String Harmonics

Module for analyzing positions of natural harmonics on a stringed instrument relative to positions of fingered notes.

AUTHOR:

• David Zelinsky dsz@dedekind.net

COPYRIGHT NOTICE:

This program is free software: you can redistribute it and/or
modify it under the terms of the GNU General Public License as
published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <https://www.gnu.org/licenses/>.

DISCUSSION:

A string has multiple modes of vibration. The most common is the fundamental, or open string. But for each positive integer N there is a mode in which the string appears to vibrate in N independent segments of equal length, with a "node" that appears stationary occurring at a fraction k/N of the distance from the nut to bridge, for each of the N-1 multiples k/N of 1/N. In this mode, the string resonates at a frequency that is N times the fundamental.

The string can be induced to vibrate in its N-th harmonic mode by touching it lightly at one of the N-1 nodes (provided that node is not also a node for some lower harmonic).

Two questions arise concerning these harmonics:

1. What is the pitch of the N-th harmonic on a given string?

2. Where are the nodes? That is, what fingered note is closest to each of the nodes?

This module has functions to answer each of those questions, with the answer presented in various formats. See the methods in the "Output" section of the code for more explanation.

A word about pitch notation used in the code:

All pitches refer to chromatic scale positions in tempered tuning. It is also assumed that when multiple strings are involved, they are tuned according to tempered pitch.

• Diatonic notes are referred to by letter names "A,B,..." in upper case (with lower case also recognized).

• Sharps and flats are denoted by a following "s" or "f" respectively.

• For absolute pitches, the octave is indicated using Scientific Pitch Notation: Each octave is numbered, starting on C, with middle C being octave number 4.

For example, "Ef3" denotes the E-flat in the octave below middle C.

Note that this differs from Lilypond notation (even in "English" mode), since Lilypond requires lower case note names; and octaves are denoted by a number of punctuation marks (single quotes or commas), with an unadorned "c" representing C3.

TABLE OF CONTENTS:

• harmonics.py

  • Global Data

  • Utility Functions

    • signed
    • ordinal
    • parse_note
    • note_with_accidental
    • note_to_number
    • number_to_interval
    • number_to_note
    • relative_octave
    • instrument_strings
    • note_in_staff

  • Computation

    • harmonic_interval
    • note_positions_near_harmonic_number
    • notes_near_harmonic_on_string
    • note_positions_near_harmonic_pitch

  • Output

    • print_harmonics
    • print_harmonics_by_position
    • print_harmonics_for_notes
    • lilypond_harmonics
    • lilypond_cello_harmonics

• intro.ly

  • introductory front-matter for lilypond_cello_harmonics

• cello-harmonics.ly

  • lilypond source file produced with lilypond_cello_harmonics

• cello-harmonics.pdf

  • typeset catalog of cello harmonics