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Diminished second

From Wikipedia, the free encyclopedia
Musical interval
diminished second
Inverseaugmented seventh
Name
Other names
Abbreviationd2[1]
Size
Semitones0
Interval class0
Just interval128:125[2]
Cents
12-Tone equal temperament0
Just intonation41.1

In modern Western tonal music theory, adiminished second is theinterval produced bynarrowing aminor second by onechromatic semitone.[1] In twelve-tone equal temperament, it isenharmonically equivalent to aperfect 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 adissonant interval.[4]

Size in different tuning systems

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In tuning systems other than 12-tone equal temperament and its multiples, the diminished second is a distinct interval. It can be viewed as acomma, the minute interval between twoenharmonically 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 inquarter-comma meantone (also known aslesser 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-toneequal temperament, the diminished second is identical to theunison (play), because the chromatic and diatonic semitones have the same size. In19-tone equal temperament, which extends13-commameantone, it is identical to the chromatic semitone and is a respectable 63.16cents wide. The most commonly usedmeantone temperaments fall between these extremes, giving it an intermediate size.

However, in53-tone equal temperament, which extendsPythagorean tuning, the interval actually shows a descending direction, i.e. a ratio belowunison, 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 betweensemitones",m2 is the minor second (diatonic semitone),A1 is the augmented unison (chromatic semitone), andS1,S2,S3,S4 are semitones as defined infive-limit tuning#Size of intervals. Notice that for 5-limit tuning,16-,15-,14-, and13-comma meantone, the diminished second coincides with the corresponding commas.

Tuning systemDefinition of diminished secondSize
Difference between
semitones
Equivalent toCentsRatio
Pythagorean tuningm2A1Opposite ofPythagorean comma−23.46524288:531441
1/12-commameantonem2 − A1Opposite ofschisma−1.9532768:32805
12-tone equal temperamentm2 − A1Unison0.001:1
1/6-comma meantonem2 − A1Diaschisma19.552048:2025
5-limit tuningS3 − S2
1/5-comma meantonem2 − A128.1651222542255{\displaystyle {\frac {512}{225}}{\sqrt[{5}]{\frac {4}{225}}}}
1/4-comma meantonem2 − A1(Lesser)diesis41.06128:125
5-limit tuningS3 − S1
1/3-comma meantonem2 − A1Greater diesis62.57648:625
5-limit tuningS4 − S1
19-tone equal temperamentm2 − A1Chromaticsemitone (A1 = m2 / 2)63.16219{\displaystyle {\sqrt[{19}]{2}}}:1
31-tone equal temperamentm2 − A1Lesserdiesis38.77231{\displaystyle {\sqrt[{31}]{2}}}:1

See also

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References

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  1. ^abBruce 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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