# Diminished second

In modern Western tonal music theory, a **diminished second** is the interval produced by narrowing a minor second by one chromatic semitone.[1] It is enharmonically equivalent to a perfect unison.[3] Thus, it is the interval between notes on two adjacent staff positions, or having adjacent note letters, altered in such a way that they have no pitch difference in twelve-tone equal temperament. An example is the interval from a B to the C♭ immediately above; another is the interval from a B♯ to the C immediately above.

Inverse | augmented seventh |
---|---|

Name | |

Other names | — |

Abbreviation | d2[1] |

Size | |

Semitones | 0 |

Interval class | 0 |

Just interval | 128:125[2] |

Cents | |

Equal temperament | 0 |

24 equal temperament | 50 |

Just intonation | 41.1 |

In particular, it may be regarded as the "difference" between a diatonic and chromatic semitone. For instance, the interval from B to C is a diatonic semitone, the interval from B to B♯ is a chromatic semitone, and their difference, the interval from B♯ to C is a diminished second.

## Size in different tuning systems

In tuning systems other than twelve-tone equal temperament, the diminished second can be viewed as a comma, the minute interval between two enharmonically equivalent notes tuned in a slightly different way. This makes it a highly variable quantity between tuning systems. Hence for example C♯ is narrower (or sometimes wider) than D♭ by a diminished second interval, however large or small that may happen to be (see image below).

In 12-tone equal temperament, the diminished second is identical to the unison (

In Pythagorean tuning, however, the interval actually shows a descending direction, i.e. a ratio below unison, and thus a negative size (−23.46 cents), equal to the opposite of a Pythagorean comma. Such is also the case in twelfth-comma meantone, although that diminished second is only a twelfth of the Pythagorean one (−1.95 cents, the opposite of a schisma).

The table below summarizes the definitions of the diminished second in the main tuning systems. In the column labeled "Difference between semitones", **m2** is the minor second (diatonic semitone), **A1** is the augmented unison (chromatic semitone), and **S _{1}**,

**S**,

_{2}**S**,

_{3}**S**are semitones as defined in five-limit tuning#Size of intervals. Notice that for 5-limit tuning, 1/6-, 1/4-, and 1/3-comma meantone, the diminished second coincides with the corresponding commas.

_{4}Tuning system | Definition of diminished second | Size | ||
---|---|---|---|---|

Difference between semitones |
Equivalent to | Cents | Ratio | |

Pythagorean tuning | m2 − A1 | Opposite of Pythagorean comma | −23.46 | 524288:531441 |

1/12-comma meantone | m2 − A1 | Opposite of schisma | −1.95 | 32768:32805 |

12-tone equal temperament | m2 − A1 | Unison | 0.00 | 1:1 |

1/6-comma meantone | m2 − A1 | Diaschisma | 19.55 | 2048:2025 |

5-limit tuning | S_{3} − S_{2} | |||

1/4-comma meantone | m2 − A1 | (Lesser) diesis | 41.06 | 128:125 |

5-limit tuning | S_{3} − S_{1} | |||

1/3-comma meantone | m2 − A1 | Greater diesis | 62.57 | 648:625 |

5-limit tuning | S_{4} − S_{1} | |||

19-tone equal temperament | m2 − A1 | Chromatic semitone (A1 = m2 / 2) | 63.16 | 2^(1÷19):1 |

## Sources

- Bruce Benward and Marilyn Saker (2003).
*Music: In Theory and Practice, Vol. I*, p. 54. ISBN 978-0-07-294262-0. Specific example of an d2 not given but general example of minor intervals described. - Haluska, Jan (2003).
*The Mathematical Theory of Tone Systems*, p. xxvi. ISBN 0-8247-4714-3. Minor diesis, diminished second. - Rushton, Julian. "Unison (prime)]".
*Grove Music Online*. Oxford Music Online. - Benward and Saker (2003), p. 92.