## Background

It has been known since Pythagoras that musical intervals correspond to a whole-number ratio of frequencies between the notes. The ratios for the 12 intervals in a chromatic scale are given below.

The problem is that these cannot be used as basis for for tuning an instrument because the intervals do not add up to what you would expect. As an example, you would expect that two major seconds, which is (\frac{9}{8})^2 = \frac{81}{64} would give the same ratio as a major third, but this is clearly not the case. Famously, the difference between 12 fifths and 7 octaves, which should be the same, is known as the Pythagorean comma:

\frac{(\frac{3}{2})^{12}}{2^7} \approx 1.01364.

One could fix a base note and then use the intervals from the table above to tune all keys on a piano, but because the notes are not evenly spaces, this will not allow transposition – a melody will sound different depending on what key it is played in. The most common solution to this problem is to use equal temperament which divides the octave into 12 equally sized ratios. This permits transposition but has the caveat that all intervals will only be approximately equal to their ideal ratio.

The equal temperament has been the most commonly used tuning system for centuries, but there have been some suggestions on how to improve the intonation of instruments, especially after the invention of electronic instruments. A recent example is presented in [1], where a method to compute just tuning in real-time using the method of least-squares is presented, and there is also a brief historical overview of other approaches.

Below, I present a method for optimal tunings which is very similar to the one in [1], but it is a lot simpler to understand and implement and I believe the resulting tuning is very similar. The idea behind this method is simply to use just tuning from a based on the lead/melody line. To describe it, we first need to define some formalism about tuning systems.

## Tuning systems

A tuning system may be described as a strictly increasing function \tau: \mathbb{Z} \to (0, \infty) such that \tau(0) = 1. The equal tuning can, for example, be described by the relative tuning function

\tau_{\text{12-ET}}(n) = 2^{\frac{n}{12}}.

As an example of how to use this function, we first pick a base, for example that note n = 0 corresponds to A4, so it has frequency 440 Hz. The note a fifth (seven semi-tones) above then has frequency 440 Hz \times \tau_{\text{12-ET}}(7) = 659.255 Hz.

Given a base note, the just intervals in the table above defines a tuning \tau_{\text{Just}} because they define the tuning of the notes 0, \ldots, 12, and all other values may then be derived by the moving up and down in octaves, e.g. \tau_{\text{Just}}(n + 12) = 2 \tau_{\text{Just}}(n) .

We represent a score with v monophonic parts as a matrix of size v \times l with integer entries. The first row is assumed to be the leading part and l is the length of the score. The score is discretized such that a single column corresponds to the shortest note duration in the score and longer notes are represented by spanning multiple columns. This formalism cannot capture neither repeated notes nor rests, but since we are only interested in harmony, it will suffice for our purpose.

As an example, consider the following arrangement of the first four bars of Air on the G String by J. S. Bach / August Willhelmj.

Using standard MIDI numbers for notes where A4 corresponds to 69, this may be represented by the following matrix:

\left(\begin{matrix}
76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 76, 81, 77, \cdots \\
67, 67, 67, 67, 67, 67, 67, 67, 72, 72, 72, 72, 72, 72, 72, 72, 69, 69, 72, 72, \ldots \\
60, 60, 60, 60, 59, 59, 59, 59, 57, 57, 57, 57, 55, 55, 55, 55, 53, 53, 53, 53, \ldots \\
\end{matrix}\right)

which we will denote A = (a_{ij}). The goal of the tuning is to translate each of these notes a_{ij} into a frequency f_{ij} \in (0, \infty). This can be done using the equal tuning by setting f_{ij} = 440 \times \tau_{\text{12-ET}}(a_{ij} - 69) for all i, j. This will result in the following:

To get pure intervals we instead use the following method: Assume that frequencies f_{1,1}, \ldots, f_{1,L} for the first row has been determined. Now, the frequencies for row i is defined by

f_{ij} = f_{1j} \tau_{\text{Just}}(a_{ij} - a_{1j}).

This ensures that all intervals between the first and the j‘th row are just.

In order to tune the first row we can either use equal tuning, but to ensure just step intervals in the lead we instead pick a frequency for the first note, in this case f_{1,1} = 660 because the first note in the lead is A5, and define

f_{1,j} = f_{1,j-1} \tau_{\text{Just}}(a_{1,j} - a_{1,j-1}).

for j > 1. The resulting arrangement with dynamically adaptive just tuning as described above sounds like this.

The tuning is dynamical, meaning that the same note played in different places in the score may be tuned to different frequencies. This may even be true for sustained notes, if the melody changes in the duration of the sustained note. As an example, the F# half-note played by the third voice in the second half of the third bar changes frequency for each 8th note duration, and is tuned as 183.33, 182.52, 183.33 and 182.52 Hz resp. in the duration of the half note.

As it is presented here, the method may be applied to a fixed score where all notes are known in advance, but it could also be used in real-time assuming the program performing the tuning has a well-defined way of determining the lead, for example the highest played note, and then use this as the base.

The code used to generate the sound clips is available on GitHub.

## Literature

[1] K. Stange, C. Wick and H. Hinrichsen, “Playing Music in Just Intonation: A Dynamically Adaptive Tuning Scheme,” in Computer Music Journal, vol. 42, no. 3, pp. 47-62, Oct. 2018, doi: 10.1162/comj_a_00478.