**Basic acoustic features**

The basic physics of a vibrating string are assumed to be intuitively familiar to readers. The sound produced when a piano string is struck by the hammer can be described as a superposition of all possible modes of vibration (this accounts for the overtone spectrum). By touching the string with a finger at a certain point, all modes of vibration without a node at this point are cancelled, leaving only those with a node. For example, touching the string at a point 2/7 of the string length from the agraffe produces the 7th harmonic, approximately two octaves and a minor seventh above the fundamental.

The following table shows the first 22 harmonics and their pitch relative to their fundamental (both theoretical and measured). In reality the harmonical series is somewhat stretched; the low harmonics are slightly lower and the high harmonics slightly higher in frequency than the integer multiples of the fundamental frequency predicted by theory.

Harmonic# |
Relevant nodes (for bass strings) |
Theoretical pitch relative to fundamental |
Measured pitch (average) |
Difference in cents |

1 |
||||

2 |
1 octave above | |||

3 |
1/3 | 1 octave + perfect 5th + 2 cents | 1 oct+P5-4c | -6 |

4 |
1/4 | 2 octaves | 2 oct-5c | -5 |

5 |
1/5, 2/5 |
2 octaves + major 3rd – 14 cents | 2 oct+M3-18c | -4 |

6 |
1/6 | 2 octaves + perfect 5th + 2 cents | 2 oct+P5-3c | -5 |

7 |
2/7 | 2 octaves + minor 7th – 31 cents | 2 oct+m7-35c | -4 |

8 |
3/8 | 3 octaves | 3 oct | 0 |

9 |
1/9, 2/9 | 3 octaves + major 2nd + 4 cents | 3 oct+M2 | -4 |

10 |
1/10, 3/10 | 3 octaves + major 3rd – 14 cents | 3 oct+M3-16c | -2 |

11 |
1/11, 2/11, 3/11, 4/11 |
3 octaves + augmented 4th – 49 cents | 3 oct+A4-48c | 1 |

12 |
1/12 | 3 octaves + perfect 5th + 2 cents | 3 oct+P5+2c | 0 |

13 |
1/13, 3/13, 4/13 | 3 octaves + minor 6th + 41 cents | 3 oct+m6+42c | 1 |

14 |
1/14, 3/14 |
3 octaves + minor 7th – 31 cents | 3 oct+m7-28c | 3 |

15 |
1/15, 4/15 | 3 octaves + major 7th – 12 cents | 3 oct+M7-6c | 6 |

16 |
3/16, 5/16 | 4 octaves | 4 oct+7c | 7 |

17 |
3/17, 4/17, 5/17 |
4 octaves + minor 2nd + 5 cents | 4 oct+m2+15c | 10 |

18 |
5/18 | 4 octaves + major 2nd + 4 cents | 4 oct+M2+15c | 11 |

19 |
4/19, 5/19, 6/19 | 4 octaves + minor 3rd – 2 cents | 4 oct+m3+8c | 10 |

20 |
4 octaves + major 3rd – 14 cents | n/a | ||

21 |
4/21, 5/21 | 4 octaves + perfect 4th – 29 cents | 4 oct+P4-12c | 17 |

22 |
5/22, 7/22 | 4 octaves + augmented 4th – 49 cents | 4 oct+A4-29c | 20 |

Obviously, real harmonic nodes are not points, and harmonic sounds are usually a mixture of adjacent harmonics, especially for higher harmonics. However, the harmonics corresponding to more distant nodes are usually dampened more quickly, as can be heard on several recordings on this site. Where the same harmonic can be obtained from different nodes, the timbre is usually slightly different. For example, the 11th harmonic is available from two nodes, 2/11 and 3/11, which are subtly coloured by adjacent harmonics (17th and 16th, and 15th and 18th, respectively). Multiphonics can be produced by placing the finger between nodes. There is plenty of room for experimentation here.