Diminished second

diminished second

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 modern Western tonal music theory a diminished second is the interval between notes on two adjacent staff positions, or having adjacent note letters, whose alterations cause them, in ordinary equal temperament, to have no pitch difference, such as B and C♭ or B♯ and C. The two notes may more often be described as Enharmonic equivalents.^[3]

More specifically, in other tunings and repertoires from Western culture, a diminished second is the minute (smaller than a semitone) pitch interval produced by narrowing a minor second, or diatonic semitone, by a chromatic semitone.^[1] It is therefore the difference between the diatonic and chromatic semitones. 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]

The diminished second can be also 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 (help·info)

Diminished second  Play (help·info)

Size in different tuning systems

In 12-tone equal temperament, the diminished second is identical to the unison ( play (help·info)), because both semitones have the same size. In 19-tone equal temperament, on the other hand, it is identical to the chromatic semitone and is a respectable 63.16 cents wide. It shows a similar size in third-comma meantone, where it coincides with the greater diesis (62.57 cents). The most commonly used meantone temperaments fall between these extremes, giving it an intermediate size.

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 about 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 S₁, S₂, S₃, S₄ are semitones as defined in 5-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	21/19:1

See also

• List of musical intervals
• List of meantone intervals
• Comma (music)

Sources

1. ^ ^{a b} Benward & 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 0824747143. Minor diesis, diminished second.
3. ^ Rushton, Julian. "Unison (prime)"]. Grove Music Online. Oxford Music Online. http://www.oxfordmusiconline.com/subscriber/article/grove/music/28790. Retrieved August 2011.  (subscription needed)
4. ^ Benward & Saker (2003), p.92.