Diesis

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This article is about music. For other uses, see Diesis (disambiguation).
Diesis on C Playⓘ.
Diesis as three just major thirds.
Diesis (128:125) demonstration
The octave C-C′, the three justly tuned major thirds C-E-G♯-B♯ and the descending diesis C′-B♯ are played (see example).
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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

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

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
]

enharmonically equivalent notes (from D♭ to C♯). Playⓘ

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

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.

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

The small diesis Playⓘ is 3 125/ 3 072 or approximately 29.61 cents.[6]

Septimal and undecimal diesis

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

Footnotes

  1. ^ The Greek name 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]

See also

References

  1. ^ "diesis".
    American Heritage Dictionary
     – via ahdictionary.com.
  2. ^ a b c Benson, Dave (2006). Music: A mathematical offering. p. 171.
    ISBN 0-521-85387-7
    .
  3. ^ A. B. (2003). "Diesis". In Randel, D. M. (ed.). The Harvard Dictionary of Music (4th ed.). Cambridge, MA: Belknap Press. p. 241.
  4. ^ a b c
    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]
  5. ^ a b Ratio corrected to 125:128 in
    ISBN 0-486-22461-9
    .
    translation of Rameau (1722)[4]
  6. ^ 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
    "diesis". Tonalsoft Encyclopedia of Microtonal Music Theory.
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