Diminished second

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diminished second
Inverse augmented seventh
Name
Other names
Abbreviationd2 [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.

Contents

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 Diminished second on C.png
Diminished second Play

Size in different tuning systems

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, 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 Lesser diesis (difference m2-A1).PNG
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 13-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, 16-, 15-, 14-, and 13-comma meantone, the diminished second coincides with the corresponding commas.

Tuning system Definition of diminished secondSize
Difference between
semitones
Equivalent to Cents Ratio
Pythagorean tuning m2 A1 Opposite of Pythagorean comma 23.46524288:531441
1/12-comma meantone m2 A1Opposite of schisma 1.9532768:32805
12-tone equal temperament m2 A1 Unison 0.001:1
1/6-comma meantonem2 A1 Diaschisma 19.552048:2025
5-limit tuning S3 S2
1/5-comma meantonem2 A128.16
1/4-comma meantone m2 A1(Lesser) diesis 41.06128:125
5-limit tuning S3 S1
1/3-comma meantonem2 A1Greater diesis62.57648:625
5-limit tuning S4 S1
19-tone equal temperament m2 A1Chromatic semitone (A1 = m2 / 2)63.16:1
31-tone equal temperament m2 A1Lesser diesis 38.77:1

See also

Related Research Articles

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<span class="mw-page-title-main">Wolf interval</span> Dissonant musical interval

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<span class="mw-page-title-main">Diesis</span> An interval in classical music

In classical music from Western culture, a diesis ( DY-ə-siss or enharmonic diesis, plural dieses, or "difference"; Greek: δίεσις "leak" or "escape" is either an accidental, or a very small musical interval, usually defined as the difference between an octave and three justly tuned major thirds, equal to 128:125 or about 41.06 cents. In 12-tone equal temperament three major thirds in a row equal an octave, but three justly-tuned major thirds fall quite a bit narrow of an octave, and the diesis describes the amount by which they are short. For instance, an octave spans from C to C′, and three justly tuned major thirds span from C to B. The difference between C-C′ and C-B is the diesis. Notice that this coincides with the interval between B and C', also called a diminished second.

<span class="mw-page-title-main">Circle of fifths</span> Relationship among tones of the chromatic scale

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<span class="mw-page-title-main">Semitone</span> Musical interval

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<span class="mw-page-title-main">Comma (music)</span> Very small interval arising from discrepancies in tuning

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<span class="mw-page-title-main">Augmented second</span> Musical interval

In classical music from Western culture, an augmented second is an interval that, in 12-tone equal temperament, is sonically equivalent to a minor third, spanning three semitones, and is created by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and the interval from C to D is an augmented second, spanning three semitones.

Quarter-comma meantone, or  1 / 4 -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning ; the result is  3 / 2 × [ 80 / 81 ] 1 / 4 = 45 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

<span class="mw-page-title-main">53 equal temperament</span> Musical tuning system with 53 pitches equally-spaced on a logarithmic scale

In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 2153, or 22.6415 cents, an interval sometimes called the Holdrian comma.

<span class="mw-page-title-main">31 equal temperament</span> In music, a microtonal tuning system

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET or 31-EDO, also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Each step represents a frequency ratio of 312, or 38.71 cents.

<span class="mw-page-title-main">Diminished third</span> Musical interval

In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds, two semitones wide. Being diminished, it is considered a dissonant interval.

<span class="mw-page-title-main">Diatonic and chromatic</span> Terms in music theory to characterize scales

Diatonic and chromatic are terms in music theory that are used to characterize scales. The terms are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

<span class="mw-page-title-main">Regular diatonic tuning</span>

A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

<span class="mw-page-title-main">Augmented seventh</span> Musical interval

In classical music from Western culture, an augmented seventh is an interval produced by widening a major seventh by a chromatic semitone. For instance, the interval from C to B is a major seventh, eleven semitones wide, and both the intervals from C to B, and from C to B are augmented sevenths, spanning twelve semitones. Being augmented, it is classified as a dissonant interval. However, it is enharmonically equivalent to the perfect octave.

<span class="mw-page-title-main">Diminished sixth</span> Musical interval

In classical music from Western culture, a diminished sixth is an interval produced by narrowing a minor sixth by a chromatic semitone. For example, the interval from A to F is a minor sixth, eight semitones wide, and both the intervals from A to F, and from A to F are diminished sixths, spanning seven semitones. Being diminished, it is considered a dissonant interval, despite being equivalent to an interval known for its consonance.

References

  1. 1 2 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.