Holdrian comma
Encyclopedia
In music theory
and musical tuning
the Holdrian comma, also called Holder's comma, and sometimes the Arabian comma, is a small musical interval of approximately 22.6415 cents
, equal to one step of 53 equal temperament
, or
. The name comma
is misleading, since this interval is an irrational number and does not describe the compromise between intervals of any tuning system; it assumes this name because it is an approximation of the syntonic comma
, which was widely used as a measurement of tuning in William Holder
's time.
Mercator's comma is a name often used for a closely related interval because of its association with Nicholas Mercator
. One of these intervals was first described by Ching-Fang in 45 BCE.
(≈ 21.8182 cents) was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the prevalent sixth-comma meantone temperament
tuning system of the time). He also considered that an "artificial comma" of
might be useful, because 31 octaves could be practically approximated by a cycle of 53 just fifths. William Holder
, for whom the Holdrian comma is named, favored this latter unit because the intervals of 53 equal temperament are closer to just intonation
than that of 55. Thus Mercator's comma and the Holdrian comma are two distinct but related intervals.
For instance, the makam
rast (similar to the Western major scale
) may be considered in terms of Holdrian commas:
c d e f g a b c'
commas: 9 8 5 9 9 8 5
while in contrast, the makam nihavend (similar to the Western minor scale
):
c d e f g a b c'
commas: 9 4 9 9 4 9 9
has medium seconds between d–e, e–f, g–a, a–b, and b–c', a medium second being somewhere in between 8 and 9 commas.
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
and musical tuning
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
the Holdrian comma, also called Holder's comma, and sometimes the Arabian comma, is a small musical interval of approximately 22.6415 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
, equal to one step of 53 equal temperament
53 equal temperament
In music, 53 equal temperament, called 53-TET, 53-EDO, or 53-ET, is the tempered scale derived by dividing the octave into 53 equal steps . Each step represents a frequency ratio of 21/53, or 22.6415 cents , an interval sometimes called the Holdrian comma.- History :Theoretical interest in this...
, or
. The name commaComma (music)
In music theory, a comma is a minute interval, the difference resulting from tuning one note two different ways. The word "comma" used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning...
is misleading, since this interval is an irrational number and does not describe the compromise between intervals of any tuning system; it assumes this name because it is an approximation of the syntonic comma
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
, which was widely used as a measurement of tuning in William Holder
William Holder
William Holder FRS was an English clergyman and music theorist of the 17th century. His most notable work was his widely known 1694 publication A Treatise on the Natural Grounds and Principles of Harmony.-Life:...
's time.
Mercator's comma is a name often used for a closely related interval because of its association with Nicholas Mercator
Nicholas Mercator
Nicholas Mercator , also known by his Germanic name Kauffmann, was a 17th-century mathematician....
. One of these intervals was first described by Ching-Fang in 45 BCE.
Mercator's comma and the Holdrian comma
Mercator applied logarithms to determine that (≈ 21.8182 cents) was nearly equivalent to a syntonic comma of ≈ 21.5063 cents (a feature of the prevalent sixth-comma meantone temperamentMeantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
tuning system of the time). He also considered that an "artificial comma" of
might be useful, because 31 octaves could be practically approximated by a cycle of 53 just fifths. William HolderWilliam Holder
William Holder FRS was an English clergyman and music theorist of the 17th century. His most notable work was his widely known 1694 publication A Treatise on the Natural Grounds and Principles of Harmony.-Life:...
, for whom the Holdrian comma is named, favored this latter unit because the intervals of 53 equal temperament are closer to just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
than that of 55. Thus Mercator's comma and the Holdrian comma are two distinct but related intervals.
Arabian comma
The name "Arabian comma" may be inaccurate; the comma has been employed mainly in Turkish music theory by Kemal Ilerici, and by the Turkish composer Erol Sayan. The name of this comma is "Holder koması" in Turkish.For instance, the makam
Makam
Makam In Turkish classical music, a system of melody types called makam provides a complex set of rules for composing and performance...
rast (similar to the Western major scale
Major scale
In music theory, the major scale or Ionian scale is one of the diatonic scales. It is made up of seven distinct notes, plus an eighth which duplicates the first an octave higher. In solfege these notes correspond to the syllables "Do, Re, Mi, Fa, Sol, La, Ti/Si, ", the "Do" in the parenthesis at...
) may be considered in terms of Holdrian commas:
c d e f g a b c'
commas: 9 8 5 9 9 8 5
while in contrast, the makam nihavend (similar to the Western minor scale
Minor scale
A minor scale in Western music theory includes any scale that contains, in its tonic triad, at least three essential scale degrees: 1) the tonic , 2) a minor-third, or an interval of a minor third above the tonic, and 3) a perfect-fifth, or an interval of a perfect fifth above the tonic, altogether...
):
c d e f g a b c'
commas: 9 4 9 9 4 9 9
has medium seconds between d–e, e–f, g–a, a–b, and b–c', a medium second being somewhere in between 8 and 9 commas.
Further reading
- Holder, William, A Treatise on the Natural Grounds, and Principles of Harmony, facsimile of the 1694 edition, Broude Brothers, New York, 1967. (Original pp. 103–106.)
External links
- Monzo, Joe (2005). "Mercator's Comma", Tonalsoft.