# Diminished second

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{{Infobox Interval|
main_interval_name = diminished second|
inverse = [augmented seventh](/source/augmented_seventh)|
complement = [augmented seventh](/source/augmented_seventh)|
other_names = —|
abbreviation = d2<ref name="B&S"/>|
semitones = 0|
interval_class = 0|
just_interval = 128:125<ref name="Haluska">Haluska, Jan (2003). ''The Mathematical Theory of Tone Systems'', p. xxvi. {{ISBN|0-8247-4714-3}}. Minor diesis, diminished second.</ref>| 
cents_equal_temperament = 0|
cents_24T_equal_temperament = |
cents_just_intonation = 41.1
}}

In modern Western tonal music theory, a '''diminished second''' is the [interval](/source/pitch_interval) produced by [narrowing](/source/Diminution) a [minor second](/source/minor_second) by one [chromatic semitone](/source/chromatic_semitone).<ref name="B&S">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.</ref> In twelve-tone equal temperament, it is [enharmonic](/source/enharmonic)ally equivalent to a [perfect unison](/source/perfect_unison);<ref name=grove>{{cite web|last=Rushton|first=Julian|author-link=Julian Rushton|title=Unison (prime)]|url=http://www.oxfordmusiconline.com/subscriber/article/grove/music/28790|work=[Grove Music Online](/source/Grove_Music_Online)|publisher=Oxford Music Online<!-- |access-date=August 2011 -->|url-access=subscription }}</ref> therefore, 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{{music|flat}} immediately above; another is the interval from a B{{music|sharp}} to the C immediately above.

:<score sound>
{
\override Score.TimeSignature
#'stencil = ##f
    \relative c'' {
        \time 4/4
        \set Score.tempoHideNote = ##t \tempo 1 = 20
        <b ces>1 <bis c?>
    }
}
</score>
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{{Music|#}} is a chromatic semitone, and their difference, the interval from B{{Music|#}} to C is a diminished second.

Being diminished, it is considered a [dissonant](/source/consonance_and_dissonance) interval.<ref>Benward and Saker (2003), p. 92.</ref>

== Size in different tuning systems ==
{{unreferenced section|date=August 2011}}
In tuning systems other than 12-tone equal temperament and its multiples, the diminished second is a distinct interval. It can be viewed as a [comma](/source/Comma_(music)), the minute interval between two [enharmonically equivalent](/source/Enharmonic) notes tuned in a slightly different way. This makes it a highly variable quantity between tuning systems. Hence for example C{{Music|sharp}} is narrower (or sometimes wider) than D{{Music|flat}} by a diminished second interval, however large or small that may happen to be (see image below).{{citation needed|date=August 2011}}

{| align="center"
|-
| [[File:Lesser diesis (difference m2-A1).PNG|thumb|center|467 px|Diminished second in [quarter-comma meantone](/source/quarter-comma_meantone) (also known as ''lesser diesis''), coinciding with the interval from C{{Music|#}} to D{{Music|b}}, defined as the difference between m2 and A1 (117.1 &minus; 76.0 = 41.1 cents). {{audio|Enharmonic scale segment on C.mid|Play}}]]
|}

In 12-tone [equal temperament](/source/equal_temperament), the diminished second is identical to the [unison](/source/unison) ({{Audio|Unison on C.mid|play}}), because the chromatic and diatonic semitones have the same size. In [19-tone equal temperament](/source/19-tone_equal_temperament), which extends {{frac|1|3}}-comma [meantone](/source/meantone_temperament), it is identical to the chromatic semitone and is a respectable 63.16 [cent](/source/cent_(music))s wide. The most commonly used [meantone temperament](/source/meantone_temperament)s fall between these extremes, giving it an intermediate size.

However, in [53-tone equal temperament](/source/53_equal_temperament), which extends [Pythagorean tuning](/source/Pythagorean_tuning), the interval actually shows a descending direction, i.e. a ratio below [unison](/source/unison), and thus a negative size, going one step down. In general, this applies for all tunings with fifths wider than 700 cents.

