Install

Package manager

This program has been coded on a Mac OS environment. You will find detailed instructions for a Mac OS platform. As a first step, install brew as a package manager.

https://brew.sh/

For other platforms, use the package manager you are familiar with, given that it is purely optional (the dependencies can be installed without it).

Python3

This program is coded in Python. Install python language on your operating system

For MacOS users:

brew install python3

For other platforms, you will find instructions on the Python3 User Guide page: https://www.python.org/

Make sure the python version is correctly installed with its package manager pip3. The following commands should return no error.

python3 --version
pip3 --version

Tkinter

This program uses tkinter as user interface. It is required to install it.

For MacOS users:

brew install python-tk

For other platforms, you will instructions on the Tkinter User Guide page: https://docs.python.org/fr/3/library/tkinter.html

Python libraries

Python libraries can be installed with the help of virtualenv, which is used to install python packages on an isolated environment. It is strongly recommended to use it to avoid polluting the operating system. Normally, it should already been installed with your python installation. In case it is not check the virtualenv User Guide: https://virtualenv.pypa.io/en/latest/installation.html

Create a new virtual environment for python

python3 -m venv .venv

Activate the new virtual environment

source .venv/bin/activate

Install the python libraries

pip3 install -r requirements.txt

Run

To run the program, launch the entry file accord.py

./accord.py

Setup

To setup the program preferences, open the settings window by clicking on the "Settings" button.

Device

Choose the device to record the piano sound. Usually, laptop microphone is fine.

Sample rate

For best quality, use the highest possible sample rate (for example 192000 samples / second)

Channels

Use this program in 1 channel (mono) as it is not intended for listening purposes.

Interval

This setting is the delay between matplotlib frames in milliseconds. Use the default setting 30ms.

Duration

This is the duration in seconds of the recording window for the sound. Use a large value for low pitches (30s) and a low value for high pitches (1s).

Zoom

This is the ratio around which the axis window will be displayed. Start with the default value and play with the setting to adjust the window length. It is expressed in percentage of the frequency. (Ex: 8% zoom for a 440Hz A4 will display a window between 405 Hz and 475 Hz)

Number of harmonics

This settings defines the number of harmonics to be computed. Usually, 12 can be computed for low sounds, 6 for medium sounds, 2-3 for very high sounds.

Figure width

This is to adjust the number of windows per row in the matrix of harmonics. A figure width of 3 for 6 harmonics will display 2 rows.

Inharmonicity

Piano harmonics are not pure multiples. For example a A4 with frequency 442 Hz will have its second harmonic at around 883 Hz. In this example, the inharmonicy factor will be 400 * log2(883/882) = 0.654

This is piano dependent and will have to be computed in each situation.

For advanced users, it corresponds to the K factor in this paper. http://www.temperamentcordier.org/inharmonicite/Memoireinharmonicite%20.pdf

Inharmonicity ratio

Inharmonicity factor increases with frequency. It is often set around 10 ^ 0.04 = 1.096

This is piano dependent and will have to be computed in each situation.

For advanced users, it corresponds to the q factor in this paper. http://www.temperamentcordier.org/inharmonicite/Memoireinharmonicite%20.pdf

Advanced estimation of inharmonicity

Mathematical works have been filed in the folder called "Inharmonicite". It consists of a series of mathematical operations based on a pre-recorded of piano string samples in isolation. For a pitch with multiple strings, only 1 string has been recorded (for example 1 string out of 3 on A4).

The articles used for this computation are the following:

• https://fr.wikipedia.org/wiki/Inharmonicit%C3%A9
• http://www.temperamentcordier.org/inharmonicite/Memoireinharmonicite%20.pdf

Get harmonics

This function returns the harmonics of the recorded sounds

Get stiffness

This function estimates the stiffness of the recorded sounds. It is based on the formula

$$fn = n \sqrt{1 + B n^2} f0$$

For 2 partials p and k, the following formula can be used to estimate the stiffness B:

$$B_{k, p} = \frac{p^2f_k^2 - k^2f_p^2}{k^4f_p^2 - p^2f_k^2}$$

All combinations of (k, p) have been estimated for a given pitch, and the stiffness returned is the mean of all computations.

Model stiffness

This function models the stiffness based on a polynomial assumption (degree 4)

Estimate error

This function estimates the error between the estimated stiffness by model and the measured stiffness

Find inharmonicity

This function estimated the inharmonicity factor for the A4 string. It correspond to the Inharmonicity setting of the settings window

It is based on the formula:

$$f_2(n) = 2f_1(n)2^{\frac{3I}{1200}}$$

which gives:

$$I = 200 log_2{\frac{1 + 4B_n}{1 + B_n}}$$

Find inharmonicity ratio

This function estimated the inharmonicity ratio for the entire piano. It correspond to the Inharmonicity Ratio setting of the settings window

It is based on the formula:

$$f_2(n) = 2f_1(n)2^{\frac{3Iq^n}{1200}}$$

Dividing stiffness for adjacent pitches gives:

$$q = \frac{log_2{\frac{1 + 4B_{n+1}}{1 + B_{n+1}}}}{log_2{\frac{1 + 4B_{n}}{1 + B_{n}}}}$$