Piano Acoustics and Tuning
Introduction
Chapters are published and open for review. Citations are still being added, and the report has not yet been through outside review. Corrections and edit requests are welcome — submit one at the margin of the page or contact the team.
This is a non-introductory document that explains all required signal analysis and processing concepts required to electronically tune a piano string. The following will be covered in great detail:
- Piano string wave behavior and acoustics
- Equal temperament tuning
- Acousto-electric conversion via microphone and preamplifier circuit
- Signal digitization via ADC
- Signal processing of a piano string’s fundamental frequency and harmonics.
- Inharmonicity and harmonic partial alignment optimization.
If you have any revision requests or find issues with this analysis, please do not hesitate to submit an edit request at the margin of the page or contact the author.
- Add citations and references.
- Add and finalize Manim Animations.
- Add additional diagrams since most of these concepts would not make any sense without them.
Mathematical Concepts Covered
Although not strictly required to intuitively understand the tuning of a piano, the following mathematical concepts are required for completeness and to fully understand the Ad Nauseam Analysis.
This legend explains the color coding used throughout the document.
| Symbol | Description |
|---|---|
■ |
Required: Foundational knowledge needed for an intuitive understanding of the physics and algorithms. |
■ |
Recommended: Recommended math needed for full coverage of all physical phenomena. |
■ |
Supplemental: Advanced math required for a complete understanding of all algorithms and physical phenomena. |
Logarithms:
Crucial for understanding decibels, musical intervals, and the equal temperament scale as human hearing is approximately logarithmic.
Complex Numbers:
A corner stone of signal analysis, complex numbers are used for representations of signals (e.g. Euler’s formula) and are implicit in the full formulation of Fourier Transforms.
Series and Summations:
Arbitrary functions or signals can be represented as an infinite sum of simpler terms. This is central to the concept of the Fourier series. Discrete summations are used in almost all digital signal processing algorithms.
Multivariable & Vector Calculus:
For the derivation and understanding of all multi-dimensional physical phenomena, particularly in the case of space and time.
Ordinary Differential Equations (ODEs):
For the derivation and modeling of 1D physical phenomena, and application of system solutions.
Partial Differential Equations (PDEs):
For the complete understanding of the modeling of multi-dimensional physical phenomena, and application of system solutions.
Although the wave equation will be covered in this analysis, it is not strictly required to understand piano acoustic behavior as its plane wave solution is sufficient for most applications.
Difference Equations:
If one is interested in the derivation of ideal samplers, digital signals, and other discrete-time applications, difference equations are required.
Difference Equations are the discrete-time analogue of differential equations. Although intuitively one can understand the behavior of discrete-time systems, mathematical behavior can be surprisingly different from continuous systems. A full understanding requires the use of difference equations.
Non-Convex Optimization:
If one is interested in the workings of optimization algorithms outside the scope of least squares, non-convex optimization is required.