## Monday, 25 April 2011

### These go to 11

The closed interval $[0, r]$ of real numbers is the set $\{x \in \mathbb{R} : 0 \leq x \leq r\}$. Observation: there is an isomorphism $f : [0, 10] \rightarrow [0,11]$. For example, let $f$ be the function: $f(x) = \frac{11}{10}x$. Then $f(0) = 0$ and $f(10) = 11$. And, clearly $x < y \text{ iff } f(x) < f(y)$.
So? Well, the interval $[0,10]$ is a structure known to all guitarists, as it provides a convenient labelling of the loudness of the output of their amp. Write "$Lo_1o_2$" to mean "output setting $o_1$ is less loud than output setting $o_2$". The settings of the output can be "labelled" by real numbers, and a function which maps each setting to real numbers is a measurement scale. The sole requirement on such a function is that $m$ somehow "represents the loudness relations" of the output settings.
Suppose $m$ is such a measurement scale. Write $m(o) = r"$ to mean "$m$ assigns real number $r$ to the output setting $o$". The representation condition is then:
$m(o_1) < m(o_2) \text{ iff } Lo_1o_2$
Given the nature of the physical device itself, the possible loudness settings have a minimum and a maximum. Call these $o_{min}$ and $o_{max}$. It is convenient to choose $m$ such that:
$m(o_{min}) = 0$ and $m(o_{max}) = 10$.
All intermediate loudness settings get mapped to reals between 0 and 10.

However, it is entirely a matter of convenience that guitar amplifier manufacturers adopt this convention. (And the loudness output labelled by 10 on one amp can be very different from the loudness labelled by 10 on another amp.) Measurement scales are usually non-unqiue, and, depending on the representation conditions all such scales must satisfy, there is a class of transformations from one scale to another. So, with guitar amps, why not have a measurement scale $m^{\ast}$ such that,
$m^{\ast}(o_{min}) = 0$ and $m^{\ast}(o_{max}) = 11$?
This is possible, as noted above: there is an isomorphism $f : [0, 10] \rightarrow [0,11]$. So, we can define $m^{\ast}$ by: $m^{\ast}(o) = f(m(o))$. This scale $m^{\ast}$ is the (well, a) Spinal Tap measurement scale:

Roy mentions the xkcd cartoon on this conceptual puzzle: