# Is there a name for this exponential analog to modular arithmetic? (octave equivalency)

In music theory, there is a concept called octave equivalency: two pitches are said to have the same pitch class if the quotient of their frequencies is a power of 2, i.e. if they are an integer number of octaves apart.

This relation can be generalized to use powers of $n$ other than 2. For example, the Bohlen-Pierce Scale doesn't use octave-equivalency, but what it calls tritave-equivalency; pitches are equivalent if the quotient of their frequencies is a power of 3.

It seems to me that this is analagous to modular arithmetic, but exponential rather than multiplicative: two numbers are congruent mod $k$ if their difference is a multiple of $k$. So while the modulo operation and modular congruence can be recursively defined as:

$$n\bmod{k} = \begin{cases} (n-k)\bmod{k}, & \text{if n\ge k} \\ (n+k)\bmod{k}, & \text{if n<0} \\ n, & \text{otherwise} \end{cases} \\ a\equiv b\pmod{k} \iff a\bmod{k} = b\bmod{k},$$

octave/tritave/etc.-equivalency (more generally $k$-tave-equivalency) can be recursively defined with:

$$n \text{ tave } k = \begin{cases} (\frac{n}{k}) \text{ tave } k, & \text{if n\ge k} \\ (n\cdot k) \text{ tave } k, & \text{if n<1} \\ n, & \text{otherwise} \end{cases} \\ a\equiv b\;\;\;\;\text{(k-tave)} \iff a \text{ tave } k = b \text{ tave } k$$

Another way of looking at the two concepts:

• $a$ and $b$ are congruent mod $k$ iff the fractional part of $\frac{a}{k}$ equals the fractional part of $\frac{b}{k}$.
• $a$ and $b$ are $k$-tave-equivalent iff the fractional part of $\log_k{a}$ equals the fractional part of $\log_k{b}$.

Is there a name for the operation I call "tave" above? Is this a concept that gets use in mathematics outside music theory?

• You are looking at the quotient group $G/H$ where $G$ is the group of positive real numbers under multiplication and $H$ is the subgroup generated by $2$. – Lord Shark the Unknown Jan 24 '18 at 7:31
• I'm not very familiar with group theory; I'm assuming $H$ is the subgroup generated by $k$ in general, with $k=2$ in the case of the octave, yes? – A. Bicksler Jan 24 '18 at 7:35
• Indeed${}{}{}$. – Lord Shark the Unknown Jan 24 '18 at 7:36
• Then, by analogy, is modular arithmetic the difference group $G-H$ where $G$ is the group of positive reals under addition and $H$ is the subgroup generated by $k$? – A. Bicksler Jan 24 '18 at 7:38