Ramsey problem

2013-05-08T17:42:32Z

128.227.123.202:

In [[music theory]] and [[musical tuning]] the '''Holdrian comma''', also called '''Holder's comma''', and sometimes the '''Arabian comma''',<ref name="Touma">[[Habib Hassan Touma]] (1996). ''The Music of the Arabs'', p.23. trans. Laurie Schwartz. Portland, Oregon: Amadeus Press. ISBN 0-931340-88-8.</ref> is a small [[interval (music)|musical interval]] of approximately 22.6415 [[cent (music)|cents]],<ref name="Touma"/> equal to one step of [[53 equal temperament]], or <math>\sqrt[53]{2}</math> ({{Audio|Holdrian comma on C.mid|play}}). The name [[comma (music)|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]] (21.51 cents)({{Audio|Syntonic comma on C.mid|play}}), 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.<ref name="Touma"/>

== Mercator's comma and the Holdrian comma ==
Mercator applied logarithms to determine that <math>\sqrt[55]{2}</math> (≈ 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 <math>\sqrt[53]{2}</math> might be useful, because 31 octaves could be practically approximated by a cycle of 53 [[just fifth]]s. [[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.

== 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]] ''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{{music|b}} f g a{{music|b}} b{{music|b}} c'
commas: 9 4 9 9 4 9 9
has [[medium second]]s between d–e, e–f, g–a, a–b, and b–c', a medium second being somewhere in between 8 and 9 commas.<ref name="Touma"/>

==References==
{{reflist}}

==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). [http://www.tonalsoft.com/enc/m/mercator-comma.aspx "Mercator's Comma"], ''Tonalsoft''.

{{Intervals|state=expanded}}

{{DEFAULTSORT:Holdrian Comma}}
[[Category:Commas]]

formulasearchengine - User contributions [en]

Ramsey problem