crpurcell / fretboard_compensation

Calculating compensated nuts and saddles for correct instrument intonation.

Fret, Nut and Saddle Compensation for Correct Intonation

The Jupyter notebooks here describe how to calculate nut and saddle offsets to optimise the intonation for a stringed instrument. The error in frequency tends to increase (sharper notes) towards higher frets because of increasing distance between the string and fretboard. Almost perfect intonation can be achieved by 1) offsetting the saddle further from the nut, which tends to flatten the error curve, 2) offsetting the nut, which tends to shift the error curve down in frequency and 3) minimising the action at the nut, which prevents intonation issues at the first few frets.

The two Jupyter notebooks contain detailed explanations and python code exploring the physics of intonation offsets due to 1) depressing a string to a fret and 2) the effect of string stiffness. The results of these investigations have been used to implement a python code fretCalc.py that produces a scale diagram of a compensated fretboard for use by luthiers.

Using fretCalc.py

The fretCalc.py code defines classes that can be used to fit for optimal nut and bridge offsets that minimise the intonation error of each string of an instrument. The code is meant to be used interactively, e.g., from an iPython terminal or Jupyter notebook, and is capable of producing a scale fretboard diagram.

Values of elasticity (Young's modulus) for nylon strings range from 4.41 GPa - 5.40 GPa, with the former being the value used in the literature for Aquila branded strings (Wong 2014). However, string parameters should ideally be measured directly (notes coming soon!).

The following shows how to run fretCalc.py from an interactive python shell or notebook:

# Import the instrument and string classes
from fretCalc import *

# Define an instance of the instrument class
myUke = instrument(nFrets=18, scaleLength_m=420.0/1000, actionNut_m=1.5/1000, \
                   actionFret12_m=3.5/1000, fretHeight_m=1.0/1000, \
                   depressHeight_m=0.5/1000, depressMod=0.2, stringSep_m=7/1000)

# Add A4 string
myUke.add_string(freqOpen_Hz=440.00, massPerLength_kgm=1e-4, \
                 stringDiameter_m=0.56/1000, elasticity_Pa=4.41e9)

# Add E4 string
myUke.add_string(freqOpen_Hz=329.63, massPerLength_kgm=2e-4, \
                 stringDiameter_m=0.73/1000, elasticity_Pa=4.41e9)

# Add C4 string
myUke.add_string(freqOpen_Hz=261.63, massPerLength_kgm=4e-4, \
                 stringDiameter_m=0.89/1000, elasticity_Pa=4.41e9)

# Add G4 string
myUke.add_string(freqOpen_Hz=392.00, massPerLength_kgm=1e-4, \
                 stringDiameter_m=0.62/1000, elasticity_Pa=4.41e9)

# Fit the fretboard
myUke.fit_fretboard()