You guessed it! … 1.414 and 1.059…

These are both irrational numbers like π, and like π are both quite important. I recall my mother telling me bedtime stories about 1.414 and 1.059, although 1.059 understandably would cause nightmares.

If you haven’t guessed, this is about geometric means (and arithmetic means), and this is important for both the audiogram and for the calculation of the frequencies of musical notes.

If there is ever a reason to look back at your grade 11 (or in the United States “11th grade”) math textbook, you should look at the definition of an irrational number. An irrational number is one that cannot be expressed as a fraction a/b. There are many irrational numbers in audiology and music such as π and , and even the twelfth root of 2 (see part 2 of this blog next week) but today’s blog will concentrate on the ubiquitous square root of 2.

For musicians, the number 1.414 can help to answer the question, “What is the exact frequency of a half octave above a musical note such as C?” For audiologists, the number 1.414 can help us answer the question, “What is the exact frequency of a puretone that is one half of an octave higher?”

Audiologists specifically may want the answer to “What is the inter-octave frequency between 1000 Hz and 2000 Hz?” If you said 1500 Hz, you have mistakenly used the arithmetic mean: (1000 + 2000)/2 = 1500 Hz. Actually the inter-octave frequency between 1000 Hz and 2000 Hz is actually 1000 Hz x 1.414 = 1414 Hz. The inter-octave frequency between 1000 Hz and 2000 Hz is 1414 Hz, which is slightly lower than the arithmatic mean of 1500 Hz. And the inter-octave frequency between 2000 Hz and 4000 Hz is 2000 Hz x 1.414 = 2828 Hz (and not 3000 Hz).

This really won’t make much difference unless the audiogram slopes at more than 40 dB between octaves, in which case the difference between the geometric (true) mean and the arithmetic (false) mean can exceed 5 dB. That is, measuring the geometric mean between 1000 Hz and 2000 Hz means that we should be measuring the hearing threshold at 1414 Hz and not the higher 1500 Hz. After all, the cochlea is laid out in a geometric (and not arithmatic) pattern.

I recall in the “olden days,” when I first entered the field of audiology, that the inter-octave mean was written on the audiogram as 1414 Hz and not 1500 Hz. (I also recall audiograms where the lowest test frequency was the musical note C and audiometers were calibrated to test a tone at 262 Hz (and not 250 Hz), but that is truly ancient history).

In the lower frequency region, the difference between the geometric and the arithmetic means are minimal (e.g., 1414 Hz vs. 1500 Hz,… less than 100 Hz), but in the upper frequency regions (e.g. 5656 Hz vs. 6000 Hz) can be much greater. This 350-Hz difference between the two types of means between 4000 Hz and 8000 Hz can, depending on the severity of the hearing loss, be clinically significant. In cases of severe presbycusis or noise exposure, there may be measurable responses at 4000 Hz but not 6000 Hz, and understandably we would not want to “chase” after making any 6000-Hz sounds audible for this client. It may be, however, that their hearing sensitivity at 5600 Hz is relatively good, and perhaps specifying hearing aid gain in this frequency region may result in improved hearing communication.

Part 2 of this blog will further divide the octave into 12 parts (geometrically speaking) and this is how we can calculate the exact frequency of those notes that are off-octave… stay tuned!