Information Loss

Here are some basic facts about the sets our musical information lives in:

• Pitch is 1-dimensional: frequencies are real numbers.
  • That is: they live in $\R$.
• Meantonal’s Pitch and Interval vectors are discrete and 2-dimensional.
  • That is: they live in $\Z^2$.
• Tuning systems like 12-tone equal temperament have a finite number of pitches per octave.
  • That is, they live in $\Z$.
• Abstractions like “12 tone pitch class” or “scale degree” have a finite total number of values, and work by modular arithmetic.
  • That is, they live in $\Z_n$ for some $n$.

Most of Meantonal’s information querying is done via linear maps from $\Z^2$ to $\Z$ via multiplication with a $1\times2$ mapping matrix.

Note

To find the MIDI value of $\sf{C}\sharp_4$, represented by $(26, 9)$, we use the MIDI mapping matrix $[2, 1]$:

$$[2, 1] \begin{pmatrix} 26 \\ 9 \end{pmatrix} = (2)(26) + (1)(9) = 61$$

Notice we went from a 2d vector to a single integer: $\Z^2\rightarrow\Z$.

No linear map from $\Z^2$ to $\Z$ is invertible. If $\sf{C}\sharp_4$ and $\sf{D}\flat_4$ are both sent to the MIDI number 61, the mapping was not one-to-one. No inverse map can exist.

Modular information like pitch class can be queried by taking remainders after regular Euclidean division.

Note

To find the 12-tone pitch class of $\sf{C}\sharp_4$, which we established was represented by the MIDI number 61, we could simply divide by 12 and take the remainder:

$$61\div12 = 5\; \sf{remainder}\; 1$$

1 is therefore the 12-tone pitch class of $\sf{C}\sharp$. Since there are an infinite number of integers, but only 12 pitch classes: $\Z\rightarrow\Z_{12}$.

The previous map can also be composed with this one: $\Z^2\rightarrow\Z_{12}$.

No map from $\Z$ to $\Z_n$ is invertible. If MIDI numbers 65 and 77 are both sent to the 12-tone pitch class 5 the function was not one-to-one. No inverse function can exist.

These sorts of maps are called semantically destructive in Meantonal.

Meantonal tries to preserve semantic information. This is achieved by keeping data in vector format wherever possible, and then simply extracting lower-dimensional representations when they are required.

Caution

If you are integrating Meantonal into an existing project where musical information is stored differnetly, it may only be possible to translate in one direction: from Meantonal’s representation.

If multiple Pitch or Interval vectors map to the same value in your system, the mapping is not one-to-one, and no inverse mapping can exist.

This means you will generally want to place any 1-dimensional pitch handling downstream of vector-based data structures and operations. Query MIDI or pitch class values, and operate them directly if possible, before continuing to operate on the original Pitch or Interval vectors the information was extracted from.