classical music Classical music generally refers to the art music of the Western world, considered to be distinct from Western folk music or popular music traditions. It is sometimes distinguished as Western classical music, as the term "classical music" also ...

from

Western culture Leonardo da Vinci's ''Vitruvian Man''. Based on the correlations of ideal Body proportions">human proportions with geometry described by the ancient Roman architect Vitruvius in Book III of his treatise ''De architectura''. image:Plato Pio-Cle ...

, a diesis ( , plural dieses ( , "difference";

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

: δίεσις "leak" or "escape"Benson, Dave (2006). ''Music: A Mathematical Offering'', p.171. . Based on the technique of playing the

aulos An ''aulos'' ( grc, αὐλός, plural , ''auloi'') or ''tibia'' (Latin) was an ancient Greek wind instrument, depicted often in art and also attested by archaeology. Though ''aulos'' is often translated as "flute" or "double flute", it was usu ...

, where pitch is raised a small amount by slightly raising the finger on the lowest closed hole, letting a small amount of air "escape".) is either an accidental (see

sharp Sharp or SHARP may refer to: Acronyms * SHARP (helmet ratings) (Safety Helmet Assessment and Rating Programme), a British motorcycle helmet safety rating scheme * Self Help Addiction Recovery Program, a charitable organisation founded in 19 ...

), or a very small musical interval, usually defined as the difference between an

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

(in the

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

2:1) and three justly tuned major thirds (tuned in the ratio 5:4), equal to 128:125 or about 41.06

cent Cent may refer to: Currency * Cent (currency), a one-hundredth subdivision of several units of currency * Penny (Canadian coin), a Canadian coin removed from circulation in 2013 * 1 cent (Dutch coin), a Dutch coin minted between 1941 and 1944 * ...

s. In

12-tone equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resultin ...

(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 In modern Western tonal music theory, a diminished second is the interval produced by narrowing a minor second by one chromatic semitone.Bruce Benward and Marilyn Saker (2003). ''Music: In Theory and Practice, Vol. I'', p. 54. . Specific example ...

. The diesis is a

comma The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

. The above-mentioned 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

quarter-comma meantone Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80 ...

tuning system (a tuning system in which, by definition, major thirds are justly tuned), the

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

. The diminished second is an interval between pairs of

enharmonically equivalent In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written no ...

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 In music, unison is two or more musical parts that sound either the same pitch or pitches separated by intervals of one or more octaves, usually at the same time. ''Rhythmic unison'' is another term for homorhythm. Definition Unison or per ...

(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 third-comma meantone temperament (see below), * a

diaschisma The diaschisma (or diacisma) is a small musical interval defined as the difference between three octaves and four perfect fifths plus two major thirds (in just intonation). It can be represented by the ratio 2048:2025 and is about 19.5 cents. ...

above unison (2048:2025) for sixth-comma, * a

schisma In music, the schisma (also spelled ''skhisma'') is the interval between a Pythagorean comma (531441:524288) and a syntonic comma (81:80) and equals or 32805:32768 = 1.00113, which is 1.9537 cents (). It may also be defined as: * the differe ...

below unison (32768:32805) for twelfth-comma, and * a

Pythagorean comma In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C ...

below unison (524288:531441) for

Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2.Bruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh edition, 2 vols. (Boston: Mc ...

. In eleventh-comma meantone, the diminished second is within 1/716 (0.0014) of a

above unison, so it closely resembles the 1:1 unison ratio of twelve-tone

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

. The word ''diesis'' has also been used to describe a large number of intervals, of varying sizes, but typically around 50 cents.

Philolaus Philolaus (; grc, Φιλόλαος, ''Philólaos''; ) was a Greek Pythagorean and pre-Socratic philosopher. He was born in a Greek colony in Italy and migrated to Greece. Philolaus has been called one of three most prominent figures in the Pytha ...

used it to describe the interval now usually called a ''

limma The word limma or leimma (from Greek: λείμμα, ''leimma''; meaning "remnant") can refer to several different musical intervals, whose only common property is their small size. :More specifically, in Pythagorean tuning (i.e. 3-limit): :*The ori ...

'', 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 Jean-Philippe Rameau (; – ) was a French composer and music theorist. Regarded as one of the most important French composers and music theorists of the 18th century, he replaced Jean-Baptiste Lully as the dominant composer of French opera an ...

, in his ''

Treatise on Harmony A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the principles of the subject and its conclusions."Trea ...

'' (1722), names 125:148 (, ''

recte The Latin adverb ''sic'' (; "thus", "just as"; in full: , "thus was it written") inserted after a quoted word or passage indicates that the quoted matter has been transcribed or translated exactly as found in the source text, complete with any e ...

'' 125:128) 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. Other theorists have used it for various other intervals.

Greater and lesser diesis

Some acoustics texts use the term ''greater diesis'' for the difference between an octave and four justly tuned minor thirds (tuned in the

6:5), which is equal to three

syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...

s minus a

, equal to 648:625 or about 62.57 cents (almost one 63.16-cent division in

19 equal temperament In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...

). Being larger, this diesis was termed "greater" while the 128:125 diesis (41.06 cents) was termed "lesser". The small diesis is 3125:3072 or approximately 29.61 cents.

Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...

, ''On the

Sensations of Tone ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'' (German ), commonly referred to as ''Sensations of Tone'', is a foundational work on music acoustics and the perception of sound by Hermann von Helmholtz. The first G ...

'', 2nd English translation by

Alexander John Ellis Alexander John Ellis, (14 June 1814 – 28 October 1890), was an English mathematician, philologist and early phonetician who also influenced the field of musicology. He changed his name from his father's name, Sharpe, to his mother's maiden na ...

(1885), page 453. Quoted in the
Tonalsoft Encyclopedia of Microtonal Music Theory
'.

Septimal and undecimal diesis

The

septimal diesis In music, 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 35.7 cents wide, which is narrower than a quarter-tone but ...

(or

slendro Slendro ( jv, ꦱ꧀ꦭꦺꦤ꧀ꦢꦿꦺꦴ, ban, slendro, translit=Sléndro) ( su, salendro, translit=Saléndro) is one of the essential tuning systems used in gamelan instruments that have pentatonic scale. Based on Javanese mythology, ...

diesis) is an interval with the ratio of 49:48 , which is the difference between the

septimal whole tone In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to an 8/7 ratio of frequencies.Andrew Horner, Lydia Ayres (2002). ''Cooking with Csound: Woodwind and Brass ...

and the

septimal minor third In music, the septimal minor third, also called the subminor third (e.g., by Ellis), is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter ...

. 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 In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31- EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equ ...

's 38.71 cent interval.

References

{{Intervals, state=expanded Commas (music)

Alternative definitions

Greater and lesser diesis

Septimal and undecimal diesis

See also

References