Inclassical music fromWestern culture, adiesis (/ˈdaɪəsɪs/DY-ə-siss orenharmonic diesis, pluraldieses (/ˈdaɪəsiz/DY-ə-seez),[1] or "difference";Greek:δίεσις "leak" or "escape"[2][a]is either anaccidental (seesharp), or a very smallmusical interval, usually defined as the difference between anoctave (in theratio 2:1) and threejustly tuned major thirds (tuned in the ratio5:4), equal to 128:125 or about 41.06 cents. In12-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 adiminished second.
As acomma, the above-mentioned 128:125 ratio is also known as thelesser diesis,enharmonic comma, oraugmented comma.
Many acoustics texts use the termgreater diesis[2] ordiminished comma for the difference between an octave and four justly tuned minor thirds (tuned in theratio6:5), which is equal to threesyntonic commas minus aschisma, equal to 648:625 or about 62.57 cents (almost one 63.16 cent step-size in19 equal temperament). Being larger, this diesis was termed the"greater" while the 128:125 diesis (41.06 cents) was termed the"lesser".[3][failed verification]
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In any tuning system, the deviation of an octave from three major thirds, however large that is, is typically referred to as adiminished second. The diminished second is an interval between pairs ofenharmonically equivalent notes; for instance the interval between E and F♭. As mentioned above, the termdiesis 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 agreater 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 aunison (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♯:
| Name | Ratio | cents | Typical use |
|---|---|---|---|
| greater limma | 135 / 128 | 92.18 | ratio of two majorwhole tones to aminor third |
| greater diesis | 648 / 625 | 62.57 | third-commameantone (discussed below) |
| lesser diesis | 128 / 125 | 41.06 | (discussed below) |
| 31 EDO diesis | 2¹⁄₃₁ | 38.71 | step-size in31 equal temperament |
| Pythagorean comma | 531 441 / 524 288 | 23.46 | Pythagorean tuning |
| diatonic comma | 81 / 80 | 21.51 | ratio of 4 fifths to amajor third and 2 octaves; measure of fifth tempering inwell temperaments |
| diaschisma | 2 048 / 2 025 | 19.55 | sixth-commameantone |
| schisma | 32 805 / 32 768 | 1.95 | eleventh-commameantone; limit of acoustic tuning accuracy |
In eleventh-comma meantone, the diminished second is within1/ 716 (0.14%) of acent above unison, so it closely resembles the 1:1 unison ratio of twelve-toneequal temperament.
The worddiesis has also been used to describe several distinct intervals, of varying sizes, but typically around 50 cents.Philolaus used it to describe the interval now usually called alimma, that of a justly tuned perfect fourth (4:3) minus two whole tones (9:8), equal to 256:243 or about 90.22 cents.Rameau (1722)[4] names 148:125 ( [sic],recte 128:125)[5]as a "minor diesis" and 250:243 as a "major diesis", explaining that the latter may be derived through multiplication of the former by the ratio15 625/ 15 552 .[4]Other theorists have used it as a name for various other small intervals.
Thesmall diesisPlayⓘ is3 125/ 3 072 or approximately 29.61 cents.[6]
Theseptimal diesis (orslendro diesis) is aninterval with the ratio of 49:48playⓘ, which is the difference between theseptimal whole tone and theseptimal minor third. It is about 35.70 cents wide.
Theundecimal diesis is equal to 45:44 or about 38.91 cents, closely approximated by31 equal temperament's 38.71 cent half-sharp (
) interval.
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