查看“Superparticular數”的源代码

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'''Superparticular數'''是以下形式的[[有理數]]：
:<math> {n + 1 \over n} = 1 + {1 \over n}</math>

其中''n''為[[正整數]]。<!--
Thus:
{{quote|A superparticular number is when a great number contains a lesser number, to which it is compared, and at the same time one part of it. For example, when 3 and 2 are compared, they contain 2, plus the 3 has another 1, which is half of two. When 3 and 4 are compared, they each contain a 3, and the 4 has another 1, which is a third apart of 3. Again, when 5, and 4 are compared, they contain the number 4, and the 5 has another 1, which is the fourth part of the number 4, etc.|Throop (2006)|<ref>Throop, Priscilla (2006). ''Isidore of Seville's Etymologies: Complete English Translation, Volume 1'', p.III.6.12,n.7. ISBN 978-1-4116-6523-1.</ref>}}-->

Superparticular數是由{{link-en|尼各马科|Nicomachus}}在其著作《[[算術簡介]]》〈Introduction to Arithmetic〉中提出，應用在[[音樂理論]]及視覺和諧度的研究中。

==應用==
在[[和聲]]的研究中，許多[[音程]]可以表示為Superparticular數的比例。此時可以用{{link-en|Størmer定理|Størmer's theorem}}找出所有分母及分子均為[[光滑數]]的音程<ref name="hh"/>。<!--

In [[graph theory]], superparticular numbers (or rather, their reciprocals, 1/2, 2/3, 3/4, etc.) arise via the [[Erdős–Stone theorem]] as the possible values of the [[dense graph|upper density]] of an infinite graph.<ref>{{cite journal |last=Erdős |first=P. |authorlink=Paul Erdős |coauthors=[[Arthur Stone (mathematician)|Stone, A. H.]] |year=1946 |title=On the structure of linear graphs |journal=[[Bulletin of the American Mathematical Society]] |volume=52  |pages=1087–1091 |doi=10.1090/S0002-9904-1946-08715-7 |issue=12}}</ref>

These ratios are also important in visual harmony.  Most [[flag]]s of the world's countries have a ratio of 3:2 between their length and height.{{cn|date=January 2013}} [[aspect ratio (image)|Aspect ratios]] of 4:3 and 3:2 are common in [[digital photography]], and aspect ratios of 7:6 and 5:4 are used in [[medium format (film)|medium format]] and [[large format]] photography respectively.{{cn|date=January 2013}}
-->

==比例英文名稱及相關的音程==
{| class="wikitable"
|+例子
! 比例
! 名稱
! [[音程]]
! 聲音
|-
| 2:1 || duplex || [[八度]] || {{audio|Perfect octave on C.mid|Play}}
|- 
| 3:2 || sesquialterum || [[純五度]] || {{audio|Just perfect fifth on C.mid|Play}}
|-
| 4:3 || sesquitertium || [[純四度]] || {{audio|Just perfect fourth on C.mid|Play}}
|-
| 5:4 || sesquiquartum || [[大三度]] || {{audio|Just major third on C.mid|Play}}
|-
| 6:5 || sesquiquintum || [[小三度]] || {{audio|Just minor third on C.mid|Play}}
|-
| 7:6 ||  || [[減三度]] || {{audio|Septimal minor third on C.mid|Play}}
|-
| 9:8 || sesquioctavum || [[大二度]] || {{audio|Major second on C.mid|Play}}
|-
| 10:9 || sesquinona   || [[弧度|小全音]] || {{audio|Minor tone on C.mid|Play}}
|-
| 16:15 || || [[小二度]] || {{audio|Just diatonic semitone on C.mid|Play}}
|-
| 25:24 || || [[增一度]] || {{audio|Just chromatic semitone on C.mid|Play}}
|-
| 81:80 || || [[syntonic comma]] || {{audio|Syntonic comma on C.mid|Play}}
|-
| 126:125 || || [[septimal semicomma]] || {{audio|Septimal semicomma on C.mid|Play}}
|-
| 4375:4374 || || [[ragisma]] || {{audio|Ragisma on C.mid|Play}}
|}

<!--==See also==
* [[Superpartient number]]
-->

==參考資料==
{{reflist|refs=
<ref name="hh">{{cite journal
 | author = Halsey, G. D.; Hewitt, Edwin
 | title = More on the superparticular ratios in music
 | journal = [[American Mathematical Monthly]]
 | volume = 79
 | year = 1972
 | pages = 1096–1100
 | id = {{MathSciNet | id = 0313189}}
 | doi = 10.2307/2317424
 | issue = 10
 | publisher = Mathematical Association of America
 | jstor = 2317424}}</ref>
}}

==外部連結==
*[https://web.archive.org/web/20120716192244/http://perso.club-internet.fr/daschour/micromegas/terpstra.html An Arithmetical Rubric]：由Siemen Terpstra所著，有關Superparticular數在和声上的應用{{Dead link|date=April 2011}}
*[https://home.comcast.net/~dcanright/super/ Superparticular numbers]{{dead link|date=2017年11月 |bot=InternetArchiveBot |fix-attempted=yes }}：[https://home.comcast.net/~dcanright/ David Canright]{{dead link|date=2017年11月 |bot=InternetArchiveBot |fix-attempted=yes }}應用Superparticular數架構[[五聲調式]]
*[https://web.archive.org/web/20060805093809/http://www.music.indiana.edu/tml/6th-8th/BOEARI2_TEXT.html ''De Institutione Arithmetica, liber II'']：[[波伊提烏]]所著

[[Category:有理數]]
[[Category:音樂理論]]