{{Short description|Interval in classical music}} {{about|music}} thumb|Diesis on C {{audio|Diesis on C.mid|Play}}. thumb|Diesis as three just major thirds.
{{Listen|filename=Diesis-example.ogg|title=Diesis (128:125) demonstration|description=The octave C-C′, the three justly tuned major thirds C-E-G{{sup|{{music|#}}}}-B{{sup|{{music|#}}}} and the descending diesis C′-B{{sup|{{music|#}}}} are played (see example).}}
In classical music from Western culture, a '''diesis''' ({{IPAc-en|'|d|aɪ|ə|s|ɪ|s}} {{respell |DY|ə|siss}} or '''enharmonic diesis''', plural '''dieses''' ({{IPAc-en|'|d|aɪ|ə|s|i|z}} {{respell |DY|ə|seez)}},<ref> {{cite dictionary |dictionary=American Heritage Dictionary |title=diesis |url= https://www.ahdictionary.com/word/search.html?q=diesis |via = ahdictionary.com }} </ref> or "difference"; Greek: {{math|δίεσις}} "leak" or "escape"<ref name=Benson/>{{efn| 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".<ref name=Benson> {{cite book |last=Benson |first=Dave |year=2006 |title=Music: A mathematical offering |page=171 |publisher=Cambridge University Press |isbn=0-521-85387-7 }} </ref> }} 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{{sup|{{music|#}}}} (namely, from C, to E, to G{{sup|{{music|#}}}}, to B{{music|#}}). The difference between C-C′ (2:1) and C-B{{sup|{{music|#}}}} (125:64) is the diesis (128:125). Notice that this coincides with the interval between B{{sup|{{music|#}}}} 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'''<ref name=Benson/> 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"''.<ref> {{cite dictionary |author = A. B. |year=2003 |title = Diesis |dictionary = The Harvard Dictionary of Music |edition = 4th |editor-first=D. M. |editor-last=Randel |editor-link=Don Michael Randel |place=Cambridge, MA |publisher=Belknap Press |page=[https://books.google.com/books?id=02rFSecPhEsC&q=Interval&pg=PA415 241] }} </ref>{{Failed verification|date=May 2014}}<!-- Very little in the preceding text is verified by A.B.'s entry in the Harvard Dictionary. In particular, "three syntonic commas minus a schisma", the bit about 19 equal temperament, and the values 62.57 and 41.06 cents: A.B. gives these to just one significant figure. -->
{| align="center" |- | [[Image:Lesser diesis (difference m2-A1).PNG|thumb|center|467 px|Diesis defined in quarter-comma meantone as a diminished second ({{nobr|m2 − A1 ≈}} {{nobr|117.1 − 76.0 ≈}} 41.1 cents), or an interval between two enharmonically equivalent notes (from D{{sup|{{music|b}}}} to C{{sup|{{music|#}}}}). {{audio|Enharmonic scale segment on C.mid|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 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 !! cents !! Typical use |- | greater limma || {{sfrac| 135 | 128 }} ||align=“right”| 92.18 |align="left"| ratio of two major whole tones to a minor third |- | greater diesis || {{sfrac| 648 | 625 }} ||align=“right”| 62.57 |align="left"| third-comma meantone<br/>(discussed below) |- | lesser diesis || {{sfrac| 128 | 125 }} ||align=“right”| 41.06 |align="left"| (discussed below) |- | 31 {{sc|EDO}} diesis || 2{{sup|{{math|¹⁄₃₁}}}} ||align=“right”| 38.71 |align="left"| step-size in 31 equal temperament |- | 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 fifths to a major third and 2 octaves;<br/> measure of fifth tempering in well temperaments |- | diaschisma || {{sfrac| 2 048 | 2 025 }} ||align=“right”| 19.55 |align="left"| sixth-comma meantone |- | schisma || {{sfrac| 32 805 | 32 768 }} ||align=“right”| 1.95 |align="left"| eleventh-comma meantone;<br/> limit of acoustic tuning accuracy |}
In eleventh-comma meantone, the diminished second is within {{sfrac| 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)<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.
== Small diesis == The '''small diesis''' {{audio|Small diesis on C.mid|Play}} is {{sfrac| 3 125 | 3 072 }} or approximately 29.61 cents.<ref> {{cite book |first1=H. |last1=von Helmhotz |author1-link=Hermann von Helmholtz |year = 1885 |first2=A.J. |last2=Ellis |author2-link=Alexander John Ellis |others=Ellis, A.J. (translator / editor) author of substantial appendicies |title=On the Sensations of Tone |edition=2nd English |title-link=Sensations of Tone |page = 453 }} : as quoted and cited in {{cite web |title = diesis |series = Tonalsoft Encyclopedia of Microtonal Music Theory |url = http://tonalsoft.com/enc/d/diesis.aspx }} </ref>
== Septimal and undecimal diesis == The septimal diesis (or slendro diesis) is an interval with the ratio of 49:48 {{Audio|Septimal diesis on C.mid|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 ({{music|t}}) interval.
==Footnotes== {{notelist}}
==See also== * chromatic diesis * septimal diesis * ditone
==References== {{reflist|25em}}
{{Intervals|state=expanded}}
Category:Commas (music)