# Diesis

In classical music from Western culture, a **diesis** (/ˈdaɪəsɪs/ *DY-ə-siss* or **enharmonic diesis**, plural **dieses** (/ˈdaɪəsiz/ *DY-ə-seez)*,^{[1]} or "difference"; Greek: δίεσις "leak" or "escape"^{[2]}^{[a]}
is either an accidental (see sharp), or a very small 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.

As a comma, the above-mentioned 128:125 ratio is also known as the **lesser diesis**, **enharmonic comma**, or **augmented comma**.

Many acoustics texts use the term **greater diesis**^{[2]} or **diminished comma** 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 step-size in 19 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]}

## Alternative definitions[edit]

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♯):

Name | Ratio | Typical use |
---|---|---|

greater diesis | 648 / 625 | third-comma meantone (discussed below) |

diaschisma | 2 048 / 2 025 | sixth-comma meantone |

schisma | 32 805 / 32 768 | twelfth-comma meantone |

Pythagorean comma |
531 441 / 524 288 | Pythagorean tuning and interval budgeting in descriptions of well temperaments |

In eleventh-comma meantone, the diminished second is within 1/ 716 (0.14%) 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 several distinct 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. Rameau (1722)^{[4]} names 125:148 ( [*sic*], *recte* 125:128)^{[5]}
as a "minor diesis" and 243:250 as a "major diesis", explaining that the latter may be derived through multiplication of the former by the ratio 15 552 / 15 625 .^{[4]}
Other theorists have used it as a name for various other small intervals.

## Small diesis[edit]

The **small diesis** is 3 125/ 3 072 or approximately 29.61 cents.^{[6]}

## Septimal and undecimal diesis[edit]

The septimal diesis (or slendro diesis) is an interval with the ratio of 49:48 , 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 half-sharp () interval.

## Footnotes[edit]

## See also[edit]

## References[edit]

**^**"diesis".*American Heritage Dictionary*– via ahdictionary.com.- ^
^{a}^{b}^{c}Benson, Dave (2006).*Music: A mathematical offering*. p. 171. ISBN 0-521-85387-7. **^**A. B. (2003). "Diesis". In Randel, D. M. (ed.).*The Harvard Dictionary of Music*(4th ed.). Cambridge, MA: Belknap Press. p. 241.- ^
^{a}^{b}^{c}Rameau, J.-P. (1722).*Traité de l'harmonie réduite à ses principes naturels*[*Treatise on Harmony distilled to its natural principles*] (in French). Paris, FR: Jean-Baptiste-Christophe Ballard. pp. 26–27.- English edition Rameau & Gossett (1971).
^{[5]}

- English edition Rameau & Gossett (1971).
- ^
^{a}^{b}Ratio corrected to 125:128 in

Rameau, J.-P. (1971) [1722].*Treatise on Harmony*. Gossett, Philip: translator, introduction, notes (English ed.). New York, NY: Dover Publications. p. 30. ISBN 0-486-22461-9.- translation of Rameau (1722)
^{[4]}

- translation of Rameau (1722)
**^**von Helmhotz, H.; Ellis, A.J. (1885).*On the Sensations of Tone*. Ellis, A.J. (translator / editor) author of substantial appendicies (2nd English ed.). p. 453.- as quoted and cited in