Diminished second

Musical interval

diminished secondInverse| augmented seventh

|

Name
Other names| —
Abbreviation| d2[1]
Size
Semitones| 0
Interval class| 0
Just interval| 128:125[2]
Cents
12-Tone equal temperament| 0
Just intonation| 41.1

In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone.[1] In twelve-tone equal temperament, it is enharmonically equivalent to a perfect unison.[3]; 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♭ immediately above; another is the interval from a B♯ to the C immediately above.

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.

Being diminished, it is considered a dissonant interval.[4]

Diminished second Playⓘ

## Size in different tuning systems[edit]

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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).[citation needed]

Diminished second in quarter-comma meantone (also known as lesser diesis), coinciding with the interval from C♯ to D♭, defined as the difference between m2 and A1 (117.1 − 76.0 = 41.1 cents). Playⓘ

In 12-tone equal temperament, the diminished second is identical to the unison (playⓘ), because the chromatic and diatonic semitones have the same size. In 19-tone equal temperament, which extends third-comma meantone, it is identical to the chromatic semitone and is a respectable 63.16 cents wide. The most commonly used meantone temperaments fall between these extremes, giving it an intermediate size.

However, in 53-tone equal temperament, which extends Pythagorean tuning, the interval actually shows a descending direction, i.e. a ratio below 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 semitones", m2 is the minor second (diatonic semitone), A1 is the augmented unison (chromatic semitone), and S1, S2, S3, S4 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.

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 | S3 − S2
1/4-comma meantone | m2 − A1 | (Lesser) diesis | 41.06 | 128:125
5-limit tuning | S3 − S1
1/3-comma meantone | m2 − A1 | Greater diesis | 62.57 | 648:625
5-limit tuning | S4 − S1
19-tone equal temperament | m2 − A1 | Chromatic semitone (A1 = m2 / 2) | 63.16 |  2 19 {\displaystyle {\sqrt[{19}]{2}}} :1
31-tone equal temperament | m2 − A1 | Lesser diesis | 38.77 |  2 31 {\displaystyle {\sqrt[{31}]{2}}} :1

## See also[edit]

* List of musical intervals
* List of pitch intervals
* List of meantone intervals
* Comma (music)

## References[edit]

1. ^ a b 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.
2. ^ Haluska, Jan (2003). The Mathematical Theory of Tone Systems, p. xxvi. ISBN 0-8247-4714-3. Minor diesis, diminished second.
3. ^ Rushton, Julian. "Unison (prime)]". Grove Music Online. Oxford Music Online.
4. ^ Benward and Saker (2003), p. 92.

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* t
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Intervals

Twelve-
semitone
(post-Bach
Western)|

| (Numbers in brackets
are the number of
semitones in the
interval.)|

| Perfect|

* unison (0)
* fourth (5)
* fifth (7)
* octave (12)

|

Major|

* second (2)
* third (4)
* sixth (9)
* seventh (11)

Minor|

* second (1)
* third (3)
* sixth (8)
* seventh (10)

Augmented|

* unison (1)
* second (3)
* third (5)
* fourth (6)
* fifth (8)
* sixth (10)
* seventh (12)

Diminished|

* second (0)
* third (2)
* fourth (4)
* fifth (6)
* sixth (7)
* seventh (9)
* octave (11)

Compound|

* ninth (13 or 14)
* tenth (15 or 16)
* eleventh (17 or 18)
* twelfth (18 or 19)
* thirteenth (20 or 21)
* fourteenth (22 or 23)
* fifteenth (24)

Other
tuning
systems|

| 24-tone equal temperament
(Numbers in brackets refer
to fractional semitones.)|

Neutral

* quarter tone (1⁄2)
* second (1+1⁄2)
* third (3+1⁄2)
* major fourth (5+1⁄2)
* minor fifth (6+1⁄2)
* sixth (8+1⁄2)
* seventh (10+1⁄2)

|

Just intonations
(Numbers in brackets
refer to pitch ratios.)|

| 7-limit|

* septimal quarter tone (36:35)
* septimal third tone (28:27)
* septimal chromatic semitone (21:20)
* septimal diatonic semitone (15:14)
* supermajor second (8:7)
* subminor third (7:6)
* supermajor third (9:7)
* subminor fifth (7:5)
* supermajor fourth (10:7)
* subminor seventh (7:4)

|

Higher-limit|

* minor diatonic semitone (17-limit)

Other
intervals|

| Groups|

* Microtone
* 5-limit
* Comma
* Pseudo-octave
* Pythagorean interval
* Subminor and supermajor

|

Semitones|

* Pythagorean limma
* Pythagorean apotome
* Major limma

Quarter tones|

* Quarter tone
* Septimal quarter tone
* Undecimal quarter tone

Commas|

* Pythagorean comma (23.5 cents)
* Syntonic comma (21.5 cents)
* Holdrian comma (22.6 cents)
* Septimal comma (27.3 cents)

* Lesser diesis (41.1 cents)
* Greater diesis (62.6 cents)
* Septimal diesis (35.7 cents)

* Diaschisma (19.5 cents)
* Semicomma (10.1 cents)
* Septimal semicomma (13.8 cents)
* Kleisma (8.1 cents)
* Septimal kleisma (7.7 cents)

* Schisma (1.95 cents)
* Breedsma (0.72 cents)
* Ragisma (0.4 cents)

Measurement|

* Cent
* Centitone
* Millioctave
* Savart

Others|

* Wolf
* Ditone
* Semiditone
* Secor
* Incomposite interval

List of pitch intervals
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