Editing Diesis (section)

== 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{{sup|{{music|b}}}}. 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{{sup|{{music|#}}}}. 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{{sup|{{music|#}}}}:
:{| class="wikitable" style="text-align:center;"
! Name !! Ratio !! [[cent (music)|cents]] !! Typical use
|-
| [[limma|greater limma]] || {{sfrac| 135 | 128 }} ||align=“right”| 92.18
|align="left"| ratio of two major [[whole tone]]s to a [[minor third]]
|-
| greater diesis || {{sfrac| 648 | 625 }} ||align=“right”| 62.57
|align="left"| third-comma [[meantone temperament|meantone]]<br/>(discussed below)
|-
| lesser diesis || {{sfrac| 128 | 125 }} ||align=“right”| 41.06
|align="left"| (discussed below)
|-
| 31&nbsp;{{sc|EDO}} diesis || 2{{sup|{{math|¹⁄₃₁}}}} ||align=“right”| 38.71
|align="left"| step-size in [[31 equal temperament]]
|-
| [[Pythagorean comma|Pythagorean<br/>comma]] || {{sfrac| 531 441 | 524 288 }} ||align=“right”| 23.46
|align="left"| [[Pythagorean tuning]]
|-
| [[diatonic comma]] || {{sfrac| 81 | 80 }} ||align=“right”| 21.51
|align="left"| ratio of 4&nbsp;[[perfect fifth|fifths]] to a [[major third]] and 2&nbsp;[[octaves]];<br/> measure of fifth tempering in [[well temperament]]s
|-
| [[diaschisma]] || {{sfrac| 2 048 | 2 025 }} ||align=“right”| 19.55
|align="left"| sixth-comma [[meantone temperament|meantone]]
|-
| [[schisma]]   || {{sfrac| 32 805 | 32 768 }} ||align=“right”| 1.95
|align="left"| eleventh-comma [[meantone temperament|meantone]];<br/> limit of acoustic tuning accuracy
|}

In eleventh-comma meantone, the diminished second is within {{sfrac| 1 | 716 }} (0.14%) of a [[cent (music)|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&nbsp;cents. [[Philolaus]] used it to describe the interval now usually called a ''[[limma]]'', that of a justly tuned perfect fourth ([[sesquitertium|4:3]]) minus two whole tones (9:8), equal to 256:243 or about 90.22&nbsp;cents. [[Rameau]] (1722)<ref name=Rameau-1722/> names 148:125 ({{sic}}, ''[[recte]]'' 128:125){{refn|name=Rameau-Gossett-1971|
Ratio 148:125 corrected to 128:125 in<br/>
{{cite book
 |first = J.-P. |last = Rameau |author-link = Jean-Philippe Rameau
 |year = 1971  |orig-year = 1722
 |edition = English (reprint)
 |title = Treatise on Harmony
 |title-link = Treatise on Harmony
 |others = Gossett, Philip (translator, introduction, notes)
 |place = New York, NY
 |publisher = Dover Publications
 |page = 30
 |ISBN = 0-486-22461-9
}}
: translation of Rameau (1722)<ref name=Rameau-1722/>
}}
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 ratio {{sfrac| 15 625 | 15 552 }}.{{refn|name=Rameau-1722|
{{cite book
 |first = J.-P.  |last = Rameau  |author-link = Jean-Philippe Rameau
 |year = 1722
 |title = Traité de l'harmonie réduite à ses principes naturels  |lang = fr
 |trans-title = Treatise on Harmony distilled to its natural principles
 |title-link = Treatise on Harmony
 |place = Paris, FR
 |publisher = Jean-Baptiste-Christophe Ballard
 |pages = 26–27
}}
: English edition Rameau & Gossett (1971).{{refn|name=Rameau-Gossett-1971}}
}}
Other theorists have used it as a name for various other small intervals.