is low due to the incompatibility with "human scale". Humans are not good at big numbers and have only limited pitch perception (the Just-noticeable difference (JND) is around 6 cents).
One actual microtone (1µt) would be defined as one millionth of the tone. Alternately, one symmetric microtone or "round number" microtone can be permitted to be derived from 1/6 of an octave.
| Name | Size | Size (¢, 12 decimals) | Size (in cents (¢), scientific notation)[1] |
|---|---|---|---|
| Tone (=9/8) | 204¢ | 203.910,001,730,775 | 2.03910001730774835488973465474759621023555E+2 |
| Millitone | 204m¢ | 0.203,910,001,731 | 2.03910001730774835488973465474759621023555E-1 |
| Microtone | 204µ¢ | 0.000,203,910,002 | 2.03910001730774835488973465474759621023555E-4 |
| Tone (=2\12) | 200¢ (exactly) | 200 | 2E+2 |
| Microtone | 200µ¢ | 0.0002 | 2E-4 |
- ↑ The values were produced by High precision calculator.
The microtone challenge
A microtone is indeed a very small interval: 4,904 microtones make one cent, and 5,884,949 an octave.
Helmholtz had argued that pitches spaced less than 5 cents are generally not possible for a human to perceive as different, but the difference itself is easy to perceive as beats of constructive/destructive interference between the two slightly different waveforms.
However, two sounds different only by only 1µt produce a very slow beat; depending on the frequency one have to wait more or less to recognize it. The beat frequency is
- at the upper limit of the human hearing range (20 kHz) 7 minutes
- in the range of the highest acoustic sensibility (4 kHz) 35 minutes
- at the lower limit of the hearing range (16 Hz) 7 days
Given this, will it be ever possible to make a microtone experience at all?
According to Paul Fraisse, the slowest beat humans can subjectively distinguish is about 1800ms, which corresponds to about 33 bpm[1]. Therefore, the general lowest limit of human perception between tones is roughly 250 µt, or about 0.05 cents. A more practical approximation might be to take 33 bpm of interference beats at 4 kHz, which is about 0.25 cents, as a practical limit of perceived difference between two simultaneously performed tones. With that in mind, it seems that adjustments in tuning of practical music on the order of microtones is necessarily imperceptible even to the most veteran ears.
See also
Notes
- ↑ Fraisse, P. (1982). Rhythm and tempo. In D. Deutsch (Ed.), The psychology of music (pp. 149-180). New York: Academic Press.