Diesis
Diesis (128:125) demonstration
The octave CC', the three justly tuned major thirds CEG♯B♯ and the descending diesis C'B♯ are played (see example).  
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In classical music from Western culture, a diesis (/ˈdaɪəsɪs/ DYəsis; "difference"; Greek: δίεσις "leak" or "escape"^{[1]}) 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 12tone equal temperament (on a piano for example) three major thirds in a row equal an octave, but three justlytuned 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 CC' (2:1) and CB♯ (125:64) is the diesis (128:125). Notice that this coincides with the interval between B♯ and C', also called a diminished second.
The diesis is a comma. The abovementioned 128:125 comma is also known as the lesser diesis, as opposed to a wider comma (648:625) known as greater diesis. As shown in the picture, in the quartercomma meantone tuning system (a tuning system in which, by definition, major thirds are justly tuned), the diminished second coincides with the diesis.

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 quartercomma meantone temperament. Less frequently and less strictly, the same term is also used to refer to a diminished second of any size. In thirdcomma meantone, the diminished second is typically denoted as a greater diesis (see below).
In quartercomma meantone, since major thirds are justly tuned, the width of the diminished second coincides with the abovementioned 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 thirdcomma meantone temperament (see below),
 a diaschisma above unison (2048:2025) for sixthcomma,
 a schisma below unison (32768:32805) for twelfthcomma, and
 a Pythagorean comma below unison (524288:531441) for Pythagorean tuning.
In eleventhcomma 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 twelvetone 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. Rameau, in his Treatise on Harmony (1722), names 125:148 ( [sic], recte 125:128)^{[2]} 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 15552:15625.^{[3]} 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 63.16cent division in 19 equal temperament). Being larger, this diesis was termed "greater" while the 128:125 diesis (41.06 cents) was termed "lesser".^{[4]}
The small diesis Play is 3125:3072 or approximately 29.61 cents.^{[5]}
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 interval.
See also
References
 1 2 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".
 ↑ Corrected in the English edition, Treatise on Harmony, translated with an introduction and notes by Philip Gossett (New York: Dover Publications, Inc., 1971), p. 30. ISBN 0486224619.
 ↑ JeanPhilippe Rameau, Traité de l'harmonie réduite à ses principes naturels (Paris: JeanBaptisteChristophe Ballard, 1722), pp. 26–27.
 ↑ A. B., "Diesis", The Harvard Dictionary of Music, fourth edition, edited by Don Michael Randel. Cambridge, MA: Belknap Press, 2003, p. 241.
 ↑ Hermann von Helmholtz, On the Sensations of Tone, 2nd English translation by Alexander John Ellis (1885), page 453. Quoted in the Tonalsoft Encyclopedia of Microtonal Music Theory.