Diesis

This article is about music. For other uses, see Diesis (disambiguation).

Diesis on C  Play (help·info).

Diesis as three just major thirds.

In classical music from Western culture, a diesis (difference, Greek: "leak or "escape"^[1]) is either an accidental (see sharp), or a comma type of musical interval, usually defined as the difference between an octave (in the ratio 2:1) and three justly tuned major thirds (tuned in the ratio 5:4), equal to 128:125 or about 41.06 cents. In 12-tone equal temperament (on a piano for example) 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 (2:1) spans from C to C', and three justly tuned major thirds (5:4) span from C to B♯ (namely, from C, to E, to G♯, to B♯). The difference between C-C' (2:1) and C-B♯ (125:64) is the diesis (128:125). Notice that this coincides with the interval between B♯ and C', also called a diminished second.

The 128:125 interval is also known as a lesser diesis, as opposed to a wider comma (648:625) known as greater diesis. As shown in the picture, in the quarter-comma meantone tuning system (a tuning system in which, by definition, major thirds are justly tuned), the diminished second coincides with the diesis.

Diesis defined in quarter-comma meantone as a diminished second (m2 − A1 ≈ 117.1 − 76.0 ≈ 41.1 cents), or an interval between two enharmonically equivalent notes (from C♯ to D♭).  Play (help·info)

Alternative definitions

In any tuning system, the deviation of an octave from three major thirds, however large that is, is typically referred to as a diminished second. The diminished second is an interval between pairs of enharmonically equivalent notes; for instance the interval between E and F♭. As mentioned above, the term diesis most commonly refers to the diminished second in quarter-comma meantone temperament. Less frequently and less strictly, the same term is also used to refer to a diminished second of any size. In third-comma meantone, the diminished second is typically denoted as a greater diesis (see below).

In quarter-comma meantone, since major thirds are justly tuned, the width of the diminished second coincides with the above mentioned value of 128:125. Notice that 128:125 is larger than a unison (1:1). This means that, for instance, C' is sharper than B♯. In other tuning systems, the diminished second has different widths, and may be smaller than a unison (e.g. C' may be flatter than B♯):

• a greater diesis above unison (648:625) for third-comma meantone temperament (see below),
• a diaschisma above unison (2048:2025) for sixth-comma,
• a schisma below unison (32768:32805) for twelfth-comma, and
• a Pythagorean comma below unison (524288:531441) for Pythagorean tuning.

In eleventh-comma meantone, the diminished second is within 1/716 (0.0014) of a cent above unison, so it closely resembles the 1:1 unison ratio of twelve-tone equal temperament.

The word diesis has also been used to describe a large number of intervals, of varying sizes, but typically around 50 cents. Philolaus used it to describe the interval now usually called a limma, that of a justly tuned perfect fourth (4:3) minus two whole tones (9:8), equal to 256:243 or about 90.22 cents. Other theorists have used it for various other intervals.

Greater and lesser diesis

Some acoustics texts use the term greater diesis^[1] for the difference between an octave and four justly tuned minor thirds (tuned in the ratio 6:5), which is equal to three syntonic commas minus a schisma, equal to 648:625 or about 62.57 cents (almost one 63.16-cent division in 19 equal temperament). Being larger, this diesis was termed "greater" while the 128:125 diesis was termed "lesser".^[2]

The small diesis  Play (help·info) is 3125:3072 or approximately 28 cents.^[3]

Septimal and undecimal diesis

The septimal diesis (or slendro diesis) is an interval with the ratio of 49:48  play (help·info), which is the difference between the septimal whole tone and the septimal minor third. It is about 35.70 cents wide

The undecimal diesis is equal to 45:44 or about 38.91 cents, closely approximated by 31 equal temperament's 38.71 cent interval.

See also

• Chromatic diesis
• Septimal diesis
• Ditone

References

1. ^ ^{a b} Benson, Dave (2006). Music: A Mathematical Offering, p.171. ISBN 0521853877. Based on the technique of playing the aulos, where pitch is raised a small amount by slightly raising the finger on the lowest closed hole, letting a small amount of air "escape".
2. ^ Don Michael Randel, The Harvard Dictionary of Music, Fourth Edition. Cambridge, MA: Belknap Press, 2003, p. 241.
3. ^ John Fonville. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.111, Perspectives of New Music, Vol. 29, No. 2 (Summer, 1991), pp. 106-137.