Why the Octave of a Note Sounds the Same but Higher in Pitch

To show why the octave sounds the same as the root note, we examine its harmonics and show that they all coincide with those of the root note.

Note |
Relative Frequency of Note(Just Tuning) |
Frequencies of Audible Harmonics (in Ratios of Whole Numbers) |
Number of Audible Harmonics Coincident with First Ten of Root Tone |
Number of Listed Harmonics Different from First Ten of Root Tone |

C | f_{C} = 1 |
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | 10 | None |

D | f_{D} = 9/8 |
9/8, 9/4, 27/8, 9/2, 45/8, 27/4, 63/8, 9 |
1 | 7 |

E | f_{E} = 6/5 |
6/5, 12/5, 18/5, 24/5, 6, 36/5, 42/5, 48/5 |
1 | 7 |

F | f_{F} = 4/3 |
4/3, 8/3, 4, 16/3, 20/3, 8, 28/3 |
2 | 5 |

G | f_{G} = 3/2 |
3/2, 3, 9/2, 6, 15/2, 9 |
3 | 3 |

A | f_{A} = 5/3 |
5/3, 10/3, 5, 20/3, 25/3, 10 |
2 | 4 |

B | f_{B} = 15/8 |
15/8, 15/4, 45/8, 15/2, 75/8 | 0 | 5 |

c | f_{c} = 2 |
2, 4, 6, 8, 10 |
5 | None |

**Table 1.** Relationships of audible harmonics of naturals within a given octave, expressed as ratios of integers. Harmonics that are coincident with those of the root tone C are in boldface.

All of the harmonics of c (the octave of C) are exactly twice those of C, so they coincide with the even harmonics of C (Table 1). Thus, c sounds so similar to C that we think of it as the same note. Because the first harmonic of the root tone C is absent from c, we perceive c to be higher in pitch than C.

By comparison, none of the audible harmonics of B is the same as any of C. Also, the harmonics of B are quite disparate from those of C (Table 1), with the frequency of each harmonic 15/8 times those of C. Thus, B sounds alien to C, and C and B played together sound discordant.

Only one of the high—and therefore almost inaudible—harmonics of D is the same as a harmonic of C. But, the other harmonics of D are quite disparate from those of C (Table 1), with the frequency of each harmonic 9/8 times those of C. Thus, D sounds alien to C, and C and D played together sound discordant (but not as discordant as B and C played together.

In the case of G (the fifth of C), all of its odd harmonics coincide with ones of C (Table 1). Also, all of the harmonics of G are exactly 3/2 those of C, so the ones that do not coincide “fit” in a way that seems concordant to present-day ears (but not so to the ears of the ancient Greeks).

Note |
Relative Frequency of Note(Just Tuning) |
Frequencies of Harmonics(in Decimals) |
Number of Harmonics Coincident with First Ten of Root Tone |

C | f_{C} = 1 |
1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | All |

D | f_{D} = 9/8 |
1.125, 2.25, 3.375, 4.5, 5.625, 6.75, 7.875, 9 |
1 |

E | f_{E} = 6/5 |
1.2, 2.4, 3.6, 4.8, 6, 7.2, 8.4, 9.6 |
1 |

F | f_{F} = 4/3 |
1.333, 2.666, 4, 5.333, 6.666, 8, 9.333 |
2 |

G | f_{G} = 3/2 |
1.5, 3, 4.5, 6, 7.5, 9 |
3 |

A | f_{A} = 5/3 |
1.666, 3.333, 5, 6.666, 8.333, 10 |
2 |

B | f_{B} = 15/8 |
1.875, 3.75, 5.625, 7.5, 9.375 | 0 |

c | f_{c} = 2 |
2, 4, 6, 8, 10 |
All 5 |

**Table 2.** Relationships of harmonics of naturals within a given octave, expressed as decimals. Harmonics that are coincident with those of the root tone C are in boldface.