The table below summarizes the definitions of the diminished second in the main tuning systems. In the column labeled "Difference between [semitone](/source/semitone)s", '''m2''' is the minor second (diatonic semitone), '''A1''' is the augmented unison (chromatic semitone), and '''S{{sub|1}}''', '''S{{sub|2}}''', '''S{{sub|3}}''', '''S{{sub|4}}''' are semitones as defined in [five-limit tuning#Size of intervals](/source/five-limit_tuning). Notice that for 5-limit tuning, {{frac|1|6}}-, {{frac|1|5}}-, {{frac|1|4}}-, and {{frac|1|3}}-comma meantone, the diminished second coincides with the corresponding commas.

{| class="wikitable center-al"
! rowspan=2 | [Tuning system](/source/Tuning_system) !! colspan=2 | Definition of diminished second !! colspan=2 | Size
|-
! Difference between<br>[semitone](/source/semitone)s !! Equivalent to !! [Cents](/source/Cent_(music)) !! [Ratio](/source/Interval_ratio)
|-
| [Pythagorean tuning](/source/Pythagorean_tuning) || [m2](/source/Pythagorean_limma) &minus; [A1](/source/Pythagorean_apotome) || Opposite of [Pythagorean comma](/source/Pythagorean_comma) || &minus;23.46 || 524288:531441
|-
| 1/12-comma [meantone](/source/meantone_temperament) || m2 &minus; A1 || Opposite of [schisma](/source/schisma) || &minus;1.95 || 32768:32805
|-
| [12-tone equal temperament](/source/12-tone_equal_temperament) || m2 &minus; A1 || [Unison](/source/Unison) || 0.00 ||1:1
|-
| 1/6-comma meantone || m2 &minus; A1 || rowspan="2" | [Diaschisma](/source/Diaschisma) || rowspan="2" | 19.55 || rowspan="2" | 2048:2025
|-
| [5-limit tuning](/source/5-limit_tuning) || S{{sub|3}} &minus; S{{sub|2}}
|-
| 1/5-comma meantone || m2 &minus; A1 || || 28.16 || <math>\frac{512}{225}\sqrt[5]{\frac{4}{225}}</math>
|-
| [1/4-comma meantone](/source/Quarter-comma_meantone) || m2 &minus; A1 || rowspan="2" | (Lesser) [diesis](/source/diesis) || rowspan="2" | 41.06 || rowspan="2" | 128:125
|-
| [5-limit tuning](/source/5-limit_tuning) || S{{sub|3}} &minus; S{{sub|1}}
|-
| 1/3-comma meantone || m2 &minus; A1 || rowspan="2" | Greater diesis || rowspan="2" | 62.57 || rowspan="2" | 648:625
|-
| [5-limit tuning](/source/5-limit_tuning) || S{{sub|4}} &minus; S{{sub|1}}
|-
| [19-tone equal temperament](/source/19-tone_equal_temperament) || m2 &minus; A1 || Chromatic [semitone](/source/semitone) (A1 = m2 / 2) || 63.16 || <math>\sqrt[19]{2}</math>:1
|-
| [31-tone equal temperament](/source/31-tone_equal_temperament) || m2 &minus; A1 || Lesser [diesis](/source/diesis) || 38.77 || <math>\sqrt[31]{2}</math>:1
|}

==See also==
* [List of musical intervals](/source/Interval_(music))
* [List of pitch intervals](/source/List_of_pitch_intervals)
*[List of meantone intervals](/source/List_of_meantone_intervals)
*[Comma (music)](/source/Comma_(music))

==References==
<references />

{{Intervals}}

{{DEFAULTSORT:Diminished Second}}
Category:Diminished intervals
Category:Seconds (music)

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Adapted from the Wikipedia article [Diminished second](https://en.wikipedia.org/wiki/Diminished_second) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Diminished_second?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